Convolution theorem

Convolution theorem In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise 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). Other versions of the convolution theorem are applicable to various Fourier-related transforms.

Contents 1 Functions of a continuous variable 1.1 Periodic convolution (Fourier series coefficients) 2 Functions of a discrete variable (sequences) 2.1 Periodic convolution 3 Convolution theorem for inverse Fourier transform 4 Convolution theorem for tempered distributions 5 See also 6 Notes 7 References 8 Further reading 9 Additional resources Functions of a continuous variable Consider two functions {displaystyle g(x)} and {displaystyle h(x)} with Fourier transforms {displaystyle G} and {displaystyle H} : {displaystyle {begin{aligned}G(f)&triangleq {mathcal {F}}{g}(f)=int _{-infty }^{infty }g(x)e^{-i2pi fx},dx,quad fin mathbb {R} \H(f)&triangleq {mathcal {F}}{h}(f)=int _{-infty }^{infty }h(x)e^{-i2pi fx},dx,quad fin mathbb {R} end{aligned}}} where {displaystyle {mathcal {F}}} denotes the Fourier transform operator. The transform may be normalized in other ways, in which case constant scaling factors (typically {displaystyle 2pi } or {displaystyle {sqrt {2pi }}} ) will appear in the convolution theorem below. The convolution of {displaystyle g} and {displaystyle h} is defined by: {displaystyle r(x)={g*h}(x)triangleq int _{-infty }^{infty }g(tau )h(x-tau ),dtau =int _{-infty }^{infty }g(x-tau )h(tau ),dtau .} In this context the asterisk denotes convolution, instead of standard multiplication. The tensor product symbol {displaystyle otimes } is sometimes used instead.

The convolution theorem states that:[1][2]: eq.8  {displaystyle R(f)triangleq {mathcal {F}}{r}(f)=G(f)H(f).quad fin mathbb {R} }         (Eq.1a) Applying the inverse Fourier transform {displaystyle {mathcal {F}}^{-1}} , produces the corollary:[2]: eqs.7, 10  Convolution theorem {displaystyle r(x)={g*h}(x)={mathcal {F}}^{-1}{Gcdot H},}         (Eq.1b) where {displaystyle cdot } denotes point-wise multiplication The theorem also generally applies to multi-dimensional functions.

Multi-dimensional derivation of Eq.1 Consider functions {displaystyle g,h} in Lp-space {displaystyle L^{1}(mathbb {R} ^{n})} , with Fourier transforms {displaystyle G,H} : {displaystyle {begin{aligned}G(f)&triangleq {mathcal {F}}{g}(f)=int _{mathbb {R} ^{n}}g(x)e^{-i2pi fcdot x},dx,quad fin mathbb {R} ^{n}\H(f)&triangleq {mathcal {F}}{h}(f)=int _{mathbb {R} ^{n}}h(x)e^{-i2pi fcdot x},dx,end{aligned}}} where {displaystyle fcdot x} indicates the inner product of {displaystyle mathbb {R} ^{n}} :   {displaystyle fcdot x=sum _{j=1}^{n}{f}_{j}x_{j},}   and   {displaystyle dx=prod _{j=1}^{n}dx_{j}.} The convolution of {displaystyle g} and {displaystyle h} is defined by: {displaystyle r(x)triangleq int _{mathbb {R} ^{n}}g(tau )h(x-tau ),dtau .} Also: {displaystyle iint |g(tau )h(x-tau )|,dx,dtau =int left(|g(tau )|int |h(x-tau )|,dxright),dtau =int |g(tau )|,|h|_{1},dtau =|g|_{1}|h|_{1}.} Hence by Fubini's theorem we have that {displaystyle rin L^{1}(mathbb {R} ^{n})} so its Fourier transform {displaystyle R} is defined by the integral formula: {displaystyle {begin{aligned}R(f)triangleq {mathcal {F}}{r}(f)&=int _{mathbb {R} ^{n}}r(x)e^{-i2pi fcdot x},dx\&=int _{mathbb {R} ^{n}}left(int _{mathbb {R} ^{n}}g(tau )h(x-tau ),dtau right),e^{-i2pi fcdot x},dx.end{aligned}}} Note that {displaystyle |g(tau )h(x-tau )e^{-i2pi fcdot x}|=|g(tau )h(x-tau )|} and hence by the argument above we may apply Fubini's theorem again (i.e. interchange the order of integration): {displaystyle {begin{aligned}R(f)&=int _{mathbb {R} ^{n}}g(tau )underbrace {left(int _{mathbb {R} ^{n}}h(x-tau ) e^{-i2pi fcdot x},dxright)} _{H(f) e^{-i2pi fcdot tau }},dtau \&=underbrace {left(int _{mathbb {R} ^{n}}g(tau ) e^{-i2pi fcdot tau },dtau right)} _{G(f)} H(f).end{aligned}}} This theorem also holds for the Laplace transform, the two-sided Laplace transform and, when suitably modified, for the Mellin transform and Hartley transform (see Mellin inversion theorem). It can be extended to the Fourier transform of abstract harmonic analysis defined over locally compact abelian groups.

