Circular convolution
The circular convolution, also known as cyclic convolution, of two aperiodic functions (i.e. Schwartz functions) occurs when one of them is convolved in the normal way with a periodic summation of the other function. That situation arises in the context of the Circular convolution theorem. The identical operation can also be expressed in terms of the periodic summations of both functions, if the infinite integration interval is reduced to just one period. That situation arises in the context of the discrete-time Fourier transform (DTFT) and is also called periodic convolution. In particular, the DTFT of the product of two discrete sequences is the periodic convolution of the DTFTs of the individual sequences.
Let x be a function with a well-defined periodic summation, xT, where:
If h is any other function for which the convolution xT ∗ h exists, then the convolution xT ∗ h is periodic and identical to:
where to is an arbitrary parameter and hT is a periodic summation of h.
The second integral is called the periodic convolution of functions xT and hT and is sometimes normalized by 1/T. When xT is expressed as the periodic summation of another function, x, the same operation may also be referred to as a circular convolution of functions h and x.
Similarly, for discrete sequences and period N, we can write the circular convolution of functions h and x as:
For the special case that the non-zero extent of both x and h are ≤ N, this is reducible to matrix multiplication where the kernel of the integral transform is a circulant matrix.
A case of great practical interest is illustrated in the figure. The duration of the x sequence is N (or less), and the duration of the h sequence is significantly less. 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.
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