A NOVEL ANALYSIS OF FM DISCRIMINATORSBY TAYLOR'S EXPANSION

Choong Woong Lee,

College of EngineeringSeoul National University

Seoul, Korea

Abstract

The theory of FM discriminators is presented by expan-ding its transfer function into Taylor's series, anid theapplication of the results to the typical FM discriminators(slope discriminator, Travis discriminator, Foster-Seeleydiscriminator, and line discriminator) is added.The output signal of the FM discriminator expressed by

Taylor's expansion shows the clear relationship betweenthe fundamental and the harmonics. The design equationsobtained from this analysis enable a designer to constructFM discriminators wbich wvill provide a desired low distor-tion performance.

INTRODUCTION

The extremely low distortion in the FM wave demodula-tion has become required to eliminate cross talks in themultiplex communication system. Previous papers (1), (2) whichdealt with FM discriminator linearity attempted to showthe correlations between the fundamental and the harmonicsby expanding the magnitudes of the tuned circuits in thediscriminator with Taylor's expansion. However, the app-

roach reported here differs from those previously employedby the following: The output signal of the discriminatorwill be shown in a series form by expanding its transferfunction into Taylor's series, which shows the clear rela-tionship between the fundamental and the harmonics, aindthen the amplitude of substantially dominant harmonic inthe output waveform will be set to be minimized. Fromthis condition, an attempt will be made to obtain thedesign equations which enable us to suppress the harmonicdistortion to the desired low level.

This approach is generally applicable and will be appliedto the four typical FM discriminators(slope discriminator,Travis discriminator, Foster-Seeley discriminator, and line

discriminator).

THEORY OF FM DISCRIMINATOR

We consider the output signal from the FM discrimin-ator wxlxose anplitude and phase characteristics vs. frequ-ency are non-linenar as shown in Fig. 1, where the oper-ating point of the discriminator is at wco=w. The ratio ofthe output, eco(w), to the input signal, e,(@), i. e. thetransfer functioni of the discriminator, h((w), can be given by

(1)

Since the transfer function is analytic and non-linear,Ih(w) and 0(w) can be expanded in the neighborhood

of the angular center frequency, wu, by Taylor's formulaas follows:

"t

K o(, or 0

-w~o.

Q

0

Fig. 1. A characteristic curve of the discriminator whoseamplitude and phase characteristics vs. frequencyare non-linear.

ikh(o)) I =a+ao(o)-,w ) -a2(w -&))2+af3(Ww(D)) (+..........

and

0(w) -iSo±+1l (w-0)) + 2 (Co- ) 2

+±I3(W-w0)3±. (3)-

Received January 7, 1969 110

h (a)) = h ((o) ej8 (0))

(2)

(3)

where

ao- th(o¢)!, al-Ih(c)', a2 2!

1h(wci)ita3 ..

3!p°-0(@¢)7 p1-at((DC) 7 i=(OW) 2)''

~0 O(w0), iJ=O'GoWC), P2 2! P2 3

If an FM signilal, ec=Ecej(&ct+mfsinpt0 is applied to the

discriminator, then, under the quasi-steady-state(3x, withthe use of (2) and (3), the output waveform, eo(w), canbe written as follows:

eo(o)-h(w)ec((w)-E£Ca0 + azlOCOSpt + a24w2CoS2pt+ a3Jw3cos3pt+x ejC'ct+ MnfsinPt + PO +AI ldacosPt + P2Jw2cos2pt+ 3Ljo3CoS3pt + ...).. 4

where

J -=maximum deviation of instantaneous angularfrequency from average

p-angular modulating frequency

co,angular frequency of carrier wave

mf=modulation index

Wc-vv,*Acospt

(4) shows that the non-linear phase characteristic of the

discriminator has no effects on the AM component of the

output signal, but has effects on the FM component of it.Hence, only the AM component of (4), eA, is

necessary for the expression of the output waveform.

That is

e.4-E {ao+ a,zlcvcospt + a2zlc2cos2Pt + a3LJl3cos3pt+ a4Acv4cos4pt + a5zlcv5cos5pt +......I (5)

or

ei-E. { (ao±+ a2c2Ot 38 4

+ 1O AsJO) cospt+ (a2Jw2+ a44cv4) cos2pt,

j-4 (a3zicv' 4- Ctals") cos3pt-i a4Av4cos4pt

+ 17 a5sJW5cos5pt+. (5)'

It is clear in (5) that the less a2/a1, a3/ca1,.... are,

the less the distortion is. Since the substantially dominant

amplitude of harmonic is a2 (unbalanced type,e.g., slopediscriminator), or a3 (balanced type, e.g., Travis, Foster-Seeley, line discriminators) as it will be ?hown in theapplication section, it would be most effective for improv-ing the discriminator linearity to. set a2-0 or a3=0,(see Appendix II). From this condition, and consideringthat a, would be, in practice, shown in terms of discrim-inator circuit elements (e.g., L, C, CO, Qe), we can expectthe design equation for the desired low distortion discrim-inator.

