Multiplexing Techniques

FDM in telephony combines many voice channels onto one wideband link (coaxial cable, microwave) by giving each a separate frequency slot. It was the standard in analog long-distance telephone networks.

Principle

• Each voice signal is band-limited to 300–3400 Hz and given a 4 kHz slot (the extra 900 Hz is a guard band).
• Each channel SSB-SC modulates (usually LSB) a different subcarrier; SSB halves the bandwidth compared with DSB.
• The outputs are added and sent together; at the receiver band-pass filters separate them and synchronous demodulators recover each voice.

ch1->[SSB mod f1]->[BPF]-+
ch2->[SSB mod f2]->[BPF]-+->(sum)-> line
 ...                     |
ch12->[SSB f12]->[BPF]---+

Hierarchy (ITU-T/CCITT)

In the basic group, carriers are 64, 68, ..., 108 kHz with LSB selection.

Requirements: sharp SSB filters (crystal/mechanical), stable carriers derived from one master oscillator, pilot tones for level and frequency control, and linear amplifiers to avoid intermodulation crosstalk.

FDM telephony has largely been replaced by digital TDM (E1/T1, SDH), but the principle remains in radio and cable systems.

In an FDM system the channels are separated only by frequency, so the quality of the filters and carrier oscillators decides how much crosstalk and distortion occur.

voice->[Mod]<-osc fc1  ->[BPF]-+
voice->[Mod]<-osc fc2  ->[BPF]-+-> composite
           (from one master)   |

Filter requirements

• Sharp cut-off: SSB generation needs filters that pass one sideband and reject the other, which is only a few hundred hertz away (e.g. 300 Hz lower edge of voice). Very steep skirts are needed, so crystal, ceramic or mechanical filters with high Q are used.
• Flat passband and low ripple across 300–3400 Hz so all speech frequencies have equal gain.
• Linear phase / low group-delay distortion, especially for data and video channels.
• High stop-band attenuation (about 60 dB) to keep adjacent-channel crosstalk low.
• Guard bands (900 Hz per voice channel) relax the filter slope.
• Stability with temperature and ageing.
• Multi-stage modulation (group → supergroup → mastergroup) is used partly so that filters work at low frequencies where sharp filters are practical.

Oscillator requirements

• High frequency stability and accuracy: in SSB-SC any carrier offset between transmitter and receiver shifts every voice frequency by the same amount; only a few hertz of error is tolerable for speech and less for data.
• All carriers are therefore derived from a single crystal master oscillator by frequency multiplication/division (frequency synthesis), so they are locked to each other.
• Low phase noise and harmonics, since spurious outputs cause interchannel interference.
• Synchronization: a pilot carrier is sent so the receiver can lock its oscillators to the transmitter's.
• Carriers must have stable amplitude for constant modulator gain.

FDM

Frequency Division Multiplexing (FDM) is a technique in which several signals share one channel by being placed in different, non-overlapping frequency bands, each modulating its own carrier. All signals are transmitted simultaneously; guard bands between them prevent interference, and band-pass filters separate them at the receiver.

 |ch1|g|ch2|g|ch3|g| ... |chN|   g = guard band
 f1      f2      f3          fN   --> frequency

FDM hierarchy in telephony

In analog telephony, each voice signal (300–3400 Hz) is allotted 4 kHz and translated by SSB-SC (LSB) modulation. Channels are combined step by step into larger blocks. The ITU-T (CCITT) hierarchy:

12 voice ch --> Group (60-108 kHz)
5 Groups    --> Supergroup (312-552 kHz)
5 Supergrps --> Mastergroup (812-2044 kHz)
3 Mastergrps--> Supermastergroup (8516-12388 kHz)

1. Basic group

• 12 voice channels.
• Carriers at 64, 68, 72, ..., 108 kHz; lower sidebands selected.
• Occupies 60–108 kHz (48 kHz bandwidth).

2. Supergroup

• 5 groups = 60 channels.
• Each group modulates carriers 420, 468, 516, 564, 612 kHz (LSB).
• Occupies 312–552 kHz (240 kHz).

3. Mastergroup

• 5 supergroups = 300 channels (CCITT).
• Occupies 812–2044 kHz, with 8 kHz guard bands between supergroups.

4. Supermastergroup

• 3 mastergroups = 900 channels, occupying 8516–12388 kHz.

AT&T (North American) version: mastergroup = 10 supergroups = 600 channels (564–3084 kHz); jumbo group = 6 mastergroups = 3600 channels (0.564–17.548 MHz).

Why a hierarchy: the same standard equipment (channel banks, group and supergroup translators) is reused; filters at each stage work at practical frequencies; blocks of channels can be routed as a unit.

Requirements: sharp SSB filters, carriers derived from one master oscillator, pilot tones for gain/frequency control, linear wideband amplifiers to limit intermodulation.

FDMA (Frequency Division Multiple Access) in satellite communication lets many earth stations share one satellite transponder by giving each station its own frequency band within the transponder bandwidth. All stations transmit continuously and at the same time.

   Transponder (e.g. 36 MHz)
 |  ES-A  |g|  ES-B  |g|  ES-C  |g| ...
 <------------ frequency ------------>

Types

• MCPC (Multiple Channels Per Carrier): each earth station FDM-multiplexes many voice channels onto one carrier (FDM/FM/FDMA). Pre-assigned, suited to heavy fixed routes.
• SCPC (Single Channel Per Carrier): each voice/data channel has its own carrier. Can be demand assigned (DAMA), e.g. INTELSAT SPADE system: about 800 channels at 45 kHz spacing in a 36 MHz transponder, assigned on request.

