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?

  • $\begingroup$ 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). $\endgroup$
    – ZaellixA
    Oct 4 at 12:57
  • 1
    $\begingroup$ Hey @ZaellixA, you are right that my question lacked information. However, I find Hilmar's answer to fit my issue. Thanks $\endgroup$ Oct 4 at 13:23

1 Answer 1


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.

  • $\begingroup$ Thanks! This is precisely what I needed to understand the scenario. $\endgroup$ Oct 4 at 13:21

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