---
title: "SQNR of uniformly quantized PCM"
subject: "Communication Systems"
chapter: 4
key: "semester_6/communication_systems:sqnr-uniform-pcm-bits"
source: https://www.bctengineer.com/q/semester_6/communication_systems/sqnr-uniform-pcm-bits
site: BCT Engineering
---

# SQNR of uniformly quantized PCM

Communication Systems · Chapter 4: Pulse Modulation

- Asked in: 2071 Magh (old course), 6+2 marks
- Video: [SNR in PCM | Quantization noise in PCM | Sgnal to noise ratio in PCM](https://www.youtube.com/watch?v=NHlhFWeFebM) — Techno Tutorials ( e-Learning)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/sqnr-uniform-pcm-bits
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-4

## Question

Derive the expression for the SQNR of uniformly quantized PCM. What is the relation between SQNR value and bit used for coding?

## Short answer

Uniform PCM with L=2^n levels and peak mp: step Δ = 2mp/L, noise power Δ²/12 = mp²/(3L²). SQNR = 3(P/mp²) 2^{2n}. For a full-scale sinusoid, P=mp²/2, giving SQNR_dB = 1.76 + 6.02n dB. Thus each added bit improves SQNR by 6 dB.

## Answer

#### SQNR of uniformly quantized PCM

Let the message $m(t)$ have peak amplitude $m_p$ (range $-m_p$ to $+m_p$) and be quantized uniformly into $L$ levels.

**Step size:**

$$
\Delta = \frac{2m_p}{L}
$$

**Quantization noise.** The error $q$ is uniformly distributed over $(-\Delta/2, \Delta/2)$ with pdf $1/\Delta$:

$$
N_q = E[q^2] = \int_{-\Delta/2}^{\Delta/2} q^2\frac{1}{\Delta}\,dq = \frac{\Delta^2}{12} = \frac{m_p^2}{3L^2}
$$

**Signal-to-quantization noise ratio.** With signal power $P = \overline{m^2(t)}$:

$$
\text{SQNR} = \frac{P}{N_q} = 3L^2\frac{P}{m_p^2}
$$

**In terms of bits.** For $n$-bit binary coding, $L = 2^n$:

$$
\text{SQNR} = \frac{3P}{m_p^2}\,2^{2n}, \qquad \text{SQNR}_{dB} = 10\log_{10}\frac{3P}{m_p^2} + 6.02\,n
$$

**Sinusoidal message** with full-scale amplitude, $P = m_p^2/2$:

$$
\text{SQNR} = 1.5\times 2^{2n}, \qquad \text{SQNR}_{dB} = 1.76 + 6.02\,n\ \text{dB}
$$

(For a uniformly distributed signal, $P = m_p^2/3$ and SQNR $= 2^{2n}$, i.e. $6.02n$ dB.)

#### Relation between SQNR and bits

- SQNR grows **exponentially** with $n$ ($\propto 4^n$); in dB it grows **linearly**: each added bit improves SQNR by about **6 dB**.
- Example: 7 bits → 43.9 dB; 8 bits → 49.9 dB (for a full-scale sine).
- But bandwidth $B_T = nf_s/2$ grows linearly with $n$. So SQNR improves exponentially with bandwidth: a key trade-off in PCM.

## Check yourself

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

1. What is N_q for uniformly quantized PCM?
   A. Delta^2/12 = mp^2/(3L^2)
   B. mp^2/L^2
   C. Delta^2/3
   D. Delta/2
   Answer: A. Error uniform in (-Delta/2, Delta/2) with pdf 1/Delta. (Key point: Noise power)

2. What is SQNR in terms of L?
   A. L^2 P/(3 mp^2)
   B. 3 L P/mp^2
   C. 3 L^2 P/mp^2
   D. L P/mp^2
   Answer: C. P divided by mp^2/(3L^2). (Key point: SQNR formula)

3. How much does each extra bit improve SQNR?
   A. About 12 dB
   B. About 9 dB
   C. About 3 dB
   D. About 6 dB
   Answer: D. SQNR is proportional to 4^n, giving 6.02 dB per bit. (Key point: Per bit)

4. What is the cost of adding bits?
   A. Bandwidth n fs/2 increases linearly
   B. Nothing; it is free
   C. Bandwidth decreases
   D. Sampling rate must be halved
   Answer: A. More bits per sample raise the bit rate and bandwidth. (Key point: Trade-off)

## Related in Communication Systems

- [E1 digital hierarchy; SQNR in uniformly quantized PCM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ae1-hierarchy-sqnr-uniform-pcm) — Communication Systems, chapter 5, asked 2×
- [PCM SQNR increases 6 dB per bit](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apcm-sqnr-6db-per-bit) — Communication Systems, chapter 4
- [PCM vs DM SQNR for 6cos(5000πt) with 128 levels](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apcm-dm-6cos5000pit-128) — Communication Systems, chapter 4
- [E1 hierarchy; PCM SQNR at 32 kbps](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ae1-pcm-4khz-1-1v-32kbps) — Communication Systems, chapter 5
- [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
- [PCM bit rate for 4.5 MHz TV signal with 1024 levels](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apcm-tv-4-5mhz-1024) — Communication Systems, chapter 4

## Related in other subjects

- [Line coding in PCM and the right technique](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aline-coding-pcm-solution) — Telecommunications, chapter 3
- [Short note: Line coding principles in PCM](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Ashort-note-line-coding-pcm) — Telecommunications, chapter 3