Features and problems

• Simple: no network-wide timing or synchronization needed.
• Intermodulation: the transponder's TWT amplifier is nonlinear; many carriers produce intermodulation products, so the amplifier is operated with input/output back-off (a few dB), which reduces capacity and power.
• Guard bands between carriers waste spectrum.
• Power control needed so that a strong carrier does not suppress weak ones (capture effect).
• Capacity falls sharply as the number of carriers rises.

Comparison: FDMA is the oldest and simplest satellite access method; TDMA uses the transponder more efficiently (single carrier at saturation) but needs precise timing; CDMA allows overlapping use with spread spectrum.

Frequency Division Multiplexing (FDM) sends several signals at the same time over one channel by giving each signal a separate frequency band. Each message modulates a different carrier; the modulated signals are added and transmitted together.

Tx: m1->[Mod fc1]-+          Rx: +->[BPF1]->[Demod]->m1
    m2->[Mod fc2]-+->(sum)->ch->-+->[BPF2]->[Demod]->m2
    m3->[Mod fc3]-+              +->[BPF3]->[Demod]->m3

Working

1. Each message is band-limited by a LPF.
2. It modulates its own subcarrier (often SSB-SC to save bandwidth).
3. Band-pass filters limit each modulated signal to its slot; guard bands separate adjacent slots.
4. All are summed and sent (possibly modulating a main carrier).
5. At the receiver, band-pass filters separate the channels and demodulators recover the messages.

Total bandwidth $\approx N \times (\text{channel bandwidth}) + \text{guard bands}$; e.g. 12 voice channels in 4 kHz slots need 48 kHz (telephone group, 60–108 kHz).

Applications: analog telephone trunks (group, supergroup), AM/FM broadcasting, cable TV, FM stereo, satellite FDMA, and OFDM in Wi-Fi/4G as a digital form.

Advantages: no synchronization between channels, simple for analog signals, continuous transmission. Disadvantages: crosstalk from nonlinearity (intermodulation) and imperfect filters, needs many sharp filters and stable oscillators, wasted guard bands, and fading affects a whole channel.

FDMA

Frequency Division Multiple Access (FDMA) is a multiple-access method in which a shared medium (e.g. a satellite transponder or a cellular band) is divided into frequency sub-bands, and each user/station is assigned its own sub-band for the duration of its use. All users transmit at the same time on different frequencies. Examples: satellite SCPC/MCPC systems, 1G AMPS cellular (30 kHz channels). FDM is the multiplexing of signals at one location; FDMA is the same idea applied to users at different locations.

FDM in telephony hierarchy

Each voice channel (300–3400 Hz) is given a 4 kHz slot and frequency-translated by SSB-SC (LSB) modulation. Channels are built up in standard blocks (ITU-T/CCITT):

12 ch  -> Basic group       60-108 kHz
x5     -> Supergroup  (60)  312-552 kHz
x5     -> Mastergroup (300) 812-2044 kHz
x3     -> Supermaster (900) 8516-12388 kHz

1. Basic group: 12 voice channels modulate carriers 64, 68, ..., 108 kHz; LSBs selected; band 60–108 kHz (48 kHz).
2. Supergroup: 5 groups translated by carriers 420, 468, 516, 564, 612 kHz into 312–552 kHz; 60 channels.
3. Mastergroup: 5 supergroups, 300 channels, 812–2044 kHz (8 kHz guard bands between supergroups).
4. Supermastergroup: 3 mastergroups, 900 channels, 8516–12388 kHz.

In the AT&T (Bell) system a mastergroup is 10 supergroups (600 channels, 564–3084 kHz) and a jumbo group is 6 mastergroups (3600 channels).

Advantages of the hierarchy: standard modular equipment, filters at convenient frequencies, and whole blocks can be switched or routed together. Carriers come from one master oscillator, and pilot tones keep levels and frequencies correct.

A satellite communication system uses an artificial satellite in orbit as a repeater in space: it receives signals from an earth station, amplifies them, changes their frequency and retransmits them to other earth stations over a wide area.

          [Satellite: transponder]
          ^ uplink        \ downlink
         /  (6 / 14 GHz)   v (4 / 12 GHz)
 [Earth station A]     [Earth station B]

Main parts

• Space segment: satellite with transponders (receive antenna, LNA, frequency down-converter, power amplifier such as TWTA/SSPA, transmit antenna), solar panels, batteries, and attitude/orbit control.
• Ground segment: earth stations with large dish antennas, high-power amplifiers (uplink), low-noise amplifiers (downlink), modems and multiplexers; plus telemetry, tracking and command (TT&C) stations.

Key features

• Geostationary orbit (GEO) at about 35 786 km above the equator; the satellite appears fixed, so antennas need not track. Three GEO satellites can cover almost the whole Earth.
• Different uplink and downlink frequencies avoid interference: C-band 6/4 GHz, Ku-band 14/12 GHz, Ka-band 30/20 GHz.
• Typical transponder bandwidth: 36 MHz.
• Multiple access: FDMA, TDMA, CDMA.
• One-way earth–satellite–earth delay about 240–270 ms for GEO.
• LEO and MEO constellations give lower delay (used for mobile broadband, GPS).

Advantages: wide coverage, broadcast capability, distance-independent cost, quick setup in remote and hilly areas (useful in Nepal). Disadvantages: high launch cost, propagation delay and echo, rain fade at high frequencies, limited lifetime (about 15 years).

Applications: TV broadcasting (DTH), telephony and VSAT networks, navigation (GPS), weather and remote sensing, internet.

