1.
1
EE 463 - Wireless
CommunicationsCommunications
Dr. Ahmed Iyanda Sulyman
Associate Professor
Electrical Engineering DepartmentElectrical Engineering Department
King Saud University
Chapter 4: Mobile radio
propagation: Small-scale fading
1. Introduction: small-scale fading & multipath
2. Impulse response model of a multipath channel
3. Multipath channel parameters
4. Types of small-scale fading
5. Statistical models
2
6. Fading channel Simulations

2.
2
Chapter 4: Mobile radio
propagation: Small-scale fading
Direct path or line of sight (LOS) wave
Reflected
wave
Diffracted
wave
3
Mobile userBase
station
Scattered
wave
 Small-scale fading (or simply fading) describes the
rapid fluctuations of a radio signal over a short period
of time (few seconds) or a short travel distance (few
1 – Introductions
( ) (
wavelength).
Path 1
Path 2
Path 3
4
 The radio waves from the transmitter arrive at the
mobile from different directions, each with different
amplitude and propagation delays (multipath comp.).
Path 4

3.
3
1 – Introductions
• Multipath components may combine (vectorial addition)
either constructively or destructively at the mobiles,y y ,
and may thus cause the signal received by the mobile
to distort or fade (small-scale fading).
 Factors causing small-scale fading:
 Multipath propagation: presence of reflectors,
scatterers, etc., in the environment.
 Speed of the mobile: movement of transmitter or
receiver or both cause frequency shift in the
transmitted signal (called Doppler shift)
5
transmitted signal (called Doppler shift).
 Speed of surrounding object: movements of
surrounding objects induce time-varying Doppler shift.
 Transmission bandwidth of the signal: If the
transmitted radio signal has bandwidth greater than
the bandwidth of the multipath channel.
 Doppler shift (revision):
 Consider a mobile moving at a constant velocity v
1 – Introductions
 Consider a mobile moving at a constant velocity, v,
along a path segment of length d between points A
and B, while it receives signals from a remote
source S.
 The phase shift in the received signal due to the
difference in path length as the mobile moves from
points A to B is given by:
)(
22



dL

6
)cos(
2
)cos(






tv



4.
4
Source, S
Illustration: Doppler Shift
L )cos(
2
)cos(
22









tv
dL





• Phase shift between A and B:
7
d

v
)cos(
2
1




v
t
fDoppler 


A B
• Frequency shift between A and B
(Doppler frequency):

 Doppler shift (revision):
 The apparent change in frequency (Doppler shift or
1 – Introductions
 The apparent change in frequency (Doppler shift or
Doppler frequency) is.
 Positive Doppler shift (apparent received frequency is
increased): receiver moves towards transmitter.
 Negative Doppler shift (apparent received frequency
d d) f
)cos(
2
1




v
t
f D 



8
is decreased): receiver moves away from transmitter.
 If source transmits fc, the received frequency is fc ± fD

5.
5
 Example 4.1:
1 – Introductions
 Consider a transmitter which radiates a sinusoidal
carrier frequency of 1850MHz. For a vehicle moving
at 28m/s, compute the received carrier frequency if
the mobile is moving:
 (a) directly toward the transmitter.
 (b) directly away from the transmitter.
 (c) in a direction perpendicular to the direction of
9
 (c) in a direction perpendicular to the direction of
arrival of the transmitted signal.
 Modeling the impulse response of wireless channels
allows numerical performance evaluation of different
2 – Impulse response model
of a multipath channel
p
mobile communication systems.
 For a mobile at a fixed position, d, the received signal
can be expressed as.
d
v
Spatial position
10
p
 For a causal system ( ),



  dtdhxtdhtxtdy ),()(),(*)(),(
0,0)(  tth 

t
dtdhxtdy  ),()(),(

6.
6
 Assume the mobile moves at a constant velocity v, then d = vt,
thus.
2 – Impulse response model
of a multipath channel
or
 where x(t) = transmitted bandpass signal (modulated signal at
d
v
Spatial position
 

