The Gaussian Fading Model

The statistical description of multipath propagation was given by Clarke [Cla68] and further refinements were done by Jakes [Jak74]. We summarize these results for both static and dynamic multipath channels.

A static channel is one where the mobile station is stationary or virtually stationary and we are interested in a statistical description of the received power which does not change in time. A dynamic channel is one where the mobile station is moving with a specified velocity.

In the following chapters we will mainly be concerned with block static multipath channels. This is an approximation of a dynamic channel which changes very slowly so that it may be considered as static over blocks or intervals of time.

1 1.0001 1.0002 1.0003 1.0004 1.0005 1.0006 1.0007 1.0008 1.0009 1.001 x 10⁹

−96

−94

−92

−90

−88

−86

−84

−82

−80

P^R=P^T

f

Figure 2.6: Frequency response at

x = 870

^m

Static multipath channels

Assume we transmit a quadrature signal with complex envelope

s ~ ( t ) = s c ( t )+ js s ( t )

by modulating the in–phase and quadrature carriers at frequency

f c

. The signal fed to the transmitting antenna is given by

u ( t ) = s c ( t )cos2 f c t

^,

s s ( t )sin2 f c t:

^(2.13)

Considering only the electric field components of the electromagnetic wave at the receiving antenna, we write the received signal vector for a stationary receiver as

r

( t ) =

0

@

r ^v ( t ) r ^h ( t )

1

A

= k Re

(^X

N i=0

G

(

i )

^A

i e ^j2f

^c

^t ~ s ( t

^,

d i )

)

(2.14) where

k

is a proportionality factor depending on the antenna characteristic. The angle

i

represents the two angles of arrival (i.e. azimuthal and elevation) of the

i ^th

wave component at the receiver, ^A

i

^{is a} column vector holding its horizontal and vertical complex amplitudes and

d i

its propagation delay. The

2

2

diagonal matrix^G

(

i )

holds the horizontal and vertical field gains for the given angles of arrival.

If the antenna is polarized in either of the vertical or horizontal direction, one of the diagonal elements is zero. The path index 0 corresponds to the LOS path, so that in the case where it is not present^A

0 =

^0.

The complex envelope of^r

( t )

is given by

~

r

( t ) = k

^X

^N

i=1

G

(

i )

^A

i s ~ ( t

^,

d i )

^(2.15)

and its corresponding Fourier transform

~R( f ) = k S ^~ ( f )

^X

^N

i=1

G

(

i )

^A

i e ^j2fd

^{i (2.16)}

From this point onward we assume the antenna is omnidirectional and polarized in the vertical direction so that^G

(

) =

A. For convenience we assume that the

d i

are in increasing order. Dropping vector notation we may write

R ~ ( f ) = k S ^~ ( f ) H ( f )

^where

H ( f )

is a random frequency response (or filter) given by

H ( f ) =

^X

^N

i=0 A i e ^j2fd

^{i (2.17)}

with covariance between frequency samples

f 1

^and

f 2

( f 1 ;f 2 ) = ( f 1

^,

f 2 ) = E H ( f 1 ) H

( f 2 ) =

^X

^N

i=1 E

^j

A i

^j

2 e ^j2(f

^1,

^f

²

^)d

^{i (2.18)}

In writing this covariance functions, we have assumed that the path gains are independent zero–mean random variables which is usually known as the uncorrelated scattering assumption. This is justifiable since they are results of reflections off of rough surfaces [Lee82] whose electromagnetic properties have a certain degree of uncertainty. Provided that the number of paths is large, which is almost always the case and that a significant number of these contribute to the total received power, it is permissible to invoke the central limit theorem [Pap82] and approximate

H ( f )

as a Gaussian random variable for each

f

. We must stress, however, that the process

H ( f )

is not a Gaussian process. This follows from the fact that the Fourier transform is a linear operator and the process defined by the path strengths is not necessarily Gaussian. This approximation is very strongly corroborated by measurement [Jak74], and as a result is usually termed Rayleigh fading when no LOS path exists since the envelope of a zero–mean complex Gaussian random variable is Rayleigh distributed as

f

^j

^Rayleigh _H

^j

( a ) = 2 a

envelope has a Ricean distribution given by

f

^j

^Rice _H

^j

( a ) = 2 a

where

I 0 (

)

is the zero–order modified Bessel function of the first kind. The ratio

K =

^j

A 0

^j

2 = _2H

^is

commonly referred to as the Ricean factor. The density and distribution functions of the energy at a given frequency^j

H ( f )

^j

²

are usually more useful for calculation purposes and are given by

f

^j

^Rayleigh _H

^j2

( a ) = 1 _2H ^exp

where

Q 1 (

;

)

is the first order Marcum Q–function. Efficient numerical methods for calculating it are given in [Shn89]. In figure 2.7 we show the density functions of the received energy at a particular frequency as a function of

K

assuming the average received power is unity (i.e.

