离散时间傅里叶变换和离散傅里叶变换

$\begin{cases} X(k)=\textbf{DFT}[x(n)]=\sum\limits_{n=0}^{N-1}x(n)W_N^{nk}\quad k=0,1,...,N-1 \\ x(n)=\textbf{IDFT}[X(k)]=\frac{1}{N}\sum\limits_{n=0}^{N-1}x(n)W_N^{-nk}\quad n=0,1,...,N-1 \end{cases}$

$\begin{cases} X(e^{j\omega})=\textbf{DTFT}[x(n)]=\sum\limits_{n=-\infty}^{\infty}x(n)e^{-j\omega n}\quad k=0,1,...,N-1 \\ x(n)=\textbf{IDTFT}[X(e^{j\omega})]=\frac{1}{2\pi}\int_{-\pi}^{\pi}X(e^{j\omega})e^{j\omega n}d\omega \end{cases}$

自定义dtft()

function [w,X] = dtft(x)
    % DTFT变换
    N = length(x);
    n = 0:N-1;
    w = linspace(-2*pi,2*pi,500); %将[-2pi,2pi]分割成500份
    X = x * exp(-1j*n'*w);
end

自定义dft()

function X = dft(xn)
% 计算离散傅立叶变换（DFT）
N = length(xn);  % 获取序列长度
n = 0:N-1;  % 时间序列
X = zeros(1, N);  % 初始化频域序列
	for k = 0:N-1
	    X(k+1) = sum(xn .* exp(-1i*2*pi*k*n/N));  % 计算DFT的每个频率分量
	end
end

例子：

% DTFT
clear;
xn = [0 1 2 3 4 5 6 7];
[w,X] = dtft(xn);
N = length(xn);
n = 0:N-1;

figure('Color','w');
subplot(2 ,2 ,[1,2])
stem(xn,"filled");
subplot(223)
plot(w,abs(X));
subplot(224)
plot(w,angle(X));

% DFT

clear;
xn = [0 1 2 3 4 5 6 7 zeros(1,92)];
X = dft(xn);
N = length(xn);
n = 0:N-1;

figure('Color','w');
subplot(2 ,2 ,[1,2])
stem(xn,"filled");
subplot(223)
stem(abs(X),'filled');
subplot(224)
stem(angle(X),'filled');