UNIT : MATHEMATICAL ENVIRONMENT. Itroductio This uit itroducs som basic mathmatical cocpts ad rlats thm to th otatio usd i th cours. Wh ou hav workd through this uit ou should: apprciat that a mathmatical formalism ca b prssd i a algorithmic laguag lik Visual Basic as a procdur or fuctio. kow how arra idig is formalisd i th ots ad programs kow how compl umbrs aris ad how to prform basic arithmtic o compl umbrs udrstad th spcial charactristics of th multiplicatio of compl umbrs wh prssd i polar form kow how to rprst polomials as a cofficit sris ad as a st of roots. Rlatig algbra to algorithms Th mathmatical otatio usd to dscrib sigal procssig ivolvs scalars, compl umbrs ad vctors; it ploits opratios of additio, subtractio, multiplicatio ad divisio, as wll as pots, logarithms, idig ad summatio. Each of ths lmts of mathmatical otatio hav quivalts i th Visual Basic computr laguag implmtatio. Cosidr th formal prssio for th calculatio of th roots of a quadratic (th valus of whr th quatio is ro: a + b + c 0 ( ( whr b + b 4ac, a b b 4ac a To map such mathmatical otatio oto a computr program, w d to idtif th iputs ad costats, th outputs ad itrmdiar valus, th opratios ad procdur of calculatio. I this ampl th iputs ar th costats a b c, th outputs, th opratios + - * / sqrt, ad th procdur o of computig th prssios for th roots ad thir assigmt to th variabls ad. With a procdural wrappr: Sub FidQuadRoot(BVal a As Doubl, BVal b As Doubl, BVal c As Doubl, BRf As Doubl, BRf As Doubl EdSub (-b + Math.sqrt(b*b - 4*a*c/(*a (-b Math.sqrt(b*b - 4*a*c/(*a Whr th otatio BRf idicats th valus that ar rturd to th callig program. UCL/PALS/BASICDSP/0MATHENV/000/
Not that this vrsio will fail for quadratics whr 4ac > b. To maipulat sigals as titis, w d to rprst thm as vctors, that is as a arra of simpl scalar valus. W also d to b abl to idtif sigl lmts (sampls of th vctor b subscriptig or idig. Although i covtioal mathmatical otatio w might writ a vctor smbol as, its pasio as (,,..., N ad a lmt of th vctor as k, i this cours w choos: vctor: pasio: lmt: [] [], [],..., [N] [k] This allows a straight-forward mappig to th Visual Basic otatio for arras ad arra idig. Cosidr th covtioal mathmatical prssio for covolutio. This taks two vctors, ad rturs a third which is a kid of vctor product: whr * j k + k I this cours such a formula would b prssd i trms of a procdur that taks sigals [] ad [] ad calculats ach output sampl [j] usig: k0 k j k [ j] [ k + ] [ j k] Hr w ot ol simplif th otatio but icorporat stadard assumptios such as that sigals ar idd from tim, ad ar ro at arlir tims. This formulatio lads to a atural Visual Basic implmtatio: Public Shard Fuctio Covolv(BRf As Wavform, _ BRf As Wavform As Wavform Dim As Nw Wavform(.Cout +.Cout -,.Rat Dim v As Doubl ' for ach output sampl For j As Itgr To.Cout ' sum of product with rvrsd sigal v 0 For k As Itgr 0 To h.cout( - v + (k + * (j - k Nt (j v Nt Rtur Ed Fuctio Th ol difficult part with this traslatio is th slctio of th uppr limit for k i th summatio loop, which should ot b allowd to accss sampls bfor th start of th UCL/PALS/BASICDSP/0MATHENV/000/
[] sigal. W also assum that [] is at last as log as [].. Compl umbrs Th fact that thr ar quatios such as + 0-0 + 40 0 which ar ot satisfid b a ral valu for, lads to th itroductio of compl umbrs. A compl umbr is a ordrd pair of ral umbrs, usuall writt i th covit form + i, whr ad ar ral umbrs. Compl umbrs ar subjct to ruls of arithmtic as dfid blow. W ca of cours rfr to a compl umbr b a sigl algbraical variabl, sa : + i Th smbol i is calld th imagiar uit (i girig tts, it is somtims rfrrd to as j. Th umbr is calld th ral part of th compl umbr ad is calld th imagiar part of. Thus th solutios for th quatio - 4 + 0, ma b writt: 4+ 0( ( whr +i, i Compl umbrs ca b rprstd as poits o a pla whr th horiotal or -ais is calld th ral ais, ad th vrtical or -ais is calld th imagiar ais. Compl umbrs ar th poits i a cartsia co-ordiat sstm o this pla, which is somtims calld th compl pla. Two compl umbrs + i + i ar dfid to b qual if ad ol if thir ral parts ar qual ad thir imagiar parts ar qual, that is if ad ol if ad Rlatioal prssios btw compl umbrs, such as <, hav o maig, although th magituds of compl umbrs ma b compard, s blow. Additio. Th sum + is dfid as th compl umbr obtaid b addig th ral ad imagiar parts of ad, that is + ( + +i( + This additio is lik th additio of vctors o th Cartsia pla. Subtractio. Th sum - is just th ivrs of additio, that is - ( - + i( - UCL/PALS/BASICDSP/0MATHENV/000/