Periodic convolution (Fourier series coefficients) Consider {displaystyle P} -periodic functions {displaystyle g_{_{P}}} and {displaystyle h_{_{P}},} which can be expressed as periodic summations: {displaystyle g_{_{P}}(x) triangleq sum _{m=-infty }^{infty }g(x-mP)}   and   {displaystyle h_{_{P}}(x) triangleq sum _{m=-infty }^{infty }h(x-mP).} In practice the non-zero portion of components {displaystyle g} and {displaystyle h} are often limited to duration {displaystyle P,} but nothing in the theorem requires that. The Fourier series coefficients are: {displaystyle {begin{aligned}G[k]&triangleq {mathcal {F}}{g_{_{P}}}[k]={frac {1}{P}}int _{P}g_{_{P}}(x)e^{-i2pi kx/P},dx,quad kin mathbb {Z} ;quad quad scriptstyle {text{integration over any interval of length }}P\H[k]&triangleq {mathcal {F}}{h_{_{P}}}[k]={frac {1}{P}}int _{P}h_{_{P}}(x)e^{-i2pi kx/P},dx,quad kin mathbb {Z} end{aligned}}} where {displaystyle {mathcal {F}}} denotes the Fourier series integral.

The pointwise product: {displaystyle g_{_{P}}(x)cdot h_{_{P}}(x)} is also {displaystyle P} -periodic, and its Fourier series coefficients are given by the discrete convolution of the {displaystyle G} and {displaystyle H} sequences: {displaystyle {mathcal {F}}{g_{_{P}}cdot h_{_{P}}}[k]={G*H}[k].} The convolution: {displaystyle {begin{aligned}{g_{_{P}}*h}(x) &triangleq int _{-infty }^{infty }g_{_{P}}(x-tau )cdot h(tau ) dtau \&equiv int _{P}g_{_{P}}(x-tau )cdot h_{_{P}}(tau ) dtau ;quad quad scriptstyle {text{integration over any interval of length }}Pend{aligned}}} is also {displaystyle P} -periodic,[A] and is called a periodic convolution. The corresponding convolution theorem is: {displaystyle {mathcal {F}}{g_{_{P}}*h}[k]= Pcdot G[k] H[k].}         (Eq.2) Derivation of Eq.2 {displaystyle {begin{aligned}{mathcal {F}}{g_{_{P}}*h}[k]&triangleq {frac {1}{P}}int _{P}left(int _{P}g_{_{P}}(tau )cdot h_{_{P}}(x-tau ) dtau right)e^{-i2pi kx/P},dx\&=int _{P}g_{_{P}}(tau )left({frac {1}{P}}int _{P}h_{_{P}}(x-tau ) e^{-i2pi kx/P}dxright),dtau \&=int _{P}g_{_{P}}(tau ) e^{-i2pi ktau /P}underbrace {left({frac {1}{P}}int _{P}h_{_{P}}(x-tau ) e^{-i2pi k(x-tau )/P}dxright)} _{H[k],quad {text{due to periodicity}}},dtau \&=underbrace {left(int _{P} g_{_{P}}(tau ) e^{-i2pi ktau /P}dtau right)} _{Pcdot G[k]} H[k].end{aligned}}} Functions of a discrete variable (sequences) By a derivation similar to Eq.1, there is an analogous theorem for sequences, such as samples of two continuous functions, where now {displaystyle {mathcal {F}}} denotes the discrete-time Fourier transform (DTFT) operator. Consider two sequences {displaystyle g[n]} and {displaystyle h[n]} with transforms {displaystyle G} and {displaystyle H} : {displaystyle {begin{aligned}G(f)&triangleq {mathcal {F}}{g}(f)=sum _{n=-infty }^{infty }g[n]cdot e^{-i2pi fn};,quad fin mathbb {R} \H(f)&triangleq {mathcal {F}}{h}(f)=sum _{n=-infty }^{infty }h[n]cdot e^{-i2pi fn};.quad fin mathbb {R} end{aligned}}} The § Discrete convolution of {displaystyle g} and {displaystyle h} is defined by: {displaystyle r[n]triangleq (g*h)[n]=sum _{m=-infty }^{infty }g[m]cdot h[n-m]=sum _{m=-infty }^{infty }g[n-m]cdot h[m].