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Transactions Letters

A Phase Modulator with Variable Maximum Phase DeviationJanghan Kim

Abstract- A phase modulator, constructed with three Arm-strong modulators, is presented where the maximum phase de-viation can be extended up to 7i without frequency multiplication.Numerical results of the signal distortion are given for severalvalues of the maximum phase deviation and compared to otherArmstrong type phase modulators.

I. INTRODUCTION

HE basic indirect phase modulation generator is an Arm-T trong modulator [ l ] , where the deviationof the outputphase must be smallto obtain good phase linearity.

Ruthroff and Bodtmann[2] have described a method of

extending the linearityof the Armstrong modulator with abalanced scheme, and Kassam and Lim[3] have improved itwith the variable phase offsetof the quadrature carrier.

In this paper, the conceptof amplitude offset of the modu-lating signal is proposed instead of the phase offset of thequadrature carrier described in[2] an d [ 3 ] . o realize this con-cept in practice, a phase m odulator with three basic Armstrongmodulators is presented, and itis shown that performanceis better than the previous modulators in terms of signaldistortion.

11. MODULATORESCRIPTION ND S I G N A L DISTORTION

The Armstrong narrowband phase modulator[1] is shownin Fig. 1, where the multiplier is a balanced modulator and

the amplitude limiter is a limiter-bandpass filter combinationthat ensures a constant amplitude output,It is seen that theoutput is

where m ( t ) deno tes a modulating signal and(bl(t) s the outputphase of the modulator. The maximum phase deviation41ma xof the modulator becomes

Paper appr oved by J. J. Uhran, Jr., the Editor for Mo dulation and Signa lDesign of the IEEE Communications Society. Manuscript received February15, 1988; revised February15, 1992.

The author was with the Departmentof Electronics Engineering, Hong-ikTechnical College, Seoul 121 -791, Korea. He is nowwith the Departmentof Electronic and Computer Engineering, Hong-Ik University, Chochiwon,Chung-N am 339-800, Korea.

IEEE Log Number 9211330.

m

si n wt

@ os wtFig. 1. Armstrong phase modulator.

Ruthroff and Bodtmann[ 2 ] ,using two Armstrong m odulatorsin a balanced scheme, have described a method for extendingthe linearity of the Armstrong modulator. The output phase oftheir modulator consists of two inverse tangent functions dueto the two modulators, Le.,

(bz(t)= tan-l(Kzm + A 2 ) + tan-l(Kzm - A2 ) ( 3 )where m is the same modulating signal given in (la).

The phase function in(3) can also be realized easily bymeans of two Armstrong modulators withK2m + A2 andKz m - A2 as their input signals, respectively, as inFig. 2. This configuration is not identicalto the original system in[2] . Here, the factorsK2 an d A2 are employed to offset theamplitude of the modulating signal. For this modulator, theresulting output is givenby

: r ; z ( t )= cos[2wt + ( b 2 ( t ) ] (4)where the output phase4 2 ( t ) is given in (3). The maximumphase deviation, from(3) , is

and is derived in the Appendix A.A n ideal phase modulator with the maximum phase de-

viation &,ax, which must be restrictedby r, as the phasefunction

~ L ( W L ) 4 m a x ~ 1 . (6)

The phase cha racteristic of the modulator given in (lb ) and(3) , can be described by elimina ting the explicit time depen-dence of $( t ) . Then, the signal distortion of the modulatorwith the output phase function(b(m) hich deviates from theperfect linear phase of(6) can be represented as follows:

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Fig. 2. Balanced phase modulator with amplitude offsetsof modulating signal.

TABLE ISIGNAL DIST ORTION ?ORARMSTRONGHASF MODULATORN FIG 1

AN D BALANCEDH A S E MODULATORN FIG 2 (-42 CA N BEDERIVED FROM (5))

Armstrong modulator balanced phase modulator

E1,degrees Ez,rad* E3,radZ E1,degrees Ez,radZ E3,radZ

K 1 = 0 . 1 3 2 K 1 = 0 . 1 3 2 K 1 2 0 . 1 3 2 -_ _ ~0 .165x10- ' 0 . 4 2 9 ~ 1 0 - ~ . 3 9 1 ~ 1 0 - ~

2 1516

E3 = l14 ( m) 4maxm] .- .(I- *)* 7n, [rad?]where E1 is the maxim um phase error described in[2 ] , nd E2an d E3 are the mean square values of the distortion describedin [ 3 ] .

The minimum values of the signal distortion are shown inTable I for the Armstrong modulator inFig. 1 (left hand) andthe balanced phase mo dulator inFig. 2 (right hand) for variousvalues of the maximum phase deviation. For the Armstrongmodulator, K1 is obtained from (2). For the balanced phasemodulator,K2 is chosento minimize the distortion andA2 ca nbe obtained from( 5 ) , but is hidden in TableI (see AppendixA). Note that the situationof the balanced phase modulatorwith 4 2 m a x = 7r/2 is equivalentto Ruthroff and Bodtmann'smodulator [2] .

(9)

111. A NE W PHASEMODULATOR

The output phase function(3) composed of two inversetangent functions, shows more improved phase linearity than(lb) expressed by a single function. This can be improved,when optimized with three or more inverse tangent functions

Fig. 3. A new phase modulator.

are incorporated as follows:

where &(t ) denotes the new phase function andKJ, K4,an d A3 are the amplitude offsets of the modulating signal,as mentioned in Section11. A new modulator with the phasefunction of (10) is shown in Fig. 3. The outputs of thesumming circuits are

m ( f )= J(K3~7-1, + As)' + 1. sin[w(t) + tan-'(K3m + A3)]

p 3 ( t ) = @rL2 + 1. s in(wt + t a n - ' K4m) ,

and the resulting output becomes

2 3 ( t ) = Pl( t ) P Z ( t ) ' P 3 ( t )= A ( t ) in[3wt+ 43( t ) ]+ first orderw term. (12a)

Here, the amplitude variationA ( t ) s given by

~ ( t ) - 1 Jm.. JK2rn2 + 1 . (12b )

The am plitude variation in (12b) and first-orderw term in (12a)

can be removed by the limiter-bandpass filter. From(lo), th emaximum phase deviation is obtained as

and is derived in the Appendix B.

