Which one of the following is correct for the given system ?

y[n] = x[n] - x[n - 1]

Concept:

LTI System:

• For a time-invariant system, a time shift in input to the system produces the same time shift in the output.
• For a linear system, a linear combination of input produces a linear combination of output.

Example 

To check for causality we can replace the input with impulse and find whether the response exists for t < 0. 

If it exists for t < 0, then the system is causal, else it is non-causal. 

Analysis:

y[n] = x[n] - x[n - 1]

h(n) = δ(n) - δ(n - 1)

As hc(n) = 0; n < 0, the system is causal. 

y[n] = x[n] - x[n - 1]

y(n - n₀) = x(n - n₀) - x(n - n₀ - 1) --- (1)

y(n,n₀) = x(n - n₀) - x(n - n0 - 1) --- (2)

Hence, it is time invariant.