# conv

Convolution and polynomial multiplication

## Syntax

``w = conv(u,v)``
``w = conv(u,v,shape)``

## Description

example

````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.```

example

````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

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

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'` 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)`.

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### 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

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

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)```