一、简介
FFT(Fast Fourier transform):快速傅里叶变换,是DFT的工程化实现方法。
DFT直接求解太过于复杂,FFT方法根据DFT求解过程中旋转因子的性质并引入分治算法思想,大大简化计算过程,被广泛应用在频谱分析的工程实践中,如matlab,C,C++,CUDA等底层实现
1 DFT简介
频谱分析是信号处理中的重要环节,从傅里叶变换FT,到拉普拉斯变换LT,离散时间傅里叶变换DTFT,Z变换ZT,到我们所讲的离散傅里叶变换DFT(他们之间的联系和区别见我的其他博客)。
相比于其他变换,DFT被广泛应用的原因是其输入的时域信号是离散的,输出的频域结果也是离散的。这就极大方便了我们进行基于计算机的频谱计算,存储和分析,没办法数字信号处理是大趋势。
DFT变换的公式为:
但为了分析方便,在FFT的计算过程中,我们依然使用k = 0 ∼ N − 1 k = 0 \sim N-1k=0∼N−1的选取策略。也即,如下:
2 旋转因子WWW的性质
3 FFT蝶形计算证明
4 FFT计算过程
二、源代码
function varargout = fushidctbianhuan(varargin)
% FUSHIDCTBIANHUAN M-file for fushidctbianhuan.fig
% FUSHIDCTBIANHUAN, by itself, creates a new FUSHIDCTBIANHUAN or raises the existing
% singleton*.
%
% H = FUSHIDCTBIANHUAN returns the handle to a new FUSHIDCTBIANHUAN or the handle to
% the existing singleton*.
%
% FUSHIDCTBIANHUAN('CALLBACK',hObject,eventData,handles,...) calls the local
% function named CALLBACK in FUSHIDCTBIANHUAN.M with the given input arguments.
%
% FUSHIDCTBIANHUAN('Property','Value',...) creates a new FUSHIDCTBIANHUAN or raises the
% existing singleton*. Starting from the left, property value pairs are
% applied to the GUI before fushidctbianhuan_OpeningFunction gets called. An
% unrecognized property name or invalid value makes property application
% stop. All inputs are passed to fushidctbianhuan_OpeningFcn via varargin.
%
% *See GUI Options on GUIDE's Tools menu. Choose "GUI allows only one
% instance to run (singleton)".
%
% See also: GUIDE, GUIDATA, GUIHANDLES
% Edit the above text to modify the response to help fushidctbianhuan
% Last Modified by GUIDE v2.5 06-Apr-2014 17:16:14
% Begin initialization code - DO NOT EDIT
gui_Singleton = 1;
gui_State = struct('gui_Name', mfilename, ...
'gui_Singleton', gui_Singleton, ...
'gui_OpeningFcn', @fushidctbianhuan_OpeningFcn, ...
'gui_OutputFcn', @fushidctbianhuan_OutputFcn, ...
'gui_LayoutFcn', [] , ...
'gui_Callback', []);
if nargin && ischar(varargin{1})
gui_State.gui_Callback = str2func(varargin{1});
end
if nargout
[varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:});
else
gui_mainfcn(gui_State, varargin{:});
end
% End initialization code - DO NOT EDIT
% --- Executes just before fushidctbianhuan is made visible.
function fushidctbianhuan_OpeningFcn(hObject, eventdata, handles, varargin)
% This function has no output args, see OutputFcn.
% hObject handle to figure
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% varargin command line arguments to fushidctbianhuan (see VARARGIN)
axes(handles.axes1);
img=imread('lena.bmp');
imshow(img);
axes(handles.axes2);
imgf=double(img);
imgs=fftshift(fft2(imgf));
imshow(log(1+abs(imgs)),[]);
set(handles.dct_radiobutton,'value',0);
set(handles.fft_radiobutton,'value',1);
% Choose default command line output for fushidctbianhuan
handles.output = hObject;
% Update handles structure
guidata(hObject, handles);
% UIWAIT makes fushidctbianhuan wait for user response (see UIRESUME)
% uiwait(handles.figure1);
% --- Outputs from this function are returned to the command line.
function varargout = fushidctbianhuan_OutputFcn(hObject, eventdata, handles)
% varargout cell array for returning output args (see VARARGOUT);
% hObject handle to figure
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
% Get default command line output from handles structure
varargout{1} = handles.output;
% --- Executes on button press in fft_radiobutton.
function fft_radiobutton_Callback(hObject, eventdata, handles)
% hObject handle to fft_radiobutton (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
set(handles.dct_radiobutton,'value',0);
set(handles.fft_radiobutton,'value',1);
img = getimage(handles.axes1);
axes(handles.axes2);
imgf=double(img);
imgs=fftshift(fft2(imgf));
imshow(log(1+abs(imgs)),[]);
set(handles.text2,'string','傅立叶变换谱');
set(handles.text2,'fontsize',12);
set(handles.text2,'fontname','宋体');
set(handles.text2,'fontweight','bold');
% Hint: get(hObject,'Value') returns toggle state of fft_radiobutton
% --- Executes on button press in dct_radiobutton.
