Maximum likelihood estimation for array processing in colored noise

Direction of arrival estimation of multiple sources, using a uniform linear array, in noise with unknown covariance is considered. The noise is modeled as a spatial autoregressive process with unknown parameters. Both stochastic and deterministic signal models are considered. For the random signal case, an approximate maximum likelihood estimator of the signal and noise parameters is derived. It requires numerical maximization of a compressed likelihood function over the unknown arrival angles. Analytical expressions for the MLEs of the signal covariance and the AR parameters are given. Similar results for the case of deterministic signals are also presented     

Minimax estimation of unknown deterministic signals in colorednoise

The estimation of a deterministic signal corrupted by random noise is considered. The strategy is to find a linear noncausal estimator which minimizes the maximum mean square error over an a priori set of signals. This signal set is specified in terms of frequency/energy constraints via the discrete Fourier transform. Exact filter expressions are given for the case of additive white noise. For the case of additive colored noise possessing a continuous power spectral density, a suboptimal filter is derived whose asymptotic performance is optimal. Asymptotic expressions for the minimax estimator error are developed for both cases. The minimax filter is applied to random data and is shown to solve asymptotically a certain worst-case Wiener filter problem     

NAR estimators of spatial covariance matrices for adaptive arraydetection

The theory of noise-alone-reference (NAR) power estimation is extended to the estimation of spatial covariance matrices. A NAR covariance estimator insensitive to signal presence is derived. The SNR (signal-to-noise ratio) loss incurred by using this estimator is independent of the input SNR and is less than that encountered with the maximum likelihood covariance estimator given that the same number of uncorrelated snapshots is available to both estimators. The analysis assumes first a deterministic signal. The results are extended and generalized to signals with unknown parameters or random signals. For the random signal case, generalized and quasi-NAR covariance estimators are presented     

Computationally efficient angle estimation for signals with knownwaveforms

This paper presents a large sample decoupled maximum likelihood (DEML) angle estimator for uncorrelated narrowband plane waves with known waveforms and unknown amplitudes arriving at a sensor array in the presence of unknown and arbitrary spatially colored noise. The DEML estimator decouples the multidimensional problem of the exact ML estimator to a set of 1-D problems and, hence, is computationally efficient. We shall derive the asymptotic statistical performance of the DEML estimator and compare the performance with its Cramer-Rao bound (CRB), i.e., the best possible performance for the class of asymptotically unbiased estimators. We will show that the DEML estimator is asymptotically statistically efficient for uncorrelated signals with known waveforms. We will also show that for moderately correlated signals with known waveforms, the DEML estimator is no longer a large sample maximum likelihood (ML) estimator, but the DEML estimator may still be used for angle estimation, and the performance degradation relative to the CRB is small. We shall show that the DEML estimator can also be used to estimate the arrival angles of desired signals with known waveforms in the presence of interfering or jamming signals by modeling the interfering or jamming signals as random processes with an unknown spatial covariance matrix. Finally, several numerical examples showing the performance of the DEML estimator are presented in this paper     

Minimum variance filters and mixed spectrum estimation

This paper presents a spectral density estimator based on a normalized minimum variance (MV) estimator as the one proposed by Lagunas. With an equivalent frequency resolution, this new estimator preserves the amplitude estimation lost in Lagunas one. This proposition comes from a theoretical study of MV filters that highlights this amplitude lost. Two signal types are taken into account: periodic deterministic signals (narrow-band spectral structures) and stationary random signals (broad-band spectral structures). Without selecting a smoothing window, the proposed estimator is an alternative to Fourier-based estimator and, without modeling the signal, it is a concurrent to high-resolution estimators.     

Eigen-based receivers for the detection of random UWB signals

Wireless ultra-wideband (UWB) receivers usually employ a deterministic clean pulse or pulses derived from a Gaussian monocycle as a template for correlation. A deterministic template cannot be expected to have high correlation with random UWB signals at all times. Such a mismatch can significantly degrade the system error performance. In this letter, a deterministic template is designed to maximize the energy capture capability on ensemble average, ending up with the eigen-based correlator. The eigen-based correlator can be used alone or alongside path diversity to further enhance signal detection. The new receivers outperform their counterparts using a clean pulse, as shown by theoretical analysis, and demonstrated by numerical results based on real UWB data.     

