Aulin Modified Pathloss by 3d

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

A Modified Model for the Fading Signal at a Mobile Radio Channel

Abstmct-A generalization of an existing model for the fading signal at a mobile radio antenna has been made. The generalization lies in let- ting the scattering waves not necessarily be traveling horizontally. The effects of this generalization are investigated concerning probability 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.

T I. INTRODUCTION

YPICAL for mobile radio communication is that one sta- tion is fixed in position while the other is moving. Usually,

no direct line exists between the stations, since, e.g., buildings break this line. Therefore, the mode of propagation of the electromagnetic energy between the stations will be largely by way of scattering. In an urban environment, it could be expected that these scattered waves not necessarily travel horizontally, since by diffraction the waves propagate from the 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 measuring the envelope of a transmitted sinusoidal

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 ] -

Ossanna [ l ] was the first t o propose a statistical model in terms of a set of vertically polarized horizontally traveling plane interfering waves. His model is primarily suited for suburban areas, and the basic assumption is that just one house

[31 I ~ 5 1 .

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

Fack, Lund, Sweden.

i

Fig. 1 . A component wave,

at a time forms a flat vertical reflector. An interfererence pat- tern is formed as the direct and the reflected wave super- impose. The orientation of the reflector was random, and nonrandom parameters were speed and direction to the trans- mitter.

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 ] , includ- ing Ossanna’s as a special case. This model is most easily under- stood by looking at Fig. 1 .

N vertically polarized, horizontally traveling plane 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 statistically independent. Denoting the angular carrier frequency by o,, the resulting field in a point (XO, y o ) can be written

where

E,(t) = cos - (xo cos a, + y o sin a,) + (p, - cos act r::

and X is the wavelength.

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

AULIN: FADING SIGNAL AT MOBILE RADIO CHANNELS 183

BASESAhD FREQUENCY IN h E R T Z

Fig. 2. Comparison between theoretical (dashed) and measured power spectra (Clarke's model).

By giving the point (xo, )lo) 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 ] .

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 trans- mitter and a receiver on the street would be possible.

The new model proposed here is simply Clarke's model generalized to the case where the vertically polarized plane 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.

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

t

b'

Fig. 3. A component wave in three dimensions.

and cy, and P, have the probability density functions(pdfs) p a ( . ) and p a ( . ) , respectively.

The resulting field in a point (xo, yo, zo) can now be written

M ~ ( r ) = X E n ( [ ) (4)

n = l

where

[ x 2n

E,(t) = C, COS w,t - -- (xo COS a, COS p,

+ y o sin a, cos 0, + zo sin p,) + pn ] (5)

where w, is the angular carrier frequency and X is the wavelength.

Giving the point (xo, yo , zo) a velocity u in a direction with angle y to the x-z plane, i.e., (xo, yo, zo) -+ ( u cos y, v sin y, zo). the resulting field is

E(t) = rc(r) - COS act - T,(t) sin w,t (6)

where

N ~ , ( t ) = X c, cos ( a n t + e,)

n = l

N

T,(t) = c, sin ( a n t + 6,) n = l

and

2nv x W, = - COS (7 - 01, j COS 0,

; n = 1, 2, ..., A! (8) 27120 . e n =- x sin Pn + (Pn

1 84 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. VT-28, NO. 3, AUGUST 1979

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 ] . These processes will be regarded as purely Gaussian.

Since the process E(?) is Gaussian, it is completely charac- terized by the mean function E{E(t)} and autocorrelation function E{E(t) E(t + T ) } . From ( 6 ) we have

E(E(t)} = E(T,(t)} COS act - E{T,(t)}

sin w,t = 0 (9)

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

E(E( f ) E(? + 7 ) ) = E{T,(f) T,(t + 7)) COS Wc7

-E{T,(t) T , ( ~ + T ) } sin W,T

= U ( T ) - cos W,T - C(T) sin W,T (10)

From (1 2 ) we have

since Jo(0) = 1, so PO(.) does not affect the distributional properties of the envelope and the phase.

