Signal & System Theory Group

Professor Peter Schreier

M.S.E.E. (Notre Dame, USA), 1999
Ph.D. (Colorado at Boulder, USA), 2003

Room: P1.7.01.2
Phone: +49 (0)5251 60-2213

Student consultation hours: by prior appointment with Ms. Merle


I am Professor and Head of the Signal and System Theory Group in the Department of Electrical Engineering and Information Technology at the Universität Paderborn, Germany.

I was born in Munich, Germany, in 1975. I received a Master of Science from the University of Notre Dame, Indiana, USA, in 1999, and a Ph.D. from the University of Colorado at Boulder, USA, in 2003, both in electrical engineering. In the Fall semester of 1998, I was a visiting research student with the Coding Group at the University of Hawaii at Manoa, USA. In the Spring semester of 2004, I was a Postdoctoral Research Associate, and in the Spring semester of 2008, a Visiting Associate Professor with the Department of Electrical and Computer Engineering at Colorado State University, Ft. Collins, USA. From 2004 until 2011, I was with the School of Electrical Engineering and Computer Science at the University of Newcastle, Australia, first as Lecturer, then Senior Lecturer, and finally Associate Professor.

I have received fellowships from the State of Bavaria, the Studienstiftung des deutschen Volkes (German National Academic Foundation), and the Deutsche Forschungsgemeinschaft (German Research Foundation). From 2008 until 2012, I was an Associate Editor of the IEEE Transactions on Signal Processing, and from 2010 until 2014 a Senior Area Editor for the same Transactions. Currently, I serve as an Associate Editor for the IEEE Signal Processing Letters. I am a Senior Member of the IEEE. From 2009 until 2014, I was a member of the IEEE Technical Committee on Machine Learning for Signal Processing, and I am currently a member of the IEEE Technical Committee on Signal Processing Theory and Methods.

List of Publications

1

Peter J. Schreier and Louis L. Scharf
Statistical signal processing of complex-valued data
The theory of improper and noncircular signals
330 pages, Cambridge University Press, 2010.

Complex-valued random signals are embedded in the very fabric of science and engineering, yet the usual assumptions made about their statistical behavior are often a poor representation of the underlying physics. This book deals with improper and noncircular complex signals, which do not conform to classical assumptions, and it demonstrates how correct treatment of these signals can have significant payoffs. The book begins with detailed coverage of the fundamental theory and presents a variety of tools and algorithms for dealing with improper and noncircular signals. It provides a comprehensive account of the main applications, covering detection, estimation, and signal analysis of stationary, nonstationary, and cyclostationary processes. Providing a systematic development from the origin of complex signals to their probabilistic description makes the theory accessible to newcomers. This book is ideal for graduate students and researchers working with complex data in a range of research areas from communications to oceanography. [BibTeX]@book{SchreierScharf:2010:Statistical-Signal-Processing-of-Complex,
abstract = {Complex-valued random signals are embedded in the very fabric of science and engineering, yet the usual assumptions made about their statistical behavior are often a poor representation of the underlying physics. This book deals with improper and noncircular complex signals, which do not conform to classical assumptions, and it demonstrates how correct treatment of these signals can have significant payoffs. The book begins with detailed coverage of the fundamental theory and presents a variety of tools and algorithms for dealing with improper and noncircular signals. It provides a comprehensive account of the main applications, covering detection, estimation, and signal analysis of stationary, nonstationary, and cyclostationary processes. Providing a systematic development from the origin of complex signals to their probabilistic description makes the theory accessible to newcomers. This book is ideal for graduate students and researchers working with complex data in a range of research areas from communications to oceanography.},
author = {Schreier, Peter J. and Scharf, Louis L.},
title = {Statistical signal processing of complex-valued data: the theory of improper and noncircular signals},
year = {2010},
publisher = {{C}ambridge {U}niversity {P}ress},
}

2

Rate region boundary of the SISO Z-interference channel with improper signaling
(Christian Lameiro, Ignacio Santamaría and Peter J. Schreier)
IEEE Trans. Comm., 2016, accepted for publication. [BibTeX]
@article{Lameiro:2016ac,
author = {Lameiro, Christian and Santamar{\'i}a, Ignacio and Schreier, Peter J.},
title = {Rate region boundary of the {SISO Z}-interference channel with improper signaling},
year = {2016, accepted for publication},
journal = {{IEEE} {T}rans.\ {C}omm.},
}
[Abstract]

3

Canonical correlation analysis of high-dimensional data with very small sample support
(Yang Song, Peter J. Schreier, David Ramírez and Tanuj Hasija)
Signal Process. 128,  pp. 449–458, November 2016. [BibTeX]
@article{Song:2016aa,
author = {Song, Yang and Schreier, Peter J. and Ram{\'i}rez, David and Hasija, Tanuj},
month = {{N}ovember},
title = {Canonical correlation analysis of high-dimensional data with very small sample support},
year = {2016},
journal = {{S}ignal {P}rocess.},
pages = {449–458},
volume = {128},
}
[Abstract]

4

Sample-poor estimation of order and common signal subspace with application to fusion of medical imaging data
(Y. Levin-Schwartz, Yang Song, Peter J. Schreier, V.D. Calhoun and Tülay Adali)
NeuroImage 134,  pp. 486–493, July 2016. [BibTeX]
@article{Levin-Schwartz:2016aa,
author = {Levin-Schwartz, Y. and Song, Yang and Schreier, Peter J. and Calhoun, V.D. and Adali, T{\"u}lay},
month = {{J}uly},
title = {Sample-poor estimation of order and common signal subspace with application to fusion of medical imaging data},
year = {2016},
journal = {{N}euro{I}mage},
pages = {486–493},
volume = {134},
}
[Abstract]

5

Shrinkage of covariance matrices for linear signal estimation using cross-validation
(Jun Tong, Peter J. Schreier, Qinghua Guo, S. Tong, Jiangtao Xi and Y. Yu)
IEEE Trans. Signal Process. 64 (11),  pp. 2965–2975, June 2016. [BibTeX]
@article{Tong:2016aa,
author = {Tong, Jun and Schreier, Peter J. and Guo, Qinghua and Tong, S. and Xi, Jiangtao and Yu, Y.},
month = {{J}une},
title = {Shrinkage of covariance matrices for linear signal estimation using cross-validation},
year = {2016},
journal = {{{IEEE}} {{T}}rans.\ {{S}}ignal\ {{P}}rocess.},
number = {11},
pages = {2965–2975},
volume = {64},
}
[Abstract]

6

A unified scheme to achieve the degrees-of-freedom region of the MIMO interference channel with delayed channel state information
(Mohsen Rezaee, Peter J. Schreier, M. Guillaud and B. Clerckx)
IEEE Trans. Comm. 64 (3),  pp. 1068–1082, March 2016. [BibTeX]
@article{Rezaee:2016aa,
author = {Rezaee, Mohsen and Schreier, Peter J. and Guillaud, M. and Clerckx, B.},
month = {{M}arch},
title = {A unified scheme to achieve the degrees-of-freedom region of the {MIMO} interference channel with delayed channel state information},
year = {2016},
journal = {{IEEE} {T}rans.\ {C}omm.},
number = {3},
pages = {1068–1082},
volume = {64},
}
[Abstract]

7

Detection of multivariate cyclostationarity
(D. Ramírez, P. J. Schreier, J. Via, I. Santamaria and L. L. Scharf)
IEEE Trans. Signal Processing 63 (20),  pp. 5395–5408, October 2015. [BibTeX]
@article{RamirezSchreierVia:2015:Detection-of-multivariate-cyclostationarity,
author = {D. Ram{\'i}rez and P. J. Schreier and J. Via and I. Santamaria and L. L. Scharf},
month = {October},
title = {Detection of multivariate cyclostationarity},
year = {2015},
journal = {IEEE Trans. Signal Processing},
number = {20},
pages = {5395–5408},
volume = {63},
}
[Abstract]

8

Benefits of improper signaling for underlay cognitive radio
(Christian Lameiro, Ignacio Santamaría and Peter J. Schreier)
IEEE Wireless Comm. Lett. 4,  pp. 22–25, February 2015.
DOI:10.1109/LWC.2014.2360179.
[BibTeX]
@article{LameiroSantamariaSchreier:2015:Benefits-of-Improper-Signaling-for-Underlay,
abstract = {In this letter we study the potential benefits of improper signaling for a secondary user (SU) in underlay cognitive radio networks. We consider a basic yet illustrative scenario in which the primary user (PU) always transmit proper Gaussian signals and has a minimum rate constraint. After parameterizing the SU transmit signal in terms of its power and circularity coefficient (which measures the degree of impropriety), we prove that the SU improves its rate by transmitting improper signals only when the ratio of the squared modulus between the SU-PU interference link and the SU direct link exceeds a given threshold. As a by-product of this analysis, we obtain the optimal circularity coefficient that must be used by the SU depending on its power budget. Some simulation results show that the SU benefits from the transmission of improper signals especially when the PU is not highly loaded.},
author = {Lameiro, Christian and Santamar{\'i}a, Ignacio and Schreier, Peter J.},
month = {{F}ebruary},
title = {Benefits of improper signaling for underlay cognitive radio},
year = {2015},
journal = {IEEE Wireless Comm. Lett.},
pages = {22–25},
volume = {4},
doi = {10.1109/LWC.2014.2360179},
}
[Abstract]
In this letter we study the potential benefits of improper signaling for a secondary user (SU) in underlay cognitive radio networks. We consider a basic yet illustrative scenario in which the primary user (PU) always transmit proper Gaussian signals and has a minimum rate constraint. After parameterizing the SU transmit signal in terms of its power and circularity coefficient (which measures the degree of impropriety), we prove that the SU improves its rate by transmitting improper signals only when the ratio of the squared modulus between the SU-PU interference link and the SU direct link exceeds a given threshold. As a by-product of this analysis, we obtain the optimal circularity coefficient that must be used by the SU depending on its power budget. Some simulation results show that the SU benefits from the transmission of improper signals especially when the PU is not highly loaded.

9

Finding brain oscillations with power dependencies in neuroimaging data
(S. Dähne, V. V. Nikulin, D. Ramírez, P. J. Schreier, K.-R. Müller and S. Haufe)
NeuroImage 96,  pp. 334–348, 2014.
DOI:10.1016/j.neuroimage.2014.03.075.
[BibTeX]
@article{DahneNikulinRamirez:2014:Finding-brain-oscillations-with,
abstract = {Phase synchronization among neuronal oscillations within the same frequency band has been hypothesized to be a major mechanism for communication between different brain areas. On the other hand, cross-frequency com- munications are more flexible allowing interactions between oscillations with different frequencies. Among such cross-frequency interactions amplitude-to-amplitude interactions are of a special interest as they show how the strength of spatial synchronization in different neuronal populations relates to each other during a given task. While, previously, amplitude-to-amplitude correlations were studied primarily on the sensor level, we present a source separation approach using spatial filters which maximize the correlation between the envelopes of brain oscillations recorded with electro-/magnetoencephalography (EEG/MEG) or intracranial multichannel re- cordings. Our approach, which is called canonical source power correlation analysis (cSPoC), is thereby capable of extracting genuine brain oscillations solely based on their assumed coupling behavior even when the signal-to- noise ratio of the signals is low. In addition to using cSPoC for the analysis of cross-frequency interactions in the same subject, we show that it can also be utilized for studying amplitude dynamics of neuronal oscillations across subjects. We assess the performance of cSPoC in simulations as well as in three distinctively different analysis sce- narios of real EEG data, each involving several subjects. In the simulations, cSPoC outperforms unsupervised state-of-the-art approaches. In the analysis of real EEG recordings, we demonstrate excellent unsupervised dis- covery of meaningful power-to-power couplings, within as well as across subjects and frequency bands.},
author = {S. D{\"a}hne and V. V. Nikulin and D. Ram{\'i}rez and P. J. Schreier and K.-R. M{\"u}ller and S. Haufe},
title = {Finding brain oscillations with power dependencies in neuroimaging data},
year = {2014},
journal = {NeuroImage},
pages = {334–348},
volume = {96},
doi = {10.1016/j.neuroimage.2014.03.075},
}
[Abstract]
Phase synchronization among neuronal oscillations within the same frequency band has been hypothesized to be a major mechanism for communication between different brain areas. On the other hand, cross-frequency com- munications are more flexible allowing interactions between oscillations with different frequencies. Among such cross-frequency interactions amplitude-to-amplitude interactions are of a special interest as they show how the strength of spatial synchronization in different neuronal populations relates to each other during a given task. While, previously, amplitude-to-amplitude correlations were studied primarily on the sensor level, we present a source separation approach using spatial filters which maximize the correlation between the envelopes of brain oscillations recorded with electro-/magnetoencephalography (EEG/MEG) or intracranial multichannel re- cordings. Our approach, which is called canonical source power correlation analysis (cSPoC), is thereby capable of extracting genuine brain oscillations solely based on their assumed coupling behavior even when the signal-to- noise ratio of the signals is low. In addition to using cSPoC for the analysis of cross-frequency interactions in the same subject, we show that it can also be utilized for studying amplitude dynamics of neuronal oscillations across subjects. We assess the performance of cSPoC in simulations as well as in three distinctively different analysis sce- narios of real EEG data, each involving several subjects. In the simulations, cSPoC outperforms unsupervised state-of-the-art approaches. In the analysis of real EEG recordings, we demonstrate excellent unsupervised dis- covery of meaningful power-to-power couplings, within as well as across subjects and frequency bands.

10

Detecting directionality in random fields using the monogenic signal
(S. C. Olhede, D. Ramírez and P. J. Schreier)
IEEE Trans. Inform. Theory 60 (10),  pp. 6491–6510, October 2014.
DOI:10.1109/TIT.2014.2342734.
[BibTeX]
@article{OlhedeRamirezSchreier:2014:Detecting-Directionality-in-Random-Fields,
abstract = {Detecting and analyzing directional structures in images is important in many applications since one-dimensional patterns often correspond to important features such as object contours or trajectories. Classifying a structure as directional or non-directional requires a measure to quantify the degree of directionality and a threshold, which needs to be chosen based on the statistics of the image. In order to do this, we model the image as a random field. So far, little research has been performed on analyzing directionality in random fields. In this paper, we propose a measure to quantify the degree of directionality based on the random monogenic signal, which enables a unique decomposition of a 2D signal into local amplitude, local orientation, and local phase. We investigate the second-order statistical properties of the monogenic signal for isotropic, anisotropic, and unidirectional random fields. We analyze our measure of directionality for finite-size sample images, and determine a threshold to distinguish between unidirectional and non-unidirectional random fields, which allows the automatic classification of images.},
author = {S. C. Olhede and D. Ram{\'i}rez and P. J. Schreier},
month = {{O}ctober},
title = {Detecting directionality in random fields using the monogenic signal},
year = {2014},
journal = {{IEEE} {T}rans.\ {I}nform.\ {T}heory},
number = {10},
pages = {6491–6510},
volume = {60},
doi = {10.1109/TIT.2014.2342734},
}
[Abstract]
Detecting and analyzing directional structures in images is important in many applications since one-dimensional patterns often correspond to important features such as object contours or trajectories. Classifying a structure as directional or non-directional requires a measure to quantify the degree of directionality and a threshold, which needs to be chosen based on the statistics of the image. In order to do this, we model the image as a random field. So far, little research has been performed on analyzing directionality in random fields. In this paper, we propose a measure to quantify the degree of directionality based on the random monogenic signal, which enables a unique decomposition of a 2D signal into local amplitude, local orientation, and local phase. We investigate the second-order statistical properties of the monogenic signal for isotropic, anisotropic, and unidirectional random fields. We analyze our measure of directionality for finite-size sample images, and determine a threshold to distinguish between unidirectional and non-unidirectional random fields, which allows the automatic classification of images.

11

Optimization and estimation of complex-valued signals
(Tulay Adali and Peter J. Schreier)
IEEE Signal Processing Magazine 31 (5),  pp. 112–128, September 2014.
DOI:10.1109/MSP.2013.2287951.
[BibTeX]
@article{AdaliSchreier:2014:Optimization-and-estimation-of-complex-valued-signals,
abstract = {Complex-valued signals occur in many areas of science and engineering and are thus of fundamental interest. When developing signal processing methods in the complex domain, there are two key issues: making use of the full statistical information and optimization. In this article, we review the necessary tools to address these two key issues and provide examples in filtering and blind source separation (BSS) that utilize these tools.},
author = {Adali, Tulay and Schreier, Peter J.},
month = {{S}eptember},
title = {Optimization and estimation of complex-valued signals},
year = {2014},
journal = {{IEEE} {S}ignal {P}rocessing {M}agazine},
number = {5},
pages = {112–128},
volume = {31},
doi = {10.1109/MSP.2013.2287951},
}
[Abstract]
Complex-valued signals occur in many areas of science and engineering and are thus of fundamental interest. When developing signal processing methods in the complex domain, there are two key issues: making use of the full statistical information and optimization. In this article, we review the necessary tools to address these two key issues and provide examples in filtering and blind source separation (BSS) that utilize these tools.

