IRE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

Leakage Rejection in Beam-SwitchedCW Radars*

P. G. SMITHt, SENIOR MEMBER, IRE

Summary-This paper is directed to the problem of transmitter-receiver leakage rejection in CW and FMCW radars which employstepped-scanning antennas. An IF carrier-elimination filter has beendevised and analyzed which provides leakage-rejection capabilitieslimited only by leakage variations during stationary beam positions.Leakage rejection is provided by coherently reducing the signal tozero IF, rejecting the leakage components by a combination syn-chronous switch and high-pass filter, and remodulation onto a secondIF carrier. The filter operates in a linear manner. Positive and nega-tive senses of Doppler signals are retained. The spectrum of theinput signal is faithfully reproduced in the output with the exceptionof a sharp notch at the input carrier frequency. In one configuration,the minimum notch bandwidth is limited by the on-time of eachbeam, and is approximately equal to one-half the reciprocal of thison-time. A more complex arrangement removes this restriction,permitting a minimum notch width which is limited only by thecutoff frequency of a high-pass RC filter. The simpler configura-tion has been used with the AN/APN-118 Doppler NavigationRadar, a system which was developed under contract with the U. S.Signal Corps, Fort Monmouth, N. J., by the Sperry GyroscopeCompany.

I. INTRODUCTION

C W AND FMCW radar systems are severelylimited by the well-knowin transmitter-receiverleakage problem. In spite of special attempts to

reduce this leakage by space duplexing or by theFMICW technique,' most applications require or wouldbenefit from additional leakage rejection. This addi-tional rejection may be accomplished by RF cancella-tion methods, by passive IF filters, or by active IFfilters. Although the useful value of additional rejec-tion is dependent upoIn the particular application, itgenerally ranges between 20 to 50 db. For applicationsemploying a fixed or slowly scanning antenna, the re-quirement mnay be met by the use of a very selectivesynchronous filter. The form of this filter will be de-scribed briefly as a special form of the beam-switchedfilter to which this paper is primarily directed.Beam switching or scanning seriously complicates the

leakage rejection problem by the introduction of switch-ing or scanning variations of the leakage amplitude andphase. This problem is introduced in step-scanning CWradars, or in Doppler navigation radars of the CW orF\ICW classes where the antenna is time-shared. An

* Received August 13, 1962.t Sperry Rand Research Center, Sudbury, Mass.' K. C. M. Glegg, "A low noise CXNV Doppler technique," Proc.

IRE Nat'l Conf. on Aeronaotical Electronics, Dayton, Ohio, Ma\ 12,13, 14, 1958, pp. 133-149.

illustrative waveform of the leakage waveforni for atime-shared CW radar is shown in Fig. 1. For thosecases where the minimum Doppler frequency to bemeasured or detected is large relative to the beanswitching rate, sufficient rejection can be achieved inmany cases by the same techniques employed for non-switched systems. However, for those applications re-quiring the measurement or detection of near-zero-Doppler frequencies, complications are introduced bythe frequency overlap between switching transients andthe low-Doppler frequencies. In theory if the leakagefunction is repetitive or varies slowly with time it canl bemeasured, stored, and applied as a cancellation wave-form to reduce the actual leakage to negligible levels.This procedure is not feasible in many applications be-cause of its complexity.

Fig. 1 Variation of leakage amplitude caused by RF switchoperation. Phase variations follow similar patterns.

The leakage rejection devices to be described in thispaper apply to systems employing step-scanning of theantenna. They, are referred to as carrier-eliminationfilters (CEF), anid operate at intermediate frequencies.Use is made of the fact that the leakage variation isvery abrupt during beam switching tinmes but remainsrelatively constant during beam oni-tinmes. The anmpli-tude and phase characteristics of the leakage need notbe identical from one beanm position to the next. Leak-age rejection capability is limited primarily by varia-tions of leakage amplitude and phase during beam on-times. For one filter configuration the low-frequencyDoppler response is determined by the beam oin-time.A somewhat more complex conifiguration permits re-

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sponse to much lower Doppler frequencies. The filtersact in a linear fashion, causing the output signal spec-trum to be identical to the input spectrum with theexception of a sharp notch near-zero-Doppler frequency.The techniques described in this paper may be ap-

plied to pulsed radar systems, when it is desirable todetect and track targets at short ranges where a tinmeoverlap occurs between transmitter and receiver opera-tion.

II. DESCRIPTION OF CARRIER-ELIMINATION FILTERS

the leakage rejection, i.e., ratio of available leakagepower input to leakage power output, may be approxi-mated by

(1)

where

B0 =bandwidth of rectangular rejection filter,

f,=beam switching rate (approximately the recipro-cal of the beam on-time).

