Multipath, Fading and the Doppler Effect

To make that simulable, the standard reduces the channel to a list of taps: "1) a discrete number of taps, each determined by their time delay and their average power; 2) the Rayleigh distributed amplitude of each tap, varying according to a doppler spectrum S(f)." Its Annex C gives three such lists, for a rural area, an urban area and hilly terrain, and the program uses them as they stand.

Delay spread and intersymbol interference

The delay spread is how far apart in time the echoes arrive, measured as the power-weighted standard deviation of the tap delays. It matters because a symbol smeared by more than its own duration runs into its neighbour: intersymbol interference.

The rule of thumb is that without an equaliser a channel can carry about 1 / (10 times the delay spread) symbols a second. Above that the receiver must equalise: estimate the channel from a known training sequence and undo the smearing. GSM's normal burst carries a 26-bit training sequence in its middle for exactly this reason ([The GSM Radio Interface: Carriers, the TDMA Frame and Bursts]).

Three ways out, and all are used:

• Equalisation, as GSM does.
• Spread spectrum, where a receiver can separate the echoes by their delays and even add them together, which is what a rake receiver in UMTS does ([The UMTS Radio Interface: W-CDMA, Codes, Power Control and Soft Handover]).
• Slower symbols on many carriers, which is OFDM: each carrier's symbol is long compared with the delay spread, so the smearing is small.

Fading

Short-term (fast) fading is what happens when the copies add. Move a receiver a few centimetres and the relative phases change, so the sum swings between constructive and destructive. It varies over a fraction of a wavelength, which at 2.4 GHz is a matter of centimetres.

When many paths arrive and none dominates, the amplitude follows a Rayleigh distribution, which is what 3GPP's taps use: "the Rayleigh distributed amplitude of each tap". When one path dominates, usually a line of sight, the distribution is Rician, and the fades are shallower. 3GPP's rural profile marks its first tap "RICE" and the rest "CLASS", because in the country there is usually a direct path.

Long-term (slow) fading, or shadowing, is the slower variation of the average level as the path passes behind buildings and trees. It is usually modelled as log-normal: the loss in decibels varies normally about the path-loss prediction.

Flat against frequency-selective. If the delay spread is much smaller than a symbol, every frequency in the signal fades together: flat fading. If not, the channel's response varies across the signal's band, so some frequencies are deeply faded and others are not: frequency-selective fading. The band over which the channel is roughly constant is the coherence bandwidth, about the reciprocal of the delay spread. A narrowband signal in a selective channel can be wiped out entirely; a wideband one loses only part of its spectrum, which is another argument for spreading.

The fade margin. Because fades are random, a link is designed with extra power in hand. DECT's own link budget names the number: "Normally switched antenna diversity is implemented, whereby MF = 11 dB in a quasi-stationary environment." The program measures what margin a Rayleigh channel actually demands.

The Doppler effect

If the transmitter, the receiver or a reflector moves, each path's frequency shifts. 3GPP states the maximum, fd = v / lambda: the speed "in ms-1" divided by "the wavelength". A wave arriving from straight ahead is shifted up by fd, one from behind down by fd, and one from the side not at all, which is why a moving receiver sees a whole Doppler spectrum rather than a single shift; 3GPP defines two shapes, CLASS (the classical spectrum) and RICE.

Two consequences:

• The carrier must be tracked. A receiver's frequency reference has to follow a shift that changes as the geometry changes.
• The channel ages. The rate at which the fading pattern changes is set by fd, and the coherence time, roughly 1 / fd, is how long the channel stays much the same. A frame longer than the coherence time fades within itself, so interleaving and coding are needed to spread the damage.

Multipath, fading and Doppler, computed

The program computes the delay spread of 3GPP's three profiles and the symbol rate each allows without an equaliser; counts how many symbols an echo spills into for three systems; simulates Rayleigh fading to find the depth of fades and the margin an availability requires; and tabulates the Doppler shift and coherence time at five speeds.

