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Chapter 2: Statistical Analysis of Fading Channels. Channel output viewed as a shot-noise process Point processes in general; distributions, moments Double-stochastic Poisson process with fixed realization of its rate Characteristic and moment generating functions Example of moments - PowerPoint PPT Presentation
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Chapter 2: Statistical Analysis of Fading Channels
Channel output viewed as a shot-noise process
Point processes in general; distributions, moments
Double-stochastic Poisson process with fixed realization of its rate
Characteristic and moment generating functionsExample of moments
Central-limit theorem
Edgeworth series of received signal densityDetails in presentation of friday the 13th
Channel autocorrelation functions and power spectra
Channel Simulations Experimental Data (Pahlavan p. 52)
Chapter 2: Shot-Noise Channel Simulations
( )
1
( ) ( ; ) cos ( ; ) ( )
Need: ( ),
sN T
i i c i i l ii
y
y t r t t t s t
f y t t
Chapter 2: Shot-Noise Channel Model
( )( ; ( ))
1
( )
1
Low pass representation of received signal
( ) ( ; ( )) ( ( ))
Band pass representation of received signal
( ) ( ; ( )) cos ( ; ( )) ( ( ))
( ; ) Pha
s
i i
s
N Tj t t
l i i l ii
N T
i i c i i l ii
i
y t r t t e s t t
y t r t t t t t s t t
t
se shift
( ; ): signal attenuation coefficient, i.e. Rayleigh, Ricean
( ), ( ) : time delays and number of paths
( ; ), ( ; ), ( ) arbitrary random processes.
i
i
i i i
r t
t N t
r t t t
Channel viewed as a shot-noise effect [Rice 1944]
Chapter 2: Shot-Noise Effect
ti ti
Counting process ResponseLinear
system
Shot-Noise Process: Superposition of i.i.d. impulse responses occuring at times obeying a counting process, N(t).
Measured power delay profile
Chapter 2: Shot-Noise Effect
Shot noise processess and Campbell’s theorem
Chapter 2: Shot-Noise Definition
( )
1
A stochastic process ( ), , , is said to be a
- if it can be represented as the
superposition of impulses occuring at random times
( ) ( , ;m ( , ))
where occur ac
i
N t
m m m mm
i
X t t
shot noise process
X t h t t
cording to a counting process, ( )
i.e. a non-homogeneous Poisson process, with intensity ( ),
and ( , ;m ( , )) assumed to be independent and
identically distributed random processes, independentm m m m
N t
t
h t t
0
of
( ) .t
N t
Shot-Noise Representation of Wireless Fading Channel
Chapter 2: Wireless Fading Channels as a Shot-Noise
( )
1
( ; ( ))
( )
1
( ; ( ))
( ) ( , ;m ( , ));
( , ;m ( , )) ( ; ) ( )
( ) ( , ;m ( , ))
( , ;m ( , )) ( ; ) Re ( )
( ): Counting process
m ( , ) = ( ;
s
i i
s
i i c
N T
l l i i ii
j t tl i i i i i l i
N T
i i ii
j t t j ti i i i i l i
i i i
y t h t t
h t t r t e s t
y t h t t
h t t r t e s t e
N t
t r t
), ( ; ) : arbitrary random processes
associated with
i i i
i
t
Counting process N(t): Doubly-Stochastic Poisson Process with random rate
Chapter 2: Shot-Noise Assumption
0
0
0
22
0 0
Conditional on ( );0 ,
( ) has a Poisson law
( )( ) exp ( )
!
