For time-invariance, Sys{x(t – \tau)} = y(t – \tau). Determine if the system is time- invariant or not: 1. y(t) = 5x(t) 2. y(t) = 2 cos (x(t)) 3. y(t) = 3 cos(t)x(t
The Correct Answer and Explanation is:
To determine whether each system is time-invariant, we check if applying a time shift to the input results in an equivalent time shift in the output. Mathematically, a system is time-invariant if Sys{x(t − τ)} = y(t − τ) for any shift τ.
Let’s evaluate each system one by one.
1. y(t) = 5x(t) Let us apply a time shift τ to the input: Input: x(t − τ) Output becomes: y₁(t) = 5x(t − τ) Compare it to y(t − τ): y(t − τ) = 5x(t − τ) Since they match, this system is time-invariant.
2. y(t) = 2 cos(x(t)) Apply the time shift: Input: x(t − τ) Output becomes: y₂(t) = 2 cos(x(t − τ)) Compare to y(t − τ): y(t − τ) = 2 cos(x(t − τ)) Since they are identical, this system is time-invariant.
3. y(t) = 3 cos(t)·x(t) Apply the time shift: Input: x(t − τ) Output becomes: y₃(t) = 3 cos(t)·x(t − τ) Compare to y(t − τ): y(t − τ) = 3 cos(t − τ)·x(t − τ) These expressions differ because cos(t) is not equal to cos(t − τ), which means the system relies explicitly on the time variable t. Therefore, this system is not time-invariant.
Conclusion: Systems 1 and 2 are time-invariant because their outputs shift consistently with the input. However, system 3 is time-variant, as it contains an explicit dependence on time via the cos(t) term, which changes independently of the input shift. Time-invariance fails whenever system behavior varies with time in a fixed way not tied solely to the input signal.