} The convolution theorem for discrete sequences is:[3][4]: p.60 (2.169)  {displaystyle R(f)={mathcal {F}}{g*h}(f)= G(f)H(f).}         (Eq.3) Periodic convolution {displaystyle G(f)} and {displaystyle H(f),} as defined above, are periodic, with a period of 1. Consider {displaystyle N} -periodic sequences {displaystyle g_{_{N}}} and {displaystyle h_{_{N}}} : {displaystyle g_{_{N}}[n] triangleq sum _{m=-infty }^{infty }g[n-mN]}   and   {displaystyle h_{_{N}}[n] triangleq sum _{m=-infty }^{infty }h[n-mN],quad nin mathbb {Z} .} These functions occur as the result of sampling {displaystyle G} and {displaystyle H} at intervals of {displaystyle 1/N} and performing an inverse discrete Fourier transform (DFT) on {displaystyle N} samples (see § Sampling the DTFT). The discrete convolution: {displaystyle {g_{_{N}}*h}[n] triangleq sum _{m=-infty }^{infty }g_{_{N}}[m]cdot h[n-m]equiv sum _{m=0}^{N-1}g_{_{N}}[m]cdot h_{_{N}}[n-m]} is also {displaystyle N} -periodic, and is called a periodic convolution. Redefining the {displaystyle {mathcal {F}}} operator as the {displaystyle N} -length DFT, the corresponding theorem is:[5][4]: p.548  {displaystyle {mathcal {F}}{g_{_{N}}*h}[k]= underbrace {{mathcal {F}}{g_{_{N}}}[k]} _{G(k/N)}cdot underbrace {{mathcal {F}}{h_{_{N}}}[k]} _{H(k/N)},quad kin mathbb {Z} .}         (Eq.4a) And therefore: {displaystyle {g_{_{N}}*h}[n]= {mathcal {F}}^{-1}{{mathcal {F}}{g_{_{N}}}cdot {mathcal {F}}{h_{_{N}}}}.}         (Eq.4b) Under the right conditions, it is possible for this N-length sequence to contain a distortion-free segment of a {displaystyle g*h} convolution. But when the non-zero portion of the {displaystyle g(n)} or {displaystyle h(n)} sequence is equal or longer than {displaystyle N,} some distortion is inevitable.  Such is the case when the {displaystyle H(k/N)} sequence is obtained by directly sampling the DTFT of the infinitely long § Discrete Hilbert transform impulse response.[B] For {displaystyle g} and {displaystyle h} sequences whose non-zero duration is less than or equal to {displaystyle N,} a final simplification is: Circular convolution {displaystyle {g_{_{N}}*h}[n]= {mathcal {F}}^{-1}{{mathcal {F}}{g}cdot {mathcal {F}}{h}}.}         (Eq.4c) This form is often used to efficiently implement numerical convolution by computer. (see § Fast convolution algorithms and § Example) As a partial reciprocal, it has been shown [6] that any linear transform that turns convolution into pointwise product is the DFT (up to a permutation of coefficients).

