ConvolutionMatricesforSignalProcessingApplicationsinMATLAB.docVIP

Convolution Matrices for Signal Processing Applications in MATLAB BASIC IDEA - TRANSFER FUNCTION There are several discrete linear system, a.k.a. digital filter, representations used by the Signal Processing Toolbox. The most commonly used is the transfer function representation, consisting of the ratio of two polynomials where and X(z) and Y(z) are the Z-transforms of the input and output sequences x(n) and y(n) respectively: In MATLAB the filter H(z) is represented by vectors containing the polynomial coefficients of B(z) and A(z). Here is an example filter in the transfer function form [b,a] = butter(5,.5) b = 0.0528 0.2639 0.5279 0.5279 0.2639 0.0528 a = 1.0000 -0.0000 0.6334 -0.0000 0.0557 -0.0000 b contains the numerator coefficients, a contains the denominator coefficients. The filters Z-transform can be computed around the unit circle with the freqz function [H,w] = freqz(b,a); plot(w,abs(H)) ylabel(Magnitude) xlabel(Frequency (radians)) title(Lowpass Filter Response) CONVOLUTION MATRIX The transfer function form is general for linear, time-invariant operators on the space of complex discrete sequences. An alternative general representation for any linear system (time-invariant or not) is an infinite dimensional matrix (a convolution matrix). However, infinity is not practical for implementation in MATLAB. Can a finite truncation of this matrix be useful? IN MATLAB Representing a system with a finite convolution matrix is ideally suited to MATLAB. Consider a system matrix C, which maps Cm to Cn (the m dimensional space of complex vectors to the n dimensional). Thus C is n-by-m. Given length n input vector x and length m output vector y, the input-output relationship of the system is y = C*x Lets compute the impulse response of our filter h = impz(b,a) h = 0.0528 0.2639 0.4944 0.3607 -0.0522 -0.1904 0.00