Selection of Phase Sensitive Detectors for Space Radar-cEK

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Selection of Phase Sensitive Detectorsfor Space RadarW. J. WILLIAMS, MEMBER, IEEE

Summary-The phase-locked loop has been suggested as apossible replacement for the banks of contiguous filters used toprocess Doppler signals.Two basic systems are considered, differing only in the type ofphase sensitive detector used. Th e two types of phase sensitivedetectors are the sinusoidal type and the sawtooth type.Th e sawtooth-type phase sensitive detector is shown to besuperior in several respects. Th e sawtooth-type detector is supenorwith respect to noise rejection, acquisition time and system designconsiderations.A phase-locked loop system using the sawtooth-type phasesensitive detector is shown to meet the requirements fo r a typicalsituation.INTRODUCTION

W[HE PURPOSE of this paper is to select a phasej comparison technique which exhibits the mostdesirable characteristics for space sensor applica-tions.Contiguous filters are usually necessary for processingpulse Doppler radar signals. However, for most applica-tions, these filters are undesirable because a typicalbank of filters may weigh up to 50 pounds and occupy1000 cubic inches.For this reason, the phase-locked loop has generatedconsiderable interest. The phase-locked loop is aphasesensitive device which has the property of behavinglike a tunable adaptive filter. It tracks a given signal,but admits noise only in a small bandwidth. Once thesignal has been acquired, the phase-locked loop has avery good tracking capability even in strong noise.However, acquisition over the entire range of Doppleruncertainty is difficult for the phase-locked loop unlesssome type of frequency searching scheme is employed.Even with the searching problem, the phase-locked looplooks attractive in situations where lightweight andsmall dimensions are at a premium such as in spacesensor applications.PHASE-LOCKED LooP OPERATIONThe phase-locked loop diagram is presented in Fig. 1.Phases of the input signal and the output signal arecompared in the phase sensitive detector. At the outputof the phase sensitive detector is a voltage that is somefunction of the phase difference between the two signals.

Manuscript received by June 11, 1964. This paper was presentedat the IEEE Region Six Conference, Salt Lake CitY, Utah, April30, 1964.The author is with th e Electronics and Space Division, EmersonElectric, St. Louis, Mo.

This output is filtered by the low-pass filter H(s) andapplied to the voltage controlled oscillator (VCO). TheVCO frequency is a linear function of control voltage.Any change in signal frequency will cause a correspond-in g change in VCO frequency, thereby tending to keepthe input and output in synchronism.EQUIVALENT SYSTEM

In this case, the Fig. 1 system can be replaced by theequivalent system in Fig. 2. It has been shown that theequivalent noise may be injected just before the low-pass filter. In the no-signal case, A is zero, so that thefrequency of the VCO is moved about in some randommanner by the noise. If the signal is of sufficient ampli-tude, the system will function as a feedback controlsystem and will be able to lock on to the phase of thesignal. The equivalent system is linear except fo r thef(e) box, and resembles a type II servo.

sin tisSt+ 0 )+n(l )PHASE f (e)+n*( t) l;72s+l )SENSITIVSE nkH(s} T Is+IDETECTORf

cos ut

VOLTAGE CONTROLLEUOSCI LLATORh O+Kvoe = ir-0O=(z5-


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Williams: Phase Sensitive Detectors for Space Radartwo signals. The two signals might be

A sin (w,t + 0) (1)B co s wt .

The difference term resulting by multiplying the twosignals isAR2 sin [(co-)t +O] (2)

wheree = [(co.-w)t + 0] .

The quantity is the instantaneous phase differencebetween the two signals. However, this type of phasecomparator presents two difficulties: 1) the output isproportional to signal strength as well as being sensitiveto signal phase and 2) the gain is dependent on phaseoffset fo r any system using this type of comparison.This gain dependence is undesirable because systemdesign is optimum for a set gain. However, to reducegain dependence, an AGC control is required, which inturn adds complexity.The sawtooth characteristic is not common but maybe obtained in several ways. The signal and referencemay have a sawtooth shape, thus giving a sawtoothcharacteristic t o t he ir phase difference. The gain in thiscase is not phase sensitive, but remains sensitive toamplitude variations. An alternate method of generatingthe suggested sawtooth characteristic is to use the zero-crossing information obtained from two signals, toswitch a flip-flop in such a manner that the integral ofits output is a quasi-linear or sawtooth function of phasedifference. This type of comparator is not gain sensitiveto phase or signal amplitude. The system gain is notdependent on phase offset or signal amplitude in thiscase. The zero-crossing information may be obtained ina forward part of the loop, but the model remains thesame fo r the linear portion of the comparator char-acteristic.

correlation intervals, M is7 /2 \MPBL = 1 - ( Pn(4)d4)

PB L = 1 - P.(O)d+-X

(sinusoidal) (3)(sawtooth) (4 )

where Pn(4) is the distribution of the phase noise. Re-sults may be obtained if Pn(s) is assumed to be Gaussianin distribution with a zero mean and a variance whichis dependent on the input noise. By the approximationfor large X,1 rx ~~~~1-f(x) = g(y)dy < e-1/2x2xz xV27r (5)and the binominal expansion, the expressions for break-lock probabilities become.