However, the correlational properties of 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 unreal-

istic; 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, P o , the phase cpo, and the amplitude &are unknown constants, and from (6)-(8) we have the field caused by the specular component

and the correlational properties of the process are thus de- Ed(t) = T c d ( t ) cos w,t - Tsd(t) sin w,? (1 6 ) picted by U(T) and ~(7).

Straightforward calculations give (see Appendix I) where

EO 2

C(T) = - E{ sin U T } and

where o is the on in (8) without index.

out 0 to 2n, we have

2nv wo = - cos (y - (Yo) cos 00

Assuming that the a, are rectangularly distributed through- h

B o =X sinPo + q 0 . 2nzo

(1 2 ) The total field is

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)

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.,

+ Tsd(t)) sin w,t

= TCr(t) cos w,t - Tsr(t) sin act (19)

and again we have a Gaussian process. The only change is that the quadrature components no longer have zero mean, but instead

Again the envelope Rr(t) and the phase *(t) are defined through

AULIN: FADING SIGNAL AT MOBILE RADIO CHANNELS

ao[ r l E- " - 2

0

-7- Eo

I

I

185

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

J "= X

Fig. 5 . The specular component.

it is also familiar that the envelope has a Rician i.e.,

distribution,

r 2 0, ( 2 2 )

and again this pdf is not affected by PO(.). l o ( * ) is the zeroth- order modified Bessel function of the first kind.

111. RAYLEIGH FADING

A. Envelope Correlation Since P O ( - ) affects the correlational properties of Tc( t ) and

T,(t), this will be studied in more detail. Denoting a(7) when pp@) = S ( P ) by ao(r), i.e.,

ao(7) = 3 2 Jo (y), we have the power spectrum of T,(t) and Ts(t)

I O, elsewhere

SO the processes are strictly band-limited within the maximum Doppler frequency shift f, = v/A.

The Fourier transformation of (12) cannot in general be carried out exactly, but for

0, elsewhere,

which can be a realistic pdf for small o m , we have (see Appendix 11)

I O,

v If I>;

Eo A v v - cosp, < I f 1 <- --. - 4 sin p, v ' A A

It can be noted that limpvlh A o 0 = 00, whereas A(v/A) is finite and actually constant in the domain (v/A) cos P, Q I f 1 < v/A. Since

independent of pp( . ) , the result is that we have a much more low-frequent spectrum even when 0, is very small. This can be seen in Fig. 6 where the spectra are plotted logarithmically.

The function u(7) does not change much for small P , , and to be able to see any difference between ao(7) and a ( ~ ) , 0, has to be chosen very large. The difference can be seen in Fig. 7 where P , has been chosen nonphysically larger, just for demonstration.

The reason why ACf) changes so drastically compared to A o w , even for very small P , , is that the Fourier transform uses ~ ( 7 ) from --OQ to +w. Since U ( T ) is stretched (by a factor cos 0) and averaged with respect to p p ( - ) , replica of u0(7), u(7)

does not change much for small 7, but it certainly does when 7

is large. The mean and the autocorrelation functions of the envelope

are [4]

= -a(O)F --, --, 1, IT

2 ( ; ; (%)2)


v - T 0

V - _ A

0 X f A


33 d5

t

L

V

b 3 - _

A f f

(d)

Fig. 6 . Continued.


t

- _ Eo 1 2

Fig. 7. Comparison between envelope autocorrelations no(r) and u(r) (a(r) dashed, & = 45").

where F is the hypergeometric function. A very good approxi- When A m = A 0 0 (Po@) = S@)) this convolution can be mation is carried out exactly giving [9]

rR (7) = E{R(t) R(t + 7)) - E{R(t)) . E{R(t + 7)) So(f> = c A o ( f ) * A o ( f ) n n

=- a2(7) = - aZ(7).

