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In digital signal processing, convolution is used to map the impulse response of a real room on a digital audio signal. In electronic music convolution is the imposition of a spectral or rhythmic structure on a sound. Often this envelope or structure is taken from another sound.
Fig 1: A sequence of four plots depicts one cycle of the overlap–save convolution algorithm. The 1st plot is a long sequence of data to be processed with a lowpass FIR filter. The 2nd plot is one segment of the data to be processed in piecewise fashion.
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 sampling theory of Shannon can be generalized for the case of nonuniform sampling, that is, samples not taken equally spaced in time. The Shannon sampling theory for non-uniform sampling states that a band-limited signal can be perfectly reconstructed from its samples if the average sampling rate satisfies the Nyquist condition. [5]
The FIR convolution is a cross-correlation between the input signal and a time-reversed copy of the impulse response. Therefore, the matched filter's impulse response is "designed" by sampling the known pulse-shape and using those samples in reverse order as the coefficients of the filter. [1]
A digital signal is an abstraction that is discrete in time and amplitude. The signal's value only exists at regular time intervals, since only the values of the corresponding physical signal at those sampled moments are significant for further digital processing. The digital signal is a sequence of codes drawn from a finite set of values. [10]
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).
Discrete-time signals, used in digital signal processing, can be obtained by sampling and quantization of continuous signals. Continuous signal may also be defined over an independent variable other than time. Another very common independent variable is space and is particularly useful in image processing, where two space dimensions are used.