1IEET1. A-T NSA(TITONA` ON INSTRUSIMINT\TION AND MEASITREMEN-, VOL. lm-18 'o. 2, JUNEl 1969

Modulation Measurements-Theory and TechniqueJOHN N. WARFIELD, SENIOR MEMBER, IEEE

Abstract-This tutorial paper reviews analog modulation theory.Emphasis is upon recent advances in the theory, and the implicationof these advances upon measurement of properties of modulatedsignals. A technique for developing measurements is given.

INTRODUCTION

Tp HIS PAPER is tutorial. It focuses upon theanswers to four questions. The first is what recentdevelopments in the theory of analog modulation

appear to be most important to those concerned withmeasurements? The second is what recent experimentshave been performed that show promise for changes inmodulation methods? The third is what is the structureof modulation theory within which advances can takeplace? The fouith is what general principles are illus-trated by the theory and experiments that are importantin research on instrumentation involved in modulationineasurements?

In this paper the word "measurement" is interpretedbroadly to include such subjects as the need for inventionof new circuits and instrumentation to accommodate in-quiry and investigation in new or advanced aspects ofanalog modulation.Any tutorial paper draws heavily on the work of

others. A goal hlas been to present in a simplified way

)rincipal results and conclusions of those authors ref-erenced, while retaining the outline of the substance oftheir scholarly presentations. Another goal has been tointegrate their work and show the commonality that tiesthe work together.

Aside from the research papers upon which this paper

is based, additional background appears in several ex-

cellent books [1]-[9].

ENVELOPE AND ANGLE

If one defines an analog modulated signal as e = A (t)cos 0(t), it is natural to suppose that A (t) can be identifiedwith the envelope of the signal and 0(t) with its angle.Yet it is easy to go astray. Consider, as an example, thesignal

e = COS (AJmt COS WXt Cc»>>Wm (1)

which can be thought of as the double-sideband sup-

pressed-carrier signal that results when the carrier iseliminated from a conventional 100-percent modulatedAM wave of unit carrier amplitude. By simply looking at(1), one is tempted to call cos comt the envelope and wct

Manuscript received June 24, 1968; revised February 3, 1969.The author is with the Battelle Memorial Institute, Columbus

Laboratories, Columbus, Ohio, 43201.

a)