12

Testing blind separability of complex Gaussian mixtures
(D. Ramírez, P. J. Schreier, J. Vía and I. Santamaría)
Signal Process. 95,  pp. 49–57, February 2014.
DOI:10.1016/j.sigpro.2013.08.010.
[BibTeX]
@article{RamirezSchreierVia:2014:Testing-blind-separability-of-complex,
abstract = {The separation of a complex mixture based solely on second-order statistics can be achieved using the Strong Uncorrelating Transform (SUT) if and only if all sources have distinct circularity coefficients. However, in most problems we do not know the circularity coefficients, and they must be estimated from observed data. In this work, we propose a detector, based on the generalized likelihood ratio test (GLRT), to test the separability of a complex Gaussian mixture using the SUT. For the separable case (distinct circularity coefficients), the maximum likelihood (ML) estimates are straightforward. On the other hand, for the non-separable case (at least one circularity coefficient has multiplicity greater than one), the ML estimates are much more difficult to obtain. To set the threshold, we exploit Wilks' theorem, which gives the asymptotic distribution of the GLRT under the null hypothesis. Finally, numerical simulations show the good performance of the proposed detector and the accuracy of Wilks' approximation.},
author = {D. Ram{\'i}rez and P. J. Schreier and J. V{\'i}a and I. Santamar{\'i}a},
month = {{F}ebruary},
title = {Testing blind separability of complex {G}aussian mixtures},
year = {2014},
journal = {{S}ignal {P}rocess.},
pages = {49–57},
volume = {95},
doi = {10.1016/j.sigpro.2013.08.010},
}
[Abstract]
The separation of a complex mixture based solely on second-order statistics can be achieved using the Strong Uncorrelating Transform (SUT) if and only if all sources have distinct circularity coefficients. However, in most problems we do not know the circularity coefficients, and they must be estimated from observed data. In this work, we propose a detector, based on the generalized likelihood ratio test (GLRT), to test the separability of a complex Gaussian mixture using the SUT. For the separable case (distinct circularity coefficients), the maximum likelihood (ML) estimates are straightforward. On the other hand, for the non-separable case (at least one circularity coefficient has multiplicity greater than one), the ML estimates are much more difficult to obtain. To set the threshold, we exploit Wilks' theorem, which gives the asymptotic distribution of the GLRT under the null hypothesis. Finally, numerical simulations show the good performance of the proposed detector and the accuracy of Wilks' approximation.

13

Neue Anwendungsgebiete für Computer Assisted Surgery (CAS)
(Peter J. Schreier)
ForschungsForum Paderborn 17,  pp. 24–30, February 2014. [BibTeX]
@article{Schreier:2014:ForschungsForumPaderborn,
author = {Schreier, Peter J.},
month = {{F}ebruary},
title = {Neue {A}nwendungsgebiete f{\"u}r {C}omputer {A}ssisted {S}urgery {(CAS)}},
year = {2014},
journal = {{ForschungsForum} {Paderborn}},
pages = {24–30},
volume = {17},
}
[Abstract]

14

A unified framework for regularized linear estimation in communication systems
(Jun Tong and Peter J. Schreier)
Signal Process. 93 (9),  pp. 2671–2686, September 2013.
DOI:10.1016/j.sigpro.2013.03.002.
[BibTeX]
@article{TongSchreier:2013:A-unified-framework,
abstract = {Two concerns often arise simultaneously when applying linear estimation in communication systems: the computational complexity can be prohibitively high when the system size is large, and the performance may degrade dramatically when the presumed model is mismatched with the actual system. In this paper, we introduce a subspace expansion framework to jointly address these concerns, in which the observation is first projected onto a lower-dimensional subspace and then the solution of the projected problem is regularized. We discuss two projection methods based on eigensubspace and Krylov subspace expansions. We show that the Krylov subspace projection provides an economical solution to regularized linear estimation. We also compare different regularization methods, such as principal components and diagonal loading. We show that diagonal loading generally outperforms other alternatives and that Krylov subspace rank reduction can yield a regularization effect close to diagonal loading. Finally, we investigate the impact of preconditioning on the performance and complexity for mismatched modeling and propose a loaded preconditioner, which can reduce complexity as well as preserve the regularization effect. Under the proposed framework, various regularization schemes are studied and some guidelines for choosing the right scheme are provided.},
author = {Tong, Jun and Schreier, Peter J.},
month = {{S}eptember},
title = {A unified framework for regularized linear estimation in communication systems},
year = {2013},
journal = {{S}ignal {P}rocess.},
number = {9},
pages = {2671–2686},
volume = {93},
doi = {10.1016/j.sigpro.2013.03.002},
}
[Abstract]
Two concerns often arise simultaneously when applying linear estimation in communication systems: the computational complexity can be prohibitively high when the system size is large, and the performance may degrade dramatically when the presumed model is mismatched with the actual system. In this paper, we introduce a subspace expansion framework to jointly address these concerns, in which the observation is first projected onto a lower-dimensional subspace and then the solution of the projected problem is regularized. We discuss two projection methods based on eigensubspace and Krylov subspace expansions. We show that the Krylov subspace projection provides an economical solution to regularized linear estimation. We also compare different regularization methods, such as principal components and diagonal loading. We show that diagonal loading generally outperforms other alternatives and that Krylov subspace rank reduction can yield a regularization effect close to diagonal loading. Finally, we investigate the impact of preconditioning on the performance and complexity for mismatched modeling and propose a loaded preconditioner, which can reduce complexity as well as preserve the regularization effect. Under the proposed framework, various regularization schemes are studied and some guidelines for choosing the right scheme are provided.

15

Regularized preconditioning for Krylov Subspace Equalization of OFDM systems over doubly selective channels
(Jun Tong and Peter J. Schreier)
IEEE Wireless Comm. Lett. 2 (4),  pp. 367–370, August 2013.
DOI:10.1109/WCL.2013.042313.130096.
[BibTeX]
@article{TongSchreier:2013:Regularized-Preconditioning-for-Krylov-Subspace,
author = {Tong, Jun and Schreier, Peter J.},
month = {{A}ugust},
title = {Regularized preconditioning for {K}rylov Subspace Equalization of {OFDM} systems over doubly selective channels},
year = {2013},
journal = {{IEEE} {W}ireless {C}omm. {L}ett.},
number = {4},
pages = {367–370},
volume = {2},
doi = {10.1109/WCL.2013.042313.130096},
}
[Abstract]

16

On the instantaneous frequency of Gaussian stochastic processes
(Patrik Wahlberg and Peter J. Schreier)
Probab. Math. Statist. 32,  pp. 69–92, 2012. [BibTeX]
@article{WahlbergSchreier:2012:On-the-instantaneous-frequency-of-Gaussi,
abstract = {We study the instantaneous frequency (IF) of continuous-time, complex-valued, zero-mean, proper, mean-square differentiable, nonstationary Gaussian stochastic processes. We compute the probability density function for the IF for fixed time, which generalizes a result known for wide-sense stationary processes to nonstationary processes. For a fixed point in time, the IF has either zero or infinite variance. For harmonizable processes, we obtain as a consequence the result that the mean of the IF, for fixed time, is the normalized first-order frequency moment of the Wigner spectrum.},
author = {Wahlberg, Patrik and Schreier, Peter J.},
title = {On the instantaneous frequency of {Gaussian} stochastic processes},
year = {2012},
journal = {{P}robab.\ {M}ath.\ {S}tatist.},
pages = {69–92},
volume = {32},
}
[Abstract]
We study the instantaneous frequency (IF) of continuous-time, complex-valued, zero-mean, proper, mean-square differentiable, nonstationary Gaussian stochastic processes. We compute the probability density function for the IF for fixed time, which generalizes a result known for wide-sense stationary processes to nonstationary processes. For a fixed point in time, the IF has either zero or infinite variance. For harmonizable processes, we obtain as a consequence the result that the mean of the IF, for fixed time, is the normalized first-order frequency moment of the Wigner spectrum.

17

Linear precoding for MIMO systems with low-complexity receivers
(Jun Tong, Peter J. Schreier and Steven R. Weller)
IEEE Trans. Wireless Comm. 11 (8),  pp. 2828–2837, August 2012.
DOI:10.1109/TWC.2012.070912.110877.
[BibTeX]
@article{TongSchreierWeller:2012:Linear-precoding-for-MIMO-systems-with-l,
abstract = {This paper considers large multiple-input multiple-output (MIMO) communication systems with linear precoding and linear minimum mean-squared error (LMMSE) equalization based on the iterative conjugate gradient (CG) algorithm. Convergence of the CG algorithm is fast when the eigenvalues of the received signal's covariance matrix are clustered, suggesting that mean-squared error and receiver complexity can be managed with judicious precoder design. In order to accelerate convergence of an iterative CG receiver, we incorporate constraints on two measures of eigenvalue clustering into the precoder design. Closed-form solutions to the optimal precoders are derived using majorization theory and convex optimization techniques. We show that if there are constraints on receiver complexity, the proposed precoders can improve performance for large MIMO systems operating over slowly time-varying fading channels.},
author = {Tong, Jun and Schreier, Peter J. and Weller, Steven R.},
month = {{A}ugust},
title = {Linear precoding for {MIMO} systems with low-complexity receivers},
year = {2012},
journal = {{IEEE} {T}rans.\ {W}ireless {C}omm.},
number = {8},
pages = {2828–2837},
volume = {11},
doi = {10.1109/TWC.2012.070912.110877},
}
[Abstract]
This paper considers large multiple-input multiple-output (MIMO) communication systems with linear precoding and linear minimum mean-squared error (LMMSE) equalization based on the iterative conjugate gradient (CG) algorithm. Convergence of the CG algorithm is fast when the eigenvalues of the received signal's covariance matrix are clustered, suggesting that mean-squared error and receiver complexity can be managed with judicious precoder design. In order to accelerate convergence of an iterative CG receiver, we incorporate constraints on two measures of eigenvalue clustering into the precoder design. Closed-form solutions to the optimal precoders are derived using majorization theory and convex optimization techniques. We show that if there are constraints on receiver complexity, the proposed precoders can improve performance for large MIMO systems operating over slowly time-varying fading channels.

18

Design and analysis of large MIMO systems with Krylov subspace receivers
(Jun Tong, Peter J. Schreier and Steven R. Weller)
IEEE Trans. Signal Process. 60 (5),  pp. 2482–2493, May 2012.
DOI:10.1109/TSP.2012.2187287.
[BibTeX]
@article{TongSchreierWeller:2012:Design-and-analysis-of-large-MIMO-system,
abstract = {This paper studies large multiple-input multiple-output (MIMO) communication systems with linear precoding and reduced-rank Krylov subspace receivers. We design precoders and analyze their performance by exploiting large-dimensional random matrix theory. We first devise low-complexity precoding schemes that can improve performance of low-rank Krylov subspace receivers in the regime of high signal-to-noise ratio (SNR). We then introduce a potential theory-based method for analyzing the convergence behavior of the mean-squared error (MSE) for various transmission schemes. This method can be applied to a broader range of problems compared to previous analytical tools. The analysis reveals that the MSE decreases super exponentially with the rank of the receiver. Numerical examples show that the proposed precoders can outperform conventional precoders when low-rank Krylov subspace receivers are used, and that the performance of such receivers can be accurately predicted.},
author = {Tong, Jun and Schreier, Peter J. and Weller, Steven R.},
month = {{M}ay},
title = {Design and analysis of large {MIMO} systems with {Krylov} subspace receivers},
year = {2012},
journal = {{{IEEE}} {{T}}rans.\ {{S}}ignal\ {{P}}rocess.},
number = {5},
pages = {2482–2493},
volume = {60},
doi = {10.1109/TSP.2012.2187287},
}
[Abstract]
This paper studies large multiple-input multiple-output (MIMO) communication systems with linear precoding and reduced-rank Krylov subspace receivers. We design precoders and analyze their performance by exploiting large-dimensional random matrix theory. We first devise low-complexity precoding schemes that can improve performance of low-rank Krylov subspace receivers in the regime of high signal-to-noise ratio (SNR). We then introduce a potential theory-based method for analyzing the convergence behavior of the mean-squared error (MSE) for various transmission schemes. This method can be applied to a broader range of problems compared to previous analytical tools. The analysis reveals that the MSE decreases super exponentially with the rank of the receiver. Numerical examples show that the proposed precoders can outperform conventional precoders when low-rank Krylov subspace receivers are used, and that the performance of such receivers can be accurately predicted.

19

Locally stationary harmonizable complex improper stochastic processes
(Patrik Wahlberg and Peter J. Schreier)
J. Time Ser. Anal. 32 (1),  pp. 33–46, Blackwell Publishing Ltd, 2011.
DOI:10.1111/j.1467-9892.2010.00682.x.
[BibTeX]
@article{WahlbergSchreier:2011:Locally-stationary-harmonizable-complex-,
abstract = {This article concerns continuous-time second-order complex-valued improper stochastic processes that are harmonizable and locally stationary in Silverman's sense. We study necessary and sufficient conditions for the property of local stationarity in the time and frequency domains. A sufficient condition by Silverman is generalized and extended to the improper case. We obtain a result on the absolute continuity of the complementary spectral measure with respect to the spectral measure, which is related to a spectral characterization of improper wide-sense stationary processes.},
author = {Wahlberg, Patrik and Schreier, Peter J.},
title = {Locally stationary harmonizable complex improper stochastic processes},
year = {2011},
journal = {{J}.\ {T}ime {S}er.\ {A}nal.},
number = {1},
pages = {33–46},
volume = {32},
doi = {10.1111/j.1467-9892.2010.00682.x},
publisher = {{B}lackwell {P}ublishing {Ltd}},
}
[Abstract]
This article concerns continuous-time second-order complex-valued improper stochastic processes that are harmonizable and locally stationary in Silverman's sense. We study necessary and sufficient conditions for the property of local stationarity in the time and frequency domains. A sufficient condition by Silverman is generalized and extended to the improper case. We obtain a result on the absolute continuity of the complementary spectral measure with respect to the spectral measure, which is related to a spectral characterization of improper wide-sense stationary processes.

20

Complex-valued signal processing: the proper way to deal with impropriety
(Tülay Adali, Peter J. Schreier and Louis L. Scharf)
IEEE Trans. Signal Process. 59 (11),  pp. 5101–5125, November 2011.
DOI:10.1109/TSP.2011.2162954.
[BibTeX]
@article{AdaliSchreierScharf:2011:Complex-Valued-Signal-Processing:-The-Pr,
abstract = {Complex-valued signals occur in many areas of science and engineering and are thus of fundamental interest. In the past, it has often been assumed, usually implicitly, that complex random signals are proper or circular. A proper complex random variable is uncorrelated with its complex conjugate, and a circular complex random variable has a probability distribution that is invariant under rotation in the complex plane. While these assumptions are convenient because they simplify computations, there are many cases where proper and circular random signals are very poor models of the underlying physics. When taking impropriety and noncircularity into account, the right type of processing can provide significant performance gains. There are two key ingredients in the statistical signal processing of complex-valued data: 1) utilizing the complete statistical characterization of complex-valued random signals; and 2) the optimization of real-valued cost functions with respect to complex parameters. In this overview article, we review the necessary tools, among which are widely linear transformations, augmented statistical descriptions, and Wirtinger calculus. We also present some selected recent developments in the field of complex-valued signal processing, addressing the topics of model selection, filtering, and source separation.},
author = {Adali, T{\"u}lay and Schreier, Peter J. and Scharf, Louis L.},
month = {{N}ovember},
title = {Complex-valued signal processing: the proper way to deal with impropriety},
year = {2011},
journal = {{{IEEE}} {{T}}rans.\ {{S}}ignal\ {{P}}rocess.},
number = {11},
pages = {5101–5125},
volume = {59},
doi = {10.1109/TSP.2011.2162954},
}
[Abstract]
Complex-valued signals occur in many areas of science and engineering and are thus of fundamental interest. In the past, it has often been assumed, usually implicitly, that complex random signals are proper or circular. A proper complex random variable is uncorrelated with its complex conjugate, and a circular complex random variable has a probability distribution that is invariant under rotation in the complex plane. While these assumptions are convenient because they simplify computations, there are many cases where proper and circular random signals are very poor models of the underlying physics. When taking impropriety and noncircularity into account, the right type of processing can provide significant performance gains. There are two key ingredients in the statistical signal processing of complex-valued data: 1) utilizing the complete statistical characterization of complex-valued random signals; and 2) the optimization of real-valued cost functions with respect to complex parameters. In this overview article, we review the necessary tools, among which are widely linear transformations, augmented statistical descriptions, and Wirtinger calculus. We also present some selected recent developments in the field of complex-valued signal processing, addressing the topics of model selection, filtering, and source separation.