Before proceeding further with a description of theconfigurations of CEF's which are the main topic forthis paper, it may be of interest to miention a few alter-nate configurations. Assunme that the filter is to operatein the IF section of the radar system, where the actualcenter frequency corresponding to the leakage carriermay be selected quite arbitrarily. As a first possibility,consider a simple rejection filter which is centered at theleakage carrier, or zero-Doppler frequency. Because ofthe large changes in leakage during beam switching, theleakage spectrum contains many spectral lines at mul-tiples of the frame rate, i.e., the number of antennaposition sets per second. The power in these switchingharmonics decreases slowly with harmonic number,making rejection by passive filtering impractical. Forexample, it can be shown that 40-db leakage rejectionwould require a rejection band of about 103 times thebeam switching rate, i.e., approximately the reciprocalof the beam on-time for relatively rapid switching be-tween adjacent beam positions. This wide rejectionband is clearly intolerable if low-Doppler frequencies areto be passed by the filter.A slight improvement over the above approach is

made by preceding the passive filter with an oI-offswitch which gates out the large leakage fluctuationsduring beam switching. However, the resulting leakagesignal still has abrupt changes from zero to the steady-state values during beam oni-times. Consequently, thesanme order of rejection banidwidth is nieeded as for thepassive filter alone.

I'tz

z

s T rOFF TIMEBEAM BEAM 2

Fig. 2 Cosine switch waveform.

A cosine switching waveform of the type shown inFig. 2 has relatively rapid fall off of switching har-monics. An analysis has been made of a filter configura-tion consisting of this type of switch followed by a pas-sive rejection filter. Results of this analysis show that

This result shows that a ratio of rejection bandwidth tobeam switching rate of 10 will give a rejection of ap-proximately 50 db. In some cases this rejection band-width may be acceptable. However, the rejection of low-Doppler frequencies covers a bandwidth which is muchgreater than that imposed by the use of antenna time-sharing, as will be shown below. Furthermore, the effec-tive beam on-time is reduced by this waveform to aboutone-half its value when rectangular gating is used. Thisreduction in duty cycle is objectionable because of thecorresponding reduction in system sensitivity.

Insertion of a cancellationi IF waveform would bepractical if the steady-state leakage amplitude andphase were identical from beam-to-beam. In this casethe use of an on-off switch for gating out the leakagetransients between beam positions would be very useful.As previously mentionied, this cancellation is not gen-erally effective because the leakage amplitude andphase values are not identical for all beam positions. Ina strict sense, the two CEF's to be described below pro-vide leakage rejection by inisertion of a cancellationwaveform at zero IF, i.e., dc. These filters use the prin-ciple of coherently sampling the leakage signal, storingthis sample, and using the stored sample to cancel theleakage signal. Sampling and storage are sinmplified bytranslating the IF signal to dc. Two methods are of in-terest, and are referred to as a long-term memory cir-cuit and a short-term memory circuit. The performancecapability of the first type is superior to that of the sec-ond type, but this superiority is paid for by an increasein filter complexity.The block diagram in Fig. 3 applies to both types of

filters, as indicated in the block labeled CEF Switch,Cancellation Circuit. Referring to this figure, the inputfrom the IF amplifier is split into two channels. Thesechannels contain coherent detectors which are suppliedwith reference carriers phased at 900 relative to eachother. This reference is obtained from a stable oscillatorand is used in the RF part of the system to provide thelocal oscillator offset frequency from the transmitterfrequency. Consequently, the reference signal has avery stable phase relationship with respect to the leak-age signal during stationary-beam times.The coherent detector outputs, corresponding to the

leakage signal, consist of bipolar waveforms as illus-trated in Fig. 4(a). The amplitude of each output is

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Smith: Leakage Rejection in CW Radars

proportional to the product of the leakage amplitudeand the cosine of the phase angle between the leakageand the coherent reference signals. For many appli-cations of this type of filter, e.g., for stationary or slowlyscanning antennas, the phase detector outputs may bepassed directly into high-pass filters where the low-frequency components corresponding to the leakagesignal are rejected. However, when switching tran-sients are present, the bandwidth of this filter wouldhave to be excessively wide in order to provide signifi-cant leakage rejection. The purpose of the switch andcancellation blocks is to reject the leakage signal with-out severely rejecting low-Doppler signals.

Deferring description of the switch and cancellationblocks momentarily, after leakage rejection by theseblocks the Doppler signals are amplified and reinsertedon a carrier. This carrier frequency may be the same asthe reference carrier which is applied to the coherentdetectors, or it may be arbitrarily selected if a secondIF is desired. In a similar manner to the input referencecarriers, the reinjected carriers are phased at 900 rela-tive to each other.The relative phasing between the coherent references

and between the reinjected carriers is important to theproper recombination of the two channels. Ideally, thesignal outputs from the two balanced modulators haveequal amplitudes, the relative phases between the car-rier components and between the undesired sidebandcomponents are 1800, and the desired sideband compo-nients are in-phase. Consequently, the output spectrumfrom the Adder will be a replica of that into the CEFwith the exception that a sharp notch has been imposedat the leakage frequency. Furthermore, the input leak-age components at multiples of the beam switching ratewill have been rejected, without appreciable rejectionof Doppler components which fall near these frequen-cies. The manner in which this discrimination is ac-complished will become clear from the following de-scription of the Switch and Cancellation blocks.The Switch and Cancellation Circuits are illustrated

