0
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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, $S\{\delta(t)\}$ instanciates 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 stll wonder if this notation is clear.

$\endgroup$
  • $\begingroup$ "which takes instances", instances of what? $\endgroup$ – nbro Mar 25 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 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 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 at 14:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.