RESEARCH FOUNDATION - NASA that the loop can function properly under most fading conditions in deep space missions. 0 When fo becomes small In the limit fo = 0, kf can be ...

SOUTHEASTERN MASSACHUSETTS UNIVERSITY

S M U RESEARCH FOUNDATION

NORTH DARTMOUTH, MASS. 02747

https://ntrs.nasa.gov/search.jsp?R=19740005730 2018-05-21T14:14:53+00:00Z

Technical Report EE-73-6 Grant NASA/GSFC NGR 22-031-002

October 2, 1973

NONLINEAR ANALYSIS OF PHASE-XDCKED LOOPS WITH RAF'IDLY VARYING PHASE

by CHI-HAU CHEN, Senior Member, IE3E

and MAISIE FAN, Student Member, IEEE Southeastern Massachusetts University North Dartmouth, Massachusetts 02747

Abstract

The performance of command and telemetry systems, useful i n deep-space conmunications, is frequently affected by the radio-frequency phase error which is introduced a t t h e point of reception by means of t h e ca r r i e r t racking loop. low data rate communications, this phase e r r o r which is highly unpredictable may vary rapidiy over the duration of the s ignal ing in te rva l . phase variatLon is cvwacter ized by a sinusoidal input phase, k sin(q,t + n / 6 ) , which models a typical phase var ia t ion i n communication over turbulent media. Conditions f o r synchronization s t a b i l i t y and the acquis i t ion behavior are examined by detailed computer study of the phase-plane t r a j ec to r i e s f o r the second and third-order loops with perfect integrator . fo/AK 5 1/4 f o r system s t a b i l i t y . condition fo r s t a b i l i t y is kfo < 4, except for very small fo. *L11 usefhl under most fading conditions i n deep space missions.

In

In t h i s paper such

For k = 0.001, it is determined that Here AK is the loop gain. For given for t h e

Thus the loop is

It i s demonstrated that f o r the phase var ia t ion considered t h e th i rd-order loop has no real admtsge over the second-order loop. Final ly , it is shown tha t nonzero i n i t i a l conditions msy result i n large etoady-state phase error .

Nonlinear Analysis of Phase-Lacked Loops With

Rapidly Varying Phase

C.H. Chen and M. Fan

I. Introduction

The performance of command and telemetry systems, useful i n deep-space

communications, is frequently affected by the radio-frequency phase e r ror which

is introduced a t t h e point of reception by means of the carrier tracking loop.

In low data rate communications, t h i s phase error may vary rapidly over t he

duration of the signali :g in te rva l .

entry are turbulence, dispersion, attenuation and residual doppler.

variations cannot be tracked by r. phase-locked loop of lower bandwidth, while the

signal-to-noise r a t i o i n t h i s minimum loop bandwicth is too low.

Causes of t h i s type of behavior i n planetary

The phase

When the r a t i o of the system d ~ t a rate t o c a r r i e r t racking loop bandwidth is

less than one, t he problem of power a l loca t ion between the ca r r i e r and t h e data

has been considered by Hayes and Lindsey (11 , Thomas [2], Sergo and Hayes [3].

For channels with time-varying phase, Heller [4] examined the perfornance of a

sequential decoding system.

the phase-locked loops is given by Viterbi [ 5 ] and Lindsey 161.

An excel lent treetment of the nonlinear analysis of

I n this paper the phase var ia t ion is characterized by a sinusoidal input

phase, k sin(wot + n/6), which models a typica l *ase var ia t ion i n communication

over turbulent media. Nonlinear analysis of t h e loop i n t h e absence of noise i s

performed by extensive computer study of the phase-plane t r a j ec to r i e s .

f o r synchronization s t a b i l i t y and the acquis i t ion behvaior can Le examined from

t h e phaue-plane analysis.

with perfect integrators ere considered with zero as w e l l as nonzero i n i t i a l

conditions.

with rapidly varying phase.

Chen C71 and Fan 181.

Conditions

Both t h e second-order loop and the third-order loop

Comparison is also made between the second-and the third-order loops

Some preliminary computer r e su l t s were reported by

11. The Loop Equations

Folloving t he notations of Viterbi ( [ 5 ] , Chapter 31, w e consider first the

d i f f e ren t i a l equation of a socond-order loop w i t h perfect integrator .

