50 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

Null-Tracking Doppler-Navigation Radar*

P. G. SMITHt, SENIOR MEMBER, IRE

Summary-A Doppler navigation radar is described which canreduce sea-bias errors to a negligible degree. The technique also re-duces all other errors which result from asymmetrical distortion ofthe echo spectrum. Spectrum distortion is caused by modulationeffects (altitude holes), by angle-sensitive reflectivity of certain typesof terrain, and by asymmetric receiver responses. Error reduction isachieved by the use of monopulse-type antenna patterns, which aredirected toward the earth fore and aft of the aircraft velocity vector.The Doppler frequencies corresponding to the null planes of the splitbeams are tracked. Because the positions of the null planes relativeto the aircraft velocity vector are dependent only upon the antennaconfiguration, and not upon outside factors beyond the control of theradar designer, the system is practically insensitive to the class oferrors listed above.

A particular system configuration illustrating the null-trackingprinciple is described. Initial acquisition of the ground track is madeby a conventional lobe-tracking system. Rapid switchover betweennull tracking and conventional tracking is provided in order to permitpicking the best mode for the particular conditions encountered.

The added complexity required by the null-tracking feature de-pends somewhat upon the antenna design, but does not appear pro-hibitive for either lens antennas or planar arrays.

I. INTRODUCTIONj 9 OPPLER systems presently in operation use

antennas which generate configurations of pen-cil-beam patterns. For the measurement of three

independenit velocity components, at least three non-planar beams must be used. The advantages and dis-advantages of the many possible configuratioins havebeen well covered in the literature, as have the possibletypes of antennas, modulation waveforms, and fre-quency trackers for these pencil-beam systems.'-'

Pencil-beam Doppler radars suffer from a class oferrors which arise from the lack of well-defined beamaxes. For antennas of reasonable dimensions and for thefrequencies used in this application, beamwidths are inthe range of three to six degrees. The distance accuracyrequirements of Doppler navigation systems fall be-tween 0.1 per cent and 1.0 per cent. This accuracy rangerequires that the relative angles between the antennabeams be defined within 0.02° to 0.20. There are severalfactors which make this accurate definition difficult,and in some cases these factors cause errors whichgreatly exceed the desired accuracy. For example, sea-bias errors are of the order of 3 to 6 per cent for the

* Received November 12, 1962.t Sperry Rand Research Center, Sudbury, Mass.1 F. B. Berger, "The nature of Doppler velocity measurement,"

IRE TRANSACTIONS ON AERONAUTICAL AND NAVIGATIONAL ELEC-TRONICS, vol. ANE-4, pp. 103-112; September, 1957.

2 F. B. Berger, "The design of airborne Doppler velocity measur-ing svstems," IRE TRANSACTIONS OF AERONAUTICAL AND NAVIGA-TIONAL ELECTRONICS, vol. ANE-4, pp. 157-175; December, 1957.

3 W. R. Fried, "Principles and performance analysis of Dopplernavigation systems," IRE TRANSACTIONS ON AERONAUTICAL ANDNAVIGATIONAL ELECTRONICS, vol. ANE-4, pp. 176-196; December,1957.

range of beamwidths given above. A simple sea-bias cor-rection can reduce these errors by a factor of approxi-mately one half, givinig over-all velocity errors in theorder of 1 per cent, depending upon the areas of radaroperation.

Factors which complicate beam-axis definition in-clude antenna beam asymmmetries, sea-bias, range ef-fects, altitude holes, receiver-response asymmetries, aindtracker-response asymmetries. These factors cause dif-ficulty in two ways. First, the initial determination ofbeam axis presents a difficult measurement problem.For appreciable amounts of beam asymmetry, the defi-nition of beam axis even becomes difficult. Grounid cali-bration systems are extremely costly and the accuracyof existing systems may not be adequate for the mostaccurate navigation systems. Second, even if the iniitialcalibration is adequate, some of the above factors pro-duce varying effects depending upon the aircraft alti-tude, aircraft velocity, and land or sea coniditions.These varying effects are always difficult to compen-sate and, in some cases, accurate compensation is notfeasible.

After an accuracy study of the B-58 Doppler radar in1958, the author proposed a system which appears toalleviate many of the difficulties described above. Ananalytical study was made to evaluate the possibilitiesof the proposed system. This paper reports the resultsof the study.The proposed system employs split-beam receiving

antennas in order to establish well-defined planes inwhich the gain is essentially zero and the responise slopeis very high. Assuming proper orientation of the nullplanes relative to the velocity vector, this system hasthe theoretical capability of removing all errors whichresult from spreading or skewing of the echo spectrum.The actual amount of improvement to be realized de-pends upoIn a number of factors which will be describedbelow. The most promising applicationi of the techniqueis for high-velocity systems which require highlyaccurate Doppler navigation. If drift angles of the air-craft are to exceed approximately 10° to 150, the anteninamust be stabilized to the ground track in order to rea-lize maximum benefit from the null-tracking technique.Aircraft roll and pitch cause no difficulty when occur-ring separately. However, there is a cross-couplingeffect between roll and pitch which is expected to be-come serious for angles exceeding 200.One configuration using the null-tracking technique is

described in Section II. This section includes a descrip-tion of beam configurations, RF systems, and the fre-quency tracker. Capabilities of the null-tracking system

March

51Smith: Null- Tracking Doppler-Navigation Radar

relative to conventional lobe-tracking systems are

described in Section III.

II. DESCRIPTION OF SYSTENI

A. Over-all System Operation

One adaptation of the null-tracking technique is

shown by the block diagram in Fig. 1. This system uses

the FM\ CW technique, which permits single-apertureoperation for simultaneous transmission and reception.4

DI1RECTION AL I O AD A

KLYSTRONCOUPLERS

L T

XG IDEBANDE_MODULATOR l.MXR MIXER

LATWORK +Ea CF E

CIRCUIT,~~ ~ ~ MDE| EOCTSUMMING~ ~ ~ ~ ~ ~ ~INO FREC UNCYFO.

NETWORKSOURCES CEFDTAPR CESSO

RON FREQUEN~~~~~~~CYROSS-TRA,GROU DE VELOCITY

Fig. 1 Null-tracking Doppler radar, block diagramN.

The use of the third sideband component in the auto-

mixer output permits operation over an altitude range

from near zero to the maximum which is limited by the

system signal-to-noise ratio.5 The system illustrated in

Fig. 1 employs time-shared components to the greatest

possible extent, in order to simplify the equipment and

to reduce its size and weight.

The use of null tracking is not limited to systems

which employ the FMICW technique nor to those which

employ extensive time sharing. An analysis has shown

that the performance of systemls emuploying null track-

ing degrades with large drift angles, unless the antenna

is stabilized along ground track. For antennas which

are not ground-track stabilized, drift angles of 100 can

be tolerated rather easily, while the performance will

become unacceptable for drift angles between 150 and

20°. Also, it will become obvious that pre-detection

Janus mixing cannot be used in nulll trackers, because

this mixing would destroy the spectrum null prior to

the null tracker. These are apparently the only basic

4K. C. M. Glegg, "A low noise CXV Doppler technique," Proc.

Nati. Conf. on Aeronlautical Electronics, Dayton, Ohio, May 12-14,1958; pp. 133-144.

5 The AN/APN-118 Doppler navigation radar developed bySperry Phoenix Co. employs the FMCW technique, operating on a

third sideband. Experimental results show that this system can

operate from take-off to touchdowxn, although the zero-altitude "hole"greatly reduces the signal strength at low altitudes.

