Modelling and analysis of wireless fading channels

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Modelling and analysis of wireless fading channels

• Geir E. Øien
• 29.08.2003

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Wireless and mobile communication channels

• Will initially study single-link wireless
  communications (one transmitter, one receiver).
• For example, transmitter may be a mobile terminal
  and receiver a base station (uplink), or vice
  versa (downlink).
• Communication typically exposed to several kinds
  of impairments, some of which are unique to the
  wireless environment.

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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 (? time-varying, correlated
  random fading)
• Attenuation of signal power from large objects ?
  (slow) shadowing
• Interference between different wireless carriers,
  transmitters, and systems ? inter-channel/inter-ce
  ll/inter-system interference
• Spreading of radiated electromagnetic power in
  space as function of distance ? path loss
• Thermal noise and background noise ? additive
  noise

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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-interfere
  nce can be compensated for).
• Perfect synchronization in time and frequency is
  available no timing errors or oscillator drift.

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Assumptions, contd

• 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
  corresponds to quadrature (Q) component.

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Relative transmitter-receiver motion

• 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 3108 m/s is the speed of light.
• Also results in randomly time-varying fading
  envelope as reflection and scattering conditions
  change with time as transmitter and receiver
  move.
• 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.

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Random flat-fading models

• 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 transmitter-receiver motion assumptions.

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Random flat-fading models, contd

• 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.

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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.

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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

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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.

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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 loaded cellular network).

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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 -.

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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.