# Deriving time-scaling property for Fourier Series

thanks for taking the time to help with this problem!

I have to prove the time-scaling property:

$$x_{(m)}[n] = \begin{cases} x[n/m], & n=0,\pm m, \pm 2m,...\\ 0, & otherwise \end{cases}$$

First, for part (a), Show that $$x_{(m)}[n]$$ has a period of $$mN$$

Then, for part (b), Show that if $$\\x[n]=v[n]+w[n]\\$$ then$$\\x_{(m)}[n]=v_{(m)}[n]+w_{(m)}[n]$$

There's a bit more to the problem after this, but I'm mostly concerned with these first two pieces. I know that for a periodic function $$x[n]=x[n+N]$$

Am I supposed to somehow prove that $$x[n/m+mN]=x[n/m]$$ using the Discrete Fourier Series analysis equation? What is a rigorous way to prove this.

For part (b), I'm unsure of how to prove the linearity in a way that doesn't seem trivial.

Thanks!

• Hint: you are supposed to show that xm[k] = xm[k+mN] for all k – Hilmar Nov 3 '14 at 23:47
• I understand this, but I'm unsure how to formally show that it is true. – jephex Nov 4 '14 at 0:42