( ) ( ) ,
( ) ( ) ( ) ,
s
s
S
s
S
s
S s
s
T s
s
kT
T
s T
T
s T
T T
s T
s s T
N T
t dtProb N T k t dt
k
N T k t dt
N T k t dt t dt
E
E
Conditional Joint Characteristic Functional of y(t)
Chapter 2: Joint Characteristic Function
Conditional moment generating function of y(t)
Conditional mean and variance of y(t)
Chapter 2: Joint Moment Generating Function
1
1 1
y 1 1 01 1
1 m0
2
2 m0
( ) ( )
( ) , ; ; ,
( ) ( ) ( ) ( , ; ( , )) ,
( ) ( ) ( ) ( , ; ( , ))
i i
s
i ini ii
i
s
s
s
n nk m
i i Ti i
k mn n
k m
n ni ii i
T
T
T
T
E y t y t
j j t j t
E y t t E h t m t d
Var y t t E h t m t d
s
Conditional Joint Characteristic Functional of yl(t)
Chapter 2: Joint Characteristic Function
†
†y 1 1
Re h t, ;m t,
m0
,*
1
*, m0
1 1
, ; ; , exp Re y (t)
exp ( ) 1
( ), ln exp Re (t)
!
( ) ( ) Re , ;m ,
y (t) ( ), , ( ) , ,
l s
s l
l s
s
n n l T
T j
l kky l T
k
kT
l k l
nl l l n
t t E j
E e d
tt E j y j
k
t E h t t d
y t y t
1 1
, ,
h t, ;m t, , ; , , , , ; ,
nn
l l l n nh t m t h t m t
Chapter 2: Joint Moment Generating Function
1
1 1
y 1 1 01 1
,1
,22 2
( ) ( )
( 2 ) , ; ; ,
( ) ( 2 ) ( ),
( )( ) ( 2 )
2!
1
2
i i
l s
iini ii
li
i
s
s
i
n nk m
i l i Ti i
mkn n
k m
n ni ii
l T l
ll T
i R
E y t y t
j t t
E y t j j t
tVar y t j j
j
1
; 2
i i i iI R I
j
Conditional moment generating function of yl(t)
Conditional mean and variance of yl(t)
Conditional correlation and covariance of yl(t)
Chapter 2: Correlation and Covariance
1 2
*1 2 1 2
21 1 2 2 0
1 2
*1 2 1 2 1 2
*1 1 2 2m0
, ( ) ( )
( 2 ) , ; ,
, , ( ) ( )
( ) ( , ; ( , )) ( , ; ( , ))
l l l s
l
l l l s l s
s
l
y T
y
y y T T
T
l
R t t E y t y t
j t t
Cov t t R t t E y t E y t
E h t m t h t m t d
Central Limit Theorem
yc(t) is a multi-dimensional zero-mean Gaussian process with covariance function identified
Chapter 2: Central-Limit Theorem
y 1 1
2
m01
Let ( , ) ( , ), where is deterministic
( ) ( )and define ( ) then
( )
lim , ; ; ,
exp ( )2
( , ;m( , ))( )
s
ld
s
d c d
i i T
c iy i
n n
nTd i
li y
ic ii
h
t t
y t E y ty t
t
t t
E dt
t t
Channel density through Edgeworth’s series expansion
First term: Multidimensional GaussianRemaining terms: deviation from Gaussian density
Chapter 2: Edgeworth Series Expansion
Channel density through Edgeworth’s series expansion
Constant-rate, quasi-static channel, narrow-band transmitted signal
Chapter 2: Edgeworth Series Simulation
Channel density through Edgeworth’s series expansion
Parameters influencing the density and variance of received signal depend on
Propagation environment Transmitted signal
(t) (t) Ts Ts (signal. interv.)