21

On Wiener filtering of certain locally stationary stochastic processes
(Patrik Wahlberg and Peter J. Schreier)
Signal Process. 90 (3),  pp. 885–890, March 2010.
DOI:10.1016/j.sigpro.2009.09.013.
[BibTeX]
@article{WahlbergSchreier:2010:On-Wiener-filtering-of-certain-locally-s,
abstract = {We study linear minimum mean squared error filters for continuous-time second-order stochastic processes that are locally stationary in Silverman's sense. We show that the optimal filter is rarely locally stationary even when the covariance functions have Gaussian shape. Using Mehler's formula we derive series expansions of the filter kernel for locally stationary covariances that are determined by Gaussians.},
author = {Wahlberg, Patrik and Schreier, Peter J.},
month = {{M}arch},
title = {On {Wiener} filtering of certain locally stationary stochastic processes},
year = {2010},
journal = {{S}ignal {P}rocess.},
number = {3},
pages = {885–890},
volume = {90},
doi = {10.1016/j.sigpro.2009.09.013},
}
[Abstract]
We study linear minimum mean squared error filters for continuous-time second-order stochastic processes that are locally stationary in Silverman's sense. We show that the optimal filter is rarely locally stationary even when the covariance functions have Gaussian shape. Using Mehler's formula we derive series expansions of the filter kernel for locally stationary covariances that are determined by Gaussians.

22

Gabor discretization of the Weyl product for modulation spaces and filtering of nonstationary stochastic processes
(Patrik Wahlberg and Peter J. Schreier)
Appl. Comput. Harmon. Anal. 26 (1),  pp. 97–120, 2009.
DOI:10.1016/j.acha.2008.02.005.
[BibTeX]
@article{WahlbergSchreier:2009:Gabor-discretization-of-the-Weyl-product,
abstract = {We discretize the Weyl product acting on symbols of modulation spaces, using a Gabor frame defined by a Gaussian function. With one factor fixed, the Weyl product is equivalent to a matrix multiplication on the Gabor coefficient level. If the fixed factor belongs to the weighted Sj{\"o}strand space M ω ∞ , 1 , then the matrix has polynomial or exponential off-diagonal decay, depending on the weight ω. Moreover, if its operator is invertible on L 2 , the inverse matrix has similar decay properties. The results are applied to the equation for the linear minimum mean square error filter for estimation of a nonstationary second-order stochastic process from a noisy observation. The resulting formula for the Gabor coefficients of the Weyl symbol for the optimal filter may be interpreted as a time–frequency version of the filter for wide-sense stationary processes, known as the noncausal Wiener filter.},
author = {Wahlberg, Patrik and Schreier, Peter J.},
title = {Gabor discretization of the {Weyl} product for modulation spaces and filtering of nonstationary stochastic processes},
year = {2009},
journal = {{A}ppl.\ {C}omput.\ {H}armon.\ {A}nal.},
number = {1},
pages = {97–120},
volume = {26},
doi = {10.1016/j.acha.2008.02.005},
}
[Abstract]
We discretize the Weyl product acting on symbols of modulation spaces, using a Gabor frame defined by a Gaussian function. With one factor fixed, the Weyl product is equivalent to a matrix multiplication on the Gabor coefficient level. If the fixed factor belongs to the weighted Sj"ostrand space M ω ∞ , 1 , then the matrix has polynomial or exponential off-diagonal decay, depending on the weight ω. Moreover, if its operator is invertible on L 2 , the inverse matrix has similar decay properties. The results are applied to the equation for the linear minimum mean square error filter for estimation of a nonstationary second-order stochastic process from a noisy observation. The resulting formula for the Gabor coefficients of the Weyl symbol for the optimal filter may be interpreted as a time–frequency version of the filter for wide-sense stationary processes, known as the noncausal Wiener filter.

23

Bounds on the degree of impropriety of complex random vectors
(Peter J. Schreier)
IEEE Signal Process. Lett. 15,  pp. 190–193, 2008.
DOI:10.1109/LSP.2007.913134.
[BibTeX]
@article{Schreier:2008:Bounds-on-the-Degree-of-Impropriety-of-C,
abstract = {A complex random vector is called improper if it is correlated with its complex conjugate. We introduce a measure for the degree of impropriety, which is a function of the canonical correlations between the vector and its complex conjugate (sometimes called the circularity spectrum). This measure is invariant under linear transformation, and it relates the entropy of an improper Gaussian random vector to its corresponding proper version. For vectors with given spectrum, we present upper and lower bounds on the attainable degree of impropriety, in terms of the eigenvalues of the augmented covariance matrix.},
author = {Schreier, Peter J.},
title = {Bounds on the degree of impropriety of complex random vectors},
year = {2008},
journal = {{IEEE} {S}ignal {P}rocess.\ {L}ett.},
pages = {190–193},
volume = {15},
doi = {10.1109/LSP.2007.913134},
}
[Abstract]
A complex random vector is called improper if it is correlated with its complex conjugate. We introduce a measure for the degree of impropriety, which is a function of the canonical correlations between the vector and its complex conjugate (sometimes called the circularity spectrum). This measure is invariant under linear transformation, and it relates the entropy of an improper Gaussian random vector to its corresponding proper version. For vectors with given spectrum, we present upper and lower bounds on the attainable degree of impropriety, in terms of the eigenvalues of the augmented covariance matrix.

24

Polarization ellipse analysis of nonstationary random signals
(Peter J. Schreier)
IEEE Trans. Signal Process. 56 (9),  pp. 4330–4339, September 2008.
DOI:10.1109/TSP.2008.925961.
[BibTeX]
@article{Schreier:2008:Polarization-Ellipse-Analysis-of-Nonstat,
abstract = {We present a novel way of extending rotary-component and polarization analysis to nonstationary random signals. If a complex signal is resolved into counterclockwise and clockwise rotating phasors at one particular frequency only, it traces out an ellipse in the complex plane. Rotary-component analysis characterizes this ellipse in terms of its shape and orientation. Polarization analysis looks at the coherence between counterclockwise and clockwise rotating phasors and whether there is a preferred rotation direction of the ellipse (counterclockwise or clockwise). In the nonstationary case, we replace this ellipse with a time-dependent local ellipse that, at a given time instant, gives the best local approximation of the signal from a given frequency component. This local ellipse is then analyzed in terms of its shape, orientation, and degree of polarization. A time-frequency coherence measures how well the local ellipse approximates the signal. The ellipse parameters and the time-frequency coherence can be expressed in terms of the Rihaczek time-frequency distribution. Under coordinate rotation, the ellipse shape, the degree of polarization, and the time-frequency coherence are invariant, and the ellipse orientation is covariant. The methods presented in this paper provide an alternative to ellipse decompositions based on wavelet ridge analysis.},
author = {Schreier, Peter J.},
month = {{S}eptember},
title = {Polarization ellipse analysis of nonstationary random signals},
year = {2008},
journal = {{{IEEE}} {{T}}rans.\ {{S}}ignal\ {{P}}rocess.},
number = {9},
pages = {4330–4339},
volume = {56},
doi = {10.1109/TSP.2008.925961},
}
[Abstract]
We present a novel way of extending rotary-component and polarization analysis to nonstationary random signals. If a complex signal is resolved into counterclockwise and clockwise rotating phasors at one particular frequency only, it traces out an ellipse in the complex plane. Rotary-component analysis characterizes this ellipse in terms of its shape and orientation. Polarization analysis looks at the coherence between counterclockwise and clockwise rotating phasors and whether there is a preferred rotation direction of the ellipse (counterclockwise or clockwise). In the nonstationary case, we replace this ellipse with a time-dependent local ellipse that, at a given time instant, gives the best local approximation of the signal from a given frequency component. This local ellipse is then analyzed in terms of its shape, orientation, and degree of polarization. A time-frequency coherence measures how well the local ellipse approximates the signal. The ellipse parameters and the time-frequency coherence can be expressed in terms of the Rihaczek time-frequency distribution. Under coordinate rotation, the ellipse shape, the degree of polarization, and the time-frequency coherence are invariant, and the ellipse orientation is covariant. The methods presented in this paper provide an alternative to ellipse decompositions based on wavelet ridge analysis.

25

A unifying discussion of correlation analysis for complex random vectors
(Peter J. Schreier)
IEEE Trans. Signal Process. 56 (4),  pp. 1327–1336, April 2008.
DOI:10.1109/TSP.2007.909054.
[BibTeX]
@article{Schreier:2008:A-Unifying-Discussion-of-Correlation-Ana,
abstract = {The assessment of multivariate association between two complex random vectors is considered. A number of correlation coefficients based on three popular correlation analysis techniques, namely canonical correlation analysis, multivariate linear regression, and partial least squares, are reviewed and connected to performance measures in signal processing and communications, such as mean-squared estimation error, mutual information, and signal-to-noise ratio (SNR). For complex data, there are three types of correlation coefficients, which account for rotational, reflectional, and total (i.e., rotational and reflectional) dependencies between two random vectors. These three types are defined and analyzed for different correlation coefficients, and a numerical example is given. It is often required to compare two complex random vectors in a lower-dimensional subspace. For the large class of increasing, Schur-convex correlation coefficients, it is shown that the low-rank approximations of two random vectors maximizing a particular correlation coefficient are determined only by the constraints imposed on the correlation analysis technique. In this context, the correlation spread is defined as a normalized measure of how much of the overall correlation is contained in a low-dimensional subspace.},
author = {Schreier, Peter J.},
month = {{A}pril},
title = {A unifying discussion of correlation analysis for complex random vectors},
year = {2008},
journal = {{{IEEE}} {{T}}rans.\ {{S}}ignal\ {{P}}rocess.},
number = {4},
pages = {1327–1336},
volume = {56},
doi = {10.1109/TSP.2007.909054},
}
[Abstract]
The assessment of multivariate association between two complex random vectors is considered. A number of correlation coefficients based on three popular correlation analysis techniques, namely canonical correlation analysis, multivariate linear regression, and partial least squares, are reviewed and connected to performance measures in signal processing and communications, such as mean-squared estimation error, mutual information, and signal-to-noise ratio (SNR). For complex data, there are three types of correlation coefficients, which account for rotational, reflectional, and total (i.e., rotational and reflectional) dependencies between two random vectors. These three types are defined and analyzed for different correlation coefficients, and a numerical example is given. It is often required to compare two complex random vectors in a lower-dimensional subspace. For the large class of increasing, Schur-convex correlation coefficients, it is shown that the low-rank approximations of two random vectors maximizing a particular correlation coefficient are determined only by the constraints imposed on the correlation analysis technique. In this context, the correlation spread is defined as a normalized measure of how much of the overall correlation is contained in a low-dimensional subspace.

26

Spectral relations for multidimensional complex improper stationary and (almost) cyclostationary processes
(Patrik Wahlberg and Peter J. Schreier)
IEEE Trans. Inform. Theory 54 (4),  pp. 1670–1682, April 2008.
DOI:10.1109/TIT.2008.917626.
[BibTeX]
@article{WahlbergSchreier:2008:Spectral-Relations-for-Multidimensional-,
abstract = {We study continuous-time multidimensional wide- sense stationary (WSS) and (almost) cyclostationary processes in the frequency domain. Under the assumption that the correlation function is uniformly continuous, we prove the existence of a unique sequence of spectral measures, which coincide with the restrictions to certain subdiagonals of the spectral measure in the strongly harmonizable case. Moreover, the off-diagonal measures are absolutely continuous with respect to the diagonal measure. As a consequence, for strongly harmonizable scalar improper almost cyclostationary processes, we obtain representation formulas for the components of the complementary spectral measure and the off-diagonal components of the spectral measure, in terms of the diagonal component of the spectral measure. We apply these results to analytic signals, which produces sufficient conditions for propriety for almost cyclostationary analytic signals.},
author = {Wahlberg, Patrik and Schreier, Peter J.},
month = {{A}pril},
title = {Spectral relations for multidimensional complex improper stationary and (almost) cyclostationary processes},
year = {2008},
journal = {{IEEE} {T}rans.\ {I}nform.\ {T}heory},
number = {4},
pages = {1670–1682},
volume = {54},
doi = {10.1109/TIT.2008.917626},
}
[Abstract]
We study continuous-time multidimensional wide- sense stationary (WSS) and (almost) cyclostationary processes in the frequency domain. Under the assumption that the correlation function is uniformly continuous, we prove the existence of a unique sequence of spectral measures, which coincide with the restrictions to certain subdiagonals of the spectral measure in the strongly harmonizable case. Moreover, the off-diagonal measures are absolutely continuous with respect to the diagonal measure. As a consequence, for strongly harmonizable scalar improper almost cyclostationary processes, we obtain representation formulas for the components of the complementary spectral measure and the off-diagonal components of the spectral measure, in terms of the diagonal component of the spectral measure. We apply these results to analytic signals, which produces sufficient conditions for propriety for almost cyclostationary analytic signals.

27

Causal Wiener filter banks for periodically correlated time series
(Mark S. Spurbeck and Peter J. Schreier)
Signal Process. 87 (6),  pp. 1179–1187, Amsterdam, The Netherlands, Elsevier North-Holland, Inc., June 2007.
DOI:10.1016/j.sigpro.2006.10.008.
[BibTeX]
@article{SpurbeckSchreier:2007:Causal-Wiener-filter-banks-for-periodica,
abstract = {A causal filter bank implementation of the cyclic Wiener filter for periodically correlated (PC) time series is developed. By converting a PC time series into a vector-valued wide-sense stationary (WSS) time series, the existing literature on factorization of spectral density matrices may be utilized. However, because PC analytic and equivalent baseband signals are generally complex improper, spectral factorization algorithms must be adapted to the improper case. Based on the factorization of the spectral density matrix for the equivalent WSS vector process, causal synthesis and whitening filters for PC time series can be built. These techniques are exploited to implement a causal cyclic Wiener filter as a multirate filter bank or an equivalent polyphase structure. This filter bank is shown to be an efficient equivalent implementation of a frequency shift (FRESH) filter. Therefore, the results derived in this paper also show how to build a causal FRESH cyclic Wiener filter.},
address = {{A}msterdam, {T}he {N}etherlands},
author = {Spurbeck, Mark S. and Schreier, Peter J.},
month = {{J}une},
title = {Causal {Wiener} filter banks for periodically correlated time series},
year = {2007},
journal = {{S}ignal {P}rocess.},
number = {6},
pages = {1179–1187},
volume = {87},
doi = {10.1016/j.sigpro.2006.10.008},
publisher = {{E}lsevier {N}orth-{H}olland, {I}nc.},
}
[Abstract]
A causal filter bank implementation of the cyclic Wiener filter for periodically correlated (PC) time series is developed. By converting a PC time series into a vector-valued wide-sense stationary (WSS) time series, the existing literature on factorization of spectral density matrices may be utilized. However, because PC analytic and equivalent baseband signals are generally complex improper, spectral factorization algorithms must be adapted to the improper case. Based on the factorization of the spectral density matrix for the equivalent WSS vector process, causal synthesis and whitening filters for PC time series can be built. These techniques are exploited to implement a causal cyclic Wiener filter as a multirate filter bank or an equivalent polyphase structure. This filter bank is shown to be an efficient equivalent implementation of a frequency shift (FRESH) filter. Therefore, the results derived in this paper also show how to build a causal FRESH cyclic Wiener filter.

28

Higher-order spectral analysis of complex signals
(Peter J. Schreier and Louis L. Scharf)
Signal Process. 86 (11),  pp. 3321–3333, November 2006.
DOI:10.1016/j.sigpro.2006.02.027.
[BibTeX]
@article{SchreierScharf:2006:Higher-order-spectral-analysis-of-comple,
abstract = {Even though higher-order spectral analysis is by now a mature field, complex signals are still not routinely used, as they are in second-order analysis. The reason is the complexity of the complex case: nth order moment functions of a complex signal can be defined in 2 n different ways, depending on the placement of complex conjugate operators. It is demonstrated that only a few of these different moments are required for a complete nth order description. Properties of nth order moments and spectra with different conjugation patterns are investigated. For the special case of analytic signals, it is shown how spectra with different conjugation patterns provide different information about the signal. Both energy and power signals and deterministic and stochastic signals are discussed. A major focus lies on extending results from continuous-time signals to their sampled versions. Such an extension is not straightforward due to a phenomenon called higher-order or dimension-reduction aliasing. It is demonstrated why spectra of sampled nonstationary signals may suffer from dimension-reduction aliasing unless they are sufficiently oversampled.},
author = {Schreier, Peter J. and Scharf, Louis L.},
month = {{N}ovember},
title = {Higher-order spectral analysis of complex signals},
year = {2006},
journal = {{S}ignal {P}rocess.},
number = {11},
pages = {3321–3333},
volume = {86},
doi = {10.1016/j.sigpro.2006.02.027},
}
[Abstract]
Even though higher-order spectral analysis is by now a mature field, complex signals are still not routinely used, as they are in second-order analysis. The reason is the complexity of the complex case: nth order moment functions of a complex signal can be defined in 2 n different ways, depending on the placement of complex conjugate operators. It is demonstrated that only a few of these different moments are required for a complete nth order description. Properties of nth order moments and spectra with different conjugation patterns are investigated. For the special case of analytic signals, it is shown how spectra with different conjugation patterns provide different information about the signal. Both energy and power signals and deterministic and stochastic signals are discussed. A major focus lies on extending results from continuous-time signals to their sampled versions. Such an extension is not straightforward due to a phenomenon called higher-order or dimension-reduction aliasing. It is demonstrated why spectra of sampled nonstationary signals may suffer from dimension-reduction aliasing unless they are sufficiently oversampled.

