# What does this notation $g(x) = S\{\delta(t) \}(x)$ mean?

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.

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.
• 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: $\int_0^x f(t)dt$ – Laurent Duval Mar 26 '19 at 14:13