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# An Introduction to Mathematics of Digital Signal Processing

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An

Introduction to of Digital Signal

the

Mathematics Processing

and PartI:Algebra,Trigonometry, The Most Beautiful Formulain Mathematics? 1978 F. R. Moore F. R. Moore Bell Laboratories Murray Hill, New Jersey 07974

Introduction As it says in the front of the ComputerMusic Journal number 4, there are many musicians with an interest in musicalsignalprocessingwith computers,but only a few have much competence in this area. There is of course a huge amount of literaturein the field of digitalsignalprocessing, including some first-rate textbooks (such as Rabiner and Gold's Theory and Application of Digital SignalProcessing, or Oppenheimand Schafer'sDigital SignalProcessing), but most of the literatureassumesthat the readeris a graduate student in engineeringor computerscience (why else would he be interested?),that he wants to know everythingabout digital signalprocessing,and that he alreadyknows a greatdeal about mathematics and computers. Consequently, much of this information is shrouded in mathematicalmystery to the musical reader,making it difficult to distinguishthe wheat from the chaff, so to speak. Digitalsignalprocessingis a very mathematicalsubject, so to make past articles clearerand future articlespossible, the basic mathematicalideas needed are presentedin this two-part tutorial. In order to prevent this presentationfrom turninginto severalfat books, only the main ideas can be outlined; and mathematicalproofs are of course omitted. But keep in mind that leamingmathematics is much like leaming to play a piano: no amount of reading will suffice-it is necessaryto actuallypracticethe techniques described(in this case, by doing the problems)before the concepts become useful in the "real"world. Thereforesome problemsare provided(without answers)to give the motivated readeran opportunityboth to test his understanding to and acquiresome skill.Page 38

Part I of the tutorial(this part) providesa generalreview of algebraand trigonometry,includingsuch areasas equations, graphs,polynomials,logarithms,complex numbers,infinite series,radianmeasures,and the basic trigonometricfunctions. PartII will discussthe applicationof these concepts and others in transforms, such as the Fourierand z-transforms,transfer functions, impulse response, convolution, poles and zeroes, and elementaryfiltering.Insofaras possible, the mathematical treatmentalwaysstopsjust short of usingcalculus,though a deep understandingof many of the concepts presented of requiresunderstanding calculus.But digitalsignalprocessing inherentlyrequiresless calculusthan analogsignalprocessing, since the integralsignsare replacedby the easier-tounderstanddiscretesummations.It is an experimentalgoal of this tutorialto see how far into digitalsignalprocessingit is possible to explorewithout calculus. Algebra To most people, mathematicsmeans formulas and equations, which are expressionsdescribingthe relationships do amongquantities.As long as the relationships not use the integrationor differentiationideas of calculus,they usually fall into the generaldomainof algebra,namedafter the arabic best-seller of the 9th century, Kitab al jabr w'al-muqabala ("Rules of Restoration and Reduction") by Abu Ja'far Mohammed Musaal-Khowarizmi ibn (from whose name the word algorithmis derived). Algebrais, in fact, merely a systematicnotation of quantitative relationshipsamong numericalquantities, usually called variables,since with algebrawe can manipulatetheVolume II Number 1

Computer Music Journal, Box E, Menlo Park, CA 94025

into variousforms without specifyingthe particurelationships lar quantitieswe aremanipulating.For example, the equation: y=x+l name givento a quantitywhich is "says"that y is an arbitrary one greaterthan anotherquantity,x. If we were to write y -1 =x we would be "saying"exactly the same thing,just as we would if we wrote any of the following: 16y = 16 + 16x y/2 = (x + 1)7r(y -ir)

a = 3x+4 b= -x-2

and

but the latter form doesn't show explicitly where these come from. relationships What do we mean when we say that x can have any value?In fact, what does valuemean?Withoutgoing too far afield into the theory of numbers,we should note that in many cases, the value of the independentvariablein a particular function is restrictedto the set of all naturalnumbers,or integers,or reals.Briefly,the set of naturalnumbers(denoted here as N) is the set of numbersused for counting: N= o0,1,2,3...}

= 7r(l -7r) +7rx

The basic notion here is that whateveris on the left hand side of the equal sign (=) is just anothername for what is on the righthand side. Of course, as the last example above shows, there are simpleways and complicatedways to say the same to thing, and it is usuallythe task of the algebraist find the simplest way of expressinga relationshipso that it can be easily understood. Functions,Numbers,and Graphs Sometimes it is desirableto give a name to an entire in relationship,ratherthanjust to the variables a relationship. have a keen sense of brevity, so these names Mathematicians are usually singleletters as well, but they servequite a different purpose.For example, the notation f(x) = x + 1 meansthat "f" is being defined as a function of x, wherex is called the independent variable,since it can take on any value whatsoever.Wecan now write (read: "y equalsf of x") to mean that the value of y (which is called a dependent variablesince its value dependson the value chosen for x) is a function of x, and the function is namedf. Rememberthat f(x) is just anothername for x + 1, so the last equation above is still sayingthe same thing as all of the previousexamples. The advantages the function notation are that it a) explicitof ly states the name of the varyingquantity (the independent variableor argumentof the function), and b) it givesa short name to what may be a complicatedexpression,allowingits furthermanipulation. For example: let Wemight now define: a = f(x)+g(x) b = f(x)-g(x) Of course, this "says"the same thing as and f(x) = x + 1 g(x) = 2x + 3 (as above), and

(the curly braces { } " denote a set, and the ellipsis "..." meanshere that the set has an infinite numberof elements). To indicate that the independentvariablemust be chosen from this set, we write f(x)=x-1 x N

where "EN" means "is an element of N", the set of all naturalnumbers.Suppose we choose x equal to 0; what is f(x) equal to? Our Pavlovianresponseis, of course, minus one, but note that this number is not a naturalnumber as defined above. So even though x might always be a naturalnumber, might not be. Othersets of numbersfrequentlyencounf(x) tered are I, the set of all integernumbers, I = 40,?1,+2,+3,...} and R, the set of all real numbers.Real numbersare those which can be written as a (possibly unending) decimal expression, such as ir, 2, and 1/3, since ir = 3.14159.... 2 = 2.000 .., and 1/3 = .333.... Sometimes R* is used to denote the positive reals,R2 for the set of all orderedpairs of real numbers,etc. Just as the integers include all of the naturalnumbers,the realsinclude the integers,as well as the rationals(numbersformed by the ratio of two integers,such as like 1/3 or 22/7), andthe irrationals, ir (whichis approximately equal to 22/7, but is not exactly equal to any ratio of two integers). It is a fundamentalmystery that the ratio of the should so transcend diameterof a circle to its circumference our ability to compute it exactly on any numberof fingers, but that's just the way our particularuniverseis arranged! irand e are also called trancendental numbersfor such metaphysicalreasons(more about e later). So if we are permittedto use the integers,we can completely solvef(x) = x - 1,x E N for all allowedvaluesof x. It is clearthat the equation 3x= 2 xEI

y = f(x)

has no solution, since no integerhas the value 2/3. Thereis another type of numberneeded to solve such equations as x2 + 1 = 0 , since no realnumberwhen multipliedby itself is equal to - 1. Mathematicians simply define the squareroot of minus one as i, the imaginary use unit. (Engineers j, since i was alreadyused to stand for currentin the engineeringPage 39

F. R. Moore: An Introduction to the Mathematics of Digital Signal Processing, Part I

literature.In Part I of this tutorial we shall stick with i; Part II will use j, since signal processing is a branch of numberis any realnumbertim

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