the vector space formulation of the rain crosspolarization problem and its compensations

502 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-27, NO. 2 , FEBkUARY 1979

The. method and data presented support the feasibility of cross-polarization interference cancellation. They are appli- cable to the determination of SNR budget in the transmission system with interference cancellation.

ACKNOWLEDGMENT

Many stimulating and helpful discussions with Messrs. D. 0. Reudink, A. S. Acampora, W. T. Barnett, E. E. Muller, V. K. Prabhu, and Y. S. Yeh are gratefully acknowledged.

1.

2.

’ 3. 4 .

5.

6 .

7.

8.

9.

10.

REFERENCES Transmission Systems for Communications, Western Electric Co., Technical Publications, pp. 444-449, February, 1970. Widrow et al., “Adaptive Noise Canceling: Principles and Applica- tions,’’ Proc. IEEE, Vol. 63, No. 12, pp. 1692-1716, December, 1975. W. T. Bamett, private communication. T. S. Chu, “Restoring the Orthogonality of Two Polarizations in Radio Communication Systems,” I. B.S.T.J., Voi. 50, NO. 9, pp. 3063-3071, November, 1971.11. B.S.T.J., Vol. 52, No. 3, pp. 319- 329, March, 1973. N. Amitay, “Method and Apparatus for Reducing Crosstalk on Cross-Polarized Communication Links,”U.S. Patent No. 3,735,266, May 22,1973. A. F. Culmone, “Polarization Diversity with Adaptive Channel Decoupling,” National Telecommunications Conference, New Orleans, pp. 25-22 to 25-27, December, 1975. B. D. Cullen, et al., “Spectrum-Reuse by Adaptive Polarization Separation,” National Telecommunications Conference, New Orleans, pp. 43-18 to 43-25, December, 1975. S . H. Lin, “Impact of Microwave Depolarization During Multipath Fading on Digital Radio Performance,” B.S.T.J., Vol. 56, No. 5, May-June, 1977. J. F. Bekerich and .I. H. Ingram, “Is Cros-Pol the Way to Go for 11 GHz Digital LOS Radio,” TelephoneEngineerandManagement, pp. 54-57, November 15, 1976. V. K. Prabhu, private communication.

The Vector Space Formulation of the Rain Crosspolarization Problem and Its Compensations

LIN-SHAN LEE

Abstruct-The rain crosspolarization problem is formulated in a vector space. In this formulation signals are vectors, and the crosspolarization effect is an operator. The compensators currently designed for satellite communications become the inverse operator approach in this formulation. A new approach using eigenvectors is developed. The result indicates that use of two orthogonal linear polarizations and a rotation of their directions can eliminate the crosspolarization. A feedback loop can be used to control the rotation angle, and only one control variable is sufficient. Even if no satellite applications have been found, good potential in terrestrial systems is expected.

Paper approved by the Editor for Radio Communication of the IEEE Communications Society for publication after presentation at the IEEE International Symposium on Antennas and Propagation, Stanford University, Palo Alto, CA, June 1977. Manuscript received May 10, 1978; revised September 18, 1978. Parts of this work were done in the Communication Satellite Planning Center, Department of Electrical Engineering, Stanford University, Stanford, CA, as part of a Ph. D. dissertation.

The author is with EDUTEL Communications and Development, lnc., Palo Alto, CA 94304.

I. INTRODUCTION

Due to an expected rapid growth in demand for communication capacity, the allocated spectrum for communication chan- nels is becoming more and more crowded. Substantial efforts have thus been made to try to reuse the same channel frequency by orthogonal polarizations. However, the rain crosspolarization severely limits the use of this technique, especially when the lower frequency band is crowded and the designers are trying to use higher frequencies.

The rain crosspolarization effect is due primarily to the nonspherical nature of the raindrops. It transfers some of the signal energy from one polarization to the orthogonal one, thus causing interference when there is a second signal in the orthogonal polarization. If xl, x2 are two orthogonal signals transmitted sharing the same frequency, yl, y2 are the signals after they propagate through the rain. The rain crosspolarization effect can be modeled by

or

which is a matrix relationship. The diagonal elements, A , B, re- present the wanted signals. The off-diagonal elements, c, d , re- present the crosspolarization or interference. The matrix, M , in (1) is a general form; a specific matrix model will be intro- duced later in this paper to illustrate the concept developed here.

