DSP-First, 2/e. LECTURE # CH2-3 Complex Exponentials & Complex Numbers TLH MODIFIED. Aug , JH McClellan & RW Schafer

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1 DSP-First, / TLH MODIFIED LECTURE # CH-3 Complx Exponntials & Complx Numbrs Aug 016 1

2 READING ASSIGNMENTS This Lctur: Chaptr, Scts. -3 to -5 Appndix A: Complx Numbrs Complx Exponntials Aug 016

3 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

4 Aug 016 4

5 LECTURE OBJECTIVES Phasors = Complx Amplitud Complx Numbrs rprsnt Sinusoids Tak Ral or Complx part A cos( t ) {( A ) t } Aug 016 5

6 Phasors Not Phasrs Captain Kirk & Sally Kllrman STAR TREK Aug 016 6

7 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

8 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

9 PLOT COMPLEX NUMBERS { 5} 5 { 5} Ral part: 5 { 5 0} x {z} Imaginary part: y {z} Aug 016 9

10 COMPLEX ADDITION = VECTOR Addition z 3 z1 z (4 3) ( 5) (4 ) ( 3 5) 6 Aug

11 *** 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

12 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

13 Eulr s FORMULA Complx Exponntial Ral part is cosin Imaginary part is sin Magnitud is on cos( ) sin( ) r rcos() rsin() Aug

14 Cosin = Ral Part Complx Exponntial Ral part is cosin Imaginary part is sin r r cos( ) r sin( ) { r } r cos( ) Aug

15 Common Valus of xp() Changing th angl p n / 1 p p p p 1/ ) ( / / n p p p 1) ( n p p p p 1/ ) ( / / 3 / 3 n Aug a + b = c θ = arctan( b a ) IF a 0

16 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

17 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

18 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

19 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

20 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

21 Not atan (3/ 3) is atan( 3) = π/3 Rmmbr 60 dgr angl Cos = 1/ Sin = 3/ Aug 016 1

22 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

23 COMPLEX CONJUGATION Flips vctor about th ral axis! Aug 016 3

24 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

25 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

26 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

27 COMPLEX MULTIPLY = VECTOR ROTATION Multiplication/division scals and rotats vctors Aug 016 7

28 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

29 MORE POWERS Aug 016 9

30 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

31 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

32 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

33 Ndd latr to dscrib priodic signals in trms of sinusoids (Fourir Sris) Espcially ovr on priod d a b b a b a / 0 / dt T T T T t p p Intgrat Complx Exp Aug

34 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

35 Rviw Appndix A and TLH Ch on Cours Wbsit Harman Chaptr Pags Complx Numbrs and MATLAB Complx Numbrs Aug

Announce. ECE 2026 Summer LECTURE OBJECTIVES READING. LECTURE #3 Complex View of Sinusoids May 21, Complex Number Review

Announce. ECE 2026 Summer LECTURE OBJECTIVES READING. LECTURE #3 Complex View of Sinusoids May 21, Complex Number Review ECE 06 Summr 018 Announc HW1 du at bginning of your rcitation tomorrow Look at HW bfor rcitation Lab 1 is Thursday: Com prpard! Offic hours hav bn postd: LECTURE #3 Complx Viw of Sinusoids May 1, 018 READIG