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.

|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:

(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).

T1 and E1 are the two standard TDM-PCM digital carrier systems used in telephony. Each one interleaves several 64 kb/s PCM voice channels into one serial stream. T1 is the North American (Bell) standard. E1 is the European (ITU-T) standard, and Nepal Telecom uses it too. Both sample speech at 8 kHz with 8 bits per sample, so one frame lasts $1/8000 = 125$ µs.

T1 system

• 24 voice channels × 8 bits + 1 framing bit = 193 bits per frame.
• $R_b = 193 \times 8000 = 1.544$ Mb/s.
• Signalling is "robbed bit": the LSB of each channel in frames 6 and 12 of a superframe.

E1 system

• 32 time slots × 8 bits = 256 bits per frame. TS0 carries frame sync, TS16 carries signalling, and the other 30 slots carry voice.
• $R_b = 256 \times 8000 = 2.048$ Mb/s.

Hierarchies

Higher levels are made by multiplexing lower-level streams. Extra bits are added for framing and bit stuffing, so each higher rate is a little more than the sum of its inputs.

T1 uses µ-law companding ($\mu = 255$) and the AMI/B8ZS line code. E1 uses A-law ($A = 87.6$) and the HDB3 line code.

Frequency division multiplexing (FDM) sends several signals at the same time over one channel by giving each signal a different frequency band. Time division multiplexing (TDM) sends several signals over one channel by giving each signal a different time slot, one after another.

FDM

• Each message is band-limited, then modulates its own sub-carrier (usually SSB in telephony).
• Guard bands between adjacent channels let the receiver separate them with band-pass filters.
• Example: 12 telephone channels in the 60–108 kHz basic group, and AM/FM broadcasting.

TDM

• Each message is sampled. A commutator takes one sample from each channel in turn to form a frame.
• The receiver separates the channels with a synchronised decommutator.
• Example: a T1 frame carries 24 PCM voice channels at 1.544 Mb/s.

 FDM: frequency axis          TDM: time axis
 | ch1 | ch2 | ch3 |  f       |1|2|3|1|2|3|1|2|3|  t
  all at the same time         one band, shared in time

Comparison

FDM suits analog signals. TDM suits digital signals, and it is used in modern telephone networks because digital hardware is cheap and regenerative repeaters stop noise from building up along the link.

Significance of constellation diagram

A constellation diagram plots every possible symbol of a digital modulation scheme as a point in the I–Q (in-phase vs quadrature) plane. The distance of a point from the origin shows the symbol's amplitude, and its angle shows the symbol's phase.

      Q                      Q
      |                  o   |   o
  o---+---o   I          ----+----  I
      |                  o   |   o
     BPSK                   QPSK

Why it matters:

• It shows the number of symbols M, and so the bits per symbol ($\log_2 M$), at a glance.
• The minimum distance between points sets the noise margin. Points that are farther apart give a lower bit error rate for the same noise.
• It shows the average and peak power of the scheme: the mean and maximum squared distance from the origin.
• On a received-signal display, spread-out clouds show noise. A rotated pattern shows phase error, and a squeezed pattern shows amplitude non-linearity. This makes it a practical test tool.
• It lets us compare schemes such as BPSK, QPSK and 16-QAM by bandwidth efficiency and noise immunity.

Comparison of E1 and T1 hierarchy

Frame structures:

|<--------- 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)

|<-------- one frame = 125 us = 256 bits -------->|
+-----+-----+-----+-----+------+------+-----+------+
| TS0 | TS1 | ... |TS15 | TS16 | TS17 | ... | TS31 |
+-----+-----+-----+-----+------+------+-----+------+
 sync  voice       voice signal voice        voice
 each time slot = 8 bits; 32 slots per frame
 voice channels: TS1-TS15 and TS17-TS31 (30)

Comparison of TDM and FDM

Voice application

Reading: "show the comparison for voice channels". Take $N = 24$ telephone channels, each band-limited to 3.4 kHz.

FDM (SSB, 4 kHz slot per channel including guard band):

$$B_{FDM} = N \times 4\ \text{kHz} = 24 \times 4 = 96\ \text{kHz}$$

TDM-PCM (8 kHz sampling, 8 bits/sample, T1 frame):

$$\begin{aligned} \text{Bits per frame} &= 24 \times 8 + 1 = 193\ \text{bits} \\ \text{Frames per second} &= f_s = 8000\ \text{frames/s} \\ R_b &= 193 \times 8000 = 1\,544\,000\ \text{b/s} = 1.544\ \text{Mb/s} \end{aligned}$$

The minimum (Nyquist) bandwidth is $R_b/2 = 772$ kHz.

So for voice, FDM needs about 8 times less bandwidth. TDM-PCM is still preferred because:

• regenerative repeaters remove noise at each hop,
• digital switching and circuits are cheap,
• it can carry speech and data together.

FDM was used in older analog trunks. TDM (T1/E1) is used in today's digital telephone networks.

Comparison of E1 and T1 hierarchies

FDMA vs TDMA

FDMA (frequency division multiple access) gives each user its own carrier frequency band for the whole call. TDMA (time division multiple access) gives all users the same band, and each user transmits in bursts during its own time slot.

