Int. J. Electron. Commun. (AEÜ) 59 (2005) 491–494

www.elsevier.de/aeue

LETTER

Newmodulation scheme based on elliptic functions

Klaus Huber

Deutsche Telekom AG, T-Systems, ITC-Security, Am Kavalleriesand 3, 64295 Darmstadt, Germany

Received 29 January 2004; received in revised form 16 November 2004

Abstract

A new modulation scheme is proposed which modulates the signal shape. It is based on elliptic functions and retains twomain features from amplitude and angular-modulation. Methods for modulation and demodulation are also given.� 2005 Elsevier GmbH. All rights reserved.

Keywords:Modulation; Elliptic functions

1. Introduction

In this letter a new modulation scheme based on ellipticfunctions is proposed. Letm(t) be a function of time whichcontains the information to be transmitted (either analog ordigital). We develop the principal ideas of the new schemefrom properties of amplitude and angular modulation. As iswell known, using amplitude modulation, we transfer theinformation contained inm(t) onto the carrier signal usingessentially the equations(t)=(a0+c m(t)) sin(2�f0t). Thesignals(t) is transmitted over the channel,f0 is the carrierfrequency anda0 andc are constants which are selected ac-cording to specific requirements.A characteristic property ofamplitude modulation is that the signals(t) passes regularlythrough zero, namely at the zeros of sin(2�f0t). As a typicalrepresentant of angular modulation, frequency modulationinfluences the carrier frequency directly. The signals(t)

to enter the channel iss(t) = a0 sin(2�f t), where the fre-quencyf is a function ofm(t), usuallyf (m(t))=f0+c m(t).Frequency modulation has as one of its main characteristicsthe property of constant amplitudea0, which is particularlygood for noise immunity. The modulation scheme which we

E-mail address:klaus.huber@online.de(K. Huber).

1434-8411/$ - see front matter� 2005 Elsevier GmbH. All rights reserved.doi:10.1016/j.aeue.2005.01.009

present in this contribution based on elliptic functions re-tains both properties, a constant amplitude and regular zerocrossings. The modulated parameter is the signal shape.

2. The new modulation scheme

We assume knowledge of elliptic functions (see e.g.[1–3]). As essential functions we use the Jacobi ellipticfunctionssn(x, k), cn(x, k), anddn(x, k). The second ar-gumentk of sn, cn, and dn is called the modulus of theelliptic function and will be omitted for convenience. Theinformationm(t) is packed into the modulusk = k(m(t)).We assume that the frequency of change ofk is small com-pared to the carrier frequencyf0 = 1/T . The signal for thechannel is given by

s(t) = a0 · sx(2�(k)f0 · t). (1)

We first consider the use ofsx(x) = sn(x). The role of� inthe classical sine function is taken by� which is a functionof k and can be easily computed using� = �/M(1, k′),whereM(1, k′) is the arithmetic geometric mean of 1 andk′ = √

1− k2. The valueM(u, v) of the two numbersu andv is given by the limit of the recursionui+1 = (ui + vi)/2,vi+1=√

uivi with starting valuesu0=u, v0=v (see e.g.[1]).In the theory of elliptic functions one usually uses the real

492 K. Huber / Int. J. Electron. Commun. (AEÜ) 59 (2005) 491–494

Fig. 1. sn(2�x/T , k) for k = 0, k = 0.8, k = 0.95, andk = 0.99

Fig. 2. cn(2�(x − T/4)/T , k) for k = 0, k = 0.8, k = 0.95, andk = 0.99

quarterperiodK(k) which is the complete elliptic integralof the first kind and related to� by

�

2= K(k) =

∫ �/2

0

d�√1− k2 sin2(�)

.

According to the value ofk the functionsn(2�f0 · t) changesits shape, but the zero crossings and the amplitude remainfixed. In Fig. 1a quarterperiod of the functionsn(2�x/T )

is displayed for some values ofk from the interval[0,1].Note that fork = 0 we get the sine function and at the otherextremek = 1 we have the hyperbolic tangent. The periodof tanh is infinity, however the scaling used then leads to apulse. Hence for practical purposes the case 0�k�1 is ofinterest.With the sn(2�x/T , k) functions we get signal shapes

which are above the sine function for 0< x < T/4. To exploitthe signal shapes below the sine function we can use thecn(x, k) function. To place this function in the same phasewith the sn(x) function we considercn(x − K) which canbe expressed as

cn(x − K, k) = k′ · sn(x, k)

dn(x, k)= k′ · sd(x, k). (2)

In Fig. 2 the functioncn(2�(x − T/4)/T ) is displayedfor some values ofk from the interval[0,1]. For k = 0 wealso get the sine function.Thus, using thesnandcn functions we can cover quite a

great variety of signal shapes ranging from a pulse over thesine function to zero. We thus define the functionsx(x) as

sx(x) ={

sn(x, k) for 0�k�1k′ · sd(x, |k|) for − 1�k�0

(3)

and define a shape modulation scheme according to equa-tion (1). The parameterk=k(m(t)) carries the message. Thevalues ofk are from the interval[−1,1]. Using suitable pre-coding or data translation techniques we can adaptk(m(t))

to desired source signals. For example for an analog signalm(t) a simple encoding, adequate for values ofk2 not tooclose to unity, is to setk2 = |c m|, with c a suitable normal-izing constant, and the sign ofk equal to the sign ofm. Wealso briefly mention two examples of digital modulation.

Example 1. A simple binary shape shift keying (SSK)scheme can e.g. be defined to senda0 sn(2�f0t) if a ‘1’ istransmitted anda0k′ sd(2�f0t) if a ‘0’ is transmitted.

Example 2. A combined (gray coded) binary phase bi-nary shape shift keying (BPBSSK) scheme can be definedby agreeing to transmit ‘11’ bya0 sn(2�f0t), ‘01’ bya0k

′ sd(2�f0t), ‘00’ by −a0k′ sd(2�f0t), and ‘10’ using

−a0 sn(2�f0t).

3. Generation of elliptic time functions

The elliptic functionssn(x), cn(x), and dn(x) can begenerated as functions of time using the following threewell-known differential equations:

d

dtsn(t) = cn(t) · dn(t) (4)

d

dtcn(t) = −sn(t) · dn(t) (5)

d

dtdn(t) = −k2sn(t) · cn(t), (6)

which can be implemented using standard hardware devices.For k close to unity the tanh characteristic of a differentialamplifier can be used as we havesn(x, k → 1) = tanh(x).Differential amplifiers which closely approximate the hy-perbolic tangent are an efficient and inexpensive means forgeneratingsn(x, k) with k close to unity. Further details aswell as circuits for generating� from k can be found in twopatent submissions[4,5].

4. Demodulation

In this section we give several methods to recoverk froms(t). The first is by sampling and the second and third areby integrations (Eqs. (7) and (8)) which are standard withincommunications.

4.1. First method

As the frequencyf0 = 1/T is fixed and the signals(t)passes through zero twice everyT seconds we can eas-ily take reference points. We haves(0) = s(T /2) = 0 and

K. Huber / Int. J. Electron. Commun. (AEÜ) 59 (2005) 491–494 493

s(T /4) = a0, s(3T/4) = −a0. We gets(T /8) = s(3T/8) =a0 sx(K/2) and s(5T/8) = s(7T/8) = −a0 sx(K/2). Thevalue ofsx(K/2) equals 1/

√1+ k′ for the shapes above the

sine and√

k′/√1+ k′ for the shapes below the sine. Hence

the slowly varyingk(m(t)) can be recovered froms(t) bysampling at odd multiples ofT/8. With k = √

�4 − 1/�2,for the binary SSK above we gets(T /8) = a0�/

√�2 + 1 or

a0/√

�2 + 1, i.e. a ratio of� : 1 in amplitudes.

4.2. Second method

The next demodulation method is by integration over aquarter periodT/4 or half a periodT/2. By rectifying thesignal we can also consider a whole periodT. From theintegrals∫

sn(x, k)dx = − ln(dn(x) + kcn(x))

k,

∫cn(x, k)dx = arcsin(k sn(x))

k

(see[6, p. 630, 5.133]) we get∫ T/2

0s(t)dt

=

∫ T/20 a0 sn

(2�

Tt

)dt

∫ T/20 a0 cn

(2�

T(t − T/4)

)dt

=

a0T

2 · �(k)kln

1+ k

1− k= a0T

�(k)ktanh−1(k)

a0T

�(k)karcsink.

(7)

Integration over a quarter period, respectively, gives half ofthe values. Fork small the integral is well approximated by(a0T/� ) (1+sign(k) (k2/12)). For the casek2=|c m| abovewe get a0T� (1+ (cm/12)).

4.3. Third method

A further demodulation works by integration of the squareof s(t), which follows from the integral

∫sn(x, k)2 dx =

(x − E(am(x), k))/k2, ([6, p. 630, 5.134]) and the identitycn(x)2 + sn(x)2 = 1. This yields∫ T

0s(t)2dt =

{∫ T

0 (a0 sn(2�t/T ))2 dt∫ T

0 (a0 cn(2�t/T ))2 dt

=

a20TK(k) − E(k)

k2K(k)

a20TE(k) − k′2K(k)

k2K(k).

(8)

E(k) is the complete elliptic integral of the second kind.Respective integrations over a half (or quarter) period givehalf (or a quarter) times these values. Fork small the integralis well approximated by(a20T/2) (1+ sign(k) (k2/8)). Forthe casek2 = |c m| above we get(a20T/2) (1+ (cm/8)).

