--- title: "Companding; T1 hierarchy of TDM-PCM telephony" subject: "Communication Systems" chapter: 5 key: "semester_6/communication_systems:companding-t1" source: https://www.bctengineer.com/q/semester_6/communication_systems/companding-t1 site: BCT Engineering --- # Companding; T1 hierarchy of TDM-PCM telephony Communication Systems · Chapter 5: Multiplexing Techniques - Asked in: 2071 Shrawan (CS II), 2+4 marks - Video: [T1 Digital Systems](https://www.youtube.com/watch?v=o4OOA9LkXPk) — Ashutosh Rasttogi - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/companding-t1 - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-5 ## Question What do you mean by companding? Explain T1 hierarchy of TDM-PCM telephony. ## Short answer Companding compresses weak signals and compresses strong ones before uniform quantization to improve SQNR. T1 hierarchy: 24 voice channels at 1.544 Mb/s, then DS2 6.312 Mb/s, DS3 44.736 Mb/s, DS4 274.176 Mb/s. ## Answer #### Companding **Companding** = **comp**ressing + exp**anding**. Before uniform quantisation, the signal passes through a compressor that boosts weak amplitudes and compresses strong ones. At the receiver, an expander applies the inverse curve. The result works like non-uniform quantisation: small step sizes for weak signals and large steps for loud ones. This gives a nearly constant SQNR over a wide range of speech levels, using only 8 bits. ```text Tx: m(t) -> [Compressor] -> [Uniform quantizer + encoder] ~~> channel Rx: channel ~~> [Decoder] -> [Expander] -> m(t) ``` Standard laws: **µ-law** ($\mu = 255$, North America/T1) and **A-law** ($A = 87.6$, Europe/E1). $$ y = \frac{\ln(1 + \mu |x|)}{\ln(1 + \mu)} \operatorname{sgn}(x), \quad |x| \le 1 $$ #### T1 hierarchy of TDM-PCM telephony **T1 (DS1)** multiplexes 24 PCM voice channels: ```text |<--------- one frame = 125 us = 193 bits --------->| +---+--------+--------+--------+-------+--------+ | F | Ch 1 | Ch 2 | Ch 3 | ... | Ch 24 | +---+--------+--------+--------+-------+--------+ 1 bit 8 bits 8 bits 8 bits 8 bits F = framing bit (pattern spread over 12 frames) ``` $$ \begin{aligned} \text{Bits per frame} &= 24 \times 8 + 1 = 193\ \text{bits} \\ \text{Frames per second} &= f_s = 8000\ \text{frames/s} \\ R_b &= 193 \times 8000 = 1\,544\,000\ \text{b/s} = 1.544\ \text{Mb/s} \end{aligned} $$ Higher levels are formed by multiplexing T1 streams and adding framing and stuffing bits: ```text 24 voice ch x 64 kb/s | [ DS1 / T1 ] 1.544 Mb/s 24 ch | x4 [ DS2 / T2 ] 6.312 Mb/s 96 ch | x7 [ DS3 / T3 ] 44.736 Mb/s 672 ch | x6 [ DS4 / T4 ] 274.176 Mb/s 4032 ch ``` | Level | Made of | Channels | Rate | |---|---|---|---| | DS1 (T1) | 24 voice | 24 | 1.544 Mb/s | | DS1C | 2 DS1 | 48 | 3.152 Mb/s | | DS2 (T2) | 4 DS1 | 96 | 6.312 Mb/s | | DS3 (T3) | 7 DS2 | 672 | 44.736 Mb/s | | DS4 (T4) | 6 DS3 | 4032 | 274.176 Mb/s | ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does companding achieve in PCM for speech? A. It removes the need for sampling B. It reduces the bandwidth of the channel to zero C. It makes quantisation effectively non-uniform, giving nearly constant SQNR over a wide range of signal levels D. It increases the number of bits to 16 Answer: C. The compressor boosts weak signals before uniform quantisation, so weak signals get small steps; the expander undoes it at the receiver. (Key point: Purpose of companding) 2. What is the role of the expander at the receiver? A. Applies the inverse of the compressor curve B. Applies the same curve as the compressor C. Adds noise to hide quantisation D. Samples the signal again Answer: A. The expander has the inverse characteristic of the compressor, restoring the original signal amplitudes. (Key point: Receiver side) 3. How is the T1 rate of 1.544 Mb/s obtained? A. 24 x 8 x 8000 B. (24 x 8 + 1) x 8000 C. 32 x 8 x 8000 D. (24 x 7 + 1) x 8000 Answer: B. A T1 frame has 24 x 8 + 1 = 193 bits (the extra bit is the framing bit), sent 8000 times per second = 1.544 Mb/s. (Key point: T1 frame and rate) 4. Which is the correct T1 hierarchy build-up of DS2, DS3 and DS4? A. DS2 = 4 DS1, DS3 = 7 DS2, DS4 = 6 DS3 B. DS2 = 2 DS1, DS3 = 4 DS2, DS4 = 4 DS3 C. DS2 = 4 DS1, DS3 = 4 DS2, DS4 = 4 DS3 D. DS2 = 6 DS1, DS3 = 7 DS2, DS4 = 4 DS3 Answer: A. DS2 = 4 DS1 (96 ch, 6.312 Mb/s), DS3 = 7 DS2 (672 ch, 44.736 Mb/s), DS4 = 6 DS3 (4032 ch, 274.176 Mb/s). (Key point: T1 hierarchy) ## Related in Communication Systems - [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 - [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 - [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× - [Short note: T1 and E1 hierarchy](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashort-note-t1-e1-hierarchy) — Communication Systems, chapter 5 - [FDM and FDM hierarchy in telephony](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afdm-hierarchy-telephony) — Communication Systems, chapter 5, asked 4× - [TDM vs FDM for voice; E1 and T1 hierarchies](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Atdm-fdm-voice-e1-t1) — Communication Systems, chapter 5 ## Related in other subjects - [TDM principle and T1 carrier](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Atdm-principle-t1-carrier) — Telecommunications, chapter 3 - [Pulse stuffing and complete TDM PCM system](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Amultiplexing-pulse-stuffing-tdm-pcm) — Telecommunications, chapter 3 - [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× - [TDM and T1/E1 carrier standards](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Atdm-t1-e1-carriers) — Telecommunications, chapter 3