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An Introduction to the Mathematics of Digital Signal Processing: Part I: Algebra, Trigonometry,and the Most Beautiful Formula in MathematicsAuthor(s): F. R. MooreSource: Computer Music Journal, Vol. 2, No. 1 (Jul., 1978), pp. 38-47Published by: MIT PressStable URL: http://www.jstor.org/stable/3680137Accessed: 03-02-2016 11:57 UTCYour use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at s.jspJSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of contentin a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship.For more information about JSTOR, please contact support@jstor.org.MIT Press is collaborating with JSTOR to digitize, preserve and extend access to Computer Music Journal.http://www.jstor.orgThis content downloaded from 165.123.228.54 on Wed, 03 Feb 2016 11:57:34 UTCAll use subject to JSTOR Terms and Conditions

ssingPartI:Algebra,Trigonometry,andThe Most Beautiful Formulain Mathematics? 1978F. R. MooreF. R. MooreBell LaboratoriesMurrayHill, New Jersey 07974IntroductionAs it says in the front of the ComputerMusicJournalnumber 4, there are many musicians with an interest inmusicalsignalprocessingwith computers,but only a few havemuch competence in this area. There is of course a hugeamount of literaturein the field of digitalsignalprocessing,including some first-rate textbooks (such as Rabiner andGold's Theory and Application of Digital SignalProcessing,or Oppenheimand Schafer'sDigital SignalProcessing), butmost of the literatureassumesthat the readeris a graduatestudent in engineeringor computerscience (why else would hebe interested?),that he wants to know everythingabout digitalsignalprocessing,and that he alreadyknows a greatdeal aboutmathematics and computers. Consequently, much of thisinformation is shrouded in mathematicalmystery to themusical reader,making it difficult to distinguishthe wheatfrom the chaff, so to speak. Digitalsignalprocessingis a verymathematicalsubject, so to make past articles clearerandfuture articlespossible, the basic mathematicalideas neededare presentedin this two-part tutorial. In order to preventthis presentationfrom turninginto severalfat books, only themain ideas can be outlined; and mathematicalproofs are ofcourse omitted. But keep in mind that learningmathematicsis much like learningto play a piano: no amount of readingwill suffice-it is necessaryto actuallypracticethe techniquesdescribed(in this case, by doing the problems)before theconcepts become useful in the "real"world. Thereforesomeproblemsare provided(without answers)to give the motivatedreaderan opportunityboth to test his understandingand toacquiresome skill.Page 38Part I of the tutorial(this part) providesa generalreviewof algebraand trigonometry,includingsuch areasas equations,graphs,polynomials,logarithms,complex numbers,infiniteseries,radianmeasures,and the basic trigonometricfunctions.PartII will discussthe applicationof these concepts and othersin transforms,such as the Fourierand z-transforms,transferfunctions, impulse response,convolution, poles and zeroes,and elementaryfiltering.Insofaras possible, the mathematicaltreatmentalwaysstopsjust short of using calculus,though adeep understandingof many of the concepts presentedrequiresunderstandingof calculus.But digitalsignalprocessinginherentlyrequiresless calculusthan analogsignalprocessing,since the integralsignsare replacedby the easier-tounderstanddiscretesummations.It is an experimentalgoal ofthis tutorial to see how far into digitalsignalprocessingit ispossible to explore without calculus.AlgebraTo most people, mathematicsmeans formulas andequations, which are expressionsdescribingthe relationshipsamongquantities.As long as the relationshipsdo not use theintegrationor differentiationideas of calculus,they usuallyfall into the generaldomainof algebra,namedafter the arabicbest-seller of the 9th century, Kitab al jabr w'al-muqabala("Rules of Restoration and Reduction") by Abu Ja'farMohammedibn Musaal-Khowarizmi(from whose name theword algorithmis derived).Algebrais, in fact, merely a systematicnotation of quantitative relationshipsamong numericalquantities, usuallycalled variables,since with algebrawe can manipulatetheComputer Music Journal, Box E, Menlo Park, CA 94025This content downloaded from 165.123.228.54 on Wed, 03 Feb 2016 11:57:34 UTCAll use subject to JSTOR Terms and ConditionsVolume II Number 1

