DSP-First, / TLH MODIFIED LECTURE # CH-3 Complx Exponntials & Complx Numbrs Aug 016 1
READING ASSIGNMENTS This Lctur: Chaptr, Scts. -3 to -5 Appndix A: Complx Numbrs Complx Exponntials Aug 016
LECTURE OBJECTIVES Introduc mor tools for manipulating complx numbrs Eulr Eq. Conugat Multiplication & Division Powrs N-th Roots of unity r rcos() rsin() Aug 016 3
Aug 016 4
LECTURE OBJECTIVES Phasors = Complx Amplitud Complx Numbrs rprsnt Sinusoids Tak Ral or Complx part A cos( t ) {( A ) t } Aug 016 5
Phasors Not Phasrs Captain Kirk & Sally Kllrman STAR TREK Aug 016 6
WHY? What do w gain? Sinusoids ar th basis of DSP, but trig idntitis ar vry tdious Abstraction of complx numbrs Rprsnt cosin functions Can rplac most trigonomtry with algbra Avoid (Most) all Trigonomtric manipulations Aug 016 7
COMPLEX NUMBERS To solv: z = -1 z = Math and Physics us z = i Complx numbr: z = x + y y x z Cartsian coordinat systm Aug 016 8
PLOT COMPLEX NUMBERS { 5} 5 { 5} Ral part: 5 { 5 0} x {z} Imaginary part: y {z} Aug 016 9
COMPLEX ADDITION = VECTOR Addition z 3 z1 z (4 3) ( 5) (4 ) ( 3 5) 6 Aug 016 10
*** POLAR FORM *** Vctor Form Lngth =1 Angl = Common Valus has angl of 0.5p 1 has angl of p has angl of 1.5p also, angl of could b 0.5p1.5pp bcaus th PHASE is AMBIGUOUS Aug 016 11
POLAR <--> RECTANGULAR Rlat (x,y) to (r,) r r x Tan 1 y y x Most calculators do Polar-Rctangular x y x r cos r sin y Nd a notation for POLAR FORM Aug 016 1
Eulr s FORMULA Complx Exponntial Ral part is cosin Imaginary part is sin Magnitud is on cos( ) sin( ) r rcos() rsin() Aug 016 13
Cosin = Ral Part Complx Exponntial Ral part is cosin Imaginary part is sin r r cos( ) r sin( ) { r } r cos( ) Aug 016 14
Common Valus of xp() Changing th angl p n 0 0 1 1 0 4 / 1 p p p p 1/ ) ( / / n p p p 1) ( 0 1 1 n p p p p 1/ ) ( / / 3 / 3 n Aug 016 15 a + b = c θ = arctan( b a ) IF a 0
COMPLEX EXPONENTIAL t cos( t) sin( t) Intrprt this as a Rotating Vctor t Angl changs vs. tim x: 0prad/s Rotats 0.p in 0.01 scs cos( ) sin( ) Aug 016 16
Cos = REAL PART Ral Part of Eulr s cos( t) { t } Gnral Sinusoid x( t) Acos( t ) So, Acos( t ) { A ( t ) } { A t } Aug 016 17
COMPLEX AMPLITUDE Gnral Sinusoid x( t) Acos( t ) { A t } Sinusoid = REAL PART of complx xp: z(t)=(a f ) t t x( t) { X } { z( t)} Complx AMPLITUDE = X, which is a constant X A whn z( t) X t Aug 016 18
POP QUIZ: Complx Amp Find th COMPLEX AMPLITUDE for: x( t) 3 cos(77p t 0.5p ) Us EULER s FORMULA: x( t) { 3 (77pt0.5p ) } { 3 0.5p 77pt } X 3 0.5p Aug 016 19
POP QUIZ-: Complx Amp Dtrmin th 60-Hz sinusoid whos COMPLEX AMPLITUDE is: Convrt X to POLAR: X x( t) {( 3 3) 3 3 (10p t) } { 1 p /3 10p t } x( t) 1 cos(10p t p / 3) Aug 016 0
Not atan (3/ 3) is atan( 3) = π/3 Rmmbr 60 dgr angl Cos = 1/ Sin = 3/ Aug 016 1
COMPLEX CONJUGATE (z*) Usful concpt: chang th sign of all s RECTANGULAR: If z = x + y, thn th complx conugat is z* = x y POLAR: Magnitud is th sam but angl has sign chang z r z* r Aug 016
COMPLEX CONJUGATION Flips vctor about th ral axis! Aug 016 3
USES OF CONJUGATION Conugats usful for many calculations Ral part: z z * ( x y) ( x y) x { z} Imaginary part: z z * y y { z} Aug 016 4
Invrs Eulr Rlations Cosin is ral part of xp, sin is imaginary part Ral part: Imaginary part: ) cos( } {, } { * z z z z ) sin( } {, } { * z z y z z Aug 016 5
Mag & Magnitud Squard Magnitud Squard (polar form): zz * ( r )( r ) r z Magnitud Squard (Cartsian form): zz* (x y) (x y) x y x y Magnitud of complx xponntial is on: cos sin 1 Aug 016 6
COMPLEX MULTIPLY = VECTOR ROTATION Multiplication/division scals and rotats vctors Aug 016 7
POWERS Raising to a powr N rotats vctor by Nθ and scals vctor lngth by r N z N r N r N N Aug 016 8
MORE POWERS Aug 016 9
ROOTS OF UNITY W oftn hav to solv z N =1 How many solutions? k N N N r z p 1 N k k N r p p, 1 1 0,1,,, N k z N k p Aug 016 30
ROOTS OF UNITY for N=6 Solutions to z N =1 ar N qually spacd vctors on th unit circl! What happns if w tak th sum of all of thm? Aug 016 31
Sum th Roots of Unity Looks lik th answr is zro (for N=6) N1 p k / N 0? k 0 Writ as gomtric sum N1 r k k 0 Numrator 1 rn 1 r 1 r N 1 thn lt r p / N ( p / N ) N 1 p 0 Aug 016 3
Ndd latr to dscrib priodic signals in trms of sinusoids (Fourir Sris) Espcially ovr on priod d a b b a b a 0 1 1 0 / 0 / dt T T T T t p p Intgrat Complx Exp Aug 016 33
BOTTOM LINE CARTESIAN: Addition/subtraction is most fficint in Cartsian form POLAR: good for multiplication/division STEPS: Idntify arithmtic opration Convrt to asy form Calculat Convrt back to original form Aug 016 34
Rviw Appndix A and TLH Ch on Cours Wbsit Harman Chaptr Pags 55-60 Complx Numbrs and MATLAB Complx Numbrs Aug 016 43