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670 IEEERANSACTIONSONCOUSTICS,PEECH, AND SIGNALROCESSING,OL. ASSP -27, NO. 6 , DECEMBER

nal ProcessinJAMES H . JUSTICE

AbNtruct-Digital synthesis of music has led to the consideration ofmodels ot her than the usual additive (Fourier) synthesis of waveforms.One of these method s, based on the FM equation,has been found to beof particular value due to its easy implem entation and the richness andevolutionary character of its harmonic structure in time. Any usefulsynthesis procedure should be accompanied by a corresponding analyticprocedure.In this paper we lay the found ations for such a procedure based onthe discrete Hilbert transform.

I . N T R O D U C T I O NSTIMULATING, challenging, and relatively new appli-cation of digital signal processing is t o be foun d in thearea of music generation via digital com pute r. By music gen-

eration, we mean the production of purposefully structuredsounds in the broad sense. Tha t is, wewish to contro l notonly the means of sound synthesis itself, including the shapingof timbre and othe r tone characteristics, but we wish to pro-vide the means to orchestrate sounds into larger structuresor compositions with a minimum of restrictions in achievingwhatever sort ofgoal we may have in min d. nshort, ourgoal s nothing less than complete co ntrol of the sound en-vironment including the warp and woof of sound textures,rhythms, .and pitches; the entire spectrum of sound charac-teristics from which the patterned cloth of music s woven.Indeed, tomorrows composer/performer,he weaver, mayfind himself as much at home with CRT terminals, graphictablets, and joy sticks as he now does with his instruments,paper, pen, and baton.Since the creation of the first music generation program in1958 by M. Mathews of Bell Labs [ 6 ] , n ever growing num-ber of engineers, mathematicians, computer scientists, andartists, all with a common goal, have been intent on convertingthe digital computer in to a responsive and sensitive instrumen tcapable of enhancing and extending our musical powers.The setting was eminently suited for, and indeed demandedthe use of d igital signal processing principles and techniqu esof all kinds, and few stones are being left untur ned. And, justas any new area of inquiry draws heavily on the tools of thepast, its own unique challenges will hopefully always con-tribute something new which may find use in other areas aswell, or perhaps pave a road to new and as yet unseen areasof application.We do not need t o look very deeply int o the complexitiesof digital music generation to appreciate the need to find

Manuscript received January 10, 1979; evised June 17,1979.The author is with the Department of Mathematical Sciences, Univer-sity of Tulsa, Tulsa, OK 74 104 , and with the Am oco Production Com-pany, Tulsa, OK 7 4 1 0 4 .

algorithms and data base siructures which provide maxiversatility with minimal externalcontrol. To provide vthe privileges that traditional music enjoys presents a comtational burden that without sufficient forethought mbecome almo st unmanageable or at best impractical.My purp ose here is not t o deal with the larger issue of digproduction of music, but with only on e, and perhaps the mbasic aspect of it: the question of synthesis itself. How domake a sound like a piano, clarinet, or perhaps some herfore unknown hybrid? I read the same math, physics, acotics, and music boo ks yo u did, and it really was not all hard. One simply takes the ingredients; a proper collectiofundamentals and harmonics, a dash of proper weights, perhaps anenvelope for flavor, and it is done. It is at point that we learn how immensely complex the real woreally s and how ad ept he ear s in discerningevery laspect of its complexity. Very soon we learned that recipe was oversimplified in amyriad of ways.It is the nature ofmusical instruments tha tno two monics of a single tone willevolve dynam ically in exathe same way; much less is this structure preserved as move from pitch to pitch on a given instrument. The mplication of the complexities of simple Fourier-type addisynthesis requiring different envelopes on each harmowhich must changeas wepass from pitch to pitch, alrebegins to become overwhelming, particularly if real-generation is being contem plated, and these ob servations oscratch the surface of the underlying reality. Those oconcerned with problems of synthesis, then, have lookedbetter and more compact means to mimic nature and in pticular t o fool our ears into thinking tha t we really did all tand more.In 1973, J . Chowning of Stanford University [2] madeobservation that truly complex spectra w hich could evolvcomplex ways could be relatively easily obtained using theequation with suitable parameters and envelopes. T he amaquality of the sounds obtainable in this way, and the compmeans of their production, led many of us to look to FMviable synthesis procedure.There was just one problem that we had to face, and I thit is summed up by the fact that when a student in my mclass approaches me with the inevitable question, HOWdmake a ____ (instrument) sound?, 1 am afraid I comclose as perhaps I will ever be to t he ministry as I place hon shoulder and dispense the best advice I can sometigive. . .Indeed FM works very well, if we just knew how to conall those little parameters, an d that is what this paper [5about.

0096-3518/79/1200-0670$00.75 979 IEEE

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JUSTICE: ANALYTICIGNALROCESSINGN 67 111. BASICs

The goal of research in sound synthesis is t o store a minimalamount of information to be combined with minimal compu-tation to produce sufficiently complex sound structures to beof interest. The FM equation, in itsoriginal form, requires thestorage of only a few parameters, some coarsely sampled en-velopes, and a single cycle of a sine or cosine waveform, suit-ably sampled.The success of Chownings FM techriique [2] for generatinga wide v ariety o f sounds with soph isticated spectra which evolvein time and w ith great econom y of means led us to investigatethe underlying principles to determine how far we might goin analyzing and synthesizing sound by this technique.In particular, we were interested in two questions. First, istherean analysis technique which will yield the parameterswhich generate a givenFM signal? S econ d, we ask the ques-tion, to what extent might FM synthesis be used to regen-erate or to approximate a given signal? The latt er question isof great interest as it is extremely difficult to synthesize cer-tain types of signals using technique s th at are practical givenlimited core space and compu tation time.A . The Analytic Signal

Let us suppose that we are given a real signal which we de-note by s (t). The Fourier transform of S s , (~) ill beconjugate symmetric since S s real. If q ( t ) s another signalthen the Fourier transform of SR t )+ s l ( t ) is given by s a)i S I (o )where SI is the Fourier transform of SI. If the Fouriertransform S w) = S, W) f iSz w) vanishes for w

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If the m odulating angle is determined b y a n equation of theform@m t )= Im ( t ) OS Pm t (11 )

then, substituting into (10) we obtain the first equation con-sidered by Chow ning [2]f ( t )= I & ) cos 0, -I- Im ( t ) os p t ) (12 )

where we choose to use the cosine in place of the sine in viewof Section 11.It is clear that not a l l possible modulating functions could beeasily represented in the form 1 l ) , ut this is avery simple andnatu ral form to w ork with , particularly in view of Section 11.A. Analysis of FM Signals

Let us begin by supposing th at we have a signal +(t) gen-erated by an FM process given by (12). We shall consider theproblem of determining the parameters in the equation.Applying the results of Section I1we may obtain sR ( t ) inthe formS R( t )= I ( t ) COS (6m(t)). (13)

If we now extract the linear trend from 6 , we obtain thecarrier angular frequency w,. This may be doneby ittinga least-squares straight line through the phase @,(t),and cal-culating its slope.Subtracting the linear trend from @ leaves us with a newsignal, the modulator s m f )= @,(t) - a, .Applying the analysis of Section I1 again to the modulatoryieldsSrn ( t )= Im ( t ) OS Y m (t). ( 14)

If now , he original signal obeyed 1 2), the function y, t)wi be a straight line whoseslope will be 0 In he eventtha t this is no t so, our signal simply did not obey 1 2) , and weare ready for further analysis.Under our original assumption that the signal obeyed (12 ) ,we have now determined al l the parameters in that equationby a four ste p process which we may sum marize here.

1 ) Obtain I , ( t ) and @,(t) from Se ction I1 analysis.2) Extract linear trend from Gm ( t ) nd obtain w, .3) Obtain I,(t) and y,(t) from @,(t) - w,t using Section4) Extract linear trend from ym to obtain p .I1 analysis.

