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TRANSMITTEDFREQUENCY

TIME

The Institution of ElectricalEngineersPaper No. 3264EJuly 1960

INSTANTANEOUSSIGNAL

FREQUENCY

(2) PRINCIPLE OF THE SYSTEMIn the system under investigation, the transmitter produces aC.w signal of constant ampli tude, whose frequency is varied insawtooth fashion (see Fig. 1).The receiver picks up some of the power from the transmitter(a short distance away) and also the echo signal after delay oftime 7"due to its travel to and from the target. The direct signalfrom the t ransmit te r is known as the ground wave. The twooscillations are made to beat together in a non-linear device (e.g,a diode detector) and the beat note is found to conta in two distinct tones (see Fig. 1).

Fig. 1.-Production of the beat note.

at length by them. In par ticula r, it has been shown that maximum sensitivity (and hence range) dependson mean power, andthat the f.m. system has the advantage that it avoids the necessityfor the high peak powers in the pulse system. The f.m. systemsuffers, however, from the indirect method of obtaining rangeinformation, and the receiver becomes complicated and expensiveif a simultaneous presentation analogous to the A-scan isrequired. The quest ion of mean power required is determinedby the noise in the system, and the possibility of using acoherent system for frequency modulation was pointed ou t byGnanalingam;' who produced a vertical-incidence ionospheresounder of great sensitivity. In all cases of practical applicationto date, the maximum range of targets of interest has been quiteshort (e.g. a irborne alt imeters). By this, we mean that the timeof travel of the signal is small compared with the repetition timeof the sweep. Gnana lingam based his analysis on an approximat ion which was val id only in the limit of zero range. In this.paper , the more general case is examined, and a complete ,expression for the beat note and its spectrum is obtained.The present paper , which arose from design studies for an.ionospheric sounder, re-examines the mathematical basis of f.m..ranging and goes on to propose a ranging system which avoids;some of the ambiguities of earlier systems, but retains coherencein the method of detection. The equipment concerned is in theprocess of construction and results of measurements will bemade available in due etmrse. .In the analysis, cons iderable a ttention is given to the exactmathematical formulation of the functions concerned, so thatmisleading approximations may be avoided.

[ 365 ]

(The paper was first received 4th September, 1959, and ill revised form 4th February, 1960.)

OF A FREQUENCY-MODULATED CONTINUOUSWAVE RANGINGSYSTEMBy A. J. HYMANS, M.Sc., Alnst.P., Graduate, and J. LAIT, M.A. .SUMMARYSome aspects of an f.m,c.w. radar with a sawtooth frequency sweepThe exact beat note for a discretetarget is calculatedits Fourier transform is obtained. A schemepreviouslygivenbyGnanalingam for producing a coherent system is shown to be onlypproximately valid, and an alternative method is proposed. Thefectof Doppler shift on the return is discussed. Range discriminan is examined critically.

.LIST OF SYMBOLSc = Velocity of propagation, m/s.

F(w) = Complex Fourier transform.pew) = Complex conjugate of F(w).t , Fz, F 3, F4 = Individual terms in Fourier transform of beatnote.flo f2 = First and second intermediate frequencies, cis.G = Numerical constant.k = Order of spectral line.m = Order of zero in envelope of beat-notespectrum. !n = Serial number of sweep intervaLP = Echo power, watts.PA , PB = Echo powers received from targe ts at rangesRA , RB, watts.R = Target range, m.\ Ro= Target range at zero time, m,RA, Rs = Ranges of targets producing complementarybeat notes, m,Ts = Sweep duration time, sec.t = Time variable, sec.tn = Time variable measured from mid-point of nthsweep interval, sec.Ve, Vg = Amplitudes of echo and ground-wave voltages,volts.Vs' V,1o Vs2 = Detector outputs, volts.v = Radial component of target velocity, m/s.20:= Angular-frequency sweep rate, rad/sz.o(w) = Dirac delta function.

7" = Time delay of echo, sec.

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.

.

c2R2 Range of target7'- XV I ' f .e OCI ty 0 propagation

+coF(w) = Lt2VeVg cos (1)g - rfe)r}W1dtThis integral may be reduced to

In most cases we may assume that Vg ?> Ve Theseoscillations are 'mixed', 'heterodyned', or made to 'beat' togein some non-linear device. The resulting signal or bea twill contain a d.c, (zero-frequency) term and a p roductGVeVgsin 1>e sin 1>g, where G is a numerical constant, andhigher-order products.In general, only the lowest-order productwill have a signifamplitude, apart from the large d.c. term.The product may be expanded as a difference, namely

+co !(2n+l)T,F(w) = WVeVg f cos g - rfe)s-}W1dt- co !(2n-I)T,or by use of the substitution ttl = t - fiT;,+co { -tT,+TF(w) = tGVeVg s-}"wT, Jcos [WOT - ex(Ts - r)2-c o -tT, + 2ex(T _ T.)t,.]s-}W+ L ~ ~ l ~ o T - exT2 + 2ctTtn)S-}Wlndtll

(5) FREQUENCY ANALYSIS OF THE BEAT SIGNAThe oscillation tG VeVgcos (1)g - 1>e) is the beat note betthe echo and the reference wave. This oscil lation is not asinusoid and can be analysed into separate harmonic compoby the use of the Fourier transform. If w is the general v ~ r in the transform, we define the Fourier transform as

The phase-sum term is an oscillation at radio frequency aremoved by filtering. The difference term is the desired ostion at video frequency, and contains all the range informaWe are thus interested in the function tGVeVgcos (1)g - 1>

I t will be seen that 1>g - 1>e has two forms in the nth inteThey are(i) During time - fTs < tn< - -tTs + T,

when 1>e = WO(t,,-I - T) + ex(t"-l - T)2+ (11 - l)woTsand 1>g = wotn+ e x t ~ + nwoTs

1>g - 1>e = WOT - ex(Ts - T)2 + 2ex(T - Ts)tn .(ii) During time - tTs + T e = wO(tn - T) + ex(t" - T)2 + lIwoTsand 1>g is as in (i),

1>g - 1>e = WOT - exT 2+ 2exTt"

(2)

366 HYMANS AND LAIT: ANALYSIS OF A FREQUENCY-MODULATED CONTINUOUS-WAVE RANGING SYSTEMThis beat note is the equivalent of the video-frequency pulse (4) PRODUCTION OF THE BEAT NOTEin a pulse radar and contains all the information. Since the Let the reference or ground wave, which passes directbeat note is a repet it ive waveform with repet it ion time T. its t ransmitter to receiver, induce an oscil latory voltage in the

Fourier transform must consist of a spectrum of lines spaced Vg sin 1>g, where 1>g is the expression in eqn. (6), with pat intervals w. = 27T/T.. arbitrarily zero at t = O. The echo from a stationary refleIn the analysis that follows two approximations are made for target will induce a voltage Ve sin 1>e' where 1>e is a funsimplicity. They are as follows: similar to 1>g but delayed in time by T, given by(a) The frequency increases linearly with time and has instan

taneous flyback, although this is not, of course, possiblein practice.With the type of radar considered, however, the assumption ofnegligible fiyback time is thought to be a reasonable approach tothe truth.(b) The exact phase of both signals is calculated, taking intoaccount the fact that the echo is merelythe direct signal delayed bya timeT. The mediumthrough which the signal travels is assumedto be non-dispersive.(3) MATHEMATICAL FORMULATION OF THE SIGNALThe instantaneous frequency, WI> is given by the following setof expressions:WI= Wo + Zca,where -iTs < t < + tT .

= Wo + 2ex(t - T.), where -iT. < t < + fT .

so that the expressions can be generalized to give, in the zthinterval,

WI = Wo + 2exto10

and cPt = I (Wo + 2exto)dtoo= woto+ extij (5)

Similarly, the general expression will be found to be1>1 = wotn+ o : t ~ + lIwoTs . (6)

An examination of eqn. (6) shows that 1>1 is continuous at theends of each sweep.

= Wo + 2ex(t - lITs),where t(21I - l) Ts< t < t(211 + l) Ts(1)Fo r convenience in handling the expressions, the sweep ratehas been taken as 2ex rather than 0:, and the orig in of time is set

atlthe centre of one sweep cycle. ..".wIt has been found useful to make the substitution

WI = Wo + 2ext,,; - tTs < tn< + fTs (3)Ins tantaneous frequency is no t a physically measurablevariable; only voltages and currents can be so regarded. Thevoltage in the t ransmi tter is proport ional to the sine or cosineof the phase angle 1>1' where

11>1 = f wldt +cQnstant (4)

Thus we must obtain this integral of the instantaneous fre.quency by substitution from eqn, (3) into eqn. (4). The constantof integration can be evaluated by arbitrarily putting 1>1 = 0 atJ = 0 (there is no loss of generality here) and then stipulatingthat phase shall be a continuous funct ion. Such a res tr ic tion is:necessarily imposed by the continuity of currents and voltages;in real circuits.Taking the interval of zero order as an example, we then have,from eqn, 0),

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HYMANS AND LAIT: ANALYSIS OF A FREQUENCY-MODULATED CONTINUOUS-WAVE RANGING SYSTEM 367Fo r our purpose, it will t hus be sufficient to consider thefunctions for positive values of w, since the negative valuescontribute nothing new.

Fig. 2 il lus trates by sketches the way in which the distributionchanges with T.Fig. 2(0) shows the case where T is small compared with Ts ;the lower beat note then has a centr al maximum which is onlyslightly wider than two line intervals. The amplitudes of each

peak in successive maxima on either side of the central oneare, to a first approximation, in the ratio2 2 2 231T : 51T : 71T (2n + 1)1T

to the central maximum which is a result independent of T.Thus the first peak on either side of the maximum is only 21%

of the central peak.Fig. 2(b) shows the case when T = iTs. Although there is an

(20)

(21)

(22)

W = 2rx(Ts - T) 2nmT

(6.2) Amplitudes of LinesInspection of eqns, (13) and (15) shows that the spectrumhas two maxima on the posit ive axisat (i) (JJ = 2etT = WBI (say),

and at (ii) w = 2et(Ts - T) = wB2 (say).Consider for the presen t the case when WBl and WB2 are sofar apart that the contr ibutions of F I and F 3 can be consideredindependen tly. The maxima at Will and WB2 mayor may notcoincide with a parti cular line, depending on the value of 7'relative to Ts The zeros of the ampli tude functions F I andF 3 occur when

(w - 2r:n)(Ts - T)/2 = nnr }[ for m = 1,2, 3, . . .and W - 2rx(T - T)]T/2 = + nnr b t t 0- uno m=

" 2nmW ='2rxT + --Ts-Ti.e. atand atNow, let WBl or WB2 be an integra l mul tiple of Ws' Oneline is now situated at the peak of a central maximum andsuccessive lines are spaced at intervals W s (=27l'/T,)about it.The zeros on the other hand are spaced at intervals 27T/(Ts "'T)or 27T/T about the central maxima, respectively. Thus, only in

the cases when T = 0 or T = T, do the lines and zeros coincide,and these cases are of little int erest in a radar. Gnanalingamused a scheme for adjusting T, at each range step, claiming thatby reducing the beat ,note to the nearest whole number of cyclesa single line spectrum'was produced. I t is easily seen that thisresult is due to the neglect of the period during each cycle whenone of the tones (upper or lower) is not received. Fo r shortranges (i.e, T --+0)some value may accrue from using the methodof synchronization proposed by Gnanalingam, but otherwisefor values of 7'of the same order of magnitudeas T, no advantageis obtained.A scheme is proposed later to obtain some of the advantagesof a coheren t system, a lthough it is no t possible to reduce thebeat-note spectrum to a single line.The relative amplitudes of the two peaks are obtained by

putting w = WBl and w = WB2 in theappropriate functions F Iand F 2, and one hasAmplitude of lower beat-note = r. - TAmplitude of upper beat-note T

(6) EXAMINATION OF THE SPECTRUM(6.1) General Nature of the Solutioninspection of eqns. (13)-(16) and (18) it will be

sin [w + 2et(Ts - T)]; ,=; T expj[woT+1W(Ts -T)][w + 2et(Ts - T)]2 (14)

( T - T)T _ T sin (w + 2etT)= T _ T exp - j(WOT + 1WT)

(w + 2etT)( ) ... (16)remaining factor in F(w) is a delta function,

+co e-jnwT = wso(w - kw s), k = 0, 1, +2. . . (17)-co where W s = 21T/Ts

the full expression for the Fourier transform of the beatisF(w) = tGVeVgWs o(w - kws)kx [FI(kw s) + F2(kws) + F3(kws) + Fikw.)] (18)

e first integral in the outer bracket gives two terms:

a similar real harmonic component arises from the termsand F4(w).

T[kws - 2et(Ts - T)]2 cos {kwsf - [wo7' - !kwsCTs - 7')]}- 2et(Ts - T)]i" (19)

integral in the outer bracket gives two terms:r,_ T sin ( - 2etT)(Ts ; : T) .

= -- eXPJ(woT - ,!WT)2 (w - 2etT)C:' ;: T) . (15)

F(w) = P ( -w )F"'(w) is the complex conjugate of F(w). This is to besince we have here taken the Fourier t ransform of afunction. The general form of the spectrum is a set of linespal:eo along the t scale at interval ws'function shows four distinct maxima, at w = 2etT and

w = 2et(T, - T).variable to is, however, merely a mathematical tool whichone to examine the properties of real electrical functions.obtain the phases and amplitudes of these real componentsmust combine the positive- and negative-frequency terms inwhen it will be seen that the real par ts of a pair of comple

y terms (such as F I and F 2) reinforce and their imaginarycartcel, as pointed out by Woodward.Pthe terms FI(w) and F2(w ) of eqns. (13) and (14) therea single real harmonic component

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271'jTs the phase difference is seen to be just (2wo'TUnless the transmitter possesses a high order of phase stabilitythe two components for each line will beat together and produca fluctuating appearance which could be misinterpreted asfading signal. This topic will be dealt with more fully in lateSections since it raises fundamental questions of the design othe system.Fig. 2(d) shows the situation at still greater ranges.upper and lower frequencies have changed places.

The corresponding positive values of the beat notes wB lwB2 thus both contain Doppler shifts, and these dependexpected on the instantaneous angular transmitted frequenin fact, WBl and WB2 are, respectively,

and during time -tT, + 'T< til< t T,d 2v-, (

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(27)

(9) RANGE AMBIGUITYAn f.m, c.w. radar ranging system of the type discussedsuffers from a difficultynot encountered with pulse-type radars.Reference to Fig. 5 shows that for every range RA there is acomplementary range RB , such that the upper and lower beatnotes are identical. Thus the echo from range A has beatfrequencies 4RAIX lc and 2IXTs - 4RAlXlc, and echoes from rangeB have beat frequencies4RnIXlcand 21XT. - 4RBlXlc. Ambiguityin range measurement wiJI occur when

The obvious way of resolving this problem is to have availabledifferent sweep r a t e s ~ . a method which is equivalent to achoice of values of the pulse-repetition frequency in a pulseradar. Another method of attack would be to use what may becalled 'channel switching' to distinguish it from the first method,which is 'range switching'.Fig. 5has been drawn to show the special casewhenRA = tRBIn the general case, the two tones overlap for a fraction( I - 4RA) of each cycle. Thus, there is in theory theer,possibility of confusion during virtually the whole cycle, but for

Fig. 4.-Schematic of a coherentsystem.

,I---( -,

L ~ I J iI

T.

. I ;T., ,, ,: ', 'I II Ir II 'I I- - -_1 _

rII

o

o

HYMANS AND LAIT: ANALYSIS OF A FREQUENCY-MODULATED CONTINUOUS-WAVE RANGING SYSTEM 369amplifiers have been omitted. The beat-note frequencies andthe output from the variable-frequency selective amplifier arepassed into a balanced modulator, which produces the differencefrequencies. Only one of these wiJI be at exactly 100kc/s, andthis can be selectedby the use of a phase-sensitive detector whoseswitching frequency is the unmodulated intermediate frequency.With an integration time as long as desired, one has the equivalent of a narrow-band filter, tuned to exactly lookc/s as suggestedby Gnanalingam.! It will be noted that great stability of theintermediate frequency is not required, since it enters the systemonly as a 'carrier' of information, and is subtracted out againat the phase-sensitive detector. It is suggested that the intermediate frequency should be lookc/s, since that is a standardvalue for this type of work and techniques are therefore wellknown. Moreover, for the projected system, it falls welloutsidethe possiblerange of beat-notefrequencies so that second-channelinterferenceis minimizedand does not lead to confusion between,say, echoes from targets at ranges 3R, 5R and higher-ordermodulation products arising from the beat note of a target atrange R. This was a difficultyon which Gnanalingam remarkedin Section 5.3 of his paper.

3.-Production of the beat notewhen theechocomes fromamoving target.or a target receding at Mach 1, 'Tn+l - 'Tn is of the order ofnly 2 X IO-6Ts' so that 'T will increase from 0 to T. (therebympleting one cycle of spectral repetition) in a time of approxi-ately 5 X 105Ts ; and at lower line-of-sight velocities theeriodic time for 'T will be proportionately greater.

TRIGGER PULSES c::.::=;::':':-:::.:..JAT INTERVALS T,I I I I I

(8) SCHEMATIC FOR A COHERENT SYSTEMAs shown above, the scheme proposed by Gnanalingam will

110t produce the desired result of a single-line beat note exceptin the trivial case of zero range. I t is, however, possible toproduce a coherent system in which each line of the beat-notespectrum is examined separately with an arbitrarily long time ofintegration equivalent to an ideally narrow pass-band filter.Fig.4 shows a block diagram of an apparatus for doing this.The swept frequency is generated by a sawtooth voltage whichitself triggered by a train of clock pulses at intervals T.n thussuring that the duration of one sweep is accurately defined.The same train of pulses is used to amplitude-modulate anexternally generated intermediate frequency (for convenience thishas been taken as 100kc/s), thus producing side frequenciesspaced at intervals of I ITs cycles per second. Each sidefrequency can be selected in turn by a variable-frequencyhigh-Q-factor tuned amplifier, so that one has available a setoffrequencies separated by the same intervals as the beat-notespectrum.The receiver is shown in Fig. 4 only in outline. All linear

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'l.OWER' CHANNEl.

SWITCHINGFREQUENCY ( I,)

Hence, if, for example, RA = iRs, the power ratio is 'so that the ambiguous signal is approximately 19dB downthe signal from range A. For smaller values of RA this rawill be even greater. In a long-range equipment such asionosphere sounder, atmospheric attenuation will also incresignal difference by a considerable factor. For twomentary signals at nearly half range (RA = tcTs)' however,amplitudes will be more nearly comparable. As '1'-?-the time of overlap between the two tones, which is ( 1 of each cycle, tends to zero, i.e. the ambiguous tones fromrange occur at different times during the cycle; and thismakes it possible to employ 'channel switching' to discriminagainst the unwanted range in a way which is analogous topulse range gate. The analogy is not perfect, however,some power from the complementary signal will get uuou

RANGE - SEl.ECTI NGFREQUENCY (UPPER)f " f l + ~ - ~7r 'If C

( RA4= (_'1' )4Rs T, - '1'

values of RA small and tending to zero one musttake into accouthe difference in amplitudes between the two signals. Evenonly the normal radar range equation is applicable this confusiis not likely to arise for small values of RA , since thesponding value of Rn is at extreme range. The rat io of thepowers received will be

RANGE SEl.ECTIONCONTROl.

RANGE SELECTING. - -_ l l I -_ - .FREQUENCY (l.OWER)

1- 1 , +

VIDEO - FREQUENCYBEAT NOTES

T,5.-Production of complementary beat notesby ambiguousechoes.

2RAIc = T,/3(a) Instantaneous frequency Wi.(b) Instantaneous beat-note frequency (echo A).(c) Instantaneous beat-note frequency (echo B).

Fig. 6.-Schematic for a complete receiver with 'channel switching'.

(e)

(b)

(a)

4 P , ~ "L......I.-- 2T, 3T,

2"T, ---f-----------------------

anc INTERMEDIATE FREQUENCY(t 21, + ~

P Ul .S ES F RO MTRANSMITTER(*)

Fig.

I"INTERMEDIATEFREQUENCY (1,)

370 HYMANS AND LAIT: ANALYSIS OF A FREQUENCY-MODULATED CONTINUOUS-WAVE RANGING SYSTEM

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HYMANS AND LAIT: ANALYSIS OF A FREQUENCY-MODULATED CONTINUOUS-WAVE RANGING SYSTEM 371t the improvement in ratio of desired to undesired signalength will be greatest at the range where it is most needed.Fig. 6 is a schematic of a receivercombining channel switchingith a two-tone system for making use of the whole beat-noteectrum for one range. Once again, all linear amplifiers haveen omitted for clarity.In this scheme two 'intermediate frequencies' are required;e one at II can well be l00kc/s as before. The other, /2' at+ rxTsl7T, will be required to beat with the upper maximumthe spectrum. In order to preserve coherence, both upperd lower range-selection frequencies are generated by the sameodulation and selection process, using the principle of freency-changing to provide separation in the frequencies. Theo selecting frequencies are then made to heterodyne with thedeo-frequency beat note in the two balanced modulators. Fo rgiven range R two l00kc/s tones wJJ1 thus be produced, thee by difference between selecting frequency II and the lowert note, and the other by differencebetween selectingfrequencyand the upper beat note. The two frequencies are gated by aswitch (with waveforms as shown in Fig. 7) and are then

t.

I -----+-----------1(c) o(d) o l - - ~ - - - - - - - ~ : : : _ - + _ - - o ; -

Fig. 7.-Gating waveforms for channel switching.(a) Beat-note voltage; echo from range R.A.(b) Beat-note voltage; echo from complementary range RD.(c) Lower-frequency gating waveform.(d ) Upper-frequency gating waveform.

to two separate narrow-band filters followed by phasedetectors. The channel switch is controlled by theof the variable-frequency selective amplifier. Thus, ifrange selection control is set to receive signals from a rangebeat-note WBl = 4rxRAlc), the switch will change overeach sweep cycle at time tn = - n:, and back at time+ 2RA Ic. Reference to Fig. 7 shows that thesignal power from the ambiguous range will beby the factor (1 - ~ ~ : ) .

in the worst case when RA {cTs and the two interferingare of comparable magnitude, the unwanted one ist completely eliminated.The realization of the channel switch should not prove difficult,small errors in the switching instant will cause only a smalliation from the ideal case, and the degradation of the signalnot be greatly increased.

(10) RANGE ACCURACY AND DISCRIMINATION(10.1) Range Accuracy

a radar systemof the type under consideration, the problemasuring range is essentially that of measuring the time delay

-r, for -r = 2Rlc. Fo r a stationary target, the two tones WEIand WB2 which comprise the beat note between echo and groundwave have angular frequencies 2rxrand 2rx(Ts - '7'); for a movingtarget these frequencies are, respectively,2v 2v2rx'7' + - (WI - Zln) and 2rx(T. - '7') - -[WI + 2rx(T. - r)]c c

Taking the first tone in each case,cR = 4rxWBlfor a stationary targetand R = ! - (WBI - 2Vw;/c)4rx 1 - 2v/cor, to a first order of approximation

c vR = 4rxWm - 2rx(Wi - WEI)for one receding with line-of-sight velocity v. I f WB t could bemeasured precisely, the effect of target movement would stillvappear as an apparent range decrease 2rx (W I - WBI)'The remainder of the discussion will concern stat ionarytargets. An ideally linear frequency sweep has been assumedthroughout the paper and the analysis has been developed onthat hypothesis. As 2rx is the ratio of the total frequencyexcursion to the sweep duration, any inaccuracy in either ofthese parameters will give rise to a percentage error in rx andtherefore in range. Any departure from linearity would introduce perturbations in the modulation which would modify thespectral envelope.A contribution to range error also arises from the fact that,whereas a change in or causes the spectral envelope to movecontinuously in frequeffey, the lines are located at frequencieswhich are integral multiples of 277IT.. Thus WBI> the peak ofthe envelope, will, in general, occur between two lines, and theproblem to be solved is one of interpolation.By increasing the bandwidth of the filter so that it passestwo adjacent lines instead of a single line, the accuracy of interpolation may be improved by scanning the lines first singly andthen in pairs. Nevertheless, a fixed-magnitude 'reading error'of the order of 77 ITs will always be present in WEI, and therefore in the derived range, in addit ion to the percentage errorarising from any uncertainty in ex.

00.2) Range DiscriminationIn certain applications, a radar system deals primarily withextended targets, and range discrimination will be important onlyin so far as it is essential (as in an airborne altimeter) to assessaccurately the range of the boundary of an extended target.Within the target itself, the echo range R and the corresponding

delay time or will be continuous, so that the spectrum will covermore or less uniformly a band of frequencies whose widthdepends on the difference between the maximum and minimumecho ranges for the target.In other applications, the system may be called upon to discriminate between echoesfrom discrete objects at adjacent ranges,and it is the purpose of this Section to extend Gnanalingam'streatment of this topic.He considered two targets whose echoes were of equal amplitude and assumed that the relative phases of their spectral lineswere sufficiently random to justify the assumption that theaverage response Vs of two components VsI and Vs2 was given

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372 HYMANSAND LAIT: ANALYSIS OF A FREQUENCY-MODULATED CONTINUOUS-WAVE RANGING SYSTEM

(11) CONCLUSIONSThe foregoing analysis of a frequency-modulated continuwave radar system shows that, except in the trivial case