in Fig. 5. For the long-term memory circuit in Fig. 5(a),the switches move synchronously with beam switching,from one position to the next. Thus, during the on-times of a given beam, the two coupling condensers cor-responding to this beam (one for each channel) charge tothe average leakage level. Because of the large dischargetime-constant between on-times of this beam, the con-denser maintains its charge from one on-time to thenext. Thus, after initially reaching its steady-state volt-age, the stored voltage on this condenser cancels thesteady-state leakage signal during the beam on-time. Ifleakage alone is present in the input and if this leakagehas constant amplitude and phase during the switchon-times, no disturbance is observed by the output cir-cuit as the beam is switched sequentially to all positionsin the frame. Therefore, perfect leakage rejection hasoccurred. The response of this circuit to Doppler signalsis derived in Section III. It is apparent that the cir-

Fig. 3-Zero IF carrier-elimination filters (CEF).

(a)0

TIME,tBEAM BEAM 2

.I

(b) On

T

Of fPosition I Position 2 TIME,

4 ~~~ ~I II

(c) OT

Off I11 1110 tl -At2 TIME,t

Fig. 4 Waveforms associated with CEF switch and cancellationcircuits. (a) Coherent detector output. (b) Switch position long-term memory circuit. (c) Switch position short-term memorycircuit.

cc

From Coherent 2- To Balanced

Detector 3 ModulatorN )

Tc0

(Duplicate Switch in Other Channel)

(a)From Coherent Rs cc On To Balanced

iDetector , iModulator

e (t) Off C RI e*(t)

(Duplicate Switch in Other Channel)

(b)Fig. 5-CEF switch and cancellation circuits for use with Fig. 3.

(a) Long-term memory circuit. (b) Short-term memory circuit.

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cuit has good low-frequency response, since signal inputvariations during and between beami on-times arereadily passed by the large coupling capacitor inlto theoutput circuit.

Fig. 5(b) illustrates the short-term memory circuit.Because of the use of a single switch and storage coIn-denser, the sample and storage function must be ac-complished at the beginniing of each beam on-tinme.Ideally this function would be acconmplished instanta-neously, at the beginning of each beanm on-time, byshorting the storage capacitor to ground. After thisrapid charge, the coupling capacitor is switched to theoutput circuit. Assuming that the leakage amplitudeand phase remain constant for the remaining on-timeafter sampling and storage, perfect rejection is accomp-lished. There are practical reasons why this ideal per-formance cannot be realized. These reasons will becomeapparent during the discussion of Doppler response. Asfor the long-term memory circuit, it can be seen thatthe short-term memory circuit has good Doppler re-sponse because of its sensitivity to changes in inputvoltage during beam on-times.The comparative performance of the long-term and

short-term memory caincellation circuits is important.Section III will show that the long-term circuit is sig-nificantly better with regard to leakage rejection andDoppler response. The advantage of the short-ternmswitch is solely one of simplicity. This advaiitage isprobably unimportant for ground-based systems, how-ever, for airborne or space systems it can be quite im-portant. This is especially true of systems having gim-balled antennas. It is important that the switchesshown in Fig. 5 be high quality, with low impedanceand negligible pedestal. Commutator or relay switchesare best suited for this purpose, but semiconductorswitches are feasible if the signal level in the circuit is

maintained high, relative to switch pedestal levels. Theshort-term memory circuit is of interest because itoffers a satisfactory solutioin in many cases. The rela-tive simplicity of this circuit is particularly importantfor those applications where commutator type switchesare not convenient. For these cases, the synchronizationbetween the antenna switch and the switch in Fig. 5(a)is considerably more complex than it is for the simplerswitch in Fig. 5(b).

III. ANALYSIS OF CARRIER-ELIMINATION FILTERS

A. Long-Term Mlemory CircuitThis circuit, Fig. 5(a), is relatively easy to analyze

because of its simplicity from an operational standpoint.Assume that the coupling condensers in Fig. 5(a) are

initially uncharged at t = 0, and that only leakage is

present in the IF input to the CEF. The ratio of energydissipated in R, during the nth on-time period of a givenbeam to the available leakage energy, i.e., that whichwould be dissipated in R1 if the input were connecteddirectly to this resistor during the beam on-tinme, can

be shown to be

1

L

L 2T(n - 1)-r 1 e 2 1- e R1C. Jo T R1Cc dcr (2)

exp - R1C

where L=leakage rejection factor, T=frame time, of isa dummy variable of integration, and the other svm-bols are defined in Figs. 5(a) and 4(b). The approximateexpression in (2) results from assuming that r<<RiC,.In anly case, 1 /L decreases exponentially with t, and thetheoretical leakage rejection capability is infinite. Froma practical standpoint there should be no difficulty indesigning the circuit so that only on-time leakage varia-tions are passed to the load resistor. Thus, the leakagerejection capability for each beam position would beequal to the ratio of iniput energy per beam on-time tothe energy in the leakage variations during this time.These variations are caused by AMJ noise in the trans-mitter tube, and by amplitude anid phase modulationiimposed by the RF and IF sections which precede theCEF. FM inoise in the transmitter tube is not generallya problem, because the IF leakage component is ob-tained by mixing the RF leakage component with thelocal oscillator output. Both of the mixer inputs arederived from the same transmitter and therefore containthe same FM noise. The heterodyning action removesthis noise component from the IF output.The Doppler response of the long-term memory filter

is determined by the tinme constant R,Cl, the beam oni-time T, and the frame time T. A precise, general analysisof this circuit response in the low-frequency range israther complex and has not beein made. However, rea-sonably accurate response curves can be obtained byconsidering separate frequency ranges where simplify-ing assumptions can be made. For example, the curve inFig. 6 shows the sketched response for the case whereRIC>>»T. This curve is normalized to the availablepower per beanm. The circuit acts as a narrow-bandrejection filter for frequencies which are multiples of11T. Because of the gating effect of the switch, the re-sponse of the circuit is extended to frequencies wellbelow the RiC, cutoff frequency. The rejection bandsare approximately r17rTRIC, cy'cles wide. This resultfollows from the fact that the effective averaging timeof the gated RiC, circuit is approximately TRIC,IT.Fill-in of the rejection bands is caused by variations ofinput voltage during beam on-times, which becomesmore and more significant for increasing frequency untilthe response becomes essentially uniform for frequenciesabove 1 r. This fill-in is also a function of the ratio TIT.For cases where the R,CG time constant falls between Tand T, a variety of response curves can be obtained, fall-ing between that showni in Fig. 6 and a response whichfollows the envelope of the mininma of this curve. The

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Smith: Leakage Rejection in CW Radars

z0

0

Fig. 6 Response for long-term memory circuitfor case where R1C>»T.

latter case would occur for the case where RUC~is ap-proximately equal to r.

B. Short- Term Memory Circ U1i.The circuit tliagram of this switch anad cancellationl

circuit is shown in Fig. 5(b). R,,, Cc, R1, and Cl are thedriving source impedance, the storage capacitor, theload resistor, and the load capacitor, respectively. Forall leakage and Doppler frequencies which are affectedsignificantly by beam switching, the effects of Cl arenegligible because this capacitor is used to control theupper cutoff frequency of the Doppler response. In thediscussion of the circuit in Fig. 5(b), reference will bemade to the time symbols defined in Fig. 4(c).The design of the switching circuit is somewhat com-

plicated by the presence of conflicting requirements.The important characteristics of the filter are leakagerejection capability, signal response vs Doppler fre-quency, and spurious responses introduced by the filter.The must serious spurious response is referred to as apedestal. This pedestal is an inherent characteristic ofthe short-term memory CEF and is introduced by theinput signal. For most applications, the pedestal can bereduced to acceptable levels byr trade offs with otherfilter characteristics. In the analysis, the following sim-plifying assumptions have been made:

1) Initial charges on Cc and Cl are zero at referencetime zero, as defined in Fig. 4(c).

2) No receiver saturation occurs in receiver sectionspreceding the CEF during beam on-times; ifsaturation occurs during beam off-times the re-ceiver recovers rapidly enough to prevent satura-tion after-effects during the following beam on-time.

3) R,<<Rl4) The load capacitance Cl is negligible over the band

of low-Doppler frequencies which are niear the re-jection band of the CEF.

In order to satisfy assumption 1), an additional switchfunction could be added which shorts Cc immediatelyprior to the beam on-time, or charge time. However,this complication is generally not required, as will bediscussed further.The analysis is made for each beam as if it were the

only beam being energized. Because of the normaliza-tion of the leakage and Doppler-signal outputs to theleakage and signal inputs, respectively, the results ap-ply equally well to all beams anid to the average valuesof leakage and signal inputs. The assumption of zeroinitial charge on the two capacitors removes the influ-ence of leakage and signal levels from preceding beams.

Response characteristics of the CEF are determinedindependently for leakage and Doppler signals. Thisprocedure is justified by the fact that the circuit is atime-variant linear system, responding linearly to theleakage plus signal for both positions of the switch.Therefore, superposition applies and permits additionof the individual responses to leakage and signal to ob-tain the response to the composite input signal. In orderto simplify the analysis, the input signal to the CEF isassumed to be gated on during the beam on-time and offotherwise. In practice this gating should not be neces-sary. Due to the short time constant RXC, the effects ofinitial charge on Cc at t=O (as would appear withoutgating) would have relatively little effect on the leakagerejection capability or on the Doppler response of thecircuit.

Considering first the leakage rejection capability,refer to Figs. 4(c) and 5(b). The coupling condenser C(charges toward the leakage level at the beginninig of thebeam on-time. Failure to reach this level after a chargetime ti results in continued charging during the switchon-time, thus coupling leakage to the load. In addition,leakage variations following the charge time are coupledinto the load. The two effects are analyzed separately byfirst determining the leakage rejection as limited onlyby the charge time, and theni by adding the effect of on-time variations of leakage.Assuming an input leakage component of one volt,

the storage condenser Cc starts charging through thesource impedance toward this level at t=0. At ti theswitch moves to the "on" position, and the voltage onCc is expressed by

V,(t1) 1 - exp (- .C (3)

During switch on-time r the storage condenser continuescharging through the series combination of R, and Ri.Assuming R,<<Rl, the energy dissipated in R, duringthe on-time can be expressed by

W= -Vc(t)]2exp -RC di)

exp RIC)[ (--)]x (4)

The ratio of leakage power transferred to the load tothe available leakage power, i.e., the power which wouldbe transferred to the load in the absence of the storageand cancellation provided by Cc, is given by

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1 AWWRlL r

RKC, 21) - 2T -

2T Rt RCcJ 1- RC-) (5)

where T and T are the switch on-time and the period ofone complete scan period, respectively, and L is definedas the leakage rejection factor. This result shows thatthe leakage rejection capability is determined primarilyby the ratio of charge time ti to charge time constantR8C,, with the ratio of switch on-time T to time con-stant RIC, playing a minor role. Eq. (5) is plotted inFig. 7 for several sets of circuit parameters.The value of t2 indicated in Fig. 4(c) represents the

time between switching to the "off " period and therapid change in leakage waveform caused by switchingto the next beam position. This time is not critical, butthe switching must be accomplished soon enough to pre-vent appreciable leakage variation to couple into theload.

Considering the leakage rejection capability aslimited by on-time leakage variations, define the nor-malized mean square variations by the symbol OL2.Notice that these variations apply to the phase detectoroutput and consequently arise from amplitude andphase variations of the leakage input to the CEF. Be-cause of the relatively large time constant RICc, theseleakage variations appear across the load with verylittle attenuation. Therefore, the ratio of leakage powertransferred to the load to the input leakage power, aslimited by on-time variations, is expressed by

cTL2 (6)

where L' is the leakage factor as determined by on-timeleakage variations. Because of the independence of theleakage output power represented by the two leakagefactors L and L', combination of these two factors is astraightforward procedure.An analysis of the short-term memory CEF has been

made in order to determine the frequency response ofthis circuit as limited by the switching function and bythe circuit parameters. For each channel of the sche-matic circuit in Fig. 5(b), the following differentialequations must be satisfied:

dei di id -?Rsdt+- (7)di diCt

for 0 < t < 11

z0

C-)ww

cowCD4

w-j

Fig. 7 Leakage rejection vs charge time.

where e-(t) and i(t) are the inlput voltage and loop cur-rent waveforms, respectively. The effect of Cl has beenomitted, and zero initial charge has been assumed, inaccordance with the assumptions listed above. For asinusoidal input signal of amplitude Es, the solution to(7) and (8) (as detailed in the Appendix) gives a volt-age wave form to the load during T

el(t)Ej

R,-I R K2((W) sin (cot + 2)

- K3(o) exp F_L(RI ±R,)C,j)

where functions K2(c), K3(CO), and k2 are defined in theAppendix. The first term of this waveform is the steady-state response which would be reached if the on-time Twere allowed to become infinite. The exponiential termresults from the charge placed on the coupling condenserCc by the signal during charge time ti. This decayingpulse will appear with varying height and polarity forsuccessive beam times, where the height anid polarityfor a specific on-time depends upon the phase of theinput signal relative to the reference time t =O. Theaverage pedestal height decreases with inicreasing inputfrequency, because of the decrease in response of theintegrator circuit RsCc during charge time t1.The average power delivered to RI, normalized to the

available power input, i.e., the power which would bedelivered to the load for the case of simple on-gatingduring T, off-gating at other times, is expressed by

Pi(w) = Average over-all {-T f 2dt} (10)

(9)

de di i= (R, + -Rj) - +-

di dt Cc

for t1 < t < ti +

where k, is the input phase of the signal relative to the(8) reference time t=O. This equation is solved in the Ap-

pendix. The results for several sets of circuit parametersare shown in Fig. 8.

246

247Smith: Leakage Rejection in CW Radars

z200

cn 1.4 o: 0/ 6

4~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~.

0.4 -stl Corner 04

0.21.2

01 0 00 000 1000NORMALIZED DOPPLER FREQUENCY,

Fig. 8-Doppler response of short-term memory circuit.

IV. DISCUSSION OF RESULTS

The fact that the performance of the long-term mem-

ory circuit is superior to that of the short-term circuitis apparent fronm the results of Section III. The formercircuit permits theoretically infinite leakage rejectioncapability. From a practical standpoint this resultmeans that the leakage rejection capability can bemade as good as necessary to cause other factors thanbeam switching to limit the achievable rejection. Thesefactors include klystron AM noise and leakage ampli-tude and phase variations during beam on-times. Theleakage variations are caused by RF beam switchingand by additive and multiplicative noise which is intro-duced by the IF section preceding the CEF.The Doppler response of the long-term memory CEF

can be shaped to have almost any desired low-frequencyDoppler response. By suitable choice of the couplingtime constant RiCG, the rejection bands shown in Fig.6 can be made as narrow as desired in order to rejectonly the average leakage and the low-frequency varia-tions of this leakage for each beam. Beam on-time r

need not be a limitation on the ability of this circuit torespond to any part of the Doppler frequency band.As stated previously, the advantage of the short-term

memory CEF is simplicity and consequently greaterreliability. The disadvantage of poorer trade off betweenleakage rejection capability and Doppler response is notserious for many applications. The remainder of thissection is devoted to a discussion of the short-termmemory CEF.

Referring to Fig. 7, it is seen that the leakage rejec-tion capability of the short-term memory CEF is deter-mined primarily by the two parameters, t1Tr andR,C,Ir. More precisely, this rejection capability is de-termined primarily by the parameter t1 R,C,. From (5)

tLL(db) ;z-9 ~db.

The 6-db difference between the solid and dottedcurves of Fig. 7 results from the fact that the RIC, cir-cuit becomes less effective in coupling leakage power asthis time constant decreases. Notice that the normali-zation factor is the available leakage power to Ri,which would be coupled completely to this load resistoronly as R,Ccc.The curve labeled R1C/-= cc in Fig. 8 shows the

limiting response of the short-term memory circuit asdetermined by the beam on-time. Notice that the low-frequency part of this curve contains a rejection notchfor 0 <XT<2 followed by a high response region forwhich the output power is approximately twice thegated input power. This high response at intermediatefrequencies is a consequence of the rapid charge of C,during time ti to the input signal voltage (as well as tothe leakage voltage). Signal energy is rapidly stored inthis condenser during charge time t1 and discharged intothe load during the remaining part of the on-time r. Theadded response due to this effect is clearly undesirablebecause the contribution it makes to the power de-livered to R, is not at the input signal frequency. Thisresponse introduces spurious components near-zero-Doppler frequency and is referred to as the pedestalresponse. It will contribute carrier and low-frequencysideband components to the output of the CEF. Thesecomponents appear as signal-induced, zero-Dopplerpulses.As shown by (9), the CEF output will contain two

components. The first term in this output is the desiredsignal, and has an amplitude which is simply the steady-state response of the RIC, circuit. The other term is thepedestal. For low-frequency inputs, e.g., cor < 1, the twoterms interact strongly, giving nearly zero resultantoutput power. However, for higher frequency inputs,e.g., cor>r, the interaction diminishes rapidly and theterms become nearly independent. Thus, the part of theresponse curve above unity output may be thought of aspedestal response, which introduces spurious, low-Doppler components in the CEF output.The other curves in Fig. 8 correspond to finite values

of R,C,Ir as labeled. It can be seen that a value ofR-C,Ir near unity reduces the output pedestal powerto approximately 40 per cent of the desired signal power,with little sacrifice in low-frequency response. Stillfurther reduction of R,C,IT to 0.5 reduces the outputpedestal power to approximately 20 per cent of thedesired output signal power. However, the rejectionnotch is broadened by a factor of two relative to thecase for R,C,T= cc. Assuming that this level of pedestalpower is acceptable, the low-frequency cutoff of theDoppler response, as limited by switch on-time (ap-proximately beam on-time), is approximately '27.The primary effects of varying RIC, on the circuit

performance are included in the curves of Fig. 8. Reduc-tion of RIC, permits more rapid discharge of the initial

1962

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voltage on Cc at t1 and consequentlyr reduces the pedestalpower coupled to the load. As shown by (5) and Fig. 7,the variation in leakage rejection factor over the rangeof RiC, values of interest is less than a factor of four,for given values of t1 R8Cc.The effects of vary,ing the charge time constant R8Cc

are illustrated in Figs. 7 and 8. As described above, theleakage rejection factor is a critical function of tIR,C,.A second effect of varying R,C, is the variation in band-width over which the pedestal remainis significant. Theupper cutoff frequency for which this pedestal is presentis equal to the RXC, cutoff frequenicy. Therefore, thechoice of this charge time constant is a compromise be-tween leakage rejection capabilitx (or required chargetime tH) and bandwidth of input signial over which thepedestal output appears. The effects of changing R,C,from the value illustrated in Fig. 8 may be approxi-mated bv shifting the pedestal cornier frequency (i.e.,the frequency for which wR C,= 1) to the left or right asrequired. As long as R,C,<<T, the niew response curvesmay be sketched by moving the decay part of the re-sponse curves in Fig. 8 to the right or left by the ratioof the new value of R,C,I r to the value used for obtain-ing these curves.

