In this work we consider a multistatic passive detection problem, which is motivated by a multiple-input multiple-output (MIMO) passive radar application. Specifically, we consider a single illuminator of opportunity (IO) and several surveillance and reference arrays for the detection of a Gaussian cyclostationary signal in temporally colored and spatially correlated noise. Concretely, 1) we derive the generalized likelihood ratio test (GLRT) for this problem and 2) provide a stochastic representation of the test statistic under the null hypothesis, which allows us to set the threshold for a constant probability of false alarm. Monte Carlo simulations are carried out to investigate the performance of the proposed GLRT.

Data fusion—the joint analysis of multiple datasets—through coupled factorizations has the promise to enable enhanced knowledge discovery, and hence is an active area. Various formulations of coupled matrix factorizations have been proposed, each with its own modeling assumptions. In this paper, we study two such methods, namely Independent Vector Analysis (IVA), i.e., extension of Independent Component Analysis (ICA) to multiple datasets, and PARAFAC2, a tensor factorization approach. We demonstrate the modeling assumptions of IVA and PARAFAC2 using simulations, revealing that both methods can accurately capture the latent components, albeit with certain differences in capturing the corresponding subject scores. By making use of a rich multi-task functional Magnetic Resonance Imaging (fMRI) dataset, we show how the two methods can be used for achieving two important goals at once, namely capturing group differences between patients with schizophrenia and healthy controls with interpretable components, as well as understanding the relationship across multiple tasks. This is achieved through the definition of source component vectors across datasets.

This paper addresses the problem of testing whether a covariance matrix can be expressed by an unknown linear combination of a set of known matrices or by another unknown linear combination of a set of different, but known, matrices. This problem is of interest in a wide range of real-world applications, such as radar, sonar, and spectrum sensing. We study the problem under the Gaussian assumption and derive the generalized likelihood ratio test (GLRT). Since there is no general closed-form solution for the maximum likelihood (ML) estimates of the covariance matrices, which are required for the GLRT, we resort to a powerful inverse iteration algorithm. Finally, an example, along with numerical results, is given to illustrate the methodology.

This paper considers improper Gaussian signaling (IGS) in an overlay cognitive radio scenario. We follow a protocol in which the secondary user (SU) uses part of its power to relay the message for the primary user (PU) and consider a simple yet illustrative 2-user scenario. We analyze two communication schemes depending on whether or not the PU cooperates with the SU and derive closed-form expressions for the optimal transmission parameters that maximize the SU rate while ensuring a specified minimum performance of the PU. Our numerical results show that IGS may significantly outperform proper signaling and that, interestingly, the cooperative approach provides negligible performance gains over its non-cooperative counterpart.

Estimating the number of correlated components between two data sets is a challenging task in the case of small sample support. Typically, a rank-reduction preprocessing step based on principal component analysis (PCA) is carried out on each data set individually to reduce the dimensionality before analyzing correlation between the data sets. However, PCA retains the components with the largest variance within a data set, and therefore fails when these components are not the ones that account for the correlation between the data sets. To overcome this, we propose an alternative technique that, instead of projecting the data into a single subspace, uses a large number of random projections.

Improper Gaussian signaling (IGS) has been shown to enlarge the rate region achievable by conventional proper Gaussian signaling (PGS) schemes in several interference-limited multiuser networks. In this work, we consider the 2-user broadcast channel (BC) when treating interference as noise "TIN" at every receiver. For this scenario, we derive a closed-form characterization of the rate region boundary with IGS. The Pareto-optimal points are achieved when at least one of the users employs maximally improper (rectilinear) signals. Differently from other interference-limited networks, our results show that IGS always outperforms PGS for the 2-user BC with TIN. Furthermore, IGS also enlarges the PGS rate region with time-sharing for this scenario.

Improper Gaussian signaling (IGS) has been used as an effective interference management tool in interference limited systems. Improper Gaussian signals are correlated with their complex conjugates. In this paper, we investigate the optimality of IGS from an energy efficiency (EE) perspective. First, we obtain closed form optimality conditions for IGS. We then leverage these conditions to devise a bisection method that finds the optimal transmission parameters. Our results show that IGS can improve the EE of an underlay cognitive radio system.

