Multiplexing: Space, Frequency, Time and Code

Space can also be divided by direction rather than distance: a sectorised antenna serves three or six sectors from one site, and a smart antenna can aim a beam per user, which is SDMA in one cell ([Directional Antennas, Sectorisation, Diversity and Spatial Reuse]).

Frequency

The band is cut into carriers, each one used by a signal all the time. This is the scheme of broadcasting: every station has its own frequency and never stops.

Its guard space is the guard band. A modulated carrier is not a line but a band, and its spectrum has skirts; the guard band keeps a strong neighbour out of a weak receiver.

The 2.4 GHz band shows the trade. 802.15.4 puts sixteen 2 MHz channels 5 MHz apart, spending 3 MHz of every 5 as guard and leaving room at the band edges. Bluetooth puts 79 channels of 1 MHz at 1 MHz spacing, with no guard at all, and relies on hopping and short packets to survive the collisions. 802.11b's channels are 22 MHz wide but only 5 MHz apart, so they overlap: thirteen are defined and only three can be used at once, which is the arrangement [The 802.15.4 Physical Layer] had to plan around.

Advantages: simple, each signal is continuous, no synchronisation between users is needed. Costs: a user's carrier is idle when it has nothing to send; the guard bands are wasted spectrum; and a receiver needs a filter sharp enough to reject its neighbours.

Time

All signals use the whole band, one after another, in slots of a repeating frame. GSM's frame is 4.615 ms in 8 slots; a mobile transmits in one slot and is free for the other seven, which is how a handset can also listen to its neighbours and save power.

Its guard space is the guard time. Two bursts must not overlap even though the mobiles are at different distances and their clocks are not perfect. GSM's slot is 156.25 bit periods long and its normal burst carries 148 of them, leaving 8.25 bit periods, about 30.5 microseconds, as guard. In that time a radio wave travels 9.1 km, which is why a mobile further away than that needs timing advance, an instruction to transmit early ([The GSM Radio Interface: Carriers, the TDMA Frame and Bursts]).

Advantages: the whole band is available to each user in turn, so a bursty user can be given more slots; only one transmitter is active, so no intermodulation between users. Costs: everyone must be synchronised; the guard time is wasted; and a user waits for its slot, which is latency.

Code

All signals use the whole band at the same time, each multiplied by its own code. If the codes are orthogonal, a receiver that multiplies the sum by its own code and adds up the result recovers its own signal and sees zero from the others. W-CDMA uses exactly this: orthogonal variable spreading factor codes to separate the channels of one cell, and a scrambling code to separate the cells.

Its guard space is the distance between the codes, and it is paid for in bandwidth: a spreading factor of 8 means 8 chips per bit, so the bit rate is one eighth of the chip rate. In return, code division brings gifts the other dimensions do not: a signal below the noise can still be recovered, multipath copies can be separated by their delays and combined, and there is no hard limit on the number of users, only a rising noise floor as each one is added (soft capacity).

Costs: every transmitter must be heard at about the same power, or a near one drowns the far ones (the near-far problem), so fast power control is essential; and the codes must be kept orthogonal, which needs synchronisation.

Combining them

No real system uses one dimension alone.

• GSM: space (cells), frequency (200 kHz carriers) and time (8 slots), plus frequency hopping.
• UMTS: space (cells), code (W-CDMA), and frequency (5 MHz carriers).
• 802.15.4: frequency (16 channels), time (the superframe and its slots), and code (32 chips per symbol, for robustness rather than for sharing users).
• Wi-Fi: frequency (channels), time (CSMA/CA), and, in later versions, space (MIMO streams).

The order in which they are applied is part of the design: GSM first divides the world into cells, then the band into carriers, then each carrier into slots, and only then hops the carrier from frame to frame.

Multiplexing, computed

The program counts carriers in the 2.4 GHz band against what each standard defines; computes GSM's guard time and the distance a burst can travel within it; separates four users sent at once with Walsh codes of length 8; counts the calls a city can carry as its cells shrink; and states the guard space each dimension costs.

# The four dimensions a medium is shared in, each measured: space, frequency,
# time and code. The guard space each needs, and what it costs.
import math
import random

# 1. Frequency division: a band cut into carriers, with a guard band between
#    them and at the edges. How many fit in the 83.5 MHz of the 2.4 GHz band,
#    and how many each standard actually defines.
print("Frequency division in the 2 400 to 2 483.5 MHz band:")
print("  system      carrier   spacing   gap between carriers   fit in 83.5 MHz   the standard defines")
for name, width, spacing, defined in (("802.15.4", 2.0, 5.0, 16), ("Bluetooth", 1.0, 1.0, 79),
                                      ("802.11b", 22.0, 5.0, 13)):
    fit = int((83.5 - width) // spacing) + 1
    gap = spacing - width
    print("  %-11s %5.1f MHz %6.1f MHz %19s %14d %19d"
          % (name, width, spacing, "%.1f MHz" % gap if gap >= 0 else "they overlap", fit, defined))
print("  802.15.4 takes 16 of the 17 that would fit, leaving 5 MHz clear at each end of the band.")
print("  802.11b's channels are wider than their spacing, so only %d of its 13 can be used at once."
      % int(83.5 // 22.0))

