---
title: "Quantization noise and quantization error"
subject: "Digital Control System"
chapter: 1
key: "semester_6/digital_control_system:quantization-noise-error"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/quantization-noise-error
site: BCT Engineering
---

# Quantization noise and quantization error

Digital Control System · Chapter 1: Introduction to Discrete Time Control System

- Asked in: 2076 Bhadra, 4 marks
- Video: [Quantization (Basics, Working Principle, Waveforms, Quantization Error & Quantization Noise)](https://www.youtube.com/watch?v=cUdnGkymAWU) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/digital_control_system/quantization-noise-error
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-1

## Question

What is quantization noise and quantization error in digital control system? Explain with appropriate example.

## Short answer

Quantization error is the bounded difference between actual and quantized sample values, within ±Q/2. Quantization noise models this error as uniform random noise with zero mean and variance Q²/12. For a 3-bit ADC with 8V range, Q=1V giving noise power 0.0833 V².

## Answer

**Quantization** in an A/D converter maps each sampled value to the nearest of $2^n$ discrete levels. The step size (quantum) is

$$
Q = \frac{FSR}{2^n}
$$

where $FSR$ is the full-scale range and $n$ the number of bits.

#### Quantization error

The difference between the actual sample and its quantized value:

$$
e(kT) = x(kT) - x_q(kT)
$$

With rounding, $-\frac{Q}{2} \le e \le \frac{Q}{2}$ (with truncation, $0 \le e < Q$). It is a fixed, bounded error for each sample.

#### Quantization noise

When the signal varies enough, the error changes randomly from sample to sample and can be treated as **noise** added to the signal: uniformly distributed over $\pm Q/2$, zero mean, with variance (noise power)

$$
\sigma^2 = \frac{Q^2}{12}
$$

The signal-to-quantization-noise ratio improves by about 6 dB for each extra bit.

#### Example

A 3-bit ADC with $FSR = 8$ V has $Q = 8/2^3 = 1$ V.

| Input (V) | Quantized level (V) | Code | Error (V) |
|---|---|---|---|
| 2.3 | 2 | 010 | +0.3 |
| 4.6 | 5 | 101 | -0.4 |
| 6.5 | 6 or 7 | 110/111 | ±0.5 |

Noise power $= 1^2/12 = 0.0833\ \text{V}^2$ (rms $\approx 0.289$ V). Using an 8-bit ADC gives $Q = 31.25$ mV, so the error falls to at most $\pm 15.6$ mV.

## Check yourself

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

1. With rounding, the quantization error lies between:
   A. -Q/2 and +Q/2
   B. 0 and Q
   C. -Q and +Q
   D. 0 and Q/2
   Answer: A. Rounding to the nearest level gives error up to half a step either side; truncation gives 0 to Q. (Key point: Error range)

2. Quantization noise power (variance) is:
   A. Q^2/4
   B. Q^2/12
   C. Q/12
   D. Q^2/2
   Answer: B. The error is uniform over +-Q/2, giving variance Q^2/12. (Key point: Noise power)

3. A 3-bit ADC with FSR = 8 V has Q = :
   A. 2 V
   B. 8 V
   C. 1 V
   D. 0.125 V
   Answer: C. Q = 8/2^3 = 1 V, so noise power is 1/12 = 0.0833 V^2. (Key point: Example)

4. Quantization error becomes usefully treated as noise when:
   A. The signal is constant
   B. The ADC has only one bit
   C. The sampling period is zero
   D. The signal varies enough that the error changes randomly sample to sample
   Answer: D. A fixed signal gives a fixed error; a varying signal makes it behave randomly. (Key point: Error vs noise)

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