29

A generalized likelihood ratio test for impropriety of complex signals
(Peter J. Schreier, Louis L. Scharf and Alfred Hanssen)
IEEE Signal Process. Lett. 13 (7),  pp. 433–436, July 2006.
DOI:10.1109/LSP.2006.871858.
[BibTeX]
@article{SchreierScharfHanssen:2006:A-generalized-likelihood-ratio-test-for-,
abstract = {A complex random vector is called improper if it is correlated with its complex conjugate. We present a hypothesis test for impropriety based on a generalized likelihood ratio (GLR). This GLR is invariant to linear transformations on the data, including rotation and scaling, because propriety is preserved by linear transformations. More specifically, we show that the GLR is a function of the squared canonical correlations between the data and their complex conjugate. These canonical correlations make up a complete, or maximal, set of invariants for the Hermitian and complementary covariance matrices under linear, but not widely linear, transformation},
author = {Schreier, Peter J. and Scharf, Louis L. and Hanssen, Alfred},
month = {{J}uly},
title = {A generalized likelihood ratio test for impropriety of complex signals},
year = {2006},
journal = {{IEEE} {S}ignal {P}rocess.\ {L}ett.},
number = {7},
pages = {433–436},
volume = {13},
doi = {10.1109/LSP.2006.871858},
}
[Abstract]
A complex random vector is called improper if it is correlated with its complex conjugate. We present a hypothesis test for impropriety based on a generalized likelihood ratio (GLR). This GLR is invariant to linear transformations on the data, including rotation and scaling, because propriety is preserved by linear transformations. More specifically, we show that the GLR is a function of the squared canonical correlations between the data and their complex conjugate. These canonical correlations make up a complete, or maximal, set of invariants for the Hermitian and complementary covariance matrices under linear, but not widely linear, transformation

30

Canonical coordinates for transform coding of noisy sources
(Peter J. Schreier and Louis L. Scharf)
IEEE Trans. Signal Process. 54 (1),  pp. 235–243, January 2006.
DOI:10.1109/TSP.2005.861085.
[BibTeX]
@article{SchreierScharf:2006:Canonical-coordinates-for-transform-codi,
abstract = {Based on the additive white quantization noise model, linear transform coders are derived for Gaussian sources corrupted by noise. There are two alternative design objectives: minimizing the trace of the error correlation matrix and thus minimizing the mean-squared error, or minimizing the determinant of the error correlation matrix and thus maximizing information rate. It is shown that a solution to both problems is to first transform the noisy observations into canonical coordinates, quantize and apply a Wiener filter in this coordinate system, and then transform the result back to the original coordinates. Canonical coordinates are uncorrelated, and quantization and Wiener filtering are applied to each component independently. The type of canonical coordinate system depends on the design objective: Quantization in half-canonical coordinates minimizes the mean-squared error and quantization in full-canonical coordinates maximizes information rate. Finally, it is also demonstrated in this paper that majorization is the fundamental principle underlying proofs of optimal transform coding.},
author = {Schreier, Peter J. and Scharf, Louis L.},
month = {{J}anuary},
title = {Canonical coordinates for transform coding of noisy sources},
year = {2006},
journal = {{{IEEE}} {{T}}rans.\ {{S}}ignal\ {{P}}rocess.},
number = {1},
pages = {235–243},
volume = {54},
doi = {10.1109/TSP.2005.861085},
}
[Abstract]
Based on the additive white quantization noise model, linear transform coders are derived for Gaussian sources corrupted by noise. There are two alternative design objectives: minimizing the trace of the error correlation matrix and thus minimizing the mean-squared error, or minimizing the determinant of the error correlation matrix and thus maximizing information rate. It is shown that a solution to both problems is to first transform the noisy observations into canonical coordinates, quantize and apply a Wiener filter in this coordinate system, and then transform the result back to the original coordinates. Canonical coordinates are uncorrelated, and quantization and Wiener filtering are applied to each component independently. The type of canonical coordinate system depends on the design objective: Quantization in half-canonical coordinates minimizes the mean-squared error and quantization in full-canonical coordinates maximizes information rate. Finally, it is also demonstrated in this paper that majorization is the fundamental principle underlying proofs of optimal transform coding.

31

The Hilbert space geometry of the Rihaczek distribution for stochastic analytic signals
(Louis L. Scharf, Peter J. Schreier and Alfred Hanssen)
IEEE Signal Process. Lett. 12 (4),  pp. 297–300, April 2005.
DOI:10.1109/LSP.2005.843772.
[BibTeX]
@article{SchreierScharfHanssen:2005:The-Hilbert-space-geometry-of-the-Rihacz,
abstract = {The Rihaczek distribution for stochastic signals is a time- and frequency-shift covariant bilinear time-frequency distribution (TFD) based on the Crame acute;r-Loe grave;ve spectral representation for a harmonizable process. It is a complex Hilbert space inner product (or cross correlation) between the time series and its infinitesimal stochastic Fourier generator. To this inner product, we may attach an illuminating geometry, wherein the cosine squared of the angle between the time series and its infinitesimal stochastic Fourier generator is given by the Rihaczek distribution. The Rihaczek distribution also determines a time-varying Wiener filter for estimating a time series from its infinitesimal stochastic Fourier generator and measures the resulting error covariance. We propose a factored kernel to construct estimators of the Rihaczek distribution that are contained in Cohen's class of bilinear TFDs.},
author = {Scharf, Louis L. and Schreier, Peter J. and Hanssen, Alfred},
month = {{A}pril},
title = {The {Hilbert} space geometry of the {Rihaczek} distribution for stochastic analytic signals},
year = {2005},
journal = {{IEEE} {S}ignal {P}rocess.\ {L}ett.},
number = {4},
pages = {297–300},
volume = {12},
doi = {10.1109/LSP.2005.843772},
}
[Abstract]
The Rihaczek distribution for stochastic signals is a time- and frequency-shift covariant bilinear time-frequency distribution (TFD) based on the Crame acute;r-Loe grave;ve spectral representation for a harmonizable process. It is a complex Hilbert space inner product (or cross correlation) between the time series and its infinitesimal stochastic Fourier generator. To this inner product, we may attach an illuminating geometry, wherein the cosine squared of the angle between the time series and its infinitesimal stochastic Fourier generator is given by the Rihaczek distribution. The Rihaczek distribution also determines a time-varying Wiener filter for estimating a time series from its infinitesimal stochastic Fourier generator and measures the resulting error covariance. We propose a factored kernel to construct estimators of the Rihaczek distribution that are contained in Cohen's class of bilinear TFDs.

32

Detection and estimation of improper complex random signals
(Peter J. Schreier, Louis L. Scharf and Clifford T. Mullis)
IEEE Trans. Inform. Theory 51 (1),  pp. 306–312, January 2005.
DOI:10.1109/TIT.2004.839538.
[BibTeX]
@article{SchreierScharfMullis:2005:Detection-and-estimation-of-improper-com,
abstract = {Nonstationary complex random signals are in general improper (not circularly symmetric), which means that their complementary covariance is nonzero. Since the Karhunen-Loeve (K-L) expansion in its known form is only valid for proper processes, we derive the improper version of this expansion. It produces two sets of eigenvalues and improper observable coordinates. We then use the K-L expansion to solve the problems of detection and estimation of improper complex random signals in additive white Gaussian noise. We derive a general result comparing the performance of conventional processing, which ignores complementary covariances, with processing that takes these into account. In particular, for the detection and estimation problems considered, we find that the performance gain, as measured by deflection and mean-squared error (MSE), respectively, can be as large as a factor of 2. In a communications example, we show how this finding generalizes the result that coherent processing enjoys a 3-dB gain over noncoherent processing.},
author = {Schreier, Peter J. and Scharf, Louis L. and Mullis, Clifford T.},
month = {{J}anuary},
title = {Detection and estimation of improper complex random signals},
year = {2005},
journal = {{IEEE} {T}rans.\ {I}nform.\ {T}heory},
number = {1},
pages = {306–312},
volume = {51},
doi = {10.1109/TIT.2004.839538},
}
[Abstract]
Nonstationary complex random signals are in general improper (not circularly symmetric), which means that their complementary covariance is nonzero. Since the Karhunen-Loeve (K-L) expansion in its known form is only valid for proper processes, we derive the improper version of this expansion. It produces two sets of eigenvalues and improper observable coordinates. We then use the K-L expansion to solve the problems of detection and estimation of improper complex random signals in additive white Gaussian noise. We derive a general result comparing the performance of conventional processing, which ignores complementary covariances, with processing that takes these into account. In particular, for the detection and estimation problems considered, we find that the performance gain, as measured by deflection and mean-squared error (MSE), respectively, can be as large as a factor of 2. In a communications example, we show how this finding generalizes the result that coherent processing enjoys a 3-dB gain over noncoherent processing.

33

Stochastic time-frequency analysis using the analytic signal: why the complementary distribution matters
(Peter J. Schreier and Louis L. Scharf)
IEEE Trans. Signal Process. 51 (12),  pp. 3071–3079, December 2003.
DOI:10.1109/TSP.2003.818911.
[BibTeX]
@article{SchreierScharf:2003:Stochastic-time-frequency-analysis-using,
abstract = {We challenge the perception that we live in a ``proper world'', where complex random signals can always be assumed to be proper (also called circularly symmetric). Rather, we stress the fact that the analytic signal constructed from a nonstationary real signal is, in general, improper, which means that its complementary correlation function is nonzero. We explore the consequences of this finding in the context of stochastic time-frequency analysis in Cohen's class. There, the analytic signal plays a prominent role because it reduces interference terms. However, the usual time-frequency representation (TFR) based on the analytic signal gives only an incomplete signal description. It must be augmented by a complementary TFR whose properties we develop in detail. We show why it is still advantageous to use the pair of standard and complementary TFRs of the analytic signal rather than the TFR of the corresponding real signal.},
author = {Schreier, Peter J. and Scharf, Louis L.},
month = {{D}ecember},
title = {Stochastic time-frequency analysis using the analytic signal: why the complementary distribution matters},
year = {2003},
journal = {{{IEEE}} {{T}}rans.\ {{S}}ignal\ {{P}}rocess.},
number = {12},
pages = {3071–3079},
volume = {51},
doi = {10.1109/TSP.2003.818911},
}
[Abstract]
We challenge the perception that we live in a ``proper world'', where complex random signals can always be assumed to be proper (also called circularly symmetric). Rather, we stress the fact that the analytic signal constructed from a nonstationary real signal is, in general, improper, which means that its complementary correlation function is nonzero. We explore the consequences of this finding in the context of stochastic time-frequency analysis in Cohen's class. There, the analytic signal plays a prominent role because it reduces interference terms. However, the usual time-frequency representation (TFR) based on the analytic signal gives only an incomplete signal description. It must be augmented by a complementary TFR whose properties we develop in detail. We show why it is still advantageous to use the pair of standard and complementary TFRs of the analytic signal rather than the TFR of the corresponding real signal.

34

Second-order analysis of improper complex random vectors and processes
(Peter J. Schreier and Louis L. Scharf)
IEEE Trans. Signal Process. 51 (3),  pp. 714–725, March 2003.
DOI:10.1109/TSP.2002.808085.
[BibTeX]
@article{SchreierScharf:2003:Second-order-analysis-of-improper-comple,
abstract = {We present a comprehensive treatment of the second-order theory of complex random vectors and wide-sense stationary (WSS) signals. The main focus is on the improper case, in which the complementary covariance does not vanish. Accounting for the information present in the complementary covariance requires the use of widely linear transformations. Based on these, we present the eigenanalysis of complex vectors and apply it to the problem of rank reduction through principal components. We also investigate joint properties of two complex vectors by introducing canonical correlations, which paves the way for a discussion of the Wiener filter and its rank-reduced version. We link the concepts of propriety and joint propriety to eigenanalysis and canonical correlation analysis, respectively. Our treatment is extended to WSS signals. In particular, we give a result on the asymptotic distribution of eigenvalues and examine the connection between WSS, proper, and analytic signals.},
author = {Schreier, Peter J. and Scharf, Louis L.},
month = {{M}arch},
title = {Second-order analysis of improper complex random vectors and processes},
year = {2003},
journal = {{{IEEE}} {{T}}rans.\ {{S}}ignal\ {{P}}rocess.},
number = {3},
pages = {714–725},
volume = {51},
doi = {10.1109/TSP.2002.808085},
}
[Abstract]
We present a comprehensive treatment of the second-order theory of complex random vectors and wide-sense stationary (WSS) signals. The main focus is on the improper case, in which the complementary covariance does not vanish. Accounting for the information present in the complementary covariance requires the use of widely linear transformations. Based on these, we present the eigenanalysis of complex vectors and apply it to the problem of rank reduction through principal components. We also investigate joint properties of two complex vectors by introducing canonical correlations, which paves the way for a discussion of the Wiener filter and its rank-reduced version. We link the concepts of propriety and joint propriety to eigenanalysis and canonical correlation analysis, respectively. Our treatment is extended to WSS signals. In particular, we give a result on the asymptotic distribution of eigenvalues and examine the connection between WSS, proper, and analytic signals.

35

Cross-validation techniques for determining the number of correlated components between two data sets when the number of samples is very small
(Christian Lameiro and Peter J. Schreier)
Proc. Asilomar Conf. Signals Syst. Computers, Pacific Grove, CA, USA, November 2016. [BibTeX]
@inproceedings{Lameiro:2016aa,
address = {Pacific Grove, CA, USA},
author = {Lameiro, Christian and Schreier, Peter J.},
booktitle = {{P}roc.\ {A}silomar {C}onf.\ {S}ignals {S}yst.\ {C}omputers},
month = {November},
title = {Cross-validation techniques for determining the number of correlated components between two data sets when the number of samples is very small},
year = {2016},
}
[Abstract]

36

Determining the number of signals correlated across multiple data sets for small sample support
(Yang Song, Tanuj Hasija, Peter J. Schreier and David Ramírez)
Proc. Eur. Signal Process. Conf., Budapest, Hungary, September 2016. [BibTeX]
@inproceedings{Song:2016ab,
address = {Budapest, Hungary},
author = {Song, Yang and Hasija, Tanuj and Schreier, Peter J. and Ram{\'i}rez, David},
booktitle = {{P}roc.\ {E}ur.\ {S}ignal {P}rocess.\ {C}onf.},
month = {{S}eptember},
title = {Determining the number of signals correlated across multiple data sets for small sample support},
year = {2016},
}
[Abstract]

37

Detecting the dimension of the subspace correlated across multiple data sets in the sample poor regime
(Tanuj Hasija, Yang Song, Peter J. Schreier and David Ramírez)
Proc. IEEE Work. Stat. Signal Process., Palma de Mallorca, Spain, June 2016. [BibTeX]
@inproceedings{Hasija:2016aa,
address = {Palma de Mallorca, Spain},
author = {Hasija, Tanuj and Song, Yang and Schreier, Peter J. and Ram{\'i}rez, David},
booktitle = {{P}roc.\ {IEEE} {W}ork.\ {S}tat.\ {S}ignal {P}rocess.},
month = {{J}une},
title = {Detecting the dimension of the subspace correlated across multiple data sets in the sample poor regime},
year = {2016},
}
[Abstract]

38

Detection of cyclostationarity in the presence of temporal or spatial structure with applications to cognitive radio
(Aaron Pries, David Ramírez and Peter J. Schreier)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process.,  pp. 4249–4253, Shanghai, China, March 2016.
DOI:10.1109/ICASSP.2016.7472478.
[BibTeX]
@inproceedings{pries2016,
abstract = {One approach to spectrum sensing for cognitive radio is the detection of cyclostationarity. We extend an existing multi-antenna detector for cyclostationarity proposed by Ram{\'i}rez et al. [1], which makes no assumptions about the noise beyond being (temporally) wide-sense stationary. In special cases, the noise could be uncorrelated among antennas, or it could be temporally white. The performance of a general detector can be improved by making use of a priori structural information. We do not, however, require knowledge of the exact values of the temporal or spatial noise covariances. We develop an asymptotic generalized likelihood ratio test and evaluate the performance by simulations.},
address = {Shanghai, China},
author = {Aaron Pries and David Ram{\'i}rez and Peter J. Schreier},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{M}arch},
title = {Detection of cyclostationarity in the presence of temporal or spatial structure with applications to cognitive radio},
year = {2016},
pages = {4249–4253},
doi = {10.1109/ICASSP.2016.7472478},
}
[Abstract]
One approach to spectrum sensing for cognitive radio is the detection of cyclostationarity. We extend an existing multi-antenna detector for cyclostationarity proposed by Ramírez et al. [1], which makes no assumptions about the noise beyond being (temporally) wide-sense stationary. In special cases, the noise could be uncorrelated among antennas, or it could be temporally white. The performance of a general detector can be improved by making use of a priori structural information. We do not, however, require knowledge of the exact values of the temporal or spatial noise covariances. We develop an asymptotic generalized likelihood ratio test and evaluate the performance by simulations.

