FDM demodulation characteristics of nonlinear phase locked loop

Carden, F. F.

Hintz, T. B.

Kelly, L. R.

English

NASA Technical Reports Server

General Disclaimer 
One or more of the Following Statements may affect this Document 
 
 This document has been reproduced from the best copy furnished by the 
organizational source. It is being released in the interest of making available as 
much information as possible. 
 
 This document may contain data, which exceeds the sheet parameters. It was 
furnished in this condition by the organizational source and is the best copy 
available. 
 
 This document may contain tone-on-tone or color graphs, charts and/or pictures, 
which have been reproduced in black and white. 
 
 This document is paginated as submitted by the original source. 
 
 Portions of this document are not fully legible due to the historical nature of some 
of the material. However, it is the best reproduction available from the original 
submission. 
 
 
 
 
 
 
 
Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19690004204 2020-03-12T08:03:32+00:00Z
XI
A
-11, 2
X 1
-12.1
y,
•
	 w
T119 FbM 1a W)VULAT1h'C CHAW.701STICS
UY
NON—LINVAR, P11ASK-1 4CK40 LOOPS
Frank F, Cordon
Lawruncu K, Kelly
Thos At Flints
Commcfnicatlon Rosoarch Croup
llectriC41 Engineering Wpartmunt
New Mexico ItAte University
Lai Cruces, New Mexico
Aber
.,.. * t r t
The projections of the state space canto the
various hypo r-planv* are used to datermina the
minimum noise bandwidth and maximum information
bandwidth for noel ino. «Ir phaso -locked loc.ps used
to demodulate FM multiplexed subcarriers,
Introduction
In many space-communications F.4 systems,
where rsccivers must operate Ill low signal to
nol*e environments, Lite pilaw-Lock loop (I'LL) is
utilized because of its thrdshold extending capa-
bilities, Since threshold is determine by the
loop bandwidth (ca) for A given ing,ut signal to
noise ratio, it is desirable to have a minimum
loop bandwidth, yet process the modulation in an
acceptable manner, The constraints on w for a
minimum of modulation distortion and fornlinelar
operation of the loop have been determined for
single sine-rave demodulation (l ) zJ. Those con-
straints require w to be much larger than var-
ious s!,gnal params ' -ers; however, how much larger
is not indiccted and a minimum w cannot be de-
t+erminad from the constraints, Rince in certain
application it is desirable to exchange signal
fidelity for a smaller w and hence, alower
threshold, it is desirabla to know the minimun wn,
This paper is concerned with the problem of de-
termining the minimum loop bandwidth before Loop
TIME
unlock occurs bucuuau of Yli mwItlpluxud su4c4r-
rivrs, In particular, thu prublur,, of dur.<rdtalAt-
Ins, a widuband video signal And oovvr4l aruLcer-
riur* is considurad, The luwur bvi4W on w to der-
tarmined As A function of ?h0dul,4tiurc lnd1Pvu, mod-
ulating froquancles and puAk frvquuncy duviaation•,
This work consider* the prub vrl of c,at4bII*b1ng
A boundary on tiro paramutar region that Insurers
locked operation of the PU, Wpunding upon the
signal to noise environment, the optimum w will
be in the region between a largo w nuodadn for 4
minimum of signal distortion and tfle OM411 wn be-
low which loop unlor.k will occur.
StateElu, tiong
In order to determine the minimum w before
the loop breaks lock, that nonlinear differential
equation of the bsseband 1'U model was tormulated
in terms of state variables, A lone filter of
the form F(p) w 1 + a/p was used, The resulting
state variable equations are (3).
X1	 0	 1	 X1	 0	 0
^► 	 +
X2	 0	 »2;cos 311 K2	 -sinxi 411	 tl)
Figure 1, State space projection showing a 	 Figure 2, State space projection showing an
properly locked PLL.	 unlocked PLL,
N69 s* 13 5 36_
IACCK>f;*I/Yf/ NU	 I'll)
	