t
dtvthxtvty  ,)(),(   ),(*)(,)()(  thtxdthxty
t
 
11
 where x(t) = transmitted bandpass signal (modulated signal at
carrier frequency fc)
 y(t)=received signal,
 h (t -) = impulse response of the multi-path radio channel,
t= time variable,  =channel multipath delay for fixed t.
 The impulse response of a multipath channel can be
expressed as.
2 – Impulse response model
of a multipath channel
1N
p
 ak(t, ) is the amplitude of the kth multipath component at time
t.
 k(t) is the excess delay of the kth multipath component at
time t.
 k(t,) is the phase shift of the kth multipath component, which
is a function of the delay  and time t




1
0
))(()),(exp(),(),(
N
k
kkk ttjtath 
12
is a function of the delay  and time t.
 N is the number of paths.
 If the channel is assumed to be time invariant, or stationary
over a small-scale time or distance interval (quasi-static
fading), then




1
0
)()exp(),(
N
k
kkk jath 

7.
7
),( th
t4
Illustration: time-varying impulse response
t2
t3
4
10 2 3
10 2 3




1
0
2222 ))(()),(exp(),(),(E.g.
N
k
kkk ttjtath 
13

t1
1
0
0 2
0k
 Cellular operators develop channel impulse response for
different environments to allow prior planning and design.
~ typical entry-level duties given wireless Engineers
2 – Impulse response
Measurements
~ typical entry level duties given wireless Engineers
 Small-scale Multipath channel Measurements
techniques:
 Small scale channel state for an environment is typically
d
v
Spatial position
14
 Small-scale channel state for an environment is typically
recorded in the form of average power delay profiles,
defined as P()  Ek[|h(tk, )|2 ], where Ek[] denotes
ensemble average over samples taken at different times tk.
 Different channel sounding techniques for estimating
the power delay profiles of wireless systems are:

8.
8
 Direct RF pulse system:
transmits a repetitive pulse of
width Tb , and uses a receiver
2 – Impulse response
Measurements
cf Tx
RF
b ,
with a wide bandpass filter to
collect all multipath signal
received from each of these
pulses and take the ensemble
average.
 Advantage: low complexity
(easily implemented).
 Disadvantage: it is subject

Pulse Generator
Tb
Trep
Pulse width = Tb, Pulse period = Trep
BPF
Envelope
Detector
Ampl.
Oscillos-
cope
RF
15
 Disadvantage: it is subject
to interference and noise in
the environment during the
test which may not represent
the actual channel being
measured.
 Spread Spectrum based channel
sounding: rather than transmitting
ordinary pulses, the transmitted
l i thi h d d
2 – Impulse response
Measurements
cf Tx
RF
pulses in this approach are encoded
using binary PN (pseudo-noise)
sequence, which spreads the pulse
over a wide band causing noise &
interference rejections.
 Advantage: moderate complexity,
and noise/interference rejection.
 Processing gain of the SS system
allows much lower power than the

PN sequence
Generator

Sequence length
RF PN sequence
Generator
BPF
16
allows much lower power than the
direct RF pulse system.
 Widely used for indoor and outdoor
channel sounding (3G and 4G).
 Disadvantage: instantaneous
measurements (real time) are not
made, but over PN sequence length
BPF
(wide)
Ampl.
Oscillos-
cope
Envelope
Detector

BPF
(narrow)

9.
9
 Frequency domain channel
sounding: this method measures
channel impulse response in
f d i d
2 – Impulse response
Measurements
frequency-domain and use
Inverse Discrete Fourier Transform
(IDFT) to convert from frequency
to time domain measurements.
 The S parameter measured in
frequency domain is proportional
to H (j)= FT{h(t)}.
 Advantage: provides amplitude &
phase information of the time-
RF
Inverse
DFT processor
S parameter
Test set
Vector Network Analyzer
with
Swept Freq. Oscillator
RF
)( jX
)()(21  jHjS 
)(
)(
)(



jX
jY
jH 
1
)( jY
Port1 Port2
17
phase information of the time-
domain channel (complex).
 Disadvantage: hard wired sync.
needed between transmitter &
receiver, making it useful only for
very close measurement.
 Used in experiments on indoor
channel measurements.
)}({
1
)( jHFTth