Multidimensional maximum-entropy covariance extension

A universal characterization of maximum-entropy covariances for multidimensional signals is presented. It is shown that the maximum-entropy extension of an arbitrary partial covariance of a nonstationary multidimensional signal always has a banded inverse, i.e the inverse is sparse and has the same support as the given partial covariance. A dual formulation of the problem that makes it possible to approximate maximum-entropy extensions with models selected from suitably constrained model sets is introduced. It is proved that the best approximation in terms of multidimensional recursible autoregressive models can be determined by solving a set of linear equations. A simple graph-theoretic criterion is introduced to characterize those partial covariances whose maximum-entropy extension coincides with its autoregressive approximation, as in the conventional (one-dimensional stationary) maximum-entropy problem     

On the blind estimation of the parameters of continuous phase modulated signals

In this paper, a blind estimator of the technical parameters of continuous phase modulated (CPM) signals is proposed. It consists in estimating jointly the modulation index, the symbol period and the frequency offset. It is based on the following observations. First, the inverse of the index is the smallest positive real number a CPM signal should be raised to in order to generate a deterministic harmonic signal; second, the frequencies of the harmonic signal are simply related to the symbol period and the carrier frequency. The practical implementation of this joint estimator is described and the asymptotic behavior of the estimation error is studied. If N is the number of signaling intervals, the estimate of the modulation index is shown to converge to a non-Gaussian distribution at rate 1/N, while the estimate of the frequency offset and the estimate of the symbol period converge at rate 1/N/sup 3/2/. We also investigate the case where the modulation index and the symbol period are available at the receiver side. An estimator of the frequency offset is proposed by adapting the above joint estimator to the latter case. The asymptotic behavior of this estimator is studied and compared with the case where all of the parameters are unknown, so as to evaluate the possible degradation of the performance due to the ignorance of certain technical parameters. Simulations results sustain our theoretical claims.     

Array processing for nonstationary interference suppression inDS/SS communications using subspace projection techniques

Combined spatial and time-frequency signatures of signal arrivals at a multisensor array are used for nonstationary interference suppression in direct-sequence spread-spectrum (DS/SS) communications. With random PN spreading code and deterministic nonstationary interferers, the use of antenna arrays offers increased DS/SS signal dimensionality relative to the interferers. Interference mitigation through a spatio-temporal subspace projection technique leads to reduced DS/SS signal distortion and improved performance over the case of a single antenna receiver. The angular separation between the interference and desired signals is shown to play a fundamental role in trading off the contribution of the spatial and time-frequency signatures to the interference mitigation process. The expressions of the receiver signal-to-interference-noise ratio (SINR) implementing subspace projections are derived, and numerical results are provided     

On the asymptotic distribution of the correlation receiver output for time-hopped UWB signals

In ultra-wideband (UWB) communications based on time-hopping (TH) impulse radio, one of the most frequently studied receivers is the correlation receiver. The multiuser interference (MUI) at the output of this receiver is sometimes modeled as a Gaussian random variable. In order to justify this assumption, the conditions of validity of the Central Limit Theorem (CLT) have to be studied in an asymptotic regime where the number of interferers and the processing gain grow toward infinity at the same rate, with the channel degree being kept constant. An asymptotic study is made in this paper based on the so-called Lindeberg's condition for the CLT for martingales. Nonsynchronized users sending their signals over independent multipath channels are considered. These users may also have different powers. It is shown that when the frame length grows and the repetition factor is kept constant, then the MUI does not converge in distribution toward a Gaussian random variable. On the other hand, this convergence can be established if the repetition factor grows at the rate of the frame length. In this last situation, closed-form expressions for the signal-to-interference-plus-noise ratio (SINR) are given for TH pulse amplitude modulation (PAM) and pulse position modulation (PPM) UWB transmissions.     

