【 MATLAB 】DFT性质讨论（一）线性、循环反转、共轭与时序列的对称性的MATLAB实现

上篇博文通过在理论上讨论了DFT的三个性质：【 MATLAB 】DFT性质讨论（一）线性、循环反转与共轭

分别讨论：

一、线性

给出一个例子，给出x1和x2，x3 = 0.3*x1+0.8*x2;

之后我们求x3的DFT，和x1和x2的DFT的线性组合是否一致，即可验证线性性质。

clc,clear,close all;

%signal 1
n1 = 0:3;
x1 = [1,1,1,1];

%signal 2
n2 = 0:5;
x2 = ones(1,6);

% signal 3 = signal 1 +signal 2
n3 = 0:max(length(n1),length(n2))-1;
% modify x1 and x2
x1 = [ones(1,4),zeros(1,length(n3)-4)];
x2 = [ones(1,6),zeros(1,length(n3)-6)];

x3 = 0.3*x1+0.8*x2;

N = length(n3);

% DFT of x1
X1 = dft(x1,N);

% DFT of x2
X2 = dft(x2,N);

% DFT of x3
X3 = dft(x3,N);

% Linear property
X3_l = 0.3*X1 + 0.8*X2;

subplot(2,1,1);
stem(n3,X3);
title('DFT of x3');

subplot(2,1,2);
stem(n3,X3_l);
title('DFT of  0.3*x1+0.8*x2');

二、循环反转

给出例子：

设

求：

a. 求出并画出 

b. 验证循环反转性质。

题解：

---

a

clc,clear,close all;

n = 0:10;
x = 10 * (0.8).^n;
N = 11;

y = x( mod(-n,N)+1 );

subplot(2,1,1)
stem(n,x);
title('Original Sequence');
xlabel('n');ylabel('x(n)');

subplot(2,1,2)
stem(n,y);
title('Circularly folded Sequence');
xlabel('n');ylabel('x(-n mod 10)');

---

b

clc,clear,close all;

n = 0:10;
x = 10 * (0.8).^n;
N = 11;

y = x( mod(-n,N)+1 );

% subplot(2,1,1)
% stem(n,x);
% title('Original Sequence');
% xlabel('n');ylabel('x(n)');
% 
% subplot(2,1,2)
% stem(n,y);
% title('Circularly folded Sequence');
% xlabel('n');ylabel('x(-n mod 10)');
k = n;
X = dft(x,N);
Y = dft(y,N);
RealX = real(X);
ImagX = imag(X);

RealY = real(Y);
ImagY = imag(Y);

subplot(2,2,1);
stem(k,RealX);
title('DFT Real Part of x(n)');
xlabel('k');ylabel('Real{ DFT(x(n)) }');

subplot(2,2,2);
stem(k,ImagX);
title('DFT Imag Part of x(n)');
xlabel('k');ylabel('Imag{ DFT(x(n)) }');

subplot(2,2,3);
stem(k,RealY);
title('DFT Real Part of x(-n mod N)');
xlabel('k');ylabel('Real{DFT(x(-n mod N))}');

subplot(2,2,4);
stem(k,ImagY);
title('DFT Imag Part of x(-n mod N)');
xlabel('k');ylabel('Imag{DFT(x(-n mod N))}');

---

三、共轭与时序列的对称性

我们讨论时序列的对称性，采用的例子是这篇博文中的：【 MATLAB 】模拟信号采样及离散时间傅里叶变换（DTFT）案例分析

使用的模拟信号：

 在 fs = 5000 对信号进行采样，可以得到离散时间序列。

之后讨论它的循环共轭对称性。

clc
clear
close all
 
 
% Discrete-time signal
Ts = 0.0002;
n = 0:25;
x = exp(-1000*abs(n*Ts));
N = length(n);

x1 = x( mod(-n,N) + 1 );

% DFT of x
X = dft(x,N);

X1 = dft(x1,N);

X1_c = conj(X1);

subplot(2,2,1);
stem(n,x);
title('Original sequence');
xlabel('n');ylabel('x(n)');

subplot(2,2,2);
stem(n,X);
title('DFT of x(n)');
xlabel('k');ylabel('DFT{x(n)}');

subplot(2,2,3);
stem(n,x1);
title('Circularly folded sequence');
xlabel('n');ylabel('x(-n mod N)');

subplot(2,2,4)
stem(n,X1);
title('DFT of x((-n mod N))');
xlabel('k');ylabel('DFT{x(-n mod N)}');


figure
stem(n,X);
title('DFT of x(n)');
xlabel('k');ylabel('DFT{x(n)}');


figure

stem(n,X1_c);
title('Conjugation{DFT of x((-n mod N))}');
xlabel('k');ylabel('Conj{DFT{x(-n mod N)}}');