IYM11U1
ow
MAGA CN on TMx on AD NUMo *ml► 	 IaiAT 
011yl
	
r^^
+	 P	
..	 fxt r	 )_j	 f q
AL
	
{^ . 4. g + i"i^'1 ^ if ^	 T1^ f t=
11L
f.l
X2
-1,8
Xt
Figure )r Plt,a♦ ee plane showing A properly locked
I'LL,
where
X  a instantraneous phase error
X 1 a instantaneous frequency error
vi (t) a input phase duet to M modulation
In equation (1), w was normalixed to unity:
Digital computer algorihms were used to obtain
solutions to equation (1) and to obtain projec-
tions of the state space onto the various hyper-
planes, Analog computer simulations wore also
used to determine a minimum w , Inspaction of
the •tats projections indicatV when the loop is
skipping cy,:los, For example figure I mhows X
versus time, for a PLL properly lockoi onto thi
modulation, Figure 2 shows A PLL unlocked and
skipping, cycles because w is too small, Figures
3 ono 4 are the pro,)ectiofls of the state space on
the X versus X piano, These correspond to
figuris 1 and 2 i respectively ,and may also be used
to determine if tit# loop breaks Jock or not, If
the trajectories are contours Around rnw(N a 0)2,
w, : ^*} then the PLL is iockod and tracking the
modulation, Trajectories continuously increas-
ing, such as figure k, indicate cycle skipping,
If it is assumed that the PLL is initially
locked onto the carrier and that the subcarriers
Are applied at t a 0, the initial conditions +era
given by
X1 {0*) " - E
	
Awl,
  cos	 (2)iml W 	 i
K
X1 (0*} 
a 1,01
4wisin(ed -24s1nX(0*),	 (3)
Equations (2) and (3) Are the result of letting
0(0 of equation (1) represent, that multiplexed
subcarriaxrs and expressing 0(t) as
N !!w.0(t) " -1I1
	cos (wit+61 )	 (4)
i
Figure 4,, Phase plane ► !,owing An unlocked PLL.
Varying 0 of each rubcAirri •` r shrews variolA
combinatihs of rinuauida to bu urud to duturnino
tlta minimum w for the worst caara, It might be
expacted that n 0 #^ eq ual to aithur "/1 or to sero
would give the Aorst case since thu m,,xin.un. rate
of frequency change occurs puriodicaily tor
0 .: "t/2 and that peak phase deviation occura parr
odically for 0 n 0, Thin was found to bu true,
and further ) tit* minimum w was approxlmatuly the
same for 0 a tt/2 or xero, n `bite initial conditions
do dstermthe what singular point the stable con-
tours will enclose, +For example ) if V a U, equa-
tion (2) becomas X(0 ) .: -Z 4W1WI and If the mod-
ulation parameters Ara such that the loop locks
quickly, then the singular paint, nearest the
value of the sum of the modulation indices will
be enclosed in the contour, Figure 3 show 's this
care for a single subcarrier,
Pajamate 'r legions
That minimum w was determined At a func-
tion of peak frequency deviations, modulation in-
dicer► and the subcsrrter frequenc y r, for one,
two And three subcarriers, The minimum w for a
given set of signal or modulation paramours was
determined rasing the criterion of cycle skipping,
In general, the condition X a n/2 doer not re-
sult in the lrop breaking tuck and hence, cannot
be used as to equivalent criterion, The condi-
tion X x T42 is a sufficient condition for un-
lock iA a restricted paMramrtor .ogi,ong however,
For example, in the single subcatrrier came the
condition Xl k 7t/2 doer cause the loop to un-
lock if the parameters Are such that tho in-
equality w s .Sw is satisfied. lluwovor, in the
parameter Toaion c*arined by .6-# A j:  w S w ins tan-tanoous phase errors graaa ly in :tx'480 9f */a can
occur without the loop brooking lock. This is
Analogue to the response of a PLL to a frequency
stop with a magnitude close to the value of the
se,partix, With A frequency step of this sixat the
trAjec,tory in the phase plane will oncloue that
paint it/2 by a considerable margin, ror the
multi-subcarrier case pAraacmot+ar regions Also ox-
tot where the condition X k I`/2 is a sufficient
condition for unlock= however, tha pt regions are
not as well, def'i'ned,
i
31
	