T1 and E1 telephone hierarchy

T1 hierarchy (Bell / North America):

 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

E1 hierarchy (ITU-T / Europe):

 30 voice ch x 64 kb/s (+ 2 overhead slots)
        |
   [ E1 ]  2.048 Mb/s     30 ch
        | x4
   [ E2 ]  8.448 Mb/s     120 ch
        | x4
   [ E3 ]  34.368 Mb/s    480 ch
        | x4
   [ E4 ]  139.264 Mb/s   1920 ch
        | x4
   [ E5 ]  564.992 Mb/s   7680 ch

Both systems use 125 µs frames. A T1 frame carries 193 bits, giving 1.544 Mb/s. An E1 frame carries 256 bits, giving 2.048 Mb/s.

Frequency division multiplexing (FDM) in telephony places many voice channels, each about 4 kHz wide, side by side in frequency on one cable or radio link. Each voice channel is shifted to its own slot by single-sideband (SSB) modulation. SSB is used because it needs only 4 kHz per channel.

Structure (CCITT FDM hierarchy)

Channels are combined in stages. Each stage takes the previous blocks, SSB-modulates them onto new carriers, and stacks them in frequency.

 Group: 12 channels, carriers 64, 68, ..., 108 kHz
 (LSB of each channel kept)

 |ch12|ch11|ch10| ch9| ch8| ... | ch2| ch1|
 60   64   68   72   76         100  104  108 kHz
 |<----------------- 48 kHz ----------------->|

 Supergroup: 5 groups, carriers 420, 468, 516,
 564, 612 kHz (LSB kept)

 | G5 | G4 | G3 | G2 | G1 |
 312  360  408  456  504  552 kHz
 |<-------- 240 kHz -------->|

Working

1. Each speech signal is band-limited to 300–3400 Hz by a low-pass/band-pass filter.
2. It SSB-modulates a channel carrier (64, 68, …, 108 kHz). The lower sideband is kept, giving the 60–108 kHz basic group.
3. Five groups are shifted to 312–552 kHz to form a supergroup, and five supergroups form a mastergroup.
4. At the receiver, the same steps are reversed. Band-pass filters pick out each block, and SSB demodulators with locally generated carriers recover the speech.
5. Pilot tones are sent for carrier synchronisation and level control.

Uses

• Long-distance analog telephone trunks on coaxial cable and microwave links.
• Satellite FDMA (SCPC) links.
• The same idea is used in radio/TV broadcasting, FM stereo, and ADSL on telephone lines.

Uses of filters in FDM

• Input low-pass filters band-limit each message so that it fits its 4 kHz slot.
• Band-pass filters after the modulators remove the unwanted sideband and the carrier leakage (SSB generation). They keep each channel inside its own band.
• Receiver band-pass filters separate each channel or group from the composite signal (demultiplexing).
• Output low-pass filters after demodulation remove high-frequency products and recover the baseband speech.
• Sharp filters keep guard bands small and reduce adjacent-channel crosstalk.

Frequency division multiplexing

FDM is a technique in which several message signals are sent at the same time over one channel by shifting each one, with modulation, to a different non-overlapping frequency band. Guard bands separate the bands, and the receiver uses band-pass filters to separate the channels.

Method of FDM in telephony

Telephone FDM uses SSB-SC modulation, so each voice channel takes only 4 kHz (0.3–3.4 kHz speech plus guard band). Channels are built up in a hierarchy:

Step 1: Basic group. Twelve voice channels modulate carriers at 64, 68, 72, …, 108 kHz. Band-pass filters keep only the lower sidebands, which fill 60–108 kHz.

Step 2: Supergroup. Five basic groups modulate carriers at 420, 468, 516, 564 and 612 kHz. The lower sidebands fill 312–552 kHz (60 channels).

Step 3: Mastergroup and above. Five supergroups form a 300-channel mastergroup (812–2044 kHz). Three mastergroups form a 900-channel supermastergroup.

 Group: 12 channels, carriers 64, 68, ..., 108 kHz
 (LSB of each channel kept)

 |ch12|ch11|ch10| ch9| ch8| ... | ch2| ch1|
 60   64   68   72   76         100  104  108 kHz
 |<----------------- 48 kHz ----------------->|

 Supergroup: 5 groups, carriers 420, 468, 516,
 564, 612 kHz (LSB kept)

 | G5 | G4 | G3 | G2 | G1 |
 312  360  408  456  504  552 kHz
 |<-------- 240 kHz -------->|

Receiver. The demultiplexer reverses each step. Band-pass filters pick out each supergroup, then each group, then each channel. Each is SSB-demodulated with a local carrier locked to transmitted pilot tones and then low-pass filtered to get the speech back.

Advantages are that many calls share one cable, the hardware is analog and simple, and no sampling is needed. The drawbacks are crosstalk from non-linearity and noise that builds up over long links, which is why FDM was later replaced by TDM-PCM (T1/E1).

In the T1 (DS1) TDM-PCM system, 24 voice channels are sampled, quantised, encoded and interleaved in time into one frame.

Given standard values

• Voice band limited to about 3.4 kHz, so the sampling rate used is $f_s = 8$ kHz (above the Nyquist rate of 6.8 kHz).
• Each sample is coded with $n = 8$ bits (256 levels, µ-law).
• Number of channels $N = 24$.
• One framing bit is added per frame for synchronisation.

Frame structure

|<--------- 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)

Proof

One frame holds one sample from every channel, so frames repeat at the sampling rate:

$$T_{frame} = \frac{1}{f_s} = \frac{1}{8000} = 125\ \mu\text{s}$$

Bits in one frame:

$$\begin{aligned} \text{Information bits} &= N \times n = 24 \times 8 = 192\ \text{bits} \\ \text{Framing bit} &= 1\ \text{bit} \\ \text{Total} &= 192 + 1 = 193\ \text{bits/frame} \end{aligned}$$

Signalling rate:

$$\begin{aligned} R_b &= \frac{\text{bits per frame}}{T_{frame}} = \frac{193}{125 \times 10^{-6}} \\ &= 193 \times 8000 = 1\,544\,000\ \text{b/s} \end{aligned}$$

Without the framing bit, the rate would be $24 \times 64\ \text{kb/s} = 1.536$ Mb/s. The framing bit adds 8 kb/s of overhead.

