DSP-First, 2/e - EMU Academic Staff Directory · License Info for SPFirst Slides §This work released under a Creative Commons License with the following terms: §Attribution §The

DSP-First, 2/e

LECTURE #3Complex Exponentials& Complex Numbers

Aug 2016 1© 2003-2016, JH McClellan & RW Schafer

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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 3

READING ASSIGNMENTS

§ This Lecture:§ Chapter 2, Sects. 2-3 to 2-5

§ Appendix A: Complex Numbers

§ Appendix B: MATLAB§ Next Lecture: Complex Exponentials

LECTURE OBJECTIVES

§ Introduce more tools for manipulating complex numbers§ Conjugate§ Multiplication & Division§ Powers§ N-th Roots of unity

1,For /2 == NNkj zez p


LECTURE OBJECTIVES

§ Phasors = Complex Amplitude§ Complex Numbers represent Sinusoids

§ Next Lecture: Develop the ABSTRACTION:§ Adding Sinusoids = Complex Addition§ PHASOR ADDITION THEOREM

}){()cos( tjj eAetA wjjw Â=+


WHY? What do we gain?

§ Sinusoids are the basis of DSP, § but trig identities are very tedious

§ Abstraction of complex numbers § Represent cosine functions§ Can replace most trigonometry with algebra

§ Avoid all Trigonometric manipulationsAug 2016 6© 2003-2016, JH McClellan & RW Schafer

COMPLEX NUMBERS

§ To solve: z2 = -1§ z = j§ Math and Physics use z = i

§ Complex number: z = x + j y

x

y zCartesiancoordinatesystem


PLOT COMPLEX NUMBERS

}{zx Â=

}{zy Á=

Real part:

Imaginary part:}05{5 j+-Â=-

5}52{ =+Á j2}52{ =+Â j


COMPLEX ADDITION = VECTOR Addition

26)53()24()52()34(

213

jj

jjzzz

+=+-++=++-=

+=


*** POLAR FORM ***

§ Vector Form§ Length =1§ Angle = q

§ Common Values§ j has angle of 0.5p§ -1 has angle of p§ - j has angle of 1.5p § also, angle of -j could be -0.5p = 1.5p -2p§ because the PHASE is AMBIGUOUS


POLAR <--> RECTANGULAR

§ Relate (x,y) to (r,q) rqx

y

Need a notation for POLAR FORM

( )xyyxr1

222

Tan-=

+=

q

qq

sincosryrx

==

Most calculators doPolar-Rectangular


Euler’s FORMULA

§ Complex Exponential§ Real part is cosine§ Imaginary part is sine§ Magnitude is one

re jq = rcos(q)+ jrsin(q)

)sin()cos( qqq je j +=


Cosine = Real Part

§ Complex Exponential§ Real part is cosine§ Imaginary part is sine

)sin()cos( qqq jrrre j +=

)cos(}{ qq rre j =ÂAug 2016 13© 2003-2016, JH McClellan & RW Schafer

Common Values of exp(jq)

§ Changing the anglepq njj eej 200110 ==+=®=

4/21 pjej ±=±

pppq )2/12(2/2/ +==®= njj eej

pppq )12(011 +==+-=-®= njj eej

ppppq )2/12(2/2/32/3 -- ===-®= njjj eeej

?1 =+± jAug 2016 14© 2003-2016, JH McClellan & RW Schafer

COMPLEX EXPONENTIAL

§ Interpret this as a Rotating Vector§ q = wt§ Angle changes vs. time§ ex: w=20p rad/s§ Rotates 0.2p in 0.01 secs

)sin()cos( tjte tj www +=

)sin()cos( qqq je j +=Aug 2016 15© 2003-2016, JH McClellan & RW Schafer

Cos = REAL PART

}{}{)cos( )(

tjj

tj

eAeAetA

wj

jwjw

Â=

Â=+ +So,

Real Part of Euler’s

}{)cos( tjet ww Â=General Sinusoid

)cos()( jw += tAtx


COMPLEX AMPLITUDE

}{)cos()( tjj eAetAtx wjjw Â=+=General Sinusoid

)}({}{)( tzXetx tj Â=Â= w

Sinusoid = REAL PART of complex exp: z(t)=(Aejf)ejwt

Complex AMPLITUDE = X, which is a constant

tjj XetzAeX wj == )(whenAug 2016 17© 2003-2016, JH McClellan & RW Schafer

POP QUIZ: Complex Amp§ Find the COMPLEX AMPLITUDE for:

§ Use EULER’s FORMULA:

p5.03 jeX =

)5.077cos(3)( pp += ttx

}3{

}3{)(775.0

)5.077(

tjj

tj

ee

etxpp

pp

Â=

Â= +


POP QUIZ-2: Complex Amp§ Determine the 60-Hz sinusoid whose

COMPLEX AMPLITUDE is:

§ Convert X to POLAR:33 jX +=

)3/120cos(12)( pp +=Þ ttx

}12{

})33{()(1203/

)120(

tjj

tj

ee

ejtxpp

p

Â=

+Â=


COMPLEX CONJUGATE (z*)

§ Useful concept: change the sign of all j’s

§ RECTANGULAR: If z = x + j y, then the complex conjugate is z* = x – j y

§ POLAR: Magnitude is the same but angle has sign change

z = re jq Þ z*= re- jqAug 2016 20© 2003-2016, JH McClellan & RW Schafer

COMPLEX CONJUGATION

§ Flips vector about the real axis!


