EDFT扩展离散傅里叶变换算法附matlab代码

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⛄ 内容介绍

快速傅里叶变换(Fast Fourier Tranformation,FFT)是将一个大点数N的DFT分解为若干小点的DFT的组合,将用运算工作量明显降低,从而大大提高了离散傅里叶变换(DFT)的计算速度.因各个科学技术领域广泛的使用了FFT技术,它大大推动了信号处理技术的进步,现已成为数字信号处理强有力的工具.本论文将比较全面地叙述各种快速傅里叶变换算法原理,特点,并完成了基于MATLAB的实现.

⛄ 部分代码

% Demoedft.m demonstrates performance of function EDFT by iterations

% and provide comparison with Nonuniform DFT (Matlab NUFFT) output.

% EDFT call line is [F,S,Stopit]=edft(X,N,1,W,tk),

% where DEMO data X are calculated as sum of the following components:

% - real sinusoid at normalized frequency 0.137, magnitude 2,

% - complex exponent at frequency 0.137+1/(4*K), magnitude 1, where K=64

% is the length of sequence X,

% - mean value (frequency 0), magnitude 1,

% - pulse in frequency ranges [-0.137 0],

% - noise generated by rand function (SNR~35 dB).

% Number of frequencies N=512 (the length of DFT).

% Number of samples in sequence with missing data KK=32.

% So, the true spectrum of DEMO sequence (without noise part) is ploted

% in red color and looks like greeting 'HI', where 'I' actually are two

% close peaks at normalized frequencies 0.137 and 0.137+1/(4*K). 

% In the first, demoedft.m plot real parts of the input sequences:

% - Subplot(511) display DEMO data X1 consisting of 64 uniform samples;

% - Subplot(512) outputs an example of missing data X2, where 32 randomly 

%   taken samples in X1 replaced by NaN;

% - Subplot(513) shows gapped data X3, where two gaps by 16 NaN each 

%   inserted in X1. 

% - Subplot(514) outputs nonuniformly sampled DEMO data X4, 

%   tkn2=[1:K]+0.5*(rand(1,K)-0.5);

% - Subplot(515) showing sparse (32 samples) nonuniformly sampled data X5, 

%   tkn3=tkn2(1:2:K); 

% NaN are not displayed in sublopts 512 and 513. 

% To proceed you should select one of these sequences displayed in figure. 

% In the next, fifteen (I=15) NEDFT iterations are performed. You will be 

% asked to strike any key after 1,2,3,5 and 10 iteration. Plots in blue 

% colour are equal to NUFFT and can be used for comparison with EDFT 

% output (green). To resolve two close sinusoids (at 1/4 of Nyquist), EDFT

% should increase resolution at least 4 times in comparison with NUFFT.

%

% EDFT and NUFFT outputs are displayed in four subplots:

% -Subplot(221) show Power Spectral Density as 10*log10(abs(F).^2/N)) in 

%   normalized frequencies range [-0.5 0.5[.

% -Subplot(222) display Power Spectrum as 10*log10(abs(S).^2). The plot 

%   indicate power of sinusoids in X.

% -Subplot(413) plot division F./S/K titled as Relative frequency resolution

%   and expose how the frequency resolution of EDFT algorithm is changed 

%   along the frequency axes in respect to NUFFT. The first iteration 

%   showing that NUFFT and EDFT have almost equal resolution on all 

%   frequencies in range [-0.5 0.5[. During the next EDFT iterations we

%   can discover that the relationship F./S/K for EDFT analysis is not that

%   simple. Although sum(F./S/K)=N and remains constant for all iterations, 

%   EDFT have ability to increase resolution around the powerful narrowband

%   components (sinusoids) and decrease resolution at frequency range where 

%   data X have weak power components. EDFT is called as high resolution 

%   method and that's true, but with the following remark- EDFT keeping 

%   the same 'summary' resolution as NUFFT has or, in other words, squares

%   under curves for NUFFT (blue) and EDFT (green) are equal. The maximum

%   frequency resolution is limited by value of division N/K. For example, 

%   if K=64 and N=512, then EDFT can potentially improve frequency

%   resolution 512/64=8 times.

