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%  exa060302.m,  for 例6.3.2 和 图 6.3.2 
%  利用单带滤波器来设计一个半带滤波器，并显示半带滤波器H0,H1,及 H0+H1的幅频响应 
%  单带滤波器的归一化截止频率为0.84,长度＝16，半带滤波器的长度＝31。 
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clear all; 
 
N0=512; 
%First step: To design a one-band filter G(z) by Chebyshev approximation 
% set the cutoff frequency of the G(w); 
F=[0 0.84,1,1]; 
A=[1 1 0,0]; 
N=15; 
B=remez(N,F,A) 
a=1; 
wf=0:pi/N0:pi*(N0-1)/N0; 
wff=0:1/N0:(N0-1)/N0; 
Gw= freqz(B,a,wf); 
Ew=exp(i*N*wf/2); 
Gr=Ew.*Gw; 
% Gr is a zero phase one-band filter,so Gr is real 
 
figure(1); 
subplot(2,1,1) 
plot(wff,real(Gr));grid on; 
title('single-band filter  G(jw)'); 
subplot(2,1,2) 
stem(B,'filled') 
title('single-band filter   g(n)') 
hold on 
plot((0:10),zeros(11),'b');grid on; 
 
% To obtain half-band filter F(z) from G(z),  
% F(z)=[G(z^2)+(N-1)/2]/2 
s=length(B) 
B2=zeros(1,2*s-1); 
for k=1:s, 
   B2(k*2-1)=B(k); 
  end,  
N2=length(B2) 
B2(N+1)=1+B2(N+1); 
B2=B2/2 
Fw=freqz(B2,a,wf); 
Ew=exp(i*N*wf); 
Fr=Ew.*Fw; 
Fg=real(Fr); 
 
figure(2); 
subplot(2,1,1) 
plot(wff,Fg);grid on; 
title('Half-band filter H(jw)'); 
subplot(2,1,2) 
stem(B2,'filled');grid on; 
title('Half-band filter h(n)') 
 
hold on 
plot((0:20),zeros(21),'b');grid on; 
 
for i=0:N0-1 
   Fh(i+1)=Fg(N0-i); 
   Fhg(i+1)=Fg(i+1)+Fh(i+1); 
end 
 
figure(3) 
plot(wff,Fg,'r-',wff,Fh,'g.',wff,Fhg,'b-');;grid on;