[Show abstract][Hide abstract] ABSTRACT: Recently, there has been an increased awareness that simplistic adaptation of performance analysis developed for random real-valued signals and parameters to the complex case may be inadequate or may lead to intractable calculations. Unfortunately, many fundamental statistical tools for handling complex-valued parameter estimators are missing or scattered in the open literature. In this paper, we survey some known results and provide a rigorous and unified framework to study the statistical performance of complex-valued parameter estimators with a particular attention paid to properness (i.e., second order circularity), specifically referring to the second-order statistical properties. In particular, some new properties relative to the properness of the estimates, asymptotically minimum variance bound and Whittle formulas are presented. A new look at the role of nuisance parameters is given, proving and illustrating that the noncircular Gaussian distributions do not necessarily improve the Cramer–Rao bound (CRB) with respect to the circular case. Efficiency of subspace-based complex-valued parameter estimators that are presented with a special emphasis is put on noisy linear mixture.
[Show abstract][Hide abstract] ABSTRACT: Arrays of sensors freely located along an axis are considered in this paper that are used to localize a near-field emitting source. Using Taylor expansion and a suitable coordinate system, simple, yet rich to interpret, Cramer-Rao bounds relative to the direction and range parameters are derived. Our analysis allows in particular to unveil a family of non-uniform linear arrays with better near field estimation capabilities, compared to the well-established uniform linear arrays.
[Show abstract][Hide abstract] ABSTRACT: This paper is devoted to the Cramer Rao bound (CRB) on the azimuth, elevation and range of a narrow-band near-field source localized by means of a uniform circular array (UCA), using the exact expression of the time delay parameter. After proving that the conditional and unconditional CRB are generally proportional for constant modulus steering vectors, we specify conditions of isotropy w.r.t. the distance and the number of sensors. Then we derive very simple, yet very accurate non-matrix closed-form expressions of different approximations of the CRBs.
[Show abstract][Hide abstract] ABSTRACT: In the plethora of second-order statistics (SOS) based blind channel equalization techniques, only two algorithms are able to perform equalization with a pre-specified delay. Delay selection is a compelling feature in order to reduce noise enhancement of Zero-Forcing (ZF) equalizers. We show that channel output shifted correlation matrices with different time lags can be combined to obtain a rank-deficient SOS-based matrix whose kernel is made of ZF equalizers of a pre-determined delay. Contrarily to existing algorithms, such a ZF equalizer is obtained at a low complexity: It involves a single Eigen Vector Decomposition (EVD) and does not require prior knowledge nor estimation of the noise power. Such a straightforward estimation translates, also, into better equalization performance at low channel SNR, as confirmed by simulations.
[Show abstract][Hide abstract] ABSTRACT: An SOS-based blind equalization algorithm for the SIMO channel has recently been proposed that has an unprecedented quadratic complexity in the channel memory, compared to all existing SOS-based techniques whose complexity is cubic in this regard. In this paper, we show that this technique can be adapted to the MIMO channel and we prove that its complexity is maintained independently from the number of input/output channels. Simulation tests are reported that sustain the feasibility of this technique in practical observation conditions.
[Show abstract][Hide abstract] ABSTRACT: This paper derives the optimal single input multiple output (SIMO) maximum likelihood sequence estimation (MLSE) receiver for the detection of quadrature amplitude modulations corrupted by potentially noncircular, stationary white or colored zero-mean Gaussian noise. It is proved that this receiver is composed by a widely linear (WL) filter followed by a modified version of the Viterbi algorithm. This WL linear filter is interpreted for complex-valued signal of interest (SOI) symbols as two WL multidimensional matched filter (WL MMF) that reduce to a single WL MMF for real-valued SOI symbols. The performance improvements of this receiver with respect to the standard SIMO MLSE are proved and illustrated.
