Question

Which of the following systems is time invariant?

• A. \( y(t) = x(2t) + x(-t) \)
• B. \( y(t) = x(t) + x(1 - t) \)
• C. \( y(t) = -x(t) + x(1 - t) \)
• D. \( y(t) = x(t) + x(t - 1) \)

Hint:

Check time invariance by applying a time shift to the input and seeing if output shifts identically.

Answer 1

The Correct Option is D

Step 1: Time invariance concept
A system is time invariant if a shift in the input signal causes an identical shift in the output.
Step 2: Test each system
For time invariance, check: If input is shifted by \( t_0 \), does output also shift by \( t_0 \)?
Option (1): \( y(t) = x(2t) + x(-t) \)
- If \( x(t) \to x(t - t_0) \), then \( y(t) \to x(2t - 2t_0) + x(-t - t_0) \).
- This is not equal to \( y(t - t_0) = x(2(t - t_0)) + x(-(t - t_0)) \).
- Not time invariant.
Option (2) and (3): terms like \( x(1 - t) \)
- These include fixed time shifts and reflections, which vary the output differently.
- Not time invariant.
Option (4): \( y(t) = x(t) + x(t - 1) \)
- Shift input: \( x(t) \to x(t - t_0) \), output becomes \( y(t) = x(t - t_0) + x(t - t_0 - 1) \)
- This is same as \( y(t - t_0) = x(t - t_0) + x(t - t_0 - 1) \).
- Time invariant!
Therefore, Option (4) is correct.