I 
11 
41
gi
r101 * a
A
,sa COMPUTER RESULTS
POINTS WKtk9 #,, PEAK 4 V/1
REFERENCE [4)
«..«. AW , WM of
0
	
	
110
WM/WA
Figuro 5. Partitioned parameter apace for single oubcarrier,
ror the single subcarrior, figure 5 shows
the minimum - as a function of Aw, w and U, the
modulation inDox. That is, the paramlter space
is divided into two regions. For all values of
Z.,, w in t1w region below the smooth continuous
curve locked operation of the PLL will result, if
the parameters are selected from the region be-
low the curve indicated in figure 5 1 then phase
errors in *%coos of nl2 will not occur. The re-
gion below the Smooth curve of figure 5 agrees
closely with region defined by the inequality
Gw •w S w1 for w < .5w . This in a condition for
locked operationW doval,pod in an oarlior paper
(3). Locked: operation can also occur in the re-
gion above the smooth curve and therefore, it
appears that the smooth curve dofineo a narrow
region of instability. Therefore, the signal and
system parameters should beat choson such that this
region is avoided, in particular, the parameters
of a F!i video signal and the parameters of the
demodulating PLL should be sol octod from the to.
gion below the continuous curve.
Another recent raper [ 4 1, assuming X 2 it/2
canoes unlock, derived an expression for the mini-
mum w using describing function. After norms-
lixinj, w  to unity the e xpression becomes.
wW
	
..r12 1W	 ..j w
m max	 (	 wm	 m
This equation &grass very closely with the points
of ;Figure 5 that indicstc :1 - it/2.
The conditions on the parameters for a
amnia= of distortion are t102j,
wn
-m 4< 1
	
(5.11)
Wn
Those conditions are satisfied by the pararwtuto
In the cross hatched region of figure 5. "'Lis
assumes ttao much lairs than requirement is oatis-
Pied by ratios 'less than .l. Thar** inequa'ities
require too largo an w for low signal to noise,
ration. For operation nof a PLL in a low 4ijn4l
to noise environment the optimum w should be in
the region bounded above by the omBoth cvrvo de-
fining, the absolute minimum w and bounded below
by the region defined by the Inequalities of
equation (5)•
Tips boundaries For the parameter region in-
suring lock for two and three subcarri,cars or*
shown in figure 6 t 71 8 and p ruspoctivoly. The
modulation indices for the family of curves are
plotted inversely along than w and w axis.
l?arsmetors #elected from the 1paca below the
smooch curves will giva locked PLL operation.
For a carrier tracking PLL the parameters should
be sol*cted from the region far above the smooth
Qurvsl however, this region was not investigated
in this paper. The region in the vicinity of the
smooth curve should be avoided. In particular,
if one of the We repre#tnts the highest fro-
,0
^k
r
32
r
40,
0 " A(a2 -N-S
1.5 LW,2 . 1.2
3.2 ^ 2.4 q 1.6 1 10
0
Y
Awl Awl v I
4.4A.4 - 4.., .6
Awl s AW2
 
n 1.21
Aw, A Awl 1.6
Awl Awl 2,
Or	 ,	 I
JL I I
W2 W2
AX	
VW2
W	
74
2	
1-2/W2 
Aw/w,
2
1.
4:	 2-/W2
F4,ury 6# Partitioned psrAmot#r opmes for twq subc&rriors,
1.24 .6 4 1.4 W2 	 1
.L. I 1 .1 L	
I I -	 -R.- I
W2 'W9 w9
2.
I
1.5
-
1 10 1. 1	 9. /W
4,
rigure 7. P&VtitiOnOd PArA'I*ttr Opus for two submriers,
33
Irs W— 
to 1	 r,4`
.
w3 $0
t
a ^ A
4!A
W3 X .6
W^
A^
Irk Wl
Filura 8, parti.ti,onod parmoter spec• for thraa aubmirriora.
4.	
9. wl WI
^;Lgura` 9. parti,ti,onad paramatsr apaco for thr•a aubcarriora!
A
f	 t
;q,^+,^ x a^,
	 #,	
r	 °	
^^ ^
	