39

Maximally improper interference in underlay cognitive radio networks
(Christian Lameiro, I. Santamaría, W. Utschick and Peter J. Schreier)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process., Shanghai, China, March 2016. [BibTeX]
@inproceedings{Lameiro:2016aa,
address = {Shanghai, China},
author = {Lameiro, Christian and Santamar{\'i}a, I. and Utschick, W. and Schreier, Peter J.},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{M}arch},
title = {Maximally improper interference in underlay cognitive radio networks},
year = {2016},
}
[Abstract]

40

Choosing the diagonal loading factor for linear signal estimation using cross validation
(Jun Tong, Qinghua Guo, Jiangtao Xi, Y. Yu and Peter J. Schreier)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process., Shanghai, China, March 2016. [BibTeX]
@inproceedings{Tong:2016ab,
address = {Shanghai, China},
author = {Tong, Jun and Guo, Qinghua and Xi, Jiangtao and Yu, Y. and Schreier, Peter J.},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{M}arch},
title = {Choosing the diagonal loading factor for linear signal estimation using cross validation},
year = {2016},
}
[Abstract]

41

A simple DoF-achievable scheme for the Gaussian interference channel with delayed CSIT
(M. Rezaee, P. J. Schreier, M. Guillaud and B. Clerckx)
Proc. IEEE Global Communications Convference: Communication Theory, San Diego, CA, December 2015. [BibTeX]
@inproceedings{RezaeeSchreierGuillaud:2015:A-simple-DoF-achievable-scheme-for-the-Gaussian,
address = {San Diego, CA},
author = {M. Rezaee and P. J. Schreier and M. Guillaud and B. Clerckx},
booktitle = {Proc. IEEE Global Communications Convference: Communication Theory},
month = {{D}ecember},
title = {A simple {DoF}-achievable scheme for the {Gaussian} interference channel with delayed {CSIT}},
year = {2015},
}
[Abstract]

42

Analysis of maximally improper signalling schemes for underlay cognitive radio
(Christian Lameiro, Ignacio Santamaría and Peter J. Schreier)
Proc. IEEE Int. Conf. Comm., London, UK, June 2015. [BibTeX]
@inproceedings{LameiroSantamariaSchreier:2015:Analysis-of-maximally-improper-signalling,
abstract = {In this paper, the impact of improper Gaussian signaling is studied for an underlay cognitive radio (CR) scenario comprised of a primary user (PU), which has a rate constraint, and a secondary user (SU), both single-antenna. We first derive expressions for the achievable rate of the SU when it transmits proper and maximally improper Gaussian signals (assuming that the SU is solely limited by the CR constraint). These expressions depend on the channel gains to and from the SU through a single variable. Thereby, we observe that improper signaling is beneficial whenever the SU rate is below a threshold, which depends on the signal-to-noise ratio (SNR) and rate requirement of the PU. Furthermore, we provide bounds on the achievable gain that also depend only on the PU parameters. Then, the achievable rate is studied from a statistical viewpoint by deriving its cumulative distribution function considering a constant received SNR at the PU. In addition, we specialize this expression for the Z interference channel, for which the expected achievable rate is also derived. Numerical examples illustrate our claims and show that the SU may significantly benefit from using improper signaling.},
address = {London, UK},
author = {Lameiro, Christian and Santamar{\'i}a, Ignacio and Schreier, Peter J.},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {C}omm.},
month = {{J}une},
title = {Analysis of maximally improper signalling schemes for underlay cognitive radio},
year = {2015},
}
[Abstract]
In this paper, the impact of improper Gaussian signaling is studied for an underlay cognitive radio (CR) scenario comprised of a primary user (PU), which has a rate constraint, and a secondary user (SU), both single-antenna. We first derive expressions for the achievable rate of the SU when it transmits proper and maximally improper Gaussian signals (assuming that the SU is solely limited by the CR constraint). These expressions depend on the channel gains to and from the SU through a single variable. Thereby, we observe that improper signaling is beneficial whenever the SU rate is below a threshold, which depends on the signal-to-noise ratio (SNR) and rate requirement of the PU. Furthermore, we provide bounds on the achievable gain that also depend only on the PU parameters. Then, the achievable rate is studied from a statistical viewpoint by deriving its cumulative distribution function considering a constant received SNR at the PU. In addition, we specialize this expression for the Z interference channel, for which the expected achievable rate is also derived. Numerical examples illustrate our claims and show that the SU may significantly benefit from using improper signaling.

43

An asymptotic LMPI test for cyclostationarity detection with application to cognitive radio (invited paper)
(D. Ramírez, P. J. Schreier, J. Vía, I. Santamaría and L. L. Scharf)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process., Brisbane, Australia, April 2015. [BibTeX]
@inproceedings{RamirezSchreierVia:2015:An-asymptotic-LMPI-test-for-cyclostationarity,
abstract = {We propose a new detector of primary users in cognitive radio networks. The main novelty of the proposed detector in comparison to most known detectors is that it is based on sound statistical principles for detecting cyclostationary signals. In particular, the proposed detector is (asymptotically) the locally most powerful invariant test, i.e. the best invariant detector for low signal-to-noise ratios. The derivation is based on two main ideas: the relationship between a scalar-valued cyclostationary signal and a vector-valued wide-sense stationary signal, and Wijsman's theorem. Moreover, using the spectral representation for the cyclostationary time series, the detector has an insightful interpretation, and implementation, as the broadband coherence between frequencies that are separated by multiples of the cycle frequency. Finally, simulations confirm that the proposed detector performs better than previous approaches.},
address = {Brisbane, Australia},
author = {D. Ram{\'i}rez and P. J. Schreier and J. V{\'i}a and I. Santamar{\'i}a and L. L. Scharf},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{A}pril},
title = {An asymptotic {LMPI} test for cyclostationarity detection with application to cognitive radio (invited paper)},
year = {2015},
}
[Abstract]
We propose a new detector of primary users in cognitive radio networks. The main novelty of the proposed detector in comparison to most known detectors is that it is based on sound statistical principles for detecting cyclostationary signals. In particular, the proposed detector is (asymptotically) the locally most powerful invariant test, i.e. the best invariant detector for low signal-to-noise ratios. The derivation is based on two main ideas: the relationship between a scalar-valued cyclostationary signal and a vector-valued wide-sense stationary signal, and Wijsman's theorem. Moreover, using the spectral representation for the cyclostationary time series, the detector has an insightful interpretation, and implementation, as the broadband coherence between frequencies that are separated by multiples of the cycle frequency. Finally, simulations confirm that the proposed detector performs better than previous approaches.

44

Model-order selection for analyzing correlation between two data sets using CCA with PCA preprocessing
(N. Roseveare and P. J. Schreier)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process., Brisbane, Australia, April 2015. [BibTeX]
@inproceedings{RoseveareSchreier:2015:Model-order-selection-for-analyzing-correlation,
address = {Brisbane, Australia},
author = {N. Roseveare and P. J. Schreier},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{A}pril},
title = {Model-order selection for analyzing correlation between two data sets using {CCA} with {PCA} preprocessing},
year = {2015},
}
[Abstract]

45

Determining the number of correlated signals between two data sets using PCA-CCA when sample support is extremely small
(Y. Song, P. J. Schreier and N. Roseveare)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process., Brisbane, Australia, April 2015. [BibTeX]
@inproceedings{SongSchreierRoseveare:2015:Determining-the-number-of-correlated-signals,
address = {Brisbane, Australia},
author = {Y. Song and P. J. Schreier and N. Roseveare},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{A}pril},
title = {Determining the number of correlated signals between two data sets using {PCA-CCA} when sample support is extremely small},
year = {2015},
}
[Abstract]

46

A regularized maximum likelihood estimator for the period of a cyclostationary process
(D. Ramírez, P. J. Schreier, J. Vía, I. Santamaría and L. L. Scharf)
Proc. Asilomar Conf. Signals Syst. Computers, Pacific Grove, USA, November 2014. [BibTeX]
@inproceedings{RamirezSchreierVia:2014:A-Regularized-Maximum-Likelihood-Estimator,
abstract = {We derive an estimator of the cycle period of a univariate cyclostationary process based on an information- theoretic criterion. Transforming the univariate cyclostationary process into a vector-valued wide-sense stationary process allows us to obtain the structure of the covariance matrix, which is block-Toeplitz, and its block size depends on the unknown cycle period. Therefore, we sweep the block size and obtain the ML estimate of the covariance matrix, required for the information- theoretic criterion. Since there are no closed-form ML estimates of block-Toeplitz matrices, we asymptotically approximate them as block-circulant. Finally, some numerical examples show the good performance of the proposed estimator.},
address = {Pacific Grove, USA},
author = {D. Ram{\'i}rez and P. J. Schreier and J. V{\'i}a and I. Santamar{\'i}a and L. L. Scharf},
booktitle = {{P}roc.\ {A}silomar {C}onf.\ {S}ignals {S}yst.\ {C}omputers},
month = {{N}ovember},
title = {A regularized maximum likelihood estimator for the period of a cyclostationary process},
year = {2014},
}
[Abstract]
We derive an estimator of the cycle period of a univariate cyclostationary process based on an information- theoretic criterion. Transforming the univariate cyclostationary process into a vector-valued wide-sense stationary process allows us to obtain the structure of the covariance matrix, which is block-Toeplitz, and its block size depends on the unknown cycle period. Therefore, we sweep the block size and obtain the ML estimate of the covariance matrix, required for the information- theoretic criterion. Since there are no closed-form ML estimates of block-Toeplitz matrices, we asymptotically approximate them as block-circulant. Finally, some numerical examples show the good performance of the proposed estimator.

47

Optimizing spatial filters for the extraction of envelope-coupled neural oscillations
(S. Dähne, V. V. Nikulin, D. Ramírez, P. J. Schreier, K.-R. Müller and S. Haufe)
Proc. Int. Work. Pattern Recognition In Neuroimaging, Tübingen, Germany, June 2014.
DOI:10.1109/PRNI.2014.6858514.
[BibTeX]
@inproceedings{DahneNikulinRamirez:2014:Optimizing-spatial-filters-for-the-extraction,
abstract = {Amplitude-to-amplitude interactions between neural oscillations are of a special interest as they show how the strength of spatial synchronization in different neuronal populations relates to each other during a given task. While, previously, amplitude-to-amplitude correlations were studied primarily on the sensor level, we present a source separation approach using spatial filters which maximize the correlation between the envelopes of brain oscillations recorded with electro-/magnetencephalography (EEG/MEG) or intracranial multichannel recordings. Our approach, which is called canonical source power correlation analysis (cSPoC), is thereby capable of extracting genuine brain oscillations solely based on their assumed coupling behavior even when the signal-to-noise ratio of the signals is low.},
address = {T{\"u}bingen, Germany},
author = {S. D{\"a}hne and V. V. Nikulin and D. Ram{\'i}rez and P. J. Schreier and K.-R. M{\"u}ller and S. Haufe},
booktitle = {Proc.\ Int. Work. Pattern Recognition In Neuroimaging},
month = {{J}une},
title = {Optimizing spatial filters for the extraction of envelope-coupled neural oscillations},
year = {2014},
doi = {10.1109/PRNI.2014.6858514},
}
[Abstract]
Amplitude-to-amplitude interactions between neural oscillations are of a special interest as they show how the strength of spatial synchronization in different neuronal populations relates to each other during a given task. While, previously, amplitude-to-amplitude correlations were studied primarily on the sensor level, we present a source separation approach using spatial filters which maximize the correlation between the envelopes of brain oscillations recorded with electro-/magnetencephalography (EEG/MEG) or intracranial multichannel recordings. Our approach, which is called canonical source power correlation analysis (cSPoC), is thereby capable of extracting genuine brain oscillations solely based on their assumed coupling behavior even when the signal-to-noise ratio of the signals is low.

48

Regularized successive interference cancellation (sic) under mismatched modeling
(Jun Tong, Qinghua Guo, Peter J. Schreier and Jiangtao Xi)
Proc. IEEE Work. Stat. Signal Process., Gold Coast, Australia, June 2014.
DOI:10.1109/SSP.2014.6884642.
[BibTeX]
@inproceedings{TongGuoSchreier:2014:Regularized-successive-interference-cancellation,
abstract = {Successive interference cancellation (SIC) has been extensively applied to estimate transmit signals in communication systems. When the channel state information (CSI) and noise statistics are imperfectly estimated, the standard SIC estimators that ignore the model mismatch may perform poorly. This paper introduces regularized SIC estimation to provide robustness against the model mismatch. Suboptimal, low-complexity implementations using (sorted) QR decomposition and approximate choice of regularization parameters are also introduced. Simulation examples demonstrate that the regularized SIC estimators can significantly outperform the standard version.},
address = {Gold Coast, Australia},
author = {Tong, Jun and Guo, Qinghua and Schreier, Peter J. and Xi, Jiangtao},
booktitle = {{P}roc.\ {IEEE} {W}ork.\ {S}tat.\ {S}ignal {P}rocess.},
month = {{J}une},
title = {Regularized successive interference cancellation (sic) under mismatched modeling},
year = {2014},
doi = {10.1109/SSP.2014.6884642},
}
[Abstract]
Successive interference cancellation (SIC) has been extensively applied to estimate transmit signals in communication systems. When the channel state information (CSI) and noise statistics are imperfectly estimated, the standard SIC estimators that ignore the model mismatch may perform poorly. This paper introduces regularized SIC estimation to provide robustness against the model mismatch. Suboptimal, low-complexity implementations using (sorted) QR decomposition and approximate choice of regularization parameters are also introduced. Simulation examples demonstrate that the regularized SIC estimators can significantly outperform the standard version.

49

An asymptotic GLRT for the detection of cyclostationary signals
(D. Ramírez, L. L. Scharf, J. Vía, I. Santamaría and P. J. Schreier)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process., Florence, Italy, May 2014.
DOI:10.1109/ICASSP.2014.6854234.
[BibTeX]
@inproceedings{RamirezScharfVia:2014:An-asymptotic-GLRT-for-the-detection-of-cyclostationary,
abstract = {We derive the generalized likelihood ratio test (GLRT) for detecting cyclostationarity in scalar-valued time series. The main idea behind our approach is Gladyshev's relationship, which states that when the scalar-valued cyclostationary sig- nal is blocked at the known cycle period it produces a vector- valued wide-sense stationary process. This result amounts to saying that the covariance matrix of the vector obtained by stacking all observations of the time series is block-Toeplitz if the signal is cyclostationary, and Toeplitz if the signal is wide- sense stationary. The derivation of the GLRT requires the maximum likelihood estimates of Toeplitz and block-Toeplitz matrices. This can be managed asymptotically (for large num- berofsamples)exploitingSzego ̈'stheoremanditsgeneraliza- tion for vector-valued processes. Simulation results show the good performance of the proposed GLRT.},
address = {Florence, Italy},
author = {D. Ram{\'i}rez and L. L. Scharf and J. V{\'i}a and I. Santamar{\'i}a and P. J. Schreier},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{M}ay},
title = {An asymptotic {GLRT} for the detection of cyclostationary signals},
year = {2014},
doi = {10.1109/ICASSP.2014.6854234},
}
[Abstract]
We derive the generalized likelihood ratio test (GLRT) for detecting cyclostationarity in scalar-valued time series. The main idea behind our approach is Gladyshev's relationship, which states that when the scalar-valued cyclostationary sig- nal is blocked at the known cycle period it produces a vector- valued wide-sense stationary process. This result amounts to saying that the covariance matrix of the vector obtained by stacking all observations of the time series is block-Toeplitz if the signal is cyclostationary, and Toeplitz if the signal is wide- sense stationary. The derivation of the GLRT requires the maximum likelihood estimates of Toeplitz and block-Toeplitz matrices. This can be managed asymptotically (for large num- berofsamples)exploitingSzego ̈'stheoremanditsgeneraliza- tion for vector-valued processes. Simulation results show the good performance of the proposed GLRT.

