Understanding BIBO Stability in LTI Systems

Linear Time-Invariant (LTI) systems are fundamental in signal processing and control theory. A key characteristic used to describe their behavior is stability. Specifically, Bounded-Input Bounded-Output (BIBO) stability is a crucial concept. An LTI system is defined as BIBO stable if, for every input signal that remains bounded over time, the corresponding output signal also remains bounded over time.

Condition for BIBO Stability

For an LTI system, BIBO stability is directly related to its impulse response, denoted as $h(t)$ for continuous-time systems or $h[n]$ for discrete-time systems. The condition for BIBO stability is that the impulse response must be absolutely integrable (for continuous-time) or absolutely summable (for discrete-time). Mathematically, this means:

• For continuous-time systems:

  $$ \int_{-\infty}^{\infty} |h(t)| dt < \infty $$

• For discrete-time systems:

  $$ \sum_{n=-\infty}^{\infty} |h[n]| < \infty $$

If this condition is met, the system is guaranteed to be BIBO stable.

Analysis of Options

Let's analyze each statement in the context of BIBO stable LTI systems:

• Statement 1: The impulse response will be integral, but may not be absolutely integrable

  This statement is incorrect. As explained above, the core condition for BIBO stability is that the impulse response must be absolutely integrable or summable. Simple integrability is not sufficient.

• Statement 2: The unit impulse response will have finite support

  This statement is not always true. Systems whose impulse response has finite support are known as Finite Impulse Response (FIR) systems. All FIR systems are indeed BIBO stable. However, BIBO stable systems are not limited to FIR systems; Infinite Impulse Response (IIR) systems can also be BIBO stable, provided their impulse response meets the absolute integrability/summability condition. Therefore, stating the impulse response *will* have finite support is too restrictive.

• Statement 3: The unit step response will be absolutely integrable

  This statement is not necessarily true. The unit step response, $y_s(t)$, is the convolution of the impulse response $h(t)$ with the unit step function $u(t)$, i.e., $y_s(t) = h(t) * u(t)$. While BIBO stability ensures the output is bounded for a bounded input like the unit step, it doesn't guarantee that the resulting step response itself is absolutely integrable. The requirement is on the impulse response, not the step response.

• Statement 4: The unit step response will be bounded

  This statement is TRUE. The unit step function, $u(t)$ (or $u[n]$), is a classic example of a bounded input signal. By the definition of BIBO stability, if the input is bounded, the output must also be bounded. Therefore, for a BIBO stable LTI system, applying a unit step input will result in a bounded unit step response.

• Statement 5: (No content provided)

Conclusion

Based on the definition and conditions for BIBO stability in LTI systems, the only statement that is universally true is that the unit step response will be bounded, as the unit step function itself is a bounded input.