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