Circular convolution: Difference between revisions

Revision as of 20:28, 26 February 2020

Latest revision as of 12:26, 17 December 2024

(47 intermediate revisions by 14 users not shown)

Line 1:

{{Short description|Mathematical operation}}

The '''circular convolution''', also known as '''cyclic convolution''', of two aperiodic functions (i.e. [[Schwartz functions]]) occurs when one of them is [[convolution|convolved in the normal way]] with a [[periodic summation]] of the other function. That situation arises in the context of the [[Discrete Fourier transform#Circular convolution theorem and cross-correlation theorem|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.<ref>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]].)</ref>

'''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 is the periodic convolution of the DTFTs of the individual sequences. And each DTFT is a [[periodic summation]] of a continuous Fourier transform function (see {{slink|Discrete-time Fourier transform|Relation to Fourier_Transform}}). Although DTFTs are usually continuous functions of frequency, the concepts of periodic and circular convolution are also directly applicable to discrete sequences of data. In that context, circular convolution plays an important role in maximizing the efficiency of a certain kind of common filtering operation.

==Definitions==

Let ''x'' be a function with a well-defined periodic summation, ''x''_''T'', where:

The ''periodic convolution'' of two T-periodic functions, <math>h_{_T}(t)</math> and <math>x_{_T}(t)</math> can be defined as''':'''

:<math>x_T(t) \ \triangleq \ \sum_{k=-\infty}^\infty x(t - kT) = \sum_{k=-\infty}^\infty x(t + kT).</math>

:<math>\int_{t_o}^{t_o+T} h_{_T}(\tau)\cdot x_{_T}(t - \tau)\,d\tau,</math> &nbsp; <ref name=Jeruchim/><ref name=Udayashankara/>

If ''h'' is any other function for which the convolution ''x''_''T'' ∗ ''h'' exists, then the convolution ''x''_''T'' ∗ ''h'' is periodic and identical to''':'''

where <math>t_o</math> is an arbitrary parameter.&nbsp; An alternative definition, in terms of the notation of normal ''linear'' or ''aperiodic'' convolution, follows from expressing <math>h_{_T}(t)</math> and <math>x_{_T}(t)</math> as [[periodic summation|periodic summations]] of aperiodic components <math>h</math> and <math>x</math>, i.e.''':'''

:<math>

\begin{align}

(x_T * h):<math>h_{_T}(t) \quad &\triangleq \ \int_sum_{k=-\infty}^\infty h(\tau)\cdot x_T(t - \taukT)\,d\tau = \sum_{k=-\infty}^\infty h(t + kT).</math>

&\equiv \int_{t_o}^{t_o+T} h_T(\tau)\cdot x_T(t - \tau)\,d\tau,

Then''':'''

\end{align}

</math><ref>Proof:

{{Equation box 1

|indent=:|cellpadding=0|border=0|background colour=white

|equation={{NumBlk||

<math>

\int_{t_o}^{t_o+T} h_{_T}(\tau)\cdot x_{_T}(t - \tau)\,d\tau = \int_{-\infty}^\infty h(\tau)\cdot x_{_T}(t - \tau)\,d\tau\ \triangleq\ (h *x_{_T})(t) = (x * h_{_T})(t).</math> &nbsp; &nbsp;

|{{EquationRef|Eq.1}} }} }}

{{Collapse top|title=Derivation of Eq.1}}

:<math>\begin{align}

&\int_{-\infty}^\infty h(\tau)\cdot x_Tx_{_T}(t - \tau)\,d\tau \\

&={} &\sum_{k=-\infty}^\infty \left[\int_{t_o+kT}^{t_o+(k+1)T} h(\tau)\cdot x_Tx_{_T}(t - \tau)\ d\tau\right] \quad t_0 \text{ is an arbitrary parameter}\\

&=\sum_{k=-\infty}^\infty \left[\int_{t_o}^{t_o+T} h(u + kT)\cdot \underbrace{x_{_T}(t-u-kT)}_{x_{_T}(t-u), \text{ by periodicity}}\ du\right] \quad \text{substituting } u\triangleq \tau-kT\\

\stackrel{\tau \rightarrow \tau+kT}{=}{}

&=\int_{t_o}^{t_o+T} \left[\sum_{k=-\infty}^\infty \left[\int_{t_o}^{t_o+T} h(\tauu + kT)\cdot x_Tx_{_T}(t - \tau -kTu)\ d\tau\right]\ du\\

&={} &\int_{t_o}^{t_o+T} \underbrace{\left[\sum_{k=-\infty}^\infty h(\tauu + kT)\cdotright]}_{\triangleq \underbrace h_{x_T_T}(t - \tau-kTu)}_\cdot x_{X_T_T}(t - \tauu), \text{ by periodicity}}\right]\ d\taudu\\

