4'7 Statistical Models for Multipath Fading Channels

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4.7 Statistical Models for Multipath Fading
Channels
several models have been suggested to explain
observed statistical nature of multipath channel

• OssanaOss64 presented 1st model
• based on interference of waves incident
  reflected from flat
• sides of randomly located buildings
• assumes existence of LOS path
• predicts flat fading spectra that agrees with
  measurements in
• suburban areas
• limited to restricted range of reflection angles
• inflexible inappropriate for urban areas
  there is ususaly not
• not an LOS path

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4.7.1 Clarkes Model for Flat Fading

• statistical characteristics of electromagnetic
  fields of received
• signal are deduced from scattering
• model assumes fixed transmitter with vertically
  polarized antenna
• field incident on mobile antenna is assumed to
  consist of N
• azimuthal plane waves with
• - arbitrary carrier phases
• - arbitrary angles of arrival
• - equal average amplitude
• with no LOS path ? scattered components arriving
  at a receiver
• will experience similar attenuation over small
  scale distances

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• e.g. mobile receiver moving with velocity v
  along x-axis
• signals angle of arrival in x-y plane measured
  with respect to
• mobiles direction
• every wave incident on the mobile undergoes
  Doppler shift
• arrives at receiver at the same time
• flat fading assumption no excess delay due to
  MPCs

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Electromagnetic fields of vertically polarized
plane waves arriving at the mobile given by

• E0 real amplitude of local average E-field
  (assumed constant)
• Cn real random variable representing the
  amplitude of each wave
• ? intrinsic impedence of free space (377?)
• fc carrier frequency
• ?n random phase of nth arriving component that
  includes Doppler Shift

?n 2?fn ?n (4.61)
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Amplitude of electric magnetic fields are
normalized such that ensemble average of Cns is

• since fn ltlt fc ? field components Ez, Hx, Hy can
  be approximated as
• Gaussian random variables if N is sufficiently
  large
• phase angles are assumed to have uniform PDF on
  interval (0,2?

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Received E-field, Ez(t) can be expressed in terms
of in-phase and quadrature components Rice48

• Tc(t) Ts(t) are Gaussian RPs denoted by Tc
  Ts at time t
• Tc Ts are uncorrelated 0-mean Gaussian RVs
  with equal variance
• given by

(4.66)
- overbar denotes ensemble average
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r(t) is envelope of Ez(t) given by

• since Tc Ts are Gaussian random variables ?
  Jacobean transform
• Pap91 shows that r has has a Rayleigh
  Distribution
• - r random received signal envelope

where ? 2 E02/2
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4.7.1.1 Spectral Shape Due to Doppler Spread in
Clarkes Model
Spectrum Analysis for Clarkes model Gans72

• Let
• p(?)d? fractional part of total incoming power
  in d? of angle ?
• A average receive power for isotropic antenna
• G(?) azimuthal gain pattern of mobile antenna
  as a function of ?
• As N ?? then p(?)d? goes from discrete
  distribution to continuous
• distribution
• Total received power can be expressed as

AG(?)p(?)d? differential variation of received
power with angle
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PSD, S(f), found by substitution of (4.73),
(4.75) into (4.72)

• S(f) is centered on fc and 0 outside of limits
  of fc ? fm
• each arriving wave has carrier slightly offset
  from fc due to angle of
• arrival

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Assume vertical ?/4 antenna (G(?) 1.5) and
incoming power p(?) 1/2?, uniformly distributed
over 0,2?
output spectrum from 4.76

• output at fc ? fm ?
• Doppler components arriving at exactly 0 180
  have ? PSD
• since ? is continuously distributed ?
  probability of components arriving
• at 0 180 0

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• Baseband signal recovered after envelope
  detection of Doppler shifted
• signal
• resulting baseband spectrum has maximum
  frequency of 2fm
• Jak 74 showed that electric field produces
  baseband PSD of

• (4.79) is result of temporal correlation of
  received signal when
• passed through nonlinear envelope detector
• K() complete elliptical integral of 1st kind

• Spectral shape of Doppler spread
• determines time domain fading waveform
• dictates temporal correlation fade slope
  behaviors
• Rayleigh fading simulators must use fading
  spectrum (e.g. 4.78)
• to produce realistic fading waveforms

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Baseband PSD of CW received signal after envelope
detection
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• Mobile Terrestrial Channel
• Clarke Model for fast fading in assumes all rays
  are arriving from
• horizontal direction
• more sophisticated models account for possible
  vertical components
• ? conclusions are similar
• empirical measurements tend to support Clarke
  model with a Doppler
• bandwidth related to transmission frequency
  velocity
• measured spectra usually show peaks near Doppler
  Frequencies

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• Mobile Satellite Communications
• Empirical measurements indicate Clarkes Model is
  not always valid
• For aeronautical terminals satellites ?
  Gaussian spectrum is
• a better model of spectrum fading process
• - maximum fD is not proportional to aircraft
  speed, but ranges between
• 20Hz 100Hz
• - factors include nearby reflections from slowly
  vibrating fuselage
• and wings
• For maritime mobile terminal satellite ?
  Gaussian fading spectrum
• with Doppler bandwidth lt 1Hz more accurately
  reflect empirical results
• - due to slower motion of ship
• - distant reflections from sea surface tend to be
  directional (not
• omni directional)
• - effects of ocean waves as reflective surfaces

