Introduction / Context:
In discrete-time systems, “dynamic” implies memory (output depends on past values), while “causal” implies dependence only on present and past—not future—inputs/outputs. The question asks which given relation best represents a dynamic, causal behavior.
Given Data / Assumptions:
- We assess each option for memory (dynamics) and causality.
- No explicit input is provided in some relations; autonomous (homogeneous) dynamics are permissible for classification.
Concept / Approach:
A dynamic system must involve dependence on at least one delayed term (past y or x). A causal system cannot involve future terms like y(n + 1) or x(n + 1). A static (memoryless) mapping such as y(n) = x(n) is causal but not dynamic. An equation with future terms is noncausal and not physically realizable in real time.
Step-by-Step Solution:
Option (a): y(n) + y(n − 1) + y(n + 1). Contains y(n + 1) (future output) → noncausal.Option (b): y(n) + y(n − 1). Only present and past output; dynamic (has memory) and causal. (Implicitly homogeneous autonomous recursion.)Option (c): y(n) = x(n). Memoryless (static) though causal; not dynamic.Option (d): y(n) + y(n − 1) + y(n + 3) = 0. Involves y(n + 3) (future output) → noncausal.Thus, (b) is the most appropriate dynamic and causal relation.
Verification / Alternative check:
(b) can be rearranged as y(n) = −y(n − 1) showing explicit recursion on past output—classic dynamic, causal behavior.
Why Other Options Are Wrong:
(a) and (d) include future terms → noncausal.(c) is memoryless → not dynamic.Added (e) y(n) = x(n + 1) would be noncausal if considered.
Common Pitfalls:
Equating “causal” with “has input”; causality concerns time indexing, not the presence of explicit input in the relation.
Final Answer:
y(n) + y(n − 1)
Discussion & Comments