wmt

(a)

~~~~~~~~T

,,- --1rr/2 3r/2 5i-i?

(¾t

(b)Fig. 1. Envelope and angle of cos w,,t Cos w,t.

le

/IvI (t) = stnw

(Lit

u1(t) = Cosw t

A (t)e -= u2+v 2e JTn u

v(t ) - H [u(t)]

(t)

u(t 1

(a) (b)

Fig. 2. Defining envelope and angle of a(t). (a) Special case. (b)General case.

the angle of the signal. These identifications are in error,since an envelope can never be negative.The envelope and angle for (1) are shown in Fig. 1,

where it is seen that the envelope is nionnegative an(d theangle is a discontinuous time function.This elementary example illustrates the desirability of

a precise definition for envelope and angle. Such a defi-nition is found by extension of a familiar concept. Fig. 2(a)shows a conventional phasor of unit amplitude rotatingwith constant angular velocity X in a counterclockwisedirection. The length is found in the usual way by takingthe square root of the sum of the squares of the two sides,while the angle is found as the inverse tangent of the ratioof the projection on the imaginary axis to the projectionon the real axis. Clearly the intuitive concept of theenvelope and angle of either the real part u, (t) or of theimaginary part v, (t) coincides with the length and angle ofthe phasor itself. The envelope and angle of both the realand the imaginary parts are the same, namely 1 and cot.

Fig. 2(b) shows the extension of the concept. Supposethat a modulated signal u(t) is given, and it is desiredto find its envelope and angle. By analogy with Fig. 2 (a),if a function v (t) can be found such that u (t) and v (t)located at right angles to each other define a "time-v-arying phasor" of the form A (t) ei0(t). as shown inFig. 2(b), the time-varying length can be identified as

the envelope of the original signal u(t), and the time-varying angle 0(t) can be identified as the angle of themodulated signal u(t). Whether the definition then makessense depends upon whether the chosen v (t) yields

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, JUNE 1969

envelopes and angles that agree with comiinonly acceptedexpressions.The function v (t) that has been found suitable as a

basis for the definition is the Hilbert transform of u(t)[10]-[15]. When u(t) is expressible as a constant plusthe sum of any number of sinusoidal components, thetransform exists and a unique envelope and angle canbe defined for u(t) in the manner stated.As an example, the signal given in (1) is taken as

u(t) . The Hilbert transform is (see Appendix)

v(t) = sin wo,t cos comt.

The length of the phasor is

A(t) = [u2(t) + v2(t)]1"2 = Icos WmtIand the angle is

@(t) = tan-, [v(t)/u(t)] = tan-, sin cowt cos WmtCos co0t Cos comt

(2)

SPECTRUM OF ANTALYTIC SIGNAL

A satisfying property of an analytic signal is that itsspectrum has no components at negative frequencies.By contrast, even the simple signal cos wt has spectrumcomponents at both positive and negative frequencies.The simple phasor e"', however, has only a spectralcomponent at co. The additional complexity entailed byworking with analytic signals is thus partially compensatedby the concentration of the spectral components at non-negative frequencies.

ENVELOPES OF STOCHASTIC SIGNALSStochastic signals generated in nature are often the

subject of measurement. Dugundji [13] has used the(3) concept of analytic signals to show that when a stochastic

signal is band-limited and representable, therefore, inthe form

(4)n

u(t)= E Ci cos (Woit + 0i)i-I

(10)

which shows clearly that as time increases, the anglejumps by x radians as cos wmt changes sign. [This prop-erty is obscured if cancellation of cos wmt is carried outin (4). ]

ANALYTIC SIGNALS [10]-[15]Because of the importance of the functions u(t) and

v(t) in defining envelope and angle of u(t), the time-varying phasor of Fig. 2(b) is given a special name,analytic signal. Mathematically an analytic signal is acomplex function of a real variable. However, the nameis derived from the close connection to analytic functionsof a complex variable.To explore the connection, let the complex variable be

z = t + jar. (5)

Then an analytic function of this complex variable can

be written as

F(z) = F(t + ja) = x(t, a) + jy(t, a). (6)

When the imaginary part of z = = 0,

F(z) F(t) = x(t, 0) + jy(t, 0)

= u(t) + jv(t) (7)

AA(t)ei'o t)

where

A(t) = [u2(t) + v2(t)] 12, (8)

