What does this notation mean $g(x) = S\{\delta(t) \}(x)$ mean, where $S$ is a "system", $\delta(t)$ is the impulse function and $g(x)$ the output function of the system? I am really not familiar with signals and signal processing. I know the definition of the impulse function

$$ \delta(t) = \begin{cases} 1, \; t = 0,\\ 0, \; t \neq 0 \end{cases} $$

I understood that a system is a function that gets as input a function and returns another function (so it should be a functional). However, I do not get this notation $g(x) = S\{\delta(t) \}(x)$? For example, why the parentheses $\{ \}$? Why do we apply the $\delta(t)$ inside $S\{ \cdot \}$? What is the precedence of those function applications? How should I read that formula? Why do we even use an impulse here, what is the intuition behind this?

In my case, also have that

$$g(x) = S\{\delta(t) \}(x) = (A\delta(t))(x) = A_{x, 0}$$

Why are the notations $S\{\delta(t) \}(x)$ and $(A\delta(t))(x)$ equivalent? What is this $A_{x, 0}$? $A$ should be a Toeplitz matrix, but I don't understand why. How should this matrix $A$ look like?

$A_{x, 0}$ should represent coefficient associated to instant $0$ for computing the output at instant $x$. I also don't get this, even though I think we are trying to convert a continuous signal to a discrete one.

You can assume that $S$ is a linear shift-invariant system.

  • $\begingroup$ Do not hesitate to validate an answer, as appropriate $\endgroup$ – Laurent Duval Dec 20 '19 at 22:27

My understanding is that a system $S$ is a notation for a generic object, which takes instances depending on its arguments, given between $\{\cdot\}$. Here, $S\{\delta(t)\}$ instantiates it as its effect on the zero-centered impulse response, with free variable $t$, taken on variable $x$.

Now, if this system $S$ is linear and shift-invariant, the operator given by $S\{\delta(t)\}$ defines $S$ it completely.

If the system applies on discretized inputs (that we can call $x$), it can be rewritten as a linear operator, represented by an infinite Toeplitz matrix. Each row implements the (linear) convolution, and the Toeplitz structure emulates the shift-invariance. I suggest to read Toeplitz and circulant matrices by R. Gray. The $0$ in $A_{x,0}$ may refer to the way an infinite matrix is matched with a signal centered at $0$, but I still wonder if this notation is clear.

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  • $\begingroup$ "which takes instances", instances of what? $\endgroup$ – nbro Mar 25 '19 at 20:37
  • $\begingroup$ I am refering to instantiation, as a way to create an object from a model, where the object inherits properties from the model. In other words, an instance of a system is a mathematical object that reproduces the effect of the abstract system $\endgroup$ – Laurent Duval Mar 25 '19 at 20:57
  • $\begingroup$ What do you mean by "free variable $t$, taken on variable $x$"? Why there are two variables? $\endgroup$ – nbro Mar 26 '19 at 11:26
  • $\begingroup$ The work you mention (what is the source?) seems to be very careful about notations. Since I don't know how far you want to go, consider that $t$ does not play a great role. It is a bit like in integrals: $\int_0^x f(t)dt$ $\endgroup$ – Laurent Duval Mar 26 '19 at 14:13

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