逆快速傅里叶变换matlab,[转载]Matlab 实现快速傅里叶变换（FFT）
从Matlab的帮助文档中，可以提取出这样一个可以用于实际的快速fourier变换程序。
function [Yf,f] = Spectrum_Calc(yt,Fs)
%功能：实现快速fourier变换
%输入参数：yt为时域信号序列，Fs为采样频率
%返回值：Yf为经过fft的频域序列，f为相应频率
L = length(yt);
NFFT = 2^nextpow2(L);
Yf = fft(yt,NFFT)/L;
Yf = 2*abs(Yf(1:NFFT/2+1));
f = Fs/2 * linspace(0,1,NFFT/2+1);
end
调用方法：
Fs =
1000; % Sampling frequency
T =
1/Fs; % Sample time
L =
1000; % Length of signal
t =
(0:L-1)*T; % Time vector
% Sum of a 50 Hz sinusoid and a 120 Hz sinusoid
x = 0.7*sin(2*pi*50*t) + sin(2*pi*120*t);
y = x +
2*randn(size(t)); % Sinusoids plus noise
[Yf,f] = Spectrum_Calc(y,Fs);
plot(f,Yf);
备注：
(1)关于Fs/2的说明：这个其实是采样定理的规定，即当采样频率fs.max大于信号中最高频率fmax的2倍时(fs.max>=2fmax)，采样之后的数字信号完整地保留了原始信号中的信息，所以，这里f的最大值为Fs/2。如果大于此值，比如取Fs，那么频域上的图像会关于Fs/2对称，而此时靠右侧的尖峰频率实际上是不存在的。
(2)这个Spectrum_Calc是用矩形窗截断的数据，在某些情况下可以加Hanning窗或其他窗，方法很简单
function [Yf,f] = Spectrum_Calc(yt,Fs)
%功能：实现快速fourier变换
%输入参数：yt为时域信号序列，Fs为采样频率
%返回值：Yf为经过fft的频域序列，f为相应频率
L = length(yt);
% 加Hanning窗
w = hann(L);
yt = yt(:).*w(:);
NFFT = 2^nextpow2(L);
Yf = fft(yt,NFFT)/L;
Yf = 2*abs(Yf(1:NFFT/2+1));
f = Fs/2 * linspace(0,1,NFFT/2+1);
end
Matlab帮助文档关于fft的说明
fft
Fast Fourier transform
Syntax
Y = fft(x)
Y = fft(X,n)
Y = fft(X,[],dim)
Y = fft(X,n,dim)
Definitions
The functions Y = fft(x) and y = ifft(X) implement the transform
and inverse transform pair given for vectors of length N by:
where
is an Nth root of unity.
Description
Y = fft(x) returns the discrete Fourier transform (DFT) of
vector x, computed with a fast Fourier transform (FFT)
algorithm.
If the input X is a matrix, Y = fft(X) returns the Fourier
transform of each column of the matrix.
If the input X is a multidimensional array, fft operates on the
first nonsingleton dimension.
Y = fft(X,n) returns the n-point DFT. fft(X) is equivalent to
fft(X, n) where n is the size of X in the first nonsingleton
dimension. If the length of X is less than n, X is padded with
trailing zeros to length n. If the length of X is greater than n,
the sequence X is truncated. When X is a matrix, the length of the
columns are adjusted in the same manner.
Y = fft(X,[],dim) and Y = fft(X,n,dim) applies the FFT operation
across the dimension dim.
Examples
A common use of Fourier transforms is to find the frequency
components of a signal buried in a noisy time domain signal.
Consider data sampled at 1000 Hz. Form a signal containing a 50 Hz
sinusoid of amplitude 0.7 and 120 Hz sinusoid of amplitude 1 and
corrupt it with some zero-mean random noise:
Fs =
1000; % Sampling frequency
T =
1/Fs; % Sample time
L =
1000; % Length of signal
t =
(0:L-1)*T; % Time vector
% Sum of a 50 Hz sinusoid and a 120 Hz sinusoid
x = 0.7*sin(2*pi*50*t) + sin(2*pi*120*t);
y = x +
2*randn(size(t)); % Sinusoids plus noise
plot(Fs*t(1:50),y(1:50))
title('Signal Corrupted with Zero-Mean Random Noise')
xlabel('time (milliseconds)')
It is difficult to identify the frequency components by looking
at the original signal. Converting to the frequency domain, the
discrete Fourier transform of the noisy signal y is found by taking
the fast Fourier transform (FFT):
NFFT = 2^nextpow2(L); % Next power of 2 from length of y
Y = fft(y,NFFT)/L;
f = Fs/2*linspace(0,1,NFFT/2+1);
% Plot single-sided amplitude spectrum.
plot(f,2*abs(Y(1:NFFT/2+1)))
title('Single-Sided Amplitude Spectrum of y(t)')
xlabel('Frequency (Hz)')
ylabel('|Y(f)|')
The main reason the amplitudes are not exactly at 0.7 and 1 is
because of the noise. Several executions of this code (including
recomputation of y) will produce different approximations to 0.7
and 1. The other reason is that you have a finite length signal.
Increasing L from 1000 to 10000 in the example above will produce
much better approximations on average.