Oguchi [ 11 -[3] was the first who investigated the effect of rain on orthogonal polarizations. Theoretical analyses are then developed and experimental results are observed and compared by Saunders [41, Thomas [SI, Watson, Arbabi, Eng [ 61, [71, Taur [8], Chu [ 9 ] , Semplak [ 10-131, Lin [ 141, etc. All of these results suggest consistently the necessity of a method to compensate for this effect if frequency is to be reused by orthogonal polarizations. Chu [ 151, [ 161 was the first who proposed a theoretical model of such a compensator, which was then realized by various designs [ 171 -[ 2 1 1 . Some new control schemes were also proposed recently [22] , [23] to solve the problem in satellite communications in particular.

In this.paper, however, a different approach is used. The matrix equations (1) and (2) make it possible to formulate the problem in a vector space and solve it with vector space tech- niques.

11. THE SIGNAL SPACE AND ITS BASES

Because the frequency reuse systems transmit two signals, say xl, x2, simultaneously sharing the same frequency, it is reasonable to think of the two signals as a two-dimensional vector x, as in ( l ) ,

x= (1;). If we define the vector addition and scalar multiplication of

U900-6778/79/0020-0502$00.75 0 1979 IEEE

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-27, NO. 2, FEBRUARY 1979 503

these vectors as

x+y= (;;) +(;;) = ( x1 + Y 1 ) x 2 + Y 2

A The set of these vectors, {X}, do satisfy the definition of a I vector space in linear algebra. Such a vector space is named the (a) “signal space” here.

Two fundamental bases to be used in this signal space can be defined as follows. A linear basis { e l , e 2 } is formed by two “.@*

orthogonal linear polarizations with unit. amplitude, one hori- zontal and one vertical. When signals x l , x i are transmitted in @ I

this form, the signal vector is represented by rc ” x 1 ,..,< 3 signals: X I . x? vector: x =

x = x l e l + x 2 e 2 = (’l) -

( 6 ) X2 e

where the subscript, e , means the coefficients, XI, x 2 , are with respect to the basis { e l , e 2 ) . A circular basis {Zl, F2} is formed by two opposite-sensed circular polarizations with unit amplitude, one right-handed and one left-handed. When signals are transmitted in this form, the signal vector is represented simi- larly (and both bases are shown in Fig. 1):

Because each linear polarization is a superposition of two circular polarizations, and each circular polarization is a superposition of two linear polarizations with 90’ out of phase, each basis vector defined above can be represented in terms of the other basis:

Zl = (,1)= z2 = (-;)e

and the transformation between the two fundamental bases is straightforward using ( 8 ) , i.e., if a vector x has a representation [X], with respect to the linear basis, and [x]; to the circular basis, then

[X] e = N[?I;

[X]: = N--1 [XI e

where

(b) Fig. 1. The mapping from real signals to the vectors in signal space:

(a) using linear basis; (b) using circular basis.

and the columns of matrices N, N-l in (10) are simply the vectors in (8).

111. THE CROSSPOLARIZATION OPERATOR AND MATRIX REPRESENTATIONS

Equation (2) shows that the rain crosspolarization effect can be modeled by

Y = M ? ( 2 )

which, in the language of linear algebra, means that this effect is a linear operator defined on the signal space. This operator maps each vector, X, in the space representing a pair of transmitted signals to a new vector, 7, in the space representing the pair of crosspolarized signals. This operator simply defines a particular mapping relationship among the vectors in the signal space. From theory of linear algebra, this relationship can be described by a matrix so long as a fixed basis is used as reference. For the crosspolarization operator discussed here, say operator T , if T maps vector X into V , and the linear basis { e l , e2} is used as reference, we can describe the relationship between [X],, [71e by a matrix [TI,,

[UIe = [Tle[?le. (1 1)

This is exactly the form of ( 2 ) , in which the matrix, M, is simply the operator represented in some basis. [TI , is called the matrix representation of the operator with respect to the basis {e l , e 2 } .

From a theory of linear algebra, the same operator has different representations with respect to different bases, but all of these matrix representations must be related by similarity transformations. For instance, for the same operator, T , described in (1 l ) , if the circular basis {Fl, z2} is used as a reference, the relationship between [X]z and [Jlz must be expres- sed, which becomes

[?la = [Tldaz (1 2 )

504 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-27, NO. 2, FEBRUARY 1979

where [TI ;is the representation of T in a circular basis, and

[ TI; = N - l [ TI eN. (13)

The matrices N , N - l are those defined in ( lo) , and (13) is simply a result of ( 9 ) . Equation (13) is called a simi2arity transformation, and [ T I , , [ T I ; are said to be similar. These matrix representations are shown in Fig. 2.

