DSP-First, 2/e. This Lecture: LECTURE #3 Complex Exponentials & Complex Numbers. Introduce more tools for manipulating complex numbers
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1 DSP-Fis, / LECTURE #3 Compl Eponnials & Compl umbs READIG ASSIGMETS This Lcu: Chap, Scs. -3 o -5 Appndi A: Compl umbs Appndi B: MATLAB Lcu: Compl Eponnials Aug , JH McClllan & RW Schaf 3 LECTURE OBJECTIVES Inoduc mo ools fo manipulaing compl numbs Conuga Muliplicaion & Division Pows -h Roos of uniy Fo k /, 1 LECTURE OBJECTIVES Phasos = Compl Ampliud Compl umbs psn Sinusoids Acos( {( A Lcu: Dvlop h ABSTRACTIO: Adding Sinusoids = Compl Addiion PHASOR ADDITIO THEOREM Aug , JH McClllan & RW Schaf Aug , JH McClllan & RW Schaf
2 WHY? Wha do w gain? Sinusoids a h basis of DSP, bu ig idniis a vy dious Absacion of compl numbs Rpsn cosin funcions Can plac mos igonomy wih algba Avoid all Tigonomic manipulaions COMPLEX UMBERS To solv: = -1 = Mah and Physics us = i Compl numb: = + y y Casian coodina sysm Aug , JH McClllan & RW Schaf PLOT COMPLEX UMBERS 5 { 5 0 { 5 5 { 5 Ral pa: { Imaginay pa: y { Aug , JH McClllan & RW Schaf Aug , JH McClllan & RW Schaf COMPLEX ADDITIO = VECTOR Addiion 3 1 (4 3 ( 5 (4 ( Aug , JH McClllan & RW Schaf 9
3 *** POLAR FORM *** POLAR <--> RECTAGULAR Vco Fom Lngh =1 Angl = Common Valus has angl of has angl of has angl of 1.5 also, angl of could b bcaus h PHASE is AMBIGUOUS Rla (,y o (, T 1 Tan 1 y y Mos calculaos do Pola-Rcangula cos y sin d a noaion fo POLAR FORM y Aug , JH McClllan & RW Schaf Aug , JH McClllan & RW Schaf Eul s FORMULA Cosin = Ral Pa Compl Eponnial Ral pa is cosin Imaginay pa is sin Magniud is on cos( sin( Compl Eponnial Ral pa is cosin Imaginay pa is sin cos( sin( cos( sin( { cos( Aug , JH McClllan & RW Schaf Aug , JH McClllan & RW Schaf
4 Common o Valus of p( COMPLEX EXPOETIAL Changing g h angl n cos( sin( 1 1 / 0 / (n 1/ (n1 3 / / 3 / 1 / 4 (n1/ 1? Aug , JH McClllan & RW Schaf Inp his as a Roaing Vco Angl changs vs. im : ad/s Roas in 0.01 scs cos( sin( Aug , JH McClllan & RW Schaf Cos = REAL PART Ral Pa of Eul s Gnal Sinusoid So, cos( { ( A cos( A cos( { A ( ( { A COMPLEX AMPLITUDE Gnal Sinusoid ( Acos( { A Sinusoid = REAL PART of compl p: (=(A ( { X { ( Compl AMPLITUDE = X, which is a consan X A whn ( X Aug , JH McClllan & RW Schaf Aug , JH McClllan & RW Schaf
5 POP QUIZ: Compl Amp Find h COMPLEX AMPLITUDE fo: ( 3 cos( Us EULER s FORMULA: ( { { X 3 3 ( Aug , JH McClllan & RW Schaf 18 COMPLEX COJUGATE (* Usful concp: chang h sign of all s RECTAGULAR: If = + y, hn h compl conuga is * = y POP QUIZ-: Compl Amp Dmin h 60-H sinusoid whos COMPLEX AMPLITUDE is: Conv X o POLAR: ( {( { 3 1 X ( (10 /3 10 ( 1 cos(10 / 3 Aug , JH McClllan & RW Schaf 19 COMPLEX COJUGATIO Flips vco abou h al ais! POLAR: Magniud is h sam bu angl has sign chang * Aug , JH McClllan & RW Schaf Aug , JH McClllan & RW Schaf
6 USES OF COJUGATIO Z DRILL (Compl Aih Conugas usful fo many calculaions Ral pa: * ( y ( y { Imaginay pa: * y y { Aug , JH McClllan & RW Schaf Aug , JH McClllan & RW Schaf Invs Eul Rlaions Cosin is al pa of p, sin is imaginay pa Ral pa: * { Imaginay pa: *, { y { cos(, { sin( Aug , JH McClllan & RW Schaf 4 Mag & Magniud Squad Magniud Squad (pola fom: * ( ( Magniud Squad (Casian fom: * ( y ( y y y Magniud of compl ponnial is on: cos sin 1 1 Aug , JH McClllan & RW Schaf
7 COMPLEX MULTIPLY = VECTOR ROTATIO Muliplicaion/division scals and oas vcos POWERS Raising o a pow oas vco by θ and scals vco lngh by Aug , JH McClllan & RW Schaf Aug , JH McClllan & RW Schaf MORE POWERS ROOTS OF UITY W ofn hav o solv =1 How many soluions? 1 k 1, k k, k 0,1,, k 1 Aug , JH McClllan & RW Schaf Aug , JH McClllan & RW Schaf
8 ROOTS OF UITY fo =6 Soluions o =1 a qually spacd vcos on h uni cicl! Wha happns if w ak h sum of all of hm? Sum h Roos of Uniy Looks lik h answ is o (fo =6 1 k / 0? k 0 Wi as gomic sum 1 k 1 hn l / 1 k 0 umao / 1 1 ( 1 0 Aug , JH McClllan & RW Schaf Aug , JH McClllan & RW Schaf Inga Compl Ep dd la o dscib piodic signals in ms of sinusoids (Foui Sis T 0 Espcially ov on piod b b a / T d d a T / T b 0 a 11 Aug , JH McClllan & RW Schaf 0 BOTTOM LIE CARTESIA: Addiion/subacion is mos fficin in Casian fom POLAR: good fo muliplicaion/division STEPS: Idnify aihmic opaion Conv o asy fom Calcula Conv back o oiginal fom Aug , JH McClllan & RW Schaf
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