Causality check for discrete-time systems: Which of the following systems is causal?

Introduction / Context:
Causality is a foundational property for real-time systems: the output at time n cannot depend on future input samples. Knowing how to spot future dependencies is vital for implementability.

Given Data / Assumptions:

• Five candidate input–output relations.
• Standard discrete-time indexing with n for current time, n−k for past, and n+k for future.
• No hidden states beyond what is shown.

Concept / Approach:

A causal system’s output y[n] relies only on x[n] and past values x[n−k], k ≥ 1. Any appearance of x[n+k] with k > 0 violates causality because it requires knowledge of future input.

Step-by-Step Solution:

Verification / Alternative check:

Physical realization: (a) can be implemented with one sample of memory (a delay). Others would need prediction of unknown future inputs, which is not causal.

Why Other Options Are Wrong:

• Any x[n+k], k > 0, violates causality by definition.
• Adding current or past terms cannot fix a future dependency.

Common Pitfalls:

• Confusing time advance (future input) with time delay (past input).
• Assuming linearity or time invariance alone guarantees causality—it does not.

Final Answer:

y[n] = 3 x[n] − 2 x[n − 1]