IEEE 'TRANSACTIONS ON SPACE ELECTRONICS AND TELEMETRY

DEFINITIONS OF SYMBOLS USEDEC = collector supply voltage.E = the effective voltage across the series-connected switching

windings, essentially constant during flux reversal.Eo = output square wave amplitude.ei = the voltage across core 1 switching winding, essentially

constant during flux reversal.e2 = the voltage across core 2 switching winding, essentially

constant during flux reversal.eb = the voltage across core 1 feedback winding, essentially

constant during flux reversal.ec = the voltage developed across each core 2 control winding,

essentially constant during flux reversal.(ec)L = the voltage across core 2 control winding at the lowest

possible frequency fL-(e,)L = the voltage across core 1 switching winding at the lowest

possible frequency fL-(e2)L = the voltage across core 2 switching winding at the lowest

possible frequency fL-f = oscillator frequency, cps.fh = highest possible frequency, cps.fL = lowest possible frequency, cps.fo = oscillator frequency when control volts is 0.fv = oscillator frequency when control volts is V.m = the numerical value of the slope of the e,-to-f characteristic,

cps per volt.nli = the number of turns on each core 1 switching winding.n2 = the number of turns on each core 2 switching winding.

nb = the number of turns on each core 1 feedback winding.n. = the number of turns on each core 2 control winding.no, = the number of turns on each core 1 output winding.nO2 = the number of turns on each core 2 output winding.p = the numerical value of the slope of the v,-to-f characteristic,

cps per volt.vc = the dc input control voltage which can be set to any value

between 0 and V.Vce = the control transistor collector-to-emitter voltage drop

during the conducting half cycle.Vbe = the control transistor base-to-emitter forward voltage drop

during the conducting half cycle.Vb = the net adjustable fixed bias in the control circuit input leg.Vr = the voltage drop due to the control transistor collector

current flowing through the resistive component rc of thecontrol transistor collector circuit.

V, = the voltage drop between Ql collector and emitter plusthe drop in Ql collector circuit due to winding resistance.

(Vc)L = the value of the dc input control voltage when the oscillatoris at its lowest possible frequency fL-

V = the highest value of the dc input control voltage in thedesired operating range.

Vr = the value of Vr when v, is zero and f = fo.{l = the total flux change in core 1 switching from saturation

in one direction to saturation in the opposite direction,in webers.

{2 = the total flux change in core 2 switching from saturationin one direction to saturation in the opposite direction,in webers.

Transmitter Power Reduction with FrequencyTracking FM Receivers*

R. M. GAGLIARDIt, MEMBER, IRE

Summary-In analog communication studies much attention hasrecently been given to the apparent advantages of FM transmissionwith receivers employing feedback frequency tracking. Althoughtheoretical investigation has indicated some carrier power reductioncan be obtained over standard FM, it has been difficult to estimatethe operating capabilities of the system. In addition, little has beenpublished concerning the best parameter values for the constructionof a typical feedback receiver. This paper is devoted to a study ofthe feedback FM receiver which maximizes power savings overstandard FM for a given desired output SNR. System parameterssuch as transmitter modulation index, loop gain IF bandwidth,and baseband filter bandwidth are specified corresponding to thisoptimum design. Particular attention is given to the output SNRin the range from 15 db to 50 db where most practical analogsystems usually operate. The paper deals only with the feedbackloop containing two stages of filtering; design procedure is alsogiven when added constraints are imposed in terms of loop transientresponse. The possibility of using phase-lock loops as the dis-criminator within the FM feedback loop is also investigated. Someexperimental results are quoted which tend to verify the designtechniques.

* Received September 4, 1962. A portion of the contents of thispaper has been presented at the 1962 IRE WESCON Convention,Los Angeles, Calif., under the title "The Design and Capabilitiesof Feedback FM Receivers."

t Hughes Aircraft Company, Culver City, Calif.

I. INTRODUCTION

M\1T UCH ATTENTION'-` has recently beeni givento the apparent advantage of FM transmissionsystems with receivers employing feedback fre-

quency tracking. Such a system has been labeled "feed-back FM,"' or simply "FBFM," and a block diagramof the receiver is shown in Fig. 1(a). Recent studies4have compared its threshold carrier power to that ofstandard FM and indicated large power savings areobtainable with FBFM when operating at large modula-

1 M. Morita and S. Ito, "High Sensitivity Receiving Systemfor Frequency Modulated Wave," 1960 IRE INTERNATIONALCONVENTION RECORD, pt. 5, pp. 228-237.

2 M. 0. Felix and A. J. Buxton, "Performance of FM ScatterSystems Using Frequency Compression," Proc. NEC, vol. 14,pp. 1029-1039; 1958.

3 R. M. Gagliardi and T. Miller, "Minimum power widebandcommunication system for space vehicles," Proc. Nat'l TelemeteringConf., Chicago, Ill., May 22-24, 1961; pp. 10-11-10-29.

4 J. J. Spilker, "Threshold Comparison of Phase Lock, FrequencyLock, and Maximum-Likelihood Types of FM Discriminators,"presented at IRE WESCON, Convention, San Francisco, Calif.;August, 1961.

5 L. H. Enloe, "Decreasing the threshold in FM by frequencyfeedback," PROC. IRE, vol. 50, pp. 18-30; January, 1962.

March18

Gagliardi: Transmitter Power Reduction with Frequency Tracking FM Receivers

2) The carrier-to-noise power ratio at the IF filteroutput must exceed a threshold value for the dis-criminator to operate properly. (Throughout amajor portion of this paper a threshold ratio of12 db will be assumed.) Therefore, we can state

(a) C1 = (16)No2BIF (2)PHASE

MODULATINGSIGNAL +

f

(b)Fig. 1-(a) The feedback FM receiver. (b) Linear equivalent

FBFM receiver.

tion indexes. More recently, Enloe5 has shown analyticallyand experimentally the existence of a second thresholdeffect in FBFM which tends to decrease substantiallythe actual power savings. Nevertheless some reduction inrequired carrier power can be achieved by utilizing fre-quency tracking. The difficulty occurs in attempting todesign the best practical FBFM system using these resultsfrom the literature. The objective of this paper is toformalize a design procedure, and indicate associatedsystem capabilities, when FBFM is to be used.

In the analysis to follow it will be assumed that specifi-cation of a signal bandwidth and an acceptable outputsignal-to-noise power ratio (S/N)o are the prime con-straints on the design criteria, and it is desired to con-struct an FM system with feedback receivers requiringthe minimum amount of transmitted power to operateeffectively. For convenience, the power capability ofFBFM will be compared directly to that required in astandard FM receiver employing the same discriminatorand yielding the same output SNR.

II. FBFM RECEIVER DESIGN

The general FBFM receiver, having the form shown inFig. 1, acts so as to reduce the deviations of the IF modu-lating signal and therefore requires a smaller IF band-width than standard FM receivers. To operate effectively,the feedback receiver must satisfy the following operatingconditions:

1) The frequency deviation within the IF stage mustbe reduced by the loop gain to a value which allowspassage through the IF filter without appreciabledistortion. The relation between the transmittedmodulation index M and effective reduced indexof the IF stage MIF is given by

MTF = M/F (1)where F = 1 + (loop gain) and is called the loopfeedback factor.

where BIF is the noise bandwidth of the IF stage,No is the spectrum level of the input noise in wattsper cps, and C1 is the minimum carrier power tosatisfy the 12-db discriminator threshold.

3) If the input noise is normalized to the receivedcarrier rms signal, the mean-square phase-noisevariation at the local oscillator output must be lessthan unity to prevent interaction of input signaland feed back noise. It has been found5 that thisinteraction causes output signal degradation sinceit adds extraneous output noise to that alreadypresent due to the input noise alone.

Thus, the FBFM receiver must be designed so as torealize a given (S/N)o and simultaneously satisfy theabove three conditions with the minimum amount ofreceived carrier power. To examine this synthesis prob-lem, it is convenient to work with a low-pass equivalentof the actual receiver. Mathematically, the receiverbehaves as a linear negative feedback system as far asphase variations around the loop are concerned. It cantherefore be replaced by the equivalent system of Fig. 1 (b)provided the operating conditions are somewhat nearthose which satisfy the previously stated conditions.The input to the equivalent system is the phase variationof the modulated signal plus Gaussian noise having apower spectrum normalized to the received carrier power.The effect of the IF band-pass filter upon the IF modula-tion is represented by the equivalent low-pass filter F1 (s)acting upon the IF modulating signal. The discriminatorand VCO (voltage controlled oscillator) are consideredideal devices and therefore appear in the equivalent loopas a differentiator and integrator, respectively. The filterF2(s) is included to represent any low-pass filtering that isapplied within the loop. Its location and bandwidth forbest system performance will need to be determined fromthe design considerations. Note that the output of thesystem can be considered as either the discriminator out-put or the VCO input. In either case the output noisebandwidth will be the same provided all loop filtering isflat over the signal band and the postdetection filtering,which occurs after the feedback loop, is ideal and band-limited to the signal bandwidth. If all the loop filtering isaccomplished in the forward path, it will take less post-detection filtering to eliminate the major portion of theout-of-band noise at the output than if some filtering isdone in the return path. Therefore, for a given postloopfilter (nonideal) less out-of-band output noise will occur ifall the filtering is done in the forward path.

1963 19

IEEE TRANSACTIONS ON SPACE ELECTRONICS AND TELEMIETRY

In terms of the equivalent loop, condition 3) above canbe expressed as

NO (2BL) < 12C (3)

where C is the received carrier power, BL is the closed-loop bandwidth from phase input to VCO output, andthe left-hand side is the mean-square VCO output. If amean-square value of 0.1 is accepted as satisfactory forour purposes, and if we label C2 as the carrier powernecessary to attain this value, we can rewrite (3) as

When the above is solved, (4) is

C2- I0ION2O 1 (F -j)CO1 + CO) F (9)

It can be noted that C1 is independent of loop gain whereasC2 increases with increasing loop gain. The minimumcarrier power must therefore eventually increase aslarger loop gains are used. The operating condition whereC2 exceeds C1, which is the point where the minimumcarrier power begins increasing, occurs when

(F - 1)2 (2 0

-. _ ~> 10.F <+X _C2 = 5No(2BL). (10)(4)

The requirements of conditions 2) and 3) are equiva-lent to stating that the received carrier power must ex-ceed both C1 of (2) and C2 of (4) if the system is to operateabove threshold. Since each of these individual thresholdsdepends upon the quantities BIF and BL, they are ineffect somewhat independent and can be separatelycontrolled.For our study here, we will consider an FBFM receiver

having an equivalent linearized loop containing two stagesof low-pass filtering. This implies that the equivalent IFfilter F,(s) will be represented by the low-pass function

F,(s) = 1 +1¢/-*.(5)

If the effective modulation index in the IF stage MIFis not much larger than unity, this equivalent Fi(s) iscommensurate with that of a double pole band-passfilter having a 3-db bandwidth of 2X1. Since the loop gainwill generally be significant, this choice for F1(s) appearsreasonable.The second stage of filtering is due to F2(s) having the

transfer function

F2(S) 1 (6)

This second filter stage is generally an RC low-pass filterfollowing the discriminator but in theory can be placedanywhere in the forward or return path. We can notefrom Fig. l(b) the transfer function from input to VCOphase output is the same for either configuration. There-fore, the quantity BL of (4) will be same no matter wherethis second stage is placed and, if desired, it can correspondto a second band-pass stage in the IF. The effect of F2 (s),of course, can be reduced by increasing c2 relative to thesignal bandwidth.Having defined F1(s) and F2(s), and assuming F2(s)

follows the discriminator in the loop, (2) becomes

C1 = (32)rNoco, (7)

and the closed-loop bandwidth BL isco ~~F- 2

BL = ± ( ) | df. (8)F!F+ WI + W2)jX + (j21@

CW1W2 Wl(A)

Let us consider the significance of (10) when differentfilter bandwidths are employed. The equation indicatesthat the range of loop gains for which C1 determinesminimum carrier power can be increased by decreasilngthe co(0l2 + c,2 factor. Since this factor increases with w2,for a given choice of c1, the 3-db banidwidth of the low-pass filter should be as narrow as possible. If we wanitthe signal deviation to be reduced by the loop gain overthe full signal bandwidth, the loop filter bandwidthmust be at least as large as the signal bandwidth. There-fore, W2 should be chosen equal to the signal bandwidth.The 02/7(0 + w2 factor also varies inversely with X,.

This means we should enlarge the IF bandwidth, whichincreases C1 but requires less of a deviation reduction topass the FM signal through the IF filter. We conclude thatusing an IF bandwidth greater than its minimum valuerelaxes loop gain requirements for distortionless recep-tion of a given modulation index, and also extends therange of loop gains for which the discriminator thresholddetermines the minimum carrier power. At the same time,however, it increases this minimum value.We can represent the above statements graphically by

plotting the minimum required carrier power C,,, as afunction of the modulation index for different values ofIF bandwidth. If Co,, is the signal bandwidth, we will let

(1 a)

@1 = kcom 2 (1lb)

where k is a constant equal to or greater than two. Theabove definition defines the 3-db bandwidth of the IFstage as kO(,m. The MIF necessary for distortionless trans-mission for each value of k can be determined from Fig. 2,which is based on passage of all sidebands exceeding Iper cent of the received FM signal power. (It can benoted that this figure yields a more conservative band-width than the 2(MIF + 1)com formula normally used inFM analysis.) The loop gain must reduce the transmittedM to this required value of MTF by means of (1). Therequired Cmn is then given by (9) if (10) is satisfied, or it isgiven by (7) if it is not. The resultant curve is shown inFig. 3, where each point corresponds to the minimumreceived carrier power (normalized) necessary to transmitthe indicated modulation index for three integer values

20 Mlarch

C02 = (0J)m

Gagliardi: Transmitter Power Reduction with Frequency Tracing FM Receivers

that a replot of Fig. 3, in terms of output SNR ratherthan modulation index, would be more meaningful. Thisconversion can be obtained by the use of the well-acceptedFM improvement formula

(S/N)o - 3M2C2Now,m (12)

1 ___ 1 ___ { ___ ] ____ 1 ___ J based on sinusoidal modulation and ideal postdetection2 4 6 8 10 2 14 16 18 filtering. The result is shown in Fig. 4 throughout ranges

IF BANDWIDTH/W, of output SNR that are most common in practical appli-2-3-db IF bandwidth vs IF modulation index. cations. Clearly, the required power reduction over

standard receivers does not continue to increase for allvalues of SNR as idealized analysis predicts. Note that for

DISCRIMINATOR THRESHOLD =12db (S/N)0 < 45-db largest power savings occur for smallerMEASURED POINNDWIDTH kwm values of k while at higher (SIN), larger values of k

STANDARD yield largest power savings. This last statement is quite,___ ____ ____ FM important since it indicates that a range of (S/N)o existswhere power requirements are improved by increasing

B / sk=8k-2 the IF bandwidth. This is contrary to the popular beliefL/1+k=61$k=6 61 4}FBFM that the FBFM receiver will attain its maximum ad-

$ / @4 @ 4 _vantage when the IF filter is as narrow as possible.) 1, 2 Inspection of the FBFM curves also show that maxi-mum power saving occurs at the break point correspond-

;0 15 20 25 30 ing to the end of the horizontal portion and the beginning0 5 10 15 20 25 30 35 of the increasing portion of each curve. This is the point

at which the discriminator threshold and feedback thresh-(M = deviation/smi). old are exactly equal; that is, C5 = C2, and (10) is satisfied

with the equal sign. Such a point theoretically exists forall output SNR although only three points are indicated

included for comparison is the power require- in Fig. 4, corresponding to the IF bandwidths shown.a standard FM receiver transmitting the same This implies that given an output SNR maximum power1. savings are obtained if the FBFM receiver operates withtportion of each FBFM curve is the region values of k, F, and M at which the discriminator thresholddominates, while the increasing portion repre- exactly equals the feedback threshold. It will be shownes of loop gain for which the feedback threshold in Section III, however, that rigid application of thisant. The results indicate power savings are rule leads to poor transient response.in transmitting a given 1ik by frequency Let us label with a subscript c the values of k, F, and

;echniques. M which are associated with this maximum power savings, however, one prime difficulty encountered in point. We then have the following relations amongg to interpret and apply Fig. 3 to the design these parameters:

(F, 1)2 - 5(k, + 1)

F(

(S/N)0o = 3

(13)

(14)

These two equations, along with the requirement thatthe IF bandwidth must be sufficiently large to pass theIF modulated signal, completely identify the three param-

eters. A plot of these "optimum" parameters for eachvalue of (S/N)o is shown in Fig. 5. The normalized carrierpower, Cm/NN,,,. required to satisfy the system thresholdwhen these parameters are used is given by the expression16wlkc, obtained from (7) and (llb). The correspondingcarrier power required by a standard FM receiver yieldingthe same output SNR can be obtained from Fig. 4. Theresultant power savings of FBFM over standard re-

ceivers is approximately 6 db when these optimum

of a practical system. This difficulty arises from the factthat it is basically unfair to compare FBFM and standardFM receivers on the basis of modulation index. In mostcases, the specified data for an FM communication systeminvolves the receiver output SNR. Parameters such astransmission modulation index are to be chosen by thesystem designer. The use of Fig. 3 now becomes restrictedsince FBFM and standard FM receivers require differ-ent modulation indexes to obtain the same output SNR.This is due to the fact that output SNR of any FM re-ceiver is independent of feedback and depends onlyupon the square of the transmitted modulation index andthe received carrier power. Since FBFM uses less carrierpower, a given M will result in a smaller output SNRthan when standard receivers are used. Hence, whenemploying frequency feedback, a larger index must betransmitted to obtain a given output SNR. It follows

Fig.

4C

36

-aw

- 32

E

_ 28

24

20

16

Fig. 3-Rec

of k. Alsoment for Evalue of IV)The flal

where C5sents valunis importkobtainablefeedback t

There isattempting

1963 21

IEEE TRANSACTIONS ON SPACE ELECTRONICS AND TELEMETRY

DISCRiMINATOR THRESHOLD =12db higher than the signal bandwidth. These oscillations may3db FBFM IF BANDWIDTH =kwm not occur at the receiver output due to the lower cutoffof the low-pass postloop filter. However, violent ringing

_______ ____within the loop may lead to saturation effects and outputsignal distortion. Therefore, the application of the pre-

=2 viously derived system design leads to relatively poor

t____________________________________6_FBFM loop transient response.A more satisfactory design will usually require a higher

, than the optimum parameters yield. The dampingfactor can be improved, without changing (S/N),, byincreasing k and decreasing F. This implies moving to a20d30 40 50 60 higher horizontal curve in Fig. 4 at the same (S/N)o,with the new operating point yielding a more satisfactorydamping factor. Clearly, the power reduction over stand-ard receivers is reduced from that of the optimum operat-

6 ing point previously discussed. A plot of this power re-POWER SAVINGS OVER kc duction as a functioni of loop damping factor is shown inSTANDARD FM = 6db

5 __ Fig. 6. The abscissa is also labeled in degrees of loop phasemargin for convenience. Note that improvement of the

4 _z/C / damping factor from 0.2 to 0.5 reduces the power re-duction from 6 db to 2 db for the output SNR range

3indicated. A repeat of Fig. 5 in terms of the design param-

2 eters associated with an improved vof 0.5 is shown inFig. 7. A set of such parameters exists for each point ofFig. 6, with Figs. 5 and 7 corresponding to only two suchpoints. Parameters for other points can only be found by

020 30 40 50 60 cross plotting (13), (14), and (15).(S/N)o,decibels As a design example, suppose a signal bandwidth of

200 kc is to be transmitted by FM and recovered at a-Receiver design parameters. Damping factor = 0.2. SNR of 40 db. If a damping factor v = 0.2 is satisfactory,

then Fig. 5 yields the FBFM receiver parameters:

parameters are used, relatively independent of SNR formost of the range indicated. Note for a SNR less than38 db, k remains fixed at its minimum value of two. Thiscannot be considered an "optimum" value but rather alimitation on loop filtering, and the 6-db power savingsis not attainable in this range.

III. Loop TRANSIENT RESPONSE

So far the system design has concentrated on maximiz-ing the power savings over standard FM receivers inobtaining a given output SNR. It was found that thesystem should operate at the optimum value of M,, kc,and F,. However, one aspect of the system design problemthat has not been considered is the transient response ofthe system. The feedback loop, being of the second order,has a damping factor given by

2V+2 (15)2 \12Fk 15

Hence, the selection of IF bandwidth and loop gainuniquely determines the damping factor. Substitution ofkc and Fc from Fig. 5, for SNR in the range indicated,yields a v of approximately 0.2. This corresponds to a

.50 per cent overshoot within the loop when a frequencystep input is applied. This is accompanied by resultanthigh-frequency riinging whiclh usually occurs at frequenicies

IF bandwidth 480 kc,Loop gain = 23,Modulation index = 8,Power savings over standard FiMI = 6 db.

The FBFM receiver will be identical to Fig. 1 with a

low-pass loop filter bandwidth of 200 kc. If a bettertransient response is desired, corresponding to a v of 0.5,then Fig. 6 yields a new set of design parameters:

IF bandwidth = 1.2 Mc,

Loop gain = 4.5,Modulation index = 7,Power savings over standard FM@V = 2 db.

In summary, we can conclude the following. The maxi-mum reduction of minimum required receiver power foracceptable operation of a frequency feedback receiver ata given output (S/N), occurs if the system operates withthe discriminator and feedback thresholds exactly equal.This operating point leads to relatively poor transientresponse (= 0.2). For improvement of the transientresponse, without resorting to additional filtering com-

pensation, we must operate the receiver with the dis-criminator threshold dominating, with a subsequent re-

duction in power savings.

40

36

32

cu1D 28

E| oZ 24

20

16

F

40 r

30 F

20 F

10

0

Fig. 5

22 MIarch

LL

Gagliardi: Transmitter Power Reduction with Frequency Tracking FMf Receivers

6zo 5

z'4

0

O 0.1I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0 DAMPING FACTOR23 40 518 710 PHASE MARGIN,

Fig. 6-FBFM ploo1

12

20

15

L- 10

5

0

10

8

6

4

2

Fig. 7-Recei'

I

To verify thceiver shown irfor operation areceiver. A 5-kc70-Mc RF caria ,1r] b TV -f I

been superimposed upon Fig. 3, and are plotted relativeto the standard receiver case. It was difficult to obtainvalid data for small IF bandwidths when operating atlarge modulation indexes, since the combination of largeloop gains and extraneous loop phase shift, due to theratio discriminator, caused the system to oscillate. Hence,complete verification of Fig. 3 was not obtained, but theusable data tended to follow the trend of the theoreticalcurves.

V. THE FBFM RECEIVER WITH PHASE-LOCK LooPDISCRIMINATORS

DEGREES In Section IV, it was found that minimum carrierlower reduction over standard FM receivers vs power for operation of the FBFM receiver, with a suitabledamping factor and phase margin. transient response, was determined by the threshold re-

quirement of the discriminator. A 12-db threshold wasassumed for all previous analysis which led to a require-

POWER SAVINGS OVER ment for carrier power given by (2). Clearly, if a reductionSTANDARD FM=2db k poevyk'} eutoSTANDAR FM kof the 12-db discriminator threshold can be achieved,further reduction in power requirements over that alreadypredicted can be realized. In terms of Fig. 3, this implies

__/_/_/ a lowering of the FBFM curves, and greater power/ / /F savings over standard FM receivers. One type of FM

discriminator that obtains such a threshold reduction isthe phase-locked loop discriminator.68 In this sectionwe will investigate the possibilities of using a phase-locked loop as the discriminator within the FBFM receiver

0 30 40 50 60 loop, as shown in Fig. 8. Again our objective is to mini-(S/N)o,decibels mize required carrier power at a given output SNR, with

ver design parameters. Damping factor = 0.5. a reasonable loop transient response.The low-pass phase equivalent system will be the same

as Fig. l(b) with the ideal discriminator (differentiator)[V. EXPERIMENTAL RESULTS being replaced by the low-pass equivalent transfer func-

tion of the phase-locked loop. As before, the filter F1(s)e previous design procedures, the FM re- represents the IF equivalent filter, F2(s) is the loop filter,

Fseig.e1 wa cstandardrutedine te lanFBor and the ideal VCO is taken as a pure integrator. UnderIS either a standard receiver or an FBFM.. phase-locked conditions, the transfer function from phaseauier. ThesiFgcarr was2sedformodulan Mc input to frequency output for the phase input to fre-rier. The IF carrqer frequency was 2 Mc

011Crn '-oirlT%Q,0 -T;+ -ylquency output for the phase-locked loop is given byanut iiie 1f litelr wu:5 u tsllg,e-uut: eIuluijupum1twi)aS iulliadjustable 3-db bandwidth. The loop filter was an RCsingle stage adjusted to the 5-kc bandwidth, while thepostloop filter involved two stages of RC filtering withthe same bandwidth. The discriminator within the loopwas a simple ratio detector type.

Experimental data corresponding to Fig. 4 was difficultto obtain since each receiver configuration had to be com-pared at the same output SNR. It was found more con-venient to obtain data corresponding to Fig. 3, in whichcomparison was made at the same transmitted modula-tion index. Each system was tested with carrier deviationof 25, 50 and 100 kc, corresponding to indexes of 5, 10,and 20. At each index, the minimum carrier power atwhich thresholding occurred was determined for thestandard receiver and for FBFM receivers with IF band-widths of multiples of twice the signal band. The thresh-old point was considered to be the knee of input-outputSNR characteristic, where output SNR begins falling offfaster than input SNR. The results of the experiment have

SF(s) = 2

1 + BSO+ 2Bo Bo

(17)

where Bo is the bandwidth of F(s) in radians/sec. The mini-mum value for Bo is approximately equal to the signalbandwidth Win and we would like to operate the phase-locked loop at this minimum point to obtain the maxi-mum power advantage. Note that in employing a phase-lock loop as a detector we are effectively replacing theideal discriminator by a cascade connection of an ideal

6 C. E. Gilchriest, "Application of the phase-locked loop totelemetry as a discriminator," IRE TRANS. ON TELEMETRY ANDREMOTE CONTROL, vol. TRC-4, pp. 20-35; June, 1958.

7A. J. Viterbi, "Acquisition and Tracking Behavior of PhaseLocked Loops," Jet Propulsion Lab., Calif. Inst. of Tech., Pasadena,External Publication 673; July, 1959.

8 B. D. Martin, "Threshold improvement in FM subearriersystems," IRE TRANS. ON SPACE ELECTRONICS AND TELEMETRY,vol. SET-6, pp. 25-33; March, 1960.

1963 23

0

IEEE TRANSACTIONS ON SPACE ELECTRONICS AND TELEMIETRY

Fig. 8-A FBFM receiver with phase-lock loop FM discriminator.

discriminator and a second-order low-pass filter with abandwidth equal to the signal bandwidth. As will beshown subsequently, this extra filtering will need to becompensated if the system is to operate in a stable region.

Let us consider threshold operation of the phase-locked loop when operating with sinusoidal input modula-tions. Thresholding is obtained when the total phaseerror after the phase detector exceeds 1.57 radians. Thistotal phase error is composed of errors in tracking plusaddition errors from random noise occurring within theclosed-loop bandwidth BLO. Since this noise variationrepresents a random effect, the threshold condition canonly be estimated on a statistical basis. For example,with a 300 acceptable error allowed in tracking, an 8-dbinput carrier-to-noise ratio in the BLO bandwidth mustbe obtained if the phase-lock loop is to exceed threshold99.9 per cent of the time.8 Higher probabilities of re-maining in lock require proportionally higher BLO SNR's.This BLO bandwidth is 3.33 times the B0 bandwidthpreviously defined. Hence, under these operating con-ditions, the carrier power necessary to satisfy the phaselock threshold is given by

C1 = 6.3No(2BLo) (18)= 6.3No(6.66Bo)

For best operation, Bo will have its minimum value(= W) and the C1 indicated above is about 2 db less thanthat of (2) with its minimum value. Note that C1 aboveis independent of the IF bandwidth and depends onlyupon the phase-lock loop bandwidth and the noise levelin the receiver. Reduction of the IF bandwidth is no longera factor in decreasing power requirements necessary tosatisfy the discriminator threshold. The value of havinga feedback loop around the discriminator is manifestedwhen we consider the maximum modulation deviationthat can be detected with and without feedback. This canbe shown as follows. The sinusoidal tracking error ETis related to the transmitted deviation D by the approxi-mate expression

XT FB (19)

where F is the feedback factor of the major loop. Re-writing, we have

Bo [(0.707)D 11/2Wm LETFO)m j ~~~(20)

For an acceptable tracking error of 300, the quantityD/Fw,m is limited to a value of unity if B0 is to equal Wi.With no feedback (F - 1) this requires D = wCm as themaximum value. However when loop feedback is applied,D - Fwmfi and can therefore be substantially increased,yielding an improvemenit in output SNR. Hence, the FMfeedback loop is used to reduce a large transmitted de-viation to one that allows the phase-lock loop to operateat its minimum banidwidth. In a sense, the feedback loopis acting as a matching device between the transmitteddeviation and the phase-locked loop. The advantage ofbeing able to transmit a larger deviation is that a largeroutput SNR is attainable without altering the operatingbandwidth of the phase-locked discriminator. However,if additional constraints are imposed in terms of thereceiver transient response, a limitation still occurs onthe maximum transmission index that can be used. Thiscan be shown by considering the loop damping factor.The combination of a single stage IF filter, ideal VCO,and the phase-locked loop transfer function of (16)creates a third-order feedback loop, and therefore F2(s)must introduce some phase lead if the system is to bestable with any appreciable gain level. For maximumeffect F2(s) should contain a pole which yields Ino appreci-able effect on the loop gain function plus a zero of trans-mission at the pole of the IF stage, so that the completeopen-loop gain function is given- by

A(s) F-IA12s s2 .

I _B

_BO 0B2(21)

The damping factor of the FBFM receiver is thengiven by

t_ I .\12 (22)

Hence, the damping factor uniquely determines themajor loop feedback factor. For example, a dampingfactor of 0.35 limits the feedback factor to a value ofapproximately four. This means the deviation is re-stricted to a value of 4'Dm if the operating conditionspreviously selected for the phase-lock loop are to bemaintained. The 2-db power savings produced by thephase-lock loop is also restricted to this range, but a12-db improvement in output SNR is obtained over thephase-lock loop with no external feedback.

Suppose it is desired to transmit a larger deviationthan the maximum dictated by the loop gain. Two alter-natives are possible. If the larger deviation is transmittedwith no increase in carrier power the tracking errorincreases, and the probability of the phase-lock loopstaying in lock is reduced. If, on the other hand, the 30°tracking error is not to be exceeded, (20) indicates that Bomust be increased, with a corresponding increase in re-quired carrier power, as given by (18). Therefore, theadvantage of the phase-lock loop FBFM system begins

24 Mlarch

Shaft: Distortion of Multitone FM Signals

diminishing. This can be shown graphically in Fig. 9,where CI/Now,, is plotted as a function of D/Om. All curvesrefer to FBFM operation. The phase-locked loop dis-criminator case (labeled PLL) is shown for various valuesof major loop feedback factor. For a given F (loop transientresponse) the PLL curves begin increasing when the factorD/Fwm exceeds unity. These break points can be shiftedto the right by accepting a larger phase tracking error anda greater probability of exceeding threshold. Also in-cluded for comparison is the minimum curve of Fig. 2,using conventional FM discriminators.

It is interesting to note that the F = 1 curve (no feed-back) represents operation with a phase-lock loop dis-criminator and no external feedback. This can be com-pared to the "standard" FBFM receiver using con-ventional discriminators. For low deviations, the phase-lock loop is about 2 db better as far as power requirementsare concerned, but the FBFM becomes superior at higherdeviations. As feedback is introduced in the FBFM-PLLreceiver, the range of advantage in the phase-lock case isextended.A second factor limiting the use of larger transmitter

deviations is the threshold requirement of the VCO phase

24

22

20

18

16

14

_ ~~~~~~~~~-- F I_/ ~~~3 _-

-PLL

2 3 4 5 6 7 8 9

D/wm

Fig. 9-Required carrier power vs transmitted modulation index forFBFM operation with PLL discriminators.

noise in the major loop. Substituting into (4) yields

C2 = 7.07NoBo(F - 1)2/F.

This will require more carrier power than that demandedby CX of (18) at loop gains such that

F> 7.9.

We concluded that the data depicted in Fig. 9 can beextended to feedback factors up to approximately eight

Distortion of Multitone FM Signals Due to

Phase Nonlinearity*PAUL D. SHAFTt, MEMBER, IRE

Summary-An equation is developed for the distortion of multi-tone FM signals caused by transmission through a network withnonlinear phase characteristics. A simplified form involving onlyone term is also given and the conditions of its applicability are

discussed. The power spectrum of the distortion is derived forthe case where the simpler form is valid. Thus, the distortion-to-signal ratio in any frequency band may be obtained. Applicationsare presented where the results are used to determine equipmentlinearity specifications and to determine the distortion whichresults from propagation through the ionosphere.

I. INTRODUCTION

N FREQUENCY modulation receivers, the outputvoltage is proportional to the time rate of change ofphase. Therefore, phase changes associated with

* Received July 6, 1962. This work was supported by Air ForceContract No. AF 04(647)-829.

t Philco Corporation Western Development Laboratories,Palo Alto, Calif.

transmission through a network' can cause an incorrectoutput voltage or distortion. Previous work2 has dealtwith the harmonic distortion of a sine wave and inter-modulation distortion of two sine waves. Here, the modula-tion will be an ensemble of sine waves which is similar to aband of white noise. This is a practical case because fre-quency-division multiplexed signals can often be repre-sented in this manner.The method of attack is to expand the network phase

characteristics into a Taylor series. The modulation isintroduced and then the autocorrelation function of thedistortion is obtained. The autocorrelation function atzero time delay gives the total power of the distortion.

1 The network can be a propagation path, transmission line,etc., as well as a lumped electrical network.

2 J. J. Hupert, "A method of evaluation of the quasi-stationarydistortion of FM signals in tuned interstages," Proc. NEC, vol. 8,pp. 445-461; 1952.

1963 25