# Why is $\delta[an]=\frac 1 a \delta[n]$ in discrete time?

Why is $$\delta[an]=\frac 1 a \delta[n]$$ in discrete time? Prove.

Hi, I want to prove it but I don’t know on what facts to rely. In continuous time we have integral and dt in that integral that turns to be dt/a for continuous time delta function. But in discrete time, we don’t have dt. So how to prove it then?

• Where did you find that "identity"? – Matt L. Apr 26 at 19:39
• – Vitali Pom Apr 26 at 19:39
• This is a property of the Dirac distribution and follows from a substitution of the integration variable. It does not have an equivalent for the Kronecker unit impulse. – Jazzmaniac Apr 26 at 19:50
• Thanks! Marking as correct. – Vitali Pom Apr 26 at 19:52

It's actually good that you couldn't prove it because the claim is wrong. Note that in discrete time we have $$\delta[n]=1$$ for $$n=0$$ and $$\delta[n]=0$$ for $$n\neq 0$$. So if $$a\neq 0$$, you simply get $$\delta[an]=\delta[n]$$ (note that $$a$$ must be an integer for this to make sense). If $$a=0$$ then $$\delta[an]=1$$ (for all $$n$$).