4.7 Statistical Models for Multipath Fading Channels Ossana[Oss64] of multipath channel
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Transcript 4.7 Statistical Models for Multipath Fading Channels Ossana[Oss64] of multipath channel
4.7 Statistical Models for Multipath Fading Channels
several models have been suggested to explain observed statistical
nature of multipath channel
Ossana[Oss64] 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 Clarke’s 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
• signal’s angle of arrival in x-y plane measured with respect to
mobile’s 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
z y
v
x
Doppler Shift for the nth wave arriving at angle n to the x-axis is
given by
v
cos n (Hz) ( 4.57)
fn =
= wavelength of incident wave
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Electromagnetic fields of vertically polarized plane waves arriving at
the mobile given by
N
z
Ez = E0 Cn cos( 2f c n )
(4.58)
n 1
y
x
E0 N
v
Hx = Cn sin n cos( 2f c n )
(4.59)
Hy =
E0
n 1
N
C
n 1
n
cos n cos( 2f c n )
(4.60)
• 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 = 2fn + n
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(4.61)
4
Amplitude of electric & magnetic fields are normalized such that
ensemble average of Cn’s is
N
2
C
n 1
n 1
(4.62)
• since fn << 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]
Ez = Tc(t) cos(2fct) – Ts(t)sin(2fct)
(4.63)
N
Tc(t) = E0 Cn cos( 2f nt n )
n 1
(4.64)
N
Ts(t) = E0 Cn sin( 2f nt n )
(4.65)
n 1
• 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:
Tc2 Ts2 E z
2
E02
2
(4.66)
- overbar denotes ensemble average
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r(t) is envelope of Ez(t) given by:
r(t) = |Ez(t)| =
Tc2 (t ) Ts2 (t )
(4.67)
• since Tc & Ts are Gaussian random variables Jacobean transform
[Pap91] shows that r has has a Rayleigh Distribution
- r = random received signal envelope
p(r) =
r
r2
exp
2 2
2
0
( 0 r )
(4.68)
(r 0 )
where 2 = E02/2
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4.7.1.1 Spectral Shape Due to Doppler Spread in Clarke’s 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:
2
Pr = AG( ) p ( )d
(4.69)
0
AG()p()d = differential variation of received power with angle
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PSD of CW signal with frequency fc that is scattered
• instantaneous frequency of received signal component arriving at
angle is obtained from 4-57 and given by
f() = f = f mcos + fc
• f m=
v
(4.70)
= maximum Doppler shift
• f() is an even function of (e.g. f() = f(-))
Let S(f) = PSD of received signal differential variation of received
power with frequency is
S(f)|df |
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(4.71)
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equate S(f)|df | with AG()p()d, where
•AG()p()d = differential variation of received power with angle
•S(f)|df | = differential variation of received power with frequency
S(f)|df | = A[p()G() + p(-)G(-)] |d|
(4.72)
differentiate 4.70 with respect to yields
|df |=|d ||-sin()| fm
solve 4.70 for
4.74 implies sin =
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=
cos-1
f fc
fm
f fc
1
fm
(4.73)
(4.74)
2
(4.75)
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PSD, S(f), found by substitution of (4.73), (4.75) into (4.72)
S(f) =
A p( )G ( ) p( )G ( )
f fc
f m 1
fm
(4.76)
2
where S(f) = 0 for | f - fc | > fm
(4.77)
• 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:
1.5
SE z ( f )
f m
f fc
1
fm
(4.78)
2
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
S (f)
Ez
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fc- fm
fc
fc+ fm
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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
SbbE z =
2
f
1
K 1
8f m
2 fm
(4.79)
• (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
10log[8fmSbbEz(f)]
0dB
-1dB
-2dB
-3dB
-4dB
-5dB
-6dB
-7dB
-8dB
10-3
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10-2
10-1
1 2
10 f/fm
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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 < 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 Clarke’s & 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 = a+jb
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
independent
4.22a RF Doppler Filter
cos2fct
Baseband Gaussian
Noise Source
balanced
mixers
Baseband Gaussian
Noise Source
Doppler
Filter
s0(t)
sin2fct
4.22b Baseband Doppler Filter
independent
cos2fct
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Baseband Gaussian
Noise Source
Baseband Gaussian
Noise Source
Baseband
Doppler Filter
Baseband
Doppler Filter
balanced
mixers
s (t)
0
sin2fct
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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
(iv) random valued line spectrum is then multiplied with discrete frequency
representation of S E ( f ) (4.63)