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TABLE I1S I G N A L DISTORTIONOR NE WPHASE MODULATORN FIG 3 ('13 IS IN (14))

and large maximum phase deviation. The maximum phasedeviation of the modulator can be extended up tox radianswithout the use of a frequency multiplier.

APPENDIX

In this Appendix, we provide the computational steps neededin the derivation of(5) and A2 for the Table I. The transducercharacteristic of the m odulator can be described by elim inatingthe explicit time dependence of4 2 ( t ) . Thus, (3) becomes

4 2 ( ~ ~ )t a n - l ( K m + A2) + t a n P 1( K 2 m A z )

and this phase increases monotonically withm and reaches amaximum atm = 1. Therefore, the maximum phase deviation4 2 m a x is

4 2 m a x = 4 2 ( m = 1)

Fig. 4. Phase deviation from the ideal phase function for the new phasemodulator in Fig. 3 (IC3 = 2.6.5, 1i.l = 2.50, '13 = 2.20).

The minimum values of the signal distortion for the modu-lator of Fig. 3 for several values of43 max is given in Table 11.The factorsK3 an d K4 are chosen to minimize the distortion,an d A3 is derived from(13) as (see AppendixB)

A3 =

Comparison of Table I1 with Table I shows that the newmodulator presents low-phase distortion for the same maxi-mum phase deviation. For example, when43max x / 2 , th eminimum valuesof the distortion are as follows (from the

second row of Table 11):El = 0.0169 for K3 = 0.949, K4 = 0.768,

E2 = 0.369 x l o p 7 fo r K3 = 0.945. K 4 = 0.763,

E3 = 0.145 x for K3 = 0.936, K4 = 0.752.

The results are lower than those values of distortion forthe other Armstrong type phase modulators in[2] and [ 3 ] ,which have the same maximum phase deviationof 7 r / 2 .(For example, El = 1.05 for the system in[2], and E2 =0.96 x lo-', E3 = 0.57 x IO-' for the system in [3])

It is also seen in Table 11, that the modulator is available inthe phase range *x without a frequency multiplier. For thisrange, it is shown thatK3 = 2.85 , K4 = 2.50, and A3 = 2.20give a maximum phase errorEl of 2.26'. Fig. 4 shows the

deviation between( 6 3 ( m ) an d x . m as a functionof m , fo rthese values of K3, K4 and A3.

IV. CONCLUSION

A phase modulator with a variable maximum phase devia-tion has been described, which exhibits low-phase distortion,

2KZ1 - KZ + A i= tan-'

Solving A2 in terms of K2 an d 4 2 m a x from (9, e have

From (16), it is shown that if4 2 ma x is fixed and an optimumvalue of K2 is found,A2 is obtained immediately. W e can findthe optimum value ofKz by a binary search, and the resultsare given in Table I.

APPENDIXB

In this Appendix, w e provide the intermediate steps in (13)and (14), and explain how to choose the optimum values ofK3 and K4 of the modulator shown inFig. 3. With time-dependence omitted, (10) can be rewritten as

4 3 ( m ) = t an- l (K3m + A3 ) + t anP1(K3m A3)+tan-' K4m,

(2K3 + K4 + A:K4)m - KiK4m3= tan-'

1 + A : - ( K i + 2K3K4)mZ'

(17)

The phase increases also monotonically withm and reachesa maximum at m = 1. Thus, substituting1 into m, of (17)yields (13).

To obtain the optimum phase functionof (17) that sets theperfect linear function given in (6), first, we fix4 s m a xwhichis the maximum phase range desired. Then, from (13), onefactor of K3, Kq, an d A3 can be expressed by the others.Solving forA3 in terms ofK3, K4, an d q ! ~ ~ ~ ~ ,e have (14).

On the above condition, by a grid search, we can find theoptimum factorsK3 an d K4 that give a minimum distortion asdefined in (7)-(9). A n example of the grid search forEl when43rrlax= x , s given in Table 111. We find that K3 = 2.85and K4 = 2.50 give a minimumEl of 2.26'. Th is result isincluded in Table 11.

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TABLE 111SIGNA L ISTORTIONl (I N DEGREES) FORNE WPHASE

MODULATORN FIG. 3 WHEN Q j m a x = 7i

IEEE TRANSACTIONS ON COMMUNICATIONS,VOL. 41, NO . 10, OCTOBER 1993

ACKNOWLEDGMENT

The author wishes to thank Dr.J. J. Uhran, Jr., of Univer-sity of Notre Dame for his valuable comments and constantenco uragem ents in correcting mistakes and improving thepresentation of this paper.

REFERENCES

E. H. Armstrong, A methodof reducing disturbances in radio signallingby a system of frequency modulation,Proc. IRE, ol. 24, pp. 689-740,May 1936.C. L. Ruthroff an d W. F. Bodtmann, A new phase modulator,IEEETrans. Commun., vol. COM-25, pp. 602-604, June 1977.S A . Kassam and T.L. Lim, A n improved phase modulator with lownonlinear distortion,IEEE Trans. Commun.,vol. COM-28,pp . 11 - 115 ,Jan. 1980.