240) EO (29)

The squared envelope is also of interest, since on noisy fading - c e r ; K ( J q $ $ ) IfKy 2v channels the SNR is proportional to R2(t ) , and we have (see Appendix 111, put Ed = 0) 0, elsewhere

-

E{R2(t)} = h ( 0 ) (33)

E{R2( t ) R2( t +T)} = 4(a2(0) + ~ ~ ( 7 ) ) where K is the complete elliptic integral of the first kind. It

Sy? can be found numerically when A V , is given by (26), rR2(7) = E{R2(t) R2(t +T)) - E{R2( t ) ) 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 E{R2(t + 7)) = 4u2(7) (31) SV, for different values of 0, are plotted in a logarithmic

scale.

cos &,/sin2 Om, so for small values of P, most of the energy

small values of P,, SV, is rather flat, and to observe, finite.

B. Phase Correlation

(30)

so should be noticed that limro Soy? = 00.

&Owing us that both rR ('1 and ' R 2(') are approximately The value of SV, near f = 2vb is proportional to proportional to a2(7).

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

S ( f ) = FCC a2(7)} = C A ( f ) * A ( f ) (32)

where C equals nlEo or 4. The two-dimensional pdf for the phase is [8]

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

190

- - 2v a

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

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

t

0

10 dB

& f x


‘25 dB

191

- 25 dB

( 4 Fig. 8. Continued.

192

T 4T2

0

4s' - T

where

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

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 significantly changed even for small values of 0,. However,

Further insight of the phase correlation can be achieved Since the phase (At) is rectangularly distributed throughout by looking at the process ~ d ~ ) = dt + 7) - dt). m e pdf for

a(7) p(7) = - .

a@) (35) an expression for F{r9(7)} has not been found.

0 to 2n Adt) can be found from (34) [SI

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

E{cp(t)l = n

4n2

3 E{p2(t)} = - . (36) pdf, using a(7) instead of ao(7).

. ...

C. The Derivative of the Envelope and Phase, Level Crossing The autocorrelation function for the phase can be found from The four-dimensional pdf for R(t), R(t), dt), and cp(t) isl (34) to give [SI

[41

and it should be noticed that this expression is independent of r > O --<i,ip<- OGcpG2n (39) a(0).

To be able to see any difference between r9(7), using a(7) 1 The dot indicates derivation with respect to r, and the prime or u0(7), 0, has to be chosen comparatively large (see Fig. 9). derivation with respect to T .


-2n 0

Fig. 10. The pdf for p(t + 7) - p(t) for different 7.

271 Aq

and we at once have We know that a(0) is independent of p p ( . ) but from (12) we

Taking pp(0 ) = S (P ) gives

and 1

. e - ( 1 / ~ ) [ ( r 2 / a ( ~ ) ) - ( r i p 2 / a ” ( ~ ) ) ~ , p p ( o ) = I F l 1 ‘ ’ ‘ G o m

r>O -=<ip<-m elsewhere

from which the statistical independence between the four processes can easily be seen. Especially, we have [4] yields

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

and it is also seen that comparatively large values of 0, are required to change uo”(0) significantly.

From (41) it is easily seen that R(t ) is a Gaussian process with zero mean and variance -u”(O). Likewise, we have

so cp(t) conditioned R(t) = r is also Gaussian with zero mean and variance -a’ f (0) / rz .

From (46) it can be concluded that small values of 0, do not affect these pdf’s.

The autocorrelation function for the derivative of the phase is [4]

r + ( T ) = E { $ ( t ) - $(t + T ) }

and since it is a function of u(T), u’(T), and o”(T) it changes significantly for large 0,.~. The power spectrum of cp(r) probably undergoes significant changes even if 0, is small? but this spectrum has not been found exactly.

The expression for the expected number of times per second Nr the envelope passes through the value r with positive slope is given by [4]

X (probability density of envelope at the value r) (49)

where

Denoting N R , using pp(p ) = 6(@, by N,, it is seen that the ratio

is not significantly affected by 0, for small values of 0,. This, as can be easily seen, is also valid for the average duration of fades of the envelope R(t ) below the value r [4] , y,

where, using similar notation,

IV. RICIAN FADING

When the model contains a specular component (16)-(21) it is well known that the joint pdf for the envelope and the phase is [4]

and by integration

O < r < m

where lo(.) is the modified Bessel function of the first kind and zero order, and e(*) is the error function.2

It can be noticed that R,(t) is a stationary process since P R , ( t ) ( r ) is independent of t, but pr(t) is not. In Fig. 11 the pdf for the phase can be seen at two different occasions. As will be seen later, not even the process

is stationary, although the pdf for $ ( t ) of course is independ- en t o f t .