Occam filters for stochastic sources with application to digitalimages

An Occam filter employs lossy data compression to separate signal from noise. Previously, it was shown that Occam filters can filter random noise from deterministic signals. Here, we show that Occam filters can also separate two stochastic sources, depending on their relative compressibility. We also compare the performance of Occam filters and wavelet-based denoising on digital images     

Generalized quadratic receivers for orthogonal signals over theGaussian channel with unknown phase/fading

The orthogonal signal structure has been shown to be the superposition of an antipodal signal set and an unmodulated (pilot tone) component which can be used for channel measurement. Starting from this point of view, the quadratic receiver for orthogonal signals over the Gaussian channel with unknown phase/fading has been shown to be equivalent to a detector-estimator receiver. The estimator makes an optimum estimate of the unknown complex channel gain based on the channel measurement provided by the unmodulated component of the received signal. This channel estimate then forms a (partially) coherent reference for the detector in detecting the data carried by the antipodal signaling component of the received signal. This paper exploits this detector-estimator structure of the quadratic receiver, and generalizes it to a receiver in which the estimator makes an estimate of the channel gain in each signaling interval based on the totality of signals received over all the signaling intervals or a subset of these intervals. The generalized quadratic receiver is just as simple to implement as the conventional quadratic receiver, and theoretical and simulation results show that it can achieve substantial performance gains over the conventional receiver. A theory is presented to show that the generalized quadratic receiver is an implementable approximation to the optimum symbol-by-symbol receiver for uncoded orthogonal signals over the Gaussian channel with unknown phase/fading. The theory shows that the structure provides a unified and systematic approach to the design of coherent symbol-by-symbol receivers, and shows that the conventional carrier-loop-type receivers are ad hoc     

Batch and Sequential Bayesian Estimators of the Number of Active Terminals in an IEEE 802.11 Network

The performance of the IEEE 802.11 protocol based on the distributed coordination function (DCF) has been shown to be dependent on the number of competing terminals and the backoff parameters. Better performance can be expected if the parameters are adapted to the number of active users. In this paper we develop both off-line and online Bayesian signal processing algorithms to estimate the number of competing terminals. The estimation is based on the observed use of the channel and the number of competing terminals is modeled as a Markov chain with unknown transition matrix. The off-line estimator makes use of the Gibbs sampler whereas the first online estimator is based on the sequential Monte Carlo (SMC) technique. A deterministic variant of the SMC estimator is then developed, which is simpler to implement and offers superior performance. Finally a novel approximate maximum a posteriori (MAP) algorithm for hidden Markov models (HMM) with unknown transition matrix is proposed. Realistic IEEE 802.11 simulations using the ns-2 network simulator are provided to demonstrate the excellent performance of the proposed estimators     

Generalized moment matching for the linear combination of lognormal RVs: application to outage analysis in wireless systems

Moving from the need for a simple and versatile method for outage computation in various contexts of interest in wireless communications, in this paper we propose a lognormal approximation for the linear combination of a set of lognormal random variables (RV) with one-sided random weights. The approximation is based on a generalization of the well known moment matching approximation (MMA) for the sum of lognormal RVs, and it allows quite simple handling of the power sum of interfering signals even in rather complicated scenarios. Specifically, composite multiplicative channel models with unequal parameters can be handled, and generic (unequal) correlation patterns for some channel components can be handled with reference to any pair of signals. At this stage of the computation, only moments of the random weights are required. The probability density function of the random weight for the useful signal component may be required in computing outage probability, and numerical methods may be only required to solve a single integral at this second stage. The suitability of the approximation is examined by evaluating outage performance for various values of system parameters in some contexts of interest, namely spread spectrum systems and typical reuse-based systems with composite Rayleigh-lognormal and Nakagami-lognormal channels.     

Estimating signal parameters using the nonlinear instantaneousleast squares approach

A novel method for signal parameter estimation is presented, termed the nonlinear instantaneous least squares (NILS) estimator. The basic idea is to use the observations in a sliding window to compute an instantaneous (short-term) estimate of the amplitude used in the separated nonlinear least squares (NLLS) criterion. The effect is a significant improvement of the numerical properties in the criterion function, which becomes well-suited for a signal parameter search. For small-sized sliding windows, the global minimum in the NLIS criterion function is wide and becomes easy to find. For maximum size windows, the NILS is equivalent to the NLLS estimator, which implies statistical efficiency for Gaussian noise. A “blind” signal parameter search algorithm that does not use any a priori information is proposed. The NILS estimator can be interpreted as a signal-subspace projection-based algorithm. Moreover, the NILS estimator can be interpreted as an estimator based on the prediction error of a (structured) linear predictor. Hereby, a link is established between NLLS, signal-subspace fitting, and linear prediction-based estimation approaches. The NILS approach is primarily applicable to deterministic signal models. Specifically, polynomial-phase signals are studied, and the NILS approach is evaluated and compared with other approaches. Simulations show that the signal-to-noise ratio (SNR) threshold is significantly lower than that of the other methods, and it is confirmed that the estimates are statistically efficient. Just as the NLLS approach, the NILS estimator can be applied to nonuniformly sampled data     