0(t) = tan-' [v(t)/u(t)]. (9)

Then, following the pattern indicated in Fig. 2(b), theenvelope and angle of u(t) are defined as the magnitudeand angle of the analytic signal F(t) formed by de-termining v(t) and adding jv(t) to u(t) in the manner

described.

its envelope is definable in the manner discussed above.Moreover, the definition so obtained is independent of theassignment of carrier frequency to (10).

COMMON ENVELOPE SIGNAL SET [10], [11]

Based on the definitions given, one can speak of a''common envelope set of signals." Such a set is describedas follows: A set { 0i(t) } of angle functions, i 1, 2, * - * , nyields a set {A(t)e0i' t } of analytic signals, all having thesame envelope, with different angles and different spectra.The existence of such sets shows that options exist for

the manner of representing and generating informationto be carried in an envelope. Stated another way, thesame information can be carried in an envelope thoughdifferent electrical signals are used.

PATTERN REGION FOR AN ANALYTIC SIGNAL [10], [11]

An analytic signal F(t) is intimately related to ananalytic function F(z), as mentioned earlier. Considerthe special case where F(z) is periodic with real period T.Then the analytic signal F(t) will likewise be periodicwith period T. Any zeros of F(z) which appear in a stripof length T indicated in Fig. 3, will be repeated periodic-ally, so that the zero pattern in this strip will representthe function and the analytic signal throughout thez plane. Consequently, the region shown in Fig. 3 can becalled the "pattern region" for the analytic signal F(t).A pattern for a signal can be shown either as a collection

of zeros of the analytic function in the pattern region oras the locus of those zeros as a parameter of the analyticsignal varies. Fig. 4 illustrates the pattern for the analyticsignal F(t) = 1 - ae"i' as the parameter a varies from0 to co. The function F(t) has the real part u(t) = 1- acos wt. The analytic function F(z) has a single real zero,which is located below the real axis for a < 1 and aboveit for a > 1. The locus is not affected by multiplication

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WARFlELD: MODULATIO(N MEA\SUREMENTS

TABLE ITABLE OF RELATIONS FOR MODULATE]) SIGNALS

F:z)Analytic FUnction

1 (1 + a cos clZ')i'0'

2 [f(z) + jf(z1]cJ

3 exp Jj[f(z) + woz]

4 exp {j[f(z) + jf(z) + woZ]

t) exp I[g(z) + j(z) + jcozollg(z) a log f(t)

f(t) > 0

F(t)Anialytic Signial

(1 + a cos wot)ejwoi

[f(t) + j f(t)]eiwlt= A(t) exp (j[wot + 0(t)] I

¢(t) = tail-l=A()= [f2(t) + f2(t)]l12

exp {j[f(t) + cwot]}

exp {j[f(t) + jf(t) + wot]

exp I{[g(t) + jg9(t) + j6oot] I

ul(t),I'Veal Signal

(1 + a cos wit) cos cool

f(t) (-(S wot - f(t) sill ,ot

cos [wot + f(t)]

exp f(t)1 cos [coot + f(t)]

fa(t) cos [coot + g(t)]

Descriptioni of Real Signal

Pure-ton-ie modutlatedconventionnal amplitutdeinodulatioti if ai < 1, coo >> W

Siigle-sideban:dampIlit u(de-mno(dllatedsignal

Conventional angle-modulated sigrnal

Single-sidebandangle-moduilated signalf(t) in angle]Single-sidebandangle-modulated signal[fa(t) in esivelopel

a-

-TI2

0

Strip in. Z-plone

7

Fig. 3. Pattern region for analytic signal.

of the analytic signal either by a constant or by eil"S, thelatter operation corresponding to a frequency translationof the signal u(t) as occurs when it becomes the envelopeon a carrier. This property corresponds to the fact thatinformation is preserved in a frequency translation. Thebaseband signal and the modulated version have the samelocus of zeros.The significance of the pattern region and the pattern

of zeros is that the zeros contain substantially all of theinformation nee(led to reconstruct the signal.

DESCRIPTION OF MODULATED SIGNALS [14]Table I shows the description of various kinds of mod-

ulation signals to illustrate the generality of the defi-nitions of envelope and angle, and the concept of theanalytic signal and associated analytic function.'

Conventional amplitude and angle modulation are rep-

' In Tables I and II a carat over a letter means Hilbert trans-form.

+a-D'a

F'

0

a-I

la=O-co

T/2 t

Locus of zeroas a increases

Fig. 4. Locus of zero in pattern region for analytic function.(Analytic signal = F(t) = 1 - aei] vtIT.)