In array signal processing, the detection of the number of sources is an important step. Most approaches assume the signals to be embedded in white noise. However, this assumption is unrealistic in many scenarios. In this letter, we propose a strategy that can handle colored noise. We model the source detection as a regression problem and apply information-theoretic criteria to determine the model order of the regression. We show simulations of different scenarios, where our approach outperforms traditional techniques.

This paper considers an unlicensed multiple-access channel (MAC) that coexists with a licensed point-to-point user, following the underlay cognitive radio paradigm. We assume that every transceiver except the secondary base station has one antenna and that the primary user (PU) is protected by a minimum rate constraint. In contrast to the conventional assumption of proper Gaussian signaling, we allow the secondary users to transmit improper Gaussian signals, which are correlated with their complex conjugate. When the secondary base station performs zero-forcing, we show that improper signaling is optimal if the sum of the interference channel gains (in an equivalent canonical model) is above a certain threshold. Additionally, we derive an efficient algorithm to compute the transmission parameters that attain the rate region boundary for this scenario. The proposed algorithm exploits a single-user representation of the secondary MAC along with new results on the optimality of improper signaling in the single-user case when the PU is corrupted by an improper noise.

It has been shown that improper Gaussian signaling (IGS) can improve the performance of wireless interference-limited systems when perfect channel-state information (CSI) is available. In this paper, we investigate the robustness of IGS against imperfect CSI on the transmitter side in a two-user single-input single-output (SISO) interference channel (IC) as well as in a SISO Z-IC, when interference is treated as noise. We assume that the true channel coefficients belong to a known region around the channel estimates, which we call the uncertainty region. Following a worst-case robustness approach, we study the rate-region boundary of the IC for the worst channel in the uncertainty region. For the two-user IC, we derive a robust design in closed form, which is independent of the phase of the channels by allowing only one of the users to transmit IGS. For the Z-IC, we provide a closed-form design for the transmission parameters by considering an enlarged uncertainty region and allowing both users to employ IGS. In both cases, the IGS-based designs are ensured to perform no worse than proper Gaussian signaling. Furthermore, we show, through numerical examples, that the proposed robust designs significantly outperform non-robust solutions.

Hardware non-idealities are among the main performance restrictions for upcoming wireless communication systems. Asymmetric hardware distortions (HWD) happen when the impairments of the I/Q branches are correlated or imbalanced, which in turn generate improper additive interference at the receiver side. When the interference is improper, as well as in other interference-limited scenarios, improper Gaussian signaling (IGS) has been shown to provide rate and/or power efficiency benefits. In this paper, we investigate the rate benefits of IGS in a two-user interference channel (IC) with additive asymmetric HWD when interference is treated as noise. We propose two iterative algorithms to optimize the parameters of the improper transmit signals. We first rewrite the rate region as an pseudo-signal-to-interference-plus-noise-ratio (PSINR) region and employ majorization minimization and fractional programming to find a suboptimal solution for the achievable user rates. Then, we propose a simplified algorithm based on a separate optimization of the powers and complementary variances of the users, which exhibits lower computational complexity. We show that IGS can improve the performance of the two-user IC with additive HWD. Our proposed algorithms outperform proper Gaussian signaling and competing IGS algorithms in the literature that do not consider asymmetric HWD.

The transmission of improper Gaussian signals, instead of the conventional proper ones, has been shown to improve the performance in interference-limited networks. In this work we analyze the performance of a multiple-antenna user that transmits maximally improper signals and whose transmit covariance matrix satisfies a set of constraints that limit the harmfulness of the interference caused by this user. As opposed to the single-antenna case, there are different possible improper spatial signatures, which provide different performance. We first obtain new results for maximally improper random vectors based on majorization theory. We then apply these results to derive the improper spatial signatures that either maximize or minimize the performance. Numerical examples show that the performance difference between these two extreme cases can be surprisingly large.

We propose a technique that jointly detects the presence of almost-cyclostationary (ACS) signals in wide-sense stationary noise and provides an estimate of their cycle period. Since the cycle period of an ACS process is not an integer, the approach is based on a combination of a resampling stage and a multiple hypothesis test, which deal separately with the fractional part and the integer part of the cycle period. The approach requires resampling the signal at many different rates, which is computationally expensive. For this reason, we propose a filter bank structure that allows us to efficiently resample a signal at many different rates by identifying common interpolation stages among the set of resampling rates.