USES OF CONJUGATION

§ Conjugates useful for many calculations§ Real part:

§ Imaginary part:

}{2

)()(2* zxjyxjyxzz

Â==-++

=+

}{22

2* zy

jyj

jzz

Á===-


Inverse Euler Relations§ Cosine is real part of exp, sine is imaginary part§ Real part:

§ Imaginary part:)cos(

2}{,

}{2*

qqq

qq =+

=ÂÞ=

Â=+

- jjjj eeeez

zzz

)sin(2

}{,

}{2*

qqq

qq =-

=ÁÞ=

Á==-

-

jeeeez

zyjzz

jjjj


Mag & Magnitude Squared

§ Magnitude Squared (polar form):

§ Magnitude Squared (Cartesian form):

§ Magnitude of complex exponential is one:

22 ||))((* zrrerezz jj === - qq

|e jq |2= cos2 q( )+ sin2 q( )=1

z z*= (x + jy)´ (x - jy) = x2 - j 2y2 = x2 + y2


COMPLEX MULTIPLY = VECTOR ROTATION

§ Multiplication/division scales and rotates vectors


POWERS

§ Raising to a power N rotates vector by Nθ and scales vector length by rN

zN = re jq( )N = rNe jNq


MORE POWERS

§


ROOTS OF UNITY

§ We often have to solve zN=1§ How many solutions?

kjNjNN eerz pq 21===

NkkNr pqpq 22,1 =Þ==Þ

1,2,1,0,2 -== Nkez Nkj !p


ROOTS OF UNITY for N=6

§ Solutions to zN=1 are N equally spaced vectors on the unit circle!

§ What happens if we take the sum of all of them?


Sum the Roots of Unity

§ Looks like the answer is zero (for N=6)

§ Write as geometric sum

e j2p k /N

k= 0

N-1

å = 0?

rkk= 0

N-1

å =1- rN

1- rthen let r = e j2p /N

01)(11Numerator 2/2 =-=-=- pp jNNjN eerAug 2016 30© 2003-2016, JH McClellan & RW Schafer

§ Needed later to describe periodic signals in terms of sinusoids (Fourier Series)

§ Especially over one period

jee

jede

jajbb

a

b

a

jj -

==òq

q q

0110/2

0

/2 =-

=-

=ò jjeedtejTTjT

Ttjp

p

Integrate Complex Exp


BOTTOM LINE

§ CARTESIAN: Addition/subtraction is most efficient in Cartesian form

§ POLAR: good for multiplication/division§ STEPS:

§ Identify arithmetic operation§ Convert to easy form§ Calculate§ Convert back to original form


Harder N-th Roots

§ Want to solve zN = c§ where c is a complex number

kjjNjNN eecerz pjq r 2)(===

NkkNr N pjqpjqr 22, +

=Þ+==Þ

1,2,1,0,2/ -== Nkeez NkjNjN !pjr


Euler’s FORMULA

§ Complex Exponential§ Real part is cosine§ Imaginary part is sine§ Magnitude is one


)sin()cos( qqq je j +=


Real & Imaginary Part Plots

PHASE DIFFERENCE = p/2


COMPLEX EXPONENTIAL

§ Interpret this as a Rotating Vector§ q = wt§ Angle changes vs. time§ ex: w=20p rad/s§ Rotates 0.2p in 0.01 secs

e jjq q q= +cos( ) sin( )



Rotating Phasor

See Demo on CD-ROMChapter 2


AVOID Trigonometry

§ Algebra, even complex, is EASIER !!!§ Can you recall cos(q1+q2) ?

§ Use: real part of ej(q1+q2) = cos(q1+q2)

( ) }{}{cos 2121 )(21

qqqqqq jjj eeeee Â=Â=+ +

)}sin)(cossin{(cos 2211 qqqq jje ++Â=

(...))sinsincos(cos 2121 j+-= qqqqAug 2016 38© 2003-2016, JH McClellan & RW Schafer

MULTIPLICATION§ CARTESIAN: use polynomial algebra

§ POLAR: easier because you can leverage the properties of exponentials

§ Multiply the magnitudes and add the angles

)(212121

2121 qqqq +=´=´ jjj errererzz

z1 ´ z2 = (x1 + jy1)´ (x2 + jy2)

= (x1x2 - y1y2)+ j(x1y2 - y1x2)


DIVISION

§ CARTESIAN: use complex conjugate to convert to multiplication:

§ POLAR: simpler to subtract exponents

z1z2

=r1e

jq1

r2ejq 2

=r1r2e j(q1-q 2 )

22

*21

*22

*21

22

11

2

1

||)()(

zzz

zzzz

jyxjyx

zz

==++

=


Complex Numbers

§ WHY? need to ADD sinusoids

§ Use an ABSTRACTION§ Complex Amplitude, X, has mag & phase§ Complex Exponential§ Euler’s Formula

z(t) = Xe jw t

s(t) = k cos(120p(t -0.002k))k=1

20

å


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DSP-First, 2/e - EMU Academic Staff Directory · License Info for SPFirst Slides §This work released under a Creative Commons License with the following terms: §Attribution §The