# Physical meaning of convolution

Suppose we have 2 signals x(t) and y(t) with $$x(t)=\delta(t)$$ If $$x(t)$$ is the input to a linear and time invariant system $$S$$, lets say we get a X(t) as the output:

$$x(t) \overset{S}{\rightarrow} X(t)$$

If we have $$y(t)$$ as an input to the same system $$S$$, then we can predict the output of that system without knowing anything more about the system:

$$Y(t) = \int_{-\infty}^{\infty}X(\tau)y(t-\tau) d\tau$$

Which are different examples of convolution of 2 signals?

Let me make a couple of suggestions:

1. It is common practice to write time-domain signals such as your $$x(t)$$ with lower case letters, and frequency-domain signals with Upper case letters:
• If $$x(t)$$ is your time-domain signal, then $$X(f)$$ would be your frequency-domain signal (note you can use any name as the dependent variable, you'll sometimes see $$x[n]$$, $$X[\omega]$$, $$X[k]$$, etc.
The point is lower-case = time, UPPER-CASE = frequency.
$$X(t)$$ and $$Y(t)$$, while not wrong per-se, just look weird.
2. When talking about LTI (Linear Time Invariant) systems, it is also common practice to name the input $$x$$ and output $$y$$. Don't ask me why, it's just what you'll see over and over in textbooks and articles.
• For your example, I'd write: $$x(t) \overset{S}\rightarrow y(t)$$

With that out of the way, if you input an impulse $$\delta(t)$$ to a LTI system, the output is called the impulse response, and is usually (again, most common) written $$h(t)$$.

$$\delta(t) \overset{S}\rightarrow h(t)$$

This impulse response completely characterizes the LTI system $$S$$. What this means is that once you know the impulse response, you can predict the output of $$S$$ to any input, through convolution.

Consider a Linear Time Invariant system $$S$$, with impulse response $$h(t)$$. For an input $$x(t)$$, $$S$$ will output $$y(t)$$ as the result of the convolution of $$x(t)$$ with $$h(t)$$ (note you had a slight error in yours; it's ($$x(\tau)$$, not $$x(t)$$):
$$y(t) = x(t) * h(t) = \int_{-\infty}^{\infty}x(\tau)h(t - \tau)d\tau$$