Digital Signal Processing Basics
Let u[n] be the unit step. Which of the following LTI systems, given by their impulse responses, is BIBO stable (every bounded input gives a bounded output)?
- Ah[n] = (−0.9)ⁿu[n]
- Bh[n] = u[n]
- Ch[n] = (1.1)ⁿu[n]
Show answer and explanation
Answer: A. h[n] = (−0.9)ⁿu[n]
An LTI system is BIBO stable if and only if the sum of the absolute values of its impulse response, Σ|h[n]|, is finite. For h[n] = (−0.9)ⁿu[n], Σ|h[n]| = Σ0.9ⁿ = 1/(1 − 0.9) = 10, which is finite, so the system is stable. h[n] = u[n] is an accumulator that keeps adding its input; the sum is infinite, and a constant input makes the output grow without limit. (1.1)ⁿ grows with time, so that system is unstable.
- A. Correct. The sign alternates, but Σ|h[n]| = 10 is finite.
- B. This is the impulse response of an accumulator; Σ|h[n]| is infinite, so it is not stable. A step input makes the output grow as n + 1.
- C. Since |1.1| > 1, h[n] keeps growing and the system is unstable.