---
title: "Short note: Invertibility of LTI system"
subject: "Signals and Systems"
chapter: 5
key: "semester_4/signals_and_systems:short-note-invertibility"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/short-note-invertibility
site: BCT Engineering
---

# Short note: Invertibility of LTI system

Signals and Systems · Chapter 5: Continuous-Time Systems

- Asked in: 2073 Bhadra, 3 marks
- Video: [Invertible and Non-Invertible Systems](https://www.youtube.com/watch?v=tq353ew4gEw) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/short-note-invertibility
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5

## Question

Write a short note on invertibility of LTI system.

## Short answer

LTI system is invertible if there exists h_inv with h*h_inv=δ, equivalently H(ω)≠0 for all ω and H_inv=1/H. Examples: delay inverse advance, integrator inverse differentiator. Not invertible if information lost, e.g. ideal LPF.

## Answer

A system is **invertible** if distinct inputs always give distinct outputs, so that the input can be recovered exactly from the output. The system that recovers it is called the **inverse system**.

```text
x(t) --> [ h(t) ] --> y(t) --> [ h_inv(t) ] --> x(t)
```

**Condition for an LTI system:** the system $h(t)$ is invertible if there is an LTI system $h_{inv}(t)$ such that the cascade gives the identity system:

$$
h(t) * h_{inv}(t) = \delta(t)
$$

In the frequency domain: $H(\omega)\,H_{inv}(\omega) = 1$, so $H_{inv}(\omega) = 1/H(\omega)$. This needs $H(\omega) \ne 0$ at every frequency.

**Examples:**
- $y(t) = x(t - t_0)$, $h(t) = \delta(t-t_0)$: inverse $h_{inv}(t) = \delta(t+t_0)$.
- Integrator $h(t) = u(t)$: inverse is the differentiator, $y(t) = \frac{dx}{dt}$.
- $y[n] = x[n] - x[n-1]$: inverse is the accumulator $y[n] = \sum_{k=-\infty}^{n} x[k]$.
- **Not invertible:** $y(t) = 0$, $y(t) = x^2(t)$ (sign is lost), an ideal LPF (high frequencies are lost).

**Use:** channel equalization in communication, deconvolution, decoding.

## Check yourself

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

1. Invertible LTI: h*h_inv =
   A. u(t)
   B. 1
   C. δ(t)
   D. 0
   Answer: C. Identity. (Key point: Condition)

2. Inverse of integrator:
   A. Delay
   B. Accumulator
   C. Amplifier
   D. Differentiator
   Answer: D. Derivative undoes integral. (Key point: Integrator)

3. Not invertible:
   A. y=x^2(t)
   B. y=x(t-t0)
   C. y=2x
   D. y=x[n]-x[n-1]
   Answer: A. Sign is lost. (Key point: Non-invertible)

4. Needs H(ω):
   A. Zero at some
   B. Non-zero at every frequency
   C. Real
   D. Even
   Answer: B. 1/H must exist. (Key point: Frequency)

## Related in Signals and Systems

- [Invertibility of LTI systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-invertibility-condition) — Signals and Systems, chapter 5
- [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×
- [LTI system: linearity, stability, TI, causality](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-properties-four) — Signals and Systems, chapter 5, asked 2×
- [Causality condition for CT LTI system](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acausal-ct-lti-condition) — Signals and Systems, chapter 5

## Related in other subjects

- [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×
- [Plot sequence; properties of LTI system](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aplot-sequence-lti-properties) — Digital Signal Analysis and Processing, chapter 1
- [Parseval's theorem; output PSD of LTI system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aparseval-power-lti-psd) — Communication Systems, chapter 1
- [Hilbert transformation; properties of LTI system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-define-lti-properties) — Communication Systems, chapter 1