50

Linear equalization in communications with mismatched modeling using Krylov subspace expansion
(Jun Tong and Peter J. Schreier)
Proc. IEEE Wireless Comm. Networking Conf. (WCNC), 2013. [BibTeX]
@inproceedings{TongSchreier:2013:Linear-equalization-in-comm,
author = {Tong, Jun and Schreier, Peter J.},
booktitle = {Proc. IEEE Wireless Comm. Networking Conf. (WCNC)},
title = {Linear equalization in communications with mismatched modeling using {K}rylov subspace expansion},
year = {2013},
}
[Abstract]

51

Power-CCA: maximizing the correlation coefficient between the power of projections
(D. Ramírez, P. J. Schreier, J. Vía and V. V. Nikulin)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process., Vancouver, Canada, May 2013.
DOI:10.1109/ICASSP.2013.6638864.
[BibTeX]
@inproceedings{RamirezSchreierVia:2013:Power-CCA:-Maximizing-the-Correlation-Coefficient,
abstract = {This work presents a variation of canonical correlation analysis (CCA), where the correlation coefficient between the instantaneous power of the projections is maximized, rather than between the projections themselves. The resulting optimization problem is not convex, and we have to resort to a sub-optimal approach. Concretely, we propose a two-step solution consisting of the singular value decomposition (SVD) of a "coherence" matrix followed by a rank-one matrix approximation. This technique is applied to blindly recovering signals in a model that is motivated by the study of neuronal dynamics in humans using electroencephalography (EEG) and magnetoencephalography (MEG). A distinctive feature of this model is that it allows recovery of amplitude-amplitude coupling between neuronal processes.},
address = {Vancouver, Canada},
author = {D. Ram{\'i}rez and P. J. Schreier and J. V{\'i}a and V. V. Nikulin},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{M}ay},
title = {Power-{CCA}: maximizing the correlation coefficient between the power of projections},
year = {2013},
doi = {10.1109/ICASSP.2013.6638864},
}
[Abstract]
This work presents a variation of canonical correlation analysis (CCA), where the correlation coefficient between the instantaneous power of the projections is maximized, rather than between the projections themselves. The resulting optimization problem is not convex, and we have to resort to a sub-optimal approach. Concretely, we propose a two-step solution consisting of the singular value decomposition (SVD) of a "coherence" matrix followed by a rank-one matrix approximation. This technique is applied to blindly recovering signals in a model that is motivated by the study of neuronal dynamics in humans using electroencephalography (EEG) and magnetoencephalography (MEG). A distinctive feature of this model is that it allows recovery of amplitude-amplitude coupling between neuronal processes.

52

GLRT for testing separability of a complex-valued mixture based on the strong uncorrelating transform
(D. Ramírez, P. J. Schreier, J. Vía and I. Santamaría)
Proc. IEEE Int. Work. Machine Learning for Signal Process., Santander, Spain, September 2012.
DOI:10.1109/MLSP.2012.6349785.
[BibTeX]
@inproceedings{RamirezSchreierVia:2012:GLRT-For-Testing-Separability-Of-A-Complex-Valued,
abstract = {The Strong Uncorrelating Transform (SUT) allows blind separation of a mixture of complex independent sources if and only if all sources have distinct circularity coefficients. In practice, the circularity coefficients need to be estimated from observed data. We propose a generalized likelihood ratio test (GLRT) for separability of a complex mixture using the SUT, based on estimated circularity coefficients. For distinct circularity coefficients (separable case), the maximum likelihood (ML) estimates, required for the GLRT, are straightforward. However, for circularity coefficients with multiplicity larger than one (non-separable case), the ML estimates are much more difficult to find. Numerical simulations show the good performance of the proposed detector.},
address = {Santander, Spain},
author = {D. Ram{\'i}rez and P. J. Schreier and J. V{\'i}a and I. Santamar{\'i}a},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {W}ork.\ Machine Learning for Signal Process.},
month = {{S}eptember},
title = {{GLRT} for testing separability of a complex-valued mixture based on the strong uncorrelating transform},
year = {2012},
doi = {10.1109/MLSP.2012.6349785},
}
[Abstract]
The Strong Uncorrelating Transform (SUT) allows blind separation of a mixture of complex independent sources if and only if all sources have distinct circularity coefficients. In practice, the circularity coefficients need to be estimated from observed data. We propose a generalized likelihood ratio test (GLRT) for separability of a complex mixture using the SUT, based on estimated circularity coefficients. For distinct circularity coefficients (separable case), the maximum likelihood (ML) estimates, required for the GLRT, are straightforward. However, for circularity coefficients with multiplicity larger than one (non-separable case), the ML estimates are much more difficult to find. Numerical simulations show the good performance of the proposed detector.

53

The random monogenic signal
(S. C. Olhede, D. Ramírez and P. J. Schreier)
Proc. IEEE Int. Conf. Image Process., Orlando, Florida, USA, September 2012.
DOI:10.1109/ICIP.2012.6467404.
[BibTeX]
@inproceedings{OlhedeRamirezSchreier:2012:The-Random-Monogenic-Signal,
abstract = {The monogenic signal allows us to decompose a two-dimensional real signal into a local amplitude, a local orientation, and a local phase. In this paper, we introduce the random monogenic signal and study its second-order statistical properties. The monogenic signal may be represented as a quaternion-valued signal. We show that for homogeneous random fields, we need exactly two quaternion-valued covariance functions for a complete second-order description. We also introduce a stochastic model for unidirectional signals and a measure of unidirectionality.},
address = {Orlando, Florida, USA},
author = {S. C. Olhede and D. Ram{\'i}rez and P. J. Schreier},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {I}mage {P}rocess.},
month = {{S}eptember},
title = {The random monogenic signal},
year = {2012},
doi = {10.1109/ICIP.2012.6467404},
}
[Abstract]
The monogenic signal allows us to decompose a two-dimensional real signal into a local amplitude, a local orientation, and a local phase. In this paper, we introduce the random monogenic signal and study its second-order statistical properties. The monogenic signal may be represented as a quaternion-valued signal. We show that for homogeneous random fields, we need exactly two quaternion-valued covariance functions for a complete second-order description. We also introduce a stochastic model for unidirectional signals and a measure of unidirectionality.

54

Regularized linear equalization for multipath channels with imperfect channel estimation
(Jun Tong and Peter J. Schreier)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process.,  pp. 3009–3012, March 2012.
DOI:10.1109/ICASSP.2012.6288548.
[BibTeX]
@inproceedings{TongSchreier:2012:Regularized-linear-equalization-for-mult,
abstract = {This paper deals with different techniques for linear equalization (LE) of multipath channels with imperfect channel estimation (CE). We develop a unified framework based on Krylov subspace expansion, which allows us to compare the performance of the conjugate gradient (CG) method, diagonal loading (DL), and a hybrid scheme. Our analysis shows that the DL method generally outperforms its alternatives, but at the cost of higher complexity. However, we also demonstrate that a proper implementation of the low-complexity CG method can also approach the performance of DL. Finally, we show that preconditioning degrades performance when the CE is poor.},
author = {Tong, Jun and Schreier, Peter J.},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{M}arch},
title = {Regularized linear equalization for multipath channels with imperfect channel estimation},
year = {2012},
pages = {3009–3012},
doi = {10.1109/ICASSP.2012.6288548},
}
[Abstract]
This paper deals with different techniques for linear equalization (LE) of multipath channels with imperfect channel estimation (CE). We develop a unified framework based on Krylov subspace expansion, which allows us to compare the performance of the conjugate gradient (CG) method, diagonal loading (DL), and a hybrid scheme. Our analysis shows that the DL method generally outperforms its alternatives, but at the cost of higher complexity. However, we also demonstrate that a proper implementation of the low-complexity CG method can also approach the performance of DL. Finally, we show that preconditioning degrades performance when the CE is poor.

55

Precoder design and convergence analysis of MIMO systems with Krylov subspace receivers
(Jun Tong, Peter J. Schreier and Steven R. Weller)
Proc. IEEE Int. Symp. Inform. Theory,  pp. 2914–2918, August 2011.
DOI:10.1109/ISIT.2011.6034110.
[BibTeX]
@inproceedings{TongSchreierWeller:2011:Precoder-design-and-convergence-analysis,
abstract = {This paper studies the design and analysis of large multiple-input multiple-output (MIMO) systems with linear precoding and Krylov subspace receivers. We design precoders that can improve performance with low-rank receivers. We then introduce a tool based on potential theory to analyze the convergence behavior of the mean-squared error (MSE). The effectiveness of the proposed precoder and the superexponential convergence of the MSE are demonstrated1.},
author = {Tong, Jun and Schreier, Peter J. and Weller, Steven R.},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {S}ymp.\ {I}nform.\ {T}heory},
month = {{A}ugust},
title = {Precoder design and convergence analysis of {MIMO} systems with {Krylov} subspace receivers},
year = {2011},
pages = {2914–2918},
doi = {10.1109/ISIT.2011.6034110},
}
[Abstract]
This paper studies the design and analysis of large multiple-input multiple-output (MIMO) systems with linear precoding and Krylov subspace receivers. We design precoders that can improve performance with low-rank receivers. We then introduce a tool based on potential theory to analyze the convergence behavior of the mean-squared error (MSE). The effectiveness of the proposed precoder and the superexponential convergence of the MSE are demonstrated1.

56

Linear precoding for time-varying MIMO channels with low-complexity receivers
(Jun Tong, Peter J. Schreier, Steven R. Weller and Louis L. Scharf)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process.,  pp. 3092–3095, May 2011.
DOI:10.1109/ICASSP.2011.5946312.
[BibTeX]
@inproceedings{TongSchreierWeller:2011:Linear-precoding-for-time-varying-MIMO-c,
abstract = {This paper considers linear precoding for time-varying multiple input multiple-output (MIMO) channels. We show that linear minimum mean-squared error (LMMSE) equalization based on the conjugate gradient (CG) method can result in significantly reduced complexity compared with conventional approaches. This reduction is achieved by incorporating a condition number constraint into the precoder optimization framework, which leads to clustered eigen values of the measurement covariance matrix. The cost is a small increase in MSE compared to the optimal precoder.},
author = {Tong, Jun and Schreier, Peter J. and Weller, Steven R. and Scharf, Louis L.},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{M}ay},
title = {Linear precoding for time-varying {MIMO} channels with low-complexity receivers},
year = {2011},
pages = {3092–3095},
doi = {10.1109/ICASSP.2011.5946312},
}
[Abstract]
This paper considers linear precoding for time-varying multiple input multiple-output (MIMO) channels. We show that linear minimum mean-squared error (LMMSE) equalization based on the conjugate gradient (CG) method can result in significantly reduced complexity compared with conventional approaches. This reduction is achieved by incorporating a condition number constraint into the precoder optimization framework, which leads to clustered eigen values of the measurement covariance matrix. The cost is a small increase in MSE compared to the optimal precoder.

57

The Wiener filter for locally stationary stochastic processes is rarely locally stationary
(Patrik Wahlberg and Peter J. Schreier)
Proc. 17th European Signal Process. Conf.,  pp. 2465–2469, August 2009. [BibTeX]
@inproceedings{WahlbergSchreier:2009:The-Wiener-filter-for-locally-stationary,
abstract = {The Wiener filter (i.e., linear minimum mean squared error filter) for wide-sense stationary stochastic processes is translation-invariant, i.e., its impulse response, like the covariance function, is only a function of the time-shift. We investigate whether there is a generalization of this result to continuous-time stochastic processes that are locally stationary in Silverman's sense: Is the optimal filter for locally stationary processes locally stationary itself? The answer is surprisingly negative: Even though the optimal filter can be locally stationary in special cases, it rarely is, even when the covariance functions have Gaussian shape.},
author = {Wahlberg, Patrik and Schreier, Peter J.},
booktitle = {{P}roc.\ 17th\ {E}uropean {S}ignal {P}rocess.\ {C}onf.},
month = {{A}ugust},
title = {The {Wiener} filter for locally stationary stochastic processes is rarely locally stationary},
year = {2009},
pages = {2465–2469},
}
[Abstract]
The Wiener filter (i.e., linear minimum mean squared error filter) for wide-sense stationary stochastic processes is translation-invariant, i.e., its impulse response, like the covariance function, is only a function of the time-shift. We investigate whether there is a generalization of this result to continuous-time stochastic processes that are locally stationary in Silverman's sense: Is the optimal filter for locally stationary processes locally stationary itself? The answer is surprisingly negative: Even though the optimal filter can be locally stationary in special cases, it rarely is, even when the covariance functions have Gaussian shape.

58

On ICA of improper and noncircular sources
(Peter J. Schreier, Tülay Adali and Louis L. Scharf)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process.,  pp. 3561–3564, April 2009.
DOI:10.1109/ICASSP.2009.4960395.
[BibTeX]
@inproceedings{SchreierAdaliScharf:2009:On-ICA-of-improper-and-noncircular-sourc,
abstract = {We provide a review of independent component analysis (ICA) for complex-valued improper and noncircular random sources. An improper random signal is correlated with its complex conjugate, and a noncircular random signal has a rotationally variant probability distribution. We present methods for ICA using second-order statistics, and higher-order statistics. For ICA based on second-order statistics, we emphasize the key role played by the circularity coefficients, which are the canonical correlations between the source and the complex conjugate. For ICA based on higher-order statistics, we show how to extend algorithms for real-valued ICA to the complex domain using Wirtinger calculus.},
author = {Schreier, Peter J. and Adali, T{\"u}lay and Scharf, Louis L.},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{A}pril},
title = {On {ICA} of improper and noncircular sources},
year = {2009},
pages = {3561–3564},
doi = {10.1109/ICASSP.2009.4960395},
}
[Abstract]
We provide a review of independent component analysis (ICA) for complex-valued improper and noncircular random sources. An improper random signal is correlated with its complex conjugate, and a noncircular random signal has a rotationally variant probability distribution. We present methods for ICA using second-order statistics, and higher-order statistics. For ICA based on second-order statistics, we emphasize the key role played by the circularity coefficients, which are the canonical correlations between the source and the complex conjugate. For ICA based on higher-order statistics, we show how to extend algorithms for real-valued ICA to the complex domain using Wirtinger calculus.

59

A time-frequency formula for LMMSE filters for nonstationary underspread continuous-time stochastic processes
(Patrik Wahlberg and Peter J. Schreier)
Proc. 16th European Signal Process. Conf., August 2008. [BibTeX]
@inproceedings{WahlbergSchreier:2008:A-time-frequency-formula-for-LMMSE-filte,
abstract = {We study linear minimum mean square error (LMMSE) filters for estimating a nonstationary second-order continuous-time stochastic process from a noisy observation. The equation for the optimal filter is treated in the Weyl symbol domain, and the involved Weyl symbols are assumed to belong to certain modulation spaces. By discretizing this equation using a Gabor frame we transform it into a matrix equation and obtain a formula for the filter by matrix inversion. The inverse matrix has off-diagonal decay at a rate that increases the more underspread the process is.},
author = {Wahlberg, Patrik and Schreier, Peter J.},
booktitle = {{P}roc.\ 16th\ {E}uropean {S}ignal {P}rocess.\ {C}onf.},
month = {{A}ugust},
title = {A time-frequency formula for {LMMSE} filters for nonstationary underspread continuous-time stochastic processes},
year = {2008},
}
[Abstract]
We study linear minimum mean square error (LMMSE) filters for estimating a nonstationary second-order continuous-time stochastic process from a noisy observation. The equation for the optimal filter is treated in the Weyl symbol domain, and the involved Weyl symbols are assumed to belong to certain modulation spaces. By discretizing this equation using a Gabor frame we transform it into a matrix equation and obtain a formula for the filter by matrix inversion. The inverse matrix has off-diagonal decay at a rate that increases the more underspread the process is.

60

The degree of impropriety (noncircularity) of complex random vectors
(Peter J. Schreier)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process.,  pp. 3909–3912, April 2008.
DOI:10.1109/ICASSP.2008.4518508.
[BibTeX]
@inproceedings{Schreier:2008:The-degree-of-impropriety-noncircularit,
abstract = {A complex random vector is called improper (noncircular) if it is correlated with its complex conjugate. We consider measures for the degree of impropriety that are invariant under linear transformation. These measures are functions of the canonical correlations between the vector and its complex conjugate, which have been termed the circularity coefficients. However, we show that these circularity coefficients do not tell the whole story: Two random vectors with identical covariance matrix and identical circularity coefficients can still behave differently in second-order estimation and detection.},
author = {Schreier, Peter J.},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{A}pril},
title = {The degree of impropriety (noncircularity) of complex random vectors},
year = {2008},
pages = {3909–3912},
doi = {10.1109/ICASSP.2008.4518508},
}
[Abstract]
A complex random vector is called improper (noncircular) if it is correlated with its complex conjugate. We consider measures for the degree of impropriety that are invariant under linear transformation. These measures are functions of the canonical correlations between the vector and its complex conjugate, which have been termed the circularity coefficients. However, we show that these circularity coefficients do not tell the whole story: Two random vectors with identical covariance matrix and identical circularity coefficients can still behave differently in second-order estimation and detection.