5. Orthogonal signalling

Elliptic functions can generalize the usual orthogonal sig-naling schemes using sines and cosines. Instead of the sinefunction we use the functionsx(x) from Eq. (3), and insteadof the cosine function we use the functionsy(x) defined by

sy(x) ={

cn(x, k) for 0�k <1,cd(x, |k|) for − 1< k <0.

(9)

The functioncd(x) is the sn(x)-function translated byK,i.e. cd(x) = sn(x + K). It can also be expressed ascd(x) =cn(x)/dn(x). Then the orthogonality property holds∫ 4K

0sx(x) sy(x)dt = 0. (10)

To see this consider Eqs. (4)–(6) and (2) to get∫sn(x) cn(x)dx = −dn(x)

k2,

∫sn(x) cn(x)

dn(x)dx = − ln dn(x)

k2,

∫sn(x) cn(x)

dn2(x)dx = 1

k′

∫cn(z) sn(z + 2K)dz

= − 1

k′

∫sn(z)cn(z)dz,

which, when integrated over a full period all vanish. Thus,elliptic functions can be used for orthogonal modulation.Orthogonality can be exploited in the base- and/or the pass-band. For the above pairings (sn, cn andk′sd, cd) also theequationsx(x)2 + sy(x)2 = 1 holds.

6. Spectral properties

The spectral properties of the newmodulation follow fromthe Fourier expansion of elliptic functions (see, e.g.[2, p.222]). For the two functions ofsx(x) we get

sn(2�f t) = 2�

kK

∞∑n=0

q(2n+1)/2 sin((2n + 1)2�f t)

1− q2n+1 ,

k′sd(2�f t) = 2�

kK

∞∑n=0

(−1)nq(2n+1)/2 sin((2n + 1)2�f t)

1+ q2n+1 ,

where q = exp(−�K ′/K) with K ′ = K(k′). For the twofunctions ofsy(x) we get

cn(2�f t) = 2�

kK

∞∑n=0

q(2n+1)/2 cos((2n + 1)2�f t)

1+ q2n+1 ,

cd(2�f t) = 2�

kK

∞∑n=0

(−1)nq(2n+1)/2 cos((2n + 1)2�f t)

1− q2n+1 .

A thorough spectral analysis is beyond the scope of thisletter. To obtain some qualitative insight, inFig. 3the Fourier

494 K. Huber / Int. J. Electron. Commun. (AEÜ) 59 (2005) 491–494

Fig. 3. Fouriertransforms ofcn(2�t/T , k)rect(t/T ) for k = 0,k = 0.8, k = 0.95, andk = 0.99

transform of the on–off keyedcn-function is plotted for somevalues ofk, i.e. the spectrum ofcn((2�/T ) t) · rect(t/T ),where rect(t/T ) = 1 for |t |�T/2 and zero elsewhere. Asthese spectra are even functions only positive frequenciesare displayed. Fork =0 the main lobe has the highest value.With growingk the main lobe decreases and the side lobesincrease.

7. Conclusion

A new modulation scheme based on elliptic functions foranalog as well as digital modulation has been proposed.The shape of the signal is suggested as a third modu-lation parameter in addition to amplitude and angle. Inits basic version the scheme combines two main featuresfrom amplitude- and angular-modulation. From amplitude

modulation it retains the fixed zero-crossings which can beused for timing/integration and from frequency modulationit has the constant value of the amplitude. The essential prop-erty which is exploited for modulation is the shape expressedby the modulusk. This property fits well into existent or-thogonal modulation schemes and generalizes them. Severaldemodulation methods have also been given and questionsof implementation have been addressed. The scheme can beimplemented in a straightforward way.

References

[1] Hurwitz A. Vorlesungen über allgemeine Funktionentheorieund Elliptische Funktionen. Berlin, Heidelberg, New York:Springer Verlag; 2000.

[2] Lawden D. Elliptic functions and applications. Berlin,Heidelberg, New York: Springer; 1989.

[3] Whittaker E, Watson G. A course of modern analysis, 4th ed.,Cambridge: Cambridge, reprint 1969.

[4] Huber K. Analoge Schaltungsanordnung zur Erzeugung undBerechnung von elliptischen Funktionen, Patent submissionAkz 103 19 637.4, Deutsches Patent und Markenamt, 02.Mai, 2003.

[5] Huber K. Verfahren zur Modulation eines Trägersignals sowieVerfahren zur Demodulation eines modulierten Trägersignals.Patent submission Akz 103 19 636.6, Deutsches Patent undMarkenamt, 02. Mai, 2003.

[6] Gradshteyn IS, Ryzhik IM. Table of Integrals, Series,and Products. Corrected and Enlarged Edition. San Diego:Academic Press; 1980.