- 2074 Chaitra · 3+5 marks
Range of a, b for stability
Short answer
An LTI system is linear and time-invariant. For h(t)=e^(at)u(t)+e^(-bt)u(t), BIBO stability requires a<0 and b>0 so both exponentials decay.
Define LTI system. Determine the range of values of "a" and "b" for the stability of LTI system with impulse response h(t) = e^(at)u(t) + e^(-bt)u(t).
LTI system: a system that is linear (superposition holds) and time-invariant (a shift in input shifts the output equally). It is completely described by its impulse response h(t), with output y(t) = x(t) * h(t).
Stability range
For BIBO stability, ∫|h(t)| dt must be finite.
h(t) = e^(at) u(t) + e^(-bt) u(t), for t ≥ 0.
∫₀^∞ |e^(at) + e^(-bt)| dt ≤ ∫₀^∞ e^(at) dt + ∫₀^∞ e^(-bt) dt
Each term must converge (both terms are positive, so the sum is finite exactly when each is finite):
- ∫₀^∞ e^(at) dt = [e^(at)/a]₀^∞ converges only if a < 0, giving -1/a.
- ∫₀^∞ e^(-bt) dt = [-e^(-bt)/b]₀^∞ converges only if b > 0, giving 1/b.
Answer: the system is stable for a < 0 and b > 0 (any a < 0 and any b > 0, with ∫|h| dt = 1/b - 1/a). If a ≥ 0 or b ≤ 0, one of the exponentials does not decay and the system is unstable.
✓ Answer checked by Claude Opus 5.5 (high effort)
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