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How to prove the following properties of DT unit impulse function. If anyone got link to a proof please mention it. I have search on the web. But only found the properties, not a proper method of proof.

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Thank you.

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  • $\begingroup$ The expressions look like steps (1) and (2) to prove (3): that the impulse is the neutral of the convolution. It doesn't seem to be 3 independent "properties" that need a proof. Please include the original problem statement and your progress so far. $\endgroup$ – Juancho Oct 18 '17 at 14:15
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Hint : Use definition of $\sigma[n]$

consider 2 cases $$m=n$$ $$m\ne n$$

also, use definition of convolution

as an aside, it took me a while to do this but a good habit is to use $[n]$ (square brackets) for discrete time index, and $(t)$ (curved brackets) for continuous time

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Remember the discrete delta function:

$$ \begin{equation} \delta[n] = 1,n=0 \\ \hspace{1.5cm} =0, otherwise \end{equation} $$

The following is valid, because if $m\neq n$ both the sides evaluate to $0$ $$x[m]\delta[n-m] = x[n]\delta[n-m]$$

Extending the logic to convolution, when you have $\sum_{n=-\infty}^{\infty} x[n]\delta[m-n]$, all terms except for $m=n$ vanish because of the delta function. So you will be left with $x[m]$

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