ELEC3506

Physical LayerLecture 220 min

Multiplexing — Sharing One Link Between Many Signals

Three ways to put more than one signal onto a single link, and what each one costs to do it.

By the end of this page you should be able to

  • State what multiplexing solves and name the three techniques the lecture covers
  • Compute FDM bandwidth requirements including guard bands
  • Compute TDM slot and frame durations for a given input rate and channel count
  • Explain why STDM needs per-slot addressing that synchronous TDM does not

The idea

A physical link is expensive to lay and often carries far more bandwidth than any one signal needs. Running one cable per conversation would waste almost all of it. Multiplexing is the fix: a technique that lets multiple signals share a single data link at the same time, instead of one link per signal.

The lecture gives three ways to do it, and they differ in exactly one question: what do you divide up to make room for everyone — frequency, or time?

Three multiplexing schemes drawn as horizontal bars. Frequency-division multiplexing: four channel bands separated by narrow guard bands along the frequency axis. Time-division multiplexing: two repeating frames, each split into four equal fixed slots, one per device A to D. Statistical time-division multiplexing: slots of varying width carrying only the devices that currently have data, skipping an idle device rather than sending it an empty slot.FDM — frequency-division, guard bands between channelsfrequency →TDM — time-division, one fixed slot per device every frameABCDABCDFrame 1Frame 2Statistical TDM — variable slots, only active devices includedACDACB is idle this scan — no slot sent for it
TDM reserves device D's slot on every frame whether or not D has anything to send; STDM only spends bandwidth on a device when it is actually transmitting, at the cost of needing an address in every slot so the receiver knows whose data just arrived.

FDM — dividing the frequency

How it works

Frequency-Division Multiplexing

FDM is an analog technique. It applies when the bandwidth of the link is greater than the combined bandwidths of the signals being carried — there is spare spectrum, and FDM slices it up.

Each channel gets its own strip of frequency. Between neighbouring channels sits a guard band — a strip of unused bandwidth that stops one channel’s signal bleeding into the next. For n channels there are n − 1 guard bands, one fewer than the number of channels, because a guard band only exists between two neighbours.

Worked example

Five 100 kHz channels, 10 kHz guard bands

  1. Channel bandwidth. Five channels at 100 kHz each: 5 x 100 = 500 kHz.

  2. Guard bands. Five channels need 5 - 1 = 4 guard bands, at 10 kHz each: 4 x 10 = 40 kHz.

  3. Total.

    Btotal=(5×100)+(4×10)=540 kHzB_{\text{total}} = (5 \times 100) + (4 \times 10) = 540 \text{ kHz}

AnswerTotal bandwidth = 540 kHz

Where marks get lost

n channels need n − 1 guard bands, not n

The most common slip in this calculation is using n guard bands instead of n − 1. Picture it: three channels sitting side by side only have two gaps between them, not three. Draw the channels as boxes if the count is not obvious — the guard bands are the spaces, and there is always one fewer space than there are boxes.

Aside

Not every FDM question includes guard bands. If three 4 kHz voice channels are packed directly into a 12 kHz link with no guard bands, they simply sit edge-to-edge: 20–24 kHz, 24–28 kHz, 28–32 kHz. The formula above is for when guard bands are explicitly part of the question — read carefully for whether they are.

TDM — dividing the time

How it works

Synchronous Time-Division Multiplexing

TDM divides the usage of a high-capacity link into time slots, one per device, repeating every frame. It only works with digital signals — this is the direct counterpart to FDM being analog-only. It is also called synchronous TDM, because every device gets its slot on a fixed schedule whether or not it has anything to send.

Worked example

1 Mbps input, 4 channels, synchronous TDM

  1. Input slot duration. Each channel arrives at 1 Mbps, so one input bit takes:

    Tslot=1Rin=11,000,000=1 μsT_{\text{slot}} = \frac{1}{R_{\text{in}}} = \frac{1}{1{,}000{,}000} = 1\ \mu\text{s}

  2. Output slot duration. Four channels share the output link, so each output slot is a quarter of the input slot: 0.25 μs.