A. Envelope Correlation

As was mentioned earlier, it is often more valuable to know the autocorrelation function for the squared envelope, rather

ALLIN: FADING SIGNAL AT MOBILE RADIO CHANNELS

t

195

Fig. 11. The pdf for the phase. (a) Ed/Eo = 1 and 2 (dashed) at an instant to. (b) Ed/EO = 1 and 2 (dashed at = t , = (ni2wo) + n.(2n/wo);n = 0, f l , *2, '...

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

than for the envelope itself. From Appendix 111 we have It seems plausible that the phase $( t ) = vr(t) - wot should be a stationary process, since

and it can at once be noticed that the quantity wo, the Dop- pler frequency shift of the specular component, is entering this expression. Since

this shift is maximum 2nvlX when y - a0 = 0 or 77 and 00 = 0, i.e., when the specular component is horizontal and enters either from ahead or behind. The angle Po should be very small, thus affecting wo very little.

In Fig. 12 are plotted several rRr(7), where also the scattering components are not necessarily traveling horizontally. The pdf for the deviation from the horizontal case is

(59)

I 0, elsewhere

Now, by taking the Fourier transform of (57), the pow-er spectrum is easily found as

where Scf) is the power spectrum of the squared envelope when no specular component is present. This spectrum is given

In Fig. 13 we can compare these spectra when pp(0) = S@) or p p @ ) (59). It can be noticed that when p p ( p ) = S@), the spectrum is not finite f o r f = 0, I f 1 = (v/h) (1 * cos (a0 - y) cos P o ) , whereas the spectrum is always finite when pp(0) is given by (59).

by (3% (33).

B. Phase Correlation

From Fig. 14 where the phasors of the specular component 6 the Rayleigh component R( t ) , and the resulting envelope R,(r) are shown, some interesting facts about the resulting phase cp,(t) can be seen. As was noticed earlier, the phase function q,(t) is not stationary, since its pdf is a function of 1. This is because the phasor of the specular component is rotating with an angular frequency of wo rad/s.

is independent of f . However, from Fig. 15, which is a shifted replica of Fig. 14,i t can be seen that the “new” Rayleigh phase angle q(t) - mot must have a statistic depending on t .

Since the phase $( t ) is a function of this “new” Rayleigh phase angle, it will certainly not be a stationary process. This can be explained by the fact that (dt) is rectangularly distributed throughout 0 to 2n, and so is rp(t) - wot for all t : measured modulo 27r. The linearly increasing phase wot is thereby hidden behind the process (dt) (see Fig. 16). Since the pdf of the phase rp(t) - wot is independent of t , the pdf of the process $(t) is too.

Looking at, e.g., the two-dimensional pdf for the phases $( t ) and $(t + T ) ? we find that this pdf is not independent of t , since the shifting gave the result that the phasor of the specular component stopped rotating, but the phasor of the Rayleigh component instead began to rotate. Thus the effect of the Doppler frequency shift of the specular component cannot be removed, although wo is known, because then (dt) also has to be known.

V. CONCLUSIONS

Previous models [1]-[3] for the fading envelope at a mobile radio channel have been extended to include a multipath propagation mode? often present in the streets of a city with tall buildings. It was assumed that the waves in such an environment also could have a vertical component of propagation direction, and the effects of this extension were studied, with the previous models as a reference and special case.

It was found that the pdf of the envelope and the phase were left unchanged, whereas the correlational properties of these quantities were not.

Examining the autocorrelation of the envelope by means of the power spectrum, it was found that the shape of the spectrum was heavily affected even for small deviations from the purely horizontal mode of multipath propagation. Concerning the autocorrelation of the phase, the power spectrum could not be found analytically, but it was made plausible that this spectrum should be affected in the same way, i.e., a lower frequency spectrum should be achieved.