_2H +

^j

A 0

^j

2 = 1

^{). We}

Figure 2.7: Ricean probability density for increasing

K

and normalized received energy

see that as

K

increases the channel becomes more and more deterministic.

K

is typically zero in urban environments but may rise to 6 dB in an indoor setting [Bul87]. If the transmitted signal

s ( t )

^{lies in a}

band

[ f c

^,

W= 2 ;f c + W= 2]

^and

W

is such that

( u )

is almost constant for^,

W= 2

u

W= 2

^(i.e.

W ( d N

^,

d 0 )

1

^{) then}

r ~ ( t )

H ( f c )~ s ( t

^,

d 0 )

^(2.25)

This is a narrowband or single–path channel model.

There are other statistical models for fading which combine the shadowing and multipath phenom-ena. A popular one is the Nakagami distribution [Nak60]. The latter model is simply the square–root of Gamma distributed random variable.

Dynamic multipath channels

Let us now assume that the mobile station is moving with velocity

v

in the horizontal plane as shown in figure 2.8. This introduces a Doppler shift at each frequency

f

^{given by}

f _i ^D ( f;t ) = vf

c ^cos( i ( t ))

^(2.26)

where

i ( t )

is the time–varying angle between the

i ^th

incoming path (assumed to be traveling horizon-tally) and direction of motion of the mobile station. This 2–dimensional model is due to Clarke[Cla68]

v

^N(t⁾ 2⁽t⁾

1⁽t⁾

Path N Path1 Path2

Figure 2.8: Two–dimensional Doppler model

and Jakes [Jak74]. An extension in 3–dimensions for which the significant conclusions regarding sys-tem design remain unchanged is considered by Aulin [Aul79]. If we consider modulating signals with bandwidths significantly less than the carrier frequency, which is always the case in mobile radio com-munications, then all frequencies are affected by virtually the same Doppler shift at any instant in time.

This allows us to write the complex envelope of the received signal as

r ~ ( t ) = k

^X

^N

i=0 A ( i ( t )) e ^j2f

^Di

^(f

^c

^;t)t ~ s ( t

^,

d i ( t ))

^(2.27)

In most cases where processing of

r ~ ( t )

is done in a short timespan

d i ( t )

d i

so that the time–varying nature of the delays may be ignored. This is further justified by the fact that the amplitudes change much more quickly than do the delays. This assumes, of course, that the mobile speed is not very large, and therefore may not apply to some aeronautical or satellite channels. For simplicity let us focus on the narrowband signal case so that

r ~ ( t )

k

(^X

N

i=0 A ( i ( t )) e ^j2f

^Di

^(f

^c

^;t)t

)

s ( t

^,

d 0 )

^(2.28)

As before, provided

N

is large enough and significant number of

A ( i ( t ))

contribute to the sum, the samples of the process

( t ) =

^P

^N _i=0 A ( i ( t )) e ^jf

^Di

^(f

^c

^;t)t

are approximately Gaussian. If the paths are results of far–off reflections, the Doppler shift for each path changes very slowly (i.e.

i ( t )

^changes

slowly) so that, at least on intervals of a certain length, we may justifiably approximate the process

( t )

as a stationary Gaussian process with correlation function

R ( t; ) = E ( t )

( t + ) =

^X

^N

i=0 E

^j

A ( i )

^j

² E e ^j2f

^D

^cos

ⁱ

;

^(2.29)

where

f D = vf c =c

is the maximum Doppler shift. In the classic model by Clarke and Jakes it is assumed that the angles of arrival are uniformly distributed on

[0 ; 2 )

^{so that}

E e ^j2f

^d

^cos

ⁱ

_{= 12}

Z

2

0 e ^j2f

^d

^cos

ⁱ

d i = J 0 (2 f D ) ;

^(2.30)

where

J 0 (_)

is the zero–order Bessel function. In the frequency domain the power spectrum of the fading process is

S ( f ) =

8

>

<

>

:

²_H

2

q

1

f

^2,

f

_D^{2 j}

f

^j

< f D ;

0 elsewhere :

^(2.31)

This is commonly referred to as the land–mobile power spectrum and is strongly corroborated by mea-surement [Jak74] in urban environments. It is shown in figure 2.9 where its characteristic peaks at the maximum Doppler shift

f D

are evident. In rural or hilly urban areas the distribution of the angles of arrival may not be uniform and as a result the power spectrum will be quite different. The actual form of the Doppler spectrum is not that critical. It is rather its bandwidth which determines the extent of the

,fD ^fD

f S^(f)

Figure 2.9: Land–mobile power spectrum

time–variation of the fading process. We will see in the following chapter that for performance analysis of such channels, the bandwidth of the fading process determines the number of degrees of freedom necessary to describe it statistically.

Dans le document Coding and multiple-access over fading channels (Page 23-29)