In mathematics , the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals ) is the product of their Four...
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In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the 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.
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:
()
Applying the inverse Fourier transform produces the corollary:
()
The theorem also generally applies to multi-dimensional functions.
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 :
and as defined above, are periodic, with a period of 1. Consider -periodic sequences and :
and
These functions occur as the result of sampling and at intervals of and performing an inverse discrete Fourier transform (DFT) on samples (see
is also -periodic, and is called a periodic convolution. Redefining the operator as the -length DFT, the corresponding theorem is:
()
And therefore:
()
Under the right conditions, it is possible for this -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 impulse response.
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
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
There is also a convolution theorem for the inverse Fourier transform:
Here, "" represents the Hadamard product, and "" represents a convolution between the two matrices.
so that
Convolution theorem for tempered distributions
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.
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.
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