A Modified Model for the Fading Signal at a Mobile ...

182

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. VT-28, NO. 3, AUGUST 1979

A Modified Modelfor the Fading Signal at a Mobile Radio Channel

i

Abstmct-A generalization of an existing model for the fading signal at a mobile radio antenna has been made. The generalization lies in letting the scattering waves not necessarily be traveling horizontally. The effectsof thisgeneralization are investigatedconcerningprobability density function (pdf), correlational properties, and power spectra of the phase and envelope. The pdf is not affected, but the power spectrum of the envelope is significantly affected. This generalized spectrum is smoother than the original and always finite, which the latter is not. Thus it is assumed that the generalized model is more consistent with measured spectra, especially in urban environments.

I. INTRODUCTION

T

YPICAL for mobile radio communication is that one station is fixed in position while the other is moving. Usually, no direct line exists between the stations, since, e.g., buildings breakthisline.Therefore,themodeofpropagationofthe electromagnetic energy between the stations will be largely by wayofscattering.Inanurbanenvironment,itcouldbe expected that these scattered waves not necessarily travel horizontally,sincebydiffractionthe waves propagatefromthe top of a house down to the street. It is the aim of this report to investigate the influence of this nonhorizontal propagation. In Section I1 previous models are presented, and the modification is shown. The effects of this modification are shown in Sections 111 and IV and in the last section a specular component is included. 11. THE MODEL

A . Previous Models When measuringtheenvelopeofatransmittedsinusoidal carrier as seen by a mobile radio antenna in an urban environment, it is observed that this signal is time-varying, although the transmitted envelope is not [ l ] , [2], [SI, [ 6 ] . It is also observed that when the receiving antenna is standing still, the envelope is almost constant [ 6 ] . This is a fairly strong implication that the envelope is constant in time, but varies spatially, and a model for this spatial variation is thus called for. Of course the spatial variation of the envelope is caused by multipath propagation, and statistical models based on this assumption have been proposed [ 1 ] [31 ~ 5 1 . Ossanna [ l ] was the first t o propose a statistical model in termsofasetofverticallypolarizedhorizontallytraveling planeinterfering waves. His model is primarilysuitedfor suburban areas, and the basic assumptionis that just one house I

Manuscript received September 5 , 1978; revised April 16, 1979. The author is with Telecommunication Theory, University of Lund, Fack, Lund, Sweden.

Fig. 1 .

A componentwave,

at a time forms a flat vertical reflector. An interfererence pattern is formedasthedirectandthereflected wave superimpose.Theorientation of thereflectorwasrandom,and nonrandom parameters were speed and direction to the transmitter. Measurements and comparison with the model were made, and Clarke [2] argued that the theoretical power spectrum of the envelope disagreed at very low frequencies and at frequencies near the maximum Doppler shift from the measured spectrum. Clark used a new model suggested by Gilbert [ l o ] , including Ossanna’s as a special case. This model ismost easily understood by looking atFig. 1 . N verticallypolarized,horizontallytravelingplane waves are superimposed. Every wave has an angle of arrival a, and a phase shift (P,,,is rectangularly distributed throughout 0 to 277, and is statisticallyindependent.Denotingtheangularcarrier frequency by o,, the resulting field in a point (XO, y o ) can be written

where

E,(t) = cos

r::

- (xo cos a, + y o sin a,)

and X is the wavelength.

0018-9545/79/0800-0182$00.75 0 1979 IEEE

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- cos act

+( p ,

AULIN: FADING SIGNAL AT MOBILE RADIO CHANNELS

183

t

b '

Fig. 3.

A component wave in three dimensions.

and c y , and P, have theprobabilitydensityfunctions(pdfs) p a ( . ) and p a ( . ) , respectively. The resulting field in a point (xo, y o , zo) can now be written M

~ ( r =)

X

(4)

En([)

n=l

where

E,(t) = C,

COS

[

w,t

2n

- -- (xo COS a,

COS

x

p,

B A S E S A h D FREQUENCY IN h E R T Z

Fig. 2.

+ y o sin a, cos 0,

Comparison between theoretical (dashed) and measured power spectra (Clarke's model).

