Question

Discrete-time LTI system is BIBO stable if:

• A. The absolute sum of its impulse response converges
• B. Its impulse response is finite
• C. It has no poles
• D. Its frequency response is zero

Hint:

A discrete-time LTI system is BIBO stable if and only if its impulse response sequence is absolutely summable: \(\sum_{n=-\infty}^{\infty} |h[n]| < \infty\). In the \(z\)-domain, this matches the condition that the Region of Convergence (ROC) must contain the unit circle (\(|z| = 1\)).

Answer 1

The Correct Option is A

Concept: Bounded-Input Bounded-Output (BIBO) stability is a fundamental system property. It guarantees that as long as the input signal applied to a system does not grow to infinity, the resulting output signal is also guaranteed to remain finite. For a discrete-time Linear Time-Invariant (LTI) system, the entire system behavior is fully captured by its impulse response sequence, denoted as \(h[n]\). The input-output relationship is governed by the convolution sum: \[ y[n] = x[n] * h[n] = \sum_{k=-\infty}^{\infty} h[k]x[n-k] \]

Step 1: Formal mathematical definition of bounded input.
Let us assume the input signal \(x[n]\) is strictly bounded. By definition, this means there exists a finite real constant value \(M_x\) such that the absolute amplitude of the signal never exceeds this boundary for any integer index \(n\): \[ |x[n]| \le M_x < \infty \quad \forall \quad n \in \mathbb{Z} \]

Step 2: Applying the absolute value to the convolution sum.
To evaluate the stability bounds of the output signal \(y[n]\), we take the absolute value of both sides of the convolution equation: \[ |y[n]| = \left| \sum_{k=-\infty}^{\infty} h[k]x[n-k] \right| \] Using the mathematical triangle inequality theorem (\(|\sum a_i| \le \sum |a_i|\)), we can distribute the absolute value operation inside the summation: \[ |y[n]| \le \sum_{k=-\infty}^{\infty} \left| h[k]x[n-k] \right| \] Using the product property of absolute values (\(|a \cdot b| = |a| \cdot |b|\)), this simplifies to: \[ |y[n]| \le \sum_{k=-\infty}^{\infty} |h[k]| \cdot |x[n-k]| \]

Step 3: Substituting the input bound parameter.
Since we established in Step 1 that \(|x[n-k]| \le M_x\) holds true for all possible indices, we can substitute \(M_x\) into our inequality to find the upper bound: \[ |y[n]| \le \sum_{k=-\infty}^{\infty} |h[k]| \cdot M_x \] Since the constant term \(M_x\) is independent of the summation index \(k Embassy\), it can be factored outside the summation: \[ |y[n]| \le M_x \sum_{k=-\infty}^{\infty} |h[k]| \]

Step 4: Establishing the final stability criterion.
For the system to be BIBO stable, the output amplitude must remain strictly bounded by a finite maximum value \(M_y\) (meaning \(|y[n]| \le M_y < \infty\)). Looking at our inequality, since \(M_x\) is already a finite value, the output is guaranteed to remain bounded if and only if the remaining summation term evaluates to a finite value: \[ \sum_{k=-\infty}^{\infty} |h[k]| < \infty \] Replacing the dummy variable \(k\) with standard index notation \(n\), this condition is written as: \[ \sum_{n=-\infty}^{\infty} |h[n]| = S < \infty \] This demonstrates that the impulse response sequence must be absolutely summable, meaning its absolute sum converges to a finite value. This matches Option (A).