relationshipsinto variousforms without specifyingthe particular quantitieswe aremanipulating.For example, the equation:y x 1"says"that y is an arbitraryname givento a quantitywhich isone greaterthan anotherquantity,x. If we were to writey-l xwe would be "saying"exactly the same thing,just as we wouldif we wrote any of the following:16y 16 16xy/2 ?(x 1)- r) 7r(1 - r) 7rx(yrFunctions,Numbers,and GraphsSometimes it is desirableto give a name to an entirerelationship,ratherthanjust to the variablesin a relationship.Mathematicianshave a keen sense of brevity, so these namesare usually singleletters as well, but they servequite a different purpose.For example, the notationx 1meansthat "f" is being defined as a function of x, wherexis called the independent variable,since it can take on anyvalue whatsoever.Wecan now writey f(x)(read: "y equalsf ofx ")to mean that the value of y (which is called a dependentvariablesince its value dependson the value chosen for x) isa function of x, and the function is namedf. Rememberthatf(x) is just anothername for x 1, so the last equation aboveis still sayingthe same thing as all of the previousexamples.The advantagesof the function notation are that it a) explicitly states the name of the varyingquantity (the independentvariableor argumentof the function), and b) it givesa shortname to what may be a on. For example:letf(x) x 1g(x) 2x 3Of course, this "says"the same thing asbut the latter form doesn't show explicitly where theserelationshipscome from.What do we mean when we say that x can have anyvalue?In fact, what does valuemean?Withoutgoing too farafield into the theory of numbers,we should note that inmany cases, the value of the independentvariablein a particular function is restrictedto the set of all naturalnumbers,orintegers,or reals.Briefly,the set of naturalnumbers(denotedhere as N) is the set of numbersused for counting:(the curly braces" {4 " denote a set, and the ellipsis "."meanshere that the set has an infinite numberof elements).To indicate that the independentvariablemust be chosen fromthis set, we writef(x) x -1andx ENwhere "EN" means "is an element of N", the set of allnaturalnumbers.Suppose we choose x equal to 0; what isf(x) equal to? Our Pavlovianresponseis, of course, minusone, but note that this number is not a naturalnumber asdefined above.So even though x might always be a naturalnumber,f(x) might not be. Othersets of numbersfrequentlyencountered are I, the set of all integernumbers,I 0, 1, 2, 3,. . .and R, the set of all real numbers.Real numbersare thosewhich can be written as a (possibly unending) decimalexpression, such as ir, 2, and 1/3, since rr 3.14159.,2 2.000. ., and 1/3 .333 . Sometimes R is usedto denote the positive reals,R2 for the set of all orderedpairsof real numbers,etc. Just as the integers include all of thenaturalnumbers,the realsinclude the integers,as well as therationals(numbersformed by the ratio of two integers,such as1/3 or 22/7), andthe irrationals,like ir (whichis approximatelyequal to 22/7, but is not exactly equal to any ratio of twointegers). It is a fundamentalmystery that the ratio of thediameterof a circle to its circumferenceshould so transcendour ability to compute it exactly on any numberof fingers,but that's just the way our particularuniverseis arranged!nrand e are also called trancendentalnumbersfor such metaphysicalreasons(more about e later).So if we arepermittedto use the integers,we can completely solvef(x) x - 1, x E N for all allowedvaluesofx.It is clearthat the equation(as above), andWe might now define:a f(x) g(x)b f(x) -g(x)andb -x -2N (0,1,2,3.}The basic notion here is that whateveris on the left hand sideof the equal sign ( ) is just anothername for what is on therighthand side. Of course, as the last example above shows,there are simpleways and complicatedways to say the samething, and it is usuallythe task of the algebraistto find thesimplest way of expressinga relationshipso that it can beeasily understood.f(x) a 3x 43x 2x ElIhas no solution, since no integerhas the value 2/3. Thereisanother type of numberneeded to solve such equations asx2 1 0, since no realnumberwhen multipliedby itselfis equal to - 1. Mathematicianssimply define the squarerootof minus one as i, the imaginaryunit. (Engineersusej, sincei was alreadyused to stand for currentin the engineeringF. R. Moore: An Introduction to the Mathematics of Digital Signal Processing, Part IThis content downloaded from 165.123.228.54 on Wed, 03 Feb 2016 11:57:34 UTCAll use subject to JSTOR Terms and ConditionsPage 39

