私は最近、フーリエ ドメインでゼロ パディングを使用した補間法の簡単な例を matlab に実装しようとしました。しかし、私はこの作業を適切に行うことができません。フーリエ空間ではほとんど見えない小さな周波数シフトが常にありますが、時間空間では大きなエラーが発生します。
フーリエ空間でのゼロ パディングは一般的な (そして高速な) 補間方法であるように思われるため、何か足りないものがあると思います。
ここにmatlabコードがあります：
clc;
clear all;
close all;
Fe = 3250;
Te = 1/Fe;
Nech = 100;
F1 = 500;
F2 = 1000;
FMax = 1500;
time = [0:Te:(Nech-1)*Te];
timeDiscrete = [1:1:Nech];
frequency = (timeDiscrete/Nech)*Fe;
signal = cos(2*pi*F1*(time))+cos(2*pi*F2*(time))+cos(2*pi*FMax*(time));
%Compute the FFT
spectrum=zeros(1,Nech);
for k = timeDiscrete
for l = timeDiscrete
spectrum(k) = spectrum(k) + signal(l)*exp(-2*pi*j*l*k/Nech);
end
end
%Compute de inverse FFT
reconstruction=zeros(1,Nech);
for k = timeDiscrete
for l = timeDiscrete
reconstruction(k) = reconstruction(k) + spectrum(l)*exp(2*pi*j*l*k/Nech);
end
end
reconstruction=reconstruction/Nech;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%% Now interpolation will take place %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Finterp = 6*Fe;
Tinterp = 1/Finterp;
TimeInterp = [0:Tinterp:(Nech-1)*Te];
[m,n] = size(TimeInterp);
NechInterp = n;
TimeInterpDiscrete = [1:1:NechInterp];
%Compute original signal value without any interpolation
signalResampled = cos(2*pi*F1*(TimeInterp))+cos(2*pi*F2*(TimeInterp))+cos(2*pi*FMax*(TimeInterp));
%Compute original signal interpolation by padding the fft and performing
%inverse fft on the result
semipaddedsize=floor(NechInterp/2);
padded_spectrum0 = zeros(1,semipaddedsize);
padded_spectrum0 = padarray(spectrum(1:Nech/2),[0 semipaddedsize-(Nech/2)],0,'post');
padded_spectrum = zeros(1,NechInterp);
padded_spectrum(1:semipaddedsize) = padded_spectrum0;
padded_spectrum(semipaddedsize+1:NechInterp-1) = conj(fliplr(padded_spectrum0));
% padded_spectrum = padarray(spectrum,[0 NechInterp-Nech],0,'post');
padded_timeDiscrete = [1:1:NechInterp];
padded_reconstruction = zeros(1,NechInterp);
for k = padded_timeDiscrete
for l = padded_timeDiscrete
padded_reconstruction(k) = padded_reconstruction(k) + padded_spectrum(l)*exp(2*pi*j*l*k/NechInterp);
end
end
padded_reconstruction=padded_reconstruction/(1*Nech);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%% Let's print out the result %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
spectrumresampled=zeros(1,NechInterp);
for k = TimeInterpDiscrete
for l = TimeInterpDiscrete
spectrumresampled(k) = spectrumresampled(k) + signalResampled(l)*exp(-2*pi*j*l*k/NechInterp);
end
end
figure(2);
plot(abs(spectrumresampled)/6,'g');
hold on;
plot(abs(padded_spectrum),'b');
figure(3);
% Ground truth : deterministic signal is recomputed
plot(TimeInterp,signalResampled,'g');
hold on;
% linear interpolation between subsampled points (matlab tracing tool)
plot(time,(reconstruction),'c');
hold on;
% Padding the spectrum and reconstructing it
plot(TimeInterp,real(padded_reconstruction),'m');
hold on;
xlabel('Time in s','FontSize',16)
ylabel('Signal value (no unit)','FontSize',16)
title('\it{ Various signal reconstruction from fourier transform }','FontSize',16)
legend('True signal', 'Reconstruction with linear interpolation', 'Reconstruction with padded spectrum');
私の評判のために結果の画像を投稿することはできませんが、グラフはmatlabを介して簡単に生成できます。このコードまたは一般的な補間のためのゼロパディング fft に関するコメントをいただければ幸いです。
前もって感謝します