In order to be more realistic, we introduce an actual matrix model of rain crosspolarization here. Chu [ 9 ] developed the X 1

following matrix with respect to the linear basis here in analyzing the rain crosspolarization effect:

T : raindrops

j 1x1,

(a)

= (€6 B ) .

A €6-

where

A = T2 cos2 eeff + T~ sin2 eeff B = r1 cos2 eeff + T~ sin2 O e f f

€6 = E[ T2 - T1] sin O e f f cos B e f f

T~ = e - ( a l + j p l ) L

T - e-(a2+iP2)L 2 -

and where Q1, a2, pl, 02 are the attenuation and phase shift constants for.signals polarized in the directions of major and minor axes of the raindrops; L is the pathlength; E and 0 ef f are two empirical parameters characterizing the random variation of the raindrop canting angles.

Chu’s matrix in (14) can be substituted into (1 3) directly to obtain its similar matrix with respect t o a circular basis. We thus have

where

and T I , T S , e,,,, E are just the same as before. Although there are many different ways to model the rain

crosspolarization, and there is no proof that Chu’s matrix is better, we simply pick it up for illustration. However, Chu’s model does show agreement with measurements to some ex- tent, and this fact is good enough for the purpose here.

It should be pointed out that the rain crosspolarization effect is a stochastic phenomenon, but it is well-defined and modeled for each particular time, and only this instantaneous behavior is discussed here. The rain crosspolarization at a specific time, to , is the total effect on Y( tO) , the received signal at to , due to the stochastic phenomenon over the entire propa-

(b) Fig. 2. The linear operator T and its matrix representations:

(a) [TI e for linear basis; (b) [ TI; for circular basis.

gation path, L , and throughout the entire propagation time, h , that Y ( t o ) has traveled; i.e.,

etc. This operator, T(to), and thematrices, [ T ( t 0 ) l e , [ T ( t o ) ] ~ do exist for each time, to , and are time-varying functions of random nature, T(t) , [ T( t ) ] e , [ T ( t ) ] a . In the discussions here, we are talking about only this instantaneous behavior of the operator T( t ) , and every matrix here is understood to be an instantaneous value of the random time function, the time, t o , or to - h is thus dropped in the notation for simplicity. Also, note that even if Chu, in developing his matrix model [ 9 1 , was concerned mainly with the mean characteristics of T(t). m, thus empirically determined the mean matrix me, every parameter in his model, (14), (1 5), does have a physical meaning of instantaneous nature. Therefore, use of Chu’s matrix here is still valid.

IV. DIAGONALIZATION BY INVERSE OPERATOR

In the matrices [TI e , [TI ;, it is the off-diagonal elements that couple the two signals and cause the interference. The goal of a compensator is simply to suppress or eliminate these elements, i.e., try to diagonalize the matrices.

The left inverse operator, T - l , of an operator, T , is an operator such that

for any basis { u i ) . This offers a way to diagonalize the matrix. However, for the crosspolarization problem here, the matrices are 2 X 2 , and [rl ] has four elements. Therefore, we have, in general, eight unknowns for such a compensator since each


element has two parameters, amplitude and phase. Fortunately, the situation can be simplified significantly since the two wanted signals need not be unity as in (1 8). In other words, if [ T1 ] is of the form

The normalization into the following form

can still diagonalize the matrices. This reduces the number of unknowns to four: a l , a 2 and C l , C2. The inverse operator in (20). can be easily realized by a network in the receiving end crosscoupling the two received signals with adjusted amplitude and phase such that

[ T-l ] , [ TI , = diagonal (21)

which is demonstrated in Fig. 3. This is exactly the way in which many groups are currently designing the compensator for satellite communications [ 171 -[ 231. Note that, in general, the instantaneous operator [TI always exists as does the nor- malized inverse operator, [ T-ln] u , even if the operator [TI is singular. But, because of the random time-varying nature of [TI u , the inverse operator [ T-l,] and the four variables in it are also randomly time-varying. A complicated feedback control mechanism with pilot signals as control reference must therefore be designed to simultaneously control the four variables and continuously reflect the random variation. Some of the designs do reduce the number of variables to three or two, but those are for some special cases or particular approxima- tions.