The residual part of y m can only be a numerical error if (12)were satisfied by our signal.We note now that if y, t) were no t a straight line, then themodulator is in turn being modulated and we may continuethe above analysis ad infinitum or ad nauseum if we so desire.B. Signal Analysis

We have indica ted ho w the analysis technique discussed inSection I1 can be applied to determine the parameters of anFM signal. We should now ike t o pursue the question of moregeneral signal analysis. Specifically, we ask, what kind of in-formation can we get about more arbitrary signals and howmight it be utilized for sound synthesis? We shall reserve

questions of synthesis fora later section. In he remainof the paper we assume that al envelope s are slowly vaing so that they may be coarsely sampled and stored or tthey m ay be generated by simple algorithms. This assumtion is made for practical considerations. Without it, almany signal could be represented (in th eory , at least) in almany of the forms considered.Le t us begin by considering our analysis for a large classsignalsused nmusic compu tation. Consider a signal of formSR(t)=a(t) C k COS k a t

k=ON

1

where we have a periodic signal represented by the summatbeing amplitude modulated by the function a(t). We do nconsider this type of signal because we need ano ther means synthesizing it, but rather because it is of interest to learn hit relate s to th e class of FM signals.

Writing SR(t) n complex exponential form,sR ( t ) = 1/2a(t) Ck(exp (&at+xp (-&at)) (N

k=Owe easily obtain the Fourier ransform of sR ( t ) ,

S, W) = 1/2 k [A O - kQ) A(u -I- k a ) ]N ( 1k=Owhere A(w) s the Fourier transform of a(t ) .signal is given byOur analysis in Se ction I1 tells us tha t the desired analy

S t ) = 1/2n k [ A W - kQ)tA(w t k a ) ]k=O i-.exp (iwt)dw (1which becomes, after change of variable

N mS t ) = 1/2n Ck A(U) exp ( i (0 k a ) )d

k=O l k Ct 1/2n 2 Ck Jm( o ) xp i w - k a ) )dw.

k = O kSl

1At this po int we are fac ed with a possible aliasing problem A w). For simplicity in carrying out our analysis, let u s spose that. A(w)= 0 for w >a2/2 (this is not much of a striction, since nvelopes may be ssumed to be slowvarying).We shall further m ake the simplifying assump tion that osignal is unbiase d, which is to say tha t the term Co in our pansion is zero. If this is no t satisfied, the multiplier ofafter conversion will not be a(t) but rather a complex tewhose real part is a(t)/2.Under this simplifying assumption, (1 9 ) becomes

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1 .........~~

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...,,.,-.......,. ,,,.., ,,. ..,.,,.,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... ,.. - , ,........................... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .Example 1. Exponentiallydecaying osine.The nvelope alculatedis the true envelope.

If we now assume a t)2 so that n o phase shifts were intro-duced by the envelope, we see that he envelope calculatedby the methodsof Section I1 will be given by

where we reca ll, now, that (21) will be more complicated ifA(w) Ofor I o I > a / 2 .The angle modulator calculated in Section I1 will be givenby

( t )= arctan( c k sin k n t / x C k cos kat ) (22)which we see can be quite a complicated ter m.A number of comments based on the above analysis are nowin order. First, t is easy to see that i our original ignals R t ) had consisted of a single term with envelope a t), then(21) and (22) would yield the ex act original envelope and theexact angular frequency of the originalsignal. In any othercase this statement is false. The summation in (21) is periodic,however, and so wll oscillate between fute d values. This in-dicates that I ( t ) in (21) will follow the original envelope inmaximum deviation from zero. A multiple of a t ) could besampled simply by taking the samples 2n/a time units apart.

This fact can be illustrated b y examples given in Examples 1and 2. The signal used in th ese examp les was of the form (15),(231,

f o r O < n < 5 1 2 and N = 1 , 5 , (23)respectively. a t ) was taken to be exponentially decaying.For he purpose of graphing, the signalwas then sampledat every fifthpoint (as a result, the signal graphs in somecases do no t esemble their true form).It can be clearly seen from the examples th at the envelopefollows the original and in the case of a single cosine dupli-cates it towithin obtainable numerical accuracy.Let us now recall t hat we have obtained an analysis of thesignal (15) in the quite different form (5). If it were true thatthe phase had a simple form [recall (22)], then we might hopeto resynthesize the signal from (5).Again, let us turn to an example. Consider once more asignal of the form (15), (24):

f f )= a ( t ) ck COS k a t .Nk=O

We have show n th at the angle moduiato r of this signal is ofthe form (22), (25):