literature.In Part I of this tutorial we shall stick with i;Part II will use j, since signal processing is a branch ofengineering.)An imaginarynumberis any realnumbertimes i,and since the realsinclude the other numbersets, we can haveimaginary integers, imaginary rationals, even imaginarynaturals!The final set of numbersis just a combination of therealswith the imaginaries,which are called complex numbers.The set of all complex numbersis denoted C, and eachmemberof the set has the formx iyf(x)54 .Sx 1-(x)32 -f(x 3) 2.5x,yE Rwherex is called the "realpart",and iy is called the "imaginary part."Complexnumbersmay be added, subtracted,multiplied and dividedaccordingto the usual rules of algebra.If cl and c2 (read "c-sub-one and c-sub-2") are twocomplex numbers,with cl xl iyl and c2 x2 iy2, thenthe rules of complex arithmeticare as follows:Rule Cl (complex addition): To add two complex numbers,add the real and imaginaryparts independently,i.e.,-4-31-1--223456789-1--x-3 -Figure 1. A graphof f(x) .5x 1cl c2 (x1 iyl) (x2 iy2) (xl x2) i(y1 Y2)Rule C2 (complex subtraction)(similarto addition):c1 - c2 (xI iyl) - (x2 iy2) (x -x2) i(y 1 -Y2)Rule C3 (complex multiplication):The productis formedbythe ordinaryrules of algebra:C1c2 (xI iyl) (x2 iy2) X1X2 iY1x2 i1 y2 i2y1y2 (X x2 -Y1Y2) i (xy2 Y1X2)(Rememberthat by definition, i2 - 1)Rule C4 (complex division):Again,ordinaryalgebrais used todefine the quotient:Sc2 iy xx iyx2 iy2m, b constantsis the graphof a straightline. m is called the slope of thefunction since it is the amount by which the function changesfor a unit changein x. Setting m to .5, as in Figure 1, meansthat every time x increasesby one, f(x) will increaseby .5,hence a positive slope is associatedwith lines slopingupwardto the right.f(x) will always cross the vertical axis whenx 0, and since f(x 0) b, b is called the verticalaxisintercept of f The horizontalaxis will be crossed,of course,whenf(x) equalszero, which occursin this example at:f(x) .5x 1 0x -2x 2 Y12 i(y1x2 - XY2)x22 y2Actually, Figure 1 is not a graphof f(x) .5x 1, but morepreciselya graphof this function for the values of x between- 4 and 9, or in most propernotation:obtained by multiplyingbyiy which is equivalentto 1.x2 2-- iY2While a function is most generallystated in algebraicform, it is often enlighteningto drawgraphsin orderto get aclearidea of how a function variesas its argumentchanges.The conventionalgraphuses a horizontalline to representtheindependentvariable,and a verticalscale to representvaluesofthe function. Thus,in orderto find the value of a function forsome value of the independentvariable,say, x 3, we slideone fingeralong the horizontalaxis until we point at 3, thenmove straightup (or down) to find the valuef(x 3) (read:"the functionfat x 3").A glance at Figure 1 tells us severalthings about thefunction f(x) .5x 1. First, the graphis a straightline,sloping upwardsto the right; second, it crosses the verticalaxis at the value 1; third,it crossesthe horizontalaxis at thevalue -2. In fact, any function which has the formPage 40f (x) mx bf(x) .5x 1- 4 x 9The original function could extend for all x, that is x 0, but graphingthe entirety of such a functionwould requirea very big piece of paperindeed. Graphsare useful to get the generalpictureof a function, but they can serveother purposesas well. For example, it is often useful to addgraphs directly, especially when it is difficult to do theaddition algebraically,or when the algebraicsum of twofunctions is difficult to interpret.Graphicaladdition of twofunctions consists of carefully drawingboth functions on thesame graph,and then carefullyaddingup the verticaldistancesfor all (or many) valuesof the independentvariable,to obtaina graphof the sum function (see Figure 2). Such graphicaltechniques are, of course, only approximate,but oftensufficient to gain considerableinsight into the shape ofcomposite functions.Computer Music Journal, Box E, Menlo Park, CA 94025This content downloaded from 165.123.228.54 on Wed, 03 Feb 2016 11:57:34 UTCAll use subject to JSTOR Terms and ConditionsVolume II Number 1

