Question

Which of the following system is non-causal sytem? (Typo: "sytem" should be "system")

• A. \( y(n) = x(n) - x(n-1) \)
• B. \( y(n) = \frac{\delta y}{\delta x} \sum_{k=-\infty}^{n} x(k) \)
• C. \( y(n) = ax(n) \)
• D. \( y(n) = x(-n) \)

Hint:

Causal system: Output depends only on present/past inputs.
Non-causal system: Output depends on future inputs.
Examples of non-causal: \(y(n)=x(n+1)\), \(y(n)=x(-n)\).

Answer 1

The Correct Option is D

A system is causal if its output \(y(n)\) at any time \(n\) depends only on present and/or past inputs (\(x(k)\) where \(k \le n\)). It is non-causal if it depends on future inputs (\(x(k)\) where \(k>n\)). 

(a) \( y(n) = x(n) - x(n-1) \): Depends on current \(x(n)\) and past \(x(n-1)\). CAUSAL.

(b) \( y(n) = K \sum_{k=-\infty}^{n} x(k) \) (assuming \(\frac{\delta y}{\delta x}=K\)): Depends on present and all past inputs. CAUSAL (accumulator).

(c) \( y(n) = ax(n) \): Depends only on current input \(x(n)\). CAUSAL.

(d) \( y(n) = x(-n) \):

Let \(n = -1\). Then \(y(-1) = x(-(-1)) = x(1)\). The output at time \(-1\) depends on the input at time \(1\). Since \(1>-1\), this is a future input.

NON-CAUSAL. (This system performs time reversal).

\[ \boxed{y(n) = x(-n)} \]