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.

Functions of a continuous variable Edit

Consider two functions   and   with Fourier transforms   and  :

where   denotes the Fourier transform operator. The transform may be normalized in other ways, in which case constant scaling factors (typically   or  ) will appear in the convolution theorem below. The convolution of   and   is defined by:

In this context the asterisk denotes convolution, instead of standard multiplication. The tensor product symbol   is sometimes used instead.

The convolution theorem states that:[1][2]: eq.8 







Applying the inverse Fourier transform  , produces the corollary:[2]: eqs.7, 10 







where   denotes point-wise multiplication

The theorem also generally applies to multi-dimensional functions.


Consider functions   in Lp-space  , with Fourier transforms  :

where   indicates the inner product of  :       and    

The convolution of   and   is defined by:




Hence by Fubini's theorem we have that   so its Fourier transform   is defined by the integral formula:


Note that   and hence by the argument above we may apply Fubini's theorem again (i.e. interchange the order of integration):


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) Edit

Consider  -periodic functions   and   which can be expressed as periodic summations:


In practice the non-zero portion of components   and   are often limited to duration   but nothing in the theorem requires that. The Fourier series coefficients are:

where   denotes the Fourier series integral.
  • The pointwise product:   is also  -periodic, and its Fourier series coefficients are given by the discrete convolution of the   and   sequences:
  • The convolution:
    is also  -periodic,[A] and is called a periodic convolution. The corresponding convolution theorem is:






Derivation of Eq.2


Functions of a discrete variable (sequences) Edit

By a derivation similar to Eq.1, there is an analogous theorem for sequences, such as samples of two continuous functions, where now   denotes the discrete-time Fourier transform (DTFT) operator. Consider two sequences   and   with transforms   and  :


The § Discrete convolution of   and   is defined by:


The convolution theorem for discrete sequences is:[3][4]: p. 60 (2.169) 







Periodic convolution Edit

  and   as defined above, are periodic, with a period of 1. Consider  -periodic sequences   and  :


These functions occur as the result of sampling   and   at intervals of   and performing an inverse discrete Fourier transform (DFT) on   samples (see § Sampling the DTFT). The discrete convolution:


is also  -periodic, and is called a periodic convolution. Redefining the   operator as the  -length DFT, the corresponding theorem is:[5][4]: p. 548 







And therefore:







Under the right conditions, it is possible for this N-length sequence to contain a distortion-free segment of a   convolution. But when the non-zero portion of the   or   sequence is equal or longer than   some distortion is inevitable.  Such is the case when the   sequence is obtained by directly sampling the DTFT of the infinitely long § Discrete Hilbert transform impulse response.[B]

For   and   sequences whose non-zero duration is less than or equal to   a final simplification is:

Circular convolution






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).


A time-domain derivation proceeds as follows:


A frequency-domain derivation follows from § Periodic data, which indicates that the DTFTs can be written as:








The product with   is thereby reduced to a discrete-frequency function:

where the equivalence of   and   follows from § Sampling the DTFT. Therefore, the equivalence of (5a) and (5b) requires:


We can also verify the inverse DTFT of (5b):


Convolution theorem for inverse Fourier transform Edit

Note that in the example below " " represents the Hadamard product, and " " represents a convolution between the two matrices. There is also a convolution theorem for the inverse Fourier transform

so that

Convolution theorem for tempered distributions Edit

The convolution theorem extends to tempered distributions. Here,   is an arbitrary tempered distribution

but   must be "rapidly decreasing" towards   and   in order to guarantee the existence of both, convolution and multiplication product. Equivalently, if   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   are smooth "slowly growing" ordinary functions. If, for example,   is the Dirac comb both equations yield the Poisson summation formula and if, furthermore,   is the Dirac delta then   is constantly one and these equations yield the Dirac comb identity.

See also Edit

Notes Edit

  1. ^ Proof:
  2. ^ An example is the MATLAB function, hilbert(g,N).

References Edit

  1. ^ 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.
  2. ^ a b Weisstein, Eric W. "Convolution Theorem". From MathWorld--A Wolfram Web Resource. Retrieved 8 February 2021.
  3. ^ 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
  4. ^ 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.
  5. ^ 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.
  6. ^ Amiot, Emmanuel (2016). Music through Fourier Space. Computational Music Science. Zürich: Springer. p. 8. doi:10.1007/978-3-319-45581-5. ISBN 978-3-319-45581-5. S2CID 6224021.
  7. ^ Horváth, John (1966). Topological Vector Spaces and Distributions. Reading, MA: Addison-Wesley Publishing Company.
  8. ^ Barros-Neto, José (1973). An Introduction to the Theory of Distributions. New York, NY: Dekker.
  9. ^ Petersen, Bent E. (1983). Introduction to the Fourier Transform and Pseudo-Differential Operators. Boston, MA: Pitman Publishing.

Further reading Edit

  • 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 Edit

For a visual representation of the use of the convolution theorem in signal processing, see: