conv

Convolution and polynomial multiplication

Syntax

w = conv(u,v)

w = conv(u,v,shape)

Description

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.

Examples

Polynomial Multiplication via Convolution

Create vectors u and v containing the coefficients of the polynomials $$x^2+1$$ 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 $$2x^3+7x^2+2x+7$$.

Vector Convolution

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.

Central Part of Convolution

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

u,v — Input vectors

vectors

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

shape — Subsection of convolution

"full" (default) | "same" | "valid"

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

"full"	Full convolution (default).
"same"	Central part of the convolution of the same size as u.
"valid"	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).

Output Arguments

w — Convolved vector

vector

Convolved vector, returned as a row or column vector.

More About

Convolution

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

$$w(k)={\displaystyle \sum _{j}u}(j)v(k-j+1).$$

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

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

The conv function supports tall arrays with the following usage notes and limitations:

• The inputs u and v must be column vectors.

• If shape is "full" (default), then only one of u or v can be a tall array.

• If shape is "same" or "valid", then v cannot be a tall array.

For more information, see Tall Arrays.

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• If used, the shape argument must be a constant at code generation time.

• For input arguments u and v:

  • You must specify the input vector as a fixed-size or variable-length vector at code generation time. Either the first or the second dimension of the vector can be variable size. All other dimensions must have a fixed size of 1.

• Input vectors u and v must have the same orientation.

GPU Code Generation

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

Refer to the usage notes and limitations in the C/C++ Code Generation section. The same usage notes and limitations apply to GPU code generation.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

The conv function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The conv function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

The conv function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a

R2026a: For a full convolution, the output is a row vector unless both input vectors are column vectors

When you compute a full convolution using w = conv(u,v) or w = conv(u,v,"full"), the output w is a row vector unless both input vectors u and v are column vectors.

For example, if you convolve a row vector and a column vector, the output is a row vector.

u = [1 0 1];
v = [2; 7; 4];
w = conv(u,v)

w =
     2     7     6     7     4

If you convolve two column vectors, the output is a column vector.

u = [1; 0; 1];
v = [2; 7; 4];
w = conv(u,v)

w =
     2
     7
     6
     7
     4

For comparison, when computing a full convolution in previous releases, the conv function returned either a row or column vector depending on the orientations and lengths of u and v.

For example, in previous releases, conv returned the full convolution of a 1-by-3 row vector and a 3-by-1 column vector as a 5-by-1 column vector.

u = [1 0 1];
v = [2; 7; 4];
w = conv(u,v)

w =
     2
     7
     6
     7
     4

However, conv returned the full convolution of a 1-by-3 row vector and a 2-by-1 column vector as a 1-by-4 row vector.

u = [1 0 1];
v = [2; 7];
w = conv(u,v)

w =
     2     7     2     7

There are no changes to the output of w = conv(u,v,"same") or w = conv(u,v,"valid"), where the output still follows the orientation of the first input vector u.