if the equationhas the formax2 bx c 0h (x) f(x) g(x)4then-b/X 3.5g(x) x2If(x)\2\/The method 2 solution yields:x2 x -6 0 0(x 3)(x-2)x -3 or 2h( ) f(1) g(1)--1.5- - - -\.5x 1Themethod 3 solution, with a 1, b 1, and c - 6 alsoyieldsf(()X-3 -2.52 -1.5 -1-.5-5-.-.511.52J -4b2- 4ac2a2.53-b /b"- 4ac2a2a-1-1-1.5-?J12-4.1.(-6)2-1-1 5- -3or22What about such formulasas x2 1 0? The quadraticformulaworksjust as well on those:-1- 2-2-2TFigure2. Graphicaladdition of f(x) .5x 1 andg(x) x2to get graphof h (x) f(x) g (x) x2 .5x 1a 1, b 0, c 1,sox Polynomialsand RootsA polynomial is an algebraicexpressionwhich has theform:f(x) ao al x a2x2 a3x3 ? 2i2 ior-iwhich says that againthere are 2 roots, and that they are bothimaginary.In factorialform, we could have written anxnThe a's are constants (numbers)called coefficients, and thehighest power of x which occurs in any given polynomial(n) is called the degree of the polynomial. Thusf(x) inFigure 2 is a first-degree polynomial, since the greatestpower of x in .5x 1 is one. Both g (x) and h (x) from thesame figure are second degree, or quadratic,polynomials.Third degree polynomials are called cubic, fourth degreequartic, and so on, though after that one rarelyhears of,say, "quinticpolynomials"instead of "fifth-degreepolynomials." A polynomial is "solved"by setting it to zero, andfinding which values of the independentvariablemake theequation true. For example, to find the roots of the quadraticequationx2 x - 6 0, we can do any of at least three things:1. try every value of x and see when the formulais true,2. try to factor the polynomial,or3. use the quadraticformula, which will give the roots forany quadraticpolynomial.Method 1 may sound a bit absurd,but sometimesit is the bestwe can do. Method2 meanstryingto write the polynomialinthe form (x- zl) (x - z2) 0. zl and z2, are called the"zeroes" of the function, since if x is equal to zl, the firstfactor, and hence the product, will be zero; and similarlyforx z2. Method 3 requiresrememberingthe generalsolutionfor any second-degreepolynomial (or looking it up), calledthe quadraticformula:0?J x2 1 (x -i)(x i) 0The Fundamental Theorem of Algebra states that anynth-degreepolynomialalwayshas exactly n roots, that theymay in generalbe complex (havingboth real and imaginaryparts),and that all the roots may not be different from eachother (distinct). Also, we might have guessedthat if i is asolution to x2 1 0, then - i is also, since complex rootsalways appearin conjugatepairs if the coefficients of thepolynomial are real numbers.(If c x iy is a complexnumber,then its conjugate,written c* , is x - iy.)If the generalformulamethod works so well, why wouldwe ever use factoring,or trial and error?The answeris bothsimple and unfortunate: General formulas exist only forpolynomialswith degreeless than 5, and in fact the FrenchmathematicianGaloisprovedthat no such formulascan existfor degree5 or more. Even the generalquarticformulais verycomplicated;it is often easierto factor than to use it! Andfinally, trial and errorsolutions are often implementedwithcomputers,using specialguessingalgorithmssuch as Newton'sMethod,which work remarkablywell.Exponents, Logarithms,and the NumbereIf we say that addition and subtractionare easy, thatmultiplication and division are harder, and that taking anumberto a power is most difficult, then the rules of expo-F. R. Moore: An Introduction to the Mathematics of Digital Signal Processing, Part IThis content downloaded from 165.123.228.54 on Wed, 03 Feb 2016 11:57:34 UTCAll use subject to JSTOR Terms and ConditionsPage 41

