延 安大学数字信号处理课件Chapter 2 The Discreete-Time Fourier Transform (DTFT).pptVIP

6. Stability condition in terms of pole location On the unit circle, the above reduces to The ROC of the transfer function H(z) of a BIBO stable LTI digital filter contains the unit circle. Conversely, if the ROC of the transfer function of an LTI digital filter includes the unit circle, then the filter is BIBO stable. 6. Stability condition in terms of pole location The stability testing of an IIR transfer function is therefore an important problem. However, it is difficult to compute the sum S of Eq. (6.102) analytical in most cases. For a causal IIR transfer function, it can be computed approximately by replacing the right-hand side of Eq. (6.102) with the following finite sum 6. Stability condition in terms of pole location %Program_6_7 % Stability Test % num = input(Type in the numerator vector = ); den = input(Type in the denominator vector = ); N = max(length(num),length(den)); x = 1; y0 = 0; S = 0;zi = zeros(1,N-1); for n = 1:1000 [y,zf] = filter(num,den,x,zi); if abs(y) 0.000001, break, end x = 0; S = S + abs(y); y0 = y;zi = zf; end if n 1000 disp(Stable Transfer Function); else disp(Unstable Transfer Function); end 6. Stability condition in terms of pole location We now develop an alternate stability condition based on the location of the poles of the transfer function. We conclude that all poles of a stable causal transfer function H(z) must be strictly inside the unit circle. 6. Stability condition in terms of pole location An anticausal digital filter has a left-sided impulse response h[n] and as a result, the ROC of its transfer function H(z) is interior to the circle going through the pole that is closest to the origin. In the case, all poles of a stable anticausal transfer function H(z) must be strictly out the unit circle. Some Special Filters Allpass Filter Comb Filter Minimum-Phase and Maximum-Phase Allpass Transfer Function Definition An IIR transfer function A(z) with unity magnitude response for all freque