---
title: "E1 digital hierarchy; SQNR in uniformly quantized PCM"
subject: "Communication Systems"
chapter: 5
key: "semester_6/communication_systems:e1-hierarchy-sqnr-uniform-pcm"
source: https://www.bctengineer.com/q/semester_6/communication_systems/e1-hierarchy-sqnr-uniform-pcm
site: BCT Engineering
---

# E1 digital hierarchy; SQNR in uniformly quantized PCM

Communication Systems · Chapter 5: Multiplexing Techniques

- Asked in: 2072 Chaitra (CS II), 4+4 marks; 2071 Chaitra (CS II), 4+4 marks (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/e1-hierarchy-sqnr-uniform-pcm
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-5

## Question

Explain E1 digital hierarchy as related to telephony system. Evaluate the expression of SQNR in uniformly quantized PCM system.

## Short answer

E1 frame has 32 slots with TS0 for sync and TS16 for signalling, carrying 30 voice channels at 2.048 Mbps; higher levels multiplex 4 lower streams. For uniform PCM, quantization noise Δ²/12 gives SQNR = 3L²(̅m²/m_p²); for a full-load sinusoid it is 1.76+6.02n dB.

## Answer

#### E1 digital hierarchy
E1 is the ITU-T (CEPT) primary PCM-TDM multiplex used in Europe and most of Asia, including Nepal.

**E1 frame**
- Voice (300–3400 Hz) sampled at 8 kHz, A-law companded, 8 bits/sample: 64 kbps per channel.
- Frame duration 125 µs, **32 time slots** × 8 bits = 256 bits.
- **TS0:** frame synchronisation/alignment; **TS16:** signalling (multiframe of 16 frames); **TS1–15 and TS17–31:** 30 voice channels.
- Rate $= 256 \times 8000 = 2.048$ Mbps.

```text
|TS0|TS1 ... TS15|TS16|TS17 ... TS31|  125 us
sync   voice 1-15  sig   voice 16-30
```

**Higher orders (PDH)** multiplex 4 tributaries each, plus framing and justification bits:

| Level | Composition | Voice ch | Rate |
|---|---|---|---|
| E1 | 30 + 2 slots | 30 | 2.048 Mbps |
| E2 | 4 × E1 | 120 | 8.448 Mbps |
| E3 | 4 × E2 | 480 | 34.368 Mbps |
| E4 | 4 × E3 | 1920 | 139.264 Mbps |

(The North American counterpart is T1: 24 channels, 1.544 Mbps.)

#### SQNR of uniformly quantized PCM
Message range $\pm m_p$, $L = 2^n$ levels.

1. Step size: $\Delta = \dfrac{2m_p}{L}$.
2. Error $q$ uniform in $(-\Delta/2, \Delta/2)$, pdf $1/\Delta$:
$$
N_q = \int_{-\Delta/2}^{\Delta/2}\frac{q^2}{\Delta}\,dq = \frac{\Delta^2}{12} = \frac{m_p^2}{3L^2}
$$
3. Signal power $S = \overline{m^2}$:
$$
\text{SQNR} = \frac{S}{N_q} = 3L^2\frac{\overline{m^2}}{m_p^2}
$$
4. For a full-load sinusoid, $\overline{m^2} = m_p^2/2$:
$$
\begin{aligned}
\text{SQNR} &= \frac{3}{2}L^2 = 1.5 \times 2^{2n} \\
\text{SQNR}_{dB} &= 1.76 + 6.02n\ \text{dB}
\end{aligned}
$$

So SQNR increases by about 6 dB for each added bit. Example: for E1 channels with 8-bit uniform coding, $1.76 + 48.16 = 49.9$ dB for a full-scale tone (companding keeps it high for weak signals too).

## Check yourself

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

1. How is an E1 frame organised?
   A. 24 slots of 8 bits plus 1 framing bit
   B. 32 slots of 8 bits: TS0 sync, TS16 signalling, 30 voice slots
   C. 32 slots of 8 bits, all carrying voice
   D. 30 slots of 8 bits plus 2 framing bits
   Answer: B. E1 has 32 time slots; TS0 is for frame alignment, TS16 for signalling, leaving 30 voice channels. (Key point: E1 frame structure)

2. What is the E1 bit rate and why?
   A. 1.544 Mbps = 193 x 8000
   B. 2.048 Mbps = 256 x 8000
   C. 2.048 Mbps = 256 x 4000
   D. 8.448 Mbps = 4 x 2.048
   Answer: B. 256 bits per 125 microsecond frame at 8000 frames/s gives 2.048 Mbps. (Key point: E1 rate)

3. For uniform quantization with step size Delta, what is the mean quantization noise power?
   A. Delta^2/3
   B. Delta^2/12
   C. Delta/12
   D. Delta^2/2
   Answer: B. Error is uniform over (-Delta/2, Delta/2); integrating q^2/Delta gives Delta^2/12. (Key point: Quantization noise power)

4. For a full-load sinusoid in n-bit uniform PCM, what is SQNR in dB?
   A. 1.76 + 6.02n
   B. 6.02 + 1.76n
   C. 3 + 6n^2
   D. 1.76 + 3.01n
   Answer: A. SQNR = 1.5 x 2^(2n), which is 1.76 + 6.02n dB, i.e. about 6 dB per added bit. (Key point: SQNR in dB)

## Related in Communication Systems

- [SQNR of uniformly quantized PCM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asqnr-uniform-pcm-bits) — 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
- [E1 hierarchy; PCM of 4.2 MHz TV signal with 512 levels](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ae1-pcm-tv-4-2mhz-512) — Communication Systems, chapter 4
- [T1 and E1 TDM-PCM hierarchy with frame diagrams](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3At1-e1-frame-diagrams) — Communication Systems, chapter 5
- [Companding; T1 hierarchy of TDM-PCM telephony](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Acompanding-t1) — Communication Systems, chapter 5
- [Short note: E1 digital hierarchy](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashort-note-e1-hierarchy) — Communication Systems, chapter 5

## Related in other subjects

- [E1 telephone hierarchy](https://www.bctengineer.com/question.md?key=semester_7%2Fcomputer_networks_and_security%3Ae1-hierarchy) — Computer Networks and Security, chapter 2
- [North American T1 vs European E1 hierarchy](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3At1-e1-hierarchy-compare) — Telecommunications, chapter 3, asked 2×