61

Near-capacity turbo equalization using optimized turbo codes
(Vladimir D. Trajkovic, Minyue Fu and Peter J. Schreier)
Proc. Australasian Telecomm. Netw. Applic. Conf.,  pp. 480–484, December 2007.
DOI:10.1109/ATNAC.2007.4665276.
[BibTeX]
@inproceedings{TrajkovicFuSchreier:2007:Near-capacity-turbo-equalization-using-o,
abstract = {In this paper we analyze a turbo equalization scheme that combines maximum a posteriori probability (MAP) equalization and turbo decoding. Our aim is to optimize the turbo equalizer in order to approach the information capacity limit for channels with severe inter-symbol interference (ISI). For this purpose, we perform an extensive search for turbo codes that give an SNR-BER performance closest to the channel information capacity limit. Our results show that the optimized turbo equalizer can approach the information capacity limit to within 0.7 dB. We also optimize the turbo equalizer in terms of the minimum number of required turbo decoding iterations. Our results show that a turbo decoder within a turbo equalization loop requires only a small number of iterations. Finally, our analysis reveals that when there are turbo codes with similar extrinsic information transfer characteristics, the computational complexity can be reduced by choosing the code with the smallest constraint length with no loss in SNR-BER performance.},
author = {Trajkovic, Vladimir D. and Fu, Minyue and Schreier, Peter J.},
booktitle = {{P}roc.\ {A}ustralasian {T}elecomm. {N}etw.\ {A}pplic.\ {C}onf.},
month = {{D}ecember},
title = {Near-capacity turbo equalization using optimized turbo codes},
year = {2007},
pages = {480–484},
doi = {10.1109/ATNAC.2007.4665276},
}
[Abstract]
In this paper we analyze a turbo equalization scheme that combines maximum a posteriori probability (MAP) equalization and turbo decoding. Our aim is to optimize the turbo equalizer in order to approach the information capacity limit for channels with severe inter-symbol interference (ISI). For this purpose, we perform an extensive search for turbo codes that give an SNR-BER performance closest to the channel information capacity limit. Our results show that the optimized turbo equalizer can approach the information capacity limit to within 0.7 dB. We also optimize the turbo equalizer in terms of the minimum number of required turbo decoding iterations. Our results show that a turbo decoder within a turbo equalization loop requires only a small number of iterations. Finally, our analysis reveals that when there are turbo codes with similar extrinsic information transfer characteristics, the computational complexity can be reduced by choosing the code with the smallest constraint length with no loss in SNR-BER performance.

62

Correlation coefficients for complex random vectors
(Peter J. Schreier)
Proc. 41st Asilomar Conf. Signals Syst. Computers,  pp. 577–581, November 2007.
DOI:10.1109/ACSSC.2007.4487279.
[BibTeX]
@inproceedings{Schreier:2007:Correlation-Coefficients-for-Complex-Ran,
abstract = {We consider the assessment of multivariate association between two complex random vectors. For complex data, there are three types of correlation coefficients, which account for rotational, reflectional, and total (i.e., rotational and reflectional) dependencies. We define and analyze these three types for different correlation coefficients, based on two popular correlation analysis techniques: canonical correlation analysis and multivariate linear regression (also known as half-canonical correlation analysis).},
author = {Schreier, Peter J.},
booktitle = {{P}roc.\ 41st\ {A}silomar {C}onf.\ {S}ignals {S}yst.\ {C}omputers},
month = {{N}ovember},
title = {Correlation coefficients for complex random vectors},
year = {2007},
pages = {577–581},
doi = {10.1109/ACSSC.2007.4487279},
}
[Abstract]
We consider the assessment of multivariate association between two complex random vectors. For complex data, there are three types of correlation coefficients, which account for rotational, reflectional, and total (i.e., rotational and reflectional) dependencies. We define and analyze these three types for different correlation coefficients, based on two popular correlation analysis techniques: canonical correlation analysis and multivariate linear regression (also known as half-canonical correlation analysis).

63

Frequency-domain properties of locally stationary improper second-order stochastic processes
(Patrik Wahlberg and Peter J. Schreier)
Proc. 41st Asilomar Conf. Signals Syst. Computers,  pp. 1093–1097, November 2007.
DOI:10.1109/ACSSC.2007.4487391.
[BibTeX]
@inproceedings{WahlbergSchreier:2007:Frequency-domain-properties-of-locally-s,
abstract = {This paper concerns continuous-time second-order complex improper stochastic processes that satisfy Silverman's condition of local stationarity, which is a generalization of wide-sense stationarity (WSS). We study their spectral relations, proving a necessary condition that is related to a characterizing inequality of the pair of complementary spectral measure and spectral measure in the WSS case.},
author = {Wahlberg, Patrik and Schreier, Peter J.},
booktitle = {{P}roc.\ 41st\ {A}silomar {C}onf.\ {S}ignals {S}yst.\ {C}omputers},
month = {{N}ovember},
title = {Frequency-domain properties of locally stationary improper second-order stochastic processes},
year = {2007},
pages = {1093–1097},
doi = {10.1109/ACSSC.2007.4487391},
}
[Abstract]
This paper concerns continuous-time second-order complex improper stochastic processes that satisfy Silverman's condition of local stationarity, which is a generalization of wide-sense stationarity (WSS). We study their spectral relations, proving a necessary condition that is related to a characterizing inequality of the pair of complementary spectral measure and spectral measure in the WSS case.

64

Turbo equalization with irregular turbo codes
(Vladimir D. Trajkovic, Minyue Fu and Peter J. Schreier)
Proc. 4th Int. Symp. Wireless Comm. Syst.,  pp. 153–157, October 2007.
DOI:10.1109/ISWCS.2007.4392320.
[BibTeX]
@inproceedings{TrajkovicFuSchreier:2007:Turbo-Equalization-With-Irregular-Turbo-,
abstract = {We analyze a turbo equalization system that combines maximum a posteriori probability (MAP) equalization with irregular turbo codes. Our goal is to approach the information capacity limit for severe inter-symbol interference (ISI) channels. To this end, we optimize the degree profile of irregular turbo codes by maximizing the minimum distance between the mutual information transfer functions for the MAP equalizer and decoder. We show that turbo equalizers employing such optimized irregular turbo codes can approach the information capacity limit of some severe ISI channels within 0.75 dB.},
author = {Trajkovic, Vladimir D. and Fu, Minyue and Schreier, Peter J.},
booktitle = {{P}roc.\ 4th\ {I}nt.\ {S}ymp.\ {W}ireless {C}omm.\ {S}yst.},
month = {{O}ctober},
title = {Turbo equalization with irregular turbo codes},
year = {2007},
pages = {153–157},
doi = {10.1109/ISWCS.2007.4392320},
}
[Abstract]
We analyze a turbo equalization system that combines maximum a posteriori probability (MAP) equalization with irregular turbo codes. Our goal is to approach the information capacity limit for severe inter-symbol interference (ISI) channels. To this end, we optimize the degree profile of irregular turbo codes by maximizing the minimum distance between the mutual information transfer functions for the MAP equalizer and decoder. We show that turbo equalizers employing such optimized irregular turbo codes can approach the information capacity limit of some severe ISI channels within 0.75 dB.

65

Spectra of multidimensional complex improper (almost) cyclostationary processes
(Patrik Wahlberg and Peter J. Schreier)
Proc. IEEE Int. Symp. Inform. Theory,  pp. 971–975, June 2007.
DOI:10.1109/ISIT.2007.4557350.
[BibTeX]
@inproceedings{WahlbergSchreier:2007:Spectra-of-multidimensional-complex-impr,
abstract = {We analyze the spectral measure and complementary spectral measure for strongly harmonizable cyclostationary and almost cyclostationary multidimensional complex improper processes. We show that the off-diagonal components of the spectral measure are absolutely continuous with respect to the diagonal component, which is a generalization of a result for scalar processes. For scalar almost cyclostationary processes, we derive representation formulas for the complementary spectral measure and the off-diagonal components of the spectral measure, in terms of the diagonal component of the spectral measure. These results are similar to the cyclostationary case, with some modifications concerning the off-diagonal components of the complementary spectral measure.},
author = {Wahlberg, Patrik and Schreier, Peter J.},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {S}ymp.\ {I}nform.\ {T}heory},
month = {{J}une},
title = {Spectra of multidimensional complex improper (almost) cyclostationary processes},
year = {2007},
pages = {971–975},
doi = {10.1109/ISIT.2007.4557350},
}
[Abstract]
We analyze the spectral measure and complementary spectral measure for strongly harmonizable cyclostationary and almost cyclostationary multidimensional complex improper processes. We show that the off-diagonal components of the spectral measure are absolutely continuous with respect to the diagonal component, which is a generalization of a result for scalar processes. For scalar almost cyclostationary processes, we derive representation formulas for the complementary spectral measure and the off-diagonal components of the spectral measure, in terms of the diagonal component of the spectral measure. These results are similar to the cyclostationary case, with some modifications concerning the off-diagonal components of the complementary spectral measure.

66

A new interpretation of bilinear time-frequency distributions
(Peter J. Schreier)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process.,  pp. 1133–1136, April 2007.
DOI:10.1109/ICASSP.2007.366884.
[BibTeX]
@inproceedings{Schreier:2007:A-New-Interpretation-of-Bilinear-Time-Fr,
abstract = {Wigner's theorem states that there exists no bilinear time-frequency distribution (TFD) that has correct marginals and is nonnegative everywhere. This means that any attempt to interpret a bilinear TFD as an energy or power distribution must be fraught with problems. In this paper, an alternative perspective is proposed, which allows a local interpretation at a point in the time-frequency plane. This approach is based on analyzing the properties of a chirping ellipse that, at a given time instant, gives the best local approximation of the signal from a given frequency. This chirping ellipse is described in terms of its mean shape, orientation, and direction of polarization (counterclockwise or clockwise). A time-frequency coherence measures the quality of the approximation that this ellipse presents. The ellipse parameters and the time-frequency coherence can be expressed in terms of the Rihaczek TFD},
author = {Schreier, Peter J.},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{A}pril},
title = {A new interpretation of bilinear time-frequency distributions},
year = {2007},
pages = {1133–1136},
doi = {10.1109/ICASSP.2007.366884},
}
[Abstract]
Wigner's theorem states that there exists no bilinear time-frequency distribution (TFD) that has correct marginals and is nonnegative everywhere. This means that any attempt to interpret a bilinear TFD as an energy or power distribution must be fraught with problems. In this paper, an alternative perspective is proposed, which allows a local interpretation at a point in the time-frequency plane. This approach is based on analyzing the properties of a chirping ellipse that, at a given time instant, gives the best local approximation of the signal from a given frequency. This chirping ellipse is described in terms of its mean shape, orientation, and direction of polarization (counterclockwise or clockwise). A time-frequency coherence measures the quality of the approximation that this ellipse presents. The ellipse parameters and the time-frequency coherence can be expressed in terms of the Rihaczek TFD

67

Causal cyclic Wiener filtering
(Mark S. Spurbeck, Peter J. Schreier and Louis L. Scharf)
Proc. 40th Asilomar Conf. Signals Syst. Computers,  pp. 1425–1429, November 2006.
DOI:10.1109/ACSSC.2006.354993.
[BibTeX]
@inproceedings{SchreierSpurbeckScharf:2006:Causal-cyclic-Wiener-filtering,
abstract = {We develop a causal filter bank implementation of the cyclic Wiener filter for periodically correlated (PC, or cyclostationary) time series. By converting the PC time series into a vector-valued wide-sense stationary (WSS) time series, we may utilize the existing literature on factorization of spectral density matrices. However, because PC analytic and equivalent baseband signals are generally improper, spectral factorization algorithms must be modified for the improper case. Then, given the spectral density matrix for the equivalent WSS vector process, a causal cyclic Wiener filter can be implemented as a multirate filter bank or an equivalent polyphase structure.},
author = {Spurbeck, Mark S. and Schreier, Peter J. and Scharf, Louis L.},
booktitle = {{P}roc.\ 40th\ {A}silomar {C}onf.\ {S}ignals {S}yst.\ {C}omputers},
month = {{N}ovember},
title = {Causal cyclic {Wiener} filtering},
year = {2006},
pages = {1425–1429},
doi = {10.1109/ACSSC.2006.354993},
}
[Abstract]
We develop a causal filter bank implementation of the cyclic Wiener filter for periodically correlated (PC, or cyclostationary) time series. By converting the PC time series into a vector-valued wide-sense stationary (WSS) time series, we may utilize the existing literature on factorization of spectral density matrices. However, because PC analytic and equivalent baseband signals are generally improper, spectral factorization algorithms must be modified for the improper case. Then, given the spectral density matrix for the equivalent WSS vector process, a causal cyclic Wiener filter can be implemented as a multirate filter bank or an equivalent polyphase structure.

68

A statistical test for impropriety of complex random signals
(Peter J. Schreier, Louis L. Scharf and Alfred Hanssen)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process.,  pp. 796–799, May 2006.
DOI:10.1109/ICASSP.2006.1660774.
[BibTeX]
@inproceedings{SchreierScharfHanssen:2006:A-Statistical-Test-for-Impropriety-of-Co,
abstract = {A complex random vector is called improper if it is correlated with its complex conjugate. In this paper, we present a generalized likelihood ratio test (GLRT) for impropriety. This test is compelling because it displays the right invariances: The proposed GLR is invariant to linear transformations on the data, including rotation and scaling, just as propriety is preserved by linear transformations. Because canonical correlations make up a complete, or maximal, set of invariants for the Hermitian and complementary covariance matrices under linear transformations, the GLR can be shown to be a function of the squared canonical correlations between the data and its complex conjugate. This validates our intuition that the internal coordinate system should not matter for this hypothesis test},
author = {Schreier, Peter J. and Scharf, Louis L. and Hanssen, Alfred},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{M}ay},
title = {A statistical test for impropriety of complex random signals},
year = {2006},
pages = {796–799},
doi = {10.1109/ICASSP.2006.1660774},
}
[Abstract]
A complex random vector is called improper if it is correlated with its complex conjugate. In this paper, we present a generalized likelihood ratio test (GLRT) for impropriety. This test is compelling because it displays the right invariances: The proposed GLR is invariant to linear transformations on the data, including rotation and scaling, just as propriety is preserved by linear transformations. Because canonical correlations make up a complete, or maximal, set of invariants for the Hermitian and complementary covariance matrices under linear transformations, the GLR can be shown to be a function of the squared canonical correlations between the data and its complex conjugate. This validates our intuition that the internal coordinate system should not matter for this hypothesis test

69

A geometric interpretation of the Rihaczek time-frequency distribution for stochastic signals
(Peter J. Schreier, Louis L. Scharf and Alfred Hanssen)
Proc. IEEE Int. Symp. Inform. Theory,  pp. 966–969, September 2005.
DOI:10.1109/ISIT.2005.1523481.
[BibTeX]
@inproceedings{SchreierScharfHanssen:2005:A-geometric-interpretation-of-the-Rihacz,
abstract = {Based on the Cramer-Loeve spectral representation for a harmonizable random process, the Rihaczek distribution is a time- and frequency-shift covariant, bilinear time-frequency distribution. It can be expressed as a complex Hilbert space inner product between the time series and its infinitesimal stochastic Fourier generator. We show that we may attach an illuminating geometry to this inner product, wherein the cosine-squared of the angle between the time series and its infinitesimal stochastic Fourier generator is given by the Rihaczek distribution. We propose to construct estimators of the Rihaczek distribution using a factored kernel in Cohen's class of bilinear time-frequency distributions},
author = {Schreier, Peter J. and Scharf, Louis L. and Hanssen, Alfred},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {S}ymp.\ {I}nform.\ {T}heory},
month = {{S}eptember},
title = {A geometric interpretation of the {Rihaczek} time-frequency distribution for stochastic signals},
year = {2005},
pages = {966–969},
doi = {10.1109/ISIT.2005.1523481},
}
[Abstract]
Based on the Cramer-Loeve spectral representation for a harmonizable random process, the Rihaczek distribution is a time- and frequency-shift covariant, bilinear time-frequency distribution. It can be expressed as a complex Hilbert space inner product between the time series and its infinitesimal stochastic Fourier generator. We show that we may attach an illuminating geometry to this inner product, wherein the cosine-squared of the angle between the time series and its infinitesimal stochastic Fourier generator is given by the Rihaczek distribution. We propose to construct estimators of the Rihaczek distribution using a factored kernel in Cohen's class of bilinear time-frequency distributions

70

A note on aliasing in higher order spectra
(Peter J. Schreier)
Proc. 6th Australian Comm. Theory Works.,  pp. 184–188, February 2005.
DOI:10.1109/AUSCTW.2005.1624249.
[BibTeX]
@inproceedings{Schreier:2005:A-note-on-aliasing-in-higher-order-spect,
abstract = {There are two types of aliasing in higher order spectra: ``regular aliasing'' due to sampling below the Nyquist frequency, and ``higher order aliasing''. Spectra of discrete-time signals may suffer from higher-order aliasing if the signals are not sufficiently oversampled. By providing some insight into the cause of higher order aliasing, we show that higher order aliasing can just as well occur in second order spectra. More importantly, we demonstrate that spectra of stationary random signals defined as ensemble-averages and spectra of ergodic random signals defined as the Fourier transform of infinite time-averages never exhibit higher order aliasing},
author = {Schreier, Peter J.},
booktitle = {{P}roc.\ 6th\ {A}ustralian {C}omm.\ {T}heory {W}orks.},
month = {{F}ebruary},
title = {A note on aliasing in higher order spectra},
year = {2005},
pages = {184–188},
doi = {10.1109/AUSCTW.2005.1624249},
}
[Abstract]
There are two types of aliasing in higher order spectra: ``regular aliasing'' due to sampling below the Nyquist frequency, and ``higher order aliasing''. Spectra of discrete-time signals may suffer from higher-order aliasing if the signals are not sufficiently oversampled. By providing some insight into the cause of higher order aliasing, we show that higher order aliasing can just as well occur in second order spectra. More importantly, we demonstrate that spectra of stationary random signals defined as ensemble-averages and spectra of ergodic random signals defined as the Fourier transform of infinite time-averages never exhibit higher order aliasing

71

Polyspectra of analytic signals
(Peter J. Schreier and Louis L. Scharf)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process.,  pp. 473–476, May 2004.
DOI:10.1109/ICASSP.2004.1326297.
[BibTeX]
@inproceedings{SchreierScharf:2004:Polyspectra-of-analytic-signals,
abstract = {For complex signals, n-th order moment functions can be defined in 2n different ways, depending on the placement of complex conjugates. We demonstrate that, for stationary analytic signals, only a few of these different moments are actually required for a complete n-th order description. Which, and how many of them, depends on the signal's spectrum. We investigate properties of n-th order moments and spectra with different conjugation patterns and show how they provide different information about the signal.},
author = {Schreier, Peter J. and Scharf, Louis L.},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{M}ay},
title = {Polyspectra of analytic signals},
year = {2004},
pages = {473–476},
doi = {10.1109/ICASSP.2004.1326297},
}
[Abstract]
For complex signals, n-th order moment functions can be defined in 2n different ways, depending on the placement of complex conjugates. We demonstrate that, for stationary analytic signals, only a few of these different moments are actually required for a complete n-th order description. Which, and how many of them, depends on the signal's spectrum. We investigate properties of n-th order moments and spectra with different conjugation patterns and show how they provide different information about the signal.

