Frequency estimation of circularly shifted single tone signal

Question

I have a discrete signal $y[n] = <e^{j ~ 2 \pi f ~ n}>_J + ~w[n]$ with $n \in [0, N[$ and $w[n]$ AWGN, $<x[n]>_K$ denotes the signal $x[n]$ circularly shifted by $K$ samples. Let's define $Y_k = \mathrm{DFT}(y[n])$ the output of the $N$-point DFT of $y[n]$. By ignoring the AWGN and using well-known Fourier properties, we can deduce that $Y_k$ is a sinc function centered around $f$, multiplied by $e^{\tfrac{k J}{N} }$ due to the circular shift.

In my application, $J$ is unknown to me but I am interested in obtaining an estimation of $f$ (which is real-valued in $[0, N[$). Over the years, many estimators $\hat{f}$ for single tone signals $z[n] = e^{j ~ 2 \pi f ~ n} + ~w[n]$ have been designed, and they work fairly well. These estimators rely either on the complex values $Z_k$ (i.e. the DFT of $z[n]$, see e.g. [R1], which I can't use because I don't know $J$), or use the amplitudes of interpolated DFT bins, e.g. $\hat{f}_{res}$ which computes the residual frequency offset inside two DFT bins from [R2]:

$$ \hat{f}_{res} = \frac{|\bar{Z_1}|^2 - |\bar{Z}_{-1}|^2}{u(|\bar{Z_1}|^2 + |\bar{Z}_{-1}|^2) + v|\bar{Z_0}|^2} $$

where $\bar{Z}_k$ is the zero-padded $2N$-point DFT ($N$ points are from $z[n]$, the following $N$ points are $0$). $\bar{Z}_0$ is the bin with greatest amplitude and $\bar{Z}_1$/$\bar{Z}_{-1}$ are its neighbors. It is interesting to note that in the case of a $N$-point DFT, $|Y_k| = |Z_k|$. However this property does not hold anymore for the interpolated bins in $\bar{Y}_k$ obtained through the zero-padding, and I am unable to figure why analytically.

Here's an illustration of the difference of amplitudes between $\bar{Z}_k$ (zero-padded DFT of true single tone signal) and $\bar{Y}_k$ (zero-padded DFT of the circularly shifted signal):

and the MATLAB code used to generate the figure:

N = 256;
f = 100.4/N;
J = 125;

% Compute z[n]
n = 0:(N-1);
z = exp(2*pi*1j*f*n);

% Compute zero-padded DFT Z[k]
z2 = zeros(1, N*2);
z2(1:N) = z;
Z2 = fft(z2);

% Compute zero-padded DFT Y[k]
y2 = zeros(1, N*2);
y2(1:N) = circshift(z, J);
Y2 = fft(y2);

subplot(2, 1, 1);
stem(abs(Z2));
hold on;
stem(abs(Y2));
xlim([1 2*N])
legend('|Z(k)|', '|Y(k)|', 'Location','northwest')

subplot(2, 1, 2);
stem(abs(Z2) - abs(Y2))
xlim([1 2*N])
legend('|Z(k)| - |Y(k)|', 'Location','northwest')

Does someone have any clue on how to model analytically the interpolated DFT bins in $\bar{Y}_k$, or the DTFT of $y[n]$ ? Thanks.

[R1] https://www.researchgate.net/publication/3321864_Fast_Accurate_Frequency_Estimators_DSP_Tips_Tricks

[R2] https://ieeexplore.ieee.org/document/5910415 (unfortunately behind a paywall)

Answer 1

The clue is that without zero-padding, the circulary shifted sequence is just a shifted version of the periodic continuation of the original sequence. That's why without zero-padding the magnitudes of the DFTs of both sequences are identical.

When you use zero-padding, the two sequences cannot be obtained from each other by pure shifting. So the DFTs must be different.

You can avoid the problem by zero-padding the circularly shifted sequence between the last and the first element of the original sequence:

% this assumes that J is in [0,N]
y2 = [z(N-J+1:N),zeros(1,N),z(1:N-J)];
Y2 = fft(y2);

This will guarantee that the magnitudes of the DFTs of the two sequences are the same, even after zero-padding.