---
title: "Quantization; SQNR for linear quantization"
subject: "Communication Systems"
chapter: 4
key: "semester_6/communication_systems:quantization-sqnr-linear"
source: https://www.bctengineer.com/q/semester_6/communication_systems/quantization-sqnr-linear
site: BCT Engineering
---

# Quantization; SQNR for linear quantization

Communication Systems · Chapter 4: Pulse Modulation

- Asked in: 2080 Bhadra (CS II), 2+6 marks
- Video: [Communication Engineering - Signal to Quantization Noise Ratio (SQNR) for Linear Quantization](https://www.youtube.com/watch?v=amWuU9spXmg) — Dr. Sapna Katiyar
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/quantization-sqnr-linear
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-4

## Question

Define quantization. Derive the expression for evaluating signal to quantization noise ratio (SQNR) for linear quantization.

## Short answer

Quantization approximates samples by the nearest of L levels with step Δ = 2mp/L. Error is uniformly distributed in ±Δ/2, giving noise power Δ²/12. For a full-scale sine, SQNR = 1.5 L² or 1.76 + 6.02n dB; each extra bit adds 6 dB.

## Answer

#### Quantization
**Quantization** is the process of approximating each continuous-amplitude sample of a signal by the nearest of a finite set of allowed levels. With $L$ levels spaced $\Delta$ apart it turns a sampled (discrete-time, continuous-amplitude) signal into a discrete-amplitude signal that can be coded with $n = \log_2 L$ bits. The difference between the true sample and the quantized value is the **quantization error** (quantization noise).

#### SQNR for linear (uniform) quantization
Let the message $m(t)$ lie in the range $-m_p$ to $+m_p$ and let the quantizer have $L$ equally spaced levels.

1. **Step size**
$$
\Delta = \frac{2m_p}{L}
$$
2. **Error range**: each sample is replaced by the nearest quantization level (the centre of its step), so the error $q$ lies between $-\Delta/2$ and $+\Delta/2$.
3. **Error pdf**: for a busy signal with many levels, $q$ is assumed uniformly distributed, $f_Q(q) = 1/\Delta$ for $|q| \le \Delta/2$.
4. **Quantization noise power** (mean square error):
$$
\begin{aligned}
N_q = \overline{q^2} &= \int_{-\Delta/2}^{\Delta/2} q^2 \frac{1}{\Delta}\,dq = \frac{1}{\Delta}\left[\frac{q^3}{3}\right]_{-\Delta/2}^{\Delta/2} \\
&= \frac{\Delta^2}{12} = \frac{(2m_p/L)^2}{12} = \frac{m_p^2}{3L^2}
\end{aligned}
$$
5. **SQNR**: with signal power $S = \overline{m^2(t)}$,
$$
\text{SQNR} = \frac{S}{N_q} = 3L^2\,\frac{\overline{m^2(t)}}{m_p^2}
$$
6. **Sinusoidal message** $m(t) = m_p\cos\omega_m t$ (full load): $S = m_p^2/2$, so
$$
\begin{aligned}
\text{SQNR} &= \frac{m_p^2/2}{m_p^2/(3L^2)} = \frac{3}{2}L^2 = 1.5 \times 2^{2n} \\
\text{SQNR}_{dB} &= 10\log_{10}1.5 + 20n\log_{10}2 \approx 1.76 + 6.02n\ \text{dB}
\end{aligned}
$$

**Key results**
- SQNR rises as $L^2$, i.e. by about 6 dB for each extra bit.
- For a signal whose power equals $m_p^2$ (e.g. a square-like signal) the result is $3L^2$, i.e. $4.77 + 6.02n$ dB.
- Example: 8-bit PCM with a full-scale sine gives $1.76 + 48.16 \approx 49.9$ dB.
- Weak signals give a much lower SQNR because $N_q$ is fixed while $S$ falls; this is why non-uniform quantization (companding) is used for speech.

## Check yourself

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

1. What does quantization do to a sampled signal?
   A. It makes time discrete by sampling at regular intervals
   B. It replaces each sample by the nearest of a finite set of allowed amplitude levels
   C. It converts the signal to a lower frequency band
   D. It removes noise by low-pass filtering
   Answer: B. Quantization makes the amplitude discrete by rounding each sample to the nearest allowed level; the difference is quantization noise. (Key point: Meaning of quantization)

2. For a message in the range -mp to +mp with L equal levels, the step size is:
   A. mp/L
   B. 2mp/L
   C. mp/(2L)
   D. 2mp/L^2
   Answer: B. The total range 2mp is divided into L equal steps, so Delta = 2mp/L. (Key point: Step size)

3. Assuming the error is uniform in (-Delta/2, +Delta/2), the quantization noise power is:
   A. Delta^2/12
   B. Delta^2/3
   C. Delta/2
   D. Delta^2/4
   Answer: A. Integrating q^2 times 1/Delta over the range gives Delta^2/12, which equals mp^2/(3L^2). (Key point: Noise power)

4. For a full-load sinusoid the SQNR with n bits is approximately:
   A. 1.76 + 3.01n dB
   B. 4.77 + 6.02n dB
   C. 1.76 + 6.02n dB
   D. 6.02 + 1.76n dB
   Answer: C. S = mp^2/2 gives SQNR = 1.5L^2, i.e. 1.76 + 6.02n dB. The 4.77 figure is for a signal with power mp^2. (Key point: SQNR of full-load sine)

## Close variants set in other exams

- [SQNR of delta modulation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adm-sqnr-derive) — Communication Systems, chapter 4, asked 2×

## Related in Communication Systems

- [SQNR for uniform quantization in terms of levels and bits](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asqnr-uniform-levels-bits) — Communication Systems, chapter 4, asked 2×
- [Need for non-uniform quantization; SQNR of DM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anonuniform-quantization-dm-sqnr) — Communication Systems, chapter 4, asked 2×
- [SQNR of delta modulation for a single-tone signal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adm-sqnr-single-tone) — Communication Systems, chapter 4
- [PCM, DPCM and DM comparison; SQNR of PCM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apcm-dpcm-dm-compare-sqnr) — Communication Systems, chapter 4
- [Non-uniform quantization; PCM numerical (4 kHz, ±2.4 V, 64 kHz)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anonuniform-companding-pcm-4khz-64khz) — Communication Systems, chapter 4
- [FM expression after change of modulating signal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afm-new-waveform-500hz-2khz) — Communication Systems, chapter 3

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