Introduction to Digital Signal and System Analysis - Tutorsindia

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Introduction to Digital Signal and System Analysis Z Domain Analysis For poles, those close to the unit-circle can produce sharp peaks in the frequency response. Therefore, high gain can be achieved by placing the poles close to the unit circle. Equal number of zeros and poles are normally placed to ensure no system delay or causing non-causality, i.e. the impulse response begins at n=0. Consider the first order system with a zero the origin The first-order system Its frequency response z H ( 1 z) = z −α exp( jW) H1( W) = exp( jW) − α To achieve low-pass, adopting 0 < 1 smallest, and the maximum gain value is < α , i.e. the pole is on the positive axis. The denominator 1 −α becomes the G max = exp(0) 1 = exp(0) −α 1−α and the minimum gain is G min = exp( jp ) 1 = exp( jp ) −α 1+ α To achieve a high-pass, adopting 1 < < 0 biggest, and the maximum gain at peak value is − α , i.e. the pole is on negative axis, the denominator + α 1 becomes the G max = exp( jp ) 1 = exp( jp ) −α 1+ α , and the minimum gain is G min = exp(0) 1 = exp(0) −α 1−α When α is close to 1, the peak gain gets high, the bandwidth gets more narrow, and the impulse response decays more slowly. The following figures illustrate the above low and high pass filters. 78 Download free ebooks at bookboon.com
Introduction to Digital Signal and System Analysis Z Domain Analysis Low pass 0 < α < 1 High pass -1 < α < 0 1 1 0 0 -1 0 5 10 n -1 0 5 10 n 5 5 |H(W)| -pi/2 ≤ phase ≤ p/2 |H(W)| 0 0 1 2 3 0 ≤ W ≤ 2p 1 0 -1 0 1 2 3 0 ≤ W ≤ 2p -pi/2 ≤ phase ≤ p/2 0 0 1 2 3 0 ≤ W ≤ 2p 1 0 -1 0 1 2 3 0 ≤ W ≤ 2p 1.5 z-plane 1.5 z-plane 1 1 0.5 0.5 0 -0.5 W=p α W=0 0 -0.5 W=p α W=0 -1 -1 -1.5 -1 0 1 Low pass -1.5 -1 0 1 High pass Figure 5.9 The frequency response, poles and zeros of low and high pass filters For a second-order system, the transfer function in polar form is H 2 ( z) = z 2 2 2 ( z − r exp( jq )( z − r exp( − jq )) z − 2rz cosq + r = z 2 Its frequency response is 79 Download free ebooks at bookboon.com
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