---
title: "Invertibility; properties of y[n], y(t)"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:invertibility-system-properties"
source: https://www.bctengineer.com/q/semester_5/data_communication/invertibility-system-properties
site: BCT Engineering
---

# Invertibility; properties of y[n], y(t)

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2068 Chaitra, 6 marks
- Video: [Basic System Properties: Invertibility and Inverse System - Continuous and Discrete Time Systems](https://www.youtube.com/watch?v=06-xo-xAELs) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/invertibility-system-properties
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Define invertibility of a system. Which properties (Linearity, Causality, Time invariance, Stability and Memory) are possessed by the following systems (i) y[n] = 3x[n] - 4x[n-1] (ii) y(t) = sin(2πx(t)).

## Short answer

Invertibility: distinct inputs give distinct outputs. (i) linear, causal, time-invariant, stable, with memory. (ii) nonlinear, causal, time-invariant, stable, memoryless.

## Answer

#### Invertibility
A system is **invertible** if distinct inputs always produce distinct outputs, so that the input can be recovered uniquely from the output by an inverse system. Example: y(t) = 2x(t) is invertible (x = y/2); y(t) = x²(t) is not (x and -x give the same y).

#### (i) y[n] = 3x[n] - 4x[n - 1]
- **Linear**: yes (a weighted sum of inputs).
- **Causal**: yes (uses only the present and the previous input).
- **Time-invariant**: yes (constant coefficients).
- **Stable**: yes (an FIR system; |y| ≤ 7M for |x| ≤ M).
- **Memory**: yes (depends on x[n - 1]).
- It is also invertible: x[n] = (y[n] + 4x[n - 1])/3, though the inverse system is unstable.

#### (ii) y(t) = sin(2π x(t))
- **Linear**: no, because sin(2π(a·x₁ + b·x₂)) ≠ a·sin(2πx₁) + b·sin(2πx₂).
- **Causal**: yes (depends on the present input only).
- **Time-invariant**: yes (a delayed input gives the same delayed output).
- **Stable**: yes, since |y(t)| ≤ 1 for every input.
- **Memory**: no, it is memoryless.
- Not invertible: x(t) and x(t) + 1 give the same output.

| Property | (i) | (ii) |
|---|---|---|
| Linearity | Yes | No |
| Causality | Yes | Yes |
| Time invariance | Yes | Yes |
| Stability | Yes | Yes |
| Memory | Yes | No (memoryless) |

## Check yourself

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

1. A system is invertible if
   A. Output is bounded
   B. It is linear
   C. It is causal
   D. Distinct inputs give distinct outputs
   Answer: D. Definition. (Key point: Invertibility)

2. y[n] = 3x[n] - 4x[n-1] is
   A. Linear, causal, time-invariant, stable, with memory
   B. Memoryless
   C. Nonlinear
   D. Non-causal
   Answer: A. FIR system. (Key point: (i) properties)

3. y = sin(2 pi x(t)) is
   A. Linear and memoryless
   B. Nonlinear and memoryless
   C. Nonlinear with memory
   D. Linear with memory
   Answer: B. sin is nonlinear and no past values. (Key point: (ii) linearity)

4. Why is (ii) not invertible?
   A. It is nonlinear
   B. It is memoryless
   C. x and x+1 give the same output
   D. It is bounded
   Answer: C. Not one-to-one. (Key point: (ii) invertibility)

## Related in Data Communication

- [Basic properties of systems](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Asystem-properties) — Data Communication, chapter 2, asked 3×
- [Causality, stability; energy and power signals](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Acausality-stability-energy-power) — Data Communication, chapter 2
- [Check properties of y(t) = 2x(t+1)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Acheck-2x-t-plus-1) — Data Communication, chapter 2
- [Recursive/nonrecursive systems; stability](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Arecursive-nonrecursive-stability) — Data Communication, chapter 2
- [Stable/unstable systems; test stability](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Astability-define-test-72c) — Data Communication, chapter 2
- [Properties of continuous-time Fourier series](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afourier-series-properties) — Data Communication, chapter 3

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