-psi ^^ ^^"^^is^	 •'^.^.,^_	 ^,.n^2.a..,u	 3a	 -.^>	 ^^^fi	 .^.r-.. _,^E^. ,.^..
r...._	
_	
.....	 .
q... ,i.y prvatc ot lit 4 vt•IVO atl,yN.it tit+it 1..n itewik
ttv,iuts n. y brstlt.t.atval .ftkta7 u ." Afiivt' sakt., . t+.tttl
Witi ► uvtivt .il au^t.arrltirai ti,ttk ,i:t ii.t .►+u t "ia altutilal
by aJlat . ta j [ravok tuts tv 1. ► *t Ir.'larkif ti.v a p •,at tit .'tai`lftie
iurtt,vt' t ti ►.s tttitla, 160i
"
	*L ' ) U lal be ilk th at vitt"'tuta
atba9ve iliv %: to** 1 ► .tta'i'vab
^s}  
 
te4tvit it i'ritltgtynr +4latut*
tlttn alkoms In llt;ury '1i for tl ►v aln j,:v au1;a.lrri,isr
or for that lilghoot trs+yuvtlty lit a vittvo a41C1141,
tt	
Thal boutittlary Of t ►►a ytu#'.k ;.:vtvt rv,llun tar
it)t,' vd opgrattattk 4r411, 14Ntttr11-t11W'4 4114 It rvi;tatt ► for
4 rslntmunt vi 4laitt,rttatt wto tai ►att,nt tlt.tt rvyutrud
14raty ^ , Yor late tita1`141 101111141 to 1101110 on-
Vlraitltt►alritV tit$ Opt itt:um w 411111,1141 be a t1l uctod bo"
low tit# ialn mot wn bounAry and above the low
distortion rosiorl,
t4t,t1 941".ti ',
,ti,rA,me., its 1'. i, iutvtnt, '10 Y,, t .tr:.ii, ;l.a
i9a r. i.t..t av t.+ra •..1 : ,,,,1,
i.t4'4416u144 Wvvatt,;.,t1^1.1,. jr hr„ui, y„ 1' , .iy-
tv4'11111a *'1mtltut y Itulrurt 1► I'$u:ikl *11 1 *G"e,
.1a,t ►v l"WIS
wv.Avor, t. '.i., "A >vil J't1,l,tuarai, t,r ti.v
Littvat tual,",t a,a4 &i4iYalat of ki,aa.:-
l.arti,v4 l.uul►at, ta 1 1tt. iron* u t1 1 .i a v i iv t:,*
t #a,ti^+ ,rtr . 1111.1 ^i^ utt+u #
 ^U.^ a ill^.^^" ♦ lt$i^ii•
IIM, txp cv^t,t9t l`^^'t,
t:attivtt, 1', V, t LU wy, ti. wi, tU14 bVtrkatin, G,,
"DW t2u401 -Ut4tlat ► ,tty and Tr4nal ent g4-
isavlur of huit* l.ltial4r k'l,asp*L19+Cii O LUoptt t
Sw it"cu ltucord 1vol,
	 944,
Ttilta work wav supported by NAM
trarit MR-32-003-07
r
"rank Carden received his PhD from Oklahomaa , d 
" Mv"rlr ,lty in 190o He in ourrentV
twhinS And doin,l research in oommunioetion
theory, At Now Maxioa State University At
Crump New Moxtoo
35
R	
r	 ^{ f k j	 j A
La, 144..t4r* 	A.b 	 ^R ^tl-_ .^.A, $.:	 s3[^A	 .mss# ._,	 .


The FDM demodulating characteristics of non-linear phase-locked loops

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server