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Approximate Signal Processing
, 1997
"... It is increasingly important to structure signal processing algorithms and systems to allow for trading off between the accuracy of results and the utilization of resources in their implementation. In any particular context, there are typically a variety of heuristic approaches to managing these tra ..."
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It is increasingly important to structure signal processing algorithms and systems to allow for trading off between the accuracy of results and the utilization of resources in their implementation. In any particular context, there are typically a variety of heuristic approaches to managing these tradeoffs. One of the objectives of this paper is to suggest that there is the potential for developing a more formal approach, including utilizing current research in Computer Science on Approximate Processing and one of its central concepts, Incremental Refinement. Toward this end, we first summarize a number of ideas and approaches to approximate processing as currently being formulated in the computer science community. We then present four examples of signal processing algorithms/systems that are structured with these goals in mind. These examples may be viewed as partial inroads toward the ultimate objective of developing, within the context of signal processing design and implementation,...
Levinson and Fast Choleski Algorithms for Toeplitz and Almost Toeplitz Matrices
, 1988
"... In this paper, we review Levinson and fast Choleski algorithms for solving sets of linear equations involving Toeplitz or almost Toeplitz matrices. The LevinsonTrenchZohar algorithm is first presented for solving problems involving exactly Toeplitz matrices. A fast Choleski algorithm is derived by ..."
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In this paper, we review Levinson and fast Choleski algorithms for solving sets of linear equations involving Toeplitz or almost Toeplitz matrices. The LevinsonTrenchZohar algorithm is first presented for solving problems involving exactly Toeplitz matrices. A fast Choleski algorithm is derived by a simple linear transformation. The almost Toeplitz problem is then considered and a Levinsonstyle algorithm is proposed for solving it. A set of linear transformations converts the algorithm into a fast Choleski method. Symmetric and band diagonal applications are considered. Formulas for the inverse of an almost Toeplitz matrix are derived. The relationship between the fast Choleski algorithms and a Euclidian algorithm is exploited in order to derive accelerated &quot;doubling &quot; algorithms for inverting the matrix. Finally, strategies for removing the strongly nonsingular constraint
Design and analysis of Toeplitz preconditioners
 IEEE Trans. Acoust. Speech Signal Process
, 1992
"... AbstractThe solution of symmetric positive definite Toeplitz systems Ax = b by the preconditioned conjugate gradient (PCG) method was recently proposed by Strang and analyzed by R. Chan and Strang. The convergence rate of the PCG method depends heavily on the choice of preconditioners for the given ..."
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AbstractThe solution of symmetric positive definite Toeplitz systems Ax = b by the preconditioned conjugate gradient (PCG) method was recently proposed by Strang and analyzed by R. Chan and Strang. The convergence rate of the PCG method depends heavily on the choice of preconditioners for the given Toeplitz matrices. In this paper, we present a general approach to the design of Toeplitz preconditioners based on the idea to approximate a partially characterized linear deconvolution with circular deconvolutions. All resulting preconditioners can therefore be inverted via various fast transform algorithms with O(N log N) operations. For a wide class of problems, the PCG method converges in a finite number of iterations independent of N so that the computational complexity for solving these Toeplitz systems is O(N log N). 1.
High Performance Algorithms To Solve Toeplitz And Block Toeplitz Matrices
, 1996
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Whitening as a tool for estimating mutual information in spatiotemporal data sets
 Journal of Statistical Physics
"... We address the issue of inferring the connectivity structure of spatially extended dynamical systems by estimation of mutual information between pairs of sites. The wellknown problems resulting from correlations within and between the time series are addressed by explicit temporal and spatial model ..."
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We address the issue of inferring the connectivity structure of spatially extended dynamical systems by estimation of mutual information between pairs of sites. The wellknown problems resulting from correlations within and between the time series are addressed by explicit temporal and spatial modelling steps which aim at approximately removing all spatial and temporal correlations, i.e. at whitening the data, such that it is replaced by spatiotemporal innovations; this approach provides a link to the maximumlikelihood method and, for appropriately chosen models, removes the problem of estimating probability distributions of unknown, possibly complicated shape. A parsimonious multivariate autoregressive model based on nearestneighbour interactions is employed. Mutual information can be reinterpreted in the framework of dynamical model comparison (i.e. likelihood ratio testing), since it is shown to be equivalent to the difference of the loglikelihoods of coupled and uncoupled models for a pair of sites, and a parametric estimator of mutual information can be derived. We also discuss, within the framework of model comparison, the relationship between the coefficient of linear correlation and mutual information. The practical application of this methodology is demonstrated for
On LowComplexity Approximation of Matrices
 Lin. Alg. Appl
, 1992
"... this paper, we pursue a complementary notion of structure which we will call the state structure. The state structure applies to upper triangular matrices and is seemingly unrelated to the Toeplitz or displacement structure mentioned above. A first purpose of the computational schemes considered in ..."
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this paper, we pursue a complementary notion of structure which we will call the state structure. The state structure applies to upper triangular matrices and is seemingly unrelated to the Toeplitz or displacement structure mentioned above. A first purpose of the computational schemes considered in this paper is to perform a desired linear transformation T on some vector (`input sequence')
AN ITERATIVE LEASTSQUARES TECHNIQUE FOR DEREVERBERATION
"... Some recent dereverberation approaches that have been effective for automatic speech recognition (ASR) applications, model reverberation as a linear convolution operation in the spectral domain, and derive a factorization to decompose spectra of reverberated speech in to those of clean speech and ro ..."
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Some recent dereverberation approaches that have been effective for automatic speech recognition (ASR) applications, model reverberation as a linear convolution operation in the spectral domain, and derive a factorization to decompose spectra of reverberated speech in to those of clean speech and roomresponse filter. Typically, a general nonnegative matrix factorization (NMF) framework is employed for this. In this work 1 we present an alternative to NMF and propose an iterative leastsquares deconvolution technique for spectral factorization. We propose an efficient algorithm for this and experimentally demonstrate it’s effectiveness in improving ASR performance. The new method results in 4050 % relative reduction in word error rates over standard baselines on artificially reverberated speech.
Incremental Refinement Structures for Approximate Signal Processing
, 1997
"... This work investigates approximate signal processing as a design philosophy supporting the realization of efficient, robust, and flexible digital signal processing systems through the use of incremental refinement structures that allow tradeoffs to be easily made between the accuracy or optimality o ..."
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This work investigates approximate signal processing as a design philosophy supporting the realization of efficient, robust, and flexible digital signal processing systems through the use of incremental refinement structures that allow tradeoffs to be easily made between the accuracy or optimality of results and the utilization of computing resources such as time, power, and chip area. The value of this approach is demonstrated through the theoretical development of incremental refinement structures for signal detection using the fast Fourier transform (FFT), image decoding using the twodimensional inverse discrete cosine transform (2D IDCT), and spectral analysis using the discrete Fourier transform (DFT). Using both deterministic and probabilistic techniques, the theoretical performance of these structures under various resource constraints is quantified in terms of welldefined measures such as probability of detection, SNR, and frequency resolution. These analyses are verified for...