# understanding discrete-time convolution

I'm trying to understand the discrete-time convolution for LTIs and its graphical representation. standard explanations (like: this one) start with the idea of decomposing an input signal $x[t]$ into a sum of unit impulses. We have unit impulse function $\delta$:

$\delta[n] = 1\ if\ n = 0, 0\ if n \neq 0$

then $x[t]$ can be represented as:

$x[t] = \sum_{k = -\infty}^{+\infty}x[k]\delta[t-k]$

so the input signal at $t$, $x[t]$, can be represented this way because $\delta[t-k]$ is going to be 1 only if $t = k$, and 0 elsewhere. this makes great sense.

the confusion for me begins in Transparency 4.2 of above link. the left hand side makes perfect sense, but what does the right represent? how does one unit impulse generate a sequence of responses? what does $x[0]h[n]$ represent here? How do you go from this to Transparency 4.9 where $h$ is used instead of $\delta$?

put differently: can someone explain the intuition behind $h$ (the response of a an LTI system) being reflected and time shifted? in the above trick for representing simply the input signal $x[t]$, there's no time shifting really; $\delta[t-k]$ seems like just a mathematical trick for getting a $1$ in the right place and zero elsewhere; it's not intuitive for me to think of it as time shifted.

• Please ask the moderators to move this dsp.SE where you will find lots of answers (for example, this one) spelling out some parts of what you want to know in great detail. You can contact the moderators by clicking on the "flag" link below your question – Dilip Sarwate Sep 7 '14 at 21:59