Question

Which of the following statement(s) is/are true?

• A. If an LTI system is causal, it is stable
• B. A discrete time LTI system is causal if and only if its response to a step input $u[n]$ is 0 for $n < 0$
• C. If a discrete time LTI system has an impulse response $h[n]$ of finite duration the system is stable
• D. If the impulse response $0 < |h[n]| < 1$ for all $n$, then the LTI system is stable

Hint:

Causality depends only on whether $h[n]$ is zero for $n<0$, while stability depends on whether $h[n]$ is absolutely summable. A system can be causal but unstable, or stable but non-causal.

Answer 1

The Correct Option is B, C

Step 1: Recall definitions.
- Causality: A discrete-time LTI system is causal if its impulse response $h[n] = 0$ for all $n < 0$. Equivalently, its step response also vanishes for $n < 0$. - Stability: An LTI system is stable (BIBO stable) if $\sum_{n=-\infty}^{\infty} |h[n]| < \infty$. This requires the impulse response to be absolutely summable.

Step 2: Evaluate option (A).
(A) "If an LTI system is causal, it is stable." This is false, since causality and stability are independent properties. Example: $h[n] = 1$ for $n \geq 0$. This is causal, but $\sum_{n=0}^{\infty} |1| = \infty$, so the system is unstable.

Step 3: Evaluate option (B).
(B) "A discrete time LTI system is causal iff step response $=0$ for $n<0$." This is true. The step response is the cumulative sum of the impulse response: \[ s[n] = \sum_{k=-\infty}^{n} h[k]. \] If $h[k]=0$ for $k<0$, then $s[n]=0$ for $n<0$. Conversely, if $s[n]=0$ for $n<0$, then $h[k]=0$ for $k<0$.

Step 4: Evaluate option (C).
(C) "If $h[n]$ has finite duration, system is stable." This is true, since $\sum |h[n]|$ is a finite sum if $h[n]$ is nonzero only for a finite number of $n$.

Step 5: Evaluate option (D).
(D) "If $0<|h[n]|<1$ for all $n$, then system is stable." This is false. Even if $|h[n]|<1$, if it extends infinitely (like $h[n]=0.5$ for $n \geq 0$), then $\sum |h[n]| = \infty$, meaning unstable. % Final Answer \[ \boxed{\text{Correct statements: (B) and (C)}} \]