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Suppose we have a discrete signal $x[n]$ and $y[n] = x[2n]$. I want to find $y[n-k]$. Is it $x[2n-k]$ or $x[2n-2k]$? I'm very confused by this operations. Can someone explain what's the logic behind it and give some references?

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It is $x[2n−2k]$. No matter what expression you have for $y[n]$, $y[n−k]$ is equivalent to taking that expression and replacing $n$ by $n - k$.

So $y[n−k] = x[2(n - k)] = x[2n−2k]$.

Check out the graphs I made here: https://www.desmos.com/calculator/tkm8cqio2o I have plotted some random function $f(x)$ (equivalent to your $x[n]$), then defined $g(x)$ (equivalent to your $y[n]$). Use the slider to see that $g(x - k)$ always coincides with $f(2x - 2 k)$. I know that in this cases the graphs are continuous, not discrete, but its the same thing.

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