% -Subplot(414) plot real parts of reconstructed sequences obtained in 

%   result of Inverse Fast Fourier Transform to outputs of NUFFT and EDFT, 

%   correspondingly. It is well known that reconstruction of data by 

%   applying ifft(fft(X,N)) for N>length(X), return sequence where initial

%   data X are padded with zeros to length N. That's true also for NUFFT 

%   and for the first iteration of EDFT. The next iterations are showing

%   an ability of EDFT to obtain reconstructed data consisting of input 

%   sequence X values plus non-zero extrapolation of X to length N. The 

%   sequence X is extrapoated in both directions - backward and forward. 

%   NaNs in sequences [X2] and [X3] are interpolated and replaced with

%   reconstructed values, while for nonuniform sequences [X4] and [X5], 

%   the reconstructed data are re-sampled at uniform time points tn=1:N.

%

% References: 

% 1) Vilnis Liepins, A spectral estimation method of nonuniformly sampled

% band-limited signals.Automatic Control and Computer Sciences,Vol.28,

% No.2, pp.66-77, 1994.

% 2) Vilnis Liepins, An algorithm for evaluation a discrete Fourier

% transform for incomplete data, Automatic control and computer sciences,

% Vol.30, No.3, pp.27-40, 1996.

clear

  disp('EDFT DEMO program started...');

% Set the length of input data (K), DFT (N), number of iterations (I),

% relative central frequency (fc) in range ]0,0.5] and SNR:

    K=64;

N=512;

I=15;

fc=0.137;

    SNR=35;

% Uniformly and nonuniformly spaced time and frequency vectors

tk=0:K-1; % uniform time vector (K samples, sampling period T=1)

tn=0:N-1; % uniform time vector (N samples, sampling period T=1)

tkn2=(1:K)+0.5*(rand(1,K)-0.5); % nonuniform (mean sampling period T=1)

tkn3=tkn2(1:2:K);  % sparse nonuniform (mean sampling period T=2)

fn=(-ceil((N-1)/2):floor((N-1)/2))/N; % normalized frequencies [0.5,0.5[

ti=tk-K/2-0.5; % uniform centerred time vector for pulse

tin2=tkn2-K/2-0.5; % nonuniform centerred time for pulse

% Generate X1 - uniform sequence (K samples)

Xph=2*pi*rand(1,2);

XC1=2*sin(2*pi*fc*tk+Xph(1,1)); % sinusoid

XC2=exp(1i*(2*pi*(fc+1/4/K)*tk+Xph(1,2))); % complex exponent

XC3=sin(pi*fc*ti)./(ti+eps).*exp(-1i*pi*fc*ti); % rectangular pulse

XC4=ones(1,K);       % mean value

X1=XC1+XC2+XC3+XC4; % signal w/o noise

sigma=sqrt((2+1+1)/10^(SNR/10)); % sigma for given SNR

XC5=sigma*(randn(1,K)+1i*randn(1,K))/sqrt(2);% complex noise

SNR1=round(10*log10((X1*X1')/(XC5*XC5')));% calculated SNR

X1=X1+XC5; % X1 signal+noise (EDFT and NUFFT input)

% Generate X2 - nonuniform sequence with NaN: (K/2 samples, K/2 NaN)

MNaN=floor(K/2); % MNaN- number of NaN in X

KK=K-MNaN;     % KK- length of X2,X3 without NaN

X2=X1;XX2=[];t2=[];

KNaN=(1:2:MNaN*2)+round(rand(1,MNaN)); % KNaN indicate NaN in sequence

X2(KNaN)=NaN*ones(1,MNaN);  % sequence with NaN (EDFT input)

    XX2(1:KK)=X1(~isnan(X2)); % sequence (NUFFT input)

    t2(1:KK)=tk(~isnan(X2)); % time vector (NUFFT input)

% Generate X3 - gapped sequence (K/2 samples, 2 x K/4 NaN)

X3=X1;XX3=[];t3=[];k4=floor(K/4);

ki=round([rand*k4+(1:k4) MNaN+rand*(MNaN-k4)+(1:(MNaN-k4))]);

X3(ki)=NaN*ones(1,MNaN); % gapped data with NaN (EDFT input)

    XX3(1:KK)=X1(~isnan(X3)); % sequence (NUFFT input)

    t3(1:KK)=tk(~isnan(X3)); % time vector (NUFFT input)

% Generate X4 - nonuniform sequence (K samples)

