---
title: "Convolution of u(t) with u(t)"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:convolution-u-u"
source: https://www.bctengineer.com/q/semester_5/data_communication/convolution-u-u
site: BCT Engineering
---

# Convolution of u(t) with u(t)

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2073 Chaitra, 2+5+1 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/convolution-u-u
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Define LTI system. Find convolution between x(t) = u(t) and h(t) = u(t) and comment on the result.

## Short answer

An LTI system is linear and time-invariant; its output is the convolution of input and impulse response. Convolution of u(t) with u(t) yields the unit ramp r(t)=t u(t). The result grows unbounded, so an integrator with h(t)=u(t) is not BIBO stable.

## Answer

**LTI system**: a system that is linear and time-invariant. Its output is the convolution of the input with the impulse response, y(t) = x(t) * h(t) = ∫ x(τ) h(t - τ) dτ.

#### Convolution of u(t) with u(t)
y(t) = ∫_{-∞}^{∞} u(τ) u(t - τ) dτ

- u(τ) = 1 only for τ ≥ 0.
- u(t - τ) = 1 only for τ ≤ t.

**t < 0**: there is no overlap, so y(t) = 0.

**t ≥ 0**: the product is 1 for 0 ≤ τ ≤ t:

y(t) = ∫₀^t 1 dτ = t

**y(t) = t·u(t) = r(t)**, the unit ramp.

```text
y(t)
  |        /
  |      /
  |    /
  |  /
--+--------------> t
  0
```

#### Comment
- Convolving two unit steps gives a ramp, which grows without limit. A bounded input u(t) produces an unbounded output.
- This shows that a system with h(t) = u(t) (an ideal integrator) is **not BIBO stable**, since ∫|h(t)| dt = ∞.
- It is causal, because h(t) = 0 for t < 0.

## Check yourself

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

1. For t >= 0, over which range do u(tau) and u(t-tau) overlap?
   A. tau >= t
   B. -t <= tau <= 0
   C. 0 <= tau <= 1
   D. 0 <= tau <= t
   Answer: D. u(tau) needs tau >= 0 and u(t-tau) needs tau <= t. (Key point: Overlap limits)

2. What is u(t) * u(t)?
   A. t u(t), the unit ramp
   B. u(t)
   C. delta(t)
   D. e^(-t)u(t)
   Answer: A. The integral of 1 from 0 to t equals t. (Key point: Result)

3. What does the result show about h(t) = u(t)?
   A. It is stable because u(t) is bounded
   B. It is not BIBO stable because a bounded input gives an unbounded ramp
   C. It is non-causal
   D. It is memoryless
   Answer: B. The integral of |h| is infinite. (Key point: Stability comment)

4. Why is the system with h(t) = u(t) causal?
   A. h(t) is bounded
   B. The output is a ramp
   C. h(t) = 0 for t < 0
   D. It is stable
   Answer: C. A causal LTI system has zero impulse response for negative time. (Key point: Causality comment)

## Related in Data Communication

- [Define LTI system and impulse response](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Alti-impulse-response-define) — Data Communication, chapter 2, asked 2×
- [Convolution of e^(-at)u(t) with u(t)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aconvolution-exp-u-step) — Data Communication, chapter 2
- [Output of LTI with h(t) = e^(-at)u(t) and e^(at)u(-t)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aconvolution-exp-u-and-u-neg) — Data Communication, chapter 2, asked 3×
- [LTI output for h(t)=e^(-2t), rectangular input](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Alti-output-rect-input) — Data Communication, chapter 2, asked 3×
- [Check LTI: y(t) = 2x(t) + 3](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Alti-check-2x-plus-3) — Data Communication, chapter 2
- [LTI output for h(t)=e^(-at)u(t), x(t)=u(t)-u(t-2)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Alti-output-u-minus-u2) — Data Communication, chapter 2

## Related in other subjects

- [Periodic and non-periodic signals; LTI output is convolution](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aperiodic-nonperiodic-convolution) — Communication Systems, chapter 1
- [Frequency response of a CT LTI system](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Act-frequency-response) — Signals and Systems, chapter 5, asked 3×
- [Describe LTI system properties](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-properties-describe) — Signals and Systems, chapter 5, asked 3×
- [Condition for LTI system to be causal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-causality-condition) — Signals and Systems, chapter 6, asked 3×