Iimitations of the null-tracking technique.The principle of null tracking will be illustrated by

reference to Figs. 1 and 2. It will be assumed that thesystem is azimuth-stabilized along the ground track,because of the results of a study which is described inPart B of this section. Roll and pitch stabilization ofthe antenna are optional. The two side beams [labeledz in Fig. 2(b) ] are used to azimuth-stabilize the forwardand aft antenna beams to ground track. This stabiliza-tion is accomplished by tracking the Doppler signals inthe two side beams and requiring that the Dopplershifts in these beams be equal. This function could beaccomplished by one beam alone, if it is oriented nor-mal to the fore and aft beams and required to track thezero-Doppler isodop. However, two beams offer anadvantage for the case where the antenna is not roll andpitch stabilized (Section II-B, below).The forward and aft beams transmit and receive on

lobe patterns in the same manner as the sidebeams. Inaddition, they receive on split patterns, labeled A inFigs. 2(b) and 2(c). These latter patterns are obtainedby special design of the forward and aft antenna feeds.Antenna configurations for accomplishing this resultare described in Section Il-C. A significant character-istic of the A pattern is that a sharp 1800 phase shiftoccurs in the null plane. The importance of this featurewill be recognized during the discussion of trackeroperationi in Sectioni IJ-D.

Referring to Fig. 1, the four transmit antenna beamsare illuminated sequentially by a klystron oscillator.The proper modulation to the oscillator is supplied bythe modulator. This unit also applies beam power andAFC corrections to the klystron. The beam modulatorcauses the klystron to be frequency modulated at a rateof fm, which is the basis for the low-noise techniquedescribed in Glegg.4 In order to reduce altitude "hole"effects, fm is wobbulated, i.e., frequency modulated at arelatively low rate.The frequency sources generate all other coherent

signals which are required in the system. Since the sys-tem uses third sideband, the local oscillator signal isgenerated by offsetting the klystron output by thedesired IF, fIF, pIUS 3fm.

Isolation of the transmitter and receiver are providedby the duplexer, which may take any one of severalpossible forms. Sequential illumination of the four trans-mit beams is provided by the RF switch, which may alsotake any one of several possible types.The receiver channel labeled 2 (commonly referred

to as the sum channel in monopulse radars) receives thesame form of spectrum as do lobe-tracking radars.After mixing and IF amplification, the signals are passedthrough the CEF (carrier elimination filter), whereresidual carrier leakage is rejected by a narrow-bandsynchronous filter6. The z signal is then fed into the fre-

6 P. G. Smith, "Leakage rejection in beam-switched CWradars," IRE TRANS. ON AEROSPACE AND NAVIGATIONAL ELEC-TRONICS, vol. ANE-9, pp. 241-250; December, 1962.

1963

52 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

(a)

x

(b)

. FIELDSTREN GTH

- i_ UANGLE OFFVERTICAL

(c)

Fig. 2 (a) Beam geometry. (b) Beam intersection with ground.Contours represent approximate 3 db equi-intensity lines. (c)Forward and aft antenna patterns vs angle off vertical, Y-Z planecut.

quency tracker, which contains four trackiing oscillatorscorresponding to each of the four anitenna beams. Al-though these tracking oscillators share many commonchannel components, they track the Doppler frequen-cies at the four beam ceniters independently.The null-pattern anitenna terminals are obtained

from microwave junctions between the antenna feedsand the RF switch. As shown in Fig. 1, the fore and aftA RF signials are switched into a common receiverchannel in synchronism with the switchinig of the fourz RF beams. This nmethod of switching permits usiingcommon channel componenits in the IF amplifiers, car-rier elimination filters, and the frequency tracker.The IF amplifier and CEF blocks of the A channel

are identical to those in the I channel. As indicated bythe single-frequency tracker in Fig. 1, the operationsof the z and A trackers are interdependent. When thefrequency tracker is in the null-tracking mode, theforward and aft oscillators track about the null pointsof the corresponding spectra. When the tracker is inthe lobe-tracking mode, these oscillators track theeffective center of the respective lobe-type spectra. Eachbeam has only one tracking oscillator, regardless of themode of operatioin. The selection of tracker mode,i.e.,I or A, is made iln the tracker and is based on lock-oninformation, and oni other factors which determine themost desirable mode of tracking; e.g., drift, roll, andpitch information play a role in determining trackingmode for systems employing nonstabilized antennas.Frequency tracker operation is described more thor-oughly in Section JJ-D and in Sectioin III below.

Velocity informationi to the data processor is pro-vided by the four tracker oscillators. The data processoris also fed informatioin on the mode of tracking in orderfor it to select the proper calibration factor. In all prob-ability these factors will be different for z and Aoperation, because of minor aintenna asymmetries. Fur-thermore, the requirement for sea-bias correction de-pends upon which tracking mode is being used.The data processor derives horizontal and vertical

velocity components from the fore anid aft trackingoscillator frequencies. The principle of this computationis straightforward and will not be described in thispaper. Azimuth stabilizationi about ground track is pro-vided by a servo which uses the cross-track velocityoutput as an error signal. Consequently this velocitycomponelnt is driven to zero. The circuits for computa-tion of ground-track velocity can provide both digitaland analog data. Digital inforination in the form ofpulse rates corresponding to ground-track velocity andvertical velocity is particularly easy to derive fromii thefore and aft tracking oscillators.

For very high speed aircraft it may not be necessaryto ground-track stabilize the antenna. If null trackingis used only during those times when the drift angle isnot excessive (say, less than 10-t15°), ground-trackstabilization is not required. In this case, cross-track

March

Smith: Null- Tracking Doppler-Navigation Radar

velocities can be obtained from the side-beam trackingoscillators of the frequency tracker.At this point an alternate method of accomplishing

ground-track stabilization is mentioned. Assume thatthe antenna produces only the fore and aft antennapatterns illustrated in Fig. 2. Once the null trackerlocks-on the A spectrum null, an azimuth servo systemwhich is designed to minimize the null signal level willcause the antenna to be grounid-track stabilized. Inorder to accomplish this form of stabilization, someform of low-amplitude azimuth yawing must be pro-duced in the fore and/or aft A patterns. Sensors of well-known design could be used to detect errors from thebest null conclition. This system is attractive from thestandpoint of simplifying the RF system and for realiz-ing a higher duty cycle for each beam. However, therequirement for maintaining a highly stable angle be-tween the fore and aft null planes discourages the useof small variable devices in the feed horns for producingthe A pattern yawing. Producing this motion by move-ment of the entire antenna is unattractive because ofthe relatively large inertia of this component. For thesereasons the two-feed system will not be consideredfurther. However, the performance results described inSection III apply to this system as well as to the oneillustrated in Figs. 1 and 2.

B. Beam ConfigurationSeveral antenna configurations are possible for the

null-tracking system. One obvious restriction is thatthe null plane of the A antenna pattern must be orientednearly parallel to the isodops, as shown in Fig. 2(b), inorder to produce a Doppler spectrum which has a sharpnull. As the null plane deviates from the parallel condi-tion, no isodop will exist from which zero return is ob-tained and degradation of the A null depth occurs.

In order to determine how seriously the spectrumnull degrades for deviations of the null plane from theisodops, computations of expected Doppler spectra havebeen made. These computations are too lengthy to beincluded in this paper, but the method is described inAppendix I and some of the assumptions and resultsare shown in Figs. 3-5 (next page). As illustrated inFig. 3(a), the aircraft is assumed to be in level flightwith a horizontal velocity Vh and drift angle d.Zero roll and pitch angles are assumed. The antenna isfixed to the aircraft with the null patterns pointingdownward at an angle y from the aircraft axis. The driftangle causes the null plane to cross the isodops [Fig.3(b) ]. Simple mathematical expressions for the I and Apatterns are assumed in order to facilitate the computa-tion. For convenience, the peaks of the A pattern areassumed to fall slightly outside the 3-db points of the Ipattern, as illustrated in Fig. 2. This condition approxi-mates the case for practical monopulse antennas.

