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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). In particular, the DTFT of the product of two discrete sequences ...
The subregion [M + 1, L + M] is appended to the output stream, and the other values are discarded. The advantage is that the circular convolution can be computed more efficiently than linear convolution, according to the circular convolution theorem:
In the particular case p = 1, this shows that L 1 is a Banach algebra under the convolution (and equality of the two sides holds if f and g are non-negative almost everywhere). More generally, Young's inequality implies that the convolution is a continuous bilinear map between suitable L p spaces.
The overlap-add method involves a linear convolution of discrete-time signals, whereas the overlap-save method involves the principle of circular convolution. In addition, the overlap and save method only uses a one-time zero padding of the impulse response, while the overlap-add method involves a zero-padding for every convolution on each ...
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:
The two methods are also compared in Figure 3, created by Matlab simulation. The contours are lines of constant ratio of the times it takes to perform both methods. When the overlap-add method is faster, the ratio exceeds 1, and ratios as high as 3 are seen. Fig 3: Gain of the overlap-add method compared to a single, large circular convolution.
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).
Then, from the perspective of operator theory, a circulant matrix is the kernel of a discrete integral transform, namely the convolution operator for the function (,, …,); this is a discrete circular convolution. The formula for the convolution of the functions ():= () is