Help proving if a system is LTI?

[asdf12312]

Homework Statement

A DT system is defined by the input-output relationship y[n] = x[n] ∗ h[n],
where x[n] is input, y[n] is output, '*' is convolution, and h[n] = u[n+1].
Is this an LTI system? Explain.

Homework Equations

x[n] - input, h[n] - impulse response
y[n]=h[n]*x[n] = Σ (k=-inf to inf) x[k]h[n-k] = Σ (k=-inf to inf) h[k]x[n-k] (convolution sum for DT)
Rule for y[n]=h[n]*x[n]: if x[n]=u[n], y[n]=Σ (k=-inf to n) h[k]

The Attempt at a Solution

I'm trying to prove if it is LTI system or not. I can prove that convolution sum is linear, y[k]=Σ(k=-inf to inf) A(x[k]h[n-k])= Ay[k] but not sure its time invariant. if I use the rule in the book, I guess x[n] and h[n] are interchangeable in convolution sum so h[n] can be used as input. Then using the rule, y[n]=Σ (k=-inf to n) x[k+1] which I suppose means its time invariant? y[n-n0]=Σ (k=-inf to n) x[k+1-n0], pretty sure this is true but is this enough to prove its LTI system?

 

[jssamp]

If it is time invariant the output you get from a time shifted input is the same as if you time shifted the output the same amount.

You have system H and its output y such that the system is described by y[n] = H{ x[n] },
if you shift input x1[n] so you get a new input x2[n] = x1[n-1] and you put this new input into H you get y2[n] = H{ x1[n-1] }
and you shift the output so y1[n-1] = H{ x1[n] },
y2[n] = y1[n-1] for time invariant system.

if it is linear then if the inputs x[n] = x1[n] produce output y[n] = y1[n] and x[n] = x2[n] produces y[n] = y2[n] then
x[n] = a*x1[n] + b*x2[n] will have output y[n] = ay1[n] + by2[n]