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The input-output relationship of a causal stable LTI system is given as y[n] = α y[n−1] + β x[n]. If the impulse response h[n] of this system satisfies the condition ∑n=0∞ h[n] = 2, the relationship between α and β is

1. α = 1 − β/2
2. α = 1 + β/2
3. α = 2β
4. α = −2β

Correct answer: α = 1 − β/2

Solution

The condition ∑n=0∞ h[n] = 2 indicates that the system is stable and the sum of the impulse response is finite. For a causal stable LTI system, the relationship between the coefficients α and β must satisfy the equation derived from the system's response, leading to α = 1 - β/2 as the correct relationship.

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