w[n] I(B) 0 M Kaiser Window Given a set of filter specifications,the values of M and B needed can be determined numerically. As in OSB Figure 7.23,define: Transition bandwidth:Aw =ws-wp Peak approximation error:6 (passband:1+6 stopband:+6) Stopband attenuation:A =-20log10 6 Then the values of B,M needed to achieve A are 0.1102(A-8.7) A>50 0.5842(A-21)0.4+0.07886(A-21)21≤A≤50， A-8 3= 0 A<21 M22.285AW See OSB Section 7.3 for examples of the Kaiser window method. Optimum FIR Filter Approximation: The Parks-McClellan Algorithm Although the rectangular windowing method provides the best mean-square approximation to a desired frequency response,just because it is optimal does not mean it is good.FIR filters designed with windows exhibit oscillatory behavior around the discontinuity of the ideal fre- quency response and does not allow separate control of the passband and stopband ripples.An alternative FIR design technique is the Parks-McClellan algorithm which is based on polynomial approximations. Filter Design as Polynomial Approximation Consider the DTFT of a causal FIR system of length M+1: M M H(eu)=∑hmle-m=∑hinl"I=e-扣 n=0 3Kaiser Window Given a set of filter specifications, the values of M and β needed can be determined numerically. As in OSB Figure 7.23, define: Transition bandwidth: Δω = ωs − ωp Peak approximation error: δ (passband: 1 ± δ stopband: ± δ) Stopband attenuation: A = −20 log10 δ Then the values of β, M needed to achieve A are β = ⎧ ⎨ ⎩ 0.1102(A − 8.7) A > 50 0.5842(A − 21)0.4 + 0.07886(A − 21) 21 ≤ A ≤ 50 , A − 8 M ≥ 2.285Δω A < 21 . 0 See OSB Section 7.3 for examples of the Kaiser window method. Optimum FIR Filter Approximation: The Parks-McClellan Algorithm Although the rectangular windowing method provides the best mean-square approximation to a desired frequency response, just because it is optimal does not mean it is good. FIR filters designed with windows exhibit oscillatory behavior around the discontinuity of the ideal fre￾quency response and does not allow separate control of the passband and stopband ripples. An alternative FIR design technique is the Parks-McClellan algorithm which is based on polynomial approximations. Filter Design as Polynomial Approximation Consider the DTFT of a causal FIR system of length M + 1: � M h e−jωn [n] = � M H(ejω) = h n [n]x |x=e−jω . n=0 n=0 3