# Multipath, fading and Doppler, from 3GPP TS 45.005's own channel profiles.
import math
import random

C = 299792458.0

# 1. Delay spread. Each profile is a list of taps: (relative delay in
#    microseconds, average relative power in dB). The delay spread is the
#    power-weighted standard deviation of the delays, and a rule of thumb makes
#    the symbol rate a channel can carry without an equaliser about 1/(10 s).
PROFILES = {
    "rural (RAx), 6 taps": [(0.0, 0.0), (0.1, -4.0), (0.2, -8.0), (0.3, -12.0),
                            (0.4, -16.0), (0.5, -20.0)],
    "urban (TUx), 12 taps": [(0.0, -4.0), (0.1, -3.0), (0.3, 0.0), (0.5, -2.6), (0.8, -3.0),
                             (1.1, -5.0), (1.3, -7.0), (1.7, -5.0), (2.3, -6.5), (3.1, -8.6),
                             (3.2, -11.0), (5.0, -10.0)],
    "hilly terrain (HTx), 12 taps": [(0.0, -10.0), (0.1, -8.0), (0.3, -6.0), (0.5, -4.0),
                                     (0.7, 0.0), (1.0, 0.0), (1.3, -4.0), (15.0, -8.0),
                                     (15.2, -9.0), (15.7, -10.0), (17.2, -12.0), (20.0, -14.0)],
}

def spread(taps):
    p = [10 ** (db / 10) for _, db in taps]
    t = [us for us, _ in taps]
    total = sum(p)
    mean = sum(pi * ti for pi, ti in zip(p, t)) / total
    var = sum(pi * (ti - mean) ** 2 for pi, ti in zip(p, t)) / total
    return mean, math.sqrt(var), max(t)

print("Delay spread of the propagation profiles of 3GPP TS 45.005, Annex C:")
print("  profile                        mean delay   delay spread   last echo   symbols/s without an equaliser")
for name, taps in PROFILES.items():
    mean, rms, last = spread(taps)
    print("  %-30s %7.2f us %11.2f us %9.1f us %19.0f" % (name, mean, rms, last, 1 / (10 * rms * 1e-6)))
print("  GSM sends 270,833 symbols/s, so urban and hilly echoes cross symbol boundaries: an equaliser is needed.")

# 2. Intersymbol interference: how many symbols an echo spills into.
print("\nHow far an echo reaches, in symbols, for three systems:")
for system, rate in (("GSM, 270.833 ksymbol/s", 270833), ("802.15.4, 62.5 ksymbol/s", 62500),
                     ("a 20 Msymbol/s link", 20e6)):
    print("  %-26s" % system + "".join(
        " %s: %5.2f symbols;" % (name.split(" (")[0], spread(taps)[1] * 1e-6 * rate)
        for name, taps in PROFILES.items()))

# 3. Fading. Rayleigh amplitudes, as 3GPP's taps use: the sum of many random
#    paths. How often is the signal far below its average, and what margin does
#    a given outage need?
rnd = random.Random(79)
# a Rayleigh amplitude is the length of a vector whose two components are
# independent and normal, so the power is the sum of their squares
samples = sorted((rnd.gauss(0, 1) ** 2 + rnd.gauss(0, 1) ** 2) / 2 for _ in range(200000))
mean = sum(samples) / len(samples)
print("\nRayleigh fading: how deep the fades are (200,000 samples, mean power %.3f):" % mean)
print("   fraction of the time   the power is below   that is a fade of")
for q in (0.5, 0.1, 0.05, 0.01, 0.001):
    v = samples[int(q * len(samples))]
    print("  %20.1f%% %19.4f %17.1f dB" % (100 * q, v / mean, 10 * math.log10(v / mean)))
print("  so a link that must work 99%% of the time needs about %.0f dB of fade margin"
      % -(10 * math.log10(samples[int(0.01 * len(samples))] / mean)))

