I am having trouble understanding why the system matrix of an LTI system is Toeplitz. I am following an Edx online course by Professor Richard Baraniuk of Rice University, named discrete-time signals and systems. In the course, the instructor showed that given a system matrix $H$, $H(n,m) = H(n'+q,m'+q)$ where $n'=n-q$ and $m'=m-q$, which is obvious. But, from that, he claimed the general Toeplitz structure $H(n,m)=H(n+q,m+q)$ and this is where I am getting lost.
$\begingroup$
$\endgroup$
4
-
$\begingroup$ Take a look at this $\endgroup$– Rodrigo de AzevedoCommented Feb 26, 2023 at 12:38
-
$\begingroup$ Take a look at the first few pages of Gray's Toeplitz and Circulant Matrices $\endgroup$– Rodrigo de AzevedoCommented Feb 26, 2023 at 12:53
-
$\begingroup$ Related $\endgroup$– Rodrigo de AzevedoCommented Feb 26, 2023 at 12:54
-
$\begingroup$ Related $\endgroup$– Rodrigo de AzevedoCommented Feb 26, 2023 at 12:55
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
LTI systems are related to Toeplitz matrices via convolution: The output of an LTI system is the convolution of the input with the system impulse response.
Assuming matrix equations are understood, observe how time domain convolution can be written as a matrix product, applicable to finite duration impulse responses. The resulting matrix $A$ with each column as a shifted copy of $r[n]$ is a Toeplitz matrix.
answered Feb 26, 2023 at 12:43
-
$\begingroup$ Could someone post a link to the definition of "System matrix" in this context? I would have taken the term to mean the state transition matrix, which is clearly not the case here. I mean -- yes, it's the convolution operator, in matrix form, but it would still be nice to see the context in which it is defined. $\endgroup$ Commented Feb 26, 2023 at 16:14
-
$\begingroup$ Thanks, I understand that, but what is not convincing to me is why the professor claimed H(n, m) = H(n+q, m+q) from the fact that H(n'+q, m'+q) = H(n, m) where again n'=n-q and m'=m-q. $\endgroup$ Commented Feb 27, 2023 at 3:45