Received power density spectra are computed for

described in Section II-A above, with the fore and aftantennas transmitting on z patterns and receiving onboth 2 and A patterns. In this case, the spectra in thez channel are relatively broad. The main points ofinterest are the depth and the steepness of the nulls inthe received spectra of the A channel. The sharpness ofthese nulls determines how accurately they can betracked, and how little the frequency tracker is affectedby sea bias and other factors which introduce asym-metries into received Doppler spectra. The second caseanalyzed is the one for which the A pattern is used forboth transmission and reception. The purpose of per-forming this computation was to check on the beliefthat a deeper and sharper null in the received spectrumwould be obtained for this combination.

Although the computations are somewhat laborious,they are straightforward and give fairly simple results.These results are plotted in Figs. 4 and 5. The fre-quency coordinate is normalized to the Doppler fre-quency corresponding to the longitudinal velocity V1.Thus, for beam centers tilted at 700 from the horizontal,the normalized null frequency is cos 700=0.342. The Aspectrum peaks occur at approximately the isodops cor-responding to the XA pattern peaks, which areseparated by approximately 1/V/2 of the A patternpeaks. The power density spectra are normalized to theupper frequency peaks, although for practical casesthere is little difference between the upper and lowerpeaks. The calculations do not include the effects of dif-ferential ranges to the portions of terrain contributingto the spectral increments. Nor do they include the cos,y variation of ground illumination intensity which isinherent in Rayleigh scattering from rough surfaces.These omissions are justified on the basis of their smalleffect for practical cases where the beamwidths arerelatively narrow. Sea-bias effects are not included, butthis may easily be done by multiplying the densityspectra by an appropriate sea-bias function, as is donein Section 111-A below.A study of the results in Figs. 4 and 5 reveals several

interesting facts. First, it becomes apparent that for anunstabilized antenna, drift angles in excess of 100 sig-nificantly degrade the null depth and sharpness. Conse-quently, null tracking is most effective for systemswhich employ ground-track stabilization. An alternatepossibility is to use a fixed antenna and operate on thenull-tracking mode only when the drift angle is lessthan 10° to 150. The criterion for allowable null degrada-tion is somewhat arbitrary. The accuracy of null track-ing depends upon the average slope of the A pattern inthe null vicinity over the filter balndwidth of the nulltracker. Drift angles of 10° to 150 would reduce thisaverage slope by a factor in the order of 2-4, dependingupon the filter bandwidth.As shown by (16), (18), (19), and (21) of Appendix I,

although the spectrum null degrades for nonzero valuestwo configurations. The first configuration is that one

531963

of drift angle, the frequency of the minimum still cor-

54 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

AIRCRAFT V VtAIRCRAFT aPOSITION HEADING

0--20~~2\ / -230

o FORWARD ANTENNA Z XIBEAM CENTER -0 ll1 _ 1111U0~~~~~~1

L0DB

I-~~~~~~ 0

ISODOP STRIP2

.320 .325 .330 .335 .340 .345 .350 .355 .360

FORWAR) COSy NORMALIZED(b) FREQUENCY, 77

Fig. 3-(a) Coordinate system. (b) Ground-plane geometry. Fg oxrdniyseta 2Asse.Aiuhbawdh r

approximately "a ;" elevation : beamwidths and A null-peakseparations are approximately ax.

responds to the correct frequency for the longitudinalvelocity component V1. Thus, small drift angles causeno error in the measurement of this component. .OODB--- 1 -

Although the case for A Transmit, A Receive does X---o - - - -give deeper nulls than the2~ Transmit, 2-A\ Receive con- 2 -201- - - _+

a-.

figuration, the null widths are significantly wider (coin- F -30 - - -pare Fig. 4 with Fig. 5). Another disadvantage of the L -40 2° Xt\(A-A) comnbination is that the tracker sensitivity is low > ° zfor this case, since the received signal iS very low in the z 0S /aa 2vicinity of the null. This difiMculty does not exist in the 0, -60 ||lj=o7 ttr70°0

(a)~~~~~~~~~~~~a

(, Z-A) configuration, where the z signal acts as a ¢ -70 - t}t-T- -coherent reference for the weak A signal. Therefore, it sL I- -is concluded that the (:, :-A) configuration is superiorto the (AX, A) configuration. _DI] [|||[iII11Assuming a fixed antenna, an exact anlalysis of the E 1 IfII1 i

effects of roll and pitch on the AS spectrum shape has3 2not been made. Howrever, some general statements can {L -30 - -+ - -be made. The presence of a pitch angle in the absence , -40 - --7< - _ __of roll and drift does not degrade the spectrum null, <, -50 1 200 // -I/Tsince the null plane remainls parallel to the isodops. Dw W 5;X/I |1 =a=60Sirmilarly, the presence of a roll angle alone does not a:7| 50U - \ \\r7O°degrade the null, since the null plane would follow the it /3:004 rX tisodops as the antenna rolls about the velocity vector. .320 .325 .330 .335 .340 .345 .350 .355 .360The presence of roll and/or pitch, in conjunction with 7oCS' NRAIEdrift, will further degrade the null over that for drift F~o:OyNRMALIEDCYalone. Also, the simultaneous presence of roll aind pitch Fig. 5-Power density spectra, A-A system. Azimtith beamwidth isangles degrades the null in the absence of drift. How- approximately "a," elevation A null-peak separation is a.

March

Smi'th: Null- Tracking Doppler-Navigation Radar

ever, this degradation is rather slow and does not ap-

pear to be serious until roll and pitch angles of the orderof 200 are encountered.The considerations above point to the fact that null

tracking is applicable to fixed antennas only duringtimes of low drift, pitch and roll angles. This limitationis of the order of 10° to 150 for drift alone or in combina-tion with small roll and pitch angles, e.g., less than 10°.Higher roll and pitch angles, of the order of 200, may betolerated for drift angles less than 100.

For antennas which are azimuth-stabilized aboutthe ground track, as described in Section II-A, rollor pitch stabilization is generally not necessary to avoidA spectrum degradation. In the presence of roll alone,all antenna beam centers simply follow their isodopsand neither tracking errors nor null degradation of theA spectrum would be observed. When a pitch angle ap-

pears alone, each of the sidebeams experiences equalchanges of Doppler shift. Consequently, the forwardand aft beams are held in the vertical plane containingthe velocity vector. The null planes of these beams falltangent to different isodops, but no null degradationoccurs. Under both roll and pitch conditions, the azi-muth stabilization will hold the two side beams on thesame isodop. The A pattern null planes will deviatefrom tangency with the isodops, and null degradationwill occur as described for the fixed antenna under driftconditions. However, this degradation will be small forroll and pitch angles less than 20°. The presence ofsimultaneous roll, pitch, and drift is difficult to analyze.However, it can be seen that the deviation of the A pat-tern null plane from tangency to the isodops has a func-tional behavior which has zero slope for zero roll, pitchand drift angles. Therefore, for ground-track stabilizedantennas, it is believed that no serious degradation ofthe A pattern will occur for combiniations of roll, pitchand drift angles up to about 200. Considerably highervalues of roll, pitch and drift angles can be toleratedwhen they occur separately. Furthermore, since very

rapid aiid smooth transitions between the null-trackingand lobe-tracking modes can be made, a combinationsystem need be no worse than a conventional systemunder any combinations of roll, pitch, and drift angles.

C. Microwave Configuration

There are a wide number of possible microwave sys-

tems which will provide the desired z Transmit, 1-AReceive system described in Section lI-A. However,this number has been reduced considerably by imposingthe following constraints upon the microwave system:

1) The system must be capable of both conventionaland null-tracking modes.

2) The system must have the best possible sensitivityover low sea states, where it provides its greatestbenefit.

