E L S E V I E R Signal Processing 36 (1994) 233-237

SIGNAL PROCESSING

Short communication

Statistical remarks about multipath-fading in microwave links

Bernard Lacaze

D~partment d'Electronique, ENSEEIHT/GAPSE, 2 rue Camichel, 31071 Toulouse Cedex, France

Received 6 August 1992; revised 5 January 1993 and 22 July 1993

Abstract

Microwave muitipath propagation can be characterized by a transfer function H the parameters of which represent weakenings and delays. Due to channel changes, H can be considered as a random function of the frequency. In the SHF band, we show that the use of the uniform law permits important simplifications in the study of the statistical characteristics of H. In particular, the H unidimensional probability law is independent of the propagation delays.

Zusammenfassung

Mikrowellen-Mehrwegeausbreitung kann durch eine Uhertragungsfunktion H beschrieben werden, deren Parameter Abschw~ichungen und Verz6gerungen darstellen. Aufgrund der Kanal/inderungen kann H als Zufallsfunktion der Frequenz betrachtet werden. Ffir das SHF-Band wird gezeigt, dab die Verwendung des Gleichverteilungsgesetzes wichtige Vereinfachungen in den Untersuchungen der statistischen Eigenschaften von H erlaubt. Im besonderen ist das eindimensionale Wahrscheinlichkeitsgesetz von H unabhfingig von den Ausbreitungsverz6gerungen.

R~sum~

La propagation microonde par trajets multiples peut ~tre caract6ris~e par une fonction de transfert H dont les param&res repr6sentent les affaiblissements et les retards de propagation. H peut ~tre consid~r6e comme une fonction al~atoire de la fr6quence, ~i cause des variations impr6visibles du milieu. Dans la bande SHF, on montre que l'emploi de la loi uniforme amine des simplifications importantes dans la mod61isation de H. En particulier, sa loi de probabilit6 unidimensionnelle ne d6pend pas de celle des retards de propagation.

Keywords: Microwave propagation; Random process; Multipath propagation

1. Introduction

1.1

In line-of-sight microwave hertzian links, the de- terioration of the message transmitted is, in large part, due to the variation of the crossing media

0165-1684/94/$7.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0165-1684(93)E0088-3

refraction index [4, 7] or to reflections on land and/or sea [5]. Both may coexist [8]. As the paths taken do not have the same length, and the propagat ion speeds are different, the waves are randomly attenuated and received in distinct ran- dom phases. The various parameters that rule this phenomenon depend on prevailing atmospheric

234 B. Lacaze / Signal Processing 36 (1994) 233-237

conditions (temperature, pressure, moisture, etc), and this is why the transmitting channel can be considered as random.

In the three-path model, the channel is in general represented by the following transfer function:

H(09) = Ao + Ale -j°'~l + A2e -j~2, (1)

where f = 09/2~ is the wave frequency, the other parameters Ao, A1, A2, Zl and rz are random and represent the amplitudes and delays of the second- ary waves in comparison with the direct wave. It is possible to fix the amplitude of the latter, Ao. The propagation delays (in comparison with the princi- pal wave) zx, z2, are rather distributed according to gaussian law. Using these hypotheses, a probabilis- tic study of H(09) (considered as an co parameter random process) will be found in [1]. From the statistics point of view this study brings out the fundamental property of line-of-sight microwave

transmission; the reason for this is that 09 Vv/rV-~k is large compared to 1.

For example, in the 4 GHz band, for a transmis- sion distance no greater than tens of kilometers

(here 64), o~ V ~ R is about 15. In addition, it is

supposed that E(Zk) and X / ~ Z k are of the same order size. As shown below, this property makes it possible to dispense with all other hypotheses con- cerning the probability law of the propagation delays Zk. In particular, it is not necessary to at- tribute the gaussian law (or another) as probability law to Zk.

i \

4 x ! 0 I~

!

2i \

.8 ! , , .\ 0 2 4 6

a) case of a folded normal law M:2 .2 o~13.2

0.02

0 015

0.01

0.005

0

-0.005

-0.01 I

N.._

2 4 6

b) case of a folded uniform law M=2.2 o"=13.2

Fig. 1. Dev ia t ion to the p robab i l i ty densi ty of the uniform law.

1.2.