# 4. Doppler. 3GPP: fd = v / lambda, with v the speed and lambda the wavelength.
#    The coherence time, how long the channel stays much the same, is about
#    1/fd; a frame longer than that fades within itself.
print("\nDoppler shift fd = v / lambda, and roughly how long the channel holds still:")
print("   speed          900 MHz            2450 MHz          coherence time at 2450 MHz")
for kmh in (1.4, 5, 50, 120, 300):
    v = kmh / 3.6
    f900, f2450 = v / (C / 900e6), v / (C / 2450e6)
    print("  %5.0f km/h %10.1f Hz %17.1f Hz %22.1f ms" % (kmh, f900, f2450, 1000 / f2450))
print("  a 802.15.4 frame of 133 octets lasts 4.256 ms, so below about %.0f km/h it sees one channel"
      % (3.6 * (1 / 0.004256) * (C / 2450e6)))

Delay spread. The rural profile has a spread of 0.10 microseconds, the urban 1.03, and the hilly terrain 5.10, with echoes out to 20 microseconds, which is 6 km of extra path: a signal bouncing off a distant hillside. Without an equaliser those allow about 1,023,588, 97,466 and 19,616 symbols a second respectively. GSM sends 270,833 symbols a second, comfortably inside the rural figure and far outside the other two, which is why every GSM burst carries a training sequence and every GSM receiver has an equaliser.

How many symbols an echo crosses. At GSM's rate an urban echo spills over 0.28 of a symbol and a hilly one 1.38. At 802.15.4's 62.5 ksymbol/s the same channels give 0.06 and 0.32: the slow symbols of a sensor radio are largely immune, which is one reason the standard needs no equaliser. At 20 Msymbol/s the urban channel smears over 20 symbols and the hilly one over 102, which is why fast systems must use OFDM or a rake.

How deep fades go. In a Rayleigh channel the power is below its mean 50 per cent of the time (a shallow 1.6 dB), below a tenth of the mean 10 per cent of the time (9.8 dB down), and 20.2 dB down 1 per cent of the time. So a link that must work 99 per cent of the time needs about 20 dB of fade margin, and 99.9 per cent needs 30 dB. That is the true cost of fading, and it dwarfs the few decibels an antenna or an amplifier buys; it is also why diversity, which DECT credits with about 10 dB, is worth so much.

Doppler. At walking pace, 5 km/h, the shift at 2450 MHz is 11.4 Hz and the channel holds still for about 88 ms. In a car at 120 km/h it is 272 Hz and 3.7 ms. An 802.15.4 frame of 133 octets lasts 4.256 ms, so below about 104 km/h a whole frame sees one channel: a sensor network, whose nodes usually do not move at all, lives in a channel that changes only when the world around it does. A GSM handset in a train does not: at 300 km/h the channel is new every 1.5 ms, which is shorter than a GSM frame, and the interleaving of [GSM Logical Channels and the Frame Hierarchy] exists to spread a codeword across those changes.

Distinctions

What it does not mean

Multipath is not only harmful. Copies can be combined: a rake receiver and MIMO turn several paths into a gain.

Rayleigh is not a worst case. It is the ordinary case when no path dominates; a Rician channel with a line of sight fades less.

A fade is not a loss of power at the transmitter. The power is being sent; it is cancelling itself at that point in space at that instant.

Doppler is not only a frequency error. The shift itself is usually easy to track; the damage is that the channel changes at that rate.

A high delay spread does not always need an equaliser. It depends on the symbol rate: the same hilly channel needs one at GSM's rate and not at 802.15.4's.

