# Documentation

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# toeplitz

Toeplitz matrix

## Syntax

``T = toeplitz(c,r)``
``T = toeplitz(r)``

## Description

example

````T = toeplitz(c,r)` returns a nonsymmetric Toeplitz matrix with `c` as its first column and `r` as its first row. If the first elements of `c` and `r` differ, `toeplitz` issues a warning and uses the column element for the diagonal.```

example

````T = toeplitz(r)` returns the symmetric Toeplitz matrix where:If `r` is a real vector, then `r` defines the first row of the matrix.If `r` is a complex vector with a real first element, then `r` defines the first row and `r'` defines the first column.If the first element of `r` is complex, the Toeplitz matrix is Hermitian off the main diagonal, which means ${\text{T}}_{i,j}=\mathrm{conj}{\text{(T}}_{j,i}\right)$ for $i\ne j$. The elements of the main diagonal are set to `r(1)`.```

## Examples

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```r = [1 2 3]; toeplitz(r)```
```ans = 3×3 1 2 3 2 1 2 3 2 1 ```

Create a nonsymmetric Toeplitz matrix with a specified column and row vector. Because the first elements of the column and row vectors do not match, `toeplitz` issues a warning and uses the column for the diagonal element.

```c = [1 2 3 4]; r = [4 5 6]; toeplitz(c,r)```
```Warning: First element of input column does not match first element of input row. Column wins diagonal conflict. ```
```ans = 4×3 1 5 6 2 1 5 3 2 1 4 3 2 ```

Create a Toeplitz matrix with complex row and column vectors.

```c = [1+3i 2-5i -1+3i]; r = [1+3i 3-1i -1-2i]; T = toeplitz(c,r)```
```T = 3×3 complex 1.0000 + 3.0000i 3.0000 - 1.0000i -1.0000 - 2.0000i 2.0000 - 5.0000i 1.0000 + 3.0000i 3.0000 - 1.0000i -1.0000 + 3.0000i 2.0000 - 5.0000i 1.0000 + 3.0000i ```

You can create circulant matrices using `toeplitz`. Circulant matrices are used in applications such as circular convolution.

Create a circulant matrix from vector `v` using toeplitz.

```v = [9 1 3 2]; toeplitz([v(1) fliplr(v(2:end))], v)```
```ans = 4×4 9 1 3 2 2 9 1 3 3 2 9 1 1 3 2 9 ```

Perform discrete-time circular convolution by using `toeplitz` to form the circulant matrix for convolution.

Define the periodic input `x` and the system response `h`.

```x = [1 8 3 2 5]; h = [3 5 2 4 1];```

Form the column vector `c` to create a circulant matrix where `length(c) = length(h)`.

`c = [x(1) fliplr(x(end-length(h)+2:end))]`
```c = 1×5 1 5 2 3 8 ```

Form the row vector `r` from `x`.

`r = x;`

Form the convolution matrix `xConv` using `toeplitz`. Find the convolution using `h*xConv`.

`xConv = toeplitz(c,r)`
```xConv = 5×5 1 8 3 2 5 5 1 8 3 2 2 5 1 8 3 3 2 5 1 8 8 3 2 5 1 ```
`h*xConv`
```ans = 1×5 52 50 73 46 64 ```

If you have the Signal Processing Toolbox™, you can use the `cconv` function to find the circular convolution.

Perform discrete-time convolution by using `toeplitz` to form the arrays for convolution.

Define the input `x` and system response `h`.

```x = [1 8 3 2 5]; h = [3 5 2];```

Form `r` by padding `x` with zeros. The length of `r` is the convolution length `x + h - 1`.

`r = [x zeros(1,length(h)-1)]`
```r = 1×7 1 8 3 2 5 0 0 ```

Form the column vector `c`. Set the first element to `x(1)` because the column determines the diagonal. Pad `c` because `length(c)` must equal `length(h)` for convolution.

`c = [x(1) zeros(1,length(h)-1)]`
```c = 1×3 1 0 0 ```

Form the convolution matrix `xConv` using `toeplitz`. Then, find the convolution using `h*xConv`.

`xConv = toeplitz(c,r)`
```xConv = 3×7 1 8 3 2 5 0 0 0 1 8 3 2 5 0 0 0 1 8 3 2 5 ```
`h*xConv`
```ans = 1×7 3 29 51 37 31 29 10 ```

Check that the result is correct using `conv`.

`conv(x,h)`
```ans = 1×7 3 29 51 37 31 29 10 ```

## Input Arguments

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Column of Toeplitz matrix, specified as a scalar or vector. If the first elements of `c` and `r` differ, `toeplitz` uses the column element for the diagonal.

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

Row of Toeplitz matrix, specified as a scalar or vector. If the first elements of `c` and `r` differ, then `toeplitz` uses the column element for the diagonal.

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

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### Toeplitz Matrix

A Toeplitz matrix is a diagonal-constant matrix, which means all elements along a diagonal have the same value. For a Toeplitz matrix A, we have Ai,j = ai–j which results in the form

`$A=\left[\begin{array}{cccccc}{a}_{0}& {a}_{-1}& {a}_{-2}& \cdots & \cdots & {a}_{1-n}\\ {a}_{1}& {a}_{0}& {a}_{-1}& \ddots & \ddots & ⋮\\ {a}_{2}& {a}_{1}& {a}_{0}& \ddots & \ddots & ⋮\\ ⋮& \ddots & \ddots & \ddots & \ddots & {a}_{-2}\\ ⋮& \ddots & \ddots & \ddots & {a}_{0}& {a}_{-1}\\ {a}_{n-1}& \cdots & \cdots & {a}_{2}& {a}_{1}& {a}_{0}\end{array}\right].$`