In what follows, we consider the model

H(09) = ~ Ak e-j'°~k. (2) k = l

This generalizes (1), where all the random variables Ak and Zk taken into account are supposed indepen- dent. We study the random process H = {H(09), 09eD} where D is such that 0 9 ~ is large compared to unity for 09 e D. This hypothesis makes it possible (see Appendix A) to assimilate the ran- dom variable Bk = [09Zk] mod 2n, the remainder of

the division of 09Zk by 2n, to a random variable uniformly distributed on (0, 2n).

Figs. l(a) and l(b) represent the distribution function and the probability density of the r.v. Bk in the typical case of a 64 km link to 4.4 GHz in which E[Zk]=O.5ns and V a r z k = 9 n s 2 [5]. The two laws chosen are the normal law and the uniform law. It is clear that the Bk approximation by a uniform law is totally justified, in particular in the normal case where the error made is not quantifiable.

We are interested in the probability law of H(09) (09 fixed) and the second-order properties of H, i.e., its autocorrelation function and its power spectrum.

B. Lacaze / Signal Processing 36 (1994) 233-237 235

2. Properties of H

2.1.

The law of H(09) is defined by its bivariate char- acteristic function ~/(u, v):

~0 (u, v) = E [e j uU,,o) + j ~n~,)],

H(09) = H,(09) + jH2(09)

= ~ AkCOS09Zk--j ~ aksin09Zk. (3) k=l k= l

(c) Lastly, in what concerns Ht and H2, the real and imaginary parts of H, we have

E [H1 (09) H1 (09 - A09)]

= E[H2(09)H2(09 - A09)]

1 E(A~) [q~k(A09 ) + q~(A09)]. (8)

4 k = l

The corresponding power spectral densities are

' i s t ( x ) = s~(x) = ~ E(A~)(A(x) + A ( - x))

k=l

Let

~k (U) = E [e juAk ]

I+2 q~k(A09) = E[e j~A°~] = eJxA~fk(X) dx, (4)

Ok and ~0k are the characteristic functions of r.v.'s Ak and TR. fk is the Zk probability density.

Hypothesis

Ak, A1, "Or, "Cs, k ~ l, r ~ s, are mutually indepen- dent random variables.

09 x / ~ Zk is large compared to unity, for co ~ D.

Conclusion

(a) Bk = [COrk] modulo 2n has the uniform distri- bution for co E D.

(b) H = {H(09), 09~D} is a wide sense stationary process, with zero mean, characteristic function ~(u, v), autocorrelation function Ku(A09), and power spectral density s(x):

1 I~I f~': ~'(u,v) = ~ ~k[UCOSX - v sin x] dx, k=l

(5)

K u (A09) = E [H(09) H*(09 -- A09)]

= ~ E [-A~ ] q,g'(A~o), k=l

s(x) = ~ E(A2) fk ( - x). k=l

(6)

(7)

2.2.

(a) To prove the properties expressed in the pre- vious subsection, we successively write

~0 (u, v)

= 12I E [e JAk( . . . . . . k--vsinto,k)'], (9) k=l

since the couples of random variables (Ak, Zk) are supposed independent. Given that it is the same for r.v. Ak and Zk between them:

E [e jAu( . . . . . . . -v sin,o,.,,~ i ,.rk ]

= ~bk [u COS 09Zk - v sin COrk], (10)

where Ck(X)= E[e j~ak] is the characteristic function of AR. The carrying forward of (10) in (9) involves (5), in utilizing the fact that 09Zk is uniformly distributed (modulo 2~) on (0, 2n) (its probability density is thus 1/2n on this interval and 0 elsewhere). We also have

1 ~2~ E[e-J'°fk] = ~ j ° e-JUdu = 0,

so E[H(o9)] = 0 for ~o~D.

(b) n(09)n*(09 - A09)

= ~ AkZl e-j°~k+j(°-A°)n. k,l=l

236 B. Lacaze / Signal Processing 36 (1994) 233-237

The various r.v.'s taken into account are in- dependent, so

E [H(CO)H* (co -- ACO)

= ~ E(Ak)E(At)EEe-JO'k]EEe j(o-a°')*'] k*l

+ ~ E[A~']E[e-JA'°'~]. k = l

Hence (6) follows, since E[e -j°'*k] = 0 for co e D. So, H is a stationary random process in the wide sense, of zero mean. Its autocorrela- tion function is continuous (linear combination of characteristic functions).

(c) If fk is the probability density of Zk, the second part of (6) is the Fourier transform of

x). k = l

Hence (7). Relation (8) is obtained in the same way as (6) and (7).

2.3.

