Samir-Mohamad Omar

Variational Bayesian blind and semiblind channel estimation

Blind and semiblind channel estimation is a topic that enjoyed explosive developments throughout the nineties, and then came to a standstill, probably because of perceived unsatisfactory performance. Blind channel estimation techniques were developed and usually evaluated for a given channel realization, i.e. with a deterministic channel model. Such blind channel estimates, especially those based on subspaces in the data, are often only partial and ill-conditioned. On the other hand, in wireless communications the channel is typically modeled as Rayleigh fading, i.e. with a Gaussian (prior) distribution expressing variances of and correlations between channel coefficients. In recent years, such prior information on the channel has started to get exploited in pilot-based channel estimation, since often the pure pilot-based (deterministic) channel estimate is of limited quality due to limited pilots. In this paper we explore a Bayesian approach to (semi-)blind channel estimation, exploiting a priori information on fading channels. In the case of deterministic unknown input symbols, it suffices to augment the classical blind (quadratic) channel criterion with a quadratic criterion reflecting the Rayleigh fading prior. In the case of a Gaussian symbol model the blind criterion is more involved. The joint ML/MAP estimation of channels, deterministic unknown symbols, and channel profile parameters can be conveniently carried out using Variational Bayesian techniques. Variational Bayesian techniques correspond to alternating maximization of a likelihood w.r.t. subsets of parameters, but taking into account the estimation errors on the other parameters. To simplify exposition, we elaborate the details for the case of MIMO OFDM systems.

Bayesian and deterministic CRBs for semi-blind channel estimation in SIMO single carrier cyclic prefix systems

Traditionally, the performance of different semi-blind channel estimation algorithms has been assessed and compared to a certain lower bound. One of these famous lower bounds that has been extensively used in the literature is the Cramer Rao Bound (CRB). Depending on how we treat the symbols and the channel, different versions of CRB have been derived. There are two possible cases on how to treat the symbols and/or the channel namely, deterministic unknowns or random. Moreover, the symbols are either jointly estimated with the channel or eliminated. In other words, we have six different cases to be handled. In this paper we present the CRBs that exist in the literature and fit to some of these cases and derive the others in the context of SIMO single carrier cyclic prefix systems (SC-CP). On the top of that we present a unified framework that permits to derive all versions of CRBs in a concrete manner. All the derived CRBs are validated numerically by conducting limited Monte-Carlo simulations.

Bayesian semi-blind FIR channel estimation algorithms in SIMO systems

When the transmission scenario includes a training sequence or pilots, semi-blind channel estimation techniques have shown a tendency to fully exploit the information available from the received signal if they are correctly implemented. This feature leads semi-blind channel estimation performance to exceed that of the schemes based on the blind part or the training sequence only. Moreover, in some situations they can estimate the channel when the other techniques fail. Semi-blind channel estimation techniques were developed and usually evaluated for a given channel realization, i.e. with a deterministic channel model. On the other hand, in wireless communications the channel is typically modeled as Rayleigh fading, i.e. with a Gaussian (prior) distribution expressing variances of and correlations between channel coefficients. In recent years, such prior information on the channel has started to get exploited in pilot-based channel estimation, since often the pure pilot-based (deterministic) channel estimate is of limited quality due to limited pilots. In this paper we provide a performance comparison between ML/MAP algorithms that use Bayesian and deterministic approaches in semi-blind channel estimation.

Singular block Toeplitz matrix approximation and application to multi-microphone speech dereverberation

We consider the blind multichannel dereverberation problem for a single source. We have shown before [5] that the single-input multi-output (SIMO) reverberation filter can be equalized blindly by applying MIMO Linear Prediction (LP) to its output (after SISO input pre-whitening). In this paper, we investigate the LP-based dereverberation in a noisy environment, and/or under acoustic channel length underestimation. Considering ambient noise and late reverberation as additive noises, we propose to introduce a postfilter that transforms the MIMO prediction filter into a somewhat longer equalizer. The postfilter allows to equalize to non-zero delay. Both MMSE-ZF and MMSE design criteria are considered here for the postfilter.We also focus here on computationally efficient (FFT based) block Toeplitz covariance matrix enhancement that enforces the SIMO filtered source plus white noise structure before applying MIMO LP. A second suggested refinement is an iterative refinement between SISO and MIMO LP. Simulations show that the proposed scheme is robust in noisy environments, and performs better compared to the classic Delay-&-Predict equalizer and the Delay-&-Sum beamformer.

Performance and Complexity Analysis of Blind FIR Channel Identification Algorithms Based on Deterministic Maximum Likelihood in SIMO Systems

We analyze two algorithms that have been introduced previously for Deterministic Maximum Likelihood (DML) blind estimation of multiple FIR channels. The first one is a modification of the Iterative Quadratic ML (IQML) algorithm. IQML gives biased estimates of the channel and performs poorly at low SNR due to noise induced bias. The IQML cost function can be “denoised” by eliminating the noise contribution: the resulting algorithm, Denoised IQML (DIQML), gives consistent estimates and outperforms IQML. Furthermore, DIQML is asymptotically globally convergent and hence insensitive to the initialization. Its asymptotic performance does not reach the DML performance though. The second strategy, called Pseudo-Quadratic ML (PQML), is naturally denoised. The denoising in PQML is furthermore more efficient than in DIQML: PQML yields the same asymptotic performance as DML, as opposed to DIQML, but requires a consistent initialization. We furthermore compare DIQML and PQML to the strategy of alternating minimization w.r.t. symbols and channel for solving DML (AQML). An asymptotic performance analysis, a complexity evaluation and simulation results are also presented. The proposed DIQML and PQML algorithms can immediately be applied also to other subspace problems such as frequency estimation of sinusoids in noise or direction of arrival estimation with uniform linear arrays.