3) The z and A Receive patterns must operate on thesame polarization.

The first requirement is imposed because of the need forconventional operation to provide initial lock-on, andfor tracking operation under certain conditions, e.g., forsystems with unstabilized antennas operating underhigh angles of roll, pitch and/or drift, as describedabove. The second requirement is imposed because ofthe fact that the signal strength is inherently low forlow sea states, when sea-bias errors are most serious.This requirement probably excludes the use of receiverswhich are cross-polarized relative to the transmitter,because of the high polarization loss which appears toexist under these conditions.7 The third requirement forthe two receiver channels to operate on the same polar-ization is imposed by the necessity for these twosignals to have a large degree of phase coherence.Because of the dependence of the A tracker operationupon this coherence, the I and A received signals shouldbe derived from the same polarization characteristics ofall ground targets. Thus, the I and A receive patternsshould have approximately the same polarization char-acteristics near the A null plane. Small polarization dif-ferences result in a loss in sensitivity rather than intracking errors. The system is not restricted to linearpolarization, nor do the I Transmit and I Receive pat-terns have to be identically polarized.

Fig. 6 shows a possible microwave system which fallswithin the constraints given above. The 2 and A pat-terns for two of the feed horns (labeled 1 and 3) areobtained by the use of a simple sidewall coupler asillustrated in the cross-sectional view. A dual switch isused for sequential switching between the four z pat-terns and the two A patterns. The A receiver obtainsinformation for only two of the four beam positions cor-responding to the transmitter excitation of horns 1 and3, while the z patterns are active for Transmit andReceive conditions in all four positions. The z duplexershown in Fig. 6 uses a junction containing a ferrite, andis described in Lax and Button.8 A turnstile RF switchis illustrated in Fig. 6. The shutter window provides anopening to one feed horn at a time. If the outer arm ofFeed 3 is 900 longer than the inner arm, excitation of thez input produces in-phase excitation to the two adjacentopenings of Feed 3. Excitation of the A arm would pro-duce out-of-phase excitation to the two openings. Thus,the z channel pattern is a pencil beam, while the Achannel pattern is of the split beam type illustrated inFig. 2(c). Not shown in Fig. 6 is the fact that the lowerpart of the shutter contains four windows to permit Aoutput feed-through.

Other possible arrangements for use with lens-typeantennas may be obtained by the use of alternate-switch and duplexer types. Ferrite switches are applica-

I Very little data is available to substantiate this statement.However, some theoretical and experimental evidence exists for thisbelief.

8 B. Lax and K. J. Button, "Electromagnetic properties of ferro-magnetics and their applications from UHF to millimeter waves,"Microwave J., vol. 11, pp. 49-56; November, 1960.

1963 55

56 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

FEED 2

SWITCH BODY

TRANSMITTER \ TO AMIXER

Fig. 6 Microwave configuration, lens feeds and switches.

ble, but no simple, lightweight configurations haveevolved from these studies.The null-tracking principle can also be applied to

planar array antennas. One possible configuration con-

sists of two identical pencil-beam arrays placed side byside along the fore-aft axis. Each array alone producesfour lobe patterns and is fed by four ports in a well-known manner. By combining the forward feeds andthe aft feeds of the two arrays in hybrid junctions, foreand aft 2 and A patterns are obtained. Side 2 patternsare obtained in a similar mnanner by combining the sidefeeds of the two arrays. This method of obtaining Apatterns is referred to as a "phase comparison" systemin monopulse radar terms. Although the A pattern ofphase comparison antennas generally have relativelyhigh sidelobes, the conventional lobe-tracking part ofthe system would prevent false lock-on or tracking atan inicorrect null. An anmplitude comparison array pre-

sents no serious design problems, but there appears tobe no great advantage of such a configuration relativeto the phase comparison system. Frequency compensa-tion methods have been devised for both types of ar-

rays, and appear to be somewhat more effective foramplitude comparison systems than for phase compari-son systems.

For FMCW systems, all microwave arrangementswhich meet the three requirements above appear tohave one problem in common, which is the difficulty ofachieving adequate transmitter-receiver isolation. In

order to achieve the required isolation (approximately40 db)9 the output arm of the duplexer, looking towardthe antenna, must be matched to a VSWR less than1.02. In addition to maintaining this match over allthe environmental conditions to be encountered, theduplexer itself must maintain at least 40-db isolationunder these conditions. Degradation of the over-allisolation to 35 db can probably be tolerated, providedthe IF amplifier can operate without appreciable satura-tion at the higher leakage level. Maintenance of thisrequired isolationi is believed to be feasible, even thoughit is difficult. It is perhaps worthwhile to point out thatthis required isolation value is not imposed by the null-tracking function, but is also a requirement of conven-tional FMCW systems. Actually, the addition of thenull-tracking function does not appear to aggravate theisolation problem materially.

D. Frequency Tracker OperationA simplified block diagram of a frequency tracker

incorporating null tracking is shown in Fig. 7. Signalinputs to this tracker are the I and A IF signals, eachof which corresponds to the particular beam in opera-tion during the time-sharing sequence of the four beams.These signals are fed into the four z mixers and the twoA mixers. In order to simplify the diagram, only theforward-beam tracking loop is shown. The presence ofthe other tracker channels is indicated by the multiple-position switches. Operation of the aft-beam trackingloop is identical to that of the forward-beam loop. Leftand right beam tracking is identical to the z trackingin the forward-beam loop.The frequency-tracking loops are closed by having

the appropriate tracking oscillators feed into the mixers,as shown in Fig. 7. Each of these oscillators is driven tothe correct frequency to center the z or A spectra at thecenter of the tracker discriminators. Multi-positionswitches distribute the appropriate mixer outputs intothe conmmoin channel of the tracker, which includes allblocks from the z and A amplifiers through the erroramplifiers. Possible switch types for this function, aswell as the other switch functions in Fig. 7, include com-mutators and diode circuits. Thus, the four 2 signalsare fed sequentially and in synchronism with the an-tenna switch, into the two I tracker filters. The topfilter has a bandwidth B1 which is slightly greater thanthe maximum spectrum width to be encountered. Thisfilter is used to provide conventional lobe-type tracking.The other z filter and the A filter have bandwidth B,iwhich is much smaller than Bl. The outputs of thesefilters are used to provide null tracking. Description ofnull tracking will be deferred until the functions per-formed by the other blocks are described.

This figure is based on the requirement that wide-band noisefrom the transmitter tube must not deteriorate the receiver-noisefigure significantly. The 40-db figure given above applies for trans-mitter power in the neighborhood of one watt.

March

Smith: Null- Tracking Doppler-Navigation Radar

Fig. 7 Combination lobe and null tracker, block diagram.

-- BIAS FUNCTION,+ bv

[A SPECTRUM]2

2 NORMALIZEDFREQUENCY, v

Fig. 8-Unbiased : and A spectra and bias function usedfor computation of bias errors.

In order to provide lock-on and tracking of the Dop-pler spectra, the four tracking oscillators are initiallyswept in frequency. This sweeping causes a velocity-search function to be performed. As an oscillator passes

through the correct frequency to cause the correspond-ing z spectrum to fall in the wide-band filter (band-width B1), the appropriate lock-on detector will observean increase in signal strength. Assuming this signalstrength exceeds a predetermined threshold, the oscil-lator sweep is stopped and spectrum tracking begins.The search control is performed through the integratorcircuit which controls the frequency of each trackingoscillator. Notice that the acquisition described aboveis performed independently for the four input 2 signals.