2 d20,

dtL dt2 - + AK cos+ + ~ A K s in+ = -

where + ( t ) is the phase error , AK is the loop gain, 8 ( t ) is the phase of the

input signal, and the t ransfer function of the loop f i l ter is 1

(2 1 a S

F(s) = 1 + - Ibe loop can t rack t h e frequency ramp with zero st,-.ady state error. Now we con-

sider the important case tha t 9 ( t ) is varying several cycles over a bit in te rva l 1

of, say, 1 second which is typ ica l i n low data rate communications. The var ia t ion

is normally caused by t h e time-varying channel. Let

e,(t) = k s i n ( % t + )

ljy normalizing t h e variables with

Eq. (1) becomes 3 o r 11

koz s i n (x 0 + + + i cos+ + a' s in+ = - - (MI2

which i n turn can be wri t ten i n the state equation form as

x1 = x2 W T

0

s i n + ' AK g x,. = - 5 cos 5 - a' s i n x - - ' (AK)*

( 3 )

( 4 )

(5)

where x1 = $ ( t ) .

loop bandwidth, the smaller t h e frequency of t he forcing function given by Eq. (3).

The frequency fo is reduced by a fac tor of AK, and the amplitude kcf is reduced

by (MI2.

It is noted from Eq. (5) t h a t t he la rger the loop gain, o r t h e

I n other words, the large loop gain reduces the e f f ec t of t he tine

varying input phase +l(t).

For a t h i r d order loop with l o o p f i l t e r transfer function

a b F(s) = 1 + - + - 8 2 8

-3-

the d i f fe ren t ia l equation (Viterbi [51, p. 64) is c\

ddel 2 + (AK + aAK)sin$(t) + bAK sin$(u)du = - dt2

2

d t 2 0

which, using Eqs, (4) and ( 3 ) , can be reduced to2 0 ll + + i cos$ + a' sin$ + b' s in$ dT = - - s i n (r + g)

(MI2

..

(8)

which in t h e state equation form becomes

U T 0

4 = x2 2

s i n (- + (10) kUO s i n 5 d T - - AK = - x cos "1 - a' s i n x - b' 2 1 I (MI2 x2

where b' t - and x = e ( t ) .

1 (MI2 Phase-plane ana lys i s of Eqs. ( 6 ) and (10) is performed by using che second-

order Runge-Kutta method (see, e.g. [ 9 ] ) . The computer results are reported i n

t h e following sections.

III. Nonlinear Analysis of the Second-Order bop

Consider first k = 0.001. The loop behavior depends on the r a t i o fo/AK.

Iet a' = 1/2 and AK = 32, the phase-plane p lo t s of the loop a r e shown i n Figs. l(a),

(b) and (c ) w i t h fo = 6.4, 8.0 and 16.0 respectively.

threshold value above which the loop would not be able t o track the input phase.

For fo/AK - < 1 / 4 t he loop would sett le w i t h a stable " l i m i t cycle." It is noted

that the steaily state er ror cannot be reduced t o zero because of the continuous

fo = 8.0 appears t o be t he

input phase var ia t ion.

increasing the number of points t o 3000, Fig. l ( d ) shows that the steady state

Thus fo/Ak - 1 /4 is the condition fo r s t a b i l i t y . By

t r a j ec to ry drifts only slowly. The loop reaches the stable t ra jec tory after less

than two cycles of change i n phase.

Next we examine the threshold value of k fo r specif ied foa F i g s . 2,3,4,5,6

are the phase-plane plo ts for fo = 1.1, 1.0, 0.8, 0.5, 0.25 respectively. The

-4-

threshold here is not a c r i t i c a l value but it may be concluded tha t k f L, 4 is

the required condition f o r s t a b i l i t y with AK = 16, a =: 8.

the allowable prcduct kfo rises rapidly.

positive value.

above condition s t a t e s that the maximum allowable phase e r ro r is 4 radians.

indicates t ha t the loop can function properly under most fading conditions i n

deep space missions.

0

When fo becomes s m a l l

I n the limit fo = 0, kf can be any 0

If the phase @=Tor var ies one fu l l cycle i n one second, t h e

This

For example, based on the updated knowledge of the e f f e c t of turbulence i n

the Venus atmosphere on ratLio propagation [ 101 , we an t ic ipa te t h a t the phase-locked

loop with proper bandwidth can maintain a continuous communication between the

atnospheric probe and the Earth.

IV. Nonlinear Analysis of t h e Third-Order Loop

Consider the case k = 0.001, AK = 16, a = 8, b' = 1/32. Figs. 7(a), (b), ( c )

are the phase-plane p lo t s of 5 vs $ for fo = 3.2, 4.0 and ' .O respectively. For

the same ra t io s of fo/AK, the t r a j e c t o r i e s o f t h e third-order loop d r i f t f a s t e r .

fo/AK - < 1/4 also appears t o be t h e threshold condition f o r s t a b i l i t y .

of b' t o b' = 1/16 as shown i n Figs . 8 ( a ) , (b) , ( c ) only causes the t r a j e c t o r i e s

t o d r i f t more.

The threshold value f o r k is i n t h e region 0.1 c k < 1.0.

s t a b i l i t y is kfo< 4 which i s consis tent with t h e second-order loop results. A

carefu l comparison is made between the second-order and the third-order loops by

using exactly the same loop parameters AK = 8, a = 4, k = 0.001, fo = 1.0 with

pr intout of 1000 points.

(50 seconds) t o cover 6 cycles of trajectories. The r e s u l t s are shown i n Fig. lO(a)

f o r the second-order loop and Figs. 10(b) and ( c ) f o r t he third-order loop with

b' 1/32 and 1/8 respectively.

phase e r ro r which increases with b ' .

f o r the s inusoidal phase va r i a t io t considered i n t h i s paper, t h e third-order loop has

no real advantage over the second-order loop.

The increase

The e f f ec t of increasing k is c lear ly i l l u s t r a t e d i n Figs. 9(a)-(d).

The condition f o r

Both loops e r e s t ab le and take t h e same amount of time

It is noted t h a t t h e third-order loop has la rger

Based on the above study, we may conclude t h a t

-5-

V. Effects of Nonzero I n i t i a l Conditions

To study the loop behaviors under d i f fe ren t i n i t i a l conditions, w e consider

both the second-order and third-order loops with AK = 8, a = 4, k = 0.001 fo r

the i n i t i a l conditions (i, 9 ) = (-3.14, 61, (-3.14, 41, (-3.lh, 2), (-3.14 3 . 0 )

The results are shown i n Fig. 11 f o r the second-order and Fig. 12 f o r the third-

order loops with b' = 1/32.

mirror image of those i n t h e upper half plane with respect 50

The t r a j ec to r i e s i n the lower half ?lane are the

the 5 = 0 axis.

It is in te res t ing t o note that, i n all i n i t i d conditions considered, the loops

reach a steady-state of 4 = 0 and 4 = constant i n s p i t e of t h e sinusoidal input

phase. The steady-state phase e r ro r s may be too large, however, f r o m p rac t i ca l

viewpoint.

s t e a d y s t a t e phase e r r o r is not to le rab le .

Thus t h e nonzero i n i t i a l conditions shouldbe avoided i f t h e large

V I . Concluding Remarks

We have examined the c r i t i c a l parameter values of the second and the third-

order loops with rapidly varying phase which is modelled by a sinusoidal input

phase variation. The nonlinear analysis is performed by studying the phase-plane

t r a j ec to r i e s .

normally has i n t racking phase e r rors , some modification of t he loop s t ruc ture

Although t h e third-order loop does not have the advantages as it

appears necessary t o obtain a more e f f i c i en t t racking system. Methods of s igna l

acquis i t ion aj.ds,a$

acquis i t ion for low signal-to-noise r a t i o s should a lso be examined.

suggested by Linhey [6 3, especial ly the computer-aided

-6-

References

1. J.F. Hayes and W.C. Lindsey, "Power a l loca t ion - Rapidly varying phase error" , IEEX Trans. on Communication Technology, pp. 323-326, Apri l 1969.

C.M. Thomas, "Carrier reference power al locat ion f o r PSK at low data rates", Internat ional Communications Conference, June 1970.

