How does this periodic signal look like?

Question

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?

Answer 1

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.