The argument in this expression is clearly a periodic functionwith period 2n/Q. It follows that (t) will be periodic (werestrain $ to the interval -n/2

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674 IEEE TRAN SACTIO NS ON ACO USTICS, SPEECH, AN D SIGNAL PROCESSING , VOL. ASSP-27, NO. 6 , DECEMBER 19

(b)Example 2. (a) Exponentially decaying Fourier series 15), (23). (b) Linear hase from (a).

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As a result, @ could be represen ted by a Fourier series andthe original signal regenerated from an expression of the form

f ( t ) = I t )cos (att1 k sin ka t ) (26)where we have removed the constraint - 4 2 < @

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676 IEEE TRANSACTIONS OM ACOUSTICS, SPEECH, ANDIGNAL PROCESSING, V O L. ASSP-27, NO. 6 , DECEMBER

The expa nsion of the signalf ( t ) hen takes the form

'cos [ ( u t n l u l n 2 u 2 - . . t n N w N ) t ] . (33)(Observe that the signal (15 ) which we analyzed earlier in (26)turns ou t to be a special case of (29) and (32) given by makingthe particular choice W k = k a , Ik = f k , where N may beinfinite.)The spectrum of such a signal can clearly achieve a high levelof co mplex ity and surely rep resents a n interesting class of sig-nals. However, our suggested analysis of a signal in to the formof nested layers of modulators dep arts quickly from the aboveanalysis.Let us consider the harmonic structure of an FM signal withmodulated modula tor (where we have gone one extra step inapplying analytica l signal nalysis to a signal s uggestedabove).Our signal will be of the form

f ( t )= I t )cos (ut Il ( t) in ult , ( t ) sin w 2 )) (34)where we are using sines in place of cosines to facilitate theuse of Bessel func tions in obtain ing an exp ansion of the func-tion. Carrying ou t the expansion, we obtain

I m \f ( t ) = I COS ut Il J f l ( I z ) in a1 n u 2 ) )

- m

Whenwe realize tha t this signal (35) is similar in form (butwith infinitely many terms) to the signalgiven in (29) and(32) w hich resulted in the expansion (33), it seems clear tha tthe harmonic structure of a signal of the form (34), given by(35), m ay well defy comprehension. However, it is on thebasis of this general form th at we suggest the utility of theform (34) an d he even more complicated forms obtainedfrom i t by the m ethods of Section 111-A. I t seems reasonab leto suggest that many signals may be represen ted in th is easilyobtained and relatively compact form.IV. ANALYSISAND SYNTHESISF SIGNALS

In this section we wish to loo k at the practical applicationof the principles developed in the preceding sections. In par-ticular, we shall be conce rned with methods of analysis andsynthesis, and with ban dwidth and sampling considerations.A . General Comments on Bandwidth and Samplingquency modulation (29), (30), (36):Letus begin by recalling the standard equation or fre-

f ( t ) = I ( t ) cos (wt+ 1 1 ( t ) in 0 1 ) (3 6)which has the expansion (31), (37)

mf ( t )= I ( t ) C J, Il ( t ) ) os ( u t + n u l t)) . (37)- m

The harmonic structure is seen to consist of frequency com-ponents u n u 1 with weights J, Z1 ( t ) ) or --03

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JUSTICE: SIGN AL PROCESSINGNUSICOMPUTATION 617

amodulator to be quasi-periodic if it is of the form (32),(41):

@(t) Ik(t) sin u k t ) . (41)The modulator may become periodic if, or example, theangular frequencies o k have rational ratios to each other,and theIks are constant.

