
2/40

Fading on link between pair of Tx & Rx Antennas

Distance between Tx & Rx Antennas

C h a n n e

l G

a i n ( d B )

3


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Channel Model for Point-to-Point Communications

s(t) h(t) y(t)

w(t)

r (t)f c f c f

|S (f )|

Real bandpass channel model

s(t) h(t) y(t)

w(t)

r (t) 0 f |S (f )|

Complex baseband channel model 4


4/40

Passband-Baseband relationships (Proakis[1])

Passband to Baseband Baseband to Passband

S (f ) = S + (f + f c ) = 2u(f + f c )S (f + f c ) S (f ) = S ( f f c )+ S (f f c ) 2s(t) = s+ (t)ej 2f c t = 1 2 [s(t) + j s(t)]ej 2f

c t s(t) = Re[ 2s(t)ej 2f c t ]

h(t) = 1 2 h+ (t)ej 2f c t h(t) = 2Re[ h(t)ej 2f c t ]

w(t) = w+ (t)ej 2f c t = 1 2 [ w(t) + j w(t)]ej 2f c t w(t) = Re[ 2w(t)ej 2f c t ]

5


5/40

Properties of Complex Baseband Additive Noise w (t )

If w(t) is zero mean, Gaussian, then w(t) is zero mean, complex Gaussian

Let w(t) = wI (t) + jwQ (t). If w(t) is wide sense stationary (WSS), thenwI (t) and wQ (t) are jointly WSS, and

Rw I ( ) = Rw Q ( ) , and Rw I w Q ( ) = Rw Q w I ( )A complex process with this property is called proper complex Dene ACF of w(t) by

Rw ( ) = E[ w(t + )w(t)]

Then from proper complex property

Rw ( ) = 2 Rw I ( ) + j 2Rw Q w I ( )

6


6/40

Proper Complex Processes (Neeser & Massey [2])

Let Y = Y I + j Y Q be a complex random vector with

Y I = E[( Y I m Y I )(Y I m Y I )] , Y Q = E[( Y Q m Y Q )(Y Q m Y Q )]

Y I Y Q = E[( Y I m Y I )(Y Q m Y Q )] , Y Q Y I = E[( Y Q m Y Q )(Y I m Y I )]

Complex covariance Y = E (Y m Y )(Y m Y ) = Y I + Y Q + j Y Q Y I Y I Y Q

Complex pseudo-covariance

Y = E (Y m Y )(Y m Y ) = Y I Y Q + j Y Q Y I + Y I Y Q

7


7/40

Denition 1. Y is a proper complex vector if Y = 0 , i.e.

Y I = Y Q and Y Q Y I = Y I Y Q

For proper complex Y ,

Y = 2 Y I + j 2 Y Q Y I

The scalar case: If Y is proper complex scalar, then Y I and Y Q areuncorrelated and

2Y = E[

|Y

mY

|2] = 22Y I = 2

2Y Q

If Y is proper and Gaussian it is said to be proper complex Gaussian (PCG) or

circularly complex Gaussian8


8/40

Useful results on PCG random vectors

Result 1. If Y is a PCG vector, the pdf of Y is given by

pY (y ) := pY I Y Q (y I , y Q )

= 1n | Y |

exp (y m Y ) 1Y (y m Y )

The pdf of Y has circular symmetry.

Notation: Y CN (m Y , )Result 2. If Y is PCG, then Z = AY + b is also PCG.The circular property is preserved under linear transformations.

Result 3. (Central Limit Theorem). If {Y k } is a sequence of independentproper complex random vectors (not necessarily Gaussian), then the sum

k Y k (after appropriate normalization) converges to a PCG vector.

9


9/40

Proper Complex and PCG processes

Covariance and pseudo-covariance functions of Y (t) = Y I (t) + jY Q (t):C Y (t + , t ) = E [( Y (t + ) mY (t + ))( Y (t) mY (t))]C Y (t + , t ) = E [( Y (t + ) mY (t + ))( Y (t) mY (t))]

Denition 2. {Y (t)} is proper complex if C Y Y (t + , t ) = 0 , i.e.C Y I (t + , t ) = C Y Q (t + , t ) and C Y Q Y I (t + , t ) = C Y I Y Q (t + , t )

For proper complex {Y (t)},C Y (t + , t ) = 2 C Y I (t + , t ) + j 2C Y Q Y I (t + , t)

Denition 3. A proper complex process {Y (t)} is PCG if, for all n, and allt1 , t 2 , . . . , t n , the samples Y (t1), Y (t2), . . . , Y (tn ) are jointly PCG .