nents show us how many problemsin mathematicsmay bemade one level easier! It is important to rememberwhichkinds of numbersthese rulesapply to, so in the followinglist,we will use p and q to stand for any realnumbers(that is,p, q E R), a and b are positive reals(a, b E R?), and m andn are positive integers(m, n E N).Rule El:Rule E2:aP - aaq aP q a - qRule E3:(aP)q apqRule E4:nflF- am/nlogax K logbxwhereKlogba1Rule E5:a-pRule E6:ao 1 (if a*0)RuleE7:na 1Rule E8:(ab)P aPK is givenin the following table for base changesamong 10,2, and e:b10Rule Ll:logaxy logaX logayRule L2: logax - logayylogaylogaxY y 9.1.44270.1Thusloglox .30103 log2x .43429 Inxand so on, where In stands for "naturallogarithm"(i.e.,In x Logarithmsare defined only for positivelogex).numbers.Wheredoes the numbere come from? Unfortunately,its true originsareburieddeep within calculus,which is not apart of our subjectmatter,but some of its properties,as weshallsee, turn out to be remarkable.e is an irrationalnumberlike 7r,which meansthat its decimalexpansionis both infiniteand that it neverrepeatsitself:e 2.71828 18284 59045 23536 0287.wherex, y E R.Also, if logax p, then x is called the antilogarithmof p to the base a, writtenx antilogap, since by definitionaP x.Any numberexcept 0 or 1 may be used for the base, but infact only three numbersare used very often: 10, 2, ande 2.71828. . Logarithmsto the base 10 are used becausewe commonly use a decimal (base 10) number system foreverythingelse! Logarithmsto the base 2 are very oftenencounteredin the relativelynew fields of computerscienceand information theory, since computers typically operateusing binaryarithmetic(internally),and both computersandinformation theory define the unit of informationas a bit(short for binary digit). Logarithmsto the base e are called"natural"logarithms,and are the most used in mathematics.It is hard for us today to appreciatewhat a boonlogarithmswere to mathematiciansbefore the adventof computersand pocket calculators.Logarithmswere so useful thattwo 16th century mathematiciansliterally devoted most ofPage 422 apbpUsing these rules, we can deduce such things as 4s 2(Rule E4, since .5 ?), x'/x5 x-' 1/x (Rules E2 andES), and /-6 -2 4J3(Rule E7). In fact, the first 3 rulesareso useful in doing calculations,that the entire system oflogarithmshas been devisedto make them universallyapplicable to the more "difficult" problems of multiplication,division,and exponentiation.If ap x, wherea is not 0 or 1, then p is called thelogarithm to the base a of x, written logaX p. Thus,log28 3, since 23 8, and loglo10000 4, since10000 104. The rules for logarithmsare derivedfrom El,E2 and E 3, above:Rule L3:their lives to calculating"log tables"in orderto relievetheircolleagues of the drudgeryof multiplicationand division:Briggscalculatedthe so-called common, Briggsian,or base 10logarithms,and Napier the "natural",Naperian,base e logarithms. Base 2 logarithmsare not found in mathematicalhandbooks, and they probably never will be, since theircomputation today is largely a matter of button-pushing.Also, if the log of a numberis availablein any one base, it iseasy to changeit to anotherbase using the following relationships:If you would like to calculateit to more accuracythan this,the following formulamay be used:e 1 -I 11!2!1 I 14!3!1 "wheren! meansn factorial, which is the product of all theintegersfrom one to n (3! 6, 4! 24, 5! 120, etc.).A more useful form of this infinite expressionyields thevalue of e raisedto any powerx:2e 1 324T.Another way to write the same thing is with sum notation:eX 1 XwhichthesameThe capital"says"exactlysig lthing.which "says"exactly the same thing. The capitalsigma(Computer Music Journal, Box E, Menlo Park, CA 94025This content downloaded from 165.123.228.54 on Wed, 03 Feb 2016 11:57:34 UTCAll use subject to JSTOR Terms and Conditions) isVolume II Number 1

