toeplitz
Toeplitz matrix
Description
T = toeplitz(
returns
a nonsymmetric Toeplitz
matrix with c
,r
)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.
T = toeplitz(
returns
the symmetric Toeplitz matrix where:r
)
If
r
is a real vector, thenr
defines the first row of the matrix.If
r
is a complex vector with a real first element, thenr
defines the first row andr'
defines the first column.If the first element of
r
is complex, the Toeplitz matrix is Hermitian off the main diagonal, which means for . The elements of the main diagonal are set tor(1)
.
Examples
Create Symmetric Toeplitz Matrix
Create Nonsymmetric Toeplitz Matrix
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
Create Circulant Matrices Using toeplitz
Function
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.
Discrete-Time Convolution Using Toeplitz
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
c
— Column of Toeplitz matrix
scalar | vector
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
r
— Row of Toeplitz matrix
scalar | vector
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
More About
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
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® 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™.
The toeplitz
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™.
This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).
Version History
Introduced before R2006a
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