var. I(t),Q(trs
Chapter 2: Edgeworth Series vs Gaussianity
Chapter 2: Channel Autocorrelation Functions
c( t;)
Sc( ;)
Sc(; f)
ScatteringFunction
F
FtF
Ft
WSSUS Channel
Power DelayProfile
Power DelaySpectrum
c()
Tm
fBc
|c(f)|
F
t=0
tTc
|c(t)|
f=0
t=0
Bd
Sc( )
f=0
Ft
Doppler Power Spectrum
dS );(
dS );(
t
|c(t;f)|
f
Sc()
Consider a Wide-Sense Stationary Uncorrelated Scattering (WSSUS) channel with moving scatters
Non-Homogeneous Poisson rate: ()
ri(t,) = ri(): quasi-static channel
p()=1/2 , p()=1/2
Chapter 2: Channel Autocorrelations and Power-Spectra
, ( ) 2 cos ( )d i m it f t
Time-spreading: Multipath characteristics of channel
Chapter 2: Channel Autocorrelations and Power-Spectra
1 1 2 2, ,
c 1 2 1 1 2m
2c 0m
, ; = ( )E , ,
1. Autocorrelation in the Time-Domai
; = ( )E (2 )
nj t t t t
m
t t r t r t e
t r J f t
Time-spreading: Multipath characteristics of channel
Chapter 2: Channel Autocorrelations Power-Spectra
2c
2 2c
( ), 2
1
; ( ) ( )
3. Power Delay Spectrum
; ( ) ( )
4. Time Variations of Frequency Respons
2. Power-delay profile
e
( ; ) ( , )
( ; ) ( )
si i i
j f
N Tj t t j f
l i ii
l
t E r
t f E r e d
C t f r t e e
E C t f E
, 2( , ) j t t j fr t e e d
Time-spreading: Multipath characteristics of channelAutocorrelation in Frequency Domain, (space-frequency, space-time)
Chapter 2: Channel Autocorrelations and Power-Spectra
Time variations of channel: Frequency-spreading:
Chapter 2: Channel Autocorrelations and Power-Spectra
c
2 2
c c
2
1a. Double Fourier transform of ( ; )
; ; ;
1
1
2
l
t t
f
m m
j
C t
S f F t F t f
E r ef
F
fd
Double Fourrier transform
Time variations of channel: Frequency-spreading
Chapter 2: Channel Autocorrelations and Power-Spectra
2c c c0
2
2
c
; ;0
1
2 1
a delta func
1b. Doppler Power Spectum of channel
No time variations: o ti n
j
f
m m
tS S f t
E rf
e d
f
S
t
d
Time variations of channel: Frequency-spreading
Chapter 2: Channel Autocorrelations and Power-Spectra
2 2
1c c
2
2
2. Scattering function
; ;
1
2
1
f
j j
m m
f f
S F S f
f f
E r e d e d f
Temporal simulations of received signal
Chapter 2: Shot-Noise Simulations
K.S. Miller. Multidimentional Gaussian Distributions. John Wiley&Sons, 1964.S. Karlin. A first course in Stochastic Processes. Academic Press, New York 1969.A. Papoulis. Probability, Random Variables and Stochastic Processes. McGraw Hill, 1984.D.L. Snyder, M.I. Miller. Random Point Processes in Time and Space. Springer Verlag, 1991.E. Parzen. Stochastic Processes. SIAM, Classics in Applied Mathematics, 1999.P.L. Rice. Mathematical Analysis of random noise. Bell Systems Technical Journal, 24:46-156, 1944.W.F. McGee. Complex Gaussian noise moments. IEEE Transactions on Information Theory, 17:151-157, 1971.
Chapter 2: References
R. Ganesh, K. Pahlavan. On arrival of paths in fading multipath indoor radio channels. Electronics Letters, 25(12):763-765, 1989.C.D. Charalambous, N. Menemenlis, O.H. Karbanov, D. Makrakis. Statistical analysis of multipath fading channels using shot-noise analysis: An introduction. ICC-2001 International Conference on Communications, 7:2246-2250, June 2001.C.D. Charalambous, N. Menemenlis. Statistical analysis of the received signal over fading channels via generalization of shot-noise. ICC-2001 International Conference on Communications, 4:1101-1015, June 2001.N. Menemenlis, C.D. Charalambous. An Edgeworth series expansion for multipath fading channel densities. Proceedings of 41st IEEE Conference on Decision and Control, to appear, Las Vegas, NV, December 2002.
Chapter 2: References