For the following system: Are there conditions on no (where no is the time shift

Question

For the following system:

Are there conditions on no (where no is the time shift), for which the system is time invariant? If so, what are the conditions on no for the system to be time invariant?

My attemp at a solution:

Determine the output y1[n] of the system for the delayed input x[n-no]

y1[n] = f[x(n-no)]

Now delay the output by no samples given by the equation

y[n-no] = f[x(n-no)]

Therefore, it is reasonable to conclude that the system is TIME VARIANT. However, what conditions on no could make it not so? If the previous steps are indeed correct, then it appears that only no = 0 would yield (1) = (2). This seems a bit trivial, though, as it implies no time shift at all.

Any insights to clarify this would be greatly appreciated.

Thanks

Explanation / Answer

Firstly we shift the input only Y1[n] = X[n-no]

Second replace all n by n-no in a given function

Y2[n] = Y[n-no] = X[n-no]

If Y1[n] = Y2[n] then it is time invariant

NOTE : no can't be zero

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