Further consideration is Iow giveni to the signial-iniduced pedestal, which is givein bv the second term of(9). This pedestal term will be significant only for thoseDoppler input signals below the pedestal corner fre-quencyfs. For signals in this banid, the CEF output willcontain the desired signal plus ain exponientially-decay-ing pulse at the carrier frequency, i.e., at the zero-Dop-pler frequency. It is significanit that these two signalsoriginate from the same input signal, which results instrong correlation between the inistantaneous strengthof the two. Thus, the output pedestal componenit willalways be weaker than the correct signal output. Thisconditioni is illustrated in Fig. 9 for the case whereR,C, r=0.5. The vector additioni of the two outputcomponents results in AMI and FMJ of the correct signaloutput at the Doppler rate, as illustrated in Fig.9 (b) -

For most Doppler radar applicatioins, the frequencytracker contains a limiter followed by- a discriminator.If the resultant signal in Fig. 9(b) is limited heavily, thelimiter output will be frequency-modulated only, i.e.,the AM is removed. Therefore, the spectrum of thelimiter output is symmetrical about the correct fre-quency, as long as the input signal is above the limitlevel. For this case, no error is encounitered in trackingthe center of the correct signal spectrum. At those times

when the input signal drops below the limit level, i.e.,low signal/noise cases, the frequenicy tracker will bepulled toward zero Doppler away from the correct sig-nal frequency. The worst frequency- error occurs whenthe correct signal is near the pedestal corner frequency.f,. Assuming the frequency, tracker centers itself insuch a position to give zero first miioml-enit to the input

spectrum, the frequenc- error f, is giveni by

fc <. forfd < f, and Ps < limit level

0 forf > f, or Ps > limit level (12)

where fd is the correct ceniter frequency, and P, P, isthe ratio of signal power to pedestal power. It is impor-tant to observe that appreciable error occurs only for aset of special conditioins, causing average errors in agiven system to be much less than those expressedby (12).As an examiiple of the use of the short-term nmemory

circuit, consider the case of a Doppler navigation radarwhich time-shares the transmitter, antenna, IF ampli-fier, anid CEF betweeni four anitennla beams. Table Igives sets of specified and derived parameters. Thepedestal requiremlenits dictate the values of the timeconstanits R.Cc and R,CG and the leakage rejection re-quirement determinies the charge time ti. One of theresistors canl be chosen arbitrarily. The value of R,would probably be chosein first, with this choice being

z Doppler0 Componentsr Pedestal3 Component

(a)

(b)

Fig. 9 Effects of pedestal on CEF output. (a) Powerdensity spectra. (b) Vector combination.

TABLE ISHORT-TERM MEMORY CEF

(EXAMPLE)

Specified ParametersBeam oni-timeFrame timeBeam switching timeMaximum pedestal/signial ouLtputPedestal cutoff frequencyLeakage rejection capabilityMNaximitum Doppler frequenc7

Deri1ved ParamietersRSCC (from fs requliremenit)RSCC/Tt1 (from Fig. 7, anid r)RIC, (from Fig. 8, and r)R, (arbitrary, amplifier considerationis)R, (imposed by RI)Cc (imposed by RI)Cl (imposed by RI, f,,a)

Tr+tiO=.11 secT=0.5 sec

0.015 secPPIPS = 0. 2

f.,= 160 cpsL>105

fmax= 20 kc

0.001 sec0.01 sec0.005 sec0.05 sec50 GO,O1000 SI1.0 4f160 ,u/f

248

Smith: Leakage Rejection in CW Radars

consistent with reasonable maximum values of inputresistance to the amplifier which follows the switchingcircuit.

V. CONCLIUSIONTwo configurations of CEF's suitable for use with

beam-switched radars have been described and ana-lyzed. These filters operate at intermediate frequencies.Good filter design causes the leakage rejection to belimited by leakage variations during beam on-timesrather than by leakage variations during beam switch-ing or fronm beam-to-beam. The more cormplex designutilizes one storage capacitor per beam, and this capaci-tor is switched into the circuit only during the "on"time of its associated beam. Although the Doppler re-sponse of this configuration contains rejection notchesat integer frequencies of the frame rate, these notchesmay be made as narrow as desired by the choice of theappropriate RC time constant. For a particular appli-cation, the notch width would be limited by the band-width of low-frequency amplitude and phase variationsof the leakage signal.The simpler configuration of CEF utilizes only one

storage capacitor for all beam positions. This capacitoris allowed to charge to near leakage level during thefirst part of each beam on-time, then it is switched intothe load circuit to block the leakage during the remain-der of the beam on-time. Leakage rejection capability ofthis circuit is determined bv the charge time allowed atthe beginning of the on-time, which subtracts from theusable beam on-time. The Doppler response of this cir-cuit contains a single rejection notch at the carrier (orzero-Doppler) frequency, which has a bandwidth of ap-proximately one-half the reciprocal of the beam on-time. This filter produces a spurious output at the zero-Doppler frequency. The amplitude of this spuriousresponse is proportional to the input signial amplitudeand may be reduced to generally acceptable levels byproper circuit design. It has been shown that the pres-ence of a small spurious component will not cause errorsin tracking the correct Doppler frequency, provided thesignal delivered to the frequency tracker is well aboveits limit level. Because of the rather special conditionsrequired for the pedestal to produce errors in the fre-quency tracker, it is not expected to be a serious disad-vantage of this filter configuration.

