# conv

Convolution and polynomial multiplication

## Description

`w = conv(`

returns
the convolution of
vectors `u,v`

)`u`

and `v`

. If `u`

and `v`

are
vectors of polynomial coefficients, convolving them is equivalent
to multiplying the two polynomials.

## 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 $$2{x}^{3}+7{x}^{2}+2x+7$$.

### 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 |

`'valid'` | Only those parts of the convolution that are computed
without the zero-padded edges. Using this option, |

## 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 `k`

th
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)

## Extended Capabilities

### Tall Arrays

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

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™.

For information about C/C++ code generation limitations, see Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).

### GPU Code Generation

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

### Thread-Based Environment

Run code in the background using MATLAB® `backgroundPool`

or accelerate code with Parallel Computing Toolbox™ `ThreadPool`

.

This 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™.

This function fully supports GPU arrays. 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™.

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

**Introduced before R2006a**

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