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4'7 Statistical Models for Multipath Fading Channels

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since fn fc field components Ez, Hx, Hy can be approximated as ... independent Guassian low-pass noise sources ... implies low pass Gaussian noise components ... – PowerPoint PPT presentation

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Title: 4'7 Statistical Models for Multipath Fading Channels


1
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

2
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

3
  • 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

4
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)
5
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?

6
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
7
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
8
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
9
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11
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

12
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

13
  • 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

14
Baseband PSD of CW received signal after envelope
detection
15
  • 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

16
  • 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

17
  • 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

18
Simulator using quadrature amplitude modulation
4.22a RF Doppler Filter
4.22b Baseband Doppler Filter
19
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

20
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
21
figure 4.24 Frequency Domain Implementation of
Rayleigh fading simulator at baseband
22
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
23
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
24
To Produce Frequency Selective fading effects ?
use several Rayleigh fading simulators and
variable gains and time delays
25
  • 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)

26
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
27
  • 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

28
e.g. 4.7 Rayleigh fading signal with R Rrms
? ? 1 fm 20 Hz (maximum doppler frequency)
29
(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
30
AFD as function of ? fm is derived from 4.80
4.83 as
(4.84)
31
e.g. for fm 200Hz ? find AFD for given
normalized threshold levels
32
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

33
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34
  • vary ? ? to create wide range of FSF effects

35
  • 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)

36
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
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