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Then many of the values of the circular convolution are identical to values of x∗h, which is actually the desired result when the h sequence is a finite impulse response (FIR) filter. Furthermore, the circular convolution is very efficient to compute, using a fast Fourier transform (FFT) algorithm and the circular convolution theorem.
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 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.
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:
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.
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).
The advantage is that the circular convolution can be computed more efficiently than linear convolution, according to the circular convolution theorem: [] = ...
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 :