&={} &\int_{t_o}^{t_o+T} \underbraceh_{\left[\sum_{k=-\infty_T}^\infty h(\tau + kT)\right]}_cdot x_{\triangleq \ h_T(\tau)_T}\cdot x_T(t - \tau)\ d\tau \quad \quadtext{substituting } \scriptstyle{(QED)}tau \triangleq u

\end{align}</math>

{{Collapse bottom}}<br>

</ref>

Both forms can be called ''periodic convolution''.{{efn-la

where ''t''_o is an arbitrary parameter and ''h''_''T'' is a [[periodic summation]] of ''h''.

|[[#McGillem|McGillem and Cooper]], p 172 (4-6)

}} The term ''circular convolution''<ref name=Udayashankara/><ref name=Priemer/> arises from the important special case of constraining the non-zero portions of both <math>h</math> and <math>x</math> to the interval <math>[0,T].</math> Then the periodic summation becomes a ''periodic extension''{{efn-la

|[[#McGillem|McGillem and Cooper]], p 183 (4-51)

}}, which can also be expressed as a ''circular function''''':'''

:<math>x_{_T}(t) = x(t_{\mathrm{mod}\ T}), \quad t\in\mathbb{R}\,</math> ([[Number#Real_numbers|any real number]]){{efn-la

The second integral is called the '''periodic convolution'''<ref>Jeruchim 2000, pp 73-74.</ref><ref name="Uday">Udayashankara 2010, p 189.</ref> of functions ''x''_''T'' and ''h''_''T'' and is sometimes normalized by 1/''T''.<ref>Oppenheim, pp 388-389</ref> When ''x''_''T'' is expressed as the [[periodic summation]] of another function, ''x'', the same operation may also be referred to as a '''circular convolution'''<ref name="Uday"/><ref>Priemer 1991, pp 286-289.</ref> of functions ''h'' and ''x''.

|[[#Oppenheim|Oppenheim and Shafer]], p 559 (8.59)

}}

And the limits of integration reduce to the length of function <math>h</math>''':'''

:<math>(h *x_{_T})(t) = \int_{0}^{T} h(\tau)\cdot x((t - \tau)_{\mathrm{mod}\ T})\ d\tau.</math>{{efn-la

|[[#Oppenheim|Oppenheim and Shafer]], p 571 (8.114), shown in digital form

}}{{efn-la

|[[#McGillem|McGillem and Cooper]], p 171 (4-22), shown in digital form

}}

== Discrete sequences ==

Line 32 ⟶ 53:

:<math>

(x_Nh * hx_{_N})[n] \ \triangleq \ \sum_{m=-\infty}^\infty h[m] \cdot \underbrace{x_Nx_{_N}[n-m]}_{\sum_{k=-\infty}^\infty x[n -m -kN]}

</math>

Line 38 ⟶ 59:

== Example ==

[[Image:Circular convolution example.svg|thumb|400px|Circular convolution can be expedited by the FFT algorithm, so it is often used with an FIR filter to efficiently compute linear convolutions. These graphs illustrate how that is possible. Note that a larger FFT size (N) would prevent the overlap that causes graph #6 to not quite match all of #3.]]

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''',&nbsp; 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 [[Discrete Fourier transform#Circular convolution theorem and cross-correlation theorem|circular convolution theorem]].

Line 45 ⟶ 66:

=== Overlapping input blocks ===

This method uses a block size equal to the FFT size (1024). We describe it first in terms of normal or ''linear'' convolution. When a normal convolution is performed on each block, there are start-up and decay transients at the block edges, due to the filter ''latency'' (200-samples). Only 824 of the convolution outputs are unaffected by edge effects. The others are discarded, or simply not computed. That would cause gaps in the output if the input blocks are contiguous. The gaps are avoided by overlapping the input blocks by 200 samples. In a sense, 200 elements from each input block are "saved" and carried over to the next block. This method is referred to as '''[[Overlap-save method|overlap-save]]''',<ref> name=Rabiner 1975, pp 65–67.</ref> although the method we describe next requires a similar "save" with the output samples.

When an FFT is used to compute the 824 unaffected DFT samples, we don't have the option of not computing the affected samples, but the leading and trailing edge-effects are overlapped and added because of circular convolution. Consequently, the 1024-point inverse FFT (IFFT) output contains only 200 samples of edge effects (which are discarded) and the 824 unaffected samples (which are kept). To illustrate this, the fourth frame of the figure at right depicts a block that has been periodically (or "circularly") 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 extended blocks overlap the contributions from the original block. The last frame is the composite output, and the section colored green represents the unaffected portion.

Line 51 ⟶ 72:

=== Overlapping output blocks ===

This method is known as ''[[overlap-add method|overlap-add]]''.<ref> name=Rabiner 1975, pp 63–65.</ref> In our example, it uses contiguous input blocks of size 824 and pads each one with 200 zero-valued samples. Then it overlaps and adds the 1024-element output blocks. Nothing is discarded, but 200 values of each output block must be "saved" for the addition with the next block. Both methods advance only 824 samples per 1024-point IFFT, but overlap-save avoids the initial zero-padding and final addition.