V. DIAGONALIZATION BY EIGENVECTORS

There is a completely different approach: diagonalization by eigenvectors. This approach was first proposed by Kreutel [24], but his approach is much more general and thus more difficult to realize than the developments here. If a vector, u, can be found for an operator, T , such that

for any basis {ui}, then u is the eigenvector, and h the associated eigenvalue of the operator, T. A nice result in linear algebra is that the matrix representation of a linear operator with respect to the basis of its eigenvectors is diagonal. Thus, a simple change of basis can eliminate the coupling between the two signals if the set of eigenvector bases can be found.

For the matrix [TI e with respect to the linear basis in (14), it can be shown that the two eigenvalues are

with

J

SATELLITE

[ T ; ' l U [ T I U = diagonal

Fig. 3. Block diagram of the inverse operator approach. [ T I ] , , [TZ], are the uplink and downlink crosspolarization effect, and [TI , is the combination of them, compensated by [ T-l ] , .

and the associated eigenvectors are

with

This means that if { u l , uq} in (24) is used as a basis, we will have

with no crosspolarization at all. This basis, ( ~ 1 , UZ}, in (24) is in fact a pair of orthogonal linear polarizations exactly the same as {el , e2} except being rotated by an angle, Go. This is shown clearly in Fig. 4. Physically, it can be shown [25] that the angle, q50, corresponds to a special direction in which the crosspolarization components of the transmitted signals will be integrated to zero. In other words, this corresponds to the "characteristic mode" of the rain-filled medium such that if signals are transmitted in this mode the polarizations will not be changed at all. Therefore, if we can find such an angle, $o, and set the polarizations of the feeds of transmitting and receiving antennas in the direction of go, we can solve the rain crosspolarization problem, as shown in Fig. 5.

Just as before, the eigenvectors, or the angle $0, are time- varying for the random operator. This can also be seen from (24c) where $0 depends on E and e,,,, and both of them are random time functions ~ ( t ) and O e f f ( t ) . They depend on the stochastic distribution of the size, shape, number, and orientation of the rain drops [9] , [25] . However, a nice feature here is that there is only one unknown variable, $0, instead of four, as in the inverse operator approach. Note that this ap-

506 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-27, NO. 2, FEBRUARY 1979

e2

Fig. 4. The eigenvectors of the rain crosspolarization operator T , {VI, Uz}. They are simply {el,e2} rotated by an angle, 90.

x2 0.

, ,' -

RECEIVER 0.

x2 ,'

0

TRANSMITTER

Fig. 5. Rotation of the polarization directions of transmitter and receiver feeds to eliminate the rain crosspolarization.

proach can never be used in satellite communications because not only different ground stations have different local rain conditions and different @o angles, but also, one satellite antenna polarization orientation corresponds to different ground station polarization orientations. However, this approach can quite possibly be used in terrestrial links when frequencies are t o be reused there. In that case, two pilot spikes can be contained in the two transmitted polarizations and fil- tered out at the receiving end as the reference of a feedback control system to control the rotation angle, &, of the feeds of both transmitting and receiving.antennas. This is indicated in block diagram form in Fig. 6. Because the number of control variables is reduced from four to one, the control mechanism should be much easier. Furthermore, because two references can be used (two pilot signals contained in the two orthogonal signals) to control only one variable, the convergence and stability property of such a system should be quite satisfactory. It should also be pointed out here that in some cases of terrestrial radio links the multipath-induced crosspolarization is also significant and even more severe than the rain effect. It is not known whether the eigenvectors proposed here can reduce that kind of crosspolarization, but i t will be shown in Section VI1 that this approach at least will never make any

TRIINSNITTER RECEIVER

Fig. 6. The block diagram of the feedback system to control rotation of the feed polarizations.

the

other crosspolarization worse, and thus solve the rain problem independently. Of course, the problem of multipath-induced crosspolarization should also be studied before any applications are actually made.