Again we shall suppose that we have analyzed a signalobtaining nested modulators until finally a modulator urnsou t o be quasi-periodic. For simplicity we may takeoursignal to have only one modulator,f ( t )=I ( t ) cos (at+ Ik(t) in (cdkt)) , (42)

in wh ich case, we have o btained an analysis of th e signal inThe quasi-periodic case is clearly more general than heperiodic case, but is perhaps less imp orta nt for analysis be-cause it would be difficult to obtain he parameters foraquasi-periodic mod ulator. Being a generalization ofhe

periodic case, however, it is the authors opinion tha t it maybe importa nt for signal synthesis. The only difficulty is thatin the nonperiodic case, the modulator waveform could notbe stored well in sampled form.An inspection of (33) should convince the reader of thepossible wealth of spectral compon ents available here. Again,this could be compounded by nesting modulators.D. Signals with Nested Modulators

(33).

As a last resort in signal analysis, themethod of Section111-A can be applied rep eatedly to successive mod ulator termsof a signal until, one hopes, the process can be terminated, theremaining modulator term being neither periodic nor quasi-periodic, bu t hopefully negligible so that i t may be discarded.This type of analysis results in signals of the form (34),(43):

f ( t ) =I ( t ) cos (utt I t ) in a1+ I 2( t ) in w 2 ) ) (43)which represents two nested modulators. The analysis of thesignal given n (3.9, (44),

f ( t ) = I C O S ut t Il Jn 12)sin ol n o z ) ) , (44)indicates th at it is related to the class of signals with quasi-periodic mo dulators. The advantag e here, howeve r, is tha t weautomatically have the parameters for the modulator term in(35) from our analysis which led to (34).Again, the nested mod ulator tech nique of signal synthesis,leads to signals of a quasi-periodic nature in com pact form (wedo no t have to worry a bout storing one period, possibly ofgreat length, o f the modulator). Only one period of the cosineas well as the envelopes need to be stored in this meth od ofsynthesis.E. A Practical Approach to Synthesis

Assuming that we wish to employ analytic signal techniquesfor signal analysis and synthesis, a relatively simple approxi-

m( -m , )

mation procedure can be resorted to which may or may notyield good results depending on how it s applied.Let us suppose that we are given a sampled signal for analy-sis. Break the signal int o small successive pieces of length 2nfor some n . Carry out the analysis of Section 111-A on eachpiece. At some level of modulation, fit he phase curve ob-tained from the analysis with a straight line segment. Do thisfor each piece of the signal and store an approximate value forthe envelope and one for the modulation frequency obtainedfrom the straight line fit. If the pieces are small enough, thefit will be reasonably good. Recreate the signal using the ap -proximatevalues obtained for each piece of the signal, lettingthe mod ulator frequency slide between the successive valuesobtained in the analysis. The result should be an approxima -tion to the original signal. Exam ple 4 shows the analysis for asignal whose modulator has a continuously varying frequencyfor which this technique is eminently suited. This techniquecan be combined with m ethods discussed earlier in this paper.

V. CONCLUSIONFM synthesis techniques are enjoying widespread use andpopularity among those interested in the digital synthesis ofmusic. Because of their easily implem ented form and com-pact tructure, hey are no t likely to be soon supplanted.While intuition and tinkering with the contro ls are some-times helpful guides to using these methods, their use hasbeen hampered by he lack of suitable analytic techniqueswhich will help us to achieve more precisely our desiredgoals. Inpractice, we may wish to get into the ballparkanalytically, and then modify the parameters until the resultthat pleases us is achieved.In this paper, we haveshown that an easily implementedprocedure based on the discrete Hilbert transform leads to

an analysis procedure which is com patible w ith he conceptof FM synthesis. It further broadens our field of view beyondthe usual Fourier methods, which, while importantand ofdefinite value inm usi c generation, can n o longer be consideredto be our only impo rtant basis for synthesis.In this paper we have raised several uestions which hopefullywill find answers and perhaps light the way t o more powerfulor versatile procedures for signal analysis. Since this paperwas presented, some of the ideas contained herein haveal-ready been finding their way int o digital music synthesis.VI. EXAMPLES

