Question

Which system among the following is a time invariant system?

(a) y(n) = n x(n)

(b) y(n) = x(n) – x(n-1)

(c) y(n) = x(-n)

(d) y(n) = x(n) cos 2nf

This question was posed to me in final exam.

This interesting question is from Fourier Series and LTI Systems in division Fourier Series of Signals and Systems

Answer 1

Correct answer is (b) y(n) = x(n) – x(n-1)

The best explanation: We know that, for any system y (n) = k x (n), to be a time invariant system, it must satisfy the relation, y (n-n1) = k x (n-n1) [where k is a constant or a function of n].

For y (n) = n x (n), y (n-n1) = (n-n1) x (n-n1)

This does not satisfy the criteria as stated above. Hence not time invariant.

 For y (n) = x (-n), y (n-n1) = x (-n+n1)

This also does not satisfy the criteria as stated above. Hence not time invariant.

For y (n) = x (n) cos 2nf, y (n-n1) = x (n-n1) cos 2(n-n1) f

This also does not satisfy the criteria as stated above. Hence not time invariant.

For y (n) = x (n) – x (n-1), y (n-n1) = x (n-n1) – x (n-n1-1)

This satisfies the above criteria. Hence given system is time invariant.