We investigate the applicability of frame multipliers as compressive sensing measurements. We show that, under certain conditions, subsampled frame multipliers yield measurement matrices with desirable properties. To that end, we prove a general probabilistic nullspace property for arbitrary nonempty sets, that accounts for the special measurement structure induced by subsampled frame multipliers. Conditions for uniqueness of reconstruction of signals that are sparse with respect to dictionaries or, more generally, to non-linear locally Lipschitz mappings are obtained as special cases. Furthermore, we show that a frame multiplier matrix is full superregular, i.e., that all its minors are nonzero, for almost all frame symbol vectors, provided that the underlying frames are full spark and sufficiently redundant. Since Gabor frames are full spark for almost all windows, we study Gabor multipliers in more detail and are able to derive improved constants for some scenarios. Finally, our simulation results reveal that, in many instances, subsampled frame multiplier matrices exhibit the same ℓ1-reconstruction performance as i.i.d. Gaussian measurement matrices.

We propose an information-theoretic framework for phase retrieval. Specifically, we consider the problem of recovering an unknown vector x ∈ ℝnup to an overall sign factor from m = ⌊Rn⌋ phaseless measurements with compression rate R and derive a general achievability bound for R. Surprisingly, it turns out that this bound on the compression rate is the same as the one for almost lossless analog compression obtained by Wu and Verdú (2010): Phaseless linear measurements are “as good” as linear measurements with full phase information in the sense that ignoring the sign of m measurements only leaves us with an ambiguity with respect to an overall sign factor of x.

Estimating the spectral characteristics of a nonstationary random process is an important but challenging task, which can be facilitated by exploiting structural properties of the process. In certain applications, the observed processes are underspread, i.e., their time and frequency correlations exhibit a reasonably fast decay, and approximately time-frequency sparse, i.e., a reasonably large percentage of the spectral values are small. For this class of processes, we propose a compressive estimator of the discrete Rihaczek spectrum (RS). This estimator combines a minimum variance unbiased estimator of the RS (which is a smoothed Rihaczek distribution using an appropriately designed smoothing kernel) with a compressed sensing technique that exploits the approximate time-frequency sparsity. As a result of the compression stage, the number of measurements required for good estimation performance can be significantly reduced. The measurements are values of the ambiguity function of the observed signal at randomly chosen time and frequency lag positions. We provide bounds on the mean-square estimation error of both the minimum variance unbiased RS estimator and the compressive RS estimator, and we demonstrate the performance of the compressive estimator by means of simulation results. The proposed compressive RS estimator can also be used for estimating other time-dependent spectra (e.g., the Wigner-Ville spectrum), since for an underspread process most spectra are almost equal.

Recent research has demonstrated significant achievable performance gains by exploiting circularity/noncircularity or properness/improperness of complex-valued signals. In this paper, we investigate the influence of these properties on important information theoretic quantities such as entropy, divergence, and capacity. We prove two maximum entropy theorems that strengthen previously known results. The proof of the first maximum entropy theorem is based on the so-called circular analog of a given complex-valued random vector. The introduction of the circular analog is additionally supported by a characterization theorem that employs a minimum Kullback-Leibler divergence criterion. In the proof of the second maximum entropy theorem, results about the second-order structure of complex-valued random vectors are exploited. Furthermore, we address the capacity of multiple-input multiple-output (MIMO) channels. Regardless of the specific distribution of the channel parameters (noise vector and channel matrix, if modeled as random), we show that the capacity-achieving input vector is circular for a broad range of MIMO channels (including coherent and noncoherent scenarios). Finally, we investigate the situation of an improper and Gaussian distributed noise vector. We compute both capacity and capacity-achieving input vector and show that improperness increases capacity, provided that the complementary covariance matrix is exploited. Otherwise, a capacity loss occurs, for which we derive an explicit expression.

We propose a compressive method for tracking doubly selective channels within multicarrier systems, including OFDM systems. Using the recently introduced concept of modified compressed sensing (MOD-CS), the sequential delay-Doppler sparsity of the channel is exploited to improve estimation performance through a recursive estimation mode. The proposed compressive channel tracking algorithm uses a MOD-CS version of OMP with reduced complexity. Simulation results demonstrate substantial performance gains over conventional compressive channel estimation.

Recent research has demonstrated significant achievable performance gains by exploiting circularity/non-circularity or properness/improperness of complex-valued signals. In this paper, we investigate the influence of theses properties on important information theoretic quantities such as entropy and capacity. More specifically, we prove a novel maximum entropy theorem that is based on the so-called circular analog of a given (in general, non-Gaussian) complex-valued random vector. Its introduction is supported by a characterization theorem that employs a minimum Kullback-Leibler divergence criterion. As an application of this maximum entropy theorem, we show that the capacity-achieving input random vector is circular for a broad range of multiple-input multiple-output (MIMO) channels including coherent and noncoherent scenarios. This result does not depend on a Gaussian assumption and thus provides a justification for many practical signalling/coding strategies, regardless of the specific distribution of the channel parameters.

We propose a compressive estimator of doubly selective channels within pulse-shaping multicarrier MIMO systems (including MIMO-OFDM as a special case). The use of multichannel compressed sensing exploits the joint sparsity of the MIMO channel for improved performance. We also propose a multichannel basis optimization for enhancing joint sparsity. Simulation results demonstrate significant advantages over channel-by-channel compressive estimation.

We propose a ldquocompressiverdquo estimator of the Wigner-Ville spectrum (WVS) for time-frequency sparse, underspread, nonstationary random processes. A novel WVS estimator involving the signal's Gabor coefficients on an undersampled time-frequency grid is combined with a compressed sensing transformation in order to reduce the number of measurements required. The performance of the compressive WVS estimator is analyzed via a bound on the mean square error and through simulations. We also propose an efficient implementation using a special construction of the measurement matrix.

We consider the estimation of doubly selective wireless channels within pulse-shaping multicarrier systems (which include OFDM systems as a special case). A new channel estimation technique using the recent methodology of compressed sensing (CS) is proposed. CS-based channel estimation exploits a channel's delay-Doppler sparsity to reduce the number of pilots and, hence, increase spectral efficiency. Simulation results demonstrate a significant reduction of the number of pilots relative to least-squares channel estimation.

We consider pulse-shaping multicarrier (MC) communications over a (possibly rapidly varying) doubly selective channel with uncorrelated scattering. Assuming transmission free of intersymbol interference (while intercarrier interference is not constrained), we show that the statistical properties of the MC system are invariant under cyclic shifts of the subcarriers. This invariance is then used to derive structural properties of the capacity-achieving input covariance function. We show that capacity can be achieved by transmit symbols that are uncorrelated over time and cyclically stationary with respect to the frequency (subcarrier) index. We also show that capacity- achieving precoding can be equivalently realized by a suitable adaptation of the transmit pulse. For classical OFDM and transmission over a WSSUS channel, precoding can be completely avoided if pulse-shaping MC transmission is used instead.

This paper considers the complex random vectors and study some important properties. We develop a theory which is based on the concept of covariance and pseudo-covariance matrix in order to prove a stronger version of the maximum entropy theorem for the complex multivariate case (F.D. Neeser et al. 1993).

We consider discrete multitone (DMT, baseband OFDM) modulation and perform a detailed noise analysis which takes into account dependencies and power differences of the real and imaginary parts after the complex valued discrete Fourier transform (DFT). We show that for colored noise, rotated rectangular symbol constellations are more appropriate than the common (quadratic) QAM symbol constellations with respect to capacity and symbol error probability, and we derive formulas for the rotation angles and constellation sizes/densities.