Reconstructing time series from FFT in MATLAB - matlab

Question

I am trying to reconstruct the sunspots signal from the FFT, the time series and periodogram are in the following site http://www.mathworks.com/help/matlab/examples/using-fft.html . I wrote the following code but the result were not similar to original wave:
YY=Y(1:floor(n/2))
% magnitude
mag_fft = 2*abs(YY)/length(Y);
% phase angle
ang_fft = angle(YY);
[new_mag,new_i]=sort(mag_fft,'descend');
new_ang=ang_fft(new_i);
new_freq=freq(new_i)
wave=zeros(1,length(YY));
wave=new_mag(1);
t=1:length(YY)
for(i=1:70)
wave=wave+new_mag(i).*sin(2*pi*new_freq(i)*t+new_ang(i));
end
wave=wave-mag_fft(1)
figure;plot(year(t),wave,'-b')
hold on;plot(year(t),relNums(t),'-r')
any ideas?

Answer 1

%http://www.mathworks.com/help/matlab/examples/using-fft.html
% sunspots
% sunspots have period of 10 years
%%
clc;clear all;close all;
load sunspot.dat
year=sunspot(:,1);
relNums=sunspot(:,2);
figure;plot(year,relNums)
title('Sunspot Data')
plot(year(1:50),relNums(1:50),'b.-');
yfft = fft(relNums);%figure;plot(ifft(yfft)-data1d,'r')
%yfft = fft(data1d); iyfft=ifft(yfft);
[sum(relNums) yfft(1)]
yfft(1)=[]; % we grid rid of the first value as it corresponeding to zero frequency.
N=length(yfft)+1;
yfft=yfft.*2./N;
%%
power_fft = abs(yfft);power1_fft = sqrt(yfft.*conj(yfft));
figure;plot(power_fft,'-b');hold on;plot(power_fft,'rO')
ang_fft = angle(yfft);real_fft= real(yfft);imag_fft= imag(yfft);
figure;plot(real_fft);hold on;plot(imag_fft,'-r')
figure;plot(angle(yfft))
ph = (180/pi)*unwrap(ang_fft); % phase in degrees
% Now the total length of the per and all other powers should be N-1 because there is no
% more corresponding poweres and phases, and the number of frequencies before the nequiest is
Nneq=length(N./(1:N/2));
Nm1=N-1; per=N./(1:Nm1); freq=1./per;
[per'/12 power_fft(1:Nm1)/100 ] % so as to display the period in years
%% ytyt
ndat=length(relNums);
x=0:ndat-1;
sumharmony1(1:Nneq,1:ndat)=0;
sumharmony2(1:Nneq,1:ndat)=0;
for i=1:Nneq
% those two forms are equal, the last one is called the cos form.
% sumharmony1(i,:)=sumharmony1(i,:)+real_fft(i)*cos(2*pi*x/(per(i)))- imag_fft(i)*sin(2*pi*x/(per(i)));
sumharmony1(i,:)=sumharmony1(i,:)+power_fft(i)*cos(2*pi*x./(per(i))+ang_fft(i));
end
y1 =sum(relNums)/N+ sum(sumharmony1);
%y2 =sum(tmp)/N+ sum(sumharmony2);
figure;plot(relNums);hold on; plot( y1, 'r');
figure;plot((relNums-y1')) % However, the excellent results, we couldnot yet reach to the that of the built in function ifft.
figure;plot(relNums(1:100),'-ob');hold on; plot( y1(1:100), 'r');
% note that we multiply by 2 because of using the window hanning.enter code here