I have derived the circular convolution property for an affine Fractional Fourier Transform and I need to work on an application for my research. Any idea?
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$\begingroup$ I can't think of any. Normally one uses the FFT because it's fast, and just copes with the fact that what you really want to work with is a signal of infinite duration. You just put up with the fact that the FFT works on a set of data that's topologically circular because, well, it's fast. One of the strong downsides to using the FFT for filtering is the fact that its convolution is circular -- this limits the filters you can use to FIR filters, and means you have to use the overlap-and-add algorithm, with all its bookkeeping. $\endgroup$– TimWescottCommented Nov 10, 2021 at 19:48
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1$\begingroup$ Welcome to SE.SP! The Wikipedia page for it has an example application. $\endgroup$– Peter K. ♦Commented Nov 10, 2021 at 19:56
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$\begingroup$ Oh Peter I think I took a class with you years ago. Glad to see you here. $\endgroup$– Amir RCommented Nov 10, 2021 at 21:45
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$\begingroup$ Thank you I will check it out. $\endgroup$– Amir RCommented Nov 10, 2021 at 21:47
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2 Answers
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A circular convolution is useful for detection of repeating waveforms such as GPS: the GPS C/A code repeats 20 times for the transmission of every bit. If we were to use circular convolution with the time reverse of one sequence (which is correlation), we could be at any offset in time and still get the same maximum peak associated with that offset in the correlation result.
The optimum strategy for SNR is to convolve with a copy of the waveform that also repeats 20 times (20 ms duration) to maximize SNR. However that can only be done if there is a frequency offsets less than 1/40ms = 25 Hz (approximately, this is within the main lobe of a Sinc function). By using circular correlation we can do the initial detection with a frequency offsets up to approximately 500 Hz.
Circular convolution is also useful for OFDM given the cyclic prefix is a repetition of the start of the FFT frame for similar reasons I provide above.
I also like to use circular convolution when evaluating / plotting the performance of cyclic sequences in that the resulting noise floor in the resulting plot is flat across the entire range (similar throughout).
answered Jul 7, 2023 at 12:25
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It should be useful for the operators derived form PDE with this kind of boundary assumption.
answered Jul 7, 2023 at 11:50