resented in Table I, along with many less conventionaltypes of modulation. Of considerable interest is Type 4,a single-sideband angle-modulated signal. This type ofsignal has been the subject of recent experiments, whichwill now be discussed.

EXPERIMENTS ON SINGLE-SIDEBAND ANGLE MODULATIONThe farniliar single-sideband AM is a hybrid formn of

modulation, because its angle as well as its envelopecarries modulation information. Single-sideband FM is,likewise, a hybrid form of modulation, because its en-velope as well as its angle carries modulation informa-tion.DuBois and Aagaard [161 investigated signals of Type 4,

Table I. The specific signal information related to theirexperiment is given in Table II. The signal u(t) shown

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IEEE TRANSACTIONS ON INSTRUMENTATION ANI) MEASUREMENT, JUNE 1969

TABLE IISIGNAL INFORRMATION, D)UBoIS-AAGAARD EXPERIMIENT (1964), BRAZEAL-GLORIOSO EXPERIMENT (1965)

F(z) f(t)

exp {j[g(z) + jg(z) + wcz]lg(z) = (A lWi) sin wmz9(Z) = -(A l/@m) COS comz

exp {j[g(t) + jg(t) + co,t]I exp [,B cos wmtl] cos [w,t + sin wmt]SSB FiA\, d K A lwn7(modulation = A cos wclt)

there contains information in both the envelope and theangle, and is, therefore, called a hybrid signal. However,the signal has been called single-sideband FM, because theangle is identical with that of ordinary FM. The mathe-matics in Table II reveals, upon brief exploration, specialrequirements for generating the signal u(t). These includean approximation to a 900 phase shifter, effective across

a band of frequencies, and an exponential functiongenerator related to the portion of u(t) given by e

Four principal conclusions of the experimental in-vestigation, which employed versions of the instrumenta-tion just mentioned, are the following.

1) The spectrum required for transmission can bereduced by about compared to that for conven-

tional FM.2) The carrier power required increases rapidly with

modulation index.3) The modulation index was limited by instrumenta-

tion to about 2.4) Phase modulation might be preferable.

Glorioso and Brazeal [17] conducted experimentsbased on the same signal. Four principal conclusions ofthis work are the following.

1) Frequency multiplication using exponential gen-

erators permits a modulation index as high as 4.2) Detection can be achieved by "envelope detection,

additional nonlinear processing, and filtering."3) Threshold properties are similar to conventionalFM at reduced bandwidth.

4) "Single sideband wide-band angle modulation showspromise as a substitute for conventional wide-bandfrequency modulation."

EVALUATION OF SINGLE-SIDEBAND ANGLE MODULATION

It is too early to evaluate the practical significance ofsingle-sideband angle modulation. The history of pastadvances in modulation techniques shows that in theirearly stages there is much confusion concerning futureapplications. Radical statements concerning their utilityor lack of utility have often been made by capablepeople which, in retrospect, prove to have been unwise.However, some remarks aimed toward evaluation seem

advisable.Kahn and Thomas [18] examined the bandwidth prop-

erties and optimum demodulation of single-sideband FM.In their examination a comparison was made of the spec-

trum of ordinary FM and single-sideband (SSB) FM

when the modulating signal is a sinusoid. It was con-

cluded that for modulation indices less than about 1

the bandwidth of ordinary FM was smaller, but as themodulation index increases beyond one, the bandwidth ofSSB-FM becomes the smaller of the two.Mazo and Salz [19] likewise examined the bandwidth

properties but took as the modulating signal a samplefunction of a stationary Gaussian process. They foundthat the bandwidth of SSB angle-modulated waves be-came larger than that of ordinary angle-modulatedwaves as the modulation index increased.A radical interpretation of the Kahn and Thomas

results would lead one to suggest that SSB-FM offerstremendous possibilities for communication, while a radi-cal interpretation of the Mazo and Salz results wouldlead one to suggest that SSB-FM is a theoretical curios-ity without prospective application. The truth very likelyresides between these two extremes and not very near toeither extreme. In fact, applications of modulation are