z
- noise source and S E ( f ) have same number of points
z
(v) truncate SEz(fm) at passband edge ( fc fm = )
• compute function’s 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
*
g*N/2 g (N/2)-1
-fm
0
g(N/2)-1
gN/2
fm
S Ez ( f )
IFFT (•)2
-fm
fm
*
g*N/2 g (N/2)-1
-fm
0
g(N/2)-1
gN/2
fm
r(t)
S Ez ( f )
IFFT (•)2
-fm
fm
independent complex Guassian
samples form line spectra
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Steps to implement simulator shown in figure 4.24
1. Specify N = number of frequency points used to represent S E ( f )
z
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
5. Multiply in-phase & quadrature noise sources by fading spectrum,
S E ( f ) yields real frequency domain signal
z
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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
s(t)
(signal under test)
1
Rayleigh Fading
Simulator
a0
Rayleigh Fading
Simulator
a1
Rayleigh Fading
Simulator
a2
gains
2
delays
r(t)
Fig 4.25: Determine performance range for wide range of channel
conditions depending on gain and time-delay settings
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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 Clarke’s –
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
NR =
2
r
p
(
R
,
r
)
d
r
2
f
exp
m
(4.80)
0
fm= maximum Doppler Frequency
R = specified threshold for level crossing
r = time derivative (slope) of received signal r(t)
p(R, r ) = joint density function of r and r at r = R
= R/Rrms normalized threshold, the value of R normalized to local
rms amplitude of fading amplitude
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e.g. 4.7: Rayleigh fading signal with
R = Rrms = 1
fm = 20 Hz (maximum doppler frequency)
compute LCR:
NR=
2 f m e
2
2 20e 1
= 18.44 crossings/second
compute maximum velocity of mobile for fc = 900MHz:
fm = v/
vmax = fm = 20Hz (0.333m) = 6.66 m/s (24 km/hr)
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(2) Average Fade Duration (AFD) average time period for which r < R
1
Pr[ r R]
AFD For a Rayleigh fading signal: =
NR
1
i
Probability that r R is given by: Pr[r R] =
T i
i = duration of the fade
T = observation interval of fading signal
r = received signal
R = specified threshold level
(4.81)
(4.82)
From Rayleigh distribution probability that r R is given by:
2
p
(
r
)
dr
1
exp
R
Pr[r R] =
(4.83)
0
p(r) = pdf of Rayleigh distribution
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AFD as function of & fm is derived from 4.80 & 4.83 as
1 exp 2
=
2
2 f m e
=
exp 2 1
2 f m
(4.84)
AFD of a signal helps determine likely number of signaling bits lost
during a fade
• primarily depends on speed of the mobile
• decreases as fm becomes large
if fade margin is built into a mobile system – it is appropriate to
evaluate receiver performance by determination of
(i) NR, the rate at which input falls below R
(ii) , how long it remains below R, on average
useful for relating SNR to resulting instantaneous BER during a fade
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e.g. for fm = 200Hz find AFD for given normalized threshold
levels
2
=
e 1
2 f m
R << Rrms
R < Rrms
R = Rrms
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ρ = 0.01 =
ρ = 0.1 =
ρ=1
=
0.012
e
1
2 0.01(200)
= 19.9us
0.12
e
1
2 0.1(200)
= 200us
e1 1
2 (200)
= 3430us
31
e.g. for ρ = 0.707 and fm = 20Hz
2
0.707 2
e 1
e
1
18.3ms
2 f m
2 0.707 20
1. AFD: =
2. For binary digitial modulation with Rb = 50bps then Tb = 20ms
Tb > signal undergoes fast Rayleigh fading
3. Assume bit errors occurs when portion of bit encounters fade for
which ρ < 0.1, what is average number of bit errors/second
=
0.12
e
1
200us
2 0.1 20
Tb > only one bit will be lost on average during a fade
NR=
2 20 0.1e
0.12
4.96 crossings/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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4.7.4 2-Ray Rayleigh Fading Model
• Common multipath model & specific implementation of fig 4.25
• Clarkes model and Rayleigh fading statistics are for flat fading
- don’t consider multipath time delay
• Impulse response model given as:
hb(t) = α1exp(j1) δ(t) + α2exp(j2) δ(t-)
4.85
• αi = independent, Rayleigh distributed, generated from 2
independent waveforms (e.g. IFFT in 5.7.2)
• i= independent and uniformly distributed over [0,2]
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α1exp(j1)
input
output
α2exp(j2)
Fig 5.26: 2-Ray Rayleigh Fading Model
• set α2 = 0 special case of Rayleigh flat fading channel
hb(t) = α1exp(j1) δ(t)
4.86
• 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
~
Complex Received Voltage, V ( r )
baseband representation of summation of multipath waves impinging
on receiver
~ 2
Received Power, P(r) = V ( r )
differs by constant related to receiver input impedance
result is independent of receiver
~
Receive Envelop R(r) = V (r )
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