Topic: Signals and Systems
If System 1 and 2 are Linear Time Invariant systems and same input $x(n)$ is provided both configuration.
Statement 1: $y_1(n)=y_2(n)$
Statement 2: $f(n)=g(n)$
- A. Statement 1 is always true
- B. Statement 2 is always true
- C. Both Statement 1 and Statement 2 are always true
- D. Both Statement 1 and statement 2 are not true
Answer: A. Statement 1 is always true
Explanation: For Linear Time Invariant (LTI) systems, the overall output is the convolution of the input signal with the impulse responses of the cascaded systems. Since the convolution operation is commutative (i.e., $x(n) * h_1(n) * h_2(n) = x(n) * h_2(n) * h_1(n)$), the final outputs $y_1(n)$ and $y_2(n)$ will always be equal regardless of the order. However, the intermediate signals in the two configurations are not necessarily equal, making Statement 2 generally false.