In spectrum sensing for cognitive radio, the presence of a primary user can be detected by making use of the cyclostationarity property of digital communication signals. For the general scenario of a cyclostationary signal in temporally colored and spatially correlated noise, it has previously been shown that an asymptotic generalized likelihood ratio test (GLRT) and locally most powerful invariant test (LMPIT) exist. In this paper, we derive detectors for the presence of a cyclostationary signal in various scenarios with structured noise. In particular, we consider noise that is temporally white and/or spatially uncorrelated. Detectors that make use of this additional information about the noise process have enhanced performance. We have previously derived GLRTs for these specific scenarios; here, we examine the existence of LMPITs. We show that these exist only for detecting the presence of a cyclostationary signal in spatially uncorrelated noise. For white noise, an LMPIT does not exist. Instead, we propose tests that approximate the LMPIT, and they are shown to perform well in simulations. Finally, if the noise structure is not known in advance, we also present hypothesis tests using our framework.

A simple scheme is proposed to achieve the optimal degrees of freedom (DoF) of the temporally correlated two-user multiple-input-multiple-output interference channel with delayed channel state information at the transmitter (CSIT). The delayed CSIT is exploited for temporal alignment at the transmitters and spatial zero-forcing of interference at the receivers. The transmitters estimate the current CSIT using perfect delayed CSIT. For most antenna configurations, the proposed scheme achieves the optimal DoF-region of this channel by precoding (resp. decoding) over a finite number of time slots at each transmitter (resp. receiver). This is an improvement over existing achievable methods, which require infinite channel extensions (hence, infinite decoding delay) and involve complicated encoding and decoding procedures.

We address the problem of determining the number of signals correlated between two high-dimensional data sets with small sample support. In this setting, conventional techniques based on canonical correlation analysis (CCA) cannot be directly applied since the canonical correlations are significantly overestimated when computed from few samples. To overcome this problem, a principal component analysis (PCA) preprocessing step is usually performed to reduce the dimension of the data. However, PCA reduces the dimension of each data set individually without taking the correlation between the data sets into account. In this paper we propose a sparse CCA (SCCA) algorithm as an alternative to the PCA-CCA approach. This algorithm is based on ℓ1-norm penalization, which optimizes the weight of the ℓ1-norm to keep a prescribed number of non-zero components. The number of correlated components is then selected based on an information-theoretic criterion.

Active Shape Models (ASM) are an iterative segmentation technique to find a landmark-based contour of an object. In each iteration, a least-squares fit of a plausible shape to some detected target landmarks is determined. Finding these targets is a critical step: some landmarks are more reliably detected than others, and some landmarks may not be within the field of view of their detectors. To add robustness while preserving simplicity at the same time, a generalized least-squares approach can be used, where a weighting matrix incorporates reliability information about the landmarks. We propose a strategy to choose this matrix, based on the covariance of empirically determined residuals of the fit. We perform a further step to determine whether the target landmarks are within the range of their detectors. We evaluate our strategy on fluoroscopic X-ray images to segment the femur. We show that our technique outperforms the standard ASM as well as other more heuristic weighted least-squares strategies.

A complex-valued signal is improper if it is correlated with its complex conjugate. The dimension of the improper signal subspace, i.e., the number of improper components in a complex-valued measurement, is an important parameter and is unknown in most of the applications. In this letter, we introduce two approaches to estimate this dimension: one based on an information-theoretic criterion and the other based on hypothesis testing. We also present reduced-rank versions of these approaches that work for scenarios where the number of observations is comparable to or even smaller than the dimension of the data. Unlike other techniques for determining model orders, our techniques also work in the presence of additive colored noise.

This paper provides a complete characterization of the boundary of an achievable rate region, called the Pareto boundary, of the single-antenna Z interference channel (Z-IC), when interference is treated as noise and users transmit complex Gaussian signals that are allowed to be improper. By considering the augmented complex formulation, we derive a necessary and sufficient condition for improper signaling to be optimal. This condition is stated as a threshold on the interference channel coefficient, which is a function of the interfered user rate and which allows insightful interpretations into the behavior of the achievable rates in terms of the circularity coefficient (i.e., degree of impropriety). Furthermore, the optimal circularity coefficient is provided in closed form. The simplicity of the obtained characterization permits interesting insights into when and how improper signaling outperforms proper signaling in the single-antenna Z-IC. We also provide an in-depth discussion on the optimal strategies and the properties of the Pareto boundary.