Frequency tracking of the I spectra is provided in a

conventional manner. The z signals are passed into a

limiter-discriminator combination which determines theamplitude and sense of each error signal. The I error

amplifier receives the four error signals sequentially and

in synchronism with the appropriate antenna beambeing energized. The four-position switch, which followsthis amplifier, distributes the error signal to the correctintegrator circuit. Closure of the tracking loops is pro-vided through the tracking oscillators, each of which isdriven to the required frequency to zero the associatederror signal.

Returning now to the A tracker, reference is made toFigs. 7 and 8. Both z and A signals from the narrow-band filters (Bn) are fed into a phase detector. By appro-priate adjustment of the relative phase shift through thez and A channels, each incremental portion of the Aspectrum on one side of its null can be brought intophase with the corresponding frequency increment ofthe z spectrum. Under this condition, each incrementalportion of the A spectrum on the opposite side of the Aspectrum null will be out-of-phase with the correspond-ing frequency increment of the z spectrum. This be-havior results from a characteristic of the A antennapattern, which experiences a sharp 1800 phase shift inpassing through the null. If the A tracker filter is cen-tered on the A spectrum null, the phase detector outputwill be zero. However, if centering does not exist, thephase detector output will be positive or negative, de-pending upon the direction of misalignment. Conse-quently, the phase detector output is equivalent to thediscriminator output of the z tracker. Closure of theA tracking loop is made in an identical manner to thatdescribed for the E tracker.As indicated in Fig. 7, the outputs of the four track-

ing oscillators are fed into the data processor. In addi-tion, information regarding the status of each of thefour tracking loops is fed into this processor. This in-formation includes lock-on or search status of each ofthe four tracking oscillators and tracker-mode status,i.e., I or A mode of forward-beam and aft-beam track-ing.The foregoing description of tracker operation applies

to a particular configuration. As pointed out previously,many configurations incorporating this feature can bedevised. A particular configuration has been selectedfor convenience in illustrating the principle. Furtherdescription of the I and A trackers will be given in thefollowing section, where a comparison is made of theirperformance capabilities.

III. PERFORMANCE COMPARISON OF NULL TRACKINGWITH LOBE TRACKING

A. Bias-Reduction Capability of Null-Tracking SystemThe major advantage of the null-tracking system

over conventional lobe trackers is the reduction in biaserrors which shift the apparent center of the receivedspectrum and/or introduce asymmetries in the spec-trum. Examples of this bias are sea bias, altitude-holeeffects which occur for modulated systems (both pulseand FMCW), antenna-beam asymmetries, and non-flat circuit responses in the receiver or frequencytracker.

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58 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

Relatively simple models have been used for deter-mining the reduction in bias errors achieved by nulltracking relative to lobe tracking. Although more re-alistic models have been analyzed, the small differencesin results do not justify the more elaborate presentation.Bias errors are derived for the high signal-to-noise case,but do not depend significantly upon signal-to-noiseratio. Nor do these errors depend to a primary degreeupon A spectrum null depth. As illustrated in Fig. 8, it isassumed that the unbiased z spectra is Gaussian andthat the bias function is linear. All functions in Fig. 8are normalized to the z spectrum peak and the fre-quency variable is normalized to the 2 spectrum band-width B, and centralized to the center frequency of thetracker f,. The significance of the bipolar nature of the[A spectrum]1/2 curve is that the components of thisspectrum are in phase with z components on one sideof the null and are out of phase with z components onthe other side of the null. Sea bias is appreciably non-linear over the antenna beamwidth.3 However, themajor nonlinearity is square law, which contributes nobias error. Therefore, the linear bias term makes themajor contribution to the error.A lobe tracker is hypothesized which centers itself at

the center of gravity of the biased spectrum, resultingin a tracking error vi. This model assumes that thebandwidth of the frequency tracker is equal to, or widerthan, the spectrum width. The tracking error is easilyfound by solving for the shift in spectrum first momentwhich is caused by the bias slope b. This solution resultsin a tracking error

bpl~ , (1)

6

where the normalization factor is the spectrum widthBs.The law of operation of the frequency tracker has

only a minor influence on bias errors. For example, asmall linear bias function shifts the peak, the powercenter, and the first moment of a Gaussian spectrumapproximately equally and by the amount given in (1).Therefore, for lobe-tracking systems, the errors causedby linear bias are not critically dependent upon the fre-quency-tracker configuration. If (1) is used to computenormalized sea-bias error vs antenna beamwidth forgiven values of b. a parabolic curve is obtained. Thisfunictional behavior checks that given in Fried.10 Theactual errors obtained fronm (1), for values of b -corn-puted from experimenital data, check the results givenin Fried within a factor of two. Considering the differ-ences in experimental sea-bias data from differentsources, this appears to be a reasonable check.

Considering next the n-ull tracker, the phase-detectorcircuit of Fig. 9 serves as the discriminator. For conven-ience in the analysis, a rectangular response has been

'0 Fried, op. cit., Fig. 11.

Fig. 9-Null-tracker discriminator.

selected for the band-pass filters, a parabolic function isused for the unbiased z spectrunm, and a linear biasfuniction is again assumed. Null tracking consists ofadjusting the tracking oscillator relative to the , and Aspectra centers to equalize the in-phase anid out-of-phase components of the portionis of these spectra (Fig.8) which fall within the z and A banidpass filters (Fig. 9).Obviously, only the portions of the I and A spectra inthe vicinity of the iiull are effective in determining thetracking poinit of the null tracker.

For the high signal-to-noise case, the mean-squaresignals applied to the two-phase detector diodes of thediscriminator in Fig. 9 are expressed by

r n+Bn /2B,

E12 - (1 - 2v2)(1 + bv)(1 + K1mv)2dv (2)vn-B, 12B,

Pn+Bn/2B8E2 2 ;f+ (B - 2v2) (1 + bv)(1 - Kmv)2dv, (3)

vn-B, /2Bs

where BI/BS=ratio of null-tracker bandwidth to signalspectrum width. If (2) and (3) are set equal and solvedfor v, this solution gives the normalized bias errorcaused by the linear bias term b. Equating (2) and (3)requires that

f vn+Bn / 2B,-Y2l 2)(1 + bv)dv = 0.

n-B, 1 2B,(4)

Evaluation of (4) results in a fifth-power polynomial inVn. Because the solution of initerest is small relative tounity, higher-order terms have beeni neglected and asolution obtained to give

b BnI\2Vn ; - 1 -

12 Bs/(v5)

The use of (1) anid (5) give,a ratio of bias errors forthe fiull tracker anid lobe tracker expressed by

(6)

Because of the approximatioiis made in the derivationof (6), this equation is not valid for ratios of Br,/B,greater than about one third. The significant result of(6) is that the reduction inl bias errors varies as thesquare of this bandwidth ratio. As an example of theapplication of (6), if the niull-trackinig bandwidth is

March

V, I B 2

.:Zll

VI 2 Bs

Smith: Null- Tracking Doppler-Navigation Radar

chosen to be 0.1 the signal spectrum width, a reductionin bias errors better than 0.01 can be realized by thenull-tracking system relative to lobe-tracking systems.

B. Performance of Trackers as a Function of Signal-to-Noise Ratio

I n order to abbreviate the presentation of trackerperformance as a function of signal-to-noise ratio, sim-plified models are used for the analyses. The results aresufficiently accurate to give a good measure of the rela-tive performances of the lobe tracker and null tracker.The effects of time sharing the transmitter, receiver,and parts of the frequency tracker, as shown for thesystem in Fig. 1, are the same for both types of trackers.Therefore, in t.he interest of simplicity and because acomparison of the two systems is desired, the analysesare made for noni-time-shared CW systems.

