# toeplitz

Toeplitz matrix

## 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 (Signal Processing Toolbox) 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

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