It is assumed that this new extended model should be used in an urban environment, but of course, measurements have to be made to confirm whether this new model fits significantly better.


rRr(

3 9 15

197

Fig. 12. Autocorrelation for the envelope. (a) Ed/Eo = 0, pa(@) = 6(@) and p a @ ) given by (59), &, = 4 5 O (Qshed). (b) Ed/Eo = 2, pp03) = 6(8) and p a @ ) given by (591, Pm = 45" (dashed). 1: QO - Y = 0, bo = 0.2: QO - -, = ,,/2, = 0.


1 0 * l o g , o ( ~ ) ; K = (Eo12*L t.) *.) G v

20 dB

2

c L W

-4


Fig. 13. Continued.


- - + Fig. 14. Phasor diagram

t

Fig. 1 5 . Shifted phasor diagram

APPENDIX I

CALCULATION OF a(7) AND c(7)

From (7) we have

N

Tc(t) = x C, COS ( a n t + e,) n = l

where

2nv x w, = - cos p, cos (y -a,)

The stochastic variables a,, On, c,, and (P, are all statistically independent with p d f s p a ( . ) , p p ( . ) , p,(*),and p , ( * ) . It is assumed that the & are rectangularly distributed throughout 0

to 2n, i.e.,

, OG(PG2n

elsewhere.

As will be seen later, p c (.) is not needed, but instead

where Eo is a positive constant. u, y, X, and zo are all nonrandom constants.

It follows that

a(7) = E{Tc(t) Tc(t + 7)) = z E{C,C , N

m , n = l

Now, the stochastic variables

are rectangularly distributed throughout 0 to 271 for all m and n except ti2 when m = n. When m = n, 6 equals zero with probability 1. Thus taking expectation over (P,, (P, gives

1 N E{Tc(t) Tc(t +7)) = a(7) = x E{c,2 cos w,7)

Eo 2N n = l

= - E{ cos w,7)

where the least expression comes from carrying out the expectation over c,. Since all w, are identically distributed subscripts can be deleted yielding

a(7) = - EO E{ COS U T } . 2

This expression cannot be further simplified without specify- ing Pa(*) and Pp(.) .

AULIN: FADING SIGNAL AT MOBILE RADIO CHANNELS 20 1

In much the Same way we have and integration with respect to T gives

1 N

2 m , n = l + 0, + e,)) + - 2 E{c,c, sin (w,(t + 7) cos, - w,t + en - e,)) (70)

y = J sin e and, using the stochastic variables S1 and S 2 from (67), bring- ing out the expectation with respect to 9,, 9, gives

C(T) = - x E ( c , ~ sin O,T) = -E( sin U T )

gives the Jacobian l N EO

2

where o is the w, in ( 6 3 ) without index. d@, 0)

2 n = l (71 1 - 8 s i n o 0 d(x, Y ) -- - f i cos e

APPENDIX I1

POWER SPECTRUM A ( f ) WHEN ~ p @ ) .1 COS

The Fourier transform of and (75) can thus be written

8 x 6 (f - xy) dx dy

A d f ) = 2nsinPm Eo I" x = a c o s pm I" r = - G

(77)

The integration area can be seen in Fig. 17, where the values of x and y for which the integrand is nonzero are also plot-

4 sin p, (72) ted. Since the integrand is zero whenever xy # f, x 4 [a cos Om, m] and y 4 [*, MI, we have, after integration with respect t o y ,

a(7) = _E, f~ Jo (h 277vr cos 0) cos P

is wanted. Since

we have

(73) U

I f P x

EO 277 sin 0,

EO 2n sin p,

/m J x U dx

- .

I V If I Q- cosp,. (79) x (74)

t

Fig. 17. The integration area.