Bygiving the point (xo, a velocity u in any direction, the power spectrum of the envelope can be found analytically and compared to measured spectra [ 2 ] [ 3 ] , [SI. As seen from Fig. 2 (taken from [2]) there still is a significant disagreement between theoretical and measured spectra. For an explanation of (a) and (b), see [ 2 ] . )lo)

]

(5)

where w, is the angular carrier frequency and X is the wavelength. Giving the point (xo, y o , zo) a velocity u in a direction with angle y to the x-z plane, i.e., (xo, y o , zo) ( u cos y, v sin y, zo). the resulting field is -+

-

E(t) = rc(r) COS act- T,(t) sin w,t

B. A Modified Model The previous models have in common that the component waves are traveling horizontally. In an urban environment this is not true, since then no transmission between a distant transmitter and a receiver on the street would be possible. Thenewmodelproposedhere is simplyClarke'smodel generalized tothe casewheretheverticallypolarizedplane waves are not necessarily traveling horizontally. A typical component wave can be seen in Fig. 3. This is a plane wave and has an angle of arrival a, to the x-z plane and pn to the x-y plane, and with each wave is associated an amplitude c, and a phase shift (P, . The parameters a,, 0, pn . and c, are all random and statistically independent. Moreover.

+ zo sin p,) + pn

(6)

where N

~ , ( t= )

X

c, cos ( a n t

+ e,)

n=l

N

c, sin ( a n t + 6,)

T,(t) = n=l

and

~

2nv

W,

= - COS (7 - 01j, COS 0,

x

27120

where 1V is the fixed number of waves and Eo is a positive constant. The p,, are rectangularly distributed throughout 0 to 211,

e n =-

x

;

.

sin Pn + (Pn


n = 1, 2, ..., A!

(8)

184

IEEE TRANSACTIONS ON

According to the central limit theorem, T,(t) and T,(t)are approximately Gaussian if N is sufficiently large. Actually, this approximation is very good for N 2 6 [ 7 ] . Theseprocesses will be regarded as purely Gaussian. Since the process E(?) is Gaussian, it is completely characterizedbythemeanfunction E{E(t)} andautocorrelation function E{E(t) E(t + T ) } . From ( 6 ) we have

E(E(t)} = E(T,(t)}

COS

act- E{T,(t)} (9)

sin w,t = 0 since E{T,(t)} = E{T,(t)} = 0 and

E ( E ( f ) E(? + 7 ) ) = E{T,(f) T,(t -E{T,(t) = U(T)

- cos

+ 7 ) ) COS W c 7

T , ( ~ + T ) } sin W,T W,T

- C ( T ) sin W,T

(10)

EO

= - E{ sin

2

3, AUGUST 1979

From (1 2 ) we have

since Jo(0) = 1, so PO(.) doesnotaffectthedistributional properties of the envelope and the phase. However, thecorrelationalpropertiesof R ( t ) and (At) are affected by pa(*), since different p p ( * )give different ~ ( 7 ) . An example can be seen in Fig. 4. Of course, the p p ( * ) chosen for Fig. 4 (b) is very unrealistic; it was chosen just to demonstrate the effect. It can be noticed that pp@) = S@) corresponds to Clarke's model. The model can easily be further generalized to include a specular component (see Fig. 5). The angles ao, Po, the phase cpo, and the amplitude &are unknown constants, and from (6)-(8) we have the field caused by the specular component Ed(t) = T c d ( t ) cos w,t - Tsd(t)sin w,?

and the correlational properties of the process are thus depicted by U ( T ) and ~(7). Straightforward calculations give (see Appendix where I)

C(T)

VEHICULAR TECHNOLOGY, VOL. VT-28, NO.

(1 6 )

and

UT}

2nv

where o is the onin (8) without index. Assuming that the a, rectangularly are distributed throughout 0 to 2n, we have

wo = - cos (y

Bo

- (Yo) cos 00

h 2nzo

= XsinPo + q 0 .

