Digital Signal Processing Problems Research Articles

Some properties of singular value decomposition and their applications to digital signal processing

Singular value decomposition (SVD), one of the most basic and important tools of numerical linear algebra, is finding increasing applications in digital signal processing. In this paper, we first review some basic properties of SVD for matrices and then for linear operators between two Hilbert spaces. While SVD results for matrices are quite well known, applications of SVD on operators are less well known. Specifically, we propose the use of a representation for a stochastic linear system from the point of view of a stochastic SVD and show its relationship to the classical Karhunen-Loève expansion. Furthermore, issues related to practical use of time-averaging of realizations of random processes from the SVD point of view are also considered. Then algorithms for performing SVD are briefly discussed. These include the well known QR method, the modified Jacobi method of Nash, and an extension of the modified Jacobi method applicable to SVD of linear operators. Finally, we review various applications of finite rank approximation properties of SVD to digital signal processing problems in least-squares estimations, complexity reduction of FIR digital filters, matrix rank and system order determinations, and digital image processing.

Simulation of multiple digital signal processor systems

The high speed needed in solving digital signal processing problems in real time has often given rise to multiple processor hardware designs. Devices such as the TMS32020 digital signal processor possess features designed to support concurrent processing. Progress in this area is currently hampered by the lack of suitable multiprocessor development tools. It is suggested that an incremental approach to multiprocessor development, using several methods of simulating the signal processor, may be used. Two simulation environments specifically for the development and testing of multiple digital signal processor designs are described. Firstly a single processor simulation system where the algorithms which will be performed by other concurrent processors may be executed in a high level language but without any need to simulate the instructions of the other processors. Secondly a multiple TMS32020 digital signal processor system where processors are simulated as several communicating tasks on a host computer using the IBM AIX (UNIX derived) multitasking operating system.

A Parallel Architecture for Discrete Relaxation Algorithm

Discrete relaxation techniques have proven useful in solving a wide range of problems in digital signal and digital image processing, artificial intelligence, operations research, and machine vision. Much work has been devoted to finding efficient hardware architectures. This paper shows that a conventional hardware design for a Discrete Relaxation Algorithm (DRA) suffers from O(n2m3) time complexity and O(n2m2) space complexity. By reformulating DRA into a parallel computational tree and using a multiple tree-root pipelining scheme, time complexity is reduced to O(nm), while the space complexity is reduced by a factor of 2. For certain relaxation processing, the space complexity can even be decreased to O(nm). Furthermore, a technique for dynamic configuring an architectural wavefront is used which leads to an O(n) time highly concurrent DRA3 architecture.

Doubly pipelined CORDIC array for digital signal processing algorithms

Abstract In this paper, we present a doubly pipelined VLSI CORDIC array processor for digital signal processing computations. The basic notion of doubly pipelined CORDIC computation will be introduced first. Then, some potential applications to digital signal processing problems will be discussed. Specifically, we shall demonstrate how this novel architecture can be applied to compute discrete Fourier transform and fast Fourier transform, to implement lattice filters, to solve Toeplitz systems, as well as matrix QR/LQ factorizations. It is shown that by adopting secondary pipelining, about one third of the hardware can be saved, and under certain cases, the throughput of the entire CORDIC processor array may be doubled.

Filter banks allowing perfect reconstruction

Splitting a signal into N filtered channels subsampled by N is an important problem in digital signal processing. A fundamental property of such a system is that the original signal can be perfectly recovered from the subsampled channels. It is shown that this can always be done, and that FIR solutions exist. This is done by mapping the NM-dimensional nonlinear problem (where N is the number of channels and M the length of the FIR filters) into an M-dimensional linear problem. For N = 2, a general class of FIR solutions is derived, together with methods to find filters. The dual problem of mixing N signals into one channel upsampled by N is also addressed. Several applications are proposed. All results are obtained by looking at the N filter bank as a true N channel system, rather than N separate channels.