The accompanying figures show a number of exampleswhich illustrate some of the main ideas in th e text. In all ex-amp les, the ignal envdlopewas theone shown in Fig. 1.Where applicable, a l l modulators were iven the envelopeshown in Fig. 2 . In allcases envelopes are graphed by and signals re graphed by X. We have connectedhegraphed points for ease in pattern recognition, but in mostcases,sampling of the signalswas to o coarse to reveal theirtrue shape, which was generally irrelevant to the example.The few exceptions to this are accurately graphed.Example 2 shows various parts of the analysis of an exponen-tially decaying Fourier series just to verify the discussion inthe text. The envelope only follows the true envelope as dis-

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678 IEEE TRANSACTIONS O N ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-27, NO. 6 , DECEMBER 1

Fig. 1 . Illustrating envelope on signal in all examples.

k

cussed, and he periodic mod ulator is correctly derivedas ample 3(b) and (c). The modula tor envelope is shown in Eshown in Example 2(c). ample 3(d) and its phase in Example 3(e). The remaining erExample 3 is a standard FM equation of the form (12). Its after extracting the linear trend from the modulator is shophase and modula tor are correctly derived as shown in Ex- in Example 3(e). The error is generally aroun d k0.1 rad.

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JUSTICE: ANALYTIC SIGNAL PROCESSING IN MUSIC CO MPUTATION 679

..................................... .?.... ........................................................................................................... . . U . . ~ . * *a .SI .. .+. I , . . . ,.,,, ).,... ,- *...................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... ....\1 . . . . . . I . . . . . . . . . . * . . . . . . . . . . . . . . . . . . . , , . *,, . . , . , . , . , .,. , L . . . ~ ..,.,..,,....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * .

(a)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. I , , . .I . . . . I ,,?...., 1 1 . . ( . > . . . . , e . .. . . . . . . . . . . . . . . . . . . . . . . . . ......... . . . . . ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .b)

Example 3 . (a) Standard FM signal of the form (12). b) Linear phase extracted from (a).

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680 IEEE TRANSACTION S O N ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-27, NO. 6 , DECEMBER 1979

, r

.. .......

i

................................I.eI

0.o

(dlExample 3. (Continued.) (c) Mod ulator derived from (a ). (d) Envelope derived from c ) .

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JUSTICE: ANALYTIC SIGN AL PROCESSING IN M USIC COMPUTATION 681

~ , .. .. . ..... . .._... . .... ................................................ . ........_......-.. ..-..-...-..-..-..... .I l . .** l ..n.)P.sn . , . , . . l .n** I 8 . . , . 1 / 1 . . * . . ~ v.,, , . , , . . ( 1 I . ~ . . # ,.,,',.,, ,n .,,. ,.., l l . i ) ..,. ,.,,, lll . . . . .C . . . , .*. ,,, *.I. .............. . . l . ( ~ , I . ... ~ . ~ ~ 1 . . . 1 1 1 . . 1 ~ 1 . . . 1 ~ I 1 , ~ , . . * . , 1 . . 1 1 . 1 , . . . , . 1 , t l - - l .-.,.. -.,,#.,,?,.,.,.** .(... ....*.. :.I. \I,.I.1 .,..... .

(f 1Example 3. (Continu ed.) e) Modulator inearphase derived from (c). ( f ) Modulator error ermafter inear trend issubtracted from ( e). (Error is generally around 0.1 rad.)

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682 IEEE TRANSA CTIONS ON ACOUSTICS, SPEECH, AN D SIGNAL PROCE SSING, VOL. ASSP-27, NO. 6 , DECEMBER 19

.... ........ .......... -.. ................. I _ .............. _ . . .... ..._.............. .* ....................................................V. .* 0 ,,.. ..,n . . , ~ ~,LI.......~ .*....I.U. .*............. .,, . , I... . ......... ........... . ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................. ......................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(a)

...................... . . . . . . . . . . . . . .(b)

Example 4. (a) Envelope extracted from F M signalwith mod ulator whose freque ncy increases linearly in time. (b) Linearphase extracted from (a).Example 4 isan FM signalwith amodu lator whose fre- frequency eventually overwhelms the sampling interval, lea

quency increases linearly in time. The resulting phase of the ing to some degradation on the right-hand side of these plotmodu lator should therefore be parabolic. The modu lator Analysis f the signal correctly derives themodulatorn

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683

........................-...........................................................................................-......... ...............*. . U ? . , : l . . 1 * . , 1 . . h , .VV.,.?..*. *, * .0 , . b , :............................................................................ .. ............... ....... . . . . . . . . . . . . ...... .......... . . . . . . .. . .4Example 4. Continued.) c)Modulatorderived rom a). Sampling nterval too coarse for increasing modulator re-quency at right.) d) Envelope extracted from modulatorn c).