The total field is (1 2 )

+ Tsd(t))sin w,t where J o ( * ) is a Bessel function of the first kind of zero order. The envelope R ( t ) and the phase function (At) are defined through the transformation

T,(t) = R ( t ) cos (At)

= TCr(t)cos w,t

- Tsr(t) sin act

(19)

and again we have a Gaussian process. The only change is that thequadraturecomponents no longer have zeromean,but instead

TJt) = R ( t ) sin cp(t) and it is well known that the envelope is Rayleigh distributed, the phase is rectangularly distributed throughout 0 to 277, and that R(t) and cp(t) are statistically independent, Le.,

Again theenvelope through

Rr(t) andthephase

* ( t ) aredefined

AULIN: FADING SIGNAL AT MOBILE RADIO CHANNELS E-" 2

ao[rl

0

I I

Eo

-7-

185

186

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. ~ ~ - 2NO. 8 , 3 , AUGUST 1979

om,we have (see Appendix

which can be arealistic pdf for small 11)

I --.

If I>;

v

O,

J

Eo

A -

4 sin p,

v'

v

- cosp, A

< I f 1 = c A o ( f ) * A o ( f )

n

a2(7) = - aZ(7).

Thesquaredenvelope

(29)

EO

240)

-

is alsoofinterest,sinceonnoisyfading

channels the SNR is proportional to R 2 ( t ) ,and we have (see Appendix 111, put Ed = 0)

cer;K(Jq$$)

IfKy 2v

0,

elsewhere

(33)

E{R2(t)} = h ( 0 )

E{R2(t) R 2 ( t +T)} = 4(a2(0)

+~ ~ ( 7 ) )

so

rR2(7)= E{R2(t) R2(t +T)) - E { R 2 ( t ) )

E{R2(t + 7)) = 4u2(7)

(30)

where K is the complete elliptic integral of the first kind. It should be noticed that limro Soy? = 00. Sy? can be found numerically when A V , is given by (26), and we have that S(0) < 00 and S(k24A) = 0. A comparison between these spectra can be done in Fig. 8, where Soy) and (31) SV, for different values of 0 , are plotted logarithmic ain

scale. are approximatelyThevalue of SV, near f = 2vb is proportionalto cos &,/sin2 Om, so for small values of P, most of the energy The Power spectrum ofR(t) andR2(t) Can thus be written near f = 0 in ,Soy? has been shifted to f = 2vb. For not too small values of P,, SV, is rather flat, and to observe, finite. S ( f ) = FCC a2(7)} = C A ( f ) * A ( f ) (32) B. Phase Correlation where C equals nlEo or 4. The two-dimensional pdf for the phaseis [8] &Owing us that both proportional to a2(7).

pV(t),@(t+dPl* 92)

rR

('1 and

' R 2(')

190

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY,VOL. VT-28, NO. 3, AUGUST 1979

-

1 0 ~ l O g l O ~; ~Kl = C(=I Eo 2.i

t

- -2va

10 dB

0

&

x

f

191

AULIN: FADING SIGNAL AT MOBILE RADIO CHANNELS

‘25 dB

- 25 dB

(4 Fig. 8. Continued.


IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. VOL. VT-28, NO. 3, AUGUST 1979

192

4T2

T

0

4s'

-T

Of course, r9(7) changes very much for large 7 even for small P,, so the power spectrum of the phase F{rv(7)} is probably a(7) significantlychangedevenforsmallvaluesof .0, However, p ( 7 ) = -. a@) (35) an expression for found. been F{r9(7)} not has Further insightof the phasecorrelationcanbeachieved Since the phase (At) is rectangularly distributed throughout by looking at the process ~ d= dt ~ + )7)- dt).m e pdf for 0 to 2n from found be Adt) can (34) [SI

where

E{cp(t)l = n 4n2 E{p2(t)} = -. 3

(36)

and it is seen that for futed 7,p(7) determines the shape of the pdf.Fig.10shows (Acp) for various 7,usingaO(7).For small values of .4,...-7 there is no significant difference in this pdf, using a(7) instead of ao(7).

C. The Derivative of the Envelope and Phase, Level Crossing Theautocorrelationfunctionforthephasecanbefoundfrom (34) to give [SI

should and itnoticed be that this expression is independent of a(0). To beableto see anydifferencebetween r9(7), using a(7) or u0(7), 0, has to be chosen comparatively large (see Fig. 9).

The four-dimensional pdf for R(t),R(t), dt),and cp(t)isl [41

r>O

--