Recent insights in the Bayesian and deterministic CRB for blind SIMO channel estimation

The performance of channel estimation is often assessed by deriving the proper Cramér-Rao Bound (CRB). Depending on how to treat the symbols and the channel, we have previously derived different versions of CRB. Specifically, we have dealt with the cases where the symbols and/or the channel are assumed to be either deterministic unknowns or random. Moreover, the symbols have been considered to be either jointly estimated with the channel or marginalized. All in all, we have derived six different versions of Bayesian and deterministic CRBs. However, we have shown that many of these CRBs are too optimistic in the sense that they are not strict enough to be attained by any deterministic or Bayesian estimator. In this paper we propose modified versions of those loose CRBs in the context of SIMO FIR system that are valid at least in the moderate and high SNR regimes. The analytical formulas for the lower bounds introduced are validated by some Monte-Carlo simulations.

Receiver diversity with blind fir SIMO channel estimates

Traditionally, the performance of blind SIMO channel estimates has been characterized in a deterministic fashion, by identifying those channel realizations that are not blindly identifiable. In this paper, we focus instead on the performance of Linear Equalizers for fading channels when they are based on blind channel estimates. Our analysis shows that with Zero Forcing Linear Equalizer (ZF-LE) at least one order of the diversity is lost depending on the way by which the scalar ambiguity that results from the blind channel estimation is resolved. However, in some Tx scenarios we are able to recover the diversity with MMSE-LE. Various Tx scenarios are considered in detail.

Structured spatio-temporal sample covariance matrix enhancement with application to blind channel estimation in cyclic prefix systems

Multichannel aspect allows the introduction of blind channel estimation techniques. Most existing such techniques for frequency-selective channels are quite complex. In this paper, we consider the blind channel estimation problem for Single Input Multi Output (SIMO) cyclic prefix (CP) systems. We have shown before [1] that blind channel estimation becomes computationally much more attractive and more straight forward to analyze in terms of performance in CP systems. Inspired by the iterative sample covariance matrix (SCM) structure enhacement techniques of Cadzow and others [2], we propose here an algorithm to structure the sample block circulant covariance matrix by enforcing two essential properties: rank and FIR structure. These two properties are exhibited by the true covariance matrix in the case of FIR SIMO channels with spatially white noise and CP transmission. The proposed enhancement procedure leads to an interesting enhanced SCM, even for the single CP symbol case. A simulation study for some classical channel estimators that depend on the SCM (with and without structuring) is presented, indicating that structuring allows for considerable performance gain in terms of the channel normalized mean square error (NMSE) over a wide SNR range.

Further results on Bayesian and deterministic CRBs in the context of blind SIMO channel estimation

The performance of channel estimation is often assessed by deriving the proper Cramér-Rao Bound (CRB). However, in the blind case a special treatment is required due to the singularity of the Fisher Information Matrix (FIM). Usually a constraint is introduced to overcome the blind ambiguity and ensuing singularity. Hence, a constrained CRB has been derived in the literature since a long time ago. Although this constrained CRB has been proven to be a valid lower bound in the medium and high SNR regimes, it fails completely in the low SNR regime because unlike the MSE it does not saturate. Motivated by the shortcoming of the constrained CRB, we derive in this paper a modified constrained CRB (MCCRB) for deterministic blind channel estimation. The MCCRB is valid over the whole SNR regime. In the second part of the paper we address Bayesian blind channel estimation and explore the apparent discrepancy between channel unidentifiability with a non-singular FIM. We highlight that in the less familiar Bayesian case this relationship needs to be interpreted differently. The analytical formulas for the introduced bounds are validated by some Monte-Carlo simulations.

Receiver diversity with blind and semi-blind FIR SIMO channel estimates

Traditionally, the performance of blind SIMO channel estimates has been characterized in a deterministic fashion, by identifying those channel realizations that are not blindly identifiable. In this paper, we focus instead on the performance of Zero-Forcing (ZF) Linear Equalizers (LEs) or Decision-Feedback Equalizers (DFEs) for fading channels when they are based on (semi-)blind channel estimates. Although it has been known that various (semi-)blind channel estimation techniques have a receiver counterpart that is matched in terms of symbol knowledge hypotheses, we show here that these (semi-)blind techniques and corresponding receivers also match in terms of diversity order: the channel becomes (semi-)blindly unidentifiable whenever its corresponding receiver structure goes in outage. In the case of mismatched receiver and (semi-blind) channel estimation technique, the lower diversity order dominates. Various cases of (semi-)blind channel estimation and corresponding receivers are considered in detail. To be complete however, the actual combination of receiver and (semi-)blind channel estimation lowers somewhat the diversity order w.r.t. the ideal picture.