2.

3. J . R . Sergo, Jr. and J.F. Hayes, "Power al locat ion i.n a two way coherent communication systems", UMR - M.J. Kelly Communications Conference, paper no. 22-2-1, Rolla, Mo., October 1370.

J .A. Heller, "Sequential decoding fo r channels with time-varying phase", Ph.D. thes i s , M.I .T . , Cambridge, Mass. September 1967.

4.

5. A. J. Viterbi , "Principles of Coherent Communication", McGraw-Hill Book CO., 1966.

6. W.C. Lindsey, "Synchronization Systems i n Communication and Control", Prentice-Hall, Inc., 1972.

7. C.H. Chen, "Phase-plane analysis of phase-locked loops with rapidly varying phase", TR EE-73-4, SMU, N. Dartmouth, Mass., July 1973.

M. Fan, "Computer study of phase-locked loop behaviors with rapidly varying phase error", TR EE-73-5, SMU, N. Dartmouth, Mass., September 1973.

8.

9. D.D. McCracken and W.S. Dorn, "Numerical Methods and Fortran ?rogramning", !'iley, New York 1964.

10. J.W. Strohbehn, "The e f f ec t of t u rb ulence i n the Venus atmosphe#on rad io prop6gationt1, paper preprint , A u g u s t 1973, t o be published.

I

I

1 I I

! I ,

I

I

. I I I

I

i 1 I I

I J I

1 I

I I

I j

j I , i I

1 i

I

!

1 , I I I i j I I I I

i

I

I

,

!

I

!

I

I ! i i I I

! I i i i I I

I I i i i

i ! ! ! I I I

i

i

I I

! I

I

i i

I

I i I

I I I ,

I

I

I 1

e o 0 1 l r u

N Q .t w .II

I

--I!- .-

I I .

I

I

I

i

1 I

i i I

I i I

! i I

I

j

t

i

i I ! 1

i I 1

I

1 , I

I

1 ! 1 j

I

I

t t

[ i

I 1 i \

!

I

I I

I f I i I

!

I I

I I

1 I I

i I

I ! !

t

i

!

I

;

j

I

I

i I

I

I

I i i

I

!

I

I

i

I 1 I

I

-< .-.

I

1

I

I

I i I

i I I 1 i

i

i I

i I i ! I I

!

I

i

!I-

'I- o .-I 1 1 1

I

W z c 3 0 a m 3 VI

f iu c v) F m

'LI

...

. e ,a L cn I

r 7

4

I

I I

i 1

I r

i

I

I

I

! I

i i I I

! !

i i i

I

!

I

i

i

i i I I

I

4 I

1 I

!

I I

I

I 1

k+.O +

* .

-b U

.

3"

. .. I

N u I UA U

d

0, e

Q

I

(.( u I

W 0 J J v\ U

0

0 - * * I 0 - I - I

I :? A +------*--+-- I

? I

+ - I 0 - I

I I I e . I

I I I 0 . I I

0 . I . I * I * I I I

i - I I

e - l

. - 0

0 -

f a - .

8

m ‘0 - n

I I I I I I I I I I I

!

I O n’ od Y

.E .‘o Q

I

1 3

I ! t

. I I e . I

i --- I I I I

0 l

e . + . . 0 9 -

.- - 0 -

I 1 - I * + e - - I - *: I I I I I I I

!

+ 4b * - . * * *

N 0 I

L U 9 (c Pl N ..(

------ --- I e I I I

i I I I

I t I I I I

I I

I I I I I 1 - I I I I + I

I . I I

I I

e 1 I

I t

I 1 - - --------- +-- -

I I I

I I I I I I I I

I- -. c 0

Y

I N * . 0 I *

I

W z L I

I e . c 7 0 a m 3

N 0

I Y) 9

I

I I

- 0 0 - 0 - * 0

0 m d

E W c v) > vl

#

? I

Y) *

* u

I

e . * * * * 0 L ’ 0 0 0 0 I l l 1 1 1

. . * . * * * . . * * * e . O O C O U U O O O O O O O O l 1 1 l 1 1 1 l 1 1 1 1 1 1

I

I i f i

I

I * !

I

. s

I

, I

1 I

I

I

I

! ! I

i

--I1 I

I

i

I I

I ' ! , , ..-... ._ , r-i-

! !

L.

---

I

I ; I

I

I i

i 1

I 1

i

i I !

! I I I

1

I I

i

i

f

i I

i

I

I i I i i I 4

w 2 ....

Y) 0 a d VI d

0

0

0 0

N 0. N

0 I

I

.L 0

W 0 (P P I @

0 8

c U c C -J h

I

UI

5

f UJ c. J)

in . T a cn e

I

I i

!

I

A I I . I

w

I n

c kt.0

I

!

-bJ I

,a a

.

-----

I

I I

i

i I I I I I

' I

d I 0

!

W

I

!

1

I

I

!

i

1 $

I

I

i I

i

!

I

I

! i ! I I

!

I

!

!

!

I i

t

1 !

I

!

I

!

!

1

i

c :- c U J 0

I

w Y

c ;z 0

8 3 v)

I w c v) > Y)

CI

a

? L v)

#

I

I

! ! I

i

1

I

! I

I i I

!

(

i i

I I

i i 1 I I i I

i i

I i i

1

I

i

'I I

!

! 1

I i

I ; i

i !

+

I I-

i : i

I

I

' I

i I

i i

I

! I J

,

I

I i j

! I I

i

I

f i I i

i

t I

[

I !

I

i

i

I

I i

i

!

I !

I ,

I

! i

! i I

I

I

! : I I I I I

I

i

i I

1 t

I i I

i

I i ! I

I

I

I I

I I

I

I I I

I I I

I I

I

! : i 2 =it

G' 05

j

I

I I

!

j

!

I

f I

I

I i

I I I !

i I I

I

i i

!

I !

I i i I

f

1

I

t

i 1

! I

0 u

1 !

I i

i

j !

I

I I

I ! I

i

!

i i

A

d L

L, 03

I2

ti

I

L W c YI * wl

I

- . i

I I I I I I I I I I I I I

G +

+ I I I I I I

I I

I I I I I I I

I + I I I I I I

I I I I I I I I I 1 I I + I

I I

0 u

I I I

i o I U

. . . I I I I N

V

c L. c 0

t d a 1

W z t- a 0

m 3 Y,

- a

f Lu c in F Y,

e 3 L

.t m -4

e 0

U U

V V

0 V

e 0 I

V U

*a e u I

e v)

?

c c c.

' 3 0

I

w z .I

c 7 0 e m 3 v)

z UJ c m P VI . ? z

II % *N" g:

I

I

e rs a

e &a

c c C .J

I

UJ I I- 2 0 a m 3 v)

S W c VI * v)

c.

a

I

e

? a. v)

0

N 0 I

i I I I I I I I I I I

I I I

t I I

I

I 4

* I I

+4 I I I I I

I i I I

I I

f I

I I I I I I I

I i I I I 1 I I

I il

if I

3 1 I I I I I I

. (.

4

I I I I I I I

i

- w 0 CF In 0

(*\ 0 b Y 0 F Q) N n

e 0 I

N W I

W m q, W h)

Q I

r*

I

N 0 I I& h

b 0

m 0 I

w V IC c3 W

.n 0 I UJ u 0 yc a3

0 I

m

.

. - * 0

* 0 * --

I I I I

e * I - . * - 0 - + - -

.I - ,* I . * e - I * - I I I I I I I I I I

m u I

LU I I I + I I I I I I

V m 0 h

. - - .

-- - A _ - - I I

a2

V I

b - i . I

- 1 * - e I * I *

. I I **.+

I- 3 0 U al 3 YI

L u1 e VI a- YI .

m r( . c I

N N 0 0 I I 'U w m e h N

m m h C I

U 0 I

' 11 0 0 IC c 'r

F. 0

m? m 0 0 - I l l

u1 w ul 000

m

m l * 0 0 I 1 IU tLI 00

0 I u c, m o

I 1.) 0 m

I Ill 0

.u 0

u rr N ?

1. ri N . m

#-

c

-I h

I

W z c 2 0 a m a

CI

r U

- v)

r u c m > In

* * * 3

5

ul

m 0 I w 0 J

W 9

0 .

-a 0 I w 0 0 \- rn (u

0 .

4 I 4 4

m 4 0 - I t

* * 4 - e

0 4 * 4 - 4

I

I I I

I-' * I

4 1 - 1

I 4 * + -- ob. 0 4

. 0 -

- * e * - 0 0 - - . . I

e * I * I * *

* I * I

0 - 1

I 4

-b I * z

n -

* *

* * * P ' * .e

--E II

c

o i I I

I I I

I I I

r y l U I I I UJI 3 1

m I - 4 1

0 1 1 1

I I I I 1 I 1 I

N I 0 1 I I

W l 9 1 d 4

I

- 4

< l

: I I I I 1 4 I I I I I I I I I

I I I

* 4

- 0 .

2 . U , 9 - I * . -w-

d

go I d

W - H

-=c

i I I I I

I

I I

I * I I I I

I I I I I

+ -

!

t

.*

* 1

I I

*! * -

4 I I I I I I I I I I

I I I n 1

0

0 I

b 0 I

N 0 I ." c.

e I 1

I I 1 I I I I I I I I I I I I

4

c

.

3 z

e V I

tu 0 I tu 0 * 0 0

0 ?

I !

0 * Q 0 e 0 0

I

8 I I I I I I I I I I + -- .e -

I cs -

n' .c .! U I

w-

+ c) 0 I P) 0 v * w

. 00 I

3 c r .. .

I I I I I I I

I I I I I I I + I I

I I 0 I

t

I

w 0 V

# I

h

c3 w L

I----- .- r

e . 0 I 0

0 e e e - e -

I I I B I I I I I I I I I

- e- % - 1

I I I I l

e e

r. u I

Y)

. . e * - .

0

t

I I I I

0 I

I I I I I IC. I O

I + I '

* . e

r -

e . r

I -

N 0 I

W 0 II) N 9 .. 0

I

e

0 -- -(I * - --a

t Y) t uf * m

? B

YI

N 0 I Y, z c N P . 0

NN 0 0 I I

W W U O hh -pi InY Uh * * 0 0

N 0 I w 0 )c

rn 0 l

U h m N

0

0'

mm 0 0 I 1

W U I 0 0 o m turn e m * -

m e 0 0 I I

NN 0 0 I t

V I W O C o m * * O E -hl

0 0 I 1

* e

NN 0 0 f l

U l W 0 0 o m m m N D m F 0 0 l l

b .

N N N 000 I l l

W W W 0 0 0 e m *

O C E

0 0 0 I l l

z : z . e .

d 0

u1 m IC N N

0 I

e . LL c 0

a . 0 0 I I

* . 0 0 I I

t

0 .

? ? H

U I

c I )

L. U a a r' u) L r c 3 0 a

L'

3 H

91 - H

H . . a

5 H

I-

e .t *e

.I e l

0 - I

l ?? f f l d l

n

N l r , - 9 0 O U N I .N -

. a N(Y

I W e 1 - * 8 . I O * . I ' : t

. I . 8 .

i I I I I I I I I I I

- - . r - * I I I (I 4 1 I I I I

e . . . . I I I I I I I I I I I

i I I I * e

- o r * - -i - - - - e

I

I I I

I I I I I

I I I

* * - 0 0

I

t

5

I

, .r .* 0 9 --,e-------- I . * * * e * e . . - -

I .* . I

I I I I I I I I I I I I I I

I I ' * a - 0 -

I I

I I I k

. 0 t

.s ,- a

00 I U

0 -$ I

CI 0

- 0 -- e * I

I I

- i - 1 * I

0 1 @ I

I I k I I I . I I

0 1 U I

I W l V I w . . . * * e * I 9 1 - 1 * I

I 4 I I I I I

71

b .

I *#

* - B * * *

e - - - e * *

t * .--------- I I

B e - * * - - * . 0 -

1 - 1 '

e - I, B .* 0 - .---

e * - - - . . . . . .

I

I I

I I

m 0 I w m

1 I I I I I I l I I

I I I I I I I

! * b

* -- -P

II 0 I

w m 0 0 m

I

I I I I I I

.--+------ i

!

L cs .

I w- C

2 .c c

- 0

h

u U

0 z

I I

* * 0 - e

I. * b *

W I w c h J V J

0 I

I - - I I I I I I I I I I I I t I I I I I I +

Y W rrr b 0 I

,

s i - * - I - I e * * - I

0 1 0 * - 0 . ) n' . I

I * - I - * I * - I a - I - a I * * I * - I * I * *

m l u t * * I I 0 -

W I - * * Irr ---- (P * * - I * N * - O* * I :

e - * .

'D k' U

I I I I I I

I I I I I I I I I

I I I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I I

..

I

t . i c I s 'f,

I . - I t I

- 0

I .. - i I I I I I I I I I !

I . * I 0 I I . * I I I

I I I I I I I I

0 * I

t I I

I I I I I I I I

i 1 1 I I

* I * I

* I * I

: I

: t * I * I

I . I

c 0 J

I

a

bu 2 c

f w c v) L v)

. 3 L

cn

0 1 I I I I 1 I I I

N I 0 1

I L Y I O # V I (PI NI W l . I

0 1 I I I I I I I

N I V I

I L Y I = I aJ. C T I * I * I

- 1 0 1

I I I I I

. I I

d l 0 1

I

;

I I I I I I I I I I

G +

I 74-

n I

??

C v) .-

In 0 u

S Y u a .*-

' I I I I I

m O

W c N 9 0 N

0

rn u w In 4 CD e d

0

n 0

Y m 9 V .t d 8

0

N 0

YI Q 0 W N 4 P

0 8

N V

Iu 0 PI K P U V 8

0

r)

0

lu 0 N m

N u N V

If- # * I O I t I .

I I I I I

I I I 4 I I I I I I I I 1 I I

I I t I l

. N 0

ut m I I I I I I I I I I I I I I I I I

9

0 e

nl 0

e .. I

I I I I I I 1 I -

l=tQ t

I O

I t I I I 1 I

InJ I U I I 8

I I 8 I 1 I I I I I I I I 8 I I I I I

I I I I I I I I I I I I I

I I

m I 0 \

b -d

ut yr

' U N P m

0

9 d- I

6 . I

w- .@ u

C .- .

-+ c d I z .?: Y

f

I

W L I

W z e

-- - . - .~ *-*

REPRODUCIBIIITY OF THE ORIGINAL PAGE IS P O O R

4 0

w

0

0 I

0 V

0 W

c c. c u -J 0

I

W z c 3 0 LL al 3 v)

L

L W c v * Y)

5 L

v) .

m 0

w P- - A + C n I - 1 - I . I

V I I I I I I I I I I

n t U l

I W I

' I n I - 1 . I

V I I I I

$ 4 I I I I I I I I I

I

I I I I I I I I I I

i n I - I . I U I I I I I 4 I

I I I

I I

I I I I I I I I

f

t

I I I -

I I

i G I * 1 0

I I I

I I 1 I N I V

I I

n 4 I

I

& a

i I I i I

I . -. I I I !

I

. -.

' - 9

1.

i I ci

I w- I I

I(Y I - -----* p)

I f - I O i m

i I

H 0 %

$ 1 h

I . ! v I I I I I I I

I I I I

* I I I

IC Lu c

. e 1.

Y)

- ..-4

*: T - . . r .c.

r. 0

m 0

0 n 3

C.! u

+

0 I

m 0 WI

0 I

0 U

i I I 8 I I I I I I I I

U

I 1 I I 8 I I I I I I I

n 0

t u 1 -

I O I I I I I I I I I 1

::

I

I I I I I I I I I I I 8 8 I

i I I I I I I I I I 1 I I I

I I

e -0 N’ c ._

\v,

N- e

(CI I

\ O

I 4 .

t Y

I I I I d I U

I . n Y -=c -4

I

I

I I I 1

I I I

I .* *?4

- * I - 1 I I n Nl 0 1 I I I O t l I I I 1

I I I I I I I I I I I I I I I I I ’ I I I 1 I 1 I I I I I

’I I I I - 1 0 I 1 w

V I I O

U U

* t t I :r

0 i n

1 - I b I O I I

I I I I 1 I 1 1 - : -