I am struggling to understand the effects that partially undersampling a signal would have on its FFT. I have a complex signal consisting on a sum of sinusoids of different frequencies, which I am sampling at the Nyquist rate, except for a couple of samples that are skipped at some point. Assuming I interpolate the missing samples, how would this affect the resulting FFT?
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$\begingroup$ I believe there’s quite some information missing. Let alone the complexity of the issue, different interpolation schemes have different characteristics. Without even knowing what interpolation method you will use I believe it is very hard to answer that. Even then, I am not sure there would be a “trivial” answer (I haven’t put extensive thought into it so I may very well be mistaken though). $\endgroup$– ZaellixAOct 4 at 12:57
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1$\begingroup$ Hey @ZaellixA, you are right that my question lacked information. However, I find Hilmar's answer to fit my issue. Thanks $\endgroup$– Carlos RaveloOct 4 at 13:23
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You can model the resulting signal, $y[n]$ as
$$y[n] = x[n] + e[n]$$
where $e[n]$ is the error signal given by
$$e[n] = \sum_m a_k\cdot \delta[n-n_m] $$
$a_k$ is the residual interpolation error at the sample time $n_m$ where there are missing samples.
The DFT spectrum of the result would be
$$Y[k] = X[k] + E[k]$$
The DFT $E[k]$ of the error is
$$E[k] = \sum_m a_m \cdot e^{-j2\pi\frac{km}{N}} $$
In most cases this will simply be white noise, i.e. the total error energy ($\sum a_m^2$) will fairly equally distributed over all frequencies. That's kind of handy: you can use the zeros in $X[k]$ to estimate the noise floor.
The error will NOT be white if the missing samples follow some regular pattern in either amplitude or time index and/or if the error is correlated with the signal itself.
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$\begingroup$ Thanks! This is precisely what I needed to understand the scenario. $\endgroup$ Oct 4 at 13:21