---
title: "Recursive/nonrecursive systems; stability"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:recursive-nonrecursive-stability"
source: https://www.bctengineer.com/q/semester_5/data_communication/recursive-nonrecursive-stability
site: BCT Engineering
---

# Recursive/nonrecursive systems; stability

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2075 Asoj, 3+5 marks
- Video: [Recursive & Non-Recursive Systems(DSP Lecture-29)](https://www.youtube.com/watch?v=tjtP89qgKco) — Engineers choice tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/recursive-nonrecursive-stability
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

What are Recursive and Nonrecursive systems? Test the stability of the CT system whose impulse response is given as h(t) = e^(-t)·sin(t)·u(t).

## Short answer

Nonrecursive systems depend only on present/past inputs; recursive systems also depend on past outputs. For h(t)=e^(-t) sin(t) u(t), ∫|h| dt ≤ ∫₀^∞ e^(-t) dt = 1 < ∞, so the system is stable.

## Answer

#### Recursive and non-recursive systems
- **Non-recursive system**: the present output depends only on present and past inputs, not on past outputs. There is no feedback. Example: y[n] = x[n] + 2x[n - 1] (FIR filter, finite impulse response).
- **Recursive system**: the present output depends also on past outputs, so there is feedback. Example: y[n] = 0.5y[n - 1] + x[n] (IIR filter, impulse response of infinite length).

#### Stability test of h(t) = e^(-t) sin(t) u(t)
BIBO stability needs ∫|h(t)| dt < ∞.

∫|h(t)| dt = ∫₀^∞ e^(-t) |sin t| dt

Since |sin t| ≤ 1:

∫₀^∞ e^(-t) |sin t| dt ≤ ∫₀^∞ e^(-t) dt = 1 < ∞

(Evaluated exactly, the integral equals (1 + e^(-π)) / (2(1 - e^(-π))) ≈ 0.545.)

Also, H(s) = 1/((s + 1)² + 1), whose poles s = -1 ± j lie in the left half-plane.

**Answer: the system is stable.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Recursive output depends on
   A. only inputs
   B. only present
   C. past outputs too
   D. none
   Answer: C. Feedback, IIR. (Key point: Recursive)

2. Non-recursive example
   A. y[n]=0.5y[n-1]+x[n]
   B. IIR
   C. feedback
   D. y[n]=x[n]+2x[n-1]
   Answer: D. FIR. (Key point: Non-recursive)

3. e^(-t)sin t u(t) is
   A. stable
   B. unstable
   C. marginal
   D. non-causal
   Answer: A. Bounded by integral of e^-t = 1. (Key point: Stability)

4. H(s) poles at
   A. 1 ± j
   B. -1 ± j
   C. 0
   D. -1
   Answer: B. Left half. (Key point: Poles)

## Related in Data Communication

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- [Unit step response for h(t) = exp(-at)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aunit-step-response-exp) — Data Communication, chapter 2

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