 Quantitative multipath channel parameters have been developed
in order to compare different multipath channels and for receiver
design purposes.
3 – Parameters of mobile
multipath channel
g p p
 Time dispersion parameters:
 Mean Excess delay:






 1
0
2
1
0
2
N
k
k
N
k
kk






1
22
N
kk 
Pt (t)
t
Mobile
channel
max
TT
Channel span (max. excess delay)
t
Prec (t)
18
 RMS delay spread: where
 Maximum excess delay max (x dB): Is the time delay during
which multipath energy falls to x dB below the maximum.
 These parameters are obtained directly from power delay profile.
 22
 





 1
0
2
02
N
k
k
k



10.
10
 Example 4.3 (Ex5.4 in Ref1)
3 – Parameters of mobile
multipath channel
 Compute the RMS delay spread for the power delay profile of a
multipath channel shown below. What is the maximum symbol
rate that can be transmitted through the channel without
needing an Equalizer?.
)(recP
0 dB 0 dB
19
 Ans:  = 0.5sec, and Rs = 200k symb./sec. For BPSK, Rb=Rs.
0
1s
 Coherence bandwidth, Bc.
3 – Parameters of mobile
multipath channel
 Is the range of frequencies over which the channel can be
considered “flat” (i.e., the channel’s frequency response stays
correlated).
 Bc is inversely proportional to the RMS delay spread, but its
exact value depends on how it is defined:
 → If the coherence bandwidth is defined as the bandwidth
over which the frequency correlation function is above
0.9, then 1
B
20
0.9, then
 → If the coherence bandwidth is defined as the bandwidth
over which the frequency correlation function is above
0.5, then
50
cB
5
1
cB

11.
11
 Example 4.4 (Ex 5.5 in Ref1)
Calculate the mean excess delay and the rms delay spread for the
3 – Parameters of mobile
multipath channel
Calculate the mean excess delay, and the rms delay spread for the
multipath power delay profile shown below.
Estimate the 50% coherence bandwidth of the channel.
Would this channel be suitable for AMPS or GSM cellular service
without the use of an Equalizer ?.
 Ans:
 =4.38sec,
  = 1.37sec, and
)(recP
0 dB
21
 Bc (50%)= 146KHz.
 Bc>30kHz used in AMPS
 (no Equalizer required)
 But Bc<200kHz used in GSM
 (Equalizer required for GSM).

-20 dB
0
1s
-10 dB
2s 5s
 Doppler spread (BD) and coherence time (TC) ,.
D l d d h b d idth B d ib th
3 – Parameters of mobile
multipath channel
 Delay spread, στ, and coherence bandwidth, BC, describe the
time dispersion of the channel in a short time window.
However, they do not give information about the time-varying
nature of the channel.
 Doppler spread BD and coherence time TC describe the effect
of time-varying nature of the channel on the received signal.
 Doppler spread BD is a measure of the spectral broadening
caused by the time rate of change of the mobile radio
22
y g
channel:
 If the baseband signal bandwidth is much greater than BD,
then the effects of the Doppler spread are negligible.

v
fB DD  max

12.
12
 Coherence time is a statistical measure of time duration over
which the channel impulse response is essentially invariant:
3 – Parameters of mobile
multipath channel
p p y
 If is defined as the time over which the time correlation
function is above 0.5, then
 Example 4.5
 A measurement team traveling at 50 m/s uses a 900MHz
max
11
DD
c
fB
T 
max
16
9
D
c
f
T


