《DSP using MATLAB》Problem 3.9

利用的频移性质为：

本习题代码：

%% ------------------------------------------------------------------------
%%            Output Info about this m-file
fprintf('\n***********************************************************\n');
fprintf('        <DSP using MATLAB> Problem 3.9 \n\n');

banner();
%% ------------------------------------------------------------------------

% -----------------------------------------------------------
%                Rectangle Window sequence, and its DTFT
% -----------------------------------------------------------
%M = 5;
%M = 15;
%M = 25;
M = 100;

n1_start = 0; n1_end = M;
n1 = [n1_start : n1_end - 1]; 

x1 =  ones(1, length(n1)); 

figure('NumberTitle', 'off', 'Name', sprintf('Problem 3.9 x1(n) Rectangle, M = %d',M));
set(gcf,'Color','white');
stem(n1, x1);
xlabel('n'); ylabel('x1');
title(sprintf('x1(n) sequence, M = %d', M)); grid on;

MM = 500;
k = [-MM:MM];        % [-pi, pi]
%k = [0:M];        % [0, pi]
w = (pi/MM) * k;

[X1] = dtft(x1, n1, w);                            

magX1 = abs(X1); angX1 = angle(X1); realX1 = real(X1); imagX1 = imag(X1);

figure('NumberTitle', 'off', 'Name', sprintf('Problem 3.9 DTFT of Rm(n), M = %d', M));
set(gcf,'Color','white');
subplot(2,1,1); plot(w/pi, magX1); grid on;
title('Magnitude Part');
xlabel('frequency in \pi units'); ylabel('Magnitude');
subplot(2,1,2); plot(w/pi, angX1); grid on;
title('Angle Part');
xlabel('frequency in \pi units'); ylabel('Radians');

figure('NumberTitle', 'off', 'Name', sprintf('Problem 3.9 Real and Imag of X1(w), M = %d', M));
set(gcf,'Color','white');
subplot('2,1,1'); plot(w/pi, realX1); grid on;
title('Real Part of X1(w)');
xlabel('frequency in \pi units'); ylabel('Real');
subplot('2,1,2'); plot(w/pi, imagX1); grid on;
title('Imaginary Part of X1(w)');
xlabel('frequency in \pi units'); ylabel('Imaginary');

%% ----------------------------------------------------------------
%%                 x(n)=cos(w0*n)Rm(n), and its DTFT
%% ----------------------------------------------------------------
n2 = n1;
w0 = 0.5 * pi;

x2 = cos(w0*n2) .* x1;
%x2 = exp(j*w0*n2) .* x1;

figure('NumberTitle', 'off', 'Name', sprintf('Problem 3.9 x2(n), M = %d', M));
set(gcf,'Color','white');
stem(n2, x2);
xlabel('n2'); ylabel('x2');
title(sprintf('x1(n)*Rm(n) sequence, M = %d', M)); grid on;

MM = 500;
k = [-MM:MM];        % [-pi, pi]
%k = [0:M];        % [0, pi]
w = (pi/MM) * k;

[X2] = dtft(x2, n2, w);                            

magX2 = abs(X2); angX2 = angle(X2); realX2 = real(X2); imagX2 = imag(X2);

figure('NumberTitle', 'off', 'Name', sprintf('Problem 3.9 DTFT of x2(n), M = %d', M));
set(gcf,'Color','white');
subplot(2,1,1); plot(w/pi, magX2); grid on;
title('Magnitude Part');
xlabel('frequency in \pi units'); ylabel('Magnitude');
subplot(2,1,2); plot(w/pi, angX2); grid on;
title('Angle Part');
xlabel('frequency in \pi units'); ylabel('Radians');

figure('NumberTitle', 'off', 'Name', sprintf('Problem 3.9 Real and Imag of X2(w), M = %d', M));
set(gcf,'Color','white');
subplot(2,1,1); plot(w/pi, realX2); grid on;
title('Real Part of X2(w)');
xlabel('frequency in \pi units'); ylabel('Real');
subplot(2,1,2); plot(w/pi, imagX2); grid on;
title('Imaginary Part of X2(w)');
xlabel('frequency in \pi units'); ylabel('Imaginary');

%% --------------------------------------------------------------
%%              Direct equation
%% --------------------------------------------------------------
Real_X_direct = 0.5 * cos( (w/pi-w0) * (M-1) / 2) * ( sin( (w/pi-w0)*M/2 ) / sin( (w/pi-w0)/2 ) ) + 0.5 * cos( (w/pi+w0) * (M-1) / 2) * ( sin( (w/pi-(2*pi-w0))*M/2 ) / sin( (w/pi-(2*pi-w0))/2) );

check = sum(abs(realX2)-abs(Real_X_direct))

figure('NumberTitle', 'off', 'Name', 'Problem 3.9 Direct')
set(gcf,'Color',[1,1,1])                  % 改变坐标外围背景颜色
plot(w/pi, Real_X_direct); title('Real Part obtained by direct equation');
xlabel('n'); ylabel('Real[x(n)]') ;
grid on;

　　运行结果：

1、方波窗序列，本题中正弦序列，以及各自DTFT；

2、谱的实部和虚部；

因为ω0=0.5π，根据频移性质，相当于沿着ω轴谱搬移了0.5π（注意到DTFT是以2π为周期的，图中显示的是[-π，π]）。