Example 4(c), an d further analysis yields its parabolic phase were sampled at every fifth point for thepurposes of graphing.in Example 4(e). The signal envelope chosen is one wh ich is characteristic ofAll signals used in the exam ples contain ed 512 points and bells,gongs, or struck objects which follow this patte rn of

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684 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-27, NO. 6 , DECEMBER 19

~...................... ........ ....... ....... .. .......... . . ..... , . .. .... , . .., ... . . . . . . . . * , . . .. . .. ... . . . ...~........... . ... .. . . . T I .*.,.,. . . , I, .1 ,.,,, ..,.. ,>. ..,.,,, ..,. .-,..,,...,.. ..r, ,>. ,. . . , . , . . , . , I .,. ..,....., , _ . . . .,,,. ,.,:..,a..,.~ V........... .... , * . , . b b L , u r , . .,.,,..,,, ? , . .~. , . , . .*, , . . , . . . r , , I ..,,.. .,.... , z > , , , , , 7 . , 1 r > .....

e)Example 4. Continued.) e) Parabolic phase of modulator derived from c).

decay after being struc k. The modulator envelope was chosenmore for its characteristic shape than for any other reason. Ifused in music synthesis, the effect would be that maximummodulation would occur abou t hemidpo int of the soundduration. This is the time when the harmonic structure wouldbe the richest, and perhaps even noisy, returning to a strongsense of distinct pitch on either side.Exam ples of all the types of signals discussed were playedwhen the paper was presented and sounds ranging from musi-cal, to explosions, to effects like air bubbling through a liquidwere dem onstrated, but the possibilities are endless.

VII. FORTHE LISTENEROne of the great joys of working in this field has been themany opportunities which I have had to come t o know a greatnumber of a rare breed of people who are both artist andtechnician, compo ser and programmer-a breed of jack-o f-

all-trades innovators which I am glad to know have not diedout yet. They come from a wide variety of backgrounds andwork in many places under all sorts of conditions, generallyless than ideal.A few years ago I had the honor of meeting Aaron Copland,a great man and a great composer who was willing to take thetime to listen to the work these people are doing and to helpus place our work i nto historical perspective. A recording wasmade of computer works from around the world selected for

a concert in his honor. The recording is available for thowho w ould like to hear a variety of works representing manapproaches to this new medium. The information is cotained in [71.ACKNOWLEDGMENT

I would like to thank my associate, B. Vassaur, for his sugestions in revising this manuscrip t, and t he A moco ProductioCompany for technical assistance.

REFERENCES[ l ] R. Bracewell, The Fourier Transform and I t s Applications, NeYork: McGraw-Hill, 1965.[2] J. M. Chowning, The synthesisof omplex audio spectra means of requencymodulation, J. Audio Eng. Soc., vol. 2no. 7,1973.[3] V. Cizek, Discrete Hilbert ransform, IEEE Trans. Au di o EZ

troacoust. ,vol. AU-18, no. 4,1970.[4] A. Hund, FrequencyModulation. New York: McGraw-H1942.[5] J. H. Justice, Analytic signal processing in music computationaresented at 1st Int.Comaut. MusicConf.. Massachusetts InTech., Cambridge, 1976.161 M. Mathews. The Technology of ComputerMusic. CambridgL 1 -- ~MA: MIT Press, 1969.[7] NewDirections,2 LP album, Tulsa Studios , Box T Admi[8] A. Oppenheim and R. Schafer, Digital Signal Processing. EngStation , Tulsa, OK 74112.wood Cliffs, NJ: Prentice-Hall, 1975.