# DFT of sum of sinusoids with random zeroed samples

I have a noisy signal consists of sum of sinusoids, I had a situation where some of the time domain samples are zeroed randomly. The following figure shows the DFT before and after zeroing, the number of zeroed samples are 40 out of 128 samples.

Obviously there is no much degradation of the DFT spectrum,. What is the math behind this, and I would like to get some keywords to search any relative subject that discuss the limitations of zeroing.

The math is well known; it is the convolution theorem for the DFT.

In this specific case:

$$DFT\left\{f[n]\cdot z[n]\right\} = DFT\left\{f[n]\right\} * DFT\left\{z[n]\right\}$$

Where:

$$f[n]$$ denotes your original sequence of 128 samples

$$z[n]$$ denotes a sequence of 128 samples, each randomly selected from $$\{0,1\}$$

'$$*$$' denotes cyclic convolution, since we're talking about the DFT.

The results of this convolution are visible in your second plot, particularly at very positive and very negative frequency bins.