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I have to test whether the following system is invariant or not: $T[x[n]]=\sum_{k=n-1}^{n+2}x[k]$, so I want to verify that, if $y[n]=T[x[n]]$, then $y[n-N]=T[x[n-N]]$. $$T[x[n-N]]=\sum_{k=n-1}^{n+2}x[k-N]$$ $$y[n-N]=\sum_{k=n-N-1}^{n-N+2}x[k]$$ In the first equation, if $k'=k-N$, we have that $T[x[n-N]]=\sum_{k'=n-N-1}^{n-N+2}x[k']$, so both sums are equal and thus the system is invariant. Is this correct or am I making wrong assumptions? Thanks in advance!

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Yes your computation is correct. Indeed, you could have seen this even easier, had you considered that the output computation sum was

$$ y[n] = T\{x[n]\} = x[n-1] + x[n] + x[n+1] + x[n+2] .$$

And this is clearly an LTI (linear time-invariant), FIR (finite impulse response) system with associated impulse response

$$h[n] = \delta[n-1] + \delta[n] + \delta[n+1] + \delta[n+2] $$

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