# How are LTI systems related to Toeplitz matrices?

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.

• Take a look at this Commented Feb 26, 2023 at 12:38
• Take a look at the first few pages of Gray's Toeplitz and Circulant Matrices Commented Feb 26, 2023 at 12:53
• Related Commented Feb 26, 2023 at 12:54
• Related Commented Feb 26, 2023 at 12:55

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.