Circular Convolution
Jump to navigation
Jump to search
A Circular Convolution is a Schwartz Functions that ...
- AKA: Acyclic Convolution.
- See: Discrete-Time Fourier Transform, Periodic Summation, Schwartz Functions, Convolution.
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/circular_convolution Retrieved:2015-2-3.
- 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. [1] Let x be a function with a well-defined periodic summation, xT, where: :[math]\displaystyle{ x_T(t) \ \stackrel{\mathrm{def}}{=} \ \sum_{k=-\infty}^\infty x(t - kT) = \sum_{k=-\infty}^\infty x(t kT). }[/math] If h is any other function for which the convolution xT ∗ h exists, then the convolution xT ∗ h is periodic and identical to': :[math]\displaystyle{ \begin{align} (x_T * h)(t)\quad &\stackrel{\mathrm{def}}{=} \ \int_{-\infty}^\infty h(\tau)\cdot x_T(t - \tau)\,d\tau \\ &\equiv \int_{t_o}^{t_o T} h_T(\tau)\cdot x_T(t - \tau)\,d\tau, \end{align} }[/math] [2]
where to is an arbitrary parameter and hT is a periodic summation of h.
The second integral is called the periodic convolution [3] [4] of functions xT and hT and is sometimes normalized by 1/T. [5] When xT is expressed as the periodic summation of another function, x, the same operation may also be referred to as a circular convolution [6] of functions h and x.
- 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. [1] Let x be a function with a well-defined periodic summation, xT, where: :[math]\displaystyle{ x_T(t) \ \stackrel{\mathrm{def}}{=} \ \sum_{k=-\infty}^\infty x(t - kT) = \sum_{k=-\infty}^\infty x(t kT). }[/math] If h is any other function for which the convolution xT ∗ h exists, then the convolution xT ∗ h is periodic and identical to': :[math]\displaystyle{ \begin{align} (x_T * h)(t)\quad &\stackrel{\mathrm{def}}{=} \ \int_{-\infty}^\infty h(\tau)\cdot x_T(t - \tau)\,d\tau \\ &\equiv \int_{t_o}^{t_o T} h_T(\tau)\cdot x_T(t - \tau)\,d\tau, \end{align} }[/math] [2]
- ↑ If a sequence, x[n], represents samples of a continuous function, x(t), with Fourier transform X(ƒ), its DTFT is a periodic summation of X(ƒ). (see Discrete-time_Fourier_transform#Relationship_to_sampling)
- ↑ Proof: :[math]\displaystyle{ \int_{-\infty}^\infty h(\tau)\cdot x_T(t - \tau)\,d\tau }[/math] :::[math]\displaystyle{ \begin{align} &= \sum_{k=-\infty}^\infty \left[\int_{t_o kT}^{t_o (k 1)T} h(\tau)\cdot x_T(t - \tau)\ d\tau\right] \\ &\stackrel{\tau \rightarrow \tau kT}{=}\ \sum_{k=-\infty}^\infty \left[\int_{t_o}^{t_o T} h(\tau kT)\cdot x_T(t - \tau -kT)\ d\tau\right] \\ &= \int_{t_o}^{t_o+T} \left[\sum_{k=-\infty}^\infty h(\tau+kT)\cdot \underbrace{x_T(t - \tau-kT)}_{X_T(t - \tau), \text{ by periodicity}}\right]\ d\tau \\ &= \int_{t_o}^{t_o+T} \underbrace{\left[\sum_{k=-\infty}^\infty h(\tau+kT)\right]}_{\stackrel{\mathrm{def}}{=} \ h_T(\tau)}\cdot x_T(t - \tau)\ d\tau \quad \quad \scriptstyle{(QED)} \end{align} }[/math]
- ↑ Jeruchim 2000, pp 73-74.
- ↑ Udayashankara 2010, p 189.
- ↑ Oppenheim, pp 388-389
- ↑ Priemer 1991, pp 286-289.
1990
- (Rao & Yip, 1990) ⇒ K. Ramamohan Rao, and Ping Yip. (1990). “Discrete Cosine Transform: algorithms, advantages, applications." Academic Press.