Maximum likelihood estimation of signals in autoregressive noise

Time series modeling as the sum of a deterministic signal and an autoregressive (AR) process is studied. Maximum likelihood estimation of the signal amplitudes and AR parameters is seen to result in a nonlinear estimation problem. However, it is shown that for a given class of signals, the use of a parameter transformation can reduce the problem to a linear least squares one. For unknown signal parameters, in addition to the signal amplitudes, the maximization can be reduced to one over the additional signal parameters. The general class of signals for which such parameter transformations are applicable, thereby reducing estimator complexity drastically, is derived. This class includes sinusoids as well as polynomials and polynomial-times-exponential signals. The ideas are based on the theory of invariant subspaces for linear operators. The results form a powerful modeling tool in signal plus noise problems and therefore find application in a large variety of statistical signal processing problems. The authors briefly discuss some applications such as spectral analysis, broadband/transient detection using line array data, and fundamental frequency estimation for periodic signals     

A Bayesian estimation approach for speech enhancement using hiddenMarkov models

A Bayesian estimation approach for enhancing speech signals which have been degraded by statistically independent additive noise is motivated and developed. In particular, minimum mean square error (MMSE) and maximum a posteriori (MAP) signal estimators are developed using hidden Markov models (HMMs) for the clean signal and the noise process. It is shown that the MMSE estimator comprises a weighted sum of conditional mean estimators for the composite states of the noisy signal, where the weights equal the posterior probabilities of the composite states given the noisy signal. The estimation of several spectral functionals of the clean signal such as the sample spectrum and the complex exponential of the phase is also considered. A gain-adapted MAP estimator is developed using the expectation-maximization algorithm. The theoretical performance of the MMSE estimator is discussed, and convergence of the MAP estimator is proved. Both the MMSE and MAP estimators are tested in enhancing speech signals degraded by white Gaussian noise at input signal-to-noise ratios of from 5 to 20 dB     

Accurate CDMA BER calculations with low computational complexity

An accurate approximation for calculating bit error rates in direct sequence code division multiple access (CDMA) radio systems using binary phase shift keyed (BPSK) signaling is presented. All interfering users are assumed to employ random signature sequences, but the desired signal can be structured with either a random or a deterministic spreading code. Bit error probabilities are given for signals having carrier phase or chip offsets that are either deterministic or random. Computational complexity of all calculations is O(1)     

The structure of least-mean-square linear estimators for synchronousM-ary signals (Corresp.)

Several authors have shown that the structure of the least-mean-square linear estimator of the sequence of random amplitudes in a synchronous pulse-amplitude-modulated signal that suffers intersymbol interference and additive noise is a matched filter whose output is periodically sampled and passed through a transversal filter (tapped delay line). It is our purpose in this paper to generalize this result to synchronousm-ary signals (e.g., FSK, PSK, PPM signals). We prove that the structure of the least-mean-square linear estimator of the sequence of random parameters in a synchronousm-ary signal, which suffers intersymbol interference and additive noise, is a parallel connection ofmmatched filters followed by tapped delay lines. A similar structure is derived for the continuous waveform estimator of a synchronousm-ary signal. Finally, we present a structure for estimation-decision detection of synchronousm-ary signals, which is based on least-mean-suare linear estimates of aposterioriprobabilities.     

Robust Transmitted Waveform and Received Filter Design for Cognitive Radar in the Presence of Signal-Dependent Interference

The problem of robust transmitted waveform and received filter design for cognitive radar in a signal-dependent interference environment is considered. When estimate errors of the target impulse response (TIR) and clutter impulse response (CIR) exist, in order to improve the worst signal-to-clutter ratio (SCR) and signal-to-interference-and-noise ratio (SINR), a robust transmitted waveform and received filter are designed based on the minimax criterion by using the information fed back from the receiver. Using deterministic and random models, the waveform and filter design problem is divided into three optimization problems. The robust waveform and filter are then obtained by solving these problems. In the deterministic model, we prove that the robust waveform and filter impulse response can be generated by a pseudorandom code. In the random model, the robust waveform and filter impulse response can be obtained by an alternative projection algorithm. Numerical results indicate that the worst SINR of the robust waveform and filter is higher than that of the traditional waveform and filter.