# 2. Time division: a frame of slots, with a guard time for clock drift and
#    propagation. GSM's own numbers: a frame of 4.615 ms in 8 slots.
print("\nTime division, GSM's frame: 4.615 ms in 8 slots")
slot = 4.615 / 8
print("  one slot %.4f ms; a normal burst carries 148 bits of the 156.25 bit periods in a slot,"
      % slot)
print("  so %.2f bit periods, %.1f microseconds, are the guard time between bursts."
      % (156.25 - 148, (156.25 - 148) * slot * 1000 / 156.25))
print("  in that guard a signal travels %.1f km, which sets how far a mobile may be without timing advance"
      % ((156.25 - 148) * slot * 1e-3 / 156.25 * 299792458 / 1000))

# 3. Code division: two users separated by orthogonal codes rather than by
#    frequency or time. Walsh codes of length 8, sent at once and separated.
def walsh(n):
    if n == 1:
        return [[1]]
    small = walsh(n // 2)
    return [row + row for row in small] + [row + [-x for x in row] for row in small]

codes = walsh(8)
rnd = random.Random(80)
users = [(0, 1), (3, -1), (5, 1), (6, -1)]              # (code index, bit sent as +1 or -1)
air = [sum(bit * codes[c][i] for c, bit in users) for i in range(8)]
print("\nCode division with Walsh codes of length 8:")
print("  four users send at once; the air carries %s" % air)
for c, bit in users:
    got = sum(a * codes[c][i] for i, a in enumerate(air)) / 8
    print("  user with code %d sent %+d, and correlating recovers %+.0f" % (c, bit, got))
idle = 2
print("  a listener correlating with unused code %d recovers %+.0f: the others are invisible to it"
      % (idle, sum(a * codes[idle][i] for i, a in enumerate(air)) / 8))

# 4. Space division: the same frequency, time and code reused at a distance.
#    How many cells of radius r fit in a city of side L, and the capacity that
#    gives with a cluster of k cells sharing the band.
print("\nSpace division: a city 10 km square, cells of radius r, a cluster of 7:")
print("   cell radius   cells   channels a cell gets (of 350)   calls at once")
for r in (5.0, 2.0, 1.0, 0.5, 0.25):
    area = 2.598 * r ** 2                               # a hexagon of circumradius r
    cells = max(1, int(100 / area))
    per_cell = 350 // 7
    print("  %10.2f km %7d %30d %14d" % (r, cells, per_cell, cells * per_cell))

# 5. What each dimension costs: the guard space, as a share.
print("\nThe guard space each dimension needs:")
print("  frequency: 802.15.4 uses 2 MHz of every 5 MHz channel spacing, so %.0f%% is guard" % (100 * 3 / 5))
print("  time:      GSM leaves %.2f of 156.25 bit periods, %.1f%%, as guard" % (156.25 - 148, 100 * 8.25 / 156.25))
print("  code:      %d chips carry 1 bit here, so the rate falls to 1/%d of the chip rate" % (8, 8))
print("  space:     a cluster of 7 gives each cell 1/7 of the channels, %.0f%% of the band" % (100 / 7))

Frequency. Seventeen 2 MHz channels spaced 5 MHz apart would fit in 83.5 MHz; 802.15.4 defines sixteen, leaving 5 MHz clear at each end, because the band's edges must not be splashed ([The 802.15.4 Physical Layer]). Bluetooth's 79 channels of 1 MHz have no guard band at all. 802.11b's 22 MHz channels at 5 MHz spacing overlap, so of thirteen only three are usable at once: a standard can define more channels than the band can hold, and the count that matters is the non-overlapping one.

Time. A GSM slot is 0.5769 ms, and 8.25 of its 156.25 bit periods, 30.5 microseconds, are guard. A wave travels 9.1 km in that time, so a burst from further away arrives more than a guard period late and would land in the next slot: timing advance exists for exactly this.

Code. Four users with Walsh codes 0, 3, 5 and 6 send +1, -1, +1 and -1 at the same instant, and the air carries the sum, [0, 0, 4, 0, 0, 4, 0, 0]. Correlating with each user's code recovers exactly the bit that user sent, and correlating with an unused code recovers zero: the other users are not noise to a receiver with the right code, they are invisible. That is the whole idea of code division, and it works only while the codes stay orthogonal.

Space. With 350 channels and a cluster of 7, every cell gets 50 channels whatever its size. A single 5 km cell over a 10 km city carries 50 calls; 1 km cells carry 1,900; 250 m cells carry 30,750. Nothing about the spectrum changed. This is why operators build more sites rather than asking for more spectrum, and why [Channel Allocation, Cell Splitting, Sectorisation and Cell Breathing] is about capacity, not coverage.