For the lobe tracker, the input spectra to the limiter-discriminator of Fig. 7 are illustrated in Fig. 10. Arectangular response of bandwidth B, is assumed forthe tracker filter. A Gaussian-shaped signal spectrumwith bandwidth less than B, is assumed. The frequencycoordinate in Fig. 10 is normalized to the tracker filterbandwidth andl centralized to the tracker center fre-quency. The ordinates of the spectra are selected to givea total integrated power approximately equal to unity,which is permissible because of AGC action. Signal-to-noise ratio SIN is defined at the output of the trackerfilter and is the ratio of total signal power and totalnoise power. Price1' and Cahn" justify the assumptionthat the sum of the spectral density functions in Fig. 10is not modified significantly by the limiter which isbetween the tracker filter and discriminator in Fig. 7.Therefore, the spectral density at the discriminatorinput is also represented by the sum of the curves inFig. 10.

For this anialysis, the signal-plus-noise power densityfunction, Fig. 10, will be interpreted as the probabilitydensity function of the instantaneous frequency at thediscriminator input. This initerpretation is justified bythe fact that the signal at this point contains onlyrandom frequency modulation, and by the fact that thenormalization factor in Fig. 10 has been selected to givea total integrated power of unity.A discriminator which has an input bandwidth equal

to or greater than that of the input signal will respondto the instantaneous frequency at its input, and willaverage this frequency over the time constant of itsoutput circuit. This output will contain dc and ac com-ponents which depend upon the input signal-to-noiseratio S/N, the normalized signal bandwidth B8/B1, andthe instantaneous tracker error vP. Neglecting a propor-

POWERDENSITY

I 0_at,/-7 0.42BS)[ + ( SI-1

B2 l

(V

I\II SIGNAL

IBSBBSBN

- NCDISE

FREQUENCY, v2 (NORMALIZED TO Bk)Fig. 10 Limiter-discriminator input spectra, lobe tracker.

tioiiality constant, these components can be shown to be

P6

t v

01 + (S-1] 2

[1 + ;])(1

(7)

(s)-12 11/2

[1 + (S) 1] (8)

where filtering in the discriminator output circuit hasnot been taken into account.The low frequency components of Eo will cause the

tracking oscillator to vary in the appropriate mannerto remove these components from the discriminator out-put. The instantaneous frequency error required tobalance out these low-frequency perturbations is ob-tained by setting the dc output of the discriminatorequal to the low-frequency components of -Eo which arecontained in the closed-loop tracker bandwidth B,.Therefore, the rms frequency error of the tracker oscil-lator is obtained by solving the following equationsfor v,.

Eo = FEo, (9)

where F is a filtering factor accounting for the portion ofac output falling in the closed-loop bandwidth of thetracker. This factor may be approximated by

I1 R. Price, "A note on the envelope and phase-modulated com-ponents of narrow-band Gaussian noise, " IRE TRANSACTIONS ONINFORMATION THEORY, VOl. IT-1, pp. 9-13; September, 1955.

12 C. R. Cahn, "A note on signal-to-noise ratio in band-passlimiters," IRE TRANSACTIONS ON INFORMATION THEORY, VOl. IT-7,pp. 39-43; January, 1961.

(10)1 + (N7

0 (,-T) 1-

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60 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

Eqs. (7)-(10) have been used to compute trackerjitter for the case shown in Fig. 11. The dotted curve inFig. 11 shows the variation in discriminator slope [thecoefficient of v, in (7) ] with S/N. This curve shows thatthe open-loop gain of the tracker decreases rapidly asthe signal-to-noise ratio drops to unity and below. Thetracker-jitter curves assume that the closed-loop band-width is proportional to discriminator gain, i.e., a single-integrator servo. A maximum closed-loop bandwidthBtm of 0.01 of the tracker-filter bandwidth B1 is used tocompute these curves. For high signal-to-noise ratiosthe frequency tracker acts as a filter to the input-signalspectrum, reducing the frequency deviation by approxi-mately V\Btm/Bs.

Before discussing the curves in Fig. 11 further, thenull tracker will be analyzed. The discriminator circuitof this tracker is shown in Fig. 9 and the input powerdensity spectra are illustrated in Fig. 8. In order toachieve significant reduction in sea-bias effects, thebandpass filter will generally have a narrow band rela-tive to the signal spectrum width, i.e., BnK<<B3. Thepower density spectrum of the A signal along the beamaxis is assumed to be zero. Although this ideal pattern(implying infinite null depth) will not be achieved inpractice, the assumption will yield little error in theanalysis if the null depth is better than -30 db relativeto the A peaks. This required null depth can be obtainedeasily with practical monopulse antennas.The circuit in Fig. 9 does not include limiters. Be-

cause the A signal EA will almost always be below thethermal noise level for accurate tracking, a limiter atany point in this channel would be ineffective unless itwere set to limit on noise. In this case the limiter wouldsuppress the signal part of EA and would alter the phasecoherence between this signal and the z signal El. Ina similar manner, a limiter prior to the bandpass filterin the z channel would alter this phase coherence be-cause of intermodulation between signal componentsover the relatively wide spectrum width B,. A limitercould be introduced in the z channel after the bandpassfilter, but this would introduce a balance problem be-tween the two channels. The AGC action illustrated inFig. 7 will maintain the average power output of the Ifilter (bandwidth Bn) nearly constant.As implied in the above paragraph, stable phase bal-

ance between the entire 2i and A channels is important.Differen-tial phase variations between these two chan-nels should be held within about 200 over a bandwidthgreater than Bn. Greater differential phase shifts willreduce the sensitivity of the null-tracker discriminator.If any components of E2 and EA within Bn have phasedifferences exceeding 90°, these components will con-tribute a reverse sense component to the discriminatoroutput Eo.The analysis of the null tracker is described in Ap-

pendix II. Expressions for the normalized mean-squarevoltages applied to each of the detectors in Fig. 9 arederived [(24) and (27) ]. The relationship between the dc

cr m

-,wYcrN

-0z

2 3 4RECIPROCAL SIGNAL/NOISE, ( s )

z

P-z

C,)

Fig. 11-Tracker jitter and discriminator gain, lobe tracker.

and ac outputs of this circuit are obtained from themean-square voltages by use of the probability densityfunctions of E1 and E2.As for the lobe tracker, the assumption is made that

the tracker will respond to all components of Eo whichfall within the closed-loop bandwidth of the trackerBt. This response introduces instantaneous errors in thetracking oscillator which cause the slowly varying dcoutput of the discriminator Bo to cancel the fluctuationterms. Setting the dc term equal to the rms value of theac term and solving for the required value of frequencyerror v, gives the rms frequency error of the tracker oscil-lator. The expression for this error is given in AppendixII, (33).As shown in Appendix II, the rms frequency error of

the null tracker depends critically upon the slope m ofthe A pattern near the null. It is desirable to make thisslope as high as possible while sacrificing very little inthe 2 pattern. Practice with monopulse systems indi-cates that the peak of the A spectrum will be about 6 bdbelow the z spectrum peak. This value is used to obtainthe reciprocal relationship of slope with signal spectrumwidth, as given in Appendix II, (23).

In obtaining the final expression for rms frequencyerror, account is taken of the dependence of closed-looptracker bandwidth Bt upon signal-to-noise ratio. Thediscriminator-gain curves of Fig. 12 show that, withoutcompensation, this bandwidth will be a function of bothsignal-to-noise ratio and spectrum width. However, thisbandwidth is assumed to vary slowly with signal spec-trum width, because of the action of the block labeled"Detectors, Comparator" in Fig. 7. This block comparesthe outputs of the wide-band and narrow-band filtersand compensates for the change in discriminator gainwith signal spectrum bandwidth. For systems whichoperate over a relatively narrow range of aircraftvelocity, the compensation for spectrum width wouldnot be necessary. The symbol Btm in Fig. 12 represents

March

Smith: Null- Tracking Doppler-Navigation Radar

rcM

QrN

F- 0z

B,m =E31n =0.1

BIAS IMPROVEMENT= 100, FOR Bt =I-TRACKER JITTER

---DISCRIMINATOR GAIN

,,

0I Pt

to

Z

Er tnl

0

ZD

_ *

Fr0

-i

RECIPROCAL SIGNAL/NOISE, (h-

Fig. 12-Tracker jitter an-d discriminator gain, null tracker.

closed-loop bandwidth at the maximum gain point(S/N)-1 =0. The signal-to-noise ratio is referred to theoutput of the z tracker filter (B1) and is given by (26)in Appendix II. Although this filter does not play a rolein the null-tracker operation, the definition is used inorder to provide a common reference for a comparisonof the trackers. The rms frequency error for the nulltracker is plotted in Fig. 12 for the parameters shown.The bias improvement factor is 100 for the case ofBW/B1=1, and drops to 6 for the case B3/B1=0.25.