  3. Output bit rate. The output carries all four channels’ bits: four times the per-channel rate.

    Rout=4×1 Mbps=4 MbpsR_{\text{out}} = 4 \times 1 \text{ Mbps} = 4 \text{ Mbps}

  4. Frame duration and frame rate. One frame carries exactly one bit from each of the four channels, so a full frame completes at the same rate as a single channel’s own bit rate:

    Tframe=4×0.25 μs=1 μs⇒Frame rate=1 MbpsT_{\text{frame}} = 4 \times 0.25\ \mu\text{s} = 1\ \mu\text{s} \quad\Rightarrow\quad \text{Frame rate} = 1 \text{ Mbps}

AnswerOutput bit rate 4 Mbps, frame rate 1 Mbps

STDM — dividing the time, only when there is data

How it works

Statistical Time-Division Multiplexing

Synchronous TDM’s fixed slots waste bandwidth on any device that has nothing to send that frame. Statistical TDM (also called asynchronous TDM) fixes this: the frame is built from a statistical analysis of which input lines are likely to be transmitting at any given moment. The multiplexer scans the input lines, and only when a frame is filled with active data does it send that frame across the link.

The cost is addressing. Because a slot’s position no longer tells the receiver which device it belongs to — an idle device is simply skipped rather than sent as an empty slot — every slot needs an address, so the receiver can tell whose data just arrived.

FDMTDM (synchronous)STDM (statistical)
Domain dividedFrequencyTimeTime
Signal typeAnalogDigitalDigital
Slot/band allocationFixed strip per channelFixed slot per device, every frameVariable — only active devices get a slot
Idle deviceIts band still exists, unusedIts slot is still sent, emptyIts slot is skipped entirely
Needs addressing?NoNo — position identifies the deviceYes — a skipped slot means position alone is not enough
This table is effectively tutorial 2's Q14, Q15, Q16 and Q18 answered in one place.

Check yourself

Ten 4 kHz voice channels are combined with FDM using 500 Hz guard bands between them. What is the total bandwidth?

Ten channels: 10 x 4 = 40 kHz. Nine guard bands (10 - 1) at 0.5 kHz each: 9 x 0.5 = 4.5 kHz. Total: 40 + 4.5 = 44.5 kHz.

Putting it together with PCM

Worked example

100 voice channels in a 20 kHz passband

This is where multiplexing, PCM and spectral efficiency from the earlier topics meet in one question.

  1. Bandwidth per channel. 100 digitized voice channels share a 20 kHz passband:

    20,000 Hz100=200 Hz per channel\frac{20{,}000 \text{ Hz}}{100} = 200 \text{ Hz per channel}

  2. Bit rate per channel. Each channel is standard telephone PCM — 8000 samples/s x 8 bits/sample:

    8000×8=64,000 bps=64 kbps8000 \times 8 = 64{,}000 \text{ bps} = 64 \text{ kbps}

  3. Spectral efficiency. Bits per second, per hertz of bandwidth used:

    64,000 bps200 Hz=320 bits/s/Hz\frac{64{,}000 \text{ bps}}{200 \text{ Hz}} = 320 \text{ bits/s/Hz}

Notice this is the same spectral-efficiency idea from the very first topic in this module — more data rate for the same bandwidth — now applied across a whole multiplexed bundle rather than one signal.

Answer200 Hz per channel; 64 kbps per channel; 320 bits/s/Hz

In the exam

  • Name the three techniques. FDM, synchronous TDM, statistical (asynchronous) TDM. Tutorial 2 asks for this list directly.
  • FDM is analog, TDM is digital. State this pairing plainly — it is a one-mark distinction that gets missed under time pressure.
  • Guard bands: n − 1, not n. Draw the channels if you are unsure.
  • TDM slot and frame arithmetic. Input slot = 1/input rate. Output slot shrinks by a factor of the channel count. Output bit rate multiplies by the channel count; frame rate matches the per-channel input rate.
  • TDM vs STDM. Fixed reserved slots vs only-active-devices slots, and the addressing cost STDM pays for the efficiency it buys.
  • WDM. Recognise it as the fibre-optic name for the same idea as FDM — the lecture gives no more detail than the name and the medium it applies to.

Check yourself

  1. Five 100 kHz channels are combined with FDM, separated by 10 kHz guard bands. What is the total bandwidth needed?
  2. A synchronous TDM link multiplexes 4 channels, each arriving at 1 Mbps. What is the resulting output bit rate?
  3. What is the defining difference between synchronous TDM and statistical TDM?
  4. Why does FDM need guard bands at all?
  5. WDM shows up in tutorial 2 attached to one specific kind of link. Which?