not limited to communication but encompass navigation,instrumentation, and even computation. Some applica-tions in navigation use sinusoidal modulation exclusively,and saving of bandwidth is a desirable goal.

It is surprising that it has taken so long to arrive at a

definition of what is meant by single-sideband anglemodulation. A similar period of maturation will probablyensue before its significance will be reasonably clear.

STRUCTURE OF MODULATION THEORYModulation theory does not yet have the status

accorded network theory or signal theory. One reason

is that modulation theory is a mix of network theoryand signal theory, and advances in modulation theorytend to result from advances in these fields. Moreover,network theory and signal theory are less applicationoriented, and thus advances in these fields do not neces-

sarily hold value to modulation theory. But advances inthese fields which are applicable to modulation must beidentified and pursued with the requirements of applica-tions in mind. The relations between these fields can beclarified by considering a structure such as that shown inFig. 5. This drawing shows various specialized areas ofstudy within network theory and signal theory. By con-

structing "study chains" from Fig. 5, such as the one

shown in Fig. 6, one can develop separate fields of studyrelated to modulation theory and practice. Not all pos-

sible chains formed in the manner illustrated in Fig. 6are meaningful, but there are many meaningful chainswhich can be formed in the manner shown.

Another way of viewing modulation is in terms of di-

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\WAFIELD: MOI)DULAlTON ME'\SUREMENTS

SSB FMOther Hybrid

DigitalSSB AM

Practical Circuits

Highly Sophisticated

Moderately Complex

AM 4 SimpleConversion

Fig. 5. A structure of network theory and signal theory. With fewexceptions, each vertical path represents a specialized area ofstudy.

Elementary

Consumer Services

Navigation

Military Communication

Unknown Signal Analysis

(a)Applications

Intermediate Advanced* a

Network andSignal Theory

Signal Theory

ISynthesisl

Stochastic

Stationary

Digital

fO Man's Design

Memory

Fig. 6. One study chain.

msensions, as shown in Fig. 7. One dimension, networkand signal theory, relates to the structure of Fig. 5. Asecond dimension relates to the practical design of cir-cuits to perform the complex operations involved ingenerating and recovering information from modulatedsignals. A third dimension involves the requirements im-posed by applications. Each dimension in Fig. 7(a) is ac-

companied by illustrative "points" in the space. A sec-

ond space, Fig. 7(b), indicates a basis for selection ofresearch topics. Requirements of applications are shownhere as well, but linked to the applications along an-

other dimension is social value which accrues from theuse of the spectruiim. The third dimension, research goals,mlay be inferred fromn the applications aind the socialvalue implications. The goals can be formulated from thetechnical background suggested in Fig. 7(a) and a knowl-edige of the relations between the dimensions shown there.

TECHNIQUTE FOR MEASUREMENT DEVELOPMENT

As technology progresses, one's idea of what consti-tutes a "technique" tends to enlarge, of necessity. A tech-niique may be enlarged to be thought of as something likea flow-chart or programu. With such a generalization inimind, it is suggested that the work discussed here can

be seen as illustrative of a technique for the development

Social Value

Research GoalsApplications

(b)Fig. 7. Dimensions of modtulation.

of measurenments in modulation. The technique consistsof five steps. They are

1) development of the mathematics,2) determination of the functions to be performned,

based upon the inathematics,3) research to (levise instrumentatioin to perform the

necessary functions,4) measurements to evaluate applicability,5) reports of concluisions.

That this sequence of events has taken place in SSBangle modulation is clearly illustrated by the historical(levelopment reported in the references.

EDUCATION