Multiplicatio. Th product is dfid as th compl umbr ( + i ( + i ( - +i( + which is obtaid b applig th ormal ruls of arithmtic for ral umbrs, tratig th smbol i as a umbr, ad rplacig i ii b -. Divisio. This is dfid as th ivrs opratio of multiplicatio; that is th quotit / is th compl umbr + i which satisfis ( + i ( +i( For which a solutio ma b foud b quatig ral ad imagiar parts, assumig that ad ar ot both ro: + + + i - + Eprssios. For a compl umbrs,, w hav: + + ad Commutativ laws ( + + + ( + ad ( ( Associativ laws ( + + Distributiv law Cojugatio. If + i is a compl umbr, th - i is calld th cojugat of, ad is dotd b *. Th product of a compl umbr with its cojugat is a purl ral umbr: * ( + i( - i A compl umbr of th form + i0 is just th ral umbr. A compl umbr of th form 0 + i is calld a pur imagiar umbr. Polar form of compl umbrs. If w itroduc polar co-ordiats r, θ i th compl pla b sttig r cos θ, r si θ, th th compl umbr + i ma b writt r + cos θ + ir siθ r( cosθ + i siθ This is kow as th polar form of th compl umbr. Th valu r is calld th absolut valu or modulus of, dotd b. Thus r + Th dirctd agl masurd from th positiv ais to th dirctio of th compl umbr from th origi of th compl pla is calld th argumt of, dotd b arg. Agls ar alwas masurd coutrclockwis ad i radias. Not: arg θ arcsi r * arccos arcta r UCL/PALS/BASICDSP/0MATHENV/000/4
B applicatio of stadard additio thorms of trigoomtr w fid th followig rlatios for th polar form of th product of two compl umbrs arg( arg + arg That is th magitud of th product is th product of th iput magituds, ad th argumt of th product is th sum of th iput argumts. From ths w ca obtai th importat rsult for th polar form of th powrs of a compl umbr r ( cos θ +i siθ r ( cos θ + isi θ This rducs to th so-calld D Moivr formula for uit magitud : ( cos θ + i siθ cos θ +i si θ From this w ca s that th squc of powrs of a compl umbr of magitud ar simpl a squc of coutr-clockwis rotatios b θ aroud th origi o th compl pla. Compl potial. Th potial of a ral umbr, writt or p, has th sris pasio: p + + + +...!! Th potial fuctio for compl + i is dotd b ad dfid i trms of th ral fuctios, cos ad si, as follows: i + ( cos + isi From this w obtai th Eulr formula for imagiar : i cos + isi This i tur lads to th followig importat idtitis: i cos π i π i -π i + si π i π i i - i Th compl umbrs iθ, 0<θ<π li o th uit circl of th compl pla; ad th valus, i, -, -i ar th poits whr th uit circl crosss th as.. Polomials Th Z-trasform of a sigal, which w will mt latr, is a wa of prssig a tim sris of sampls as a sigl mathmatical objct: a polomial i th variabl whr th cofficits of th polomial ar simpl th sampl valus. Hr w cosidr som UCL/PALS/BASICDSP/0MATHENV/000/5
basic opratios o polomials. I its simplst form a polomial ma b prssd a +a + a 0 +...+ a whr th si of th polomial is calld th ordr of th polomial, ad th costats a 0, a, tc ar calld th polomial cofficits. Not that this prssio ma also b writt j0 a j j Opratios such as additio, subtractio, multiplicatio ad divisio ma b prformd o polomials b lmtar arithmtic ad th collctio of trms of quivalt powrs of. For multiplicatio, th product of a polomial p of ordr m with a polomial q of ordr is a polomial of ordr m+. Th divisio of a polomial of ordr m b a polomial of ordr (whr < m, lavs a polomial or ordr m- ad a rmaidr of dgr -. Polomials ma b rducd ito a product of factors; a polomial p( of ordr havig factors: p( a +a +a +...+a b ( -b ( -b 0 0...( -b For ma purposs it is mor covit to work with th cofficits b tha th polomial cofficits a. This is bcaus it as to s that th valus b ar also th valus of for which th polomial has th valu 0. Thus th b cofficits ar also calld th roots of th polomial p(. A polomial of ordr must hav roots, but ths ma ot all b ral. Howvr if th cofficits of p( ar ral, a compl valud roots must occur i compl cojugat pairs. For polomials of a ordr > 4, thr ists o formula which dirctl calculats th valus of th roots from th a cofficits, ad so itrativ umrical mthods d to b usd. Fiall, thr is a clos rlatioship btw th form of polomials ad th form of powr sris approimatios to fuctios. Hr w just giv som sris pasios of som fuctios that ma b of us latr i th cours: a a(a - a(a - (a - (+ + a + + +... whr <!! - ( - + + + +... whr < - ( - a + a + a + a +... whr <, a <. UCL/PALS/BASICDSP/0MATHENV/000/6
+ + + +...!! si - +! 5 7 - +... 5! 7! Ercis cos - +! 4 6 - +... 4! 6!. Us th Compl class to implmt a fuctio that will solv arbitrar quadratic quatios. I particular, writ a program to accpt th cofficits a, b ad c, ad which prits out th valus of th two roots i compl otatio. UCL/PALS/BASICDSP/0MATHENV/000/7