23
g
carrier to estimate the small-scale propagation parameter for
an urban environment:
(a) What is the Doppler spread BD for the mobile channel,
(b) What is the coherence time Tc for the mobile channel.
 Ans: BD =(3x50)Hz, Tc = 3/(50x16) sec.
 Depending on the relation between the signal parameters
(such as bandwidth, symbol period, etc) and the propagation
4 – Types of small-scale
fading
( , y p , ) p p g
channel parameters (such as RMS delay spread and Doppler
spread), transmitted signals will undergo different types of
small-scale fading.
 Basically there are two fading mechanisms, one independent
of the other:
 Small-scale fading based on multipath delay spread
24
 Small-scale fading based on Doppler spread

13.
13
 Small-scale fading based on multipath delay spread:
4 – Types of small-scale
fading
 Time dispersion of the multipath causes the transmitted
signal to undergo either flat or frequency selective fading;
 Flat fading:
Signal BW (Bs) < Channel BW (Bc)
Symbol period (Ts) > Delay spread (στ)
[note: if one is satisfied, the other will be satisfied as well]
25
 Frequency selective fading:
Signal BW > Channel BW
Symbol period < Delay spread
Illustration of flat fading:
)(tx ),( th
)(ty
0 sT
)(tx ),( th
0 max t
0 maxsT
)(*),()( txthty 
t
sTmax
X(f) H(f) Y(f)= H(f).X(f)
26
cf
f
cf
f
cf
f
X(f) H(f) Y(f) H(f).X(f)
sc BB 

14.
14
Illustration of frequency selective fading:
)(tx ),( th
)(ty
0 sT
)(tx ),( th
0 max t
0 maxsT
)(ty
t
sTmax
X(f) H(f)
27
cf
f
cf
f
cf
f
X(f) H(f)
Y(f)
sc BB 
 Small-scale fading based on Doppler spread:
F di i d t D l d B ( l ll d
4 – Types of small-scale
fading
 Frequency dispersion due to Doppler spread BD, (also called
time selective fading since ), leads to signal distortion
(either slow or fast fading):
 Fast fading:
Symbol period (Ts) > Coherence time (Tc)
Large Doppler spread
Channel variations faster than base-band signal variations
 Sl f di
D
c
B
T
1

28
 Slow fading:
Symbol period < Coherence time
Small Doppler spread
Channel variations slower than base-band signal variations

15.
15
 Rayleigh fading model:
 Baseband received signal can be written as y(t)= yI(t)+ jyQ(t).
5 – Statistical Models
Q
 The I-Q (In-Phase and Quad.) components of the received signal
follow zero-mean Gaussian distribution. Thus the envelope of the
received signal can be modeled as a zero-mean Rayleigh random
variable, when LOS is absent.
 Prob. density function (PDF):
0.4
0.5
0.6
0.7
)(rp
10|)(|,)
2
exp()( 2
2
2
 tyr
rr
rp

29
 The cum. distribution function (cdf),
is the probability that received signal
envelope does not exceed a value R:
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
0.1
0.2
0.3
r
  





  2
2
0 2
exp1)(Pr

R
drrpRr
R
 Rayleigh fading model:
 Baseband received signal power is given
b ( ) | ( )|2
5 – Statistical Models
by Prec(t)=|y(t)|2.
 Thus the received signal power for a
Rayleigh fading model, is exponentially
distributed with mean E[Prec(t)]=2σ2.
 Prob. density function (PDF) of received
power:
0|)(|,)
2
exp(
2
1
)( 222  tyr
x
xpr

t
rec
P
P
30
 Note that E[Prec(t)] is the received signal
power based on path loss and shadowing
alone, while the pdf above models the
random variations around E[Prec(t)], based
on small-scale fading (see illustrations).
)log(d

16.
16
 Example 4.6
Gi th t th i d i l lit d i l
5 – Statistical Models
 Given that the received signal amplitude r, over a wireless
channel, is a Rayleigh distributed random variable. Compute
the mean value, and the variance of r. Calculate also the
median value, what does this signify?.
 Ans:



0
2
)(][

drrrprErmean
]2[])[()var( 222
meanmeanmean rrrrErrEr 
2

31
 Median value represents the value of r for which the cdf is 0.5
 Solve (for Rayleigh ch.)
22
0
2
4292.0)
2
2(
2
)( 



  drrpr
,)(5.0
0

medianr
drrp 177.1 medianr
 Example 4.6b
C id h l ith R l i h f di l d
5 – Statistical Models
 Consider a channel with Rayleigh fading envelope and
average received power E[Prec(t)]=20dBm. Find the
probability that the received power is below 10dBm.
 Ans: note that E[Prec(t)]=20dBm=100mW. We want to
find the prob. that r 2 < 10dBm =10mW.
,095.0)
100
exp(
100
1
]10Pr[
10
2
  dx
x
r
32
,)
100
p(
100
][
0


17.
17
 Ricean fading model:
 Baseband received signal can be written as y(t)= yI(t)+ jyQ(t).
5 – Statistical Models
Q
 The I-Q (In-Phase and Quad.) components of the received signal
follow Gaussian distribution with E[y(t)]=A , when LOS is present.
Thus the envelope of the received signal can be modeled as a
Ricean random variable with E[y(t)]=A, when LOS is present.
 Prob. density function (PDF):
0,)()
2
)(
exp()( 202
2
2


 r
Ar
I
Arr
rp

)(rp
0.4
0.5
0.6
1,1  A
33
 A= amplitude of the LOS signal,
 I0 =Bessel function of the first kind ,
of order 0.
 Becomes a Rayleigh fading model
if A=0 (no LOS).
 Rice factor: dB
0
0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
0.1
0.2
0.3
r





 2
2
2
log10

A
K
 Ricean fading model:
 The average received power in Ricean fading model is given by
5 – Statistical Models
 Where 2σ2 is the average power in the non-LOS multipath
components, and A2 is the power in the LOS component.
 Thus the Rice factor K is the ratio of power in the LOS to the power
in the non-LOS components.
22
0
22
2)(][  

AdrrprrEPrec
)2( 2
22
22  PA 
34
 Note:
 Using these relations, the Ricean pdf can also be written as:
)1/(2and,
)1(
.
)1
2
(
)2(
2 2
2
2
22





 KP
K
P
K
A
A
KKA rec
rec




0,)
)1(
2()
)1(
exp
)1(2
)( 0
2







 


 r
P
KK
rI
P
rK
K
P
Kr
rp
recrecrec

18.
18
 Nakagami fading model:
 Gives more generalized fading statistics than Rayleigh and Ricean
5 – Statistical Models
g g y g
models. Practical measurements show that Nakagami distribution
models signal received over mobile channels, more accurately.
 The Nakagami pdf is given by
is the gamma function, and   



0
1
exp dyyym m
0,5.0,exp
)(
2
)(
2
)12(













 
rm
P
mr
r
P
m
m
rp
rec
m
m
rec
][ 2
rEPrec 
35
m is called the Nakagami fading parameter (usually m ≥0.5)
 m =1, Nakagami fading becomes Rayleigh fading model.
 m ≥2, Less severe fading can be modeled.
 As m → ∞, non-fading or AWGN channel, can be modeled.
 For m=(K+1)2/(2K+1), Ricean fading with factor K can be modeled.
 Nakagami fading model:
 Thus the Nakagami distribution can model Rayleigh, Ricean, AWGN
5 – Statistical Models
g y g , ,
(no fading), and more general fading in more or less severer than
Rayleigh and Ricean. Thus measurements in different environments
can fit the Nakagami distribution by choosing appropriate m.
 The power distribution for Nakagami fading, obtained by making
change of variable x=r2 in above pdf, is given by
050)(
1







