# An Introduction To The Mathematics Of Digital Signal Processing: Part I .

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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

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

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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