The Hilbert space geometry of the Rihaczek distribution for stochastic analytic signals - Scharf, Louis L.; Schreier, Peter J.; Hanssen, Alfred (IEEE Signal Process.\ Lett.)
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.

A geometric interpretation of the Rihaczek time-frequency distribution for stochastic signals - Schreier, Peter J.; Scharf, Louis L.; Hanssen, Alfred (Proc.\ IEEE Int.\ Symp.\ Inform.\ Theory)
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

A note on aliasing in higher order spectra - Schreier, Peter J. (Proc. 6th\ Australian Comm.\ Theory Works.)
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

An adaptive channel tracking method for MIMO-OFDM systems - Tong, Jun; Gong, Yaohuan; Sun, Shengxian (Int.\ Conf.\ Comm.\ Circuits Syst.)
The tracking of the time-varying channel is crucial for MIMO-OFDM wireless communication systems. Aiming at doubly-selective fading MIMO channels, we present a novel pilot pattern and the corresponding adaptive channel tracking algorithm. Compared with the traditional pilot patterns, the new pattern has higher frequency efficiency and is more suitable for large transmit arrays. The computational complexity, frequency efficiency and BER performance of the system assisted by the proposed channel tracking scheme are analyzed. Simulation results demonstrate that the proposed scheme can track the timevarying channel effectively.

Polyspectra of analytic signals - Schreier, Peter J.; Scharf, Louis L. (Proc.\ IEEE Int.\ Conf.\ Acoustics, Speech and Signal Process.)
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.

The Karhunen-Loève expansion of improper complex random signals with applications in detection - Schreier, Peter J.; Scharf, Louis L. (Proc.\ IEEE Int.\ Conf.\ Acoustics, Speech and Signal Process.)
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.

Widely-linear beamforming - McWhorter, Todd; Schreier, Peter J. (Proc. 37th\ Asilomar Conf.\ Signals Syst.\ Computers)
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.

Canonical coordinates are the right coordinate system for transform coding of noisy sources - Schreier, Peter J.; Scharf, Louis L.; Hu, Tianjian; Voran, Stephen D. (Proc.\ IEEE Works.\ Statistical Signal Proces.)
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.

A unified approach to performance comparisons between linear and widely linear processing - Schreier, Peter J.; Scharf, Louis L.; Mullis, Clifford T. (Proc.\ IEEE Works.\ Statistical Signal Proces.)
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.

Stochastic time-frequency analysis using the analytic signal: why the complementary distribution matters - Schreier, Peter J.; Scharf, Louis L. (IEEE Trans.\ Signal\ Process.)
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.

Second-order analysis of improper complex random vectors and processes - Schreier, Peter J.; Scharf, Louis L. (IEEE Trans.\ Signal\ Process.)
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.

Reducing interference in stochastic time-frequency analysis without losing information - Schreier, Peter J.; Scharf, Louis L. (Proc. 36th\ Asilomar Conf.\ Signals Syst.\ Computers)
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.

Canonical coordinates for reduced-rank estimation of improper complex random vectors - Schreier, Peter J.; Scharf, Louis L. (Proc.\ IEEE Int.\ Conf.\ Acoustics, Speech and Signal Process.)
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.

Low-rank approximation of improper complex random vectors - Schreier, Peter J.; Scharf, Louis L. (Proc. 35th\ Asilomar Conf.\ Signals Syst.\ Computers)
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.

MAP decoding of linear block codes based on a sectionalized trellis of the dual code - Schreier, Peter J.; Costello, Jr., Daniel J. (Proc.\ Int.\ Zurich Seminar Broadband Comm.)
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