time linearity of LTI system

Question

There is this following question.

Consider the transformation H {x}[n] = n x [n]. Does H define an LTI system?

What I understood from the question is:

x[n] -> H -> y[n] or according to the question, x[n] -> H -> n x[n]

to prove the time invariance, if I put n-no instead of n, it should be,

x[n-no] -> H -> y[n-no] = n x[n-no] but it turns out to be, x[n-no] -> H -> n x[n - no] - no x[n-no] which means the output depends on extra term no x[n-n0] and is not time invariant.

Incase, the trasformation was any constant * x[n] the it would have conserve the time invariance property, if I am not wrong.

But the proof mentioned in the notes is,

I have difficulty understanding the proof. If anyone could explain it to me or if I understood the question wrong, I'd appreciate it.

Answer 1

Your notes gave a counter example, you just proved it in the general case. Your note used your x[n] as delta and showed that y[n-1] unequal to H{x[n-1]} which is exactly what you showed.

In the notes example H{x}=0 for all n, as discrit delta equals zero foll all n but n=0, which means H is zero every where for all input of the kind delta[n-n0] (becouae of the n factor which zeros y at its only non zero point) Than you have zero output for delta input. But as you showed time varied output isn't zeros every where.