---
title: "System types; output as convolution"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:lti-causal-convolution-proof"
source: https://www.bctengineer.com/q/semester_6/communication_systems/lti-causal-convolution-proof
site: BCT Engineering
---

# System types; output as convolution

Communication Systems · Chapter 1: Introduction

- Asked in: 2071 Magh (old course), 4+4 marks
- Video: [Impulse Response and Convolution](https://www.youtube.com/watch?v=-oNqg6JH0eI) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/lti-causal-convolution-proof
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

Discuss linear, non linear, causal and time invariant systems used in communication. Prove that the output of any system is given by convolution of input and impulse response of the system.

## Short answer

Linear systems obey superposition; nonlinear (e.g., square-law) create new frequencies. Causal systems depend only on present/past; time-invariant systems have unchanging parameters. Convolution proof: x(t)=∫x(λ)δ(t-λ)dλ; by time invariance δ→h, and superposition, y(t)=∫x(λ)h(t-λ)dλ = x(t)*h(t).

## Answer

#### Types of systems
- **Linear system:** obeys superposition, $a x_1 + b x_2 \to a y_1 + b y_2$. Example: an ideal amplifier $y = 5x(t)$, filters, cables.
- **Non-linear system:** does not obey superposition; creates new frequencies. Example: square-law device $y = x^2(t)$ used in modulators and detectors, a diode, a saturated amplifier.
- **Causal system:** output depends only on present and past inputs; $h(t) = 0$ for $t < 0$. All real-time systems are causal, e.g. $y(t) = x(t-1)$. A system such as $y(t) = x(t+1)$ is non-causal (ideal filters are non-causal).
- **Time-invariant system:** a delay of the input causes the same delay of the output, $x(t - t_0) \to y(t - t_0)$; its parameters do not change with time, e.g. a fixed RC filter. A system like $y(t) = t\,x(t)$ or a time-varying fading channel is time-variant.

Communication channels and filters are usually modelled as **linear time-invariant (LTI)**; modulators and detectors are deliberately non-linear or time-varying.

#### Proof: output = input convolved with impulse response
Let $h(t)$ be the response of an LTI system to $\delta(t)$.

**Step 1: represent the input as a sum of impulses** (sifting property):

$$
x(t) = \int_{-\infty}^{\infty}x(\lambda)\,\delta(t - \lambda)\,d\lambda
$$

i.e. $x(t)$ is a continuous sum of impulses located at $t = \lambda$ with strength $x(\lambda)d\lambda$.

**Step 2: use time invariance:** $\delta(t - \lambda) \to h(t - \lambda)$.

**Step 3: use homogeneity:** $x(\lambda)\delta(t-\lambda)d\lambda \to x(\lambda)h(t-\lambda)d\lambda$.

**Step 4: use additivity (superposition)** over all $\lambda$:

$$
y(t) = \int_{-\infty}^{\infty}x(\lambda)\,h(t-\lambda)\,d\lambda = x(t) * h(t)
$$

This is the **convolution integral**. With $\mu = t - \lambda$ it can also be written $y(t) = \int h(\mu)x(t-\mu)d\mu = h(t) * x(t)$.

For a **causal** system with input starting at $t = 0$, the limits reduce to $y(t) = \int_0^t x(\lambda)h(t-\lambda)d\lambda$. In the frequency domain the result becomes $Y(f) = X(f)H(f)$.

## Check yourself

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

1. Which system is non-causal?
   A. y(t)=x(t-1)
   B. y(t)=5x(t)
   C. y(t)=x(t+1)
   D. y(t)=x(t)+x(t-2)
   Answer: C. x(t+1) needs a future input. (Key point: Causal system)

2. What is the first step in proving y = x * h?
   A. Take the Laplace transform
   B. Write x(t) as an integral of weighted shifted impulses using the sifting property
   C. Assume the system is non-linear
   D. Differentiate h(t)
   Answer: B. x(t)=integral x(lambda) delta(t-lambda) d lambda. (Key point: Step 1 of proof)

3. Which property turns delta(t-lambda) into h(t-lambda)?
   A. Time invariance
   B. Homogeneity
   C. Causality
   D. Additivity
   Answer: A. Time invariance makes a shifted impulse give a shifted response. (Key point: Role of time invariance)

4. Which gives the output?
   A. y(t)=x(t)h(t)
   B. y(t)=integral x(lambda) h(t-lambda) d lambda
   C. y(t)=integral x(lambda) h(lambda) d lambda
   D. y(t)=x(t)+h(t)
   Answer: B. Superposition integrates the individual responses to give the convolution integral. (Key point: Final result)

## Related in Communication Systems

- [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
- [Matched filter impulse response is reverse delayed input](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-reverse-delayed) — Communication Systems, chapter 9, asked 2×
- [Classification of systems; ideal LPF is non-causal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aclassify-systems-ideal-lpf) — Communication Systems, chapter 1
- [Hilbert transform; ideal LPF is non-causal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-ideal-lpf-noncausal) — Communication Systems, chapter 1
- [LTI output of a WSS input is WSS](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Alti-wssp-output) — Communication Systems, chapter 9
- [Analog and digital system block diagrams; linear and non-linear distortion](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aanalog-digital-block-linear-nonlinear-distortion) — Communication Systems, chapter 1

## Related in other subjects

- [Convolution sum; ideal LPF is non-causal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-sum-ideal-lpf-noncausal) — Signals and Systems, chapter 5
- [Causal, non-causal and anticausal systems](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Acausal-anticausal) — Data Communication, chapter 2, asked 2×
- [Short note: Causal and non-causal systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Ashort-note-causal-noncausal) — Signals and Systems, chapter 5
- [First-order system impulse and step response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afirst-order-system-response) — Signals and Systems, chapter 5