Quick revision

• Multipath: copies by reflection, scattering, diffraction, with "random amplitudes and delays"; modelled as taps (delay, average power, Rayleigh amplitude, Doppler spectrum).
• Delay spread: power-weighted spread of the delays. 3GPP: rural 0.10 us, urban 1.03 us, hilly 5.10 us (echoes to 20 us). Symbols a second without an equaliser, about 1/(10 times the spread): 1.02 M, 97 k, 20 k. GSM at 270.833 ksymbol/s needs an equaliser and a training sequence.
• Fading: short-term (fast), over centimetres, Rayleigh (no dominant path) or Rician (line of sight); long-term (slow), shadowing, log-normal.
• Flat against frequency-selective, decided by the delay spread against the symbol time; coherence bandwidth is about 1 / delay spread.
• Fade margin: Rayleigh is 9.8 dB down 10 per cent of the time, 20.2 dB down 1 per cent, 30.4 dB down 0.1 per cent. DECT assumes 11 dB with switched diversity.
• Doppler: fd = v / lambda; at 2450 MHz, 11.4 Hz at 5 km/h and 272 Hz at 120 km/h; coherence time about 1 / fd: 88 ms and 3.7 ms. An 802.15.4 frame (4.256 ms) sees one channel below about 104 km/h.
• Answers: fade margin, diversity, equalisation, spread spectrum, OFDM, coding and interleaving.

Test yourself

1. What is multipath propagation, and how is a multipath channel modelled? It is the arrival of the same transmitted signal by several paths of different lengths, produced by reflection, scattering and diffraction, so that copies arrive with different delays, amplitudes and phases. For simulation the channel is reduced to a discrete number of taps, each with a time delay and an average power, with the amplitude of each tap varying randomly, usually with a Rayleigh distribution, according to a Doppler spectrum. 3GPP TS 45.005 gives such tap lists for rural, urban and hilly terrain.

2. Define delay spread and explain how it limits the symbol rate. The delay spread is the spread of the arrival times of the multipath copies, computed as the power-weighted standard deviation of the tap delays. Echoes arriving later than one symbol period spill into the following symbol, causing intersymbol interference, so as a rule of thumb a channel can carry about one tenth of the reciprocal of the delay spread in symbols per second before an equaliser is needed. For 3GPP's urban profile, with a spread of 1.03 microseconds, that is about 97,000 symbols a second, well below GSM's 270,833, which is why GSM receivers equalise.

3. Distinguish short-term and long-term fading. Short-term or fast fading is the rapid variation of the received amplitude as multipath copies add and cancel; it changes over a fraction of a wavelength, so over centimetres at 2.4 GHz, and follows a Rayleigh distribution when no path dominates or a Rician one when there is a line of sight. Long-term or slow fading, also called shadowing, is the slower variation of the average received level as the path passes behind buildings, hills and trees; it is usually modelled as log-normal, that is normal in decibels.

4. What is the difference between flat and frequency-selective fading? If the delay spread is much smaller than the symbol duration, equivalently if the signal's bandwidth is smaller than the coherence bandwidth, then all the frequency components fade together and the fading is flat. If the delay spread is comparable to or larger than the symbol duration, the channel's response varies across the signal's band, so some frequencies are deeply attenuated while others are not, which is frequency-selective fading; it is combated by equalisation, spread spectrum or multi-carrier modulation.

5. How much fade margin does a Rayleigh channel require, and why? Because the received power varies at random, a link designed for the average fails whenever the signal fades below the receiver's sensitivity. In the chapter's simulation the power was 9.8 dB below the mean 10 per cent of the time, 20.2 dB below it 1 per cent of the time and 30.4 dB below it 0.1 per cent of the time, so a link that must work 99 per cent of the time needs about 20 dB of margin and one that must work 99.9 per cent about 30 dB. Diversity reduces the margin needed, which is why DECT assumes 11 dB with switched antenna diversity.

6. State the Doppler shift and explain coherence time. The maximum Doppler shift is fd = v / lambda, the speed of the mobile divided by the wavelength; paths arriving from ahead are shifted up by fd and those from behind down by fd, giving a spectrum rather than a single shift. The coherence time, roughly the reciprocal of fd, is how long the channel remains much the same. At 2450 MHz and 120 km/h, fd is 272 Hz and the coherence time about 3.7 ms, so a frame longer than that fades within itself and needs coding and interleaving; an 802.15.4 frame of 4.256 ms is inside the coherence time below about 104 km/h.