Sampling discrete functions

Question

I have the following signal: $$x[n] = \delta[n+1]+\delta[n]+\delta[n-4]$$ $$y[n] = 2x[2n-1] + x[2n+2]$$ Now, let's say I decompose $y[n]$ into $y_1[n] = x[2n-1]$ and $y_2[n] = x[2n+2]$. In this situation, why is it that $y_1[n]$ will only sample for odd values of $n$ and $y_2[n]$ will only sample for even values of $n$. It doesn't seem intuitive to only sample odd values of $n$. I can see that $(2n-1)$ will be odd for all integer values of $n$. However, I don't see why the constraint is applied just purely to $n$.

Answer 1

There is a confusion about which $n$ is meant.

$y_1$ samples $x$ at indexes $2n-1$, which are indeed odd numbers, so that the $n$ of $x[n]$ is odd. But the $n$ of $y[n]$ is unrestricted.

You can avoid the misinterpretations by using distinct identifiers,

$$x[n] = \delta[n+1]+\delta[n]+\delta[n-4]$$ $$y[m] = 2x[2m-1] + x[2m+2].$$

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By the way,

$$y_1[m]=\delta[2m]+\delta[2m-1]+\delta[2m-5],$$ $$y_2[m]=\delta[2m+3]+\delta[2m+2]+\delta[2m-2],$$ and the deltas are taken at both parities.