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Cyclic signals and FFT
Summary
| DFT (FFT) is by nature cyclic. Processing a cyclic signal with DFT methods is lossless (and usually quite straightforward). |
Introduction
Fast Fourier Transform (FFT) is defined for cyclic signals. It transforms a periodic signal into its discrete Fourier transform, and back.The process is lossless - besides limited numerical precision, no information is lost.
Most practical applications use FFT on signals that are not periodic. This requires for example windowing, which is an approximation.
Available computing power has grown over the years according to “Moore's law”, and has grown on an exponential scale. The idea of “taking the FFT of a whole audio CD” used to be a joke, but today it can be done on a low-cost PC.
Many problems simplify considerably, if one can use both time and frequency domain processing, and FFT is an efficient tool to go back and forth without introducing error.
A hardware implementation of an algorithm would still use conventional DSP techniques to minimize cost. But an FFT-based algorithm can be quite efficient in research, development or rapid prototyping.
Modern modulation methods like OFDM or single-carrier FMDA (SC-FDMA) use FFT processing. A part of the signal - the cyclic prefix - makes the signal periodic, at least “locally”.
This allows using FFT methods, ideally without loss.
Examples
  How to delay a signal by an arbitrary amount - even a fraction of a sample |
How to calculate the frequency corresponding to each FFT bin |
How to increase the sampling rate of a signal, in steps of “one extra sample per period” |
How to find the signal-to-noise ratio between a signal and a reference. Both are out of sync? Doesn't matter! |
© Markus Nentwig 2007-2008
The content of this page is provided without any warranty and may not be reproduced without permission.