Direct form implementation of a Butterworth filter may cause instability because of coef- ficient quantization.The poles may move to the outside of the unit circle,and the zeros may spread around z=-1 as shown in the next figure.A solution is to use a cascade structure of first and second order systems,with zeros and poles grouped into complex conjugate pairs.Furthermore,since all of the zeros are located at z=-1,the numerator of the transfer function is(1+1)N.No multiplication is actually needed to implement the zeros,because this is equivalent to the cascade of N "delay and add"operations. Butterworth Filter,15th Order Pole-zero plot -1-050 05 Impulse Response 02 20 30 Time.n Figure 2 Pole-Zero Plot,Response Design Specifications: 5.Maximum stophand gain =-30 dB Chebyshev Filters Type I HCH(ej)2= 1+2V tan (w/2) tan (wc/2) Three design parameters:order N,PB cut-off frequency we,allowed PB ripple e(ie.the maximum allowable passband gain is 1,and the minimum allowable passband gain is (1-e). VN(z)=cos(N cos-1z)is the Nth-order chebyshev polynomial.Here we take the con- vention that cos-1x becomes the inverse hyperbolic cosine and is imaginary if >1. 5• Direct form implementation of a Butterworth filter may cause instability because of coef­ ficient quantization. The poles may move to the outside of the unit circle, and the zeros may spread around z = −1 as shown in the next figure. A solution is to use a cascade structure of first and second order systems, with zeros and poles grouped into complex conjugate pairs. Furthermore, since all of the zeros are located at z = −1, the numerator of the transfer function is (1 + z−1)N . No multiplication is actually needed to implement the zeros, because this is equivalent to the cascade of N “delay and add” operations. Chebyshev Filters Type I 1 2 |HCH(e = jω)| � tan (ω/2) � 1 + �2V 2 N tan (ωc/2) • Three design parameters: order N, PB cut-off frequency ωc, allowed PB ripple � (ie. the maximum allowable passband gain is 1, and the minimum allowable passband gain is (1 − �)). VN (x) = cos(N cos−1 • x) is the Nth-order chebyshev polynomial. Here we take the con￾vention that cos−1 x becomes the inverse hyperbolic cosine and is imaginary if | | x > 1. 5