Some complexity issues in digital signal processing

Over the past decade a large class of problems, called NP-complete [5], have been shown to be equivalent in the sense that if a fast algorithm can be found for one, fast algorithms can be found for all. At the same time, despite much effort, no fast algorithms have been found for any, and these problems are widely regarded as intractable. This class includes such notoriously difficult problems as the traveling salesman problem, graph coloring, and satisfiability of Boolean expressions. Using FIR filter implementation as an illustration, we describe some problems in digital signal processing that are NP-complete. These include: 1) minimize the number of additions needed to implement a fixed FIR filter; 2) minimize the number of registers needed to implement a fixed FIR filter; and 3) minimize the time to perform the additions of such an FIR filter using P adders. Large-seale instances of such problems may become important with the use of programmable chips to implement signal processing. Our main purpose in this paper is to illustrate the usefulness of asymptotic complexity theory in the field of digital signal processing. The theory discriminates between tractable and intractable problems, Sometimes identifies fast algorithms for the former, and justifies heuristics for the latter.

Coding of Real-Number Sequences for Error Correction: A Digital Signal Processing Problem

Error-correcting codes defined over the real-number and complex-number fields are introduced. The possibility of utilizing realnumber arithmetic permits the codes to be implemented with operations normally available in standard programmable digital signal processors by methods which are discussed. Hadamard and discrete Fourier transform codes are presented for block coding, and the latter are seen to be cyclic and to include the class of BCH codes. It is shown that maximum distance separable real-number BCH ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N, K</tex> ) codes exist for all nontrivial values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> . A large class of block and convolutional real-number single-error-correcting codes, derived from similar codes over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(p)</tex> , are presented. Both block and convolutional codes are seen to be describable by the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> -transform in a manner which emphasizes their similarities to conventional digital signal processing structures such as digital filters and digital filter banks. Methods for correcting weight <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t + 1</tex> errors in a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> error-correcting code are demonstrated and interpreted; in particular, the use of a VLSI digital signal processor for implementation of an algorithm for correcting almost all double adjacent error patterns in a single-error-correcting convolutional code is discussed.

Incoherent optical 1-bit cross-correlators for radio antenna arrays

A new incoherent optical approach to an important digital signal processing problem in radio astronomy is presented. It greatly simplifies the many thousands of 1-bit digital cross-correlations that can be required by using a digitally controlled light panel and photodiode array.

Group character tables in discrete transform theory

It has been shown that a transform satisfying a generalized form of the convolution theorem can be described by the group character table of an appropriate finite Abelian group G. Exact correspondence between the transform properties and the underlying group properties is established and it has been demonstrated that many of the digital signal processing problems may be solved efficiently using the group theoretic approach. Sets of permutations are fully characterized such that a permutational convolution defined with respect to them can be converted into a transform product by some invertible transform.

LM-g splines

As an extension of the notion of an L-g spline, three mathematical structures called LM-g splines of types I, II, and III are introduced. Each is defined in terms of two differential operators L = D n + ∑ j=0 n−1 a j(t)D j and M = ∑ i=0 m b i(t)D i, where n ⩾ m ges; 0, n > 0, D = d dt the coefficients a j , j = 0,…, n − 1, and b i , i = 0,…, m, are sufficiently smooth; and b m is bounded away from zero on [0, T]. Each of the above types of splines is the solution of an optimization problem more general than the one used in the definition of the L-g spline and hence it is recognized as an entity which is distinct from and more general mathematically than the L-g spline. The LM-g splines introduced here reduce to an L-g spline in the special case in which m = 0 and b 0 = constant ≠ 0. After the existence and uniqueness conditions, characterization, and best approximation properties for the proposed splines are obtained in an appropriate reproducing kernel Hilbert space framework, their usefulness in extending the range of applicability of spline theory to problems in estimation, optimal control, and digital signal processing are indicated. Also, as an extension of recent results in the generalized spline literature, state variable models for the LM-g splines introduced here are exhibited, based on which existing least squares algorithms can be used for the recursive calculation of these splines from the data.

Fixed-point implementation algorithms for a class of orthogonal polynominal filter structures

In previous papers it has been demonstrated that a class of digital filter structures derived from the theory of orthogonal polynomials has practical utility in digital signal processing problems. In this paper the specific algorithms necessary to synthesize and scale these structures in fixed-point arithmetic (based upon L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> norm scaling) are presented and implemented as Fortran programs.