Each bit lasts $T_b = 1/1.544\ \text{MHz} \approx 0.648$ µs, and the minimum (Nyquist) transmission bandwidth is $R_b/2 = 772$ kHz.

Answer: Signalling rate $R_b = 193 \times 8000 = 1.544$ Mb/s (proved).

Principle of FDM

Frequency division multiplexing shares the bandwidth of one channel among many signals by giving each signal its own frequency band at the same time. Each message is band-limited and modulates a different carrier $f_{c1}, f_{c2}, \dots$. The modulated signals are added and sent together. Guard bands between bands prevent overlap. At the receiver, band-pass filters tuned to each band separate the signals, and demodulators recover the messages.

 Spectrum of composite FDM signal
   | ch1 |g| ch2 |g| ch3 |g| ch4 |
   fc1      fc2     fc3     fc4      f
   (g = guard band)

When the same idea is used to let many earth stations share one satellite transponder, it is called FDMA.

SCPC (Single Channel Per Carrier)

• Each voice or data channel gets its own carrier inside the transponder band, instead of first being grouped into a multi-channel FDM baseband.
• Example: a 36 MHz transponder divided into narrow channel slots (e.g. 45 kHz each).
• Often used with voice activation: the carrier is switched off during silence, which saves transponder power.
• Suits thin-route traffic (small stations with only a few circuits).
• In pre-assigned SCPC, channels are fixed to station pairs, so they sit idle when not in use.

DAMA (Demand Assigned Multiple Access)

• Channels are not fixed. All stations share a common pool of carrier frequencies, and a channel is given to a station only when a call is requested. It is returned to the pool when the call ends.
• A common signalling channel (central or distributed control) handles the requests.
• Example: the INTELSAT SPADE system (SCPC, PCM, multiple access, demand assignment) uses about 800 SCPC channels at 45 kHz spacing in a 36 MHz transponder.
• Advantage: far better use of the transponder when traffic is random and many stations have light traffic.

Difference between FDM and TDM

FDM telephone hierarchy and spectral details

Telephone FDM uses SSB-SC modulation. Each voice channel (300–3400 Hz) gets a 4 kHz slot that includes its guard band. The standard (CCITT) groups are:

1. Basic group (12 channels, 60–108 kHz). Channel $k$ modulates a carrier $f_k = 112 - 4k$ kHz ($k = 1 \dots 12$, i.e. 108, 104, …, 64 kHz). The lower sideband is kept, so channel $k$ occupies $f_k - 4$ to $f_k$ kHz.

2. Supergroup (60 channels, 312–552 kHz). Five groups modulate carriers at 420, 468, 516, 564 and 612 kHz. Taking the lower sideband turns each 60–108 kHz group into a 48 kHz block. For example, $420 - 108 = 312$ to $420 - 60 = 360$ kHz.

3. Mastergroup (300 channels, 812–2044 kHz). Five supergroups are placed with 8 kHz guard bands between them. (The Bell system mastergroup instead uses 10 supergroups = 600 channels in 564–3084 kHz.)

4. Supermastergroup (900 channels, 8516–12388 kHz). Three mastergroups.

 Group: 12 channels, carriers 64, 68, ..., 108 kHz
 (LSB of each channel kept)

 |ch12|ch11|ch10| ch9| ch8| ... | ch2| ch1|
 60   64   68   72   76         100  104  108 kHz
 |<----------------- 48 kHz ----------------->|

 Supergroup: 5 groups, carriers 420, 468, 516,
 564, 612 kHz (LSB kept)

 | G5 | G4 | G3 | G2 | G1 |
 312  360  408  456  504  552 kHz
 |<-------- 240 kHz -------->|

Each step needs only one set of modulators and filters for a whole block of channels. This keeps equipment simple. The wide guard bands at higher levels make the filters easy to build.

Block diagram of an analog communication system

An analog communication system carries a continuously varying message from a source to a user over a channel. The message is first converted to an electrical signal and then modulated onto a carrier.

 [Information source]
         |
 [Input transducer]
         |
 [Transmitter (modulator)]
         |
 [Channel] <---- noise, interference
         |
 [Receiver (demodulator)]
         |
 [Output transducer] --> user

• Information source and input transducer: produce the message, e.g. speech converted to voltage by a microphone.
• Transmitter: amplifies, filters and modulates (AM/FM/PM) the message onto a carrier suited to the channel, then power-amplifies it.
• Channel: a wire, coaxial cable, optical fibre or free space. It attenuates and distorts the signal and adds noise.
• Receiver: selects the wanted signal, amplifies it, demodulates it and filters it to recover the message.
• Output transducer: a loudspeaker or display that turns the electrical signal back into its original form.

FDM system for three message signals

Assume three messages $m_1(t), m_2(t), m_3(t)$, each band-limited to $W = 4$ kHz, and SSB modulation with carriers $f_{c1} = 100$ kHz, $f_{c2} = 105$ kHz and $f_{c3} = 110$ kHz (upper sideband kept, 1 kHz guard band).