 
mxxm m
m
36
0,5.0,exp
)(
)( 








 xm
PmP
xp
recrec

19.
19
5 – Level Crossing and
Average Fade durations
• Two important fading parameters are the average # of
level crossings or fades and the duration of theseg
fades.
 Level crossing rate (LCR): the expected rate at
which the Rayleigh fading envelope crosses a specified
level R, when envelope is in a positive going direction.
(i.e. r1 – r2 = +ve).
 The number of level crossing per second is given by
)exp(2)( 2
  

frdrRprN 
37
 Where , and is the joint pdf
of at r =R. Also , where Rrms is
the local rms amplitude of the fading envelope [Ref 5].
 Note also: where P0 is a target received
power (receiver sensitivity).
)exp(2),(
0
max
  DR frdrRprN
rr ofslopetheis ),( rRp 
rr and rmsRR /
recPP /0
5 – Level Crossing and
Average Fade durations
• Example: Rayleigh fading simulation, fc=900MHz,
and Mobile Speed = 120km/hrp
l id2/
38
ncorrelatio-de2/ 
Exercise: at threshold level -20dB about RMS, estimate NR for this simulation.

20.
20
5 – Level Crossing and
Average Fade durations
• Example (5.7 in Ref1): For a Rayleigh fading signal,
compute the positive-going level crossing rate for  =1,p p g g g 
when the maximum Doppler frequency (fm) is 20Hz.
What is the maximum velocity of the mobile for this
Doppler frequency if the carrier frequency is 900MHz?.
 Ans:
crossings per sec.
 The maximum velocity of the mobile can be obtained
using the Doppler relation, .
44.18)1exp()1)(20(2 2
 RN
/max
vfD 
39
 Thus v =20Hz(1/3m)=6.66m/s=24 km/hr.
max
5 – Level Crossing and
Average Fade durations
 Average fade duration (AFD): is the average period
of time for which the received signal is below aof time for which the received signal is below a
specified level R. For a Rayleigh fading signal, this is
given by
 The average fade duration helps to determine the
likely number of signaling bits that may be lost when
fading occurs



2
}1){exp(
]Pr[
1
max
2
DR
AFD
f
Rr
N


40
fading occurs.

21.
21
5 – Level Crossing and
Average Fade durations
• Example (5.9 in Ref1): (a) Find the average fade duration
for a threshold level of =0.707 when the Doppler frequency
i 20H F bi di it l d l ti ith d t t f 50is 20Hz. For a binary digital modulation with data rate of 50
bps, is this Rayleigh fading scenario slow or fast?. (b) What is
the average number of bit errors per second for the given
data rate for the case  = 0.1 [assuming that a bit error
occurs whenever any portion of a bit encounters fading]?.
 Ans: (a)
 For Rb=50bps, Tb= 20ms, which is > . Thus the signal
undergoes fast Rayleigh fading.
ms3.18
2)20)(707.0(
1)707.0exp( 2




FD
AFD
41
u de goes ast ay e g ad g
(b) For  = 0.1, we have . This is less than
the duration of one bit. Therefore, only one bit on average will
be lost when a fading event occur.
 For  = 0.1, crossings per second. Thus there are
total of 5 bits in error per sec., resulting in BER=(5/50)=0.1
ms2sec002.0 AFD
96.4RN
 Frequency Flat /Freq selective fading models:
 Two-ray frequency-selective fading model: Models
5 – Fading Channel
Simulations
the effect of multipath delay spread as well as fading.
Delay 
 

Input
(transmitted
signal)
Output
(received
signal)
 11 exp  j
42
 The impulse response of the channel model is
y 
 22 exp  j
           tjtjth expexp, 2211

