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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?

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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.

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  • $\begingroup$ 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. $\endgroup$
    – user58547
    Jul 31 at 7:09
  • $\begingroup$ Reading a bit more, it's relatively simple to calculate energy of $y$, which is $\infty$ and power of $y$, which is $14/5=2.8$. $\endgroup$
    – user58547
    Jul 31 at 8:08
  • $\begingroup$ sounds about right $\endgroup$
    – Hilmar
    Aug 1 at 1:23

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