What each costs. 802.15.4 gives up 3 MHz of every 5 to guard bands, 60 per cent of the spacing; GSM gives up 5.3 per cent of every slot to guard time; the length-8 Walsh code gives up seven eighths of the chip rate; a cluster of 7 gives each cell one seventh, 14 per cent, of the channels. Every dimension is paid for.

Distinctions

What it does not mean

There is no fifth dimension. Every scheme is one of these four or a combination; polarisation and MIMO streams are refinements of space.

Guard space is not waste to be eliminated. It is what makes the separation work; Bluetooth's zero guard band is paid for in collisions.

Code division does not give infinite capacity. Each extra user raises the noise floor for the rest, which is soft capacity, not free capacity.

TDM is not the same as packet switching. A time slot is reserved for a user whether or not it has data; a packet network gives the medium to whoever has something to send.

Defining more channels is not having more. 802.11b defines thirteen channels in a band that holds three non-overlapping ones.

Quick revision

• Four dimensions: space (SDM), frequency (FDM), time (TDM), code (CDM); as access schemes, SDMA, FDMA, TDMA, CDMA.
• Space: reuse at a reuse distance, or by direction (sectors, smart antennas). Capacity comes from smaller cells: program, 50 calls with 5 km cells against 30,750 with 250 m cells, same spectrum.
• Frequency: carriers plus guard bands. 2.4 GHz: 802.15.4 16 channels of 2 MHz, 5 MHz apart; Bluetooth 79 of 1 MHz, no guard; 802.11b 13 defined, 3 usable.
• Time: slots in a frame plus a guard time. GSM: 4.615 ms, 8 slots, slot 156.25 bit periods, burst 148, guard 8.25 (30.5 us, 9.1 km), hence timing advance.
• Code: orthogonal codes, the whole band at once. Program: four Walsh-8 users recovered exactly, an unused code sees zero. Costs bandwidth (1 bit per 8 chips) and needs power control (near-far).
• Combined: GSM = space + frequency + time (+ hopping); UMTS = space + code + frequency; 802.15.4 = frequency + time + spreading.

Test yourself

1. Name the four dimensions in which a medium can be multiplexed, and give the guard space each requires. Space, where signals are separated by being in different places or directions and the guard space is the reuse distance or the width of a beam; frequency, where the band is divided into carriers and the guard space is a guard band between them; time, where users take turns in slots of a frame and the guard space is a guard time covering propagation delay and clock error; and code, where all users occupy the whole band at once with different spreading codes and the guard space is the distance between the codes, paid for in bandwidth.

2. Compare frequency division and time division multiplexing. In frequency division each signal has its own carrier and uses it continuously, so no synchronisation between users is needed and receivers need only a filter, but guard bands are wasted, a user's carrier is idle when it has nothing to send, and the whole band is never available to one user. In time division every user has the whole band in turn, so capacity can be moved between users by giving them more slots and only one transmitter is active at a time, but all users must be synchronised, a guard time is wasted in every slot, and a user must wait for its slot.

3. How does code division multiplexing separate users, and what does it cost? Each user multiplies its data by a distinctive code, and all users transmit over the whole band at the same time; a receiver multiplies the received sum by the code of the wanted user and accumulates, which recovers that user's data while orthogonal codes contribute zero. It costs bandwidth, since a spreading factor of n means n chips per bit and a rate of one nth of the chip rate; it requires the codes to remain orthogonal, hence synchronisation; and it requires fast power control, because a nearby transmitter can drown distant ones, which is the near-far problem.

4. Why is GSM's guard time 8.25 bit periods, and what happens beyond the distance it allows? A slot is 156.25 bit periods long but the normal burst carries only 148, leaving 8.25 bit periods, about 30.5 microseconds, so that bursts from mobiles at different distances and with imperfect clocks do not overlap. A radio wave travels about 9.1 km in that time, so a mobile further away than that would have its burst arrive more than a guard period late and overlap the next slot; GSM therefore uses timing advance, instructing distant mobiles to transmit correspondingly early.

5. Why does making cells smaller increase capacity, and by how much? Because space division reuses the same channels in every cluster of cells, so the number of simultaneous calls is the number of cells times the channels per cell, and the channels per cell depend only on the cluster size, not on the cell's radius. Halving the radius quarters the area of a cell and so roughly quadruples the number of cells and the capacity. In the chapter's model of a 10 km city with 350 channels and a cluster of 7, one 5 km cell carries 50 calls and 250 m cells carry 30,750, with no change in spectrum.

6. How does GSM combine the dimensions of multiplexing? It divides space into cells, each reusing frequencies at a reuse distance; within a cell the band is divided by frequency into 200 kHz carriers; each carrier is divided by time into a frame of eight slots; and the carrier used may hop from frame to frame, which adds frequency diversity. A particular call is therefore identified by its cell, its carrier and its slot, and is separated from every other call in at least one of those dimensions.