72

Widely-linear beamforming
(Todd McWhorter and Peter J. Schreier)
Proc. 37th Asilomar Conf. Signals Syst. Computers,  pp. 753–759, November 2003.
DOI:10.1109/ACSSC.2003.1292015.
[BibTeX]
@inproceedings{McWhorterSchreier:2003:Widely-linear-beamforming,
abstract = {In this paper we describe a beamforming algorithm based on widely-linear rather than linear data models. Initially, we develop this beamformer by generalizing the Capon (MVDR) optimization problem. That is, if the objective is to minimize output power while maintaining a specified directional gain, then we show that the output power of the widely-linear beamformer is less than or equal to the output power of the Capon (MVDR) beamformer. This result is valid regardless of the ``true'' distribution of the data. We also derive the widely-linear beamformer by considering beamforming to be an estimation problem. Linear models assume that the composite covariance matrix formed from the real and imaginary parts of the array-snapshot has a particular structure. This structure is often summarized by stating that the covariance formed from the array snapshot and its transpose (not Hermitian transpose) is zero. We could also call these data ``proper'' Gaussian vectors. The beamformers in this paper are appropriate for situations in which these implicit assumptions are violated.},
author = {McWhorter, Todd and Schreier, Peter J.},
booktitle = {{P}roc.\ 37th\ {A}silomar {C}onf.\ {S}ignals {S}yst.\ {C}omputers},
month = {{N}ovember},
title = {Widely-linear beamforming},
year = {2003},
pages = {753–759},
doi = {10.1109/ACSSC.2003.1292015},
}
[Abstract]
In this paper we describe a beamforming algorithm based on widely-linear rather than linear data models. Initially, we develop this beamformer by generalizing the Capon (MVDR) optimization problem. That is, if the objective is to minimize output power while maintaining a specified directional gain, then we show that the output power of the widely-linear beamformer is less than or equal to the output power of the Capon (MVDR) beamformer. This result is valid regardless of the ``true'' distribution of the data. We also derive the widely-linear beamformer by considering beamforming to be an estimation problem. Linear models assume that the composite covariance matrix formed from the real and imaginary parts of the array-snapshot has a particular structure. This structure is often summarized by stating that the covariance formed from the array snapshot and its transpose (not Hermitian transpose) is zero. We could also call these data ``proper'' Gaussian vectors. The beamformers in this paper are appropriate for situations in which these implicit assumptions are violated.

73

Canonical coordinates are the right coordinate system for transform coding of noisy sources
(Peter J. Schreier, Louis L. Scharf, Tianjian Hu and Stephen D. Voran)
Proc. IEEE Works. Statistical Signal Proces.,  pp. 234–237, September 2003.
DOI:10.1109/SSP.2003.1289387.
[BibTeX]
@inproceedings{SchreierScharfHu:2003:Canonical-coordinates-are-the-right-coor,
abstract = {Historically, transform coding of noisy sources has been performed by first estimating the message and then quantizing this estimate. We show that it is also optimum to first transform the noisy observations into canonical coordinates, quantize, apply a Wiener filter in this coordinate system, and then transform the result back to the original coordinates. Canonical coordinates are uncorrelated, and quantizing and Wiener filtering are applied to each component independently. Optimality of this approach can be proved assuming additive white quantization noise. Half canonical coordinates minimize the mean-squared error by minimizing the trace of the error covariance matrix and full canonical coordinates maximize information rate by minimizing the determinant of the error covariance matrix.},
author = {Schreier, Peter J. and Scharf, Louis L. and Hu, Tianjian and Voran, Stephen D.},
booktitle = {{P}roc.\ {IEEE} {W}orks.\ {S}tatistical {S}ignal {P}roces.},
month = {{S}eptember},
title = {Canonical coordinates are the right coordinate system for transform coding of noisy sources},
year = {2003},
pages = {234–237},
doi = {10.1109/SSP.2003.1289387},
}
[Abstract]
Historically, transform coding of noisy sources has been performed by first estimating the message and then quantizing this estimate. We show that it is also optimum to first transform the noisy observations into canonical coordinates, quantize, apply a Wiener filter in this coordinate system, and then transform the result back to the original coordinates. Canonical coordinates are uncorrelated, and quantizing and Wiener filtering are applied to each component independently. Optimality of this approach can be proved assuming additive white quantization noise. Half canonical coordinates minimize the mean-squared error by minimizing the trace of the error covariance matrix and full canonical coordinates maximize information rate by minimizing the determinant of the error covariance matrix.

74

A unified approach to performance comparisons between linear and widely linear processing
(Peter J. Schreier, Louis L. Scharf and Clifford T. Mullis)
Proc. IEEE Works. Statistical Signal Proces.,  pp. 114–117, September 2003.
DOI:10.1109/SSP.2003.1289353.
[BibTeX]
@inproceedings{SchreierScharfMullis:2003:A-unified-approach-to-performance-compar,
abstract = {Recently, a number of papers have been published that show significant performance gains can be obtained by accounting for the fact that communication signals can be improper. In this paper, we derive a general result comparing the performance of conventional processing, which ignores the improper nature of signals, with processing that takes it into account. In particular, for an estimation and a detection problem, we find that the performance gain, as measured by mean-squared error and deflection, respectively, can be as large as a factor of 2, but no larger. In a communications example, we show how this finding generalizes the result that coherent processing enjoys a 3 dB gain over non-coherent processing.},
author = {Schreier, Peter J. and Scharf, Louis L. and Mullis, Clifford T.},
booktitle = {{P}roc.\ {IEEE} {W}orks.\ {S}tatistical {S}ignal {P}roces.},
month = {{S}eptember},
title = {A unified approach to performance comparisons between linear and widely linear processing},
year = {2003},
pages = {114–117},
doi = {10.1109/SSP.2003.1289353},
}
[Abstract]
Recently, a number of papers have been published that show significant performance gains can be obtained by accounting for the fact that communication signals can be improper. In this paper, we derive a general result comparing the performance of conventional processing, which ignores the improper nature of signals, with processing that takes it into account. In particular, for an estimation and a detection problem, we find that the performance gain, as measured by mean-squared error and deflection, respectively, can be as large as a factor of 2, but no larger. In a communications example, we show how this finding generalizes the result that coherent processing enjoys a 3 dB gain over non-coherent processing.

75

The Karhunen-Loève expansion of improper complex random signals with applications in detection
(Peter J. Schreier and Louis L. Scharf)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process.,  pp. 717–720, April 2003.
DOI:10.1109/ICASSP.2003.1201782.
[BibTeX]
@inproceedings{SchreierScharf:2003:The-Karhunen-Loeve-expansion-of-improper,
abstract = {Non-stationary complex random signals are in general improper (not circularly symmetric), which means that their complementary covariance is non-zero. Since the Karhunen-Loeve expansion in its known form is only valid for proper processes, we derive the improper version of this expansion. It produces two sets of eigenvalues and an improper internal description. We use the Karhunen-Loeve expansion to solve the problem of detecting non-stationary improper complex random signals in additive white Gaussian noise. Using the deflection criterion we compare the performance of conventional processing, which ignores complementary covariances, with processing that takes these into account. The performance gain can be as great as a factor of 2.},
author = {Schreier, Peter J. and Scharf, Louis L.},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{A}pril},
title = {The {Karhunen-Lo{\`{e}}ve} expansion of improper complex random signals with applications in detection},
year = {2003},
pages = {717–720},
doi = {10.1109/ICASSP.2003.1201782},
}
[Abstract]
Non-stationary complex random signals are in general improper (not circularly symmetric), which means that their complementary covariance is non-zero. Since the Karhunen-Loeve expansion in its known form is only valid for proper processes, we derive the improper version of this expansion. It produces two sets of eigenvalues and an improper internal description. We use the Karhunen-Loeve expansion to solve the problem of detecting non-stationary improper complex random signals in additive white Gaussian noise. Using the deflection criterion we compare the performance of conventional processing, which ignores complementary covariances, with processing that takes these into account. The performance gain can be as great as a factor of 2.

76

Reducing interference in stochastic time-frequency analysis without losing information
(Peter J. Schreier and Louis L. Scharf)
Proc. 36th Asilomar Conf. Signals Syst. Computers,  pp. 1565–1570, November 2002.
DOI:10.1109/ACSSC.2002.1197041.
[BibTeX]
@inproceedings{SchreierScharf:2002:Reducing-interference-in-stochastic-time,
abstract = {The analytic signal is commonly used in stochastic time-frequency analysis in Cohen's class to reduce interference terms. However, we show that the usual time-frequency representation (TFR) based on the analytic signal gives only an incomplete signal description. This is because the analytic signal constructed from a non-stationary real signal is in general improper, which means that it has non-zero complementary correlation. We show how to augment the standard TFR by a complementary TFR to obtain a complete second-order characterization of the signal while still reducing interference terms compared to the TFR of the real signal.},
author = {Schreier, Peter J. and Scharf, Louis L.},
booktitle = {{P}roc.\ 36th\ {A}silomar {C}onf.\ {S}ignals {S}yst.\ {C}omputers},
month = {{N}ovember},
title = {Reducing interference in stochastic time-frequency analysis without losing information},
year = {2002},
pages = {1565–1570},
doi = {10.1109/ACSSC.2002.1197041},
}
[Abstract]
The analytic signal is commonly used in stochastic time-frequency analysis in Cohen's class to reduce interference terms. However, we show that the usual time-frequency representation (TFR) based on the analytic signal gives only an incomplete signal description. This is because the analytic signal constructed from a non-stationary real signal is in general improper, which means that it has non-zero complementary correlation. We show how to augment the standard TFR by a complementary TFR to obtain a complete second-order characterization of the signal while still reducing interference terms compared to the TFR of the real signal.

77

Canonical coordinates for reduced-rank estimation of improper complex random vectors
(Peter J. Schreier and Louis L. Scharf)
Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process.,  pp. 1153–1156, May 2002.
DOI:10.1109/ICASSP.2002.5744004.
[BibTeX]
@inproceedings{SchreierScharf:2002:Canonical-coordinates-for-reduced-rank-e,
abstract = {We consider the problem of minimum mean squared error (MMSE) estimation of complex random vectors in the improper case. Accounting for the information present in the complementary covariance requires the use of widely linear transformations. Based on these, we present the eigenanalysis of improper complex random vectors. This paves the way for a study of two different rank-reduced implementations of the complex Wiener Filter that make use of canonical coordinates: one that is optimum with respect to maximizing mutual information and one that minimizes mean squared error.},
author = {Schreier, Peter J. and Scharf, Louis L.},
booktitle = {{P}roc.\ {IEEE} {I}nt.\ {C}onf.\ {A}coustics, {S}peech and {S}ignal {P}rocess.},
month = {{M}ay},
title = {Canonical coordinates for reduced-rank estimation of improper complex random vectors},
year = {2002},
pages = {1153–1156},
doi = {10.1109/ICASSP.2002.5744004},
}
[Abstract]
We consider the problem of minimum mean squared error (MMSE) estimation of complex random vectors in the improper case. Accounting for the information present in the complementary covariance requires the use of widely linear transformations. Based on these, we present the eigenanalysis of improper complex random vectors. This paves the way for a study of two different rank-reduced implementations of the complex Wiener Filter that make use of canonical coordinates: one that is optimum with respect to maximizing mutual information and one that minimizes mean squared error.

78

Low-rank approximation of improper complex random vectors
(Peter J. Schreier and Louis L. Scharf)
Proc. 35th Asilomar Conf. Signals Syst. Computers,  pp. 597–601, November 2001.
DOI:10.1109/ACSSC.2001.986993.
[BibTeX]
@inproceedings{SchreierScharf:2001:Low-rank-approximation-of-improper-compl,
abstract = {In reduced-rank signal processing for radar, sonar, and digital communications, we seek the right tradeoff between model bias and model variance for reconstructing signals from noisy data. Here, we extend the classical theory by considering the low-rank approximation of complex random vectors, which may or may not be proper. We show that, in general, widely linear approximation is superior to strictly linear approximation, unless the vector to be approximated is proper, in which case the optimum procedure is strictly linear. We analyze the case where the approximated random vector becomes proper in its internal coordinate system. This class of random vector, which we call generalized proper, possesses qualities similar to proper random vectors.},
author = {Schreier, Peter J. and Scharf, Louis L.},
booktitle = {{P}roc.\ 35th\ {A}silomar {C}onf.\ {S}ignals {S}yst.\ {C}omputers},
month = {{N}ovember},
title = {Low-rank approximation of improper complex random vectors},
year = {2001},
pages = {597–601},
doi = {10.1109/ACSSC.2001.986993},
}
[Abstract]
In reduced-rank signal processing for radar, sonar, and digital communications, we seek the right tradeoff between model bias and model variance for reconstructing signals from noisy data. Here, we extend the classical theory by considering the low-rank approximation of complex random vectors, which may or may not be proper. We show that, in general, widely linear approximation is superior to strictly linear approximation, unless the vector to be approximated is proper, in which case the optimum procedure is strictly linear. We analyze the case where the approximated random vector becomes proper in its internal coordinate system. This class of random vector, which we call generalized proper, possesses qualities similar to proper random vectors.

79

MAP decoding of linear block codes based on a sectionalized trellis of the dual code
(Peter J. Schreier and Daniel J. Costello, Jr.)
Proc. Int. Zurich Seminar Broadband Comm.,  pp. 271–278, February 2000.
DOI:10.1109/IZSBC.2000.829262.
[BibTeX]
@inproceedings{SchreierCostello:2000:MAP-decoding-of-linear-block-codes-based,
abstract = {Block codes for use in turbo coding schemes provide an alternative to punctured convolutional codes when high rate component codes are needed. Since block codes have large, time-varying trellis diagrams, full maximum a posteriori (MAP) soft-in soft-out decoders are very complex. It is shown how to modify the MAP algorithm to utilize a sectionalized trellis diagram of the dual code for decoding, which minimizes computational complexity for high rate component codes. This paper also gives simulation results for some high rate block turbo codes},
author = {Schreier, Peter J. and {Costello, Jr.}, Daniel J.},
booktitle = {{P}roc.\ {I}nt.\ {Z}urich {S}eminar {B}roadband {C}omm.},
month = {{F}ebruary},
title = {{MAP} decoding of linear block codes based on a sectionalized trellis of the dual code},
year = {2000},
pages = {271–278},
doi = {10.1109/IZSBC.2000.829262},
}
[Abstract]
Block codes for use in turbo coding schemes provide an alternative to punctured convolutional codes when high rate component codes are needed. Since block codes have large, time-varying trellis diagrams, full maximum a posteriori (MAP) soft-in soft-out decoders are very complex. It is shown how to modify the MAP algorithm to utilize a sectionalized trellis diagram of the dual code for decoding, which minimizes computational complexity for high rate component codes. This paper also gives simulation results for some high rate block turbo codes