 TRANSMITTER
 m1 ->[LPF]->[SSB mod fc1]->[BPF1]--+
 m2 ->[LPF]->[SSB mod fc2]->[BPF2]--+-->(+)--> channel
 m3 ->[LPF]->[SSB mod fc3]->[BPF3]--+

 RECEIVER
          +->[BPF1]->[demod fc1]->[LPF]-> m1
 channel -+->[BPF2]->[demod fc2]->[LPF]-> m2
          +->[BPF3]->[demod fc3]->[LPF]-> m3

Spectrum of the multiplexed signal:

  |  m1  |g|  m2  |g|  m3  |
  100    104 105  109 110   114  kHz
  |<--------- 14 kHz ---------->|

Working:

1. LPFs band-limit each message to $W$.
2. Each message modulates its own carrier. BPFs keep one sideband, so each signal sits in its own band.
3. The adder combines them. The composite signal (bandwidth $3W$ + guard bands = 14 kHz here) is sent over one channel.
4. At the receiver, BPFs centred on each band separate the three signals. Coherent demodulators with $f_{c1}, f_{c2}, f_{c3}$ and LPFs recover $m_1, m_2, m_3$.

Guard bands and good filters prevent crosstalk between the channels.

TDM-PCM telephony carries many 64 kb/s PCM voice channels on one line by interleaving their 8-bit samples in time. The two standards are T1 (North America, Japan) and E1 (Europe, ITU-T, used in Nepal). Both sample speech at 8 kHz, so each frame lasts $125$ µs.

T1 frame

|<--------- 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)

$$\begin{aligned} \text{Bits per frame} &= 24 \times 8 + 1 = 193\ \text{bits} \\ \text{Frames per second} &= f_s = 8000\ \text{frames/s} \\ R_b &= 193 \times 8000 = 1\,544\,000\ \text{b/s} = 1.544\ \text{Mb/s} \end{aligned}$$

• 12 frames form a superframe (24 in the extended superframe, ESF). The F-bits of these frames carry the frame alignment pattern (100011011100 for the superframe).
• Signalling uses the robbed-bit method: in frames 6 and 12, the LSB of every channel carries signalling (A and B bits).
• µ-law companding ($\mu = 255$) and AMI/B8ZS line coding are used.

E1 frame

|<-------- one frame = 125 us = 256 bits -------->|
+-----+-----+-----+-----+------+------+-----+------+
| TS0 | TS1 | ... |TS15 | TS16 | TS17 | ... | TS31 |
+-----+-----+-----+-----+------+------+-----+------+
 sync  voice       voice signal voice        voice
 each time slot = 8 bits; 32 slots per frame
 voice channels: TS1-TS15 and TS17-TS31 (30)

$$R_b = 32 \times 8 \times 8000 = 2.048\ \text{Mb/s}$$

• TS0 carries the frame alignment word (0011011) in alternate frames, plus alarm bits.
• TS16 carries channel-associated signalling. 16 frames form a multiframe of 2 ms, and each frame's TS16 carries the signalling of 2 channels.
• A-law companding ($A = 87.6$) and HDB3 line coding are used.

T1 hierarchy

 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

E1 hierarchy

 30 voice ch x 64 kb/s (+ 2 overhead slots)
        |
   [ E1 ]  2.048 Mb/s     30 ch
        | x4
   [ E2 ]  8.448 Mb/s     120 ch
        | x4
   [ E3 ]  34.368 Mb/s    480 ch
        | x4
   [ E4 ]  139.264 Mb/s   1920 ch
        | x4
   [ E5 ]  564.992 Mb/s   7680 ch

At each higher level, the higher rate is a little more than (number of inputs × lower rate). The extra bits are used for framing and bit stuffing (justification), which align input streams whose clocks differ slightly. For example, $4 \times 1.544 = 6.176$ Mb/s, but DS2 runs at 6.312 Mb/s.

T1 digital carrier system

T1 is the North American TDM-PCM carrier that sends 24 voice channels over one twisted-pair line at 1.544 Mb/s.

• Each voice channel is band-limited to 3.4 kHz and sampled at 8 kHz.
• Each sample is µ-law compressed ($\mu = 255$) and coded into 8 bits, giving 64 kb/s per channel.
• One frame = one 8-bit sample from each of the 24 channels + 1 framing bit = 193 bits in 125 µs.

|<--------- 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)

$$\begin{aligned} \text{Bits per frame} &= 24 \times 8 + 1 = 193\ \text{bits} \\ \text{Frames per second} &= f_s = 8000\ \text{frames/s} \\ R_b &= 193 \times 8000 = 1\,544\,000\ \text{b/s} = 1.544\ \text{Mb/s} \end{aligned}$$

• Signalling uses robbed bits (the LSB in frames 6 and 12 of a 12-frame superframe).
• The line code is bipolar AMI (B8ZS on newer lines). Repeaters are placed about every 1.8 km.

Multiplexing hierarchy:

 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

Higher levels add framing and stuffing bits, so DS2 = 6.312 Mb/s rather than $4 \times 1.544 = 6.176$ Mb/s.

Adaptive Delta Modulation (ADM)

Linear delta modulation (DM) sends 1 bit per sample, which says whether the signal is above or below its staircase approximation. A fixed step size $\Delta$ causes two problems:

• Slope overload when the signal changes faster than $\Delta f_s$.
• Granular noise when the signal is nearly flat.

