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I know this topic is more suited for Computing & Technology, but it has even more to do with general questions about Fourier transform capabilities. I have a question about sample restoration in Discrete Fourier Transform. Suppose we have a signal with the stack of frequencies from 1 Hz to 128 Hz, and the number of samples is twice the number of frequencies, i.e. 256. If for some reason half of the samples is lost, first half or last half (128 samples), can we restore completely those lost samples from the remaining half? Note that in this case the one who has to restore the signal doesn`t know what frequencies are in the signal, but does know the number of the missing samples. Can we find all the frequencies from that sparse signal? In this example, the frequencies are harmonics and all beginning at the same faze and with the same amplitude.

If we apply FFT algorithm on the time domain, I assume that the information about missing samples can be extracted from the frequency domain, because all sine waves have the same length in a time domain, and the only thing which happened is that frequencies are missing their fazes, for example, 1Hz is cut in half of its faze, 2Hz are missing a whole faze, and so on. Is it possible to use FFT to analyze the remaining samples, discover all the frequencies in a frequency domain, and use those frequencies to recreate a time domain and restore missing samples?

I ask this because I found a paper about recovering missing samples from oversampled band-limited signals. There is a statement in it which says that a band-limited oversampled signal is completely determined even if an arbitrary finite number of samples is lost. I understood this statement as described above. I just want to clarify this in the general sense of the matter, the possibility or feasibility of that kind of sample restoration.

And, if this method of sample restoration is possible, I have another question. Given the previous example, can a signal be restored if we continue to stack frequencies from 128Hz on, as f_{n}+ 1Hz = f_{n+1}, n<256? Is there an algorithm that can find all the frequencies in a partial time domain, as I described here, even if there are greater number of frequencies than the number of remaining samples?

If it`s not absolutely necessary, please don`t insert math equations because all I see is[Math Processing Error], write them in text instead. Thank you.

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# Fourier transform capabilities in reconstructing missing data

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