---
title: "Is y[n] = ax[n] + b linear"
subject: "Signals and Systems"
chapter: 6
key: "semester_4/signals_and_systems:linearity-affine-dt"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/linearity-affine-dt
site: BCT Engineering
---

# Is y[n] = ax[n] + b linear

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2080 Asoj, 3 marks; 2079 Asoj, 3 marks; 2080 Chaitra, 4 marks; 2078 Chaitra, 3 marks (asked 4 times)
- Video: [Linear and Non-Linear Systems (Solved Problems) | Part 1](https://www.youtube.com/watch?v=QcAY1oPozhw) — 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/linearity-affine-dt
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

Determine if the following system is linear y[n] = ax[n] + b

## Short answer

The system is linear only if b = 0; otherwise it fails superposition and the zero-input test. For b ≠ 0 it is incrementally linear, consisting of a linear part plus a constant offset.

## Answer

A system is **linear** if it obeys superposition: $a_1x_1[n] + a_2x_2[n] \rightarrow a_1y_1[n] + a_2y_2[n]$.

**Given:** $y[n] = a\,x[n] + b$

**Individual outputs:** $y_1[n] = a\,x_1[n] + b$ and $y_2[n] = a\,x_2[n] + b$.

**Weighted sum of outputs:**

$$
a_1y_1[n] + a_2y_2[n] = a\left(a_1x_1[n] + a_2x_2[n]\right) + (a_1 + a_2)\,b
$$

**Output for the combined input** $x_3[n] = a_1x_1[n] + a_2x_2[n]$:

$$
y_3[n] = a\left(a_1x_1[n] + a_2x_2[n]\right) + b
$$

These are equal only if $(a_1 + a_2)b = b$ for all $a_1, a_2$, i.e. only if $b = 0$.

**Zero-input check:** $x[n] = 0$ gives $y[n] = b$. A linear system must give zero output for zero input.

**Conclusion:**
- For $b \ne 0$ the system is **non-linear**. It is **incrementally linear**: the difference between two outputs, $y_1 - y_2 = a(x_1 - x_2)$, is linear in the difference of inputs. It can be seen as a linear system $a\,x[n]$ plus a fixed zero-input response $b$.
- For $b = 0$, $y[n] = a\,x[n]$ is linear.

## Check yourself

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

1. Which test is applied to decide linearity?
   A. Whether a shifted input gives a shifted output
   B. Whether the output for a1x1 + a2x2 equals a1y1 + a2y2
   C. Whether h[n] = 0 for n < 0
   D. Whether the output is bounded
   Answer: B. Superposition compares the response to the combined input with the combined responses. (Key point: Test for linearity)

2. What is a1y1 + a2y2 for y = ax + b?
   A. a(a1x1 + a2x2) + b
   B. a(a1x1 + a2x2) + (a1 + a2) b
   C. a(a1x1 + a2x2)
   D. (a1 + a2) a x + b
   Answer: B. Each output carries its own b, which are scaled by a1 and a2. (Key point: Weighted sum of outputs)

3. What does x[n] = 0 give for y = ax + b?
   A. 0
   B. a
   C. y[n] = b, which is non-zero for b != 0, violating linearity
   D. infinity
   Answer: C. A linear system must give zero output for zero input. (Key point: Zero input)

4. What is the correct classification for b not equal to 0?
   A. Non-linear but incrementally linear
   B. Linear and time varying
   C. Linear for all b
   D. Non-causal
   Answer: A. The difference of outputs is a(x1 - x2), so the system is incrementally linear only. (Key point: Conclusion)

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