4oM4PBL < -/2 (7r/2of)2BL .-- --r-\/2r2(JMPBL < e-1/2(r/a)2rV2ir

(sinusoidal case)(sawtooth case).

(6)(7)

The correlation period may be approximated as thereciprocal of the noise bandwidth. Table I presents theresults for an M of one and some selected noise powers.As noted, the probability of break-lock is much smallerfor the sawtooth comparator than for the sinusoidalcomparator for equivalent noise power.The phase noise variance may be found as a functionof input noise.2= B,2SINo (8)where

B, , is the system noise bandwidthS/No is the input signal power to noise density ratio.Rey2 shows that for the case of the sinusoidal compara-tor the system noise bandwidth is

THE BREAK-LOCK PROBLEMThe system is assumed to be locked to the phase ofthe return signal from the target. The target can belost in two ways: 1) the phase change of the returnsignal can exceed the tracking rate of the phase-lockedloop, or 2) noise can cause a break-lock condition byexceeding the positive slope range of the phase com-parator. The loop will follow any signal whose band-width is less than the loop bandwidth. Widening theloop bandwidth admits more noise which increases theprobability of break-lock due to noise. Fig. 3 illustratesthat break-lock will occur if the magnitude of e exceeds7r/2 in the case of the sinusoidal comparator, or 7r inthe case of the sawtooth comparator. Break-lock prob-ability fo r the two cases, fo r a particular number of

Bn=Kt Sec .1 + co s (e)KtT22IT, cps2 1 + KtT2 co s (e )whereK,=KA/2 in this case.

Bryne3 shows that for the sawtooth comparatorKo 1 + KoT22/T cpsBn = 2 1 + KoT2

(9)

(10)I E. Parzen, "Modern Probability Theory and Its Applications,"John Wiley and Sons, Inc., New York, N. Y; 1960.2 T. J. Rey, "Automatic phase control: theory and design,"PROC. IRE, vol. 48, pp. 1760-1771; October, 1960.3 C. J. Byrne, "Properties and design of the phase-controlledoscillator with a sawtooth comparator," Bell Sys. Tech. J., vol. 41,pp. 559-602; March, 1962.

231

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232 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS DecemberOUTPUT

[. UNLOCK

I7at iF At

(a)

fseize = flock TTo insure a good possibility of lock, the search mustplace the initial conditions within a positive lock areawhich is constrained by the seize frequency and thediscontinuities. The phase plane is a good vehicle fo rdiscussion of this problem. Since frequency difference isthe derivative of phase difference, the use of the phaseplane is convenient fo r this application.Fig. 4 illustrates the situation fo r sinusoidal and saw-tooth phase-locked loops. In order fo r locking to occur,a search trajectory must pass through one of the areasproscribed by the seize frequency and the discontinuitylimits. If the search is a ramp of frequency, the searchtrajectories will be parabolas. The phase difference insuch a case will be

_ UNLOCK

e = at2 + eOso that

(b)Fig. 3-Sinusoidal and sawtooth phase comparator characteristics.(a) Sinusoidal phase comparator. (b) Sawtooth phase comparator.

e'= 2at.This is a parametric representation of

TABLE IBREAK-LOCK PROBABILITY FOR ONE CORRELATION PERIOD

Noise Power (a') Probability of Break-LockVolts Squared Sinusoidal Comp. Sawtooth Comp.

1.OXt10 5. X 10 - 7 5. OX 10-12811.OX10-2 3.7X10-57 3.5X10-2281 .OX10-l 1.8X10-7 8.0X10-271.0 1.4X10-1 3.5X 10-7SEARCH AND ACQUISITION

Some type of search is required to obtain initial lock.Initial lock is necessary to assure positive locking in areasonable length of time. It is possible that in thenoise-free case a phase-locked loop may lock for a ratherlarge frequency offset. In the locking process, many dis-continuities may be passed, and therefore the circuitmay flicker for a long while before locking. Locking isvery unlikely when the frequency offset is large and thesignal-to-noise ratio is small. If the offset frequency isless than the seize frequency, locking will be quiteprobable even in noise. The seize frequency is themaximum mistuning or offset that cannot cause a dis-continuity to be crossed in the locking action.The seize frequency has been found to have thefollowing relationship to the maximum lock frequencyfor the systems considered here3' 4

4 D. Richman, "Color-carrier reference phase synchronizationaccuracy in NTSC color television," PROC. IRE, vol. 42, pp. 106-133;January, 1954.