[Show abstract][Hide abstract] ABSTRACT: This paper is devoted to time delay estimation for wide sense stationary complex circular or noncircular Gaussian signals. Using a theorem by Whittle that we have extended to complex data, closed-form expressions of the Cramer Rao bound (CRB) are given for the time delay alone in presence of nuisance parameters. In particular, we prove that the CRB for the time delay is weakly reduced for noncircular signals w.r.t. circular signals, except for very low signal to noise ratios (SNR), for which the CRB for rectilinear signals is half of the CRB for circular signals. Then, the maximum likelihood (ML) estimate that extends the generalized cross correlation (GCC) estimate is derived.
[Show abstract][Hide abstract] ABSTRACT: Eigenspace techniques are very popular techniques for blind channel identification, but are ones with a large complexity, cubic in the channel order. The newly introduced channel compaction is a signal processing technique that consists in using small-sized linear transformations to progressively force to zero some of the channel coefficients. As such, channel compaction was used to develop the first (and, up to now, the only) blind channel equalization technique with a quadratic complexity. In this paper, we apply blind compaction to develop a new blind identification technique, the first to have a quadratic complexity. Simulation tests show that the low-complexity compaction-based blind identification performs quite similarly to the most referenced existing eigenspace blind identification techniques.
[Show abstract][Hide abstract] ABSTRACT: In this paper, the problem of testing impropriety (i.e., second-order noncircularity) of a sequence of complex-valued random variables (RVs) based on the generalized likelihood ratio test (GLRT) for Gaussian distributions is considered. Asymptotic (w.r.t. the data length) distributions of the GLR are given under the hypothesis that RVs are proper or improper, and under the true, not necessarily Gaussian distribution of the RVs. The considered RVs are independent but not necessarily identically distributed: assumption which has never been considered until now. This enables us to deal with the practical important situations of noncircular RVs disturbed by residual frequency offsets and additive circular noise. The receiver operating characteristic (ROC) of this test is derived as byproduct, an issue previously overlooked. Finally illustrative examples are presented in order to strengthen the obtained theoretical results.
[Show abstract][Hide abstract] ABSTRACT: This paper presents a performance analysis of likelihood ratio test (LRT)-based and generalized likelihood ratio test (GLRT)-based array receivers for the detection of a known signal corrupted by a potentially noncircular interference. Studying the distribution of the statistics associated with the LRT and GLRT, expressions of the probability of detection (PD) and false alarm (PFA) are given. In particular, an exact closed-form expression of PD and PFA are given for two LRT-based receivers and asymptotic (with respect to the data length) closed-form expression are given for PD and PFA for four GLRT-based receivers. Finally illustrative examples are presented in order to strengthen the obtained results.
Preview · Article · Oct 2011 · Acoustics, Speech, and Signal Processing, 1988. ICASSP-88., 1988 International Conference on
[Show abstract][Hide abstract] ABSTRACT: This correspondence presents an asymptotic analysis of the eigenvalue decomposition (EVD) of the sample covariance matrix associated with independent identically distributed (i.i.d.) non necessarily circular and Gaussian data that extends the well known analysis presented in the literature for circular and Gaussian data. Closed-form expressions of the asymptotic bias and variance of the sample eigenvalues and eigenvectors are given. As an application of these extended expressions, the statistical performance analysis of the widely used minimum description length (MDL) criterion applied to the detection of the number of noncircular or/and non-Gaussian sources impinging on an array of sensors is considered with a particular attention paid to uncorrelated rectilinear sources.
Preview · Article · Sep 2011 · IEEE Transactions on Signal Processing
[Show abstract][Hide abstract] ABSTRACT: This paper presents an asymptotic analysis of the eigen value decomposition (EVD) of the sample covariance matrix associated with independent identically distributed (IID) non necessarily circular and Gaussian data that extends the well known analysis presented in the literature for circular and Gaussian data. Closed-form expressions of the asymptotic bias and variance of the sample eigenvalues and eigenvectors are given. As an application of these extended expressions, the statistical performance analysis of the minimum description length (MDL) criterion applied to the detection of the number of noncircular or/and nonGaussian components is considered.