A next question is: .because the operator and its eigenvectors are random and time-varying, what will be the case if the control system cannot trace exactly the @,, variation but with an error, A@, i.e., the polarizations are set at angle @ instead of Go

@ = $0 + A@. (26)

This means the basis used is not the eigenvectors { u 1 , U2) as in (24), but is(w1, w q }

w1 = ( cos@) -sin @ e

with @ as in (26). It can be shown, using a simple similarity transformation, that the rain crosspolarization operator has a matrix representation with respect to this basis, {wl, w2}, as

where A'(A@), B'(A@), and c'(A@) are all functions of original parameters and A@. In order to illustrate the result more clearly, define

I c'(A@) l2 I €6 l2

R(A@) = 10 log

which is the reduction of crosspolarization achieved by this control system in dB because the denominator or the dB reference, ~ 6 , is the original off-diagonal element in [ TI e of (14). It can be shown that

sin (2 A@) R(A@) = 20 log

This is plotted in Fig. 7. I t can be seen that even if the crosspolarization cannot be eliminated precisely, significant reduction can always be achieved so long as A@ can be kept small.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-27, NO. 2 FEBRUARY 1979 507

0

-5

R

(dB,

-10

-1 5

1

A4 (degrees) Fig. 7. Reduction of crosspolarization achieved by the control system

with control error A@, R(A@).

VI. CONSIDERATION FOR CIRCULAR BASIS

As a check, a theory of linear algebra says the same operator should have the same eigenvectors with respect to any basis. If we use the matrix [ 7'1 ;, in (1 6), in the diagonalization proc- ess, it can be shown that the two eigenvectors are

Th.ey can be shown t o be exactly the same as ul, u2 in (24), except scaling and phase factors, and they are represented in a circular basis, because the superposition of two opposite-sensed circular polarizations with the same amplitudes is a linear polarization, and its direction can be adjusted by the relative phase between the two circular components. This is shown more clearly in Fig. 8. Of course, the matrix representation of the operator with respect to the basis {Gl, ij2} in (31) must be exactly the same as in (25):

In fact, the eigenvector representations in circular basis {Fl, F2}

0 0

t

0 0 0 -+ Fig. 8. The equivalence between the eigenvectors in linear basis as

well as in circular basis.

are designed to use circular polarizations, the same control mechanism discussed above may still be used t o eliminate the crosspolarization if a network, as shown in Fig. 9, can be designed to transform the transmitted signals into eigenvector basis. Note that in Fig. 9 the control variable, Go, is no longer the polarization angle of the antenna feed but the phase ad- justments in the network; the transmitter and receiver are transmitting and receiving circular polarizations, but the signals are carried by eigenvectors owing to the reasons shown in Fig. 8 and the eigenvector representations of (3 1). As a matter of fact, the networks in Fig. 9 simply realize the similarity transformation

where 6 is the transforming matrix from basis {.?I, ;2} into { &, &} formed by the eigenvectors in (3 1).

VII. APPLICATION TO A GENERALIZED CHANNEL MODEL

Here one must check to learn whether or not the variable polarization angle will have some effect on other problems such as when the sky is clear, the channel is noisy, or other crosspolarization sources are present. A generalized channel model is used.

A generalized channel model is shown in Fig. 10 which can be represented by the following expression in linear basis, and the same arguments can be presented if circular basis is used:

(34)

or

where L is the generalized crosspolarization operator, and r? is the independent random-noise vector.

The rotation of polarizations corresponds to a change of - in (31) suggest that even when the transmitters and receivers the basis, from {el, e 2 } to {VI, UZ}. This changes (35) into

I . I

*---- Fig. 10. A generalized dual-polarization channel model.

where

are the transforming matrices formed by the new basis of eigenvectors.

CASE 1 : When the Sky is Clear

When the sky is clear there is no crosspolarization and the two wanted signals will be equal.

(39)

This is a scaled identity operator and thus remains unchanged for any change of the basis.

Therefore, the rotation has no effect on the transmission.

CASE 2: When the Channel is Noisy

The new noise vector in basis { U1, Up} is

If original noise components n l , n 2 in basis ( e l , e 2 ) are two independent white Gaussian noise components with noise density, ql, q z , it can be shown that the new noise components are again white Gaussian noise with noise density

and, thus, the worst case signal-to-noise ratio will not exceed that of basis { e l , e 2 ) .