There are several points of interest in comparingFigs. 11 and 12. Both sets of curves correspond to thesame conditions of signal spectrum width and closed-loop bandwidth of the frequency tracker. For the case

of most interest, B3 B1 1, the jitter functions are com-

parable for low signal-to-noise ratios. However, for highsignal-to-noise ratios the null tracker gives better per-formance. This effect is explained by the fact that thenull-tracker filter rejects echoes from those targetswhich are far removed from the beam axes but whichare still in the 2 and A beams. On the other hand, thelobe tracker weighs these echoes quite heavily in its dis-criminator output. Because the Doppler frequencies ofthese targets are far removed from the spectrum center,they can produce wide deviations in the discriminatoroutput, which subsequently cause large deviations in thetracker oscillator. Actually, the null tracker is not com-

pletely immune to targets off the beam axes, because itmay "wander" to the vicinity of these targets undercertain conditions, e.g., for long-duration fades of targetechoes near the null axis. However, the "beam-sharpen-ing" effect of the null tracker appears to be a significantadvantage over the lobe tracker under high signal-to-noise ratio conditions.

Calculations have been made for smaller ratios ofnull-tracker to lobe-tracker bandwidths than those illus-trated in Fig. 12. Although the bias improvement in-

creases rapidly as this ratio decreases, i.e., as the squareof the ratio, the tracker jitter changes rather slowly.For the high signal-to-noise case the jitter decreasesdirectly with null-tracker bandwidth, while for the lowsignal-to-noise case the jitter increases as the squareroot of this bandwidth.A note of caution is in order when comparing the jit-

ter functions in Figs. 11 and 12. The results in Fig. 12assume perfectly matched phase functions in the twochannels of the null tracker and perfect balance in thenull-tracker discriminator. Practical systems will giveresults which are somewhat worse than those given inFig. 12. In any event, it is expected that null-trackingsystems can be designed to reduce sea-bias errors to anegligible degree while giving low signal-to-noise per-formance which is comparable to lobe-tracking systems.The high signal-to-noise performance of null trackers isexpected to be significantly better than that of lobetrackers.

It may be somewhat surprising that the tracker-jitterfunctions in Figs. 11 and 12 increase relatively slowly asthe signal-to-noise ratio decreases below unity. It mustbe remembered that these functions represent only apartial picture of frequency-tracker operation. To com-plete the picture, consideration must be taken of thereduction in discriminator gain with signal-to-noiseratio. Notice that the gain is independent of spectrumwidth in the lobe tracker, but this is not the case withthe null tracker. With the latter, the compensationshown in Fig. 7 causes the high signal-to-noise discrimi-nator gain to be independent of spectrum width, but thelow signal-to-noise gain varies with spectrum width [asgiven by the denominator of (34) ]. For the case of mostinterest B1B1;z 1, the discriminator gains of thetrackers vary identically with signal-to-noise ratio. Onereason why the tracker jitter decreases rather slowlywith decrease in signal-to-noise ratio is the fact that thediscriminator gain (and therefore the tracker closed-loop bandwidth) also decreases with this ratio. In effect,the tracker becomes sluggish for low signal-to-noiseratios, and eventually signal "unlock" occurs. This un-lock is caused by a momentary signal fade, a severe air-craft acceleration, or by an unintentional bias in thetracker loop which drives the oscillator in a preferreddirection in the absence of a good restoring force, i.e.,the case of low discriminator gain. The gain-suppressioneffect illustrated in Figs. 11 and 12 for low predetectionsignal-to-noise ratios is fundamental and corresponds tothe well-known signal-to-noise suppression effect whichoccurs in nonlinear detection devices.13

Although the above analyses are made for CW sys-tems, the behavior for time-shared systems is similar tothe results described above. The primary effect of timesharing is to reduce the effective bandwidth (and speed

13 W. B. Davenport and W. L. Root, "Introduction to the Theoryof Random Signals and Noise," McGraw-Hill Book Co., Inc., NewYork, N. Y., chs. 12 and 13; 1958.

1963 61

62 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

of response) of the tracker. Time sharing also limits theminimunm data delay of the tracker to approximatelyone period of time sharing.

IV. CONCLUSIONSThe study in this report shows that the null-tracking

Doppler radar is theoretically capable of reducing seabias and other bias effects to a negligible degree. Theexpected improvement varies as the square of the ratioof null-tracker bandwidth to spectrum width. A band-width ratio of 0.1 at the maximum aircraft velocityappears to be a good design value because it provides theorder of 100: 1 reduction of sea-bias errors withoutappreciable reduction in tracking sensitivity. For highsignal-to-noise ratios, the null-tracking radar offers sig-nificant reduction in tracking jitter caused by the rela-tively wide spectrum of conventional lobe trackers.Another advanitage of the null-tracker system is that theantenna calibrationl should be simplified significantly,making feasible the ground calibration of high-accuracysystems. No reductioin in sea-current errors can be ex-pected for this system, because all Doppler systemsmeasure relative velocity between the aircraft andground or sea reflections.

For maximum benefit from niull trackiiig, the antennashould be azinmuth-stabilized to the ground track.Fixed antennas are expected to be usable for drift anglesup to 100 to 150. Although a precise analysis of com-bined effects of drift, roll, and pitch has not beeni made,these effects are not expected to be serious with fixed-antenna systems for drift, roll, and pitch angles up to10°. For antennas which are ground-tracked stabilizedin azimuth, roll and pitch angles up to 20° are not ex-pected to cause difficulty.The added complexity of a system employing null

tracking does not appear to be prohibitive. A rough esti-mate is that the added complexity is of the order of20 per cent for the entire radar system.

APPENDIX I

RECEIVED POWER DENSITY SPECTRA

Because of the length of the analyses of power densityspectra, only the assumptions, the method of approach,and the results are included in this Appendix. Two con-figurations are analyzed, the 1-A case and the A-Acase. The former system (as described in Section II)transmits on a single-lobe pattern, and receives on this2 pattern as well as a A pattern. The latter systemtransmits and receives on the A pattern. The geometryused in the analysis is shown in Fig. 3. All antennabeamwidths are assumed to be narrow enough to permitlinear transformations from polar coordinates to rec-

tangular coordinates over the angles of interest. Signalstrength variation caused by changes in target rangeacross the antenna beams has been neglected. Levelflight under zero pitch and roll conditions is assumed.The earth is assumed to be flat but rough iu the.sense of

causing Rayleigh scatterinig as opposed to specularreflection.Each of the field patterns is normalized to its peak.

Simple expressions are chosen to represenit these pat-terns and in terms of the coordinate system of Fig. 3are expressed by

sin'^y:(x, y) = exp si2 x2

L 2a2 i

sin4 y 2-XexpL_ 2h2'a2 tan y/j (11)

-\)e sin2 tihA(X, y) = i y ta-

sin4 zy h \2-exp _ 2h2oa2 tan ) _

X exp - x2_ it2a2

(12)

where

a=-angle between the Ainull and peak-one-half elevation beamwidth of z pattern

a; azimuth beamwidth of I and A patterns.

