It is well-known that the discrete Fourier transform (DFT) of a finite length discrete-time signal samples the discrete-time Fourier transform of the same signal at equidistant points on the unit circle. Hence, as the signal length goes to infinity, the DFT approaches the DTFT. Associated with the DFT are circular convolution and a periodic signal extension. In this paper we identify a large class of alternatives to the DFT using the theory of polynomial algebras. Each of these Fourier transforms approaches the DTFT just as the DFT does, but has its own signal extension and notion of convolution. Further, these Fourier transforms have Vandermonde structure, which enables their fast computation. We provide a few experimental examples that confirm our theoretical results.

We address the problem of robust coding in which the signal information should be preserved in spite of intrinsic noise in the representation. We present a theoretical analysis for 1- and 2-D cases and characterize the optimal linear encoder and decoder in the mean-squared error sense. Our analysis allows for an arbitrary number of coding units, thus including both under- and over-complete representations, and provides insights into optimal coding strategies. In particular, we show how the form of the code adapts to the number of coding units and to different data and noise conditions in order to achieve robustness. We also present numerical solutions of robust coding for high-dimensional image data, demonstrating that these codes are substantially more robust than other linear image coding methods such as PCA, ICA, and wavelets.

We address the problem of speech estimation as statistical estimation with “missing” data in the independent component analysis (ICA) domain. Missing components are substituted by values drawn from “similar” data in a multi-faceted ICA representation of the complete data. The paper presents the algorithm for the inference of missing data in the case of a fixed pattern of missing data. We apply our approach to the problem of bandwidth extension, or where speech is degraded by a fixed filtering process and show the capability of the algorithm to reconstruct fine missing details of the original data with little artifacts. The evaluation is done using objective distortion measures on speech samples from the NTT database.

We address the problem of Speech Enhancement in a setting where parts of the time-frequency content of the speech signal are missing. In telephony, speech is band-limited and the goal is to reconstruct a wide-band version of the observed data. Quite differently, in Blind Source Separation scenarios, information about a source can be masked by noise or other sources. These masked components are “gaps” or missing source values to be “filled in”. We propose a framework for unitary treatment of these problems, which is based on a relatively simple “spectrum restoration” procedure. The main idea is to use Independent Component Analysis as an adaptive, data-driven, linear representation of the signal in the speech frame space, and then apply a vector-quantization-based matching procedure to reconstruct each frame. We analyze the performance of the reconstruction with objective quality measures such as log-spectral distortion and Itakura-Saito distance.

Biological sensory systems are faced with the problem of encoding a high-fidelity sensory signal with a population of noisy, low-fidelity neurons. This problem can be expressed in information theoretic terms as coding and transmitting a multi-dimensional, analog signal over a set of noisy channels. Previously, we have shown that robust, overcomplete codes can be learned by minimizing the reconstruction error with a constraint on the channel capacity. Here, we present a theoretical analysis that characterizes the optimal linear coder and decoder for one- and twodimensional data. The analysis allows for an arbitrary number of coding units, thus including both under- and over-complete representations, and provides a number of important insights into optimal coding strategies. In particular, we show how the form of the code adapts to the number of coding units and to different data and noise conditions to achieve robustness. We also report numerical solutions for robust coding of highdimensional image data and show that these codes are substantially more robust compared against other image codes such as ICA and wavelets.