Let us examine the case where the h k follows

Rayleigh's law of mean E(Ak) = akX/~; then, we have E(A2)=2tr 2 and fk(x)=(x/tr2)e -x2/207,, x > 0. Since I-C~Zk] is uniformly distributed on (0, 2n) independent of Ak, Ak COS COrk and Ak sin co Zk are N(0, trk 2) (gaussian of variance ak 2) and indepen- dent. So, Hi(co) and H2(co) are also independent N(0,

This last property does not mean that the pro- cesses HI and H 2 are gaussian, because the law of Hi(co) - Hi(co - Aco) is not gaussian. For, we have, for example,

A cos coz - A cos (co - Aco)~

= - 2A sin z(co - Aco/2) sin zAco/2.

This is a product of a gaussian r.v. with another which is not degenerate (and not determined be- cause the law of z Aco is not given).

In the case where the A k does not follow a Rayleigh law but follows another specific law, (5) permits the calculation (at least numerically) of the characteristic function of the couple (H1 (co), H2(co)), and so, after an inversion (by FFT, for example), the law of a couple at issue, which does not depend on the Zk law.

On the other hand, (5) and (7) show that Zk law holds in the power spectra of the processes H, H1, H2 and is preponderant. Inversely, if the probability densities of the various Zk are separate enough, the measurement of s(x) should permit these laws to be estimated.

2.4. Measurement of the statistic parameters of H

According to (5), the univariate laws of H(co), H1 (co), H2 (co), for co E D, depend solely on the laws of the r.v. Ak.

To obtain the multivariate laws, for example, the law of (H(co), H(o) -- Aco)) according to (6), we have to dispose in addition of the laws of the zk.

After observation of this process, the model used shows Ak and Zk to be constant. Indeed, they vary according to the particular observation, that is to say as a function of a change of atmospheric condi- tions (e.g. moisture change). It is therefore clear that the processes H, H1, H2 are not ergodic: to measure the unknown characteristics (for example E(A2)), several observations will be required at different times, to give distinct values for Ak and Zk. Such estimations can be found in [2].

3. Conclusion

In the case of multipath microwave transmission, we have studied the model of transfer function:

H(co) = ~ Age -j°~k, k = l

w h e r e A k and rk are the amplitudes and delays due to various paths. In the H statistical study, it is usual to suppose I-1, 5, 7] that the delays follow a normal law and that n is limited to 3.

B. Lacaze / Signal Processing 36 (1994) 233-237 237

For a frequency of a few gigahertz (GHz) and for path length exceeding a few tens of kilometers,

~ o ~ is large compared to unity. In this case, we have shown that zk does not prevail in the unidimensional laws of these processes H, H~ and H2. We have established the general formula (5) that makes it possible to calculate the H, Ht and H2 probability laws for any value of n whatsoever; it is based on the uniform law used. This law inevi- tably applies in the case studied. At least, we have shown that the H , H ~ , H 2 second-order statistical properties for arbitrary values of n, are expressed as functions of the exact delay laws and the amplitude order 2 moments.

4. Appendix A

-Hypothes is : A is a real r.v. with a probability density.

-Conclusion: if B, = u_.AA designates the remainder of the division of uA by 2~, the law of Bu tends to the uniform law on (0, 2~) when u tends to infinity. This property follows from the Riemann-Lebes-

gue lemma concerning Fourier transform and from Levy theorem concerning the convergence of the probability laws and their characteristic functions.

Let us consider

~o(t) = E [e irA] = ei'Xf(x) dx, (8)

where f is the probability of A. The Riemann-Lebesgue lemma [3, p. 19] implies

lira ~o(t) = 0. (9)

Bu, by the way it is built, takes its values in [0, 2~]. The Levy theorem [6, p. 34] expresses itself in this case in the following way:

lim F~(x) = F(x), x e I u ~ c ~

lim E[e inS~] = E[einC], n e Z , (10) U ~ o o

where Fu(x) = P[Bu < x], F(x) = P [ C < x] are the distribution functions of Bu and C, and I is the set of continuity points of F.

As E[e i"Bu] = E[e intB~+2k~)] for any k ~ Z , we obtain after (8) and (9)

E[e inB~] = ~p(nu), n ~ Z

and

{01 f°r n ~ Z * ' (11) lim E[e in-u] = for n = 0,

In addition, as C is uniformly distributed on (0, 2~)

f.2~ dx {~ for n e Z ' , E[e inc-] = / e i n x - = (12)

,J0 2~ for n = 0,

The comparison of (11) and (12) with (10) implies the property.

.

[1]

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