# How does this periodic signal look like?

I'm very new to DSP, and I'm unsure about finding how a signal $$y[n]$$ would look like. The following is given:

1. Assume we have a finite support signal $$x[n]$$ which has the values $$1, 2, 3$$ for $$n = 1, 2, 3$$ and $$0$$ otherwise.

2. Now we have the periodic repetition of this signal, call it $$y[n] = \sum_{k=-\infty}^\infty x[n+5k]$$.

To understand how the periodic signal would look like, I'd like to write down a few samples and plot them.

First, I wrote down the values of $$x[n]$$ like so:

n x[n]
... ...
0 0
1 1
2 2
3 3
4 0
... ...

Now for $$y[n]$$, I tried the same:

n y[n]
... ...
0 $$\sum_{k=-\infty}^\infty x[0+5k] = 0$$
1 $$\sum_{k=-\infty}^\infty x[1+5k] = 1$$
2 $$\sum_{k=-\infty}^\infty x[2+5k] = 2$$
3 $$\sum_{k=-\infty}^\infty x[3+5k] = 3$$
4 $$\sum_{k=-\infty}^\infty x[4+5k] = 0$$
5 $$\sum_{k=-\infty}^\infty x[5+5k] = 0$$
6 $$\sum_{k=-\infty}^\infty x[6+5k] = 1$$
7 $$\sum_{k=-\infty}^\infty x[7+5k] = 2$$
8 $$\sum_{k=-\infty}^\infty x[8+5k] = 3$$
9 $$\sum_{k=-\infty}^\infty x[9+5k] = 0$$
... ...

Would the above list of samples for $$y[n]$$ be correct?

## 1 Answer

Would the above list of samples for y[n] be correct?

Yes

This is a standard way of turning a finite signal into a periodic signal. The $$\sum$$ means "infinite repetition" and $$5k$$ is the spacing between repetition. Since it's an infinite sum a spacing of $$+5k$$ gives the same result as a spacing of $$-5k$$ and you'll see both variants being used.

Since your signal is shorter than the spacing there is no overlap and you get just get repetitions of [1 2 3] with a spacing of 5.

If you want to learn a bit more, try it with $$2k$$ instead of $$5k$$ and see what you get.

• Thanks for validating and also the clarifications. I have a bit of a math background, but I wasn't sure if I apply this correctly. I will try to calculate energy and power of this signal, I might open a separate question for that if I get stuck. Jul 31 '21 at 7:09
• Reading a bit more, it's relatively simple to calculate energy of $y$, which is $\infty$ and power of $y$, which is $14/5=2.8$. Jul 31 '21 at 8:08
• sounds about right Aug 1 '21 at 1:23