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Let's suppose I have a discrete signal $x[n]$ with a certain frequency $f_x$.

Now, due to various unimportant reason, I want to obtain a new signal $y[m]$ with a new frequency $f_y = \frac{f_x}{k}$, with, for example $k = 10$.

Now, there are various ways to do it, I am for the moment interested in two:

Decimation:

$y_d[m] = x[m* k + (k - 1)]$

Mean:

$y_a[m] = \frac{1}{k}\sum_{i=m*k}^{m*k+k-1}{m[i]}$

What are the differences between the two, from a theoretical standpoint? What effect on the spectrum of $y_d$ and $y_a$?

I have limited knowledge of signal processing, so sorry for if something confuses you. A brief explanation with some references for further reading would be perfect.

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What you are calling decimation is in fact downsampling: keeping one of every $k$ samples from the original signal.

Proper decimation involves a digital low-pass filter in order to eliminate all components above $\pi/k$ (or $f_s/(2k)$ in continuous time) since these components cannot be represented at the new lower sampling frequency (and will result in aliasing).

After low-pass filtering, you downsample your signal keeping one out of every $k$ samples.

The second method you propose is actually a low-pass filter followed by downsampling. But in this case, the low-pass filter you use is a moving-average filter, which is very simple, but it is not a very good low-pass filter. Gain in the pass band is very variable, and attenuation in the stop band is poor.

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