FFT interpolation using zero-padding and the Chirp Z-Transform for a single tone sinusoid

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Why would FFT interpolation by zero-padding or using the Chirp Z-Transform produce a maximum at a bin that corresponds to a frequency less than the input frequency of a single tone sinusoid?
I am attempting to precisely determine the frequency of a single tone sinusoid from a data set which captures just 1-2 oscilations of the wave. To do this I have tried to use zero-padding interpolation and the Chirp Z-Transform. When I perform my interpolation the maximum in the FFT or CZT falls into a bin that does not correspond to the frequency of the input sinusoid and instead always undershoots. Further, a convolution between the input sinusoid and the complex exponential of the same frequency is smaller in magnitude than the convolution between the input and the complex exponential of the frequency coresponding to the maximum FFT bin. This obviously goes against the properties of sinusoids.
I have tried to diagnose this issue by windowing, using different interpolation factors, using different length and frequency input signals, and using other fft codes, to no success.
Below is the code that performs the interpolation of the fft of an input sinusoid, plots the fft and czt, plots the input sinusoid along with the sinusoid corresponding to the max bin, and finally performs the convolution between the input and the sinusoids of the FFT max bin and the input frequency.
%create input sinusoid
M = 100;
m = linspace(-pi,pi,M);
f = 2;
x = sin(f*m);
%set interpolation factor and run zeropadded fft
N = M*100;
FFT = fft(x,N);
%define czt parameters
r1 = 0;
r2 = .1;
a = exp(2j*pi*(r1));
w = exp(-2j*pi*(r2-r1)/N);
CZT = czt(x,N,w,a);
%Plot interpolated FFT and CZT
figure, plot(abs(FFT))
hold on
plot(abs((CZT)))
legend('fft','czt')
% Find index of maximum bin for FFT and CZT
[~,indexFFT] = max(abs(FFT(1:N/2)));
[~,indexCZT] = max(abs(CZT(1:N/2)));
% Create sinusoids from the extracted bins to compare against input
% sinusoid
a = exp(-2j*pi*(indexFFT-1)*(linspace(1,M,N))/(N));
b = real(a);
a2 = exp(-2j*pi*(indexCZT-1)*(linspace(1,M,N))/(N/(r2-r1)));
b2 = real(a2);
a3 = exp(-2j*pi*(f)*(linspace(1,M,M))/(M));
b3 = real(a3);
figure,plot(linspace(1,M,N),b,'-.')
hold on
plot(linspace(1,M,N),b2,'--')
plot(linspace(1,M,M),b3,':k')
hold off
legend('FFT','CZT','input sinusoid')
title('the sinusoid associated with the maximum FFT bin')
% Perform convolution between zeropadded input and the sinusoid from the
% fft max bin, and compare against the convolution with the true sinusoid with true frequency
x_Padded = zeros(1,N);
x_Padded(1:M) = x;
X = 0;X2 = 0;
for n=1:N
X = X + x_Padded(n)*exp(-2j*pi*(f)*(n-1)/(M));
X2 = X2 + x_Padded(n)*exp(-2j*pi*(indexFFT-1)*(n-1)/(N));
end
X2_mag = abs(X2);
X_mag = abs(X);

采纳的回答

David Goodmanson
David Goodmanson 2019-8-11
编辑:David Goodmanson 2019-8-12
Hi Matt,
I made one minor change at the beginning of your code, replacing the first four lines with
M = 100;
m = linspace(0,1,M)-1/2;
f = 2;
x = sin(2*pi*f*m);
This does the same thing, but putting the 2 pi into the sine argument is more consistent with common practice and with the rest of the code. And it's useful later.
Since you have zerofilled by a factor of 100, frequency f = 2 appears at point 200 in the FFT array. Plus one more point since Matlab is one-based and f=0 is at point 1. So, point 201 is the goal.
The problem is that the oscillations are for sine, but you are looking at the abs amplitudes of the positive frequency components and picking the largest one of those. Writing the continuous-m integral expression for simplicity [ and using f0 in place of f ] , the fourier amplitude for frequency fn is
an = Integral (1/(2*pi)) * ...
[ exp(2*pi*i*f0*m) - exp(-*pi*i*f0*m) ] * exp(-2*pi*i*fn*m) dm
The first term does have max abs value at fn = f0, just like you expect. However, the second term also makes a contribution, and you are finding max abs of the sum of the two terms. That peaks out at the wrong spot, point 195.
Replacing the signal with
x = exp(2*pi*i*f*m)
i.e. positive frequency only, gives the correct answer. Except not quite. Now it peaks out at point 203. The problem is that the linspace array does not represent the periodic wave correctly, because it includes both end points. But if you go to
M = 100;
m = ((0:M-1)/M)-1/2;
f = 2;
x = exp(2*pi*i*f*m);
which includes only the first end point, then it all works out. The max is at point 201. The max of the chirp transform is at point 2001, which sounds right. The waves in the second plot all align perfectly.
  4 个评论
Matt Southwick
Matt Southwick 2019-8-14
Thanks again David! I'll look into Shaum's.
As a further proof to your awnser and work-around for the discrete problem, I found that applying a hilbert transform to the signal to filter out the negative frequencies allows the input frequency to be determined by peak picking of the fft magnitude.
M = 100;
m = ((0:M-1)/M)-1/2;
f = 2;
x = sin(2*pi*f*m);
x2 = hilbert(x);
N = 100*M;
FFT = fft(x2,N);
figure,plot(abs(FFT))
[~,ind] = max(abs(FFT));
David Goodmanson
David Goodmanson 2019-8-15
编辑:David Goodmanson 2019-8-15
Hi Matt,
That's an interesting use of the hilbert transform. Of course what Mathworks calls the hilbert transform and what the rest of the world usually calls the hilbert transform are not the same (I wish Mathworks would call it something different), but their function gets the job done here.

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