function dct_radiobutton_Callback(hObject, eventdata, handles)
% hObject handle to dct_radiobutton (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
set(handles.dct_radiobutton,'value',1);
set(handles.fft_radiobutton,'value',0);
img = getimage(handles.axes1);
axes(handles.axes2);
imgd=img;
imgs=dct2(imgd);
imshow(log(abs(imgs)));
set(handles.text2,'string','DCT变换系数');
set(handles.text2,'fontsize',12);
set(handles.text2,'fontname','宋体');
set(handles.text2,'fontweight','bold');
% Hint: get(hObject,'Value') returns toggle state of dct_radiobutton
% --- Executes on button press in close_button.
function close_button_Callback(hObject, eventdata, handles)
% hObject handle to close_button (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
close(fushidctbianhuan);
% --- Executes on selection change in fft_dct_pop.
function fft_dct_pop_Callback(hObject, eventdata, handles)
% hObject handle to fft_dct_pop (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
% handles structure with handles and user data (see GUIDATA)
val=get(hObject,'value');
str=get(hObject,'string');
switch str{val};
case 'lena'
lena = [];
lena = imread('lena.bmp');
img = lena;
case 'Saturn',
saturn = [];
load imdemos saturn
img = saturn;
case 'Tire',
tire = [];
load imdemos tire
img = tire;
case 'Pout',
pout = [];
load imdemos pout
img = pout;
case 'Trees',
trees = [];
load imdemos trees
img = trees;
case 'Quarter',
quarter = [];
load imdemos quarter
img = quarter;
case 'Circuit',
circuit = [];
load imdemos circuit
img = circuit;
case 'Rice',
rice = [];
load imdemos rice
img = rice;
case 'Fingerprint'
fingerprint = [];
fingerprint = imread('fingerprint.jpg');
img = fingerprint;
case 'Licenceplate'
licenceplate = [];
licenceplate = imread('licenceplate.jpg');
img = licenceplate;
case 'Haze'
haze = [];
haze = imread('haze.jpg');
img = haze;
case 'Cloudy'
cloudy = [];
cloudy = imread('cloudy.tif');
img = cloudy;
end
axes(handles.axes1);
imshow(img);
control=get(handles.fft_radiobutton,'value');
switch control;
- 1.
- 2.
- 3.
- 4.
- 5.
- 6.
- 7.
- 8.
- 9.
- 10.
- 11.
- 12.
- 13.
- 14.
- 15.
- 16.
- 17.
- 18.
- 19.
- 20.
- 21.
- 22.
- 23.
- 24.
- 25.
- 26.
- 27.
- 28.
- 29.
- 30.
- 31.
- 32.
- 33.
- 34.
- 35.
- 36.
- 37.
- 38.
- 39.
- 40.
- 41.
- 42.
- 43.
- 44.
- 45.
- 46.
- 47.
- 48.
- 49.
- 50.
- 51.
- 52.
- 53.
- 54.
- 55.
- 56.
- 57.
- 58.
- 59.
- 60.
- 61.
- 62.
- 63.
- 64.
- 65.
- 66.
- 67.
- 68.
- 69.
- 70.
- 71.
- 72.
- 73.
- 74.
- 75.
- 76.
- 77.
- 78.
- 79.
- 80.
- 81.
- 82.
- 83.
- 84.
- 85.
- 86.
- 87.
- 88.
- 89.
- 90.
- 91.
- 92.
- 93.
- 94.
- 95.
- 96.
- 97.
- 98.
- 99.
- 100.
- 101.
- 102.
- 103.
- 104.
- 105.
- 106.
- 107.
- 108.
- 109.
- 110.
- 111.
- 112.
- 113.
- 114.
- 115.
- 116.
- 117.
- 118.
- 119.
- 120.
- 121.
- 122.
- 123.
- 124.
- 125.
- 126.
- 127.
- 128.
- 129.
- 130.
- 131.
- 132.
- 133.
- 134.
- 135.
- 136.
- 137.
- 138.
- 139.
- 140.
- 141.
- 142.
- 143.
- 144.
- 145.
- 146.
- 147.
- 148.
- 149.
- 150.
- 151.
- 152.
- 153.
- 154.
- 155.
- 156.
- 157.
- 158.
- 159.
- 160.
- 161.
- 162.
- 163.
- 164.
- 165.
- 166.
- 167.
- 168.
- 169.
- 170.
- 171.
- 172.
- 173.
- 174.
- 175.
- 176.
- 177.
- 178.
- 179.
- 180.
- 181.
- 182.
- 183.
- 184.
- 185.
- 186.
- 187.
- 188.
- 189.
- 190.