# Why do we assume the matrix of impulse responses to be a Toeplitz matrix during deconvolution

• $y(n)$ = output signal

• $x(n)$ = input signal

• $\mathbf H$ = system response as a toeplitz matrix

$$\mathbf H = \begin{bmatrix}h(0)&&&\\h(1)&h(0)&&\\h(2)&h(1)&h(0)&\\\vdots&&&\ddots\end{bmatrix}$$

I understand that $\mathbf H$, with its transposable (orthonormal) properties, makes it easy achieve $x(n)$ when we only know $y(n)$ during deconvolution. But how did we just assume $\mathbf H$ to be of this shape. Why not some other matrix shape when doing deconvolution? Is there a certain reason for this?

To apply de-convolution, you're assuming that $y = h * x + \epsilon$ is a reasonable model, where $*$ denotes convolution; i.e. you convolve some filter $h$ with a signal $x$ and add some noise $\epsilon$.
Now, assume you have $n$ samples of $y$. Then, write those samples of $y$ in terms of samples of $x$. When you write the convolution as a matrix, you get precisely $y=H x$ as you've written above.