ADM changes the step size to suit the signal slope:

• If several successive output bits are the same (e.g. 1 1 1), the staircase is lagging behind, so the step size is increased. A common rule is to multiply it by 1.5 or 2.
• If the bits alternate (1 0 1 0), the signal is flat, so the step size is decreased (e.g. divided by the same factor), down to a minimum $\Delta_{min}$.

 m(t) -->(+)--> [Comparator] --+--> output bits b(k)
          ^-                   |
          |                    v
          |            [Step size logic]
          |                    | delta(k)
          +--- [Accumulator] <-+

The receiver uses the same logic on the received bits, so no extra side information is sent.

Advantages: much less slope overload and granular noise, a wider dynamic range, and good speech quality at 16–32 kb/s. An example is CVSD (continuously variable slope DM), used in military and Bluetooth voice.

Given: $N = 24$ voice channels, $f_m = 3.4$ kHz.

(i) PAM-TDM, Nyquist sampling

Nyquist rate for each channel:

$$f_s = 2 f_m = 2 \times 3.4 = 6.8\ \text{kHz}$$

Total pulse rate of the TDM signal:

$$r = N f_s = 24 \times 6.8\ \text{kHz} = 163.2\ \text{k pulses/s}$$

The minimum channel bandwidth needed for TDM-PAM is half the pulse rate. This is the Nyquist bandwidth, the same as $N f_m$:

$$\begin{aligned} B_{min} &= \frac{N f_s}{2} = \frac{24 \times 6.8}{2} \\ &= 81.6\ \text{kHz} \;(= 24 \times 3.4\ \text{kHz}) \end{aligned}$$

Answer (i): Minimum channel bandwidth = 81.6 kHz.

(ii) PCM with 8-bit encoder, $R_b = 1.5 \times 10^6$ b/s

Each frame has one 8-bit sample from each channel:

$$R_b = N \, n \, f_s$$ $$\begin{aligned} f_s &= \frac{R_b}{N n} = \frac{1.5 \times 10^6}{24 \times 8} \\ &= \frac{1.5 \times 10^6}{192} = 7812.5\ \text{Hz} \end{aligned}$$

Check: $7812.5 > 6800$ Hz (the Nyquist rate), so there is no aliasing, with a guard band of $7812.5 - 6800 = 1012.5$ Hz.

Answer (ii): Sampling rate $f_s = 7812.5$ samples/s ≈ 7.81 kHz.

E1 is the ITU-T (European) standard TDM-PCM digital carrier. It multiplexes 30 voice channels of 64 kb/s plus 2 overhead channels into a 2.048 Mb/s stream. It is used in Europe, Asia and Nepal.

E1 frame

• Speech is sampled at 8 kHz, so a frame lasts 125 µs.
• Each frame has 32 time slots × 8 bits = 256 bits.
• TS0: frame alignment word and alarm bits.
• TS16: signalling (channel-associated). 16 frames form a 2 ms multiframe.
• TS1–TS15 and TS17–TS31: 30 voice channels.

|<-------- one frame = 125 us = 256 bits -------->|
+-----+-----+-----+-----+------+------+-----+------+
| TS0 | TS1 | ... |TS15 | TS16 | TS17 | ... | TS31 |
+-----+-----+-----+-----+------+------+-----+------+
 sync  voice       voice signal voice        voice
 each time slot = 8 bits; 32 slots per frame
 voice channels: TS1-TS15 and TS17-TS31 (30)

$$R_b = 32 \times 8 \times 8000 = 2.048\ \text{Mb/s}$$

E1 digital hierarchy

Four lower-level streams are combined at each step. Extra bits for framing and justification (bit stuffing) are added.

E1 uses A-law companding ($A = 87.6$) and the HDB3 line code. Compared with T1 (24 channels, 1.544 Mb/s), E1 carries more channels and has a separate slot for signalling, so all 8 bits of each voice slot stay available for speech.

T1 and E1 are TDM-PCM digital carrier systems. Both use 8 kHz sampling, 8-bit samples and 125 µs frames, but they differ in channel count, framing and rates.

E1 frame

E1 is the ITU-T TDM-PCM carrier that carries 30 voice channels. Each voice channel is sampled at 8 kHz and coded with 8 bits (A-law), giving 64 kb/s. One frame takes one byte from each of 32 time slots:

|<-------- one frame = 125 us = 256 bits -------->|
+-----+-----+-----+-----+------+------+-----+------+
| TS0 | TS1 | ... |TS15 | TS16 | TS17 | ... | TS31 |
+-----+-----+-----+-----+------+------+-----+------+
 sync  voice       voice signal voice        voice
 each time slot = 8 bits; 32 slots per frame
 voice channels: TS1-TS15 and TS17-TS31 (30)

• TS0: frame alignment word (0011011) in alternate frames, plus alarm/CRC bits.
• TS16: channel-associated signalling. 16 frames form a 2 ms multiframe.
• 30 slots carry speech.

Signalling (bit) rate:

$$\begin{aligned} \text{Bits per frame} &= 32 \times 8 = 256 \\ R_b &= 256 \times 8000 = 2\,048\,000\ \text{b/s} = 2.048\ \text{Mb/s} \end{aligned}$$

Bit duration $T_b = 1/2.048\ \text{MHz} \approx 0.488$ µs.

E1 TDM hierarchy

 30 voice ch x 64 kb/s (+ 2 overhead slots)
        |
   [ E1 ]  2.048 Mb/s     30 ch
        | x4
   [ E2 ]  8.448 Mb/s     120 ch
        | x4
   [ E3 ]  34.368 Mb/s    480 ch
        | x4
   [ E4 ]  139.264 Mb/s   1920 ch
        | x4
   [ E5 ]  564.992 Mb/s   7680 ch

Each level combines 4 streams of the level below. A few extra bits are added for framing and justification, so for example $4 \times 2.048 = 8.192$ Mb/s, but E2 = 8.448 Mb/s.