22.
22
 Two-ray Rayleigh fading model:
 α1 and α2 are independent and Rayleigh distributed.
5 – Fading Channel
Simulations
 Input
(transmitted
Output
(received
 11 exp  j
p y g
 θ1 and θ2 are independent and uniformly distributed over [0,2π].
 τ is the time delay between the two rays.
 Setting α = 0, a flat
Rayleigh fading
channel is obtained.
 By varying τ, it is
43
Delay  
signal) signal)
 22 exp  j
y y g
possible to create a
wide range of
frequency selective
fading effects.
 N-ray Rayleigh fading model:
 Example 4.7
5 – Fading Channel
Simulations
p
 Develop an (N+1) -ray mobile channel simulator as shown below:
 Input
(transmitted
signal)
Output
(received
signal)
 11 exp  j
1
44
Delay 1 
 22 exp  j
Delay N 
 NN j exp
1
N

23.
23
 Clarke’s model for flat fading (freq domain):
 While stat. models (Rayleigh, Ricean, etc.) directly predict
received signal envelope in time domain Clarke’s model predicts
5 – Fading Channel
Simulations
received signal envelope in time domain, Clarke’s model predicts
the power spectrum of the received signal first in frequency
domain. This is then used to produce a time-domain waveform.
 The model is widely used to model narrow-band channel in the
IS-54 (US digital cellular system).
 Given that the source, S,
transmits a continuous wave
signal of frequency fc.
Source, S (Base Station)
45
 Then the instantaneous
frequency of the received
signal component arriving at
an angle θ at the mobile is:

v(mobile)
cDc fff
v
f  

coscos max
 Clarke’s model for flat fading:
 Thus,



 ff
5 – Fading Channel
Simulations
 Clarke’s model uses statistical characteristic
of electromagnetic waves to show that the
received power spectral density (PSD), S(f),
is proportional to [Ref3].
 Therefore,







 
 
max
1
cos
D
c
f
ff

f

1 1
cos 













 




  c
f
ff
ff

 fS 
Doppler power Spectrum
fcfc -fDmax fc +fDmax
46
 i.e.,
 Ref1, pages 214-218, show that K = 1.5/fDmax .
2
max
max
1







 




 



D
c
D
f
ff
fff
 fS 
2
max
1
)(







 


D
c
f
ff
K
fS

24.
24
 Simulation of Clarke’s fading model using Doppler filter:
 Baseband received signal: y(t)= yI(t)+ jyQ(t), and r (t)=|y(t)|.
5 – Fading Channel
Simulations
Q
 Generate two independent, random complex Gaussian
source.
 Then use spectral filter defined by S(f) in the previous slide,
to shape the random signals in frequency domain.
 Time domain waveform of the resulting fading generated can
be produced by using an inverse fast Fourier transform
(IFFT) at the last stage of the simulator.
Baseband Baseband
Mixer
47

Gaussian
Noise Source
Doppler
Filter


Baseband
Gaussian
Noise Source
Baseband
Doppler
Filter
Independent
Cos(2fct)
Sin(2fct)
Mixer
To IFFT
Note: for
baseband
waveform
simulations,
mixers will
be omitted.
 Simulation of Clarke’s fading model using Doppler filter:
 Ref 4 provide steps for computer program to implements this:
5 – Fading Channel
Simulations
 1. Specify # of frequency points N to represent , and
the . Usually N is to power of 2, i.e. N=2k , where k=1,2,3,…
 2. Calculate frequency spacing between adjacent spectral lines as
f=2fDmax /(N-1). Time duration of the fading waveform is T=1/f.
 3. Generate complex Gaussian random variables for each of the
N/2 positive frequency component of the noise source.
4 C t t th ti f t f th i
)( fS
maxDf
48
 4. Construct the negative frequency components of the noise
source by conjugating positive frequency values & assigning
these at the negative frequency values.
 5. Multiply the in-phase and quadrature noise sources by the
fading spectrum . (Freq domain operation ends here))( fS