CASE 3: When Other Crosspolarization Sources Exist Within the Channel

In this case,

(43 1

where [TI e is due to the rain effect and [ 01 e is due to other sources. When represented in new basis, we have

[ L I u = Q - l [ T ] e Q + Q - l [ O I e Q . (44)

Note that the matrix, [ O l e , can always be decomposed into two parts, one symmetric and one skew symmetric

o c 0 Q'

( d l = (a ' O).+(-b' O ) e

0 b'

or

and a change of basis means

It can be shown that both parts in (47) will have crosspolarization terms either unchanged or reduced.

CASE 4: When Other Crosspolarization Sources Exist in Transmitting and Receiving Systems

This case is illustrated in Fig. 1 1, which can be represented by

[?le = W ~ [ L I ~ W ~ [ X I ~ + w ~ [ Z I ~ (48 1

where w1, W 2 are the crosspolarization matrix for the transmitting and receiving systems. Note that W l , W 2 are independent of the polarizations of the transmitted signals and thus should be independent of the basis chosen. Therefore, for the new basis the expression is simply

where [ L ] ,, [E], are as in (37) and W 1 , W 2 are unchanged. It can also be shown [ 2 5 ] that the crosspolarization will not be greater than before.


q+$ y, ;cmQ x *

n2

Fig. 11. The generalized channel model when other crosspolarization sources exist in transmitting and receiving systems.

VIII. DISCUSSIONS AND CONCLUSION

The vector space formulation of the rain crosspolarization problem is performed in this paper. The currently designed compensators correspond to inverse operators in this formulation, and a new approach using eigenvectors is developed. The inverse operator approach needs four variables because it is a general solution to the crosspolarization problem and not par- ticularly for rain; it can eliminate the crosspolarization no matter how it is induced and by what kind of matrix it can be represented. However, the eigenvector approach makes use of the special structure of rain crosspolarization matrix, thus only one variable is sufficient. Even though there is no proof that Chu’s matrix is better than other rain crosspolarization models, his matrix is derived from both theoretical and experimental results and every parameter has a specific physical meaning [9 ] , and therefore should be good enough for the purpose here. The variation of the two parameters, O e f f and E , is a major weak point of Chu’s model, but in the developments here the two parameters combine into one variable, Go, whose variation can be’tracked by simple feedback control. So long as such a zero crosspolarization direction does exist in some angle, $0, the approach is valid.

REFERENCES [ 11 T. Oguchi. “Attenuation of Electromagnetic Waves due to Rain

with distorted Raindrops (I).” J. of Radio Research Lab., Vol. 7, No. 33, Sept. 1960, pp. 467-483.

[2] T. Oguchi. “Attenuation of Electromagnetic Waves due to Rain with Distorted Raindrops (II).” J. of Radio Research Lab., Vol. 11, No. 53, Jan. 1964, pp. 19-37.

[3] T. Oguchi. “Attenuation and Phase Rotation of Radiowaves due to Rain: Calculation at 19.3 and 34.8 GHz.” Radio Science, Vol. 8, No. 1, Jan. 1973, pp. 31-38.

[4] M. J. Saunders. “Crosspolarization at 18 and 30 GHz due to Rain.” IEEE Transac. on Antennas and Propagation, Vol. AP- 19, No. 2, March 1971, pp. 273-277.

[5] D. T. Thomas. “Crosspolarization Distortion in Microwave Radio Transmission Due to Rain.” Radio Science, Vol. 6 , No. 10, Oct.

[6] P. A. Watson, M. Arbabi. “Rainfall Crosspolarization of Linearly and Circularly Polarized Waves at Microwave Frequencies.” Electronic Letters, Vol. 8, No. 11, 1 June 1972, pp. 283-285.

. [ 71 P. A. Watson, C. Eng, M. Arbabi. “Rainfall Crosspolarization at Microwave Frequencies.” Proceedings of IEEE, Vol. 120, No. 4, April 1973, pp. 413-418.

[8] R. R. Taur. “Rain-depolarization: Theory and Experiment.” COMSAT Tech. Review, Vol. 4, No. 1, Spring 1974, pp. 187- 190.

[9] T. S. Chu. “Rain-induced Crosspolarization at Centimeter and Millimeter Wavelength,” Bell Syst. Tech. J., Vol. 53, No. 8, Oct.

[ 101 R. A. Semplak. “Dual Frequency Measurements of Rain-induced Microwave Attenuation on a 2.6-kilometer Propagation Path.” Be11 Syst. Tech. J . , Vol. 50, No. 8, Oct. 1971, pp. 2599-2606.

