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Convolution and polynomial multiplication



w = conv(u,v) returns the convolution of vectors u and v. If u and v are vectors of polynomial coefficients, convolving them is equivalent to multiplying the two polynomials.


w = conv(u,v,shape) returns a subsection of the convolution, as specified by shape. For example, conv(u,v,'same') returns only the central part of the convolution, the same size as u, and conv(u,v,'valid') returns only the part of the convolution computed without the zero-padded edges.


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Create vectors u and v containing the coefficients of the polynomials x2+1 and 2x+7.

u = [1 0 1];
v = [2 7];

Use convolution to multiply the polynomials.

w = conv(u,v)
w = 1×4

     2     7     2     7

w contains the polynomial coefficients for 2x3+7x2+2x+7.

Create two vectors and convolve them.

u = [1 1 1];
v = [1 1 0 0 0 1 1];
w = conv(u,v)
w = 1×9

     1     2     2     1     0     1     2     2     1

The length of w is length(u)+length(v)-1, which in this example is 9.

Create two vectors. Find the central part of the convolution of u and v that is the same size as u.

u = [-1 2 3 -2 0 1 2];
v = [2 4 -1 1];
w = conv(u,v,'same')
w = 1×7

    15     5    -9     7     6     7    -1

w has a length of 7. The full convolution would be of length length(u)+length(v)-1, which in this example would be 10.

Input Arguments

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Input vectors, specified as either row or column vectors. The vectors u and v can be different lengths or data types.

When u or v are of type single, then the output is of type single. Otherwise, conv converts inputs to type double and returns type double.

Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
Complex Number Support: Yes

Subsection of the convolution, specified as 'full', 'same', or 'valid'.


Full convolution (default).


Central part of the convolution of the same size as u.


Only those parts of the convolution that are computed without the zero-padded edges. Using this option, length(w) is max(length(u)-length(v)+1,0), except when length(v) is zero. If length(v) = 0, then length(w) = length(u).

More About

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The convolution of two vectors, u and v, represents the area of overlap under the points as v slides across u. Algebraically, convolution is the same operation as multiplying polynomials whose coefficients are the elements of u and v.

Let m = length(u) and n = length(v). Then w is the vector of length m+n-1 whose kth element is


The sum is over all the values of j that lead to legal subscripts for u(j) and v(k-j+1), specifically j = max(1,k+1-n):1:min(k,m). When m = n, this gives

w(1) = u(1)*v(1)
w(2) = u(1)*v(2)+u(2)*v(1)
w(3) = u(1)*v(3)+u(2)*v(2)+u(3)*v(1)
w(n) = u(1)*v(n)+u(2)*v(n-1)+ ... +u(n)*v(1)
w(2*n-1) = u(n)*v(n)

Using this definition, conv calculates the direct convolution of two vectors, rather than the FFT-based convolution.

Extended Capabilities

GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Version History

Introduced before R2006a

See Also

| | | | (Signal Processing Toolbox) |