The discrete Fourier transform is fundamental in modern communication systems. It is used to generate and process (i.e. modulate and demodulate) the signals transmitted in 4G, 5G, and wifi systems, and is always implemented by one of the fast Fourier transforms (FFT) algorithms. It is possible to generalize the FFT to work correctly on input vectors with periodic missing values. I will consider whether this has applications, such as more general transmitted signal waveforms, or further applications such as spectral density estimation for time series with missing data. More speculatively, can we generalize to "recursive" missing values, where the non-missing blocks have gaps? If so, how do we optimally recognize such a pattern in a given time series?