Now, since

x 2x dX

x ' V

integration with respect to x finally gives

Fig. 18 . Phasor diagram.

the cosine theorem we have


I

. . . I Y ~ ~ r~ ~ - ~ - - - = aye) + 2a2(7) (89)

A 0 =

V If'>;

V V -cosp, < I f \ < - x x

' arcsin

V I f I<-cosp, . (81) x

APPENDIX 111 CALCULATION OF THE AUTOCORRELATION

FUNCTION FOR THE SQUARED ENVELOPE AT RICIAN FADING

In Fig. 18, R ( f ) is the Rayleigh envelope, (c(f) the Rayleigh phase, and R.(t) the resulting Rician fadine envelone. From

where O,(t) is the unknown but nonrandom phase of the specular component. Taking expectation we have

E{Rr2(f)} =Ed + E{R2(f)} + 2 a E { R ( t ) )

* E{cos (O,(t) - e,(t>)) ( 8 3 ) since R(t) and 0 2 ( t ) are statistically independent. This gives

using the properties of the Rayleigh envelope and phase, Le.,

E{R(t)} =JT E{R (t)} = 2 4 0 )

From (82) we also have

E{Rr2(t) - Rr2(t + 7))

where the second term is 4Eda(0), using (85). Furthermore, since (see (1 3))

we have

= 2E{Tc2(t) Tc2(t + T)} + 2E2{TC2(t)} (88)

where E{Tc2(t)) =a(O), using (lo), and

E{TC2(f) - TC2(t + T) }

= E{ Tc (t)} E{TC2 (t + 7)) + 2l? { T, ( f ) T, (t + 7))


yielding where X1 and X3 are zero-mean Gaussian random variables

( 9 3 )

we have

This integral can easily be solved, using the substitution

X, = rl COS

x2 = rl sin 0,

x 3 = r2 COS O 2

x 4 = r2 sin O 2

with correlation ~ ( 7 ) . Thus

REFERENCES [ I ] J . F. Ossanna, Jr., “A model for mobile radio fading due to

building reflections: Theoretical and experimental waveform power spectra,” Bell. Syst. Tech. J., vol. 43, pp. 2935-2971, Nov. 1964.

[2 ] R. H. Clarke, “A statistical theory of mobile radio reception,” BellSyst. Tech. J., vol. 47, pp. 957-1000, July 1968.

[3 ] M. J. Gans, “A power-spectral theory of propagation in the mobile radio environment,” IEEE Trans. Veh. Technol., vol. VT-21, pp. 27-38, Feb. 1972.

[ 4 ] S. 0. Rice, “Mathematical analysis of random noise,” BellSyst. Tech. J., vol. 23, pp. 282-332, July 1944; and Bell Syst. Tech. J., vol. 24, pp. 46-156, Jan. 1945.

Bell Syst. Tech. J., vol. 27, pp. 109-157, Jan. 1948. [5 ] W. C. Jakes ef al., Microwave Mobile Communications. New

York: Wiley, 1974. [6 ] T. Aulin, “Characteristics of a digital mobile radio channel,”

Telecommunication Theory, Univ. of Lund, Lund, Sweden, Tech. Rep. TR-93, Oct. 1977.

[ 7 ] W. R. Bennett, “Distribution of the sum of randomly phased components,” Quart. App l . Math., vol. 5 , pp. 385-393, Jan. 1948.

[8 ] E. W. Ng and M . Geller, “A table of integrals of the error functions,”J. Res. NBS-B, vol. 73B, pp. 1-20, Jan.-Mar. 1969.

(91 I. S. Gradshteyn and I. W. Ryzhik, Table of Integrals, Series and Products. New York: Academic, 1965.

[ l o ] E. N. Gilbert, “Energy reception for mobile radio,” Bell Syst. Tech. J., vol. 44, pp. 1779-1803, Oct. 1965.

- , “Statistical Properties of a sine wave plus random noise,”

Tor Aulin (S’77) was born in Malmo, Sweden, on September 12, 1948. He received the M.S. degree in electrical engineering from University of Lund, Lund, Sweden, in 1974.

During 1974-1975 he worked in industry, mostly with automated control, and currently he is preparing his doctoral thesis at the In- stitute of Telecommunication Theory, Uni- versity of Lund. His main interests are narrow- band, high-performance digital modulation sys- tems with constant envelope, and characteristics

= E { X l X,} + E{X, X 4 } = ={x1 x31 (96) of the mobile radio channel.

Aulin Modified Pathloss by 3d

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