Result 4. If a PCG process

{Y (t)

} is passed through a linear system, the

output is PCG as well.10


10/40

Part II: From Point-to-Point CommunicationsModel to Mobile Communications Channel Model

11


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Complex Baseband Model for Point-to-Point

Communications

s(t) h(t) y(t)

w(t)

r (t)

If bandpass noise {w(t)} is AWGN with PSD N 0 / 2, then baseband noise{w(t)} is a PCG white process with

Rw ( ) = E[ w(t + )w(t)] = N 0 ( )

Rw I w Q ( ) = Rw Q w I ( ) = 0 = {wI (t)}; {wQ (t)} independent Rw I ( ) = Rw Q ( ) = 12 Rw ( ) = N 02 ( )

S w I (f ) = S w Q (f ) = 12 S w (f ) =

N 02 for all f

12


12/40

Mobile Comm. Channel Model for link between Tx/Rx Pair

v

d

X

Y Mobile

BS Antenna

Causal LTI system corresponding to multipath prole at (d,)

s(t) hd, ( ) y(t)

w(t)

r (t)

13


13/40

Multipath Prole and Channel Impulse Response

At location (d,), n-th path connecting Tx and Rx antennas has

amplitude gain of n (d,) delay of n (d,) carrier phase shift of n (d,) = 2f c n (d,) + constant.

Thus y(t) =n

n (d,) ej n (d, ) s(t n (d,))

= hd, ( ) = n n (d,) e

j n (d, )

( n (d,))

As MS moves, (d,) varies with time = Time varying channel

14


14/40

Two scales of variation

Small scale variations movements of the order of few carrier wavelengths multipath prole roughly constant # paths, strengths, delays channel variations due to phase differences in paths average power gain in vicinity of (d,) is G(d,) =

n

2n

typical values: f c = 1 G Hz = c = 0 .3 m.

Large scale variations variations in G(d,) that result from changing multipath prole scale of distance between objects in environment typical values: 10s of meters (outdoor)

15


15/40

Separation of scales

Small and large scale variations separated s(t) y(t)

g(d,)

hd, ( )x(t)

hd, ( ) is hd, ( ) normalized to have average power gain of 1, i.e.hd, ( ) =

n n (d,) ej n (d, ) ( n (d,))

with

n

2n (d,) = 1

amplitude gain (real) g(d,)G(d,) = g2(d,)

16


16/40

Part III: Small Scale Variations in Gain Basics

17


17/40

Small Scale Variations

Small scale variations are captured in hd,hd, ( ) =

n

n (d,) ej n (d, ) ( n (d,))

where { n (d,)} are normalized so that n 2n (d,) = 1 As (d,) changes with t, channel becomes time varying:h(t; ) := hd ( t ) , ( t ) ( ) =

n

n (t) ej n ( t ) (

n (t))

Assume g(d,) is constant over small scales(t) y(t)

g

h(t; )x(t)

y(t) =

0

h(t; )x(t )d

18


18/40

Doppler shifts in phase

For movements of the order a few wavelengths { n (t)} and { n (t)} are roughly constant

BUT n (t) = 2f c n (t) + const. changes signicantly Doppler shift is function of angle n of path w.r.t. velocity vector

vv tn

v t cos n

n (t + t ) n (t) 2f c v t cosn

c

= 2v t cosn

c

= 2 f max t cosn

f max = v/ c is the maximum Doppler frequency

19


19/40

Delay Prole of Channel

n and n are roughly independent of t, i.e. time variations are mainly dueto changes in n= h(t; ) =

n

n ej n ( t ) ( n )

0 DS

|h(t; )|

0 (LOS)

n

n20


20/40

Part IV: Flat Fading

21


21/40

Frequency-Nonselective (Flat) Fading

Denition 4. The quantity DS = max n min n is called the delay spread of the channel.

w.l.o.g. assume min n = 0 . Then max n = DS and

y(t) = DS

0h(t; )x(t )d

If passband bandwidth of s(t), W 1 DS , then x(t) is roughly constantover time intervals of order of DS

= y(t) x(t) DS

0h(t; )d = x(t)

n

n ej n ( t ) = x(t)E (t)

= h(t; ) E (t) ( )22


22/40

Channel Model for Flat Fading

s(t) y(t)x(t)

g E (t)

