# Modelling and analysis of wireless fading channels

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```					Modelling and analysis of
Geir E. Øien
29.08.2003
Wireless and mobile communication
channels
• Will initially study single-link wireless communications
• For example, transmitter may be a mobile terminal and

• Communication typically exposed to several kinds of
impairments, some of which are unique to the wireless
environment.
Wireless and mobile channel
impairments
• Impairments result mainly from:
– Multipath transmission due to reflections from (possibly moving)
objects in the surroundings  multipath fading and inter-symbol
interference (frequency-selectivity)
– Relative transmitter-receiver motion  Doppler effect
– Attenuation of signal power from large objects  (slow)
– Interference between different wireless carriers, transmitters, and
systems  inter-channel/inter-cell/inter-system interference
function of distance  path loss
– Thermal noise and background noise  additive noise
Assumptions
• Bandwidth B [Hz] available for communications, on a
carrier frequency fc [Hz].
• Digital communications with linear modulation (e.g.
QAM, QPSK) is used.
• I.e., transmitted waveform represents a sequence of
complex-valued modulation symbols, modulated onto a
complex sinusoidal carrier.
• Communications take place at Nyquist rate, i.e. 2B channel
symbols are transmitted per second (highest possible rate
where intersymbol-interference can be compensated for).
• Perfect synchronization in time and frequency is available
[no timing errors or oscillator drift].
Assumptions, cont’d

• Thus, the complex baseband representation of transmitted
signals can be used:

Transmitted waveform x(t) is represented by a sequence of
complex-valued discrete samples x(k), where sampling has
taken place at Nyquist rate.

• Real part corresponds to in-phase (I) component of
modulation symbol/waveform, and imaginary part
• Assume: Transmitter and receiver move relative to each
other with a constant effective velocity v [m/s].
• Results in a Doppler shift in the carrier frequency fc by a
maximal Doppler frequency of

fD = vfc/c [Hz]

where c = 3•108 m/s is the speed of light.
• Also results in randomly time-varying fading envelope as
reflection and scattering conditions change with time as
• Random fading models used to describe this phenomenon.
• How fast the fading varies depends on fD. The faster the
motion, the more rapid the fading variations.
• The complex baseband model of a flat-fading channel
becomes

y(k) = (k)x(k) + w(k),   k  Z,

where y(k) is received symbol at discrete time instant k,
(k) is the fading envelope, x(k) is the transmitted
information channel symbol, and w(k) is (complex-valued)
AWGN.
• (k) is modelled as a temporally correlated random
variable.
• Distribution (pdf) given by multipath model.
• Correlation properties given by multipath model and

• Rayleigh fading: Assumes isotropic scattering conditions,
no line-of-sight [most common model]
– I- and Q-components of complex fading gain are complex, zero-
mean gaussian processes
– thus the fading envelope follows a Rayleigh distribution
• Ricean (Rice) fading: Assumes line-of-sight component is
also present.
– I- and Q-components of complex fading gain are still complex
gaussian, but not zero-mean
– thus the fading envelope follows a Rice distribution
• Nakagami-m fading: More general statistical model which
encompasses Rayleigh fading as a special case, and can
also approximate Ricean fading very well.
Multipath transmission
• As waves are radiated from a transmitter antenna, they will
be reflected from reflecting objects.
• Waves are also scattered from objects with rough surfaces.
• Thus a transmitted signal will typically travel through many
different transmission paths, and arrive at the receiver as a
sum of different paths, coming in at various spatial angles.
• Typical assumption in mobile systems (at mobile side):
Isotropic scattering  Transmitted energy arrives equally
distributed over all possible spatial angles, with uniformly
distributed phases.
• In addition, a stronger line-of-sight (LOS) component may be
present.
Mathematical modeling of multipath
transmission
• Signal components from different incoming paths to
receiver have different delays (phases) and amplitude
gains.
• Thus, mutual interference between paths results in a
channel response which is a weighted sum of complex
numbers, in general time- and frequency-dependent.
• A multipath transmission channel can then be modelled by
a time-variant, complex-valued channel frequency
response.
• Here: Will only consider frequency-independent (flat)
channel responses [channel impulse response has only one
tap; thus no inter-symbol interference]
Attenuation of signal power from large
objects
• The mean received power attenuation depends strongly
(and relatively deterministically) on the path length
undergone the transmitted signal [cf. Path loss].
• However, slow stochastic variations may also be
experienced in the mean received power attenuation, due to
shadowing imposed by large terrain features between
transmitter and receiver (e.g. hills and buildings).
• Empirical studies that these variations can be modelled by
a log-normal probability distribution.
• This means that the mean received power attenuation in dB
has a normal (i.e., gaussian) distribution.
• Cf. Stüber, Ch. 2.4 for details [self-study].
Interference
• Electromagnetic disturbances from different sources within
a frequency band may interfer with the desired information
signal.
• These disturbances may come from other users (intra- or
inter-cell), or from other systems sharing the same
frequency band (may be problem in unlicensed bands).
• In our discussion we shall either disregard such
interference, or model it simply as an increased noise floor
(appropriate if there are many independent interference
sources).
• I.e., our “additive noise” term may encompass certain
types of interference (e.g., inter-user interference in a fully
Path loss
• In free space, received signal power typically decays with the
square of the path length d [m] experienced by the signal
during transmission.
• However, real-life environments are not “free space”, since the
earth acts as a reflecting surface: Other (maybe even more
severe) models may apply.
• Power may decay even faster with increased d.
• Transmit and receive antenna gains (and heights above ground)
and carrier frequency will also influence the path loss.
• Several analytical and empirical models developed for different
environments (macro-/microcell, urban/rural…).
• We refer to [Stüber, Ch. 2.5] for details [self-study].
• In our presentation, path loss will manifest itself as a (constant)
expected power attenuation G [-].
System noise
• Noise in a communication system typically comes from a
variety of mutually independent sources:
– thermal noise in receiver equipment
– atmospheric noise
– various kinds of random interference
• Noise is typically independent of the information signal,
and of the fading characteristics of the channel.
• Thus it usually is modelled as Additive White Gaussian
Noise (AWGN). [NB: law of large numbers!]
• Constant power spectral density N0/2 [W/Hz] over the total
(two-sided) bandwidth 2B.
• I.e., total noise power N0.

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