(13)+ eO = e.4aThe frequency may be discretely searched in a linearfashion, in which case the search trajectories appear as inFig. 5.The search trajectory should pass through a lockpoint as it passes through the locking area. Since theinitial phase difference is unknown, the search trajec-tory must accommodate all possible initial phase differ-

ences so that the trajectory passes through a lockingarea for any value of phase difference. The rapid searchtrajectories may easily miss the scattered positive lockareas of the sinusoidal case so that slow search isessential. The sawtooth comparator produces a con-tinuous corridor, however, which cannot be missed.The discrete search is more attractive than the con-tinuous search since it can be digital in nature. Thevoltage controlled oscillator is simply advanced in fre-quency in discrete steps with the locus of frequency be-ing a ramp function. If the system locks onto the signalduring one of the steps, the stepping sequence is inter-rupted and the system then is free to follow the signalover the entire range of frequency and phase.The time required to search the frequency range ofinterest may be found from the search trajectory. If in(13), eo is considered to be zero,

e/2= e.4a (13a)

77; 2

UNLOCK

(11)

/

UNLOCK

(12)

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Williams: Phase Sensitive Detectors for Space RadarThe constant may be evaluated for the desired trajec-tory by choosing a typical point on the trajectory. Thus,

FAST SEARCH 2(HIT) Ws__FST SEARC (AISS) a = - . (14)0SLO SEARCH-27T ir -7ff0'; 2ff/2 57 27r L The time fo r the trajectory to pass through the area of

LOC| LOCK AREA interest is 8w WT =-zL de'. (15)(a) The time required to search the entire frequency range is

8T,T's ~~~~~~~(16FAST SEARCH T2W s(17)(H T) _ FASTC ASEARCH/ / e~~~~~~~~~HIT)ince

LOCK AREA -7r | (A s( = _max (17)'4 ~~~~~~and \ maX= 7r BW (18)8 T~2(b) T(8= (- ) (19)Fig. 4-Four search trajectories and locking areas with continuous BW 2ramp searching. (a ) Sinusoidal case. (b ) Sawtooth case. The phase-lock loop system must lock during the timethat the search trajectory is in the lock area. There mustthen be sufficient time to determine if locking hasoccurred so that searching may be stopped.The slow search necessary for positive lock in the caseof the sinusoidal comparator is not practical for typical

situations. The search time fo r a typical set of parame-ters will be in the order of several seconds, when it mustl/2 2 l l | | | tS _ be fractions of a second. A rapid search has a probabilityWs T


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234 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS December6T1tsK= *(22)1 + KT2

To clarify the situation let us consider an example.Suppose that it is necessary to acquire in 0. 2 sec, in abandwidth of 20 kc. Further, the signal-to-noise densityis 1.8 X 103. To detect satisfactorily, a filter bank of 100filters (each with a 200-cycle bandwidth) would benecessary with 8 db taken as the minimum signal-to-noise r at io f or detection at a filter.The following selection of parameters is presented asbeing satisfactory fo r the situation using the sawtoothcomparator system.T2/T1=0.04K=2rX104T2=2X10-4T, 5X10-3T search 0. 2 secFrequencystep rate 130 steps/sec.

These values produce with the equations previouslyderived, Bn= 1256 cpsPBL < 1.26 X 10-7/correlation periodPBL < 1.58 X 10 4/secTime to lock 1.9 X 10-' sec.

These values are well within the realm of possibility. Itis interesting to consider that the search time requiredfor the sinusoidal system fo r positive lock will be8 /T1 2Ts B ( ) = 1.0 sec.BW KT2JCONCLUSIONIt appears from the results that the phase-lock loopsystem using the sawtooth comparator based on zero-crossing information is considerably superior to thesinusoidal comparator system both in break-lock proba-bility and acquisition time. A suggested system for aphase-locked system with search is shown in Fig. 6.

Fig. 6 Phase-locked loop system with search.

A Unified Approach to the Error Analysis ofAugmented Dynamically Exact Inertial

Navigation SystemsSAMUEL L. FAGIN, SENIOR MEMBER, IEEE

Summary-A model fo r pure inertial navigation systems is de-fined which describes a wide class of dynamically exact systems forterrestrial and space applications. Based on this model a standarderror block diagram is described and it is shown how the conventionalsimplifying assumptions lead to the specific version used fo r slowmoving terrestrial applications.To efficiently treat augmented inertial systems, a generalizedapproach to error analysis of estimation systems is introduced. Thisapproach, discussed in an earlier paper, is amplified as it pertains toaugmented inertial systems. It is shown how the standard errorManuscript received August 14, 1964. This paper was presentedat the 1964 \Western Electronic Show and Convention, Los Angeles,Calif.The author is with the Inertial Division, Sperry Gyroscope Com-pany Division of Sperry Rand Corporation, Great Neck, N. Y.

propagation block diagram can be used to place the augmented in -ertial navigation system error analysis in the canonical form neces-sary for a generalized error analysis. Augmentations considered areposition, velocity and attitude fixes. Specific examples are given.I. INTRODUCTION

I N A RECENT paper,1 it was pointed out that,assuming optimal use of redundant data, a wideclass of "estimation systems" could be convenientlystudied employing a generalized digital computer im-1 S. L. Fagin, "Recursive Linear Regression Theory, OptimalFilter Theory, and Error Analyses of Optimal Systems," presentedat the 1964 IEEE International Conv., New York, N. Y., March23-26.

Selection of Phase Sensitive Detectors for Space Radar-cEK

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