Since the terms after the sixth in (5) are negligible incomparison with the fundamental, the percentage of thetotal harmonic distortion, D, can be given approximately by

D 164(a2+a4w) ±442(4a3±5a54w2)22 (8 aL +6a34v+ SaAoA)

+AJw04(4a42+a52MJc2) x 100% (6)

ANALYSES OF FM DISCRIMINATORS

In this section, the application examples of the theory tothe analyses of the several typical FM discriminators willbe shown:

(1) Slope Discriminator

Fig. 2 is a slope discriminator where C is the totaleffective capacitance. The transfer function(4) of the slopediscriminator can be given by

h( v) - Kr, 1reis I1+j26Q,

- Kres- ( etQ21~+ (23Qe)2 (7)

where

Kres --gmc.oLQeg =transconductance of tube

co1

1Qe LC

Qe f0/half power bandwidth

111

4W=w - W0

Since the third term of (8), a2, predominantly influencesthe non-linearity of discriminating characteristics, it wouldbe required for the better linearity to make a2-0. Fromthis condition, the following relation is obtained:

2 Q, IdW _-4W -1T (9)

Fig. 2.A circuit diagram of slope discriminator whose Cis the total effective capacitance.

The output signal of the discriminator for the inputsignal, e, - Evej (wc+mfsinPt) can be expressed as in theform of (5)' by

eo'=t 7 Ieo(,)

=7)KresEc { ( Ioa2a02+ - 3a44w')+ (asA + 3 a34W3 +- 16 a5SW) cospt

+ I(a24W2+a4a4co)cos2pt.....3 (8)

where

7=rectification efficiency of diode

1

[1+ (2Qe - ) 2j 1/2

(2QeWI ) ( 2Qc)al--[i± (2Q. JW )2 3/2

1-2 (2Qe4W )2j ( 2Qe ) 21- 2 [i± (2Qc 4W) 21/

2(2Qe4j -T3J (2Qe JW ) ( 2Qe ) 3

2F1-(2Qe W)17/2

-7(2Q0e±) )3 (24W 2 W(2)I

a3 - -27

Substituting (9) into (8), we have

eo' -EcKres 0.815-0.384 (2 Qe ) cospt

-0. 209(2 Qe )cos3Pt- 0. 141. (2QeQ ) cos4pt- -

(1()

(10) shows that the series does not converge rapidly

unless 2 Q, (K<< 1. Therefore, the slope discriminatortoo

cannot discriminate an FM signal with wide frequency

deviation without considerable amount of harmonic distor-

tion.If we consider the case that the ratio of the coefficient

of cos4pt to that of cospt is less than 1/100 to neglect the

terms after the fifth harmonic, the correspoiiding value of

2Qe- Is

2Qed

-< O. 30o

(11)

By the use of trigonometric formulas, (10) can be re-

written as

Cot-EcKres 0.0815-0.0528 (2 Qe )

- 0. 384 (2 Q, ) + 0. 157 (2 Q-) cospt

0

-0.0705t2Qc-) cos2pt-0. 05225(2 Qe , ) cos3pt

Ato4

(0.0176 (2Q)

cos4pt. (12)

When the value of 2Qe 'do)is

A=0.3

woo (13)

112

C

the percentage of total harmonic distortion is 1. 267%.By tising (13) and (9), we get the relations among fo,fc, and the desired linear bandwidth, By,,, as follows:

fe =f0± 1 . 178 Bm (14)

where

Bm --2af

(2) Travis Discriminator

Fig. 3 is a circuit diagram of Travis FM discriminatorwhich is used for the demodulation of wide band FM

signal such as in the telemetering system. The upper part

of Fig. 3(a) is shown in Fig. 3(c), where the transfer

function of the circuit(5', hl(o), can be given by

where

Kresi =gmlW1MQj

g,-- transconductance of tube

AI-mutual inductance

1

,vLs, C,

(0 -01 -

1 -,'Yc81 (

01__(0(0

Q1-f/half power bandwidth

Similarly, the transfer function of the lower part ofFig. 3(a), h2(o), iS

e K )_2Q21

(16)4+Re2Vi I jtar22I+(282QD2 2

where

{C~ ~ ~ ~~

F -, F

(b)ci P M

e,+ e,,,C

K (C)

