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