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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 ...
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.
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 :
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain ) equals point-wise multiplication in the other domain (e.g., frequency domain ).
It repeats the end of the symbol so the linear convolution of a frequency-selective multipath channel can be modeled as circular convolution, which in turn may transform to the frequency domain via a discrete Fourier transform. This approach accommodates simple frequency domain processing, such as channel estimation and equalization.
The significance of this result is explained at Circular convolution and Fast convolution algorithms. Relationship to the Z-transform is a ...
The premise behind the circular convolution approach on multidimensional signals is to develop a relation between the Convolution theorem and the Discrete Fourier transform (DFT) that can be used to calculate the convolution between two finite-extent, discrete-valued signals.