We consider the application of compressed sensing (CS) to the estimation of doubly selective channels within pulseshaping multicarrier systems (which include OFDM systems as a special case). By exploiting sparsity in the delay-Doppler domain, CS-based channel estimation allows for an increase in spectral efficiency through a reduction of the number of pilot symbols. For combating leakage effects that limit the delay-Doppler sparsity, we propose a sparsity-enhancing basis expansion and a method for optimizing the basis with or without prior statistical information about the channel. We also present an alternative CSbased channel estimator for (potentially) strongly time-frequency dispersive channels, which is capable of estimating the "offdiagonal" channel coefficients characterizing intersymbol and intercarrier interference (ISI/ICI). For this estimator, we propose a basis construction combining Fourier (exponential) and prolate spheroidal sequences. Simulation results assess the performance gains achieved by the proposed sparsity-enhancing processing techniques and by explicit estimation of ISI/ICI channel coefficients.Index Terms-channel estimation, compressed sensing, CoSaMP, dictionary learning, doubly selective channel, intercarrier interference, intersymbol interference, Lasso, multicarrier modulation, orthogonal frequency-division multiplexing (OFDM), orthogonal matching pursuit (OMP), sparse reconstruction.CP-OFDM is a simple special case of the pulse-shaping MC framework; it is obtained for a rectangular transmit pulse g[n] that is 1 for n = 0, . . . , N−1 and 0 otherwise, and a rectangular receive pulse γ[n] that is 1 for n = N −K, . . . , N −1 and 0 otherwise (N −K ≥ 0 is the CP length). Georg Tauböck (S'01-M'07) received the Dipl.-Ing. degree and the Dr.techn. degree (with highest honors) in electrical engineering and the Dipl.-Ing. degree in mathematics (with highest honors) from His research interests include wireline and wireless communications, compressed sensing, signal processing, and information theory.Franz Hlawatsch (S'85-M'88-SM'00) received the Diplom-Ingenieur, Dr. techn., and Univ.-Dozent (habilitation) degrees in electrical engineering/signal processing

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Abstract-Recent research has demonstrated significant achievable performance gains by exploiting circularity/non-circularity or propeness/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 former theorem is based on the so-called circular analog of a given complex-valued random vector. Its introduction is supported by a characterization theorem that employs a minimum Kullback-Leibler divergence criterion. In the proof of latter theorem, on the other hand, 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.

In this paper, we consider discrete multitone (DMT) or baseband orthogonal frequency-division multiplexing (OFDM) modulation and perform a detailed noise analysis which takes into account dependencies and power (variance) differences of real and imaginary part after the complex-valued discrete Fourier transform (DFT). The derivation is based on the so-called pseudocovariance matrix of a complex random vector, which was introduced by Neeser and Massey (1993). We show that the relevant pseudocovariance matrix is not the zero matrix in general, in contrast to passband OFDM, for which it can be proven (under certain assumptions) that all occurring pseudocovariance matrices are vanishing. We show that for colored noise rotated rectangular symbol constellations are more appropriate than the common quadratic quadrature amplitude modulation (QAM) symbol constellations with respect to capacity and symbol error probability, and we derive formulas for the rotation angles and constellation sizes/densities. Finally, we extend the results to a multitransceiver [multiple-input-multipleoutput (MIMO)] scenario, for which we assume a very general noise model at the inputs of the receivers, allowing correlations between the noise signals of different receivers. This requires the introduction of pseudocross-covariance matrices of complex random vectors, which are the important objects (together with cross-covariance matrices) in the MIMO situation.