Companding

Companding = compressing + expanding. 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.

 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:

|<--------- 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:

 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

Functional block diagram of PCM

Pulse code modulation (PCM) turns an analog signal into a stream of binary code words by sampling, quantising and encoding it.

 TRANSMITTER
 m(t)->[LPF]->[Sampler]->[Quantizer]->[Encoder]->
                                     PCM bits
 CHANNEL with REGENERATIVE REPEATERS
 ->[Equalizer]->[Decision/Timing]->[Regenerate]->
 RECEIVER
 ->[Regenerator]->[Decoder]->[Reconstruction LPF]->m'(t)

• Anti-aliasing LPF: limits the message to $W$ Hz.
• Sampler: takes samples at $f_s \ge 2W$ (8 kHz for speech).
• Quantiser: rounds each sample to one of $L = 2^n$ levels (with companding for speech).
• Encoder: turns each level into an $n$-bit code word, then a line code is applied.
• Regenerative repeaters: reshape and retime the pulses along the line, so noise does not build up.
• Receiver: the regenerator cleans the pulses, the decoder turns code words back into PAM samples, and the LPF rebuilds $m(t)$.

Signalling rate of T1

T1 sends 24 voice channels, each sampled at 8 kHz with 8 bits per sample, plus 1 framing bit per frame.

$$\begin{aligned} \text{Bits per frame} &= 24 \times 8 + 1 = 193\ \text{bits} \\ \text{Frames per second} &= f_s = 8000\ \text{frames/s} \\ R_b &= 193 \times 8000 = 1\,544\,000\ \text{b/s} = 1.544\ \text{Mb/s} \end{aligned}$$

Answer: T1 signalling rate = 1.544 Mb/s (frame of 193 bits every 125 µs).

T1 frame diagram

|<--------- 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)

Bit rate and bandwidth of T1 and E1

Both systems sample speech at $f_s = 8$ kHz with 8 bits per sample.

T1: 24 channels + 1 framing bit:

$$R_b = (24 \times 8 + 1) \times 8000 = 193 \times 8000 = 1.544\ \text{Mb/s}$$

E1: 32 slots (30 voice + 2 overhead):

$$R_b = 32 \times 8 \times 8000 = 256 \times 8000 = 2.048\ \text{Mb/s}$$

Bandwidth. The minimum (Nyquist) bandwidth for binary transmission is $B = R_b/2$. With practical NRZ/AMI pulses, the first-null bandwidth is about $R_b$.

DPCM encoder

Differential PCM uses the fact that speech samples next to each other are strongly correlated. It does not quantise each sample directly. It quantises the difference between the sample and a prediction made from earlier samples. The difference is smaller than the sample, so fewer bits are needed for the same quality.

 m[n] ->(+)--e[n]-->[Quantizer]--eq[n]--+-->[Encoder]-> bits
        ^-                              |
        |                               v
        |                   m^[n] ---> (+)
        |                     ^         | mq[n]
        +-------- m^[n] --[Predictor]<--+

• Prediction error: $e[n] = m[n] - \hat m[n]$.
• Quantised error: $e_q[n] = e[n] + q[n]$.
• Predictor input: $m_q[n] = \hat m[n] + e_q[n] = m[n] + q[n]$.

The predictor (a linear filter on past values of $m_q$) works on the quantised samples. The receiver has the same predictor, so encoder and decoder stay in step and quantisation errors do not build up.

The SNR improves by the prediction gain $G_p = \sigma_m^2 / \sigma_e^2$. DPCM saves about 1–2 bits per sample compared with PCM, e.g. 32–48 kb/s instead of 64 kb/s for speech.

E1 digital hierarchy

E1 is the ITU-T TDM-PCM carrier. Each frame carries 32 time slots of 8 bits in 125 µs: TS0 for frame sync, TS16 for signalling, and 30 voice slots. Its rate is $R_b = 256 \times 8000 = 2.048$ Mb/s. Higher levels multiplex 4 lower streams each, adding framing and stuffing bits.

E1 uses A-law companding and the HDB3 line code.

Numerical

Given: $f_m = 4$ kHz, amplitude $\pm 1.1$ V, $R_b = 32$ kb/s. Assumption: Nyquist-rate sampling, a uniform quantiser covering $\pm 1.1$ V, and a full-scale sinusoidal test signal.

Sampling rate:

$$f_s = 2 f_m = 2 \times 4 = 8\ \text{kHz}$$

Bits per sample:

$$n = \frac{R_b}{f_s} = \frac{32\,000}{8000} = 4\ \text{bits}, \qquad L = 2^4 = 16\ \text{levels}$$

SQNR:

$$\begin{aligned} \Delta &= \frac{2 \times 1.1}{16} = 0.1375\ \text{V} \\ N_q &= \frac{\Delta^2}{12} = \frac{0.1375^2}{12} = 1.5755 \times 10^{-3}\ \text{V}^2 \\ P_s &= \frac{(1.1)^2}{2} = 0.605\ \text{V}^2 \\ \text{SQNR} &= \frac{0.605}{1.5755 \times 10^{-3}} = 384 = \tfrac{3}{2} L^2 \\ \text{SQNR (dB)} &= 10 \log_{10} 384 = 25.84\ \text{dB} \end{aligned}$$

This agrees with the rule $\text{SQNR} = 1.76 + 6.02n = 1.76 + 24.08 = 25.84$ dB.

Answer: $n = 4$ bits per sample; SQNR = 384 ≈ 25.84 dB.