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FIR Filters - Chebyshev Filters

Impulse response of the system characterizes the filter. Impulse response of the filter is always equal to the value of the filter coefficients. Thus if designed filter produces the filter coefficients as its output for impulse input then correctness of the filter design is verified. | Chapter 4 Chebyshev Filters Chebyshev filters are designed to have an amplitude response characteristic that has a relatively sharp transition from the pass band to the stop band. This sharpness is accomplished at the expense of ripples that are introduced into the response. Specifically Chebyshev filters are obtained as an equiripple approximation to the pass band of an ideal lowpass filter. This results in a filter characteristic for which 4J where c2 10r 1 1 Tn m Chebyshev polynomial of order n r passband ripple dB Chebyshev polynomials are listed in Table 4.1. TABLE 4.1 Chebyshev Polynomials n Tn o 0 1 2 3 4 5 6 7 8 9 10 H W M o p p - 8 8 8 C C Ci 0 cn i g oo I . - 1 1 1 K o O M p M bo c u_ oo o s C C O 1 1 _J p -J . T T Eg1 1-1 31 o 1 M 8 1 1 h- w e to to o p p c C M 8 X 8 o 8 tc 1 77 78 Chapter Four 4.1 Transfer Function The general shape of the Chebyshev magnitude response will be as shown in Fig. 4.1. This response can be normalized as in Fig. 4.2 so that the ripple bandwidth yr is equal to 1 or the response can be normalized as in Fig. 4.3 so that the 3-dB frequency a 0 is equal to 1. Normalization based on the ripple bandwidth involves simpler calculations but normalization based on the 3-dB point makes it easier to compare Chebyshev responses to those of other filter types. Figure 4.1 Magnitude response of a typical lowpass Chebyshev filter. frequency Figure 4.2 Chebyshev response normalized to have pass-band end at co 1 rad s. Features are a ripple limits b pass band c transition band d stop band and e intersection of response and lower ripple limit at co 1. Chebyshev Filters 79 frequency Figure 4.3 Chebyshev response normalized to have 3-dB point at 1 rad s. Features are a ripple limits 6 pass band c transition band d stop band and e response that is 3 dB down at co 1. The general expression for the transfer function of an nth-order Chebyshev lowpass filter is given by H n 1 s-s s-sx s-s2 s-s ffH-sJ where Ho 1 1 n 10r 2 f -s i 1 si ai fai n odd n even