It is well-known that the use of improper signaling schemes can be beneficial in interference-limited networks. Here we consider an underlay cognitive radio scenario, where a multi-antenna primary user is protected by an interference temperature constraint that ensures a prescribed rate requirement. We study how the interference temperature threshold changes when the interference is constrained to be maximally improper. Since the spatial structure of the impropriety is an additional degree of freedom, we provide the maximum value of the interference threshold that ensures the rate requirement. We illustrate the potential payoffs of improper signaling with some numerical examples, which show that a secondary user can significantly improve its achievable rate with respect to the proper signaling case.

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.

Linear signal estimation based on sample covariance matrices (SCMs) can perform poorly if the training data are limited and the SCMs are ill-conditioned. Diagonal loading (DL) may be used to improve robustness in the face of limited training data. This paper introduces two leave-one-out cross-validation schemes for choosing the DL factor. One scheme repeatedly splits the training data with respect to time, while the other repeatedly splits the out-of-training data with respect to space. We derive computationally efficient implementations and compare them with the oracle choice in terms of the mean squared error.

In interference channels (ICs), channel state information (CSI) can be utilized to design transmit signals that are optimally adapted to the state of the channels. This requires feeding CSI back to the transmitters. The CSI available at the transmitters (CSIT) is usually degraded, due to the limited capacity of the feedback link and the delays involved in the channel estimation and feedback. We discuss the scenario of fast fading and delayed CSIT, which is highly relevant in mobile environments with short channel coherence times. We consider the two-user multiple-input-multiple-output (MIMO) IC where the transmitters are provided with delayed CSIT. The DoF region for this channel was characterized by Vaze and Varanasi. We devise a simple and intuitive achievable scheme, which has a unified structure for different antenna configurations. We show that the proposed scheme also achieves the DoF region of the two-user MIMO broadcast channel (BC). In the IC, our scheme does not require the knowledge of direct channels at the transmitters. Moreover, we show that the amount of feedback can be further reduced by exploiting the invariances of the problem. Our approach can be helpful when analyzing more complicated networks.

We consider the two-user Multiple-Input-Multiple-Output Interference Channel (MIMO-IC) with outdated channel state information at the transmitters (CSIT). For a fast- fading scenario, the degrees-of-freedom (DoF) region for this channel has been characterized by Vaze and Varanasi in [1]. We devise a simple achievable scheme, which has a unified structure for different antenna configurations. In the proposed scheme, matrices with random entries chosen from a continuous probability distribution are used in the process of designing the precoders. The unified structure of the precoders involving random matrices makes our scheme suitable for generalization to other networks.

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.

This paper is concerned with determining the number of correlated signals between two data sets using canonical correlation analysis (CCA) when a principal component analysis (PCA) preprocessing step is performed for initial rank reduction. In signal processing applications, it is commonplace in scenarios with large dimensions, insufficient samples, or knowledge of low-rank underlying signals to extract the principal components of the data before correlation is analyzed. While there exist information-theoretic criteria to either determine the number of signals in a single data set or the number of correlated signals between two data sets, there has yet to be a treatment of the joint order estimation of the number of dimensions which should be retained through the PCA preprocessing and the number of correlated signals. We present the likelihood and information criteria for this scenario, along with some verifying simulations.

This paper is concerned with determining the number of correlated signals between two data sets when the number of samples from these data sets is extremely small. In such a scenario, a principal component analysis (PCA) preprocessing step is commonly performed before applying canonical correlation analysis (CCA). We present a reduced-rank version of the hypothesis test based on the Bartlett-Lawley statistic, which allows jointly determining the required PCA dimension reduction and the number of correlated signals.

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.

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 signal 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 number of samples) exploiting Szegö's theorem and its generalization for vector-valued processes. Simulation results show the good performance of the proposed GLRT.

Detecting and analyzing directional structures in images is important in many applications since one-dimensional patterns often correspond to important features such as object contours or trajectories. Classifying a structure as directional or nondirectional requires a measure to quantify the degree of directionality and a threshold, which needs to be chosen based on the statistics of the image. In order to do this, we model the image as a random field. So far, little research has been performed on analyzing directionality in random fields. In this paper, we propose a measure to quantify the degree of directionality based on the random monogenic signal, which enables a unique decomposition of a 2-D signal into local amplitude, local orientation, and local phase. We investigate the second-order statistical properties of the monogenic signal for isotropic, anisotropic, and unidirectional random fields. We analyze our measure of directionality for finite-size sample images and determine a threshold to distinguish between unidirectional and nonunidirectional random fields, which allows the automatic classification of images.

