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Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the discrete-time Fourier transform (DTFT).
The most common fast convolution algorithms use fast Fourier transform (FFT) algorithms via the circular convolution theorem. Specifically, the circular convolution of two finite-length sequences is found by taking an FFT of each sequence, multiplying pointwise, and then performing an inverse FFT. Convolutions of the type defined above are then ...
The motivation behind using the circular convolution approach is that it is based on the DFT. The premise behind circular convolution is to take the DFTs of the input signals, multiply them together, and then take the inverse DFT. Care must be taken such that a large enough DFT is used such that aliasing does not occur.
which gives rise to the interpretation as a circular convolution of and . [7] [8] It is often used to efficiently compute their linear convolution. (see Circular convolution, Fast convolution algorithms, and Overlap-save) Similarly, the cross-correlation of and is given by:
However, in practice, this cannot be achieved, as real signals are always time-limited. So, to mimic the infinite behavior, prefixing the end of the symbol to the beginning makes the linear convolution of the channel appear as though it were circular convolution, and thus, preserve this property in the part of the symbol after the cyclic prefix.
Using the circular convolution theorem, we can use the discrete Fourier transform to transform the cyclic convolution into component-wise multiplication = () = so that = [((()) (()))]. This algorithm is much faster than the standard Gaussian elimination , especially if a fast Fourier transform is used.
The DTFT is often used to analyze samples of a continuous function. ... An important special case is the circular convolution of sequences s and y defined by ...
And for any parameter +, [A] it is equivalent to the -point circular convolution of [] with [] in the region [,]. The advantage is that the circular convolution can be computed more efficiently than linear convolution, according to the circular convolution theorem :