Derivations of Eq.4 A time-domain derivation proceeds as follows: {displaystyle {begin{aligned}{scriptstyle {rm {DFT}}}{g_{_{N}}*h}[k]&triangleq sum _{n=0}^{N-1}left(sum _{m=0}^{N-1}g_{_{N}}[m]cdot h_{_{N}}[n-m]right)e^{-i2pi kn/N}\&=sum _{m=0}^{N-1}g_{_{N}}[m]left(sum _{n=0}^{N-1}h_{_{N}}[n-m]cdot e^{-i2pi kn/N}right)\&=sum _{m=0}^{N-1}g_{_{N}}[m]cdot e^{-i2pi km/N}underbrace {left(sum _{n=0}^{N-1}h_{_{N}}[n-m]cdot e^{-i2pi k(n-m)/N}right)} _{{scriptstyle {rm {DFT}}}{h_{_{N}}}[k]quad scriptstyle {text{due to periodicity}}}\&=underbrace {left(sum _{m=0}^{N-1}g_{_{N}}[m]cdot e^{-i2pi km/N}right)} _{{scriptstyle {rm {DFT}}}{g_{_{N}}}[k]}left({scriptstyle {rm {DFT}}}{h_{_{N}}}[k]right).end{aligned}}} A frequency-domain derivation follows from § Periodic data, which indicates that the DTFTs can be written as: {displaystyle {mathcal {F}}{g_{_{N}}*h}(f)={frac {1}{N}}sum _{k=-infty }^{infty }left({scriptstyle {rm {DFT}}}{g_{_{N}}*h}[k]right)cdot delta left(f-k/Nright).}         (5a) {displaystyle {mathcal {F}}{g_{_{N}}}(f)={frac {1}{N}}sum _{k=-infty }^{infty }left({scriptstyle {rm {DFT}}}{g_{_{N}}}[k]right)cdot delta left(f-k/Nright).} The product with {displaystyle H(f)} is thereby reduced to a discrete-frequency function: {displaystyle {begin{aligned}{mathcal {F}}{g_{_{N}}*h}(f)&=G_{_{N}}(f)H(f)\&={frac {1}{N}}sum _{k=-infty }^{infty }left({scriptstyle {rm {DFT}}}{g_{_{N}}}[k]right)cdot H(f)cdot delta left(f-k/Nright)\&={frac {1}{N}}sum _{k=-infty }^{infty }left({scriptstyle {rm {DFT}}}{g_{_{N}}}[k]right)cdot H(k/N)cdot delta left(f-k/Nright)\&={frac {1}{N}}sum _{k=-infty }^{infty }left({scriptstyle {rm {DFT}}}{g_{_{N}}}[k]right)cdot left({scriptstyle {rm {DFT}}}{h_{_{N}}}[k]right)cdot delta left(f-k/Nright),quad scriptstyle {mathsf {(Eq.5b)}}end{aligned}}} where the equivalence of {displaystyle H(k/N)} and {displaystyle left({scriptstyle {rm {DFT}}}{h_{_{N}}}[k]right)} follows from § Sampling the DTFT. Therefore, the equivalence of (5a) and (5b) requires: {displaystyle {scriptstyle {rm {DFT}}}{{g_{_{N}}*h}[k]}=left({scriptstyle {rm {DFT}}}{g_{_{N}}}[k]right)cdot left({scriptstyle {rm {DFT}}}{h_{_{N}}}[k]right).} We can also verify the inverse DTFT of (5b): {displaystyle {begin{aligned}(g_{_{N}}*h)[n]&=int _{0}^{1}left({frac {1}{N}}sum _{k=-infty }^{infty }{scriptstyle {rm {DFT}}}{g_{_{N}}}[k]cdot {scriptstyle {rm {DFT}}}{h_{_{N}}}[k]cdot delta left(f-k/Nright)right)cdot e^{i2pi fn}df\&={frac {1}{N}}sum _{k=-infty }^{infty }{scriptstyle {rm {DFT}}}{g_{_{N}}}[k]cdot {scriptstyle {rm {DFT}}}{h_{_{N}}}[k]cdot underbrace {left(int _{0}^{1}delta left(f-k/Nright)cdot e^{i2pi fn}dfright)} _{{text{0, for}} k notin [0, N)}\&={frac {1}{N}}sum _{k=0}^{N-1}{bigg (}{scriptstyle {rm {DFT}}}{g_{_{N}}}[k]cdot {scriptstyle {rm {DFT}}}{h_{_{N}}}[k]{bigg )}cdot e^{i2pi {frac {n}{N}}k}\&= {scriptstyle {rm {DFT}}^{-1}}{bigg (}{scriptstyle {rm {DFT}}}{g_{_{N}}}cdot {scriptstyle {rm {DFT}}}{h_{_{N}}}{bigg )}.end{aligned}}} Convolution theorem for inverse Fourier transform There is also a convolution theorem for the inverse Fourier transform: {displaystyle {mathcal {F}}{g*h}={mathcal {F}}{g}cdot {mathcal {F}}{h}} {displaystyle {mathcal {F}}{gcdot h}={mathcal {F}}{g}*{mathcal {F}}{h}} so that {displaystyle g*h={mathcal {F}}^{-1}left{{mathcal {F}}{g}cdot {mathcal {F}}{h}right}} {displaystyle gcdot h={mathcal {F}}^{-1}left{{mathcal {F}}{g}*{mathcal {F}}{h}right}} Convolution theorem for tempered distributions The convolution theorem extends to tempered distributions. Here, {displaystyle g} is an arbitrary tempered distribution (e.g. the Dirac comb) {displaystyle {mathcal {F}}{f*g}={mathcal {F}}{f}cdot {mathcal {F}}{g}} {displaystyle {mathcal {F}}{alpha cdot g}={mathcal {F}}{alpha }*{mathcal {F}}{g}} but {displaystyle f=F{alpha }} must be "rapidly decreasing" towards {displaystyle -infty } and {displaystyle +infty } in order to guarantee the existence of both, convolution and multiplication product. Equivalently, if {displaystyle alpha =F^{-1}{f}} is a smooth "slowly growing" ordinary function, it guarantees the existence of both, multiplication and convolution product.[7][8][9] In particular, every compactly supported tempered distribution, such as the Dirac Delta, is "rapidly decreasing". Equivalently, bandlimited functions, such as the function that is constantly {displaystyle 1} are smooth "slowly growing" ordinary functions. If, for example, {displaystyle gequiv operatorname {III} } is the Dirac comb both equations yield the Poisson summation formula and if, furthermore, {displaystyle fequiv delta } is the Dirac delta then {displaystyle alpha equiv 1} is constantly one and these equations yield the Dirac comb identity.

