Introduction:
In the field of digital signal processing, LTI (Linear Time-Invariant) systems play a crucial role. Stability is an essential property of these systems, as it ensures that the output remains bounded for any bounded input. The stability of an LTI discrete system can be determined by analyzing its impulse response.

Impulse Response:
The impulse response of an LTI system is the output produced when the input is an impulse function. It characterizes the behavior of the system and provides valuable information about its stability.

Stability Criterion:
For an LTI discrete system to be stable, the square sum of its impulse response must be finite. This criterion is known as Bounded Input Bounded Output (BIBO) stability.

Explanation:
To understand why the square sum of the impulse response should be finite for stability, let's consider an LTI discrete system with impulse response h[n].

- Bounded Input: Suppose we have a bounded input signal x[n], which means that its amplitude is limited to a finite range. This is a common scenario in practical applications.
- System Response: When this bounded input signal is applied to the LTI system, its output y[n] is obtained by convolving the input signal with the impulse response: y[n] = x[n] * h[n].
- Bounded Output: The stability of the system ensures that the output signal y[n] remains bounded for any bounded input x[n]. In other words, the output does not grow uncontrollably or approach infinity.
- Finite Square Sum: If the square sum of the impulse response, i.e., ∑(|h[n]|^2), is finite, it guarantees that the output will also be bounded. This is because the convolution operation amplifies the input signal, and the finite sum of squared impulse response values ensures that the amplification remains within a finite range.
- Counterexample: If the square sum of the impulse response is infinite, the amplification of the input signal could also become infinite, leading to an unbounded output. This violates the stability criterion.

Conclusion:
In conclusion, for an LTI discrete system to be stable, the square sum of its impulse response must be finite. This ensures that the output remains bounded for any bounded input signal. The criterion of finite square sum is a fundamental requirement for the stability of LTI systems and is widely used in the analysis and design of digital signal processing systems.