XN1=2*sin(2*pi*fc*tkn2+Xph(1,1)); % sinusoid

XN2=exp(1i*(2*pi*(fc+1/4/K)*tkn2+Xph(1,2)));% complex exponent

XN3=ones(1,K); % mean value

XN4=sin(pi*fc*tin2)./(tin2+eps).*exp(-1i*pi*fc*tin2); % rectangular pulse

X4=XN1+XN2+XN3+XN4;     % signal w/o noise

SNR2=10*log10((X4*X4')/(XC5*XC5'));% calculated SNR

X4=X4+XC5;     % signal+noise

% Generate X5 - sparse nonuniform sequence (half of X4 samples)

X5=X4(1:2:K);K5=length(X5);

% Plot real part of sequences X1, X2, X3, X4 and X5.

figure(1)

clf

Xmin=min(real([X1 X4]))-1;

Xmax=max(real([X1 X4]))+1;

subplot(511)

stem(tk+1,real(X1))

axis([0 K+1 Xmin Xmax])

xlabel('Sample number')

ylabel('real(X1)')

title(['Input sequence X1 - uniform data: ',int2str(K),' samples, SNR=' ...

    ,int2str(round(SNR1))])

subplot(512)

stem(tk+1,real(X2))

axis([0 K+1 Xmin Xmax])

xlabel('Sample number')

ylabel('real(X2)')

title(['Input sequence X2 - missing data: ',int2str(KK),' samples and ' ...

    ,int2str(MNaN),' NaN'])

subplot(513)

stem(tk+1,real(X3))

axis([0 K+1 Xmin Xmax])

xlabel('Sample number')

ylabel('real(X3)')

title(['Input sequence X3 - gapped data: ',int2str(KK),' samples and ' ...

    ,int2str(k4),'+',int2str(MNaN-k4),'  NaN'])

subplot(514)

stem(tkn2,real(X4))

axis([0 K+1 Xmin Xmax])

         xlabel('Sample number')

ylabel('real(X4)')

title(['Input sequence X4 - nonuniform data: ',int2str(K), ...

        ' samples, SNR=',int2str(round(SNR2))])

subplot(515)

stem(tkn3,real(X5))

axis([0 K+1 Xmin Xmax])

         xlabel('Sample number')

ylabel('real(X5)')

title(['Input sequence X5 - sparse nonuniform data: ' ...

        ,int2str(K/2),' samples'])

% Selecting of sequence for NEDFT input and calculating DFT output

disp('DEMO data are calculated as sum of the following components:');

disp(['   - real sinusoid at normalized frequency ',num2str(fc), ...

        ', magnitude 2;']);

disp(['   - complex exponent at frequency ',num2str(fc),'+1/(4*' ...

        ,int2str(K),'), magnitude 1;']);

disp('   - mean value (frequency 0), magnitude 1;');

disp(['   - rectangular pulse in frequency ranges [-',num2str(fc),' 0];']);

disp(['   - noise generated by randn function (SNR~',int2str(SNR),' dB).']);

DEMO=1;

 while DEMO==1||DEMO==2||DEMO==3||DEMO==4||DEMO==5

disp(' ');

disp('Input [1] to select uniform sequence X1');

disp('Input [2] to select missing sample sequence X2');

disp('Input [3] to select gapped sequence X3');

disp('Input [4] to select nonuniformly sampled sequence X4');

disp('Input [5] to select sparse nonuniformly sampled sequence X5');

DEMO=input('Input sequence # or any other valid key to end of DEMO:');

    if DEMO==1

X=X1;XX=X1;tkk=tk;KP=K;h=2; % X1 - uniform data

    elseif DEMO==2

X=X2;XX=XX2;tkk=t2;KP=KK;h=3;   % X2 - missing data

    elseif DEMO==3

X=X3;XX=XX3;tkk=t3;KP=KK;h=4; % X3 - gapped data

    elseif DEMO==4

X=X4;XX=X4;tkk=tkn2;KP=K;h=5; % X4 - nonuniform data (K samples)

    elseif DEMO==5

X=X5;XX=X5;tkk=tkn3;KP=K5;h=6; % X5 - sparse nonuniform data

    else

disp('...end of EDFT DEMO.') % End DEMO if other valid key entered

    return

    end

   

% Calculate outputs for NUFFT used in 4 plottings.

dft_out=nufft(XX,tkk,fn);

sub1_PWD=10*log10(abs(dft_out).^2/N);

sub2_PS=20*log10(abs(dft_out/KP));

sub3_FR=ones(1,N)*KP/K;

sub4_RD=real(ifft(ifftshift(dft_out)));

% Set values for input argument W used for the first EDFT iteration.