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.

Linear equalization can be applied to combat intersymbol interference (ISI) and cross-antenna interference (CAI) for communication systems over multipath channels. If the channel estimation is imperfect, the receiver uses a mismatched model of the system. Regularized equalization based on Krylov subspace expansion can be applied to improve robustness and reduce complexity for large systems. In this paper, we study the convergence behavior, stopping criteria, and preconditioner design for Krylov subspace methods. We show that the optimal rank can be chosen using a decision-aided estimate of the mean-squared error (MSE). However, due to the semi-convergence behavior, conventional preconditioners may fail to provide gains. To overcome this issue, we introduce regularized preconditioners, which cluster only the largest eigenvalues of the system matrix in Krylov subspace methods.

This paper deals with different techniques for linear equalization 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.

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.

This paper considers large multiple-input multiple-output (MIMO) communication systems with linear precoding and linear minimum mean-squared error (LMMSE) equalization based on the iterative conjugate gradient (CG) algorithm. Convergence of the CG algorithm is fast when the eigenvalues of the received signal's covariance matrix are clustered, suggesting that mean-squared error and receiver complexity can be managed with judicious precoder design. In order to accelerate convergence of an iterative CG receiver, we incorporate constraints on two measures of eigenvalue clustering into the precoder design. Closed-form solutions to the optimal precoders are derived using majorization theory and convex optimization techniques. We show that if there are constraints on receiver complexity, the proposed precoders can improve performance for large MIMO systems operating over slowly time-varying fading channels.

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.

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.

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.

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.

A complex random vector is called improper if it is correlated with its complex conjugate. We introduce a measure for the degree of impropriety, which is a function of the canonical correlations between the vector and its complex conjugate (sometimes called the circularity spectrum). This measure is invariant under linear transformation, and it relates the entropy of an improper Gaussian random vector to its corresponding proper version. For vectors with given spectrum, we present upper and lower bounds on the attainable degree of impropriety, in terms of the eigenvalues of the augmented covariance matrix.

We study continuous-time multidimensional wide- sense stationary (WSS) and (almost) cyclostationary processes in the frequency domain. Under the assumption that the correlation function is uniformly continuous, we prove the existence of a unique sequence of spectral measures, which coincide with the restrictions to certain subdiagonals of the spectral measure in the strongly harmonizable case. Moreover, the off-diagonal measures are absolutely continuous with respect to the diagonal measure. As a consequence, for strongly harmonizable scalar improper almost cyclostationary processes, we obtain representation formulas for the components of the complementary spectral measure and the off-diagonal components of the spectral measure, in terms of the diagonal component of the spectral measure. We apply these results to analytic signals, which produces sufficient conditions for propriety for almost cyclostationary analytic signals.

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.

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.

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

A complex random vector is called improper if it is correlated with its complex conjugate. We present a hypothesis test for impropriety based on a generalized likelihood ratio (GLR). This GLR is invariant to linear transformations on the data, including rotation and scaling, because propriety is preserved by linear transformations. More specifically, we show that the GLR is a function of the squared canonical correlations between the data and their complex conjugate. These canonical correlations make up a complete, or maximal, set of invariants for the Hermitian and complementary covariance matrices under linear, but not widely linear, transformation

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

The Rihaczek distribution for stochastic signals is a time- and frequency-shift covariant bilinear time-frequency distribution (TFD) based on the Crame/spl acute/r-Loe/spl 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.

Nonstationary complex random signals are in general improper (not circularly symmetric), which means that their complementary covariance is nonzero. Since the Karhunen-Loeve (K-L) expansion in its known form is only valid for proper processes, we derive the improper version of this expansion. It produces two sets of eigenvalues and improper observable coordinates. We then use the K-L expansion to solve the problems of detection and estimation of improper complex random signals in additive white Gaussian noise. We derive a general result comparing the performance of conventional processing, which ignores complementary covariances, with processing that takes these into account. In particular, for the detection and estimation problems considered, we find that the performance gain, as measured by deflection and mean-squared error (MSE), respectively, can be as large as a factor of 2. In a communications example, we show how this finding generalizes the result that coherent processing enjoys a 3-dB gain over noncoherent processing.

For complex signals, n-th order moment functions can be defined in 2/sup n/ 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.

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.

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.