APPENDIXRESPONSE OF SHORT-T_ERM MEMORY CIRCIUIT

Referring to the circuit in Fig. 5(b) and to the switch-ing sequence in Fig. 4(c). the following differential equa-tions must be satisfied:

dei

dt

de,

dt

di i

Rs +di C l

di

(R, + R1) +dt Cc

for 0 < t < 1l (13)

for t1 < t <t1 + r (14)

where ei(t) and i(t) are the input voltage and loop cur-rent waveforms, respectively. The effect of Cl is neg-ligible over most of the frequency range of interest, anidconsequently Cl has been omitted for this analysis. Fora sinusoidal input signal of amplitude Ei and radianfrequency co, and for zero initial charge on Cc and Cl,the solutionis to the above equations are of the form

eC( ) = K1(T) [sin (wt + 01)Esj

- sin&exp( R-C )]

for 0 < t < ti

e(t) = FK2(T) sin (cwt + 02)E, Rt L-w

- K3(wT) exp ( I t)]

for t1 < / < ti +7-

(15)

(16)

where

Kl(wT) = I +( c (UT) 2 2

K2(W [) ( (

K3(w-r) = K1(w-[) sin (cw/l + i1) - sin pI exp (--RC)l

+ K2(cor) sin (wti +02) - sin (wtil + ti),41(c) = i - tan-' wRC,

02(0)) = 4i + ctn- 1CRtCc,Rt = R. + R1,

-i= signal phase relative to reference time, I = 0.

The average power delivered to R,, normalized to theavailable input power, i.e., power delivered to R, for thecase of simple gating and for Cc= c, is given by

Pl(w-T) = Average overoi4

This equation has been solved for two frequenicy rangesof interest:

Range 1:

0.5T0 < WT < 0.5-

K1(wr) --1

dn(WT) Z 0 i

K3(wT) : K2(wT) sin [L2(WT) + WtIJ.

1962 249

2 t1+ -r el 2 (t)di . (17)

T E ,2

IRE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS December

Range 2: Range 2:2r 7

C< U <<

K2(Wcr) ; 1

02(WT) OfiiK3(wr) ; Kl(wT) sin [i- tan-'1R,C, + toi].

After a considerable amount of manipulation the fol-lowing results are obtained:

Range 1:

O < CtT <0.5T

PIUT R8C.R({C-

Pj(cWT) ()K22WT) {1 + RtCc1 - exp( )]

< r <<Rtc. RIC,

PI(wrJT) (-R)R {1 +[1 - exp (- )]

2TK '(cor) rR,RtC,2(rT) 2

Rt+RC(T)' L 2

- 1

-7K12(coexp(-R )[~(1+ RsinC,T- 2K,2&co-r) exp - 11+-

\RtCjL\ Rt/l

+ ( >) 2 ( () - 1)]}. (19)

- sin coT- 2K2 (co-r)

_ cor

T COSCOT\

RtC, (W7)2 These results were used to compute the curves showin inFig. 8. Because of the relationship Rl>>R,, Rt can be

(18) replaced by RI in (18) and (19) with negligible errors

being introduced.

Amplitude Modulated CW Radar*OLE K. NILSSENt, SENIOR MEMBER, IRE AND W. D. BOYERt, MEMBER, IRE

Summary-This paper describes a radar system in which ameasure of range is obtained as the phase difference betweenamplitude modulations on the transmitter and echo signals. Sig-nificant features of this AM radar system are simplicity and capa-bility of accurately measuring very short ranges. For instance, anexperimental unit ranges from zero to 50 feet with maximum errorof six inches.

INTRODUCTIOONA N EXTREMELY SIMPLE method of radar is one

in which the transmitter signal is sinusoidallyamplitude modulated anid a measure for range is

obtained by observing the phase delay of the modula-tion on the echo signal. In spite of its simplicity, themethod has received little attention in the literature.'Presumably, this lack of attention is due to the limita-tion on maximum operating range brought on by the

* Received November 20, 1962.t Scientific Laboratory, Ford Motor Company, Dearborn, Mich.1 The only reference the authors could find was a brief mention in

L. N. Ridenour, Ed., "Radar System Engineering," Rad. Lab. Ser.,McGraw-Hill Book Co., Inc., New York, N. Y., vol. 1, ch. 5; 1947.

necessary requirement that the amplitude of the radarecho be substantially larger than that of the directtransmitter-to-receiver leakage signal. In certain appli-cations of radar to very short ranges, however, it be-comes possible to satisfy this requirement, thereby war-ranting a further investigation into the problems as-sociated with this type of amplitude modulation (AM)radar.The purpose of this paper is to present such an inves-

tigation. This is done with reference to a radar systemdeveloped specifically for use as an obstacle sensor inland vehicles and intenided to operate in the range fromzero to 50 feet.

ANALYSIS

Principle of OperationFig. 1 shows a block diagram of the radar system

under investigation. The sinusoidally amplitude modu-lated transmitter signal may be expressed as

St(t) = A1[ + M sincomtjeiwot, (1)

* exp ( RC) + T ____\RtC,/ RtC, (CWT)2 _)

250

2Tr r