Fig. 3.(a) A circuit diagram of Travis discriminator.(b) Frequency characteristic curve of Travis

discriminator.(c) The equivalent circuit diagram of the upper

part of (a).

ei I +J26j12h1Go).LL K 1

Kres 1 e-jtan bQ1 (15)Vres e

Kres2 - g,52AfQ2

1(02

L32 C2

82 (0 2 (02 )

0)2

Q2 ---f2/half power bandwidth

Using (15) and (16), the output signal of the discri-

iiinator, eo, iS

eo-27j(,ejj -- Ce2 )

)2Itc { C ('Q )2Kres2

7 2_ 3 (17)

where

at-aiplitude of input FM signal

--rectification efficiency of diodes

Considering Nv(02-cswe can put Kresj-~KrCs2 Kres,and Q1-Q2 (2e, then, (17) becomes

I~E7r~1~i1 1 1(8)eo--Ec Kres 1 -"-i1;) A/1+(282Qe) Jwhe(21Q) V,I (252Qe)2

where

Kres gmc0)VAIQe

113

Using Taylor's expansion formula, (18) can be expressedin the series from of (5)' as follows:

eo=EcYKres { (aiw + a,4w +10- a5sdo) cospt

+I (aA3l+ 45 a54W5) cos3pt + -La,Jw5cos5pt+ i

(19)

where

2 (2Q W ) ( 2Qe)

- 1 + (2 Q JW )21 3/2

D=- 1.70662

4 3.35(2 Qe c ) .

(22)

It follows from (22) that the distortion is 2.91% for

2QeQ h, _1 and for the distortion less than 2.91%, theeof

4dwvalue of 2Q, ~ should be

2Qe (.<1 (23)

From (22), we also get the relationship between theeffective Q and the distortion as follows:

2 (2Qe_))-3J (2QA) (23 2)a, - ±1+ 2 W)2j 7/2

fc_ z 62D \1/4e2af ( .706+3. 35D/

[8(2QeJW 4_40O(2QejW )2+4 [1 2(Qe 2' )2} 11/2

Cco

(2QW2))

DJW= oC- (01- (2 C)c

If the linear portion of frequency characteristic curve,B., and the distortion, D, of the discriminator are deter-mined, by the use of (20) and (22), the upper and lowerresonance frequencies, f2, fi can be given by

f2=fc+O.61248Bm( 1.706±3.35D 1/4~~ tmk 62D

ff=fc0-.6124Bm( 1 *706+ 3.35D ) 1/4 (25)

In order to improve the linearity of (19), a3/a, anda5/a, should be made as small as possible. Since a3/aj>a5/al, it would be effective for the good linearity of thediscriminator to set a3-0, from which the following rela-tion is obtained:

4W2QC - 1 (20)

Substituting (20) into (19), (19) can be given by

eo--Ec2Kres t0.62 (2Qe~) -0.0335X

(2 Q J-) 4 cospt 0.01675 (2 Q, j) cos3pt

where

Bm=2Df

(3) Foster-Seeley Dscriminator

Fig. 4 is a circuit diagram of Foster-Seeley discrimina-tor, where e,, e, can be shown(6) by

RFC

-00030 (2Q,e a )cos5pt....} (21) Fig. 4.A circuit diagram of Foster-Seeley discriminator.

Since the terms after the fifth harmonic are negligible,the distortion of (21) can be given by

(1 +j23Qe) +J ki. /L2el or e2 -gine-wOLIQ, 2 LI,I (1±+j26Q,) 2±k2Q.2

114

a5=

(24)

(26)

where

gm=transconductance of tube

ei=input FM singal

00= 1_ 1 _c0-1L1C1 L2C2

Qe = loaded Q of transformer primary or secondary

3=coo

M4L1L2

Denoting 23Qe= x kQ L2= a, kQe= b, the output

voltage of the discriminator, e0, can be expressed as follows:

eo07(Ie21-jell)