[ 111 R. A. Semplak. “The Effect of Rain on Circular Polarization at 18 GHz.” Bell Syst. Tech. J . , Vol. 52, No. 6, July-August, 1973,

1971, pp. 833-839.

I 1974, pp. 1557-1579.

pp. 1029-1031.

R. A. Semplak. “Simultaneous Measurements of Depolarization by Rain Using Linear and Circular Polarizations at 18 GHz.” Bell Syst. Tech. J . , Vol. 53, No. 2, Feb. 1974, pp. 400-454. R. A. Semplak. “Measurements of Rain-induced Polarization Rotation at 30.9 GHz.” Radio Science, Vol. 9, No. 4, April 1874, pp. 425-529. S. H. Lin. “An Occurrence of Very Heavy Rain on a 42-km Path.” IEEE Transac. on Communications, Vol. COM-22, No. 5 , May 1974, pp. 708-710. T. S. Chu. “Restoring the Orthogonality of Two Polarizations in Radio Communication Systems, I.” Bell Syst. Tech. J., Vol. 50,

T. S. Chu. “Restoring the Orthogonality of Two Polarizations in Radio Communication Systems, 11.” Bell Syst. Tech. J., Vol. 52, No. 3, March 1973, pp. 319-327. A. F. Culmone. “Polarization Diversity with Adaptive Channel Decoupling.” National Telecommunication Conference, Dec. 1975, New Orleans, pp. (25-22)-(25-27). B. D. Cullen, et al. “Spectrum Reuse by Adaptive Polarization Separation.” National Telecommunication Conference, Dec. 1975, New Orleans, pp. (43-18)-(43-25). H. Kannowade. “An Automatic Control System for Com- pensating Crosspolarization Coupling in Frequency Reuse Com-

COM-24, No. 9, Sept. 1976, pp. 968-999. munication Systems.” IEEE Trans. on Communications, Vol.

D. F. Difonzo, et al. “Adaptive Polarization Control for Satellite Frequency Reuse Systems.” COMSAT Tech. Review, Vol. 6, No.

NO. 9, NOV. 1971, pp. 3063-3069.

2, Fill 1976, pp. 253-283. 211 A. E. Williams. “A Dual-polarized 4/6 GHz Adaptive Polarization

Control Network.” COMSAT Tech. Review,- Vol. 7, No. 1, Spring 1977.

221 Lin-shan Lee. “A Polarization Control System For Satellite Communications With Multiple Uplinks”, IEEE Transactions on Communications, Vol. COM-26, pp. 1201-1211, Aug. 1978.

231 Lin-shan Lee. “New Automatic Polarization Cancelling Con- trol For Multiple-Station Satellite Communication Systems”, International Conference on Communications, June 1978, Toronto, Canada, Record, pp. 43.3.1-43.3.5.

[24] R. W. Kreutel. “The Orthogonalization of Polarized Fields in Dual Polarized Radio Transmission Systems”, COMSAT Tech- nical Reveiw. Vol. 3, No. 2, Fall 1973, pp. 375-386.

[25] Lin-shan Lee. “Use of the MinimumCrosspolarization Modes Through Rain to Reduce the RainCrosspolarization in Com- munication Systems”, Ph.D. dissertation, E.E. Dept., Stanford University, Aug. 1977.

Monitoring Mobile Packet Radio Devices

DANIEL MINOLI AND ISRAEL GITMAN

Absrruct-A store-and-forward broadcast radio network with mobile switching nodes is considered. The problems of monitoring connectivity changes in the network and updating information in switching nodes to enable routing of messages are addressed. Analytical models are developed for determining the fraction of time that switching nodes have correct information as a function of trajectory and speed for mobility,

Paper approved by the Editor for Radio Communication of the IEEE Communications Society for publication after presentation at the 28th Vehicular Technology Conference, Denver, CO, 1978. Manuscript received March 6, 1978; revised September 18, 1978. The work of D. Minoli was done prior to his joining Bell Laboratories.

D. Minoli was with Network Analysis Corporation, Great Neck, NY 11024. He is now with Bell Laboratories, Holmdel, NJ 07733.

I. Gitman is with Network Analysis Corporation, Great Neck, NY 11024.

0900-6778/79/0020-0509$00.75 Q 1979 IEEE

the vector space formulation of the rain crosspolarization problem and its compensations

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