E (t) = DS

0h(t; )d =

n

n ej n ( t )

23


23/40

Purely Diffuse (no LOS) Scattering Rayleigh fading

v

X

Y MS

BS

No LOS path or no single path dominates all other paths If we model {n } as independent random Unif[0, 2], then {E (t)} is zero

mean process

The process { n ej n ( t )} is proper complex If number of paths is large, by CLT (Result 3),

{E (t)

} is a Proper Complex Gaussian (PCG) random process

24


24/40

First Order Statistics for Purely Diffuse Scattering

For xed t, E (t) = E I (t) + jE Q (t) is PCG r.v. withE |E (t)|2 =

n

2n = 1

= E I (t) and E Q (t) are independent N (0, 1/ 2) r.v.s

Envelope and phase(t) =

|E (t)

| =

E 2I (t) + E 2Q (t) , and (t) = tan 1

E Q (t)

E I (t)

For xed t, (t) and (t) are independent, (t) has a Rayleigh pdf and(t) is Unif[0, 2].

0 1 2 30

0.2

0.4

0.6

0.8

1

p (x) = 2 xex2

u(x)

25


25/40

Autocorrelation function of {E (t )}RE (t, t + ) = E [ E

(t)E (t + )]

= En

n ej n ( t )i

i ej i ( t + )

=n

2n E ej [ n ( t + ) n ( t )]

n 2n e

j 2f max cos n

= RE ( )

{E (t)} is approximately stationary

In-phase and Quadrature components have correlation functions

RE I ( ) = RE Q ( ) = 12

Re {RE ( )} = 12

n

2n cos(2f max cosn )

RE Q E I ( ) =

RE I E Q ( ) =

1

2Im

{RE ( )

} =

1

2 n 2n sin(2f max cosn )

26


26/40

Isotropic Scattering Environment

Isotropic scattering may be approximated by continuum of paths with p() =

12

(uniform)

to getRE ( ) =

12

ej 2f max cos d = J 0(2f max )

where J 0(

) is the zeroth order Bessel function of the rst kind [4]

J 0(x) = 12

cos(x cos )d

AlsoRE I ( ) = RE Q ( ) =

12

RE ( )

RE Q E I ( ) = RE I E Q ( ) = 0

27


27/40

Accuracy of Bessel Approximation

0 0.5 1 1.5 2 2.5 30.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5BesselN=8N=16

f max or c

R E

I (

) o r

R E I

( )

Even for a few uniformly distributed discrete paths (N = 8 , 16)we get an ACF that is well approximated by a Bessel function

28


28/40

Fading Process in Distance Variable

Fading process is fundamentally a function of location To get fading as function of time we assumed MS is traveling along

constant velocity vector v.

Distance relative to location at time 0 equals vt Fading process in is given by

E ( ) = E (/v )

Fading ACF over distance variable is given byRE () = E E ( + ) E ( )

= RE

v

= RE

f max c

=

p() e

j 2 cos c d

= J 0

2 c

(isotropic Rayleigh fading)

29


29/40

Fading ACF and Coherence Distance/Time

Coherence distance c is measure of distance separation over which E remains roughly unchanged

c can be dened more precisely in terms of the ACF as (say): c = largest such that |RE () | > 0.9RE (0) = 0 .9

Coherence time T c = cv For isotropic Rayleigh fading

c 0.1 c , and T c 0.1f max .

Fading is said to be slow if T c T s , where T s is the symbol period

30


30/40

Scattering with LOS Component

If LOS (specular) path with parameters 0 , 0 and 0(t), thenE (t) = 0 ej 0 ( t ) + 1 20 E (t)

where {E (t)} is zero mean PCG, Rayleigh fading process Note: {E (t)} is zero-mean process, but not Gaussian since LOS

component dominates diffuse components in power

Rice Factor: =

power in the specular componenttotal power in diffuse components

= 201 20

From the denition of it follows that

0 =

+ 1, and 1 20 =

1( + 1)

31


31/40

Ricean Fading

For xed t, the envelope has Ricean pdf (Rice [3]) p (x) = 2 x( + 1) I 0 2x ( + 1) exp x2( + 1) u(x)

where I 0() is zeroth order modied Bessel function of 1st kind [4]

I 0(y) = 12

exp( y cos )d .