== See also ==

*[[Convolution theorem#Functions of discrete variable sequences|Convolution theorem]]

*[[Hilbert transform#Discrete Hilbert transform|Discrete Hilbert transform]]

*[[Circulant matrix]]

*[[Hilbert transform#Discrete Hilbert transform|Discrete Hilbert transform]]

== NotesPage citations ==

{{Reflistnotelist-la|1}}

== References ==

{{reflist|refs=

*Rabiner, Lawrence R.; Gold, Bernard (1975). ''Theory and application of digital signal processing''. Englewood Cliffs, N.J.: Prentice-Hall. pp 63–67. {{ISBN|0-13-914101-4}}

<ref name=Rabiner>

*Oppenheim, Alan V.; Schafer, Ronald W.; Buck, John A. (1999). ''Discrete-time signal processing''. Upper Saddle River, N.J.: Prentice Hall. {{ISBN|0-13-754920-2}}.

{{cite book

*Priemer, Roland (July 1991). ''Introductory Signal Processing (Advanced Series in Electrical and Computer Engineering) (v. 6)''. Teaneck, N.J.: World Scientific Pub Co Inc. [https://books.google.com/books?id=QBT7nP7zTLgC&printsec=frontcover&dq=Priemer,+Roland&hl=en&sa=X&ei=J2owUZzANIb_ygGex4HAAg&ved=0CC8Q6AEwAA {{isbn|9971509199}}.

|author1=Rabiner, Lawrence R.

*Jeruchim, Michel C.; Philip Balaban, K. Sam Shanmugan (October 2000). ''Simulation of Communication Systems: Modeling, Methodology and Techniques'' (2nd ed.). New York: Kluwer Academic Publishers. {{isbn|0306462672}}.

|author2=Gold, Bernard

*Udayashankara, V. (June 2010). ''Real Time Digital Signal Processing''. India: Prentice-Hall. {{isbn|8120340493}}.

|title=Theory and application of digital signal processing

*{{cite book |author1=Oppenheim, Alan V. |author2=Willsky, with S. Hamid | title=Signals and Systems | publisher=Pearson Education | year=1998 | isbn=0-13-814757-4}}.

|pages=63–67

|year=1975

|publisher=Prentice-Hall

|___location=Englewood Cliffs, N.J.

|isbn=0-13-914101-4

|url-access=registration

|url=https://archive.org/details/theoryapplicatio00rabi/page/63

}}</ref>

<ref name=Priemer>

{{cite book

|last=Priemer

|first=Roland

|title=Introductory Signal Processing

|pages=286–289

|publisher=World Scientific Pub Co Inc.

|series=Advanced Series in Electrical and Computer Engineering

|volume=6

|date=July 1991

|___location=Teaneck, N.J.

|url=https://books.google.com/books?id=QBT7nP7zTLgC&q=Priemer,+Roland

|isbn=9971-50-919-9

}}</ref>

<ref name=Jeruchim>

{{cite book

|last1=Jeruchim

|first1=Michel C.

|last2=Balaban

|first2=Philip

|last3=Shanmugan

|first3=K. Sam

|title=Simulation of Communication Systems: Modeling, Methodology and Techniques

|pages=73–74

|publisher=Kluwer Academic Publishers

|edition=2nd

|date=October 2000

|___location=New York

|isbn=0-30-646267-2

}}</ref>

<ref name=Udayashankara>

{{cite book

|last=Udayashankara

|first=V.

|title=Real Time Digital Signal Processing

|page=189

|publisher=Prentice-Hall

|date=June 2010

|___location=India

|isbn=978-8-12-034049-7

}}</ref>

}}

{{refbegin}}

#<li value="5">{{cite book

|ref=Oppenheim

|last1=Oppenheim

|first1=Alan V.

|authorlink=Alan V. Oppenheim

|last2=Schafer

|first2=Ronald W.

|author2-link=Ronald W. Schafer

|last3=Buck

|first3=John R.

|title=Discrete-time signal processing

|pages=[https://archive.org/details/discretetimesign00alan/page/548 548], 571

|year=1999

|publisher=Prentice Hall

|___location=Upper Saddle River, N.J.

|isbn=0-13-754920-2

|edition=2nd

|url-access=registration

|url=https://archive.org/details/discretetimesign00alan

}}

#{{cite book

|ref=McGillem

|last1=McGillem

|first1=Clare D.

|last2=Cooper

|first2=George R.

|title=Continuous and Discrete Signal and System Analysis

|publisher=Holt, Rinehart and Winston

|edition=2

|date=1984

|isbn=0-03-061703-0

}}

== Further reading ==

*{{cite book |author1=Oppenheim, Alan V. |author2=Willsky, with S. Hamid |title=Signals and Systems |publisher=Pearson Education |year=1998 |isbn=0-13-814757-4}}

{{refend}}

[[Category:Functional analysis]]