See also Moment-generating function of a random variable Notes ^ Proof: {displaystyle {begin{aligned}int _{-infty }^{infty }g_{_{P}}(x-tau )cdot h(tau ),dtau &=sum _{k=-infty }^{infty }left[int _{x_{o}+kP}^{x_{o}+(k+1)P}g_{_{P}}(x-tau )cdot h(tau ) dtau right]quad x_{0}{text{ is an arbitrary parameter}}\&=sum _{k=-infty }^{infty }left[int _{x_{o}}^{x_{o}+P}underbrace {g_{_{P}}(x-tau -kP)} _{g_{_{P}}(x-tau ),{text{ by periodicity}}}cdot h(tau +kP) dtau right]quad {text{substituting }}tau rightarrow tau +kP\&=int _{x_{o}}^{x_{o}+P}g_{_{P}}(x-tau )cdot underbrace {left[sum _{k=-infty }^{infty }h(tau +kP)right]} _{triangleq h_{_{P}}(tau )} dtau end{aligned}}} ^ An example is the MATLAB function, hilbert(g,N). References ^ McGillem, Clare D.; Cooper, George R. (1984). Continuous and Discrete Signal and System Analysis (2 ed.). Holt, Rinehart and Winston. p. 118 (3-102). ISBN 0-03-061703-0. ^ Jump up to: a b Weisstein, Eric W. "Convolution Theorem". From MathWorld--A Wolfram Web Resource. Retrieved 8 February 2021. ^ Proakis, John G.; Manolakis, Dimitri G. (1996), Digital Signal Processing: Principles, Algorithms and Applications (3 ed.), New Jersey: Prentice-Hall International, p. 297,, ISBN 9780133942897, sAcfAQAAIAAJ ^ Jump up to: a b Oppenheim, Alan V.; Schafer, Ronald W.; Buck, John R. (1999). Discrete-time signal processing (2nd ed.). Upper Saddle River, N.J.: Prentice Hall. ISBN 0-13-754920-2.  Also available at ^ Rabiner, Lawrence R.; Gold, Bernard (1975). Theory and application of digital signal processing. Englewood Cliffs, NJ: Prentice-Hall, Inc. p. 59 (2.163). ISBN 978-0139141010. ^ Amiot, Emmanuel (2016). Music through Fourier Space. Zürich: Springer. p. 8. ISBN 978-3-319-45581-5. ^ Horváth, John (1966). Topological Vector Spaces and Distributions. Reading, MA: Addison-Wesley Publishing Company. ^ Barros-Neto, José (1973). An Introduction to the Theory of Distributions. New York, NY: Dekker. ^ Petersen, Bent E. (1983). Introduction to the Fourier Transform and Pseudo-Differential Operators. Boston, MA: Pitman Publishing. Further reading Katznelson, Yitzhak (1976), An introduction to Harmonic Analysis, Dover, ISBN 0-486-63331-4 Li, Bing; Babu, G. Jogesh (2019), "Convolution Theorem and Asymptotic Efficiency", A Graduate Course on Statistical Inference, New York: Springer, pp. 295–327, ISBN 978-1-4939-9759-6 Crutchfield, Steve (October 9, 2010), "The Joy of Convolution", Johns Hopkins University, retrieved November 19, 2010 Additional resources For a visual representation of the use of the convolution theorem in signal processing, see: Johns Hopkins University's Java-aided simulation: Categories: Theorems in Fourier analysis

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