Question

If the impulse response of a discrete-time LTI system is finite in length, which property is always guaranteed?

• A. Causality
• B. Stability
• C. Linearity
• D. Time invariance

Hint:

All Finite Impulse Response (FIR) systems are inherently stable because a finite sum of bounded coefficients can never grow to infinity. Causality, however, depends on whether $h[n] = 0$ for all $n < 0$.

Answer 1

The Correct Option is B

Concept: A discrete-time LTI system is a Finite Impulse Response (FIR) system if its impulse response $h[n]$ is bounded within a finite index window from $n_1$ to $n_2$. The absolute condition for a discrete-time system to be Bounded-Input Bounded-Output (BIBO) stable is that its impulse response must be absolutely summable: \[ S = \sum_{n=-\infty}^{\infty} |h[n]| < \infty \]

Step 1: Formulating the summation for a finite response.
Let the system have a finite duration impulse response spanning from a starting integer index $N_1$ to an ending integer index $N_2$. Outside this interval, the values are zero. The infinite stability summation simplifies to: \[ S = \sum_{n=N_1}^{N_2} |h[n]| \]

Step 2: Checking the convergence of the sum.
Because each individual value $h[n]$ within the operational range must be finite for any real physically realizable circuit, adding a finite number of finite values ($N_2 - N_1 + 1$ terms) will always yield a finite sum: \[ S = |h[N_1]| + |h[N_1+1]| + \dots + |h[N_2]| < \infty \] Since this sum cannot diverge to infinity, the absolute summability criteria is inherently satisfied. Therefore, BIBO stability is always guaranteed for any FIR system, irrespective of whether it is causal or non-causal.