Let a > 0, b > 0, ab < 1; and let g ∈ L 2 (R). In this paper we investigate the relation between the frame operator S : f ∈ L 2 (R) → n,m (f, g na,mb ) g na,mb and the matrix H whose entries H k,l ; k ,l are given by (g k /b,l /a where I denotes the identity operator of L 2 (R) and l 2 (Z 2 ), respectively, and A ≥ 0, B < ∞. Next, when g generates a frame, we have that (g k/b,l/a ) k,l has an upper frame bound, and the minimal dual function • γ can be computed as ab k,l (H −1 ) k,l ; o,o g k/b,l/a . The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a g generating a frame are inherited by • γ . In particular, we show that • γ ∈ S when g ∈ S generates a frame (S Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula.

Abstract-The generalized entropies of Rényi inspire new measures for estimating signal information and complexity in the time-frequency plane. When applied to a time-frequency representation (TFR) from Cohen's class or the affine class, the Rényi entropies conform closely to the notion of complexity that we use when visually inspecting time-frequency images. These measures possess several additional interesting and useful properties, such as accounting and cross-component and transformation invariances, that make them natural for time-frequency analysis. This paper comprises a detailed study of the properties and several potential applications of the Rényi entropies, with emphasis on the mathematical foundations for quadratic TFRs. In particular, for the Wigner distribution, we establish that there exist signals for which the measures are not well defined.

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This paper presents an adaptive algorithm for the restoration of lost sample values in discrete-time signals that can locally be are that the positions of the unknown samples should be known and that they should be embedded in a sufficiently large neighborhood of known samples. The estimates of the unknown samples are obtained by minimizing the sum of squares of the residual errors that involve estimates of the autoregressive parameters. A statistical analysis shows that, for a burst of lost samples, the expected quadratic interpolation error per sample converges to the signal variance when the burst length tends to infinity. The method is in fact the first step of an iterative algorithm, in which in each iteration step the current estimates of the missing samples are used to compute the new estimates. Furthermore, the feasibility of implementation in hardware for real-time use is established. The method has been tested on artificially generated autoregressive processes as well as on digitized music and speech signals. described by means of autoregressive processes. The only restrictions sk t T IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING,