Phase locked-loop tracking with sine wave modulation in Apollo communication system

Lee, S. Y.

Nelson, L. D.

English

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7ECNNICAL
MEMORANDUM	
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TRACKING ANALYSIS OF A FIRST ORDER
PHASE-LOCKED LOOP WITH T'NO
SINEWAVES MODULATION
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BELLCOMM. INC.
945 L'4NPINT PLAZA NORTH, S.W., WASHINGTON, D.C. 20024
COVER SHEET FOR TECHNICAL MEMORANDUM
TITLE .- Tracking Analysis of a First Order	 TM- 71-1033-1
Phase-Locked Loop with Two Sinewaves
Modulation	 DATE-January 21, 1971
FILING CASE NO(S)— 103-6
AUTH001IM— S. Y. Lee
FI LINO SUBJECTM-
	
Communication Systems
	 L. D. Nelson
(ASSIGNED BY AUTHORM1- Phase Locked-Loop Tracking
AUSTRACT
This paper gives a tracking anal;, ,- 5is of a first order
phase-locked loop with two sinewaves modulation. By assuming the
fluctuation about the d-c value of the phase error (not necessarily
the total error) to be small, a linearized model is obtained. A
periodic solution of this model is then derived for the output
phase of the first order loop. It is shown that this periodic
solution is unique. Using this periodic solution, analytical
tracking behaviors of the output'phase of this model are obtained.
Computer plots of the phase error for some typical parameters
used in Apollo communication systems are included.
in
SEE REVERSE SIDE FOR DISTRIBUTION LIST
`	 BELLCOMM. INC.
955 VENFANT PLAZA NORTH, S.W. 	 WASHINGTON, 0. C. 20024
N
.`	 SUBJECT: Tracking Analysis of a First Order	 DATE: January 21, 1971
Phase-Locked Loop with Two Sinewaves
Modulation - Case 103-6 	 FROM; S. Y. Lee
L. D. Nelson
TM-71-1033-1
TECHNICAL MEMORANDUM
I. INTRODUCTION
Basically, a phase-locked loop is an electronic
servomechanism that operates as a coherent detector by con-
tinuously correcting the frequency of its local oscillator
according to a measurement of the error between the phase of
.he incoming signal and that of its local oscillator. The
simplest form of loop is shown in Figure 1. The precise rela-
tionship between the input and error signals is a nonlinear
integro-differential equation from which very little informa-
tion concerning the tracking behavior is analytically available.
With the assumption of the total loop phase error being small,
Viterbi 2 and others obtained a mathematically equivalent
linearized model for the analysis of a single sinewave modula-
tion. The purpose of this memorandum is to develop a method
of analysis for two sinewaves modulation, and to derive
analytically its tracking behavior using a linearized model.
Input Signal	 Error Signal
cos ^ i	K0 sin
V
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BELLCOMM, INC. 	 - 2 -
II. BACKGROUND
General Equation for the Phase Error of the Phase-Locked Loop
The basic operation of the phase-looked loop given in
Figure 1 is described in References Ill and [2]. Furthermore, in
these same references, a general nonlinear differential equation
which gives the precise relationship between the input phase
signal o f at point (1) and the phase error 0 at point (2) of the
a	 phase-locked loop is derived. In an operational notation this
equation is
SO + KF(s)sino = so  - sw 0t	 (1)
or equivalently
$ + KF(s)sino = $ i	 w0	 (2)
where s = ^ , K = K 0K1K2 is defined as loop gain, 0 denotes the
phase 0(t) is the multiplier output (i.e., the error signal),
K1F(s) is the linear transfer function of the loop filter, w0
is the center frequency of the voltage controlled oscillator (VCO),
K0
 is the phase-detector gain factor, and K2 is the gain of the
VCO. By using the result of (1) we shall proceed to derive the
effects of two sinewaves modulation on the tracking behaviors of
the phase-locked loop.
III. THE PHASE-LOCKED LOOP WITH TWO SINEWAVES MO DULATION
The differential equation for the phase-locked loop
preceded by a limiter for the case of two sinewaves modulation
can be obtained as follows: Let the input signal at point 1
be represented by cos w c t + a cos(w c + wd)t, where a < 1, and
we and w  are the singular frequencies. Then the resultant input
signal can be derived from the vectorial addition of the unmodulated
carriers as shown in Figure 2,
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y^	 y w
	
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't	 deoS^t
lL ^ ° X	 ^	 1
1
I
a `^^	 I	 ^
wit	
^
Figure 2 - Vector Addition of the Unmodulated Carriers
2	 _1 asinwdt
cosw ct + acos (wc+wd) t = 	 cos (wct+tan l
+acoswdt^	 (3)
The unit gain limiter removes the amplitude variations.
Therefore, the input signal ecomes
as in w t
cos ^ i = cos [w ct + tan -1 ^-+acosw 
d ^j	
(4)
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Hence by substituting for 0 1 from (4) and defining
assn w t
e(t) 8 =tan- 1
 l+acosw £	 (5)d
the general equation for the phase error of the phase-locked
loop with two sinewaves modulation becomes
$ + KF (s) sink = we - w O + 9	 (6 )
Instead of solving (6) as is, we first obtain 6 as an infinite
series. Expanding (5),
e (t) _	 (_1) n+1 nn sin n wdt	 (7 )
n=1
and differentiating term by term, the series obtained is uniformly
convergent and, thus,
w a (a+cos w t)
8 (t) = d	 d = wd F. (_1) n+lancosnwdt	 (8)
l+a +2acoswdt	 n[=1
Substituting (8) into ( 6) yields
sBELLCOMM, INC.	 - 5 -
_
a + KF(s)sino = we - w0 + wd 	(-l)n+lancosnwdt 	(9)
n=1
Assume the solution of (9)o 0, has the form of a
constant term plus a fluctuating term such that
0 _ 00 + 10(t) •	 (10)
Then (9) and (10) yield
^ + KF(s) [sino 0coso + coso 0sinfl _
W 	 1)n+lancosnwdt + we - w 0 	(11)
n=1
Assuming the fluctuations about the do value of the phase error
are small (this is generally true; see Section V) we have
W
^ l + KF (s) [sin^ 0+^ lcoso 0 ] = W  E ( _1) n+lancosnwdt + we - w 0	 (12)
n=1
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For the case of the first-order loop F(s) = 1 and
equation ( 12) reduces to
^ 1 + K[sin^ 0 + ^ 1coso 0 ] = we - w 0 + wd 
4 
(_1) n+1
ancosnwdt	 (13)
n==1
By noting that (13) is a linear differential equation in ^ l (t)
and ;(t) is of period LE , the steady state solution of (13) must
d
also have period w= . Thus, assume a steady state series solution
d
for 1 (t) to be of the farm of
a
[Ancosnwdt + Bnsin nwdt]	 (14)
n=1
Then
00
^, 1 (t) - ^ 1 = W  Fw n [Bncosnwdt - Ansin nwdt]	 (15)
n=1
Substituting (14) and (15) into (13) yields
00
w  
Z 
n[Bncosnwdt - Ansin nwdt] + KsinO o + w0 we
n=1
CO	 cc	 (16)
+ Kcos^o 
L 
[Ancosnwdt + Bnsin nwdt] = w  
L 
(_1 )n+lancosnwdt
n=1
	
n=1
it	 V7 w
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By taking the integral of (16) from 0 to 2w it is seen that
d
Ksin^ O + wO - we = 0 .	 (17)
Equating coefficients of cos nwdt and sin nwdt, for each n there
are two equations in two unknowns,
Y
j
•A
i^
-nwdAn + KcosO OBn = 0
KcosO OAn + nwdBn = (-l)n+lanwd
Solving for An and Bn,
A = (_1)n+lanb	
B	
(-1)n+Iann	 (18)
n	 b + n	 n
where
KcosO0
.	 b	 (19I
d
Thus substituting into (14) and the analog of (10)
yields the complete solution for the linearized model of the phase
error of the first order phase-locked loop with he effect of two
sinewaves modulation,
ancos (nw t+a )
(-1) n+l	 d	 (22)
n=l	 [b2 + n 2 )
—nw,
ancos nwdt + tan-1 ., Q
n+l	
^	
(23)
[(Kcoso 0 ) 2
 + (nwd)2]
solution 4^(t,t0 ) of the homogeneous
^ ,Kcos O 0
=a
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BELLCOMM. INC.	 - 8 -
r
(_1) n+lan (bcosnwdt + nsin nwdt)(t)	 ^ 0 + ^	 ( 20 )
	
n=1	 b + n
where b is given in (19) and from (17) ,
^ 0 = sin-1 
w 
c_ 0
	 (21)
-nw
Let a = tan-1 (b) tan-1 Kc-- ^-- and (20) can be written as0
rBFLLCOMM. INC.	 9 -
- (t-t0)Kcoso0
0 (t,t0 )	 e
The 00 is "small" (less than ff in absolute value) in practice=
hence, cost  # 0. The period of 6 being 271
d 
then forces
m(= , 0) - 1 # 0
d
Equation (12) thus satisfies the conditions of a theorem in (7) and
on page 363 in [6) which implies that the solution (20) is unique.
The preceding series (20) (or ( 22)) is uniformly convergent
for 0 s a < 1 and, hence, is the Fourier series of the function
t i (t). Thus, the output phase error for the first order loop with
two sinewaves modulation is a periodic fluctuation of zero average
value about the phase error that would be present if there were only
one sinewave modulation. It should be noted that if the fluctuation
due to two sinewaves modulation were not assumed small, then the
average effect would not be expected to be zero since the nonlinear
sine function would weigh positive and negative fluctuation unevenly.
IV. LIMITING BEHAVIOR OF THE OUTPUT PHASE EQUATION
By examining ( 23) , it can be seen that there are three
interesting special cases. These are:
Case A - When the input signal is one sinewave then the
phase error is 0 0 , which is derived in prior
literature.
Case B - As expected, when K is infinite which corresponds
to the ideal phase-locked loop (i.e., tracking
the ^ i = w ct + e(t) exactly) then the phase errors
O 1' 00, and ^ 1 (0 are zero.
Case C - When w  becomes large relative to Kcoso 0 such that
1ti	 —2	 (24)(nwd) + K cos 
^0	 (nwd)
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BELLCOMM, INC.	 - 10 -
then the solution of (23) yields
01 ti ^0 + 0 - Kcos^ o	edt
	 (wd2>>K2 Cos 2 0 0 )	 (25)
by noting that
(-l),+ 1 an sin nwdt
n=1
and
edt = - r ( _ 1) n+1 _ a cosnwdt
n=1	 n	 wd
V. COMPUTER PLOTS OF THE PHASE ERROR FOR SOME TYPICAL PARAMETERS
USED IN APOLLO COMMUNICATION SYSTEM
The tracking behaviors of the first order loop can be
seen by considering (23) when the system parameters are known.
For Apollo communication systems, we know K. is approximately equal
to 10 6 and wd is approximately equal to 10 3 . Therefore from (19),
b y 10 3 . Substituting this value into (23) and noting that the
contribution of 
^0 
only corresponds to a d-c shift, we can determine
the behavior of the phase error ^ l (t) by plotting ^1(t) versus w 
d 
t
using "a" as a parameter. Examples of these iplots and others for
different values of b follow. As a "goodness' cr.1terion for this
linear model we must examine sin* l- * l l < e for all values of t,
where a is an arbitrarily assigned small number. For the examples,
suppose we let e = . 01, then only two cases do not satisfy the
goodness criterion. These cases are where (b=5, a=.9) and (b=10,
a=.9); (sing-x, 1 1 attains the values . 16 and .05 respectively.
It should be noted that for the linear model derived in this memo-
randum the assumption of sin* 1 * is less restrictive than if the
assumption is sino ti 0
e
0	 1.00 2.00	 3.00	 4.00
6. 00 wd t
-.002
-.004
-.006
-.008
i
i
a =	 0.1
3 = 0.3
a =	 0.5
a =	 0.7
lII
1).uu
, b1 (t)
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BELLCOMM, INC.	 - 11 -
( 1 )
	
b = 1000
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(2) b = 100
Vl(t)
-.020'
A
•4 ,+M -.040'
;t
Yk
-.060
-.060
,o^ -Mwme=
I
i
a = 0.1
a = 0.3
a = 0.5
a = 0.7
a = 0.9^---
0	 1.00	 2.00	 3.00	 4.00	 5.00	 6.00 wdt
1.000 2.00	 3.00	 4.00	 5.00	 6.00	 wdt
-.300
r
—.400
:+s y
—.500
—.600
—.700
F
*Y
.t
^,^ ltt)
-.100
-.200'
>t'	 1
BELLCOMM, INC.	 — 13 —
(3) b = 10
0-0.20
'e
-0.40
} -0.60
tjl
i
-0.80
0	 1.00	 2.00	 3.00	 4.ou	 D.vv
	 d.uu wdt
BELLCOMM, INC.	 - 14 -
(4) b = 5
y;
-1.00
BELLCOMM. INC.	 15 -
ACKNOWLEDGMENT
The authors would like to thank Mr. S. C. Chu for
his helpful suggestions. The able programming assistance of
Mr. H. R. Singers is gratefully acknowledged. Thanks are due
to Department 2034 for their comments of this paper.
1
	 ,/ • 4.
S. Y. Lee
02;"0010
1033-LDN 7f 	L. D. Nelson
Attachment
References
1
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BELLCOMM. INC.
REFERENCES
1. Gardner, F. M., Phase-Lock Techniques, John Wiley 6 Sons, Inc.,
New York, 1966.
2. Viterbi, A. J., "Acquisition and Tracking Behavior of Phase-
Locked Loops", Symposium  on Active Networks and Feedback
Systems, Polytechnic Inst tute o Broo yn, Apr 	 =-F  1960,
pp 583-618.
3. Schuchman, L., "Time to Cycle Slip in First and Second Order
Phase-Lock Loops", Bellcomm Technical Memorandum, TM-69-2034-8,
December 19, 1969.
4. Van Trees, H. L., "Functional Techniques for Analysis of the
Nonlinear Behavior of Phase-Locked Loops", presented at
WESCON, San Francisco, California, August 20-23, 1963.
5. Tausworthe, R. C., "Theory and Practical Design of Phase-Locked
Receivers", NASA Technical Report No. 32-819, Vol. 1,
February 15, 1966.
6. Hahn, W., Stability of Motion,tion, Springer-Verlag Book Company,
New York, 3.^.
7. Fearnsides, J. J. and Chu, S. C., "A Minimum-Fuel Mode for
Steady-State Attitude Control of an Earth-Orbiting Spacecraft",
IEEE Trans. on Auto. Control, Vol. AC-15, June 1970, pp. 362-365.
8. Panter, P. F., Modulation Noise and Spectral Analysis, McGraw-
Hill Book Company, New Yor"rk,1965.
I
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Tracking analysis of a first order phase- locked loop with two sinewaves modulation

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