2gmECwOOL4Qe -+ (X+)2- 1+( )22(27V(1-Fb2- )+4x

where

= rectification efficiency of diodes

EC=amplitude of input signal

(27) can be expanded into a series form of (5)' by theuse of Taylor's expansion. Thus,

eo= BgmEco)LlQe4 (alI + 3 a34w3+ 10 a5js5) cospt

+ 1 (a34W3+ a5sh5) cos3Pt

(28)+ 116 a5dJW5cosspt+..........16

where

et a,(X), a3-a3 ( 2Q- ) = ) ,

2a--(i+b2) V1±a2

a[(1 -b2)2+2(1+a2)2(1- b2)]a3- (+a25/2) 1 b2)3

aa[5(3-4a2) (1 +b2)4+ 18(1-b2) (1 +a2)2(1 + b2) 2

a5- ~~~~20(l1±a2)9/2(1+b2)5-20(1 + a2)4(1+ b2)2+ 60(1 + a2)4(1- b2)2]

the linearity of the discriminator is. Since a3/al>a5/aj,it is suggested to set a4=0. By this condition, the follow-ing relationship between a and b is obtained:

a-= (b'V V2(b21-1 (29)

It is shown that (29) holds only when b is greater thanunity. When b is less than unity. there is no real a which will

b Lsatisfy a3=0. Using (29) and the relation, a_- 2 2

we have

L2 4 F b2+_IL1 b2 L 2(b2-1) (30)

When the linearity is the main requirement, it is usualto choose b=2, because in double tuned circuit, the uniformnreponse is obtained over most of the pass band for b=2.(7),(8) Putting b=2 into (30), we get the ratio of theinductance of the secondary coil to that of the primary,L2/L1L1.04. When a good compromise between sensiti-vity and linearity is required, it is common practice to

set b-i.5(8), from which and using (30), we get L21L,=1.878. All these relations give a good agreement withthose conventional data'8).From the condition of a3=0, (28) can be given by

eo= 2Ecgmw)oL1Qe { [a, (2 Q, t ) +i a (2 Qe,w )

x cospt + 16 a5 (2 Qe ) cos3pt

+ a5 2(Qe ( )5cos5Pt - ..........} (31)

Considering the terms after the fifth harmonic negligible,the total distortion of (31), D, is

D- 5.069 a516a, ( 2Q +j)+ lOa5 (32)

By inspection of (31), it is unlerstood that 2Qe .,_1(Oc

for" the low distortion. However, it is customary to let

2Qe -1, where Qe is determined(8) as below:

(28) shows that the less a3/ai, a5/a1, are, the betterQe fc

24f

115

and (32) reduces to

D- 5.0991a5I16a1+10a5

-(2n+1) 8; (n=0,1,2, -")8(34)

(4) Line Discriminator

Fig. 5 is a circuit diagram of transmission line discri-minator9), (10) where Zsh and ZOP are the impedances of

the short-circuited and the open-circuited transmission linesrespectively. The output voltage, eo, of the discriminatorresulting from the input FM singal having constant am-

plitude can be expressed as follows:

Fig. 5.A circuit diagram of transmission line discriminator.

eo=77(jell - e2D)

-3REC j 1+ r ,cot2O 1 r 1tan2O

where

1-rectification efficiency of diodes

Ec=amplitude of the input FM signal

r=coupling resistor constant

0 =(w .CI-f 22r2o-wangular frequency of the input signal

-=angular center frequency of the input signal

a-wavelength of the input signal

AC= wavelength of the center frequency of the inputsignal

L -inductance per unit length of the transmission line

C-capacitance per unit length of the transmission line

I=length of the transmission line

Referring to (5)', (35) may be written in the seriesform as below, by neglecting the terms after the fifth

harmonic since the series converges rapidly:

eo -r,Ec { (aiA + -3 a3zi) cospt+ --a3W3cospt } (36)

where

4r2(VC1) - r2(2n+1)=(J+r2)3/2 - 2(1+2)"2fc

8r2(2r4-82+ 5) ( VLCI)3a3 3(1+r2)7/2

r2(2r4- r2+5)(2n+1)3192(1 + r2)7/2fc3

(36) shows that a3 should be set to be zero for the

distortionless output signal. From this condition, the value

of r is obtained as r=0. 803 or 1.796. It is recommendedto take r-0.803 instead of r=1.796 from the viewpointof the input impedance of the discriminator."'()

Substituting r=0.803 into (36), we obtain the distor-

tionless output signal indicated by

(37)

It is shown in (37) that there is no distortion in the-(35) output waveform even under the 100% frequency devia-

tion. This means that the frequency characteristic curve

of the discriminator is a straight line.(')

However, from the vie.w point of the input impedanceof the line discriminator, it is recommended that the cou-

pling resistor constant, r, should be unity, because it

makes the input impedance to be the characteristic impe-dance of the transmision line, Z0, which is independent of

frequency.(10) When r=1, (36) gives, by neglecting the

terms after the second,

eo-Em2 [.l11-%(2n+1)-0.0858( f) (2n+1)3]cosPt

(38)

(38) indicates that it is required to set n=0 for the

minimum distortion in the output waveform. When n=o,

the percentage of the distortion is only 2.79% for =---.

