It has been suggested that this article or section be merged into convolution. (Discuss) (Actually the merge discussion is here.) Wikipedia does not have an article with this exact name. ...
For the computer science usage see convolution (computer science) . In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version...
Circular convolution is defined in articles Discrete Fourier transform and Convolution. And we note that circular boundary conditions are merely one possible boundary condition for convolution. One can also have e.g. even/odd boundary conditions of various types. (For example, every type of discrete cosine transform and discrete sine transform corresponds to convolutions with a different distinct boundary condition.) In mathematics, the discrete Fourier transform (DFT), sometimes called the finite Fourier transform, is a Fourier transform widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal, solve partial differential equations, and to perform other operations such as convolutions. ...
For the computer science usage see convolution (computer science) . In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version...
Depending on the application, circular convolution can be a useful operation or an unwanted side-effect. The intent of this article is to document specific applications and issues.
Filtering
Fast convolution  Convolution of time-domain functions is equivalent to multiplication of their Fourier transforms. Due to the efficiencies of the fast Fourier transform algorithm, the latter is often done (in digital signal processing) to maximize the speed of a signal filtering task. In the time domain, convolution filtering is inherently a continuous streaming process. But in the frequency domain, block-processing is required. The time-domain signal is divided into segments (blocks) and processed piecewise. Then the filtered segments are carefully pieced back together, mindful of edge effects, which means that a portion of each output segment must be discarded. Data loss is avoided by overlapping the initial input segments. The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...
A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. ...
Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ...
If linear convolution were performed on each block, the discarded data would represent the start-up transient (or latency) of the filter. When the DFT or FFT is used, the edge-effect has the same duration, just different values. During the start-up transient, the convolution is actually using data from both the beginning and the end of the block, as if the data were periodic or circular. To illustrate this, the fourth frame of the figure depicts a block that has been periodically extended, and the fifth frame depicts the individual components of a linear convolution performed on the entire sequence. The edge effects are where the contributions from the added blocks overlap the contributions from the original block. The last frame is the composite output, and the portion colored green represents the unadulterated output from the original input block.
Test signal generation Sometimes it is useful to generate a periodic test signal/sequence. If one cycle of the test signal is synthesized and then simply repeated indefinitely, there might be a harsh discontinuity where one cycle ends and the next begins. One way to mitigate that is to filter the repeating sequence, but that requires a dedicated, real-time filter component. So another way is to circularly filter the synthesized segment [once] before repeating it indefinitely.
Other applications
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