The relative sophistication of the mathematics of theSSB angle modulation system, and the chain of circulm-stances which must be evolved before a system of mod-ulation can become useful in practice suggests that thereis a strong role to be played in the universities inmarrying the good mathematical background now en-joyed by students to the practical instrumentation as-pects of modulation systems. Some excellent suggestionsalong this line have been displayedl by Voeleker andTrussell 1201. The spectrum is crowded. Modulation isvital to communications. Analog modulation will beused differently in the future than it is now. Trhe inewfields of digital communication, solid-state (levices, andlike subjects bring not only renewed importance to com-munications, but a new and larger challenge. This chal-lenge will be most effectively met by stronger programsin modulation within the universities.

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUIREMENT, JUNE 1969

CONCLUSIONSThe introduction of the analytic signal has provided a

base for improvement and extension of analog modula-tion theory. It has led to the concept of single-sidebandfrequency modulation, and induced experiments thatsuggest that there are benefits to be derived from thismodulation method.Modulation theory needs refinemiient and extension.

Study chains evolved as shown herein suggest ap-proaches. Such refinement and extension can be the basisfor a general technique for developing new instrumenta-tion and measurements in modulation systems.To provide a continuing source of persons equipped

to work with sophisticated communiications systems, acloser and deeper relationship between theory anid ex-periment in university communications laboratories isdesirable.

ACKNOWLEDGMENT

The comments of Drs. J. D. Hill and N. S. Nahmanwere helpful in the preparation of this paper.

APPENDIX

SHORT TABLE OF HILBERT TRANSFORMS*

Number Time Function

1 u(t)

2 eiwt (w > 0)

3 e"i't (u> 0)

4 cos ct

5 sin cot

Hilbert Transform

(1 /r) P.V. fo u(x) dx

-jeiwt, -~-

sin wt

-Cos ut

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[3] H. S. Black, Modulation Theory. Princeton, N. J.: VanNostrand, 1953, 363 pp.

[4] C. Cherry, Pulses and Transients in Communications Cir-cuits. London: Chapman and Hall, 1949, 317 pp.

[5] J. J. Downing, Modulation Systems and Noise. EnglewoodCliffs, N. J.: Prentice-Hall, 1964, 209 pp.

[6] S. Goldman, Frequency Analysis, Modulation, and Noise.New York: McGraw-Hill, 1948, 434 pp.

[7] P. Penfield and R. P. Rafuse, Varactor Applications. Cam-bridge, Mass.: M.I.T. Press, 1962, 623 pp.

[8] M. Schwarz, Information Transmission, Modulation, andNoise. New York: McGraw-Hill, 1959, 461 pp.

[9] J. M. Wozencraft and I. M. Jacobs, Principles of Com-munications Engineering. New York: Wiley, 1965, 720 pp.

[10] H. B. Voeleker, "Toward a unified theory of modulation-Part I: Phase-envelope relationships," Proc. IEEE, vol. 54,pp. 340-353, March 1966.

[11] Zr, "Toward a unified theory of modulation-Part II:Zero manipulation," Proc. IEEE, vol. 54, pp. 735-755, May1966.

[12] J. Ville, "Theory and applications of the notion of complexsignal," (transl. of the 1948 French version), Defense Docu-mentation Center, AD 636038, August 1, 1958, 34 pp.

[13] J. Dugundji, "Envelopes and pre-envelopes of real wave-forms," IRE Trans. Information Theory, vol. IT-4, pp. 53-57,March 1958.

[14] E. Bedrosian, "The analytic signal representation of modu-lated waveforms," Proc. IRE, vol. 50, pp. 2071-2076, October1962.

[15] R. Deutsch, Nonlinear Transformation of Random Proc-esses. Englewood Cliffs, N. J.: Prentice-Hall, 1962, 157 pp.

[16] J. L. Dubois and J. S. Aagaard, "An experimental SSB-FMsystem," IEEE Trans. Communications Systems, vol. CS-12,pp. 222-229, June 1964.

[17] R. M. Glorioso and E. H. Brazeal, Jr., "Experiments in'SSB FM' communication systems," IEEE Trans. Com-munication Technology, vol. COM-13, pp. 109-116, March1965.

[18] R. E. Kahn and J. B. Thomas, "Bandwidth properties andoptimum demodulation of single-sideband FM," IEEETrans. Communication Technology, vol. COM-14, pp. 113-117, April 1966.

[19] J. E. Mazo and J. Salz, "Spectral properties of single-side-band angle modulation," IEEE Trans. Communication Tech-nology, vol. COM-16, pp. 52-62, February 1968.

[20] H. Voelcker and I. Trussell, "Development of a modemcommunication Laboratory," IEEE Trans. Education, vol.E-7, pp. 166-175, December 1964.

* E. Hille, Analytic Function Theory, vol. 1. Boston: Ginn, 1959,p. 176. P.V. means "principal value," as defined in the referencedbook.

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