In the development of the minimax problem in optimal equiripple FIR linear phase
ID: 1834301 • Letter: I
Question
In the development of the minimax problem in optimal equiripple FIR linear phase filter design, we use the fact that the frequency responses of all four types of causal linear phase filters can be written in a common form as
For type I FIR linear phase filter with h[n]=h[M-n] for n=0,1,...M, where M is even, show
where
In the development of the minimax problem in optimal equiripple FIR linear phase filter design, we use the fact that the frequency responses of all four types of causal linear phase filters can be written in a common form as where M+1 is the filter length For type I FIR linear phase filter with h[n]=h[M-n] for n=0,1,...M, where M is even, show whereExplanation / Answer
H(ejw)= e-jwM/2M/2n=0 a[n]cosn a[n]=h[M/2] for n=0 a[n]=2h[M/2-n] for 1<=n<=M/2now let us consider that H(ejw)= e-jwM/2M/2n=0 a[n]cosn = Hr(w)e-jwM/2 H(ejw)= (M-1)/2n=0 h[n]e-jwn+Mn-(M+1)/2 h[n]e-jwn = (M-1)/2n=0 h[n]e-jwn+(M-1)/2n=0 h[M-n]e-jw(M-n) = (M-1)/2n=0 h[n]( e-jwn+e-jw(M-n)) = e-jwM/2(M-1)/2n=0 2h[n]cos{w(M/2-n)} consider here n= (M+1/2-m) = e-jwM/2(M+1)/2m-1 2h[M+1/2-m] cos{w(m-1/2)} = [(M+1)/2n-1b(n)cos{w(n-1/2)}]e-jwM/2 = Hr(w)e-jwM/2 = Hr(w)= (M+1)/2n-1b(n)cos{w(n-1/2)} = real even function of w b(n)=2h(M+1/2-n) n=1,2,....M+1/2
h(n)=-h[M-n] h[M/2]=0 H(ejw)=[ M/2n=1 c(n) sin wn]e-jwM/2+j/2 = jHr(w)e-jwM/2 c(n)= 2h[M/2-n] Hr(w)= M/2n=1 c(n) sin wn = A0(w) =real odd function