Consider the discrete time system (S) with input x[n] and output y[n] as shown in the Figure. The two sub-systems represented by their impulse responses h₁[n] and h₂[n] are linear and time invariant. Which of the following statements is necessarily TRUE?

The correct answer is: S is linear and time invariant.

To determine the properties of the overall discrete-time system (S), we can analyze the components and their connections as shown in the diagram.

1. Analysis of System Components:
According to the given information and the block diagram:
- The sub-systems with impulse responses $h_1\left[n\right]$ and $h_2\left[n\right]$ are linear and time-invariant (LTI) systems.
- The delay elements represented by the transfer function $z^{-1}$ introduce a shift in time, which is a linear and time-invariant operation.
- The summation blocks (indicated by the circles with $\Sigma$ and sign labels "+" or "-") perform addition and subtraction, which are linear and time-invariant operations.

2. Mathematical Formulation:
Let us trace the signals through each branch of the system:
Let $w_1\left[n\right]$ be the output of the sub-system $h_1\left[n\right]$:

$$w_1\left[n\right]=x\left[n\right]*h_1\left[n\right]$$

The signal going through the first delay block $z^{-1}$ becomes $w_1\left[n-1\right]$. At the bottom-left adder, the input signal $x\left[n\right]$ is added to the negative of this delayed signal, producing:

$$v\left[n\right]=x\left[n\right]-w_1\left[n-1\right]$$

This combined signal $v\left[n\right]$ then passes through the sub-system $h_2\left[n\right]$, giving:

$$w_2\left[n\right]=v\left[n\right]*h_2\left[n\right]$$

This output is then delayed by the second delay element to yield $w_2\left[n-1\right]$. Finally, at the top-right summation block, the overall output $y\left[n\right]$ is formed by subtracting this delayed signal from $w_1\left[n\right]$:

$$y\left[n\right]=w_1\left[n\right]-w_2\left[n-1\right]$$

3. Conclusion of Linearity and Time-Invariance:
Since every individual component in the diagram is linear and time-invariant, and the overall system is constructed solely using connections of these components (parallel branches, cascades, additions, and subtractions), the combined system is guaranteed to be linear and time-invariant.

4. Why Other Statements Are Not Necessarily True:
- Causality: The question states only that the sub-systems are linear and time-invariant. It does not state that the impulse responses $h_1\left[n\right]$ and $h_2\left[n\right]$ are causal. If either of these sub-systems is non-causal (i.e., depends on future inputs), the overall system (S) will also be non-causal. Therefore, we cannot conclude that S is necessarily causal.