Question

In a causal and stable LTI system, the impulse response $h(t)$ must satisfy which condition ?

• A. $h(t)$ is always positive
• B. $h(t)$ is periodic
• C. $h(t)$ must be absolutely integrable
• D. $h(t)$ must be bounded

Hint:

For an LTI system to be stable: - Continuous-time: $h(t)$ must be absolutely integrable ($\int |h(t)|dt < \infty$). - Discrete-time: $h[n]$ must be absolutely summable ($\sum |h[n]| < \infty$).

Answer 1

The Correct Option is C

Concept: For any general Linear Time-Invariant (LTI) system, stability is classified under the Bounded-Input Bounded-Output (BIBO) standard. A system is defined as BIBO stable if every bounded input produces a bounded output. The necessary and sufficient criterion to guarantee this in the time domain relates directly to the absolute integrability of its impulse response $h(t)$.

Step 1: Derivation of the stability criterion.
Let the input signal to an LTI system be $x(t)$, bounded by a finite real maximum value $M_x$: \[ |x(t)| \le M_x < \infty \quad \forall \quad t \] The output $y(t)$ of the system is given by the convolution integral of the input and the impulse response: \[ y(t) = \int_{-\infty}^{\infty} h(\tau) x(t - \tau) \, d\tau \] Taking the absolute value of both sides: \[ |y(t)| = \left| \int_{-\infty}^{\infty} h(\tau) x(t - \tau) \, d\tau \right| \] Applying the triangle inequality for integrals, the absolute value of an integral is less than or equal to the integral of the absolute value: \[ |y(t)| \le \int_{-\infty}^{\infty} |h(\tau)| \cdot |x(t - \tau)| \, d\tau \] Since $|x(t-\tau)| \le M_x$, we can substitute this upper bound into the inequality: \[ |y(t)| \le M_x \int_{-\infty}^{\infty} |h(\tau)| \, d\tau \]

Step 2: Imposing the absolute integrability constraint.
For the output $y(t)$ to remain strictly bounded ($|y(t)| < \infty$), given that $M_x$ is a finite non-zero constant, the remaining integral factor must evaluate to a finite value: \[ \int_{-\infty}^{\infty} |h(\tau)| \, d\tau < \infty \] This mathematically demonstrates that the impulse response $h(t)$ must be absolutely integrable. Additionally, causality implies $h(t) = 0$ for $t < 0$, which restricts the limits from $0$ to $\infty$, but absolute integrability remains the core prerequisite for stability.