W=ones(1,N);

% Get EDFT outputs F and S for iteration it.

  for it=1:I

if DEMO==4||DEMO==5

        [F,S,Stopit]=edft(X,N,1,W,tkk); %Basic algorithm

else

[F,S,Stopit]=edft(X,N,1,W); %Faster algorithm

F=fftshift(F);     %Adjust size 

S=fftshift(S);     %of output

end

% Calculate outputs for EDFT iteration it used in 4 plottings.

sub1=10*log10(abs(F).^2/N);

sub2=20*log10(abs(S));

sub3=real(F./S/K);

sub4=real(ifft(ifftshift(F)));

frsig=[-fc 0 fc fc+1/4/K];

% Plots Power Spectral Densities in subplot221.

figure(h)

clf

subplot(221)

plot([frsig;frsig],[10*log10(N)*[1 1 1 1];-100*[1 1 1 1]],'r:');

hold on

plot([-fc;0],[10*log10(pi^2/N);10*log10(pi^2/N)],'r:');

plot(fn,sub1,'g-',fn,sub1_PWD,'b-')

hold off

axis([-0.5 0.5 -100 50])

xlabel('Normalized frequency')

ylabel('10*log[abs(F).^2/length(F)]')

title(['Power Spectral Density (iter.',int2str(it),')'])

% Plots Power Spectrums in subplot222.

subplot(222)

plot([frsig;frsig],[[0 0 0 0];-100*[1 1 1 1]],'r:');

hold on

plot([-fc;0],[20*log10(pi/K);20*log10(pi/K)],'r:');

         plot(fn,sub2,'g-',fn,sub2_PS,'b-')

         hold off

axis([-0.5 0.5 -100 50])

xlabel('Normalized frequency')

ylabel('10*log[abs(S).^2]')

title(['Power Spectrum (iter.',int2str(it),')'])

% Plots Frequency Resolutions in subplot413.

subplot(413)

plot([frsig;frsig],[[0 0 0 0];N/K*[1 1 1 1]],'r:');

         hold on

plot(fn,sub3,'g-',fn,sub3_FR,'b-')

         plot([-fc;0],[1;1],'r:');

hold off

axis([-0.5 0.5 0 N/K])

xlabel('Normalized frequency')

     ylabel(['(F./S)/length(X',int2str(DEMO),')'])

title(['Relative frequency resolution (iter.',int2str(it),')'])

% Plots orriginal (red) and Reconstructed Data in subplot414.

subplot(414)

plot(tk,real(X1),'r-',tn,sub4,'g-',tn,sub4_RD,'b-')

axis([1 N Xmin Xmax])

xlabel('Sample number')

ylabel('real(ifft(F))')

     title(['Sequence X',int2str(DEMO),': True (red) and reconstructed' ...

        ' by EDFT (green, iter.',int2str(it),') and NUFFT (blue)'])

% Waiting for keyboard action

    drawnow

    if it==1

disp(' ');

    disp(['Plots True Spectrum [red color], EDFT [green colour] and' ...

        ' NUFFT [blue colour].']);

    end

    if it==1||it==2||it==3||it==5||it==10

    disp(['Calculate [F,S]=edft(X',int2str(DEMO),',',int2str(N),',' ...

        '',int2str(it),',...) and ifft(F). Strike any key to continue.']);    

    pause

    end

% Set EDFT input argument W for next iteration.

if DEMO==4||DEMO==5

     W=S;

else

     W=ifftshift(S); %Adjust size of output

end

  end

    disp(['Calculate [F,S]=edft(X',int2str(DEMO),',' ...

        '',int2str(N),',',int2str(it),',...) and ifft(F).']);

  end

⛄ 运行结果

​

⛄ 参考文献

[1]谭子尤, 张雅彬. 离散傅里叶变换快速算法的研究与MATLAB算法实现[J]. 中国科技信息, 2006(22):3.

[2]马维祯. 多维离散傅里叶变换DFT(2n;k)算法[C]// 中国第十届电路与系统学术年会. 1992.

⛳️ 完整代码

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