25.
25
5 – Fading Channel
Simulations
 Simulation of Clarke’s fading model:
 6. Perform an IFFT on the resulting frequency6. Perform an IFFT on the resulting frequency
domain signals from the I-Q parts to get two N-
length time series, & add the squares of each
signal point in time to create an N-point time
series.
 7. Take the square root of the sum obtained in step
6 to obtain an N-point time series of a simulated
Rayleigh fading signal, to model the expression:
r (t)=|y(t)|=sqrt(yI(t)2+ yQ(t)2), with proper Doppler
d d ti l ti
49
spread and time correlation.
 By making a freq component dominant in
amplitude within , and at f=0, the fading is
changed from Rayleigh to Ricean.
)( fS
 Simulation of Clarke’s fading model:
 Block diagram:
5 – Fading Channel
Simulations
50

26.
26
 Simulation of Clarke’s fading model:
 Example: Fig 5.15 in [Ref 1, 6] – typical Rayleigh fading envelope
at 900MHz and 120km /hr
5 – Fading Channel
Simulations
at 900MHz, and v =120km /hr.
51
 Other models: Jake’s model [Ref 5], Young and Beaulieu’s
method for computational efficient Rayleigh fading sim.[Ref7].
 Jakes’ fading model [5]:
 Clarke’s model (time domain): defines the complex channel
gain for non LOS frequency flat and 2 D isotropic scattering
5 – Fading Channel
Simulations
gain, for non-LOS, frequency flat, and 2-D isotropic scattering
assumptions as [3]
 N= number of multipaths, nUniform(-,), and nUniform(-,)
are the phase and amplitudes of the nth multipath component.
 Jakes’ model: the high degree of randomness in equation above



N
n
tfj nnD
e
N
th
1
])cos(2[ max
2
)(

52
is not desirable for efficient simulation. Thus, Jakes proposed the
following sum of sinusoid model to simulate h(t), & is widely used.









M
n
DDI t
M
n
f
M
n
tfth
1
max
]
24
2
cos[2cos)
2
cos(2)2cos(2)( max













M
n
DQ t
M
n
f
M
n
th
1
]
24
2
cos[2cos)
2
sin(2)( max




27.
27
 Jake’s fading model [5]:
5 – Fading Channel
Simulations
0 4
0.6
se
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
-0.2
0
0.2
0.4
Time
ImpulseRespons
1
1.5
x 10
-3
53
 Other recent models: Zheng et al. [8], Young and Beaulieu’s
method for computational efficient Rayleigh fading sim.[Ref7].
-800 -600 -400 -200 0 200 400 600 800
0
0.5
Frequency
PSD
References
1. T.S. Rappaport, “Wireless Communications: principles and practice,” Second
Ed., Prentice HallPTR, Upper Saddle River, NJ 07458, USA.
2 A Goldsmith “Wireless Communications ” Cambridge University Press New2. A. Goldsmith, Wireless Communications, Cambridge University Press, New
York, NY 10013-2473, USA.
3. R. H. Clarke, “A Statistical Theory of Mobile-Radio Reception,” Bell Systems
Technical Journal, vol. 47, pp. 957-1000, 1968.
4. J. I. Smith, “A Computer Generated Multipath Fading Simulation for Mobile
Radio,” IEEE Trans. Vehicular Technology, vol. VT-24, no.3, pp.39-40, Aug 1975.
5. W. C. Jakes, “Microwave Mobile Communications,” Wiley-IEEE Press, May
1994.
6. V. Fun, T.S. Rappaport, and B. Thoma, “Bit Error Simulation of /4-DQPSK
54
Mobile Radio Communication using Two-ray and Measurement-based Impulse
Response Models,” IEEE Journal on Selected Areas in Commun., Apr. 1993.
7. D. J. Young and N. Beaulieu, “The Generation of Correlated Rayleigh Random
Variates by Inverse Discrete Fourier Transform,” IEEE Transactions on
Communications, vol 48, no.7, July 2000.
8. Zheng and Xiao, “Simulation Models With Correct Statistical Properties for
Rayleigh Fading Channels,” IEEE Trans. Commun. June 2003.