University of Illinois, Urbana Champaign
ECE 310
Use the windowing method to design a length-N low-pass, generalized linear phase FIR filter with cut-off frequency π/3. (a) Find an expression for the filter coefficients {hn} N−1 n=0 if the rectangular window is used for the design. (b) Find an expression for the filter coefficients {hn} N−1 n=0 if the Hamming window is used for the design. (c) Discuss the main differences of the two resulting filters. Solution: The impulse response d[n] of an ideal low-pass filter with cutoff frequency ωc is d[n] = sin(ωcn) πn . which is of infinite length. Next, in order to obtain a causal, GLP filter with finite length N, we shift d[n] by N−1 2 to the right and only focus on the value with n = 0, 1, . . . , N − 1, then we obtain g[n] = d n − N − 1 2 = sin ωc n − N−1 2 π n − N−1 2 , n = 0, . . . , N − 1 (1) which has even symmetry. After that, we apply the window function w[n] on g[n] which gives h[n] = g[n]w[n] = w[n] · sin ωc n − N−1 2 π n − N−1 2 , n = 0,