0 1 2 30

0.5

1

1.5

2

RayleighRicean =1Ricean =5Ricean =10

32


32/40

Fading Example

v = 72 km/hr = 20 m/s; f c = 900 MHz c = 1/3 m f max = 60 Hz

0 50 100 150 200 25020

15

10

5

0

5

10

15

t (ms)

C h a n n e

l G a

i n ( d B )

RayleighRicean =5Ricean =10

33


33/40

Part V: Frequency-Selective Fading

34


34/40

Frequency-Selective Fading

s(t) y(t)

g

h(t; )x(t)

s(t) has passband bandwidth of W

If W 1 DS , then fading is at

If W > 1 DS , then fading is frequency selective

From Paulraj et al [5]

Environment SpreadSuburban 20 s

Urban 5 s

Mall 0.3 sIndoors 0.1 s

s(t) has baseband bandwidth of W/ 2 = x(t) has bandwidth W/ 2 By Sampling Theorem (sinc interpolation formula)

x(t ) =

=

x (t /W ) sinc[W ( /W )]35


35/40

y(t) =

DS

0h(t; ) x(t

)d

=

= x (t /W )

DS

0h(t; ) sinc [W ( /W )] d

E (t) = DS

0h(t; ) sinc [W ( /W )] d

E (t) 0 for < 0 and for /W > DS . If L =

DS W then

y(t) L 1

=0

x (t /W ) E (t)

= h(t; ) L 1

=0

E (t) ( /W )

36


36/40

Tapped Delay Line Model

... ...x(t) 1W 2W W L 1W

E 0(t) E

1(t) E

2(t) E (t) E L

1(t)

y(t)

Recall that h(t; ) =n

n ej n ( t ) ( n )

= E (t) = DS

0h(t; ) sinc [W ( /W )] d

=n

n ej n ( t ) sinc[W ( n /W )]

37


37/40

Sinc Mask

0 DS

|h(t; )|

0 (LOS)

n

n

38


38/40

Statistical Model for {E (t )} If {E (t)} includes a dominant LOS component, then it has Riceanenvelope; else it has Rayleigh envelope Autocorrelation function of {E (t)}

RE ( ) = E [( t)E (t + )E ]

n

2n ej 2f max cos n sinc2 [W ( n /W )]

Check: If the fading is at, i.e. DS 1W , E (t) 0 for = 0 , andRE 0 ( )

n

2n ej 2f max cos n sinc2 [W n ]

n

2n ej 2f max cos n RE ( )

Nature of RE ( ) depends on angular location and spread of pathscontributing to tap

39


39/40

Cross-correlation Between Taps

RE k E ( ) = E [ E k (t + )E

(t)]

n 2n e

j 2f max cos n sinc[W ( n

/W )] sinc[W ( n

k/W )]

Frequency diversity depends on cross-correlation between taps

Fading in neighboring taps can be highly correlated

If tap delays are chosen to match cluster centers in delay prole, then tapswill be less correlated

Cluster model for channel

h(t; ) L c 1

=0

E (t) ( )

where Lc is number of clusters; delay of cluster 40


40/40

References

[1] J. G. Proakis. Digital Communications . Mc-Graw Hill, New York, 3rd edition, 1995.

[2] F. D. Neeser and J. L. Massey. Proper complex random processes with applications to information theory. IEEE Trans. Inform. Th. , 39(4),July 1993.

[3] S. Rice. Statistical properties of a sine wave plus noise. Bell Syst. Tech. J. , 27(1):109157, January 1948.[4] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions . Dover, New York, 1964.

[5] A. J. Paulraj and C. B. Papadias. Space-time processing for wireless communications. IEEE Signal Processing Magazine , pages 4983,November 1997.

[6] P. A. Bello. Characterization of randomly time-variant linear channels. IEEE Trans. Commun. Systems , pages 360393, December 1963.

[7] R. Clarke. A statistical theory of mobile radio reception. Bell Syst. Tech. J. , 47(6):9571000, July-August 1968.[8] M. Gudmundson. Correlation model for shadow fading in mobile radio systems. Electron. Lett. , 27(23):21452146, 1991.

[9] W. C. Jakes, Jr. Microwave Mobile Communications . Wiley, New York, 1974.

[10] E. Wong and B. Hajek. Stochastic Processes in Engineering Systems . Springer-Verlag, New York, 1985.

[11] D. Tse and P. Viswanath. Fundamentals of Wireless Communication . Cambridge University Press, 2005.

41

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