Title: 4'7 Statistical Models for Multipath Fading Channels
14.7 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
24.7.1 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
4Electromagnetic 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)
5Amplitude 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?
6Received 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
7r(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
84.7.1.1 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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11PSD, 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
12Assume 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
14Baseband 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
19Smith 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
20Simulations 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
21figure 4.24 Frequency Domain Implementation of
Rayleigh fading simulator at baseband
222. 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
236a. 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
24To 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)
264.7.3 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
28e.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
30AFD as function of ? fm is derived from 4.80
4.83 as
(4.84)
31e.g. for fm 200Hz ? find AFD for given
normalized threshold levels
32e.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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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)
364.8.1.2 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