116

eo-0.958(2n+ 1)iEc7-cospt

-0.0286 (AJ)3(2n+13cos3pt}

=10. 7X 106+1. 178x 2X 75x103=10. 876MHZ.CONCLUSION

The theory proposed here is of general applicability andwas applied only to the four typical FM discriminators(slope discriminator, Travis discriminator, Foster-Seeleydiscriminator, and line discriminator). The results obtainedfrom the analyses of those discriminators are the designequations which will allow a discriminator a desired lowdistortion performance and the followings:

(1) There are both even and odd harmonics in the

output signal of unbalanced type discriminator (e.g., slopediscriminator), while there are only odd harmonics in thebalanced type (e.g., Travis, Foster-Seeley, and line dis-criminator).

(2) In designing the tuned type discriminators, the follo-wing conditions are required for the low distortion:

(a) Slope discriminator; 2Qe fc << 1,

(b) Travis and Foster-Seeley discriminators; 2Qe ff <1

In both cases, the less the value of 2Qef. is, the less

the distortion is, and the relationship between the effectiveQ of the tuned circuit of FM discriminator and the fre-quency deviation, Jf, is of inverse proportionality.

(3) In designing the line discriminator, it is requiredthat n should be set to be zero for the minimum distortion,

r should be 0.803 for the best linearity, and r should beunity for the constant input impedance, Z0.

APPENDIX I

In this appendix I section, the examples of the applica-tion of the derived design equations of FM discriminatorswill be shown:

(a) Design a slope discriminator with the total distortion1. 267%. The center frequency and the maximum frequencydeviation of the input signal are 10.7 MHz and + 75 kH,respectively. Assume that the Q of coil is 100 and theloading effect of diode is negligible.

(Sol.)From (14), the resonance frequency of the tuned circuit

of slope discriminator is

ffc+f±. 178Bm

and the loaded Q is calculated by (13)

QeO. 3 2f . 3 10876 = 21.7532,df 0.15

Assuming that the stray capacitance including the outputcapacitance of tube is 40 pF, the total shunt resistive loadon the tuned circuit at the resonant frequency can be givenby

R,h Q 21.752=796K

27rfoC 6.28X 10.867X106X40x -7961KQ

and the inductance of the tuned circuit is given by

L R~h 7.961X10'3 - 5 36pH21rfoQe 6.28X10. 876 x106 X 21.752-The equivalent parallel resistance due to the effective

series resistance of the inductor, Rp is

Rp - Q0woLL-10 x 27r x 10. 876 x06 x 5. 36 x10-6=36. 61KQ

Then, the parallel resistance R, can be obtained by

R_ RshRp 7.961 x 103 x 36.61X 103-10 17KQRp-Rsh 36.61x10'-7.961 X10 K-OKQ

(b) Determine the effective Q and the resonance frequ-encies, fl, f2 of Travis discriminator which satisfies thefollowing specifications:

Main carrier frequency: f, 30 MHzBandwidth of FM receiver: Bm=300 KH,Distortion of discriminator: D-1%

(Sol.)

Since the linear portion of the frequency characteristiccurve of the discriminator must be equal to the bandwidthof FM receiver, the linear portion of the characteristiccurve should be set 'f ±150kHR.The effective Q isobtained by (24)

Qe 23f ( 1. 706+3. 35D )

30X106 62X 0.01 1/42x150x103 \ 1.7063-8.35x0.01 I

=100 X 0. 7727-77.27

117

and we can obtain fi, f2 by (25)

fi=fc-o.6124B ( 1.706+3.35D 1/4

=30X 106-0.6124 X2 X 150 X1(1 670+.3XO.01) 1/4

=29.76225 MHz

and

0.0286( 10)1. 11 xiO1 -°0858

100 0058%=0.0258%

APPENDIX II

fi-fc+0.6124Bm( 1i.706+3.35D ) 1/4

=30X106+ 6i24x2Xi X103 (1.706+ 35x0.01) 1/4

30.23775 MH,

(c) A Foster-Seeley discriminator having 1% distortionis required for use with a 10.7 MH, FM I-F channel.The maximum frequency deviation of the carier is + 75

kH,. Determine Q, and L2/L, of the discriminator.