The equations for constant velocity contours ofground reflectioins relative to the aircraft are expressedas hyperbolas in the x-y coordinate set. Over the narrowbeamwidths of interest, these conitours (i.e., isodops) areapproximated by straight lines

dyy = y(x = 0) + x (x = 0).

dx(13)

The expression for y in (13) was thein substituted into(11) and (12) to give equations for z and A field intensi-ties along the isodops.

Referring to Fig. 3(b), the power density reflectedfrom the isodop strip corresponds approximately to thepower density in a corresponding Doppler frequencyband dfd or a velocity banid dv. After some manipula-tion, the following relationship caii be derived

dy(x = 0) = h(l -q2)-312dq, (14)

whereV fd

VI ft

fl = Doppler frequency corresponding to a targetwhich is in line with the velocity vector Vl.

fd = general Doppler frequency corresponding tovelocity variable v.

Because the relationship in (14) applies approxi-mately over the range of x values encountered acrossthe antenna beamwidths, the normalized power re-flected from the isodop strip can be expressed by

P(n7)d-q ~- kdyS 2:2(x,y = t )A2(x)dx, (15)

March

Smith: Null-Tracking Doppler-Navigation Radar

(16)

where P(,7) is the normalized power density function andk is a proportionality constant which is selected as anormalizing factor to cause P(-q) to be unity at the upperpeak of the A power-density spectrum.

After a considerable amount of algebraic manipula-tion, (15) can be reduced to the following form:

F1-_fv "1112F[ tan%T1P(7) 2A2-1 II2232-B2+ta 2#

L1 - 72iL 2C2j

Near the null frequency

Na ef n a

Near the peak frequencies, q, and 'q2

where

/Ve sin2 yA=

a (l-772)

B = (1 2q)I ta y

tan -y (r1-2 1 )1 /2

2 sin y

a

=71 QS + V2)

/ a\'112 CoS Y - i).

The minimum value of P(Qq) occurs for B =0, where7-7o= -cos y. Therefore, the Doppler frequency at the

minimum of the A spectrum is independent of driftangle and depends only upon the fixed value of y andthe longitudinal component of aircraft velocity vi.From (16) the minimum value of P(,q) can be ex-pressed by

e 2 tan2,3Pmin(7)= P(7O) -( (18)4 a sin2 -y

The development for the A Transmit, A Receive casefollows a similar path as for the I-A case, alnd reducesto the form

A -tn 22[ 3/2- B4P() A 4 tan' B_1_N tn

3BI 31

C2 tan2 i3 4C4_Near the null frequency

I)-[1 f22 3]/2POO) l n2

Near peak frequencies t7, and 2.

Again, the minimum value of this spectrum occurs atjo= cos y and is expressed by

3e2 a 4 tan4fPmin(q) = P(,q0) 64 \ s

64 ax sin' oy(21)

Eqs. (16), (17), (20), and (21) are used to plot thecurves in Fig. 4 and Fig. 5 for special cases of a and a,as defined after (12).

APPENDIX I IANALYSIS OF NULL TRACKER

Referring to Fig. 9, expressions are desired for thenmean-square voltages E1 and E2 which are applied to

(17) the two detectors of the discriminator. Because of theincoherence between the different frequency compo-nents of the I and A spectra, they add incoherently on amean-square basis. However, account must be taken ofthe coherence between common frequency componentsof the z and A spectra in deriving the resultant values ofE12 and E22. Observing these conditions permits writing

E12 = k2f Gn(') j[1 + Kim(v + V')]2-oo

+ [1 + K, ]N0} dv, (22)

where K1 is the transformer coupling defined in Fig. 9,k2 is a convenient normalization factor, Gn(v) is thepower response of the narrow-band filter of Fig. 9, v isthe normalized frequency variable, Vf is the normalizedfrequency error, and No is the thermal noise density inthe I and A channels, normalized to the peak of the 2spectrum density. The slope m of the [A spectrum]1/2,as illustrated in Fig. 8, is normalized to the peak of the2 spectrum and for practical antennas is estimated tofollow the relationship

2BI

B,¢(23)

The normalization factor for all frequency variables isthe tracker-filter bandwidth B1. This latter factor playsno direct role in null-tracker operation, but is used forconvenience in comparing the two trackers. The cou-pling coefficient K1 is made small enough to insure thatEs is almost always greater than KjEA. Assuming aGaussian function for Gn(v), (22) is evaluated to give

E12= k2' _+ I + 2) No + K B2+ 2Kimv]

= 1 + k3v, (24)

for PK<<B,jBi. The constanits k2 and k2' have beeni used(19) to give the normnalization for the latter fornm of (24).

Notice that this normalization is accomplished by theAGC action of Fig. 1 and the gain control of the I andA amplifiers in Fig. 7. From (23) and (24),

2K,mk3~

1 + No

BK LB1( S )-]-1

(20)

(25)

631963

64 IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

where the following relationship has been used in ob-taining the latter form

S B.

N B No(26)

Similarly,

E22 = 1 - k3ve. (27)

Voltages E1 and E2 in Fig. 9 have a noise-like charac-ter. The signal contribution to these voltages resultsfrom a summing of reflections from many targets scat-tered randomly about the axis of the antenna beam,while the noise contribution results from thermal noisewhich is passed by the tracker filter. Consequently, E1and E2 have normal probability density functions, andtheir amplitudes have Rayleigh density functions. Useis made of the following relationships between the firsttwo moments m1 and m2 and the variance ¢rR2 of Ray-leigh density functions

2Ml2 =M2

4

crR2 =M2-Ml2 (28)

terms contributing to fluctuation to total output is ob-tained from (24) and is given by

m2Bn21K12 FN ± ,

L 2.8B,2_2[1 + No] (30)

This fraction is essentially independent of v, for smallfrequency errors. Therefore, by the use of (25)-(30), theac compoinent of Eo may be expressed by

k4k5EO- [2E12±+E22j1/2 (31)

k4k5

/2

Assuming that the z spectrum width is always widerthan filter bandwidth Bn, the discriminator output coII-tains a noise voltage which falls within the trackerclosed-loop bandwidth Bt of

-Pot BB t (32)

to relate the average amplitude and the variance of theamplitude El to E12(v,). In this manner, the dc andac components of the discriminator output are obtained.The dc component is given by

Eo =k4[ E2+ /E22 ], (29)

where k4 is a constant depending upon the gain of thedetector circuit and upon the statistical characteristicsof the signal. In computing the ac component, accountmust be taken of the correlation between the fluctua-tions on ElI and those on E2 . The phase detector ofFig. 9 is balanced to the Ey, signal. Thus, if the couplingcoefficient K1 is chosen small enough to insure that E2, islarge relative to K1EA for a large percentage of the time,the fluctuation on the ET signal will not appear on out-put Eo. Therefore, in determining the fluctuation part ofthis output, only fluctuations on the EA input need beconsidered. At zero error signal, the ratio of the output

Following the procedure described for the lobe tracker(Section III-B), the expressions for Eo and Eot areequated and solved for ', to obtain the rms trackingerror due to signal and noise fluctuations. For smallerrors, the following solution is obtained:

B. /BtmFBJ S -1rms v, ;-z:/--

8B3 V Bil -B\ NI

B 2- 1/2

+ 1.4Bs_2-

(33)

To obtain this solution, it is assumed that the closed-loop tracker bandwidth is proportional to discriminatorgain. Fig. 12 shows how this gain varies with S7N andwith B8/B1. However, the circuit in Fig. 7 reduces thelatter variations. Therefore, (33) applies to the casewhere

B tan B t7n1+ N0o ( )

B, N

(34)

March