This paper considers the problem of estimating an unknown input (bias) by means of the augmented-state Kalman (AKF) filter. To reduce the computational complexity of the AKF, [12] recently developed an optimal two-stage Kalman filter (TS-AKF) that separates the bias estimation from the state estimation, and shows that his new two-stage estimator is equivalent to the standard AKF, but requires less computations per iteration. This paper focuses on the derivation of the optimal two-stage estimator for the square-root covariance implementation of the Kalman filter (TS-SRCKF), which is known to be numerically more robust than the standard covariance implementation. The new TS-SRCKF also estimates the state and the bias separately while at the same time it remains equivalent to the standard augmented-state SRCKF. It is experimentally shown in the paper that the new TS-SRCKF may require less flops per iteration for some problems than the Hsieh's TS-AKF [12]. Furthermore a second, even faster (single-stage) algorithm has been derived in the paper by exploiting the structure of the least-squares problem and the square-root covariance formulation of the AKF. The computational complexities of the two proposed methods have been analyzed and compared the those of other existing implementations of the AKF.

This paper presents a novel adaptive equalization algorithm for time-varying MIMO systems with ISI channel conditions. The algorithm avoids channel estimation before equalization and leads to a direct QR-based procedure for updating the equalizer coefficients to track the time-varying channel characteristics. Our approach does not require precise channel estimation and needs relatively few pilot symbols for satisfactory equalization. The theoretical foundations of the proposed algorithm are rooted in signal recovery results derived from the generalized Bezout identity and the finite alphabet property inherent in digital communication schemes. Concerning the convergence behavior of the algorithm, we address the following three issues: existence of fixed points, exclusiveness of fixed points, and robustness under noise disturbance and parameter selection. The equalizer demonstrates promising capability in achieving low symbol error rates for a very broad range of SNRs. Simulation results are presented confirming that this approach outperforms the more traditional recursive least squares (RLS) adaptive equalizer for this application and rivals the performance of MMSE equalizers requiring channel knowledge.

Mobile communication channels are often modeled as linear time-varying filters or, equivalently, as time-frequency integral operators with finite support in time and frequency. Such a characterization inherently assumes the signals are narrowband and may not be appropriate for wideband signals. In this paper time-scale characterizations are examined that are useful in wideband time-varying channels, for which a time-scale integral operator is physically justifiable. A review of these time-frequency and time-scale characterizations is presented. Both the time-frequency and time-scale integral operators have a two-dimensional discrete characterization which motivates the design of time-frequency or time-scale rake receivers. These receivers have taps for both time and frequency (or time and scale) shifts of the transmitted signal. A general theory of these characterizations which generates, as specific cases, the discrete time-frequency and time-scale models is presented here. The interpretation of these models, namely, that they can be seen to arise from processing assumptions on the transmit and receive waveforms is discussed. Out of this discussion a third model arises: a frequency-scale continuous channel model with an associated discrete frequency-scale characterization.

We present a statistical framework for the fixed-frequency computational time-reversal imaging problem assuming point scatterers in a known background medium. Our statistical measurement models are based on the physical models of the multistatic response matrix, the distorted wave Born approximation and Foldy-Lax multiple scattering models. We develop maximum likelihood (ML) estimators of the locations and reflection parameters of the scatterers. Using a simplified single-scatterer model, we also propose a likelihood time-reversal imaging technique which is suboptimal but computationally efficient and can be used to initialize the ML estimation. We generalize the fixed-frequency likelihood imaging to multiple frequencies, and demonstrate its effectiveness in resolving the grating lobes of a sparse array. This enables to achieve high resolution by deploying a large-aperture array consisting of a small number of antennas while avoiding spatial ambiguity. Numerical and experimental examples are used to illustrate the applicability of our results.

We consider the problem of estimating a discrete signal $X^n = (X_1, \ldots, X_n)$ based on its noise-corrupted observation signal $Z^n = (Z_1, \ldots, Z_n)$. The noise-free, noisy, and reconstruction signals are all assumed to have components taking values in the same finite $M$-ary alphabet $\{0, \ldots, M-1 \}$. For concreteness we focus on the additive noise channel $Z_i = X_i + N_i$, where addition is modulo-$M$, and $\{N_i\}$ is the noise process. The cumulative loss is measured by a given loss function. The distribution of the noise is assumed known, and may have memory restricted only to stationarity and a mild mixing condition. We develop a sequence of denoisers (indexed by the block length $n$) which we show to be asymptotically universal in both a semi-stochastic setting (where the noiseless signal is an individual sequence) and in a fully stochastic setting (where the noiseless signal is emitted from a stationary source). It is detailed how the problem formulation, denoising schemes, and performance guarantees carry over to non-additive channels, as well as to higher-dimensional data arrays. The proposed schemes are shown to be computationally implementable. We also discuss a variation on these schemes that is likely to do well on data of moderate size. We conclude with a report of experimental results for the binary burst noise channel, where the noise is a finite-state hidden Markov process (FS-HMP), and a finite-state hidden Markov random field (FS-HMRF), in the respective cases of one- and two-dimensional data. These support the theoretical predictions and show that, in practice, there is much to be gained by taking the channel memory into account.