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• 4.7.2 Simulation of Clarkes Gans Fading Model
• design process includes simulation of multipath
  fading channels
• simulate in-phase quadrature modulation paths
  to represent Ez
• as given in (4.63)
• - in-phase quadrature fading branches produced
  by 2
• independent Guassian low-pass noise sources
• - each noise source formed by summing 2
  independent, orthogonal
• Guassian random variables
• e.g. g ajb
• a b are Gaussian random variables
• g is complex Gaussian
• - spectral filter in (4.78) used to shape random
  signals in frequency
• domain
• - allows production of accurate time domain
  waveforms of Doppler
• fading using IFFT at last stage of simulator

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Simulator using quadrature amplitude modulation
4.22a RF Doppler Filter
4.22b Baseband Doppler Filter
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Smith 75 demonstrated simple computer program
that implements baseband Doppler filter
(figure4.22b) (i) complex Gaussian random number
generator (noise source) produces baseband
line spectrum with complex weights in positive
frequency band - fm maximum frequency component
of line spectrum (ii) from properties of real
signals ? negative frequency components obtained
as complex conjugate of Gaussian values for
positive frequencies (iii) IFFT of each complex
Gaussian signal should be purely real Gaussian RP
in time domain - used in each of the
quadrature arms in figure 5.24

• (v) truncate SEz(fm) at passband edge ( fc? fm
  ? )
• compute functions slope at sampling frequency
  just before passband
• edge increase slope to passband edge

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Simulations usually implemented in frequency
domain using complex Gaussian line spectra -
leverages easy implementation of 4.78 - implies
low pass Gaussian noise components are a series
of frequency components (line spectrum from fm
to fm ) - equally spaced and each with complex
Gaussian weight
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figure 4.24 Frequency Domain Implementation of
Rayleigh fading simulator at baseband
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2. Compute frequency spacing between adjacent
spectral lines ?f 2fm/(N-1) ? defines T time
duration of fading waveform T 1/ ?f
3. Generate complex Gaussian random variable for
each N/2 positive frequency components of
noise source
4. Construct negative frequency components of
noise source by conjugate of positive
frequency values
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6a. Perform IFFT on resulting frequency domain
signals in (5) ? yields two N-length time
series 6b Add squares of each signal point in
time to create N-point time series under
radical ? 5.67
7. Take square root of sum in 6 ? obtain N- point
time series of simulated Rayleigh fading
signal with Doppler Spread and Time
Correlation
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To Produce Frequency Selective fading effects ?
use several Rayleigh fading simulators and
variable gains and time delays
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• To create Ricen Fading channel
• make single frequency component dominant in
  amplitude within
• fading spectrum at f 0
• To create multipath fading simulator with many
  resolvable MPCs
• alter probability distribution of individual
  multipath components in
• simulator

• IFFT must be implemented to produce real
  time-domain signal
• given by Tc(t) and Ts(t) (5.64 and 5.65)
• To determine impact of flat fading on s(t) ?
  compute s(t) ? r(t)
• s(t) applied signal
• r(t) output of fading simulator
• To determine impact of several MPCs use
  convolution (figure 4.25)

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4.7.3 Level Crossing Fading Statistics
2 important statistics for designing error
control codes and diversity schemes for Rayleigh
fading signal in mobile channel (1) Level
Crossing Rate (LCR) average number of level
crossings (2) Average Fade Duration (AFD)
mean duration of fades

• Makes it possible to relate received signals time
  rate of change to
• received signal level
• velocity of mobile

Rice computed joint statistics for fading model
similar to Clarkes that provided simple
expressions for computing LCR AFD
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• LCR expected rate at which Rayleigh Fading
  envelope crosses
• specified level in a positive-going direction
• Rayleigh Fading normalized to local rms signal
  level
• NR level crossings per second at specified
  threshold level of R

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e.g. 4.7 Rayleigh fading signal with R Rrms
? ? 1 fm 20 Hz (maximum doppler frequency)
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(2) Average Fade Duration (AFD) average time
period for which r lt R
?i duration of the fade T observation
interval of fading signal r received signal
R specified threshold level
From Rayleigh distribution ? probability that r
? R is given by
(4.83)
Prr ? R
p(r) pdf of Rayleigh distribution
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AFD as function of ? fm is derived from 4.80
4.83 as
(4.84)
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e.g. for fm 200Hz ? find AFD for given
normalized threshold levels
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e.g. for ? 0.707 and fm 20Hz
1. AFD
2. For binary digitial modulation with Rb 50bps
then Tb 20ms
3. Assume bit errors occurs when portion of bit
encounters fade for which ? lt 0.1, what is
average number of bit errors/second

• if 1 bit error occurs during a fade ? 5 bits
  errors occur per second
• BER bit errors per second/Rb 5/50 10-1

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• vary ? ? to create wide range of FSF effects

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• Other Small Scale Fading Models
• 4.7.5 Saleh Valenzuela Indoor Statistical
  Model wideband
• model where resolvable MPCs arrive in clusters
• 4.7.6 SIRCIM SMRICM indoor and outdoor
  statistical models
• based on empirical measurements in 5 factory
  buildings at 1.3GHz
• statistical model generates measured channel
  based on discrete
• impulse response of channel model
• developed computer programs that generate small
  scale channel
• impulse response
• (i) Simulation of Mobile Radio Channel Impulse
  Response Model (SMRICM)
• (ii) Simulation of Indoor Radio Channel Impulse
  Response Model (SIRCIM)

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4.8.1.2 Fading Rate Variance Relationships
baseband representation of summation of multipath
waves impinging on receiver
differs by constant related to receiver input
impedance result is independent of receiver