used to denote that we should add up all valuesof xn/n!startingwith n 1, then n 2, etc. (It is read: "the sum over nfrom one to infinity of x to the n dividedby n factorial".)Sums and SeriesSuch formulasas the one above for ex are called infiniteseries, or infinite sequences, since there are infinitely manytermsin the sum, even though we know what any one of themwould be. Such sums need not be infinite, of course. Forexample,the following formulaillustratesa finite sum:their beautiful princesswas known only to initiates, for itconveyed all of their secrets at once. The name of theirprincesswas Sohcahtoa.]If we label a right triangle (one which contains aright angle) with respect to an anglee- (see Figure3), sideO is opposite the angle, side A is adjacent to anglea, andside H is, of course,the hypotenuse of the triangle.n 1 2 3 - nkk 1Owhich is just the sum of the first n integers.It is both interesting and useful that many such sumshave a general,or "closedform" formula,makingit unnecessaryto carryout the lengthyaddition sequence.For example:nk1)n (n 2n1 2 3 . The closed form is clearlymore useful if n is greaterthan 3 or4 or so. Other sum formulasoften crop up in digital signalprocessing.For exampleo00Earkk O248 E k 0 sina HA cos e Hsineof acosine of o2-k -OAtangentof a tanaoo1 A The 3 basic trigonometricfunctions are defined asfollows:rThis sum exists only if r 1, since otherwisethe sum will beinfinite, a is the first term in the sequence,and r is called theratio, since it is multipliedby any term to get the next term inthe sequence. Thus, we see thatICa a ar ar2 1Figure3. A right trianglewith inscribedanglee and sidesO, A, and Hnk 1a2Clearly,the size of the righttriangledoesn't matter,since, fora givenanglea, if we double the length of one of the sides, theotherswill double as well. Only the ratios of their lengths areneeded to define the trigonometricfunctions.The 3 remainingtrigonometricfunctions are definedin termsof the first 3:l-MIf there is not an infinite number of terms, we can removethe restrictionthat r be less than 1:cosecant of e cscesecant of a seccotangentofan-1ark rnalr(1-r)1-rk OIf r # 1, and the last term 1 ar n -1 , then this sum is alsoequal toa - rl1 -rTrigonometry[It has been said that a tribe called the TrigonometricIndians once roamed the earth, that they spoke in sinelanguage,and never used wrong angles. The secret name of1- sinH- OH1Acos e- - A1A cot atanl- O Radians,Degrees,and GradsAs almosteveryoneknows, if you slice a pie into 360equal wedges,you have not only very smallslices to eat, butthe angleat the tip of each slice will be one degree(1 0). Ifyou are very hungry,however,and slice the pie into 4 equalpieces, the angleat the tip of each slice will be 900, which isexactly right.Anothermeasureis to dividethe circularpie into 400equal pieces, or 400 grads.But by far the most commonmeasureof anglesused in mathematicsis the radian. Sincethe ratio of the circumferenceof a circle to its diameteris7r 3. 14159 26535 ., and since the radiusof a circleF. R. Moore: An Introduction to the Mathematics of Digital Signal Processing, Part IThis content downloaded from 165.123.228.54 on Wed, 03 Feb 2016 11:57:34 UTCAll use subject to JSTOR Terms and ConditionsPage 43

is exactly one half its diameter, the circumferenceof acircle is exactly 2nrtimes the length of its radius,and we saythere are 21rradiansin a circle. A right angleis then any of900, 100 grads,or n/2 radians,dependingon which measurewe areusing.If we choose a circle with radiusequal to one unit,and we inscribe our right trianglesinside the circle (seeFigure4),is also equal to one, as is sin - 3n/2. In fact sin-11 e hasinfinitely many solutions, all of the form e i/2 k2n,where k is any integer. The principle values of the inversetrigonometricfunctions are chosen to be close to e 0,and these are used to resolvethe problemof which answertochoose. Thus:sin-' x ? 22 02 cos-'x nr , and tan-1x? 2Inspection of Figure 5 also shows that the sine andcosine functions are also identical to each other, except fortheir startingplace at e 0, i.e., they differ only in phase:2H2 Osin(-- %., e) , andcosocos(&- ) sinesin e/IIIICOS&IIII/IIFigure4. A unit circle with inscribedrighttriangleswe can "solve" the triangles conveniently with thePytha orean theorem: 02 A2 H', or 02 A2 1,. Anglesare conventionallyO 1 - A , and A 7J1--Ufrom the right hand horizontalmeasuredcounter-clockwiseaxis (see a1, and a2 in the figure). Angles measuredin aclockwise directionare considerednegative.We can treat the anglee as an independentvariableandgraphthe basic functions as shown in Figure5.The inversetrigonometricfunctions are defined in asimilarway to the antilogarithm:if sin a x, then the arcsineof x sin-1x 0, and so on, for each of the six trigonometricfunctions.We can see from the graphof sin e that the function isperiodic, that is, it repeatsitself over and over againas e getslargeror smallerby 27r,which is called the period of sin e.Furthermore,sin & always has a value between 1 and - 1inclusive,so we say that the domainof the sine function is theset of all realnumbersbetween I and - 1, or in more mathematical form:sinE R , - 1 sin e 1Because of this restricteddomain, it is meaninglessto writesin-12 o, since no anglee has a sine equal to 2. But whatabout sin-' 1 e? From the graph,it is clear that sin 7r/2 1,so, o 7r/2 is one solution to this equation. But sin 5nT/2Page 44IIIIIIIItan eiIIi/i1.IiIIII007r27rr223r237r27r 57r5 2227r31r37T71r 44rr2271rFigure5. Graphsof sin e-, cos e, and tan e as functions ofe,& in radians.TrigonometricIdentitiesMany formulasmay be derivedfrom the basic definitions of the trigonometricfunctionswhich are often useful inthe manipulation of equations involving trigonometricfunctions. They are called identities since, like all equations,the expressionson either side of the equal sign "say"exactlythe same thing, but in a useful way. In the followingidentities,A and B are any angles:Computer Music Journal, Box E, Menlo Park, CA 94025This content downloaded from 165.123.228.54 on Wed, 03 Feb 2016 11:57:34 UTCAll use subject to JSTOR Terms and ConditionsVolume II Number 1

(T1): sin2A 2 sin AcosA(T2): cos 2A cos2 A - sin2 A(T3): sin2 A ? - /2cos 2A(T4): cos2 A 1/2cos 2A(TS): sin A sin B 2 sin 1 (A B) cos ? (A - B)(T6): sin A - sin B 2 cos x (A B) sin ? (A - B)(T7): cos A cos B 2 cos ? (A B) cos 1 (A - B)(T8): cos A - cos B 2 sin (A B) sin a (B - A)(T9): sin A sin B 1/2[cos(A - B) - cos (A B)](T10): cos A cos B ? [cos (A - B) cos (A B)](T11): sin A cos B 1/2[sin (A - B) sin (A B)](T12): sin (A ? B) sin A cos B cos A sin B(T13): cos (A B) cos A cos B T sin A sin BThese identities are fairly easy to derivefrom each other, and,of course,many more exist.Like ex, the sine and cosine functions may be represented as summationseries:sinx x-x3!2cosx 1-2!2! ----5!X44!4 !" --7!X66!"-'" wherex is an anglemeasuredin radians.other words, pitched soundshave periodsrangingfrom about1/20 to 1/20, 000 of a second. The amplitude,or strength,ofthe vibrationis a measureof how far the pressuredeviatesfrom the atmosphericmean. One could measurethe peakdeviation from the mean, or possibly the averagedeviation,but the word amplitudegenerallyrefersto the peak deviation,unless stated otherwise,and is relatedto our perceptionof theloudness of a sound. Finally, the general shape of thewaveformdeterminesits tone quality,or timbre.All of thesefactors interact perceptually.For instance, the pitch can beaffected by the amplitudeand the shape as well as the periodof a waveform.Hence it is importantto distinguishbetweenfrequency, which is a measureof the repetition rate of aperiodic waveform,and pitch, which is our perception ofsomethinglike the "tonalheight" of a sound.An importantmathematicaltool which will be describedin Part II of this tutorial is Fourier's theorem, which statesthat any periodic waveformcan be describedas the sum ofa number,possibly an infinite number,of sinusoidalvariations,each with a particularfrequency, amplitude, and phase.Futhermore,there is a method for determiningexactly whatthese frequencies,amplitudes,and phasesmust be in ordertore-construct the waveformby addingtogether sine waves,which are seen to be the basic "buildingblocks" of periodicwaveforms.Actually there are a few other requirementsaswell as periodicity;suffice it to be said that any waveformwhich could exist in the physicalworld will obey these otherconditions (called the Dirichletconditions).Stated mathematically,the waveformmust obey theconditionf(t) f(t T), wheref is the periodicwaveform,t is time, and T is the period of the waveform.ThenUsing Tr

requires understanding of calculus. But digital signal processing inherently requires less calculus than analog signal processing, since the integral signs are replaced by the easier-to- understand discrete summations. It is an experimental goal of this tutorial to see how far into digital signal processing it is possible to explore without .

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