T = 0 and T = Ts ' the beat-note spectrum cannot be reduto a single line.An alternative method has been proposed which does pethe detection, with adequately long time-constant, of indivilines in the beat-note spectrum. As a refinement, chaswitching enables the additional ambiguity which occurs inregion of T = 1-T, to be minimized, and the use of a suitchosen carrier frequency eliminates the possibility of confubetween echoes from targets at harmonically-related ranges.Examinat ion of the beating between nearby target ecdemonstrates the effect of carrier instability and suggestseven under optimum conditions, the separation betweenbeat-note tones for two such targets must exceed l / (Ts cycles per second by a significant factor before the positionthe individual targets become clearly defined.

targets but gives little help towards their resolution. Thetion must exceed 21T/(Ts - T) by a significant factor beforepositions of the individual targets can be confidently identand even then the line st ructure of the spect ra may give onrelatively inaccurate estimate of the actual separation betwthe targets concerned [cf. Figs. 8(c) and (d). The probabof separation does not appear to be markedly altered iamplitude of one target is reduced by half. In the presencnoise, integration times must be increased if the resolutionot to suffer.The existence of the term T, - T in each denominatorthat range discrimination will deteriorate with increasing rathus a separation in wBi of 21T/(Ts - T) is equivalent to aseparation DR = 1 T c / 2 c x ~ , at T = 0, rising to 21TC /3cxTT = iTs and to ttcjo/T, at T = 1-Ts.

(12) ACKNOWLEQGMENTThe authors gratefully acknowledge the facilities made aable to them at the Royal Military College of Science. Twish to thank the Dean for permission to publish the paper.

(13) REFERENCES(1) GNANALINGAM, S.: 'An Apparatus for the DetectionWeak Ionospheric Echoes' , Proceedings I.E.E.,No. 1670, July, 1954 (101, Part III, p. 243).(2) KEEP, D. N.: 'Frequency-Modulation Radar for Use inMercantile Marine', ibid., Paper No. 1940R, Novem1955 (103 B, p. 519).(3) TUCKER, D. G.: 'Underwater Echo-Ranging', JournalBritish Institution of Radio Engineers, 1956, 16, p. 243.(4) KAY, L,,: 'A Comparison between Pulse andModulation Echo-Ranging Systems', ibid., 1959,19, p,(5) WOODWARD, P. M.: 'Probabi li ty and Informationwith applications to Radar' (Pergamon Press, 1953),p,

: I 'u1LLLt t

TARGET AMPLITUDES IN RATIO 2: IQUAL TARGETS

Cd) ,I i I ,Iii i! Jl,.uu. ~ 1 L J L u iLLt t t t t t t t~ S E P A R A T I O N i ~ l ? " T

I I :

';',iUi" J1lil L J1LH

by Vl = Vsl +Vs1 Eqn. (15)shows that, for stationary targets,at ranges having delays T, T + (h, the corresponding phase oflines at frequency ware (wo - tW)T and (wo - tW)(T + OT);thus there will be a phase difference (wo - tW)OT between theinterfer ing components . Beating will thus not be entirelyrandom, but will be more or less severe according to the rapidityand extent of carrier fluctuations; pronounced beating will alsooccur, for a v.h.f, or h.f, carrier, for quite small differences OTbetween the delays of the targets. Even in the idealized casewhen noise is absent, the problem is not readily susceptible of amathematical analysis which is at the same time simple andrewarding; instead, typical cases are presented in Fig. 8, whichshows alternative line spectra, with the upper and lower limits ofbeat ing, for target pairs of equal and unequal amplitude. Inthe left-hand spectrum of each pair , the range of the nearertarget is such that a line occurs exactly at the peak of the spectralenvelope; the right-hand spectrum of each pair shows an intermediate case.Figs. 8(a)-(d) show, respectively, separations in WBI of

Fig. 8.-Typicalline spectra for adjacent targets, showingupperand lower limitsof beating.The arrowheads below each line spectrum show the precise posit ions of WBI forthe targets. "

i21T/(Ts - T), 21T/(Ts - T), 127T/(T - T), and t27T/(Ts - T);.Fig. 8(b) corresponds to Gnanalingam's 'critical separation'.I t is legit imate to conclude from the Figures that there is littleprobability of resolution for separation in WB I up to andincluding 21T/(Ts - T); the presence of beating indicates multiple