(Sol.)If we take b= .5, then from (30), we get

L2 = 1. 878

and from (32), we can express Qe as follows:

Q = ife ( 165aD ) 1/4

On the other hand, we obtain a-1. 0274 by puttingb=1.5 into (29), and then we have a,=0.44,a5=-0.02071 with the substitution of b=1. 5 anda= 1.0274 into a, and a5. Then by substituting all knownconstants into the above equation, we obtain Q. as

10.7 x 106Q`-2X75X10'x< ( 16X0.44X0.01 1/4x5.099X I-0.02071-10X-0.0207X 0.i /

-73.7775

(d) Find the distortion of line discriminator when the

center frequency and the' maximum frequency deviation of

the input signal are 100 MHR and + 10 MH, respectively.Assume that n=0, r-1.

It seems to be better for the good linearity of the FMdiscriminator to set the amplitude of the second harmonic,

1 -(a2JW2+a4JW4), or the amplitude of the third harmonic,

1 (a3z13+ 5 a54W5), to be zero. If we do so, the

following equations will be obtained:

a2+ a4,4z2=0 (1)or

a3+ 4-5a5W2-0 (2)

From (1) or (2), aw can be shown by

A-j0= [for (1)]a4 (3)

or

4dw=j 4a3 [for (2)] (4)(4)

(3) and (4) show that (1) holds only for Jc=jJa2

and (2) holds only for cw=j 4a3 . However, the ang-

ular frequency deviation expressed in complex number hasno physical meaning since it is physically unrealizable.

Hence, we cannot use a2+a4Aw2=0 or a3+ -4- a54w2=04

instead of a2=0 or a3=0.

REFERENCES

(Sol.)From (38), we get the distortion, D, as

0.0286( f' )(2n+1)31.117-(2n+1)-0,0858 (fc (2n+1

(1) L.B. Arguimbau, "Discriminator Linearity," Electro-nics, vol. 18, pp. 142-144, March 1945.

(2) A.R. Vallarino and M.S. Buyer, "Harmonic Distortionin Frequency-Modulation Off-Resonance Discriminator",Elec. Communication, vol. 26, pp. 167-172, June1949.

118

(3) F.E. Terman, Electronic and Radio Engineering, 4 thed. New York: McGraw-Hill, 1958, pp. 596-60.

(4) Samuel Seely, Electron-Tube Circuits, 2nd ed. NewYork: McGraw-Hill, pp. 339-342.

(5) Seely, op. cit., pp. 342-344.(6) John. D. Ryder, Electronic Fundamentals, and

Applications, Englewood Cliffs: Prentice-Hall, 1954,pp. 538-544.

(7) C.L. Alley and K.W. Atwood, Electronic Engineering,New York, Wiley, 1962, p. 333.

(8) F. Langford-Smith, Radiotron Designer's Handbook,4th ed., RCA, Harrison, N.J.; RCA, p. 1090.

(9) C.W. Lee and W.Y. Seo, "Super Wide-Band FM Line

Discriminator", proc. IEEE(Correspondence), vol. 51,pp. 1975-1676, Nov. 1963.

(10) Choong Woong Lee, "in Analysis of a Super Wide-Band FM Line Discriminator," ibid, vol. 52, pp. 1034-1038, Sept. 1964.

ACKNOWLEDGEMENT

The author wishes to express his sincere thanks to Prof.Keh Kun Choi at Dept. of Electronic Engineering, Collegeof Eng. Seoul National University, Seoul, Korea forhis helpful discussions and advice in preparing the paper.

BIOGRAPHY

CHOONG WOONG LEE (M'63) was born in Phungan-pukdo Province, Korea on May 3, 1935.He received the B.E. and M.E. degrees in electronic

engineering in 1958 and 1960 respectively from SeoulNational University, Korea. During 1963 he attendedSydney University, Australia, as a Fellow on the Co-lombo Scholarship Plan.

From 1958 to 1961 he worked at the Scientific Re-search Institute, Ministry of National Defense, Korea,and from then until March, 1964, he was a researchmember of the Electronics Division of the Army Re-search and Testing Laboratory, which originated fromthe Institute. After that he joined the staff of theCollege of Engineering, Seoul National University,where he presently is the Head of Electronics Courseat the Department of Industrial Education..

His special interests include FM communicationsystems, electronic circuit designs, and microwavetechniques.

Mr. Lee is a member of the Korea Institute of Elec-tronics Engineers.

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