# Partially undersampling effect on FFT

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?

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

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.

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