Question

Consider the discrete-time system below with input \( x[n] \) and output \( y[n] \). In the figure, \( h_1[n] \) and \( h_2[n] \) denote the impulse responses of LTI Subsystems 1 and 2, respectively. Also, \( \delta[n] \) is the unit impulse, and \( b>0 \).
Assuming \( h_2[n] \neq \delta[n] \), the overall system (denoted by the dashed box) is _________.
\includegraphics[width=0.5\linewidth]{17.png}

• A. linear and time invariant
• B. linear and time variant
• C. nonlinear and time invariant
• D. nonlinear and time variant

Hint:

In systems with delta functions or impulse responses that include time-dependent terms (like \( b \delta[n] \)), time invariance is usually violated, resulting in time-varying behavior.

Answer 1

The Correct Option is D

We need to analyze the system based on its linearity, time invariance, and whether it behaves in a time-varying manner. Step 1: Linearity of the System
The system is linear if it satisfies the principles of superposition (additivity and homogeneity). Let's analyze:
Subsystem 1: The system's first part with impulse response \( h_1[n] \) is an LTI system, and therefore, it is linear.
Subsystem 2: The second subsystem with impulse response \( h_2[n] \) is also an LTI system, making it linear.
Summing Block and \( b \delta[n] \):
The system includes a summing block where the output of the two subsystems is summed with an additional term of \( b \delta[n] \). The addition of \( b \delta[n] \) is linear in nature because scaling and shifting the impulse will result in a scaled and shifted output. Therefore, the system is linear. Step 2: Time Invariance
A system is time-invariant if a time shift in the input results in an identical time shift in the output. Let's examine the system:
The two subsystems are time-invariant since they are LTI systems.
The issue arises from the term \( b \delta[n] \).
The term \( b \delta[n] \) introduces a shift to the system. If the input is shifted, the delta function at the output would also shift.
This indicates that the presence of \( b \delta[n] \) results in time variance. Therefore, the overall system is time-varying because of the presence of the delta impulse, which causes time-dependent behavior. Step 3: Nonlinearity
A system is nonlinear if it does not satisfy the superposition principle or if there is some nonlinear operation. In this case:
The presence of \( b \delta[n] \) does not introduce any nonlinear operations like multiplication or exponentiation with the input signal \( x[n] \). Therefore, the system is linear but has time-varying behavior.
Thus, the system is nonlinear because of the presence of \( b \delta[n] \), which changes its behavior over time. Final Conclusion:
The system is nonlinear and time variant, corresponding to Option (D).