[Show abstract][Hide abstract] ABSTRACT: It isnow well knownthat time invariant(TI)linearbeamformers, such as the Capon’s beamformer, are only optimal for stationary Gaussian observations whose complex envelope is necessarily second order (SO) circular. However in many applications such as in radiocommunications, most of the signals are nonGaussian and their complex envelope presents very often some SO and/or higher order (HO) non circularity properties. For this reason we proposein this paper a third order widely non linear Volterra minimum variance distortionnless response (MVDR) beamformer taking into account both the potentialnonGaussian characterand the potentialHO non circularity (up to sixth order) of interferences. Some properties, performance and adaptive implementation of this new beamformer are presented in the paper. Illustrations shows the great interest, with respect to the existing beamformers, of this new beamformer for non Gaussian and HO non circular interferences, omnipresent in practice.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we address the problem of the sensor placement for estimating the direction of a narrow-band source, randomly located in the far-field of a planar antenna array. Estimation performance is evaluated by means of the expectation of the conditional Cramer Rao bound (ECRB), which depends on the prior probabilistic distribution of the DOA angles. We study the particular, but practical, case where the azimuth angle is uniformly distributed. Surprisingly, it turns out that the optimal arrays are not isotropic, i.e. they do not have the same accuracy in all possible look directions. In fact, optimal arrays computed here increase performance by about 10% compared to optimal isotropic arrays computed in a previous work.
[Show abstract][Hide abstract] ABSTRACT: This paper presents a performance analysis of the single antenna interference cancellation (SAIC) and the multiple antenna interference cancellation (MAIC) techniques in the presence of non null frequency offsets of both the signal of interest (SOI) and interference, that are neither corrected nor compensated, in terms of signal to interference plus noise ratio (SINR) and bit error rate (BER). General theoretical expressions of the SINR and BER are given for BPSK SOI and interference. To obtain engineering insight, the particu-lar case of the SAIC with no return to zero (NRZ) pulse shape filters is considered, where simple expressions are given and analyzed. It is proved in particular that the performance more deteriorates for non null frequency offset of the interference than for non null frequency offset of the SOI. Finally illus-trative examples are presented in order to specify the validity domain of our approximations and to quantify the obtained results in the context of the global system for mobile com-munication (GSM) standard.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we consider the problem of testing impropriety (i.e., second-order noncircularity) of a complex-valued random variable (RV) based on the generalized likelihood ratio test (GLRT) derived under Gaussian distributions. Asymptotic (w.r.t. the data length) distributions of the GLR are given under the hypothesis that data are proper or improper, and under the true, not necessarily Gaussian distribution of the data. This enables us to derive in particular the receiver operating characteristics (ROC) of this test, an issue previously overlooked. Finally illustrative examples are presented in order to strengthen the obtained theoretical results.
[Show abstract][Hide abstract] ABSTRACT: In this correspondence we mainly consider the asymptotic distribution of the estimator of circularity coefficients of scalar and multidimensional complex random variables. A particular attention is paid to rectilinear RV. After deriving new properties of the circularity coefficients, the maximum likelihood estimate of the circularity coefficients in the Gaussian case and asymptotic distribution of this estimate for arbitrary distributions are given. Finally, an illustrative example is presented in order to strengthen the obtained theoretical results.
Full-text · Article · Dec 2009 · Signal Processing
[Show abstract][Hide abstract] ABSTRACT: In many detection applications, the main performance criterion is the signal to interference plus noise ratio (SINR). After linear filtering, the optimal SINR corresponds to the maximum value of a Rayleigh quotient, which can be interpreted as the largest generalized eigenvalue of two covariance matrices. Using an extension of Szegö's theorem for the generalized eigenvalues of Hermitian block Toeplitz matrices, an expression of the theoretical asymptotic optimal SINR w.r.t. the number of taps is derived for arbitrary arrays with a limited but arbitrary number of sensors and arbitrary spectra. This bound is interpreted as an optimal zero-bandwidth spatial SINR in some sense. Finally, the speed of convergence of the optimal wideband SINR for a limited number of taps is analyzed for several interference scenarios.
Full-text · Article · Oct 2009 · Signal Processing