ECE 598: The Speech ECE 598: The Speech ChainChain

Lecture 3: PhasorsLecture 3: Phasors

A Useful One-Slide Idea: A Useful One-Slide Idea: LinearityLinearity

Derivatives are “linear,” meaning that, for Derivatives are “linear,” meaning that, for any functions f(t) and g(t),any functions f(t) and g(t),

x(t) = A f(t) + B g(t)x(t) = A f(t) + B g(t)

ImpliesImplies

dx/dt = A df/dt + B dg/dtdx/dt = A df/dt + B dg/dt

dd22x/dtx/dt22 = A d = A d22f/dtf/dt22 + B d + B d22g/dtg/dt22

Example: Example:

x(t)=cos(x(t)=cos(t-t-) ) dx/dt = –dx/dt = –sin(sin(t-t-))

x(t)=Acos(x(t)=Acos(t-t-) ) dx/dt = –dx/dt = –Asin(Asin(t-t-))

Review: Spring Mass SystemReview: Spring Mass System

Newton’s Second Law:Newton’s Second Law:f(t) = m df(t) = m d22x/dtx/dt22

Force of a Spring:Force of a Spring:f(t) = – k x(t)f(t) = – k x(t)

The Spring-Mass System Equation The Spring-Mass System Equation with no external forces:with no external forces:

dd22x/dtx/dt22 = - (k/m) x(t) = - (k/m) x(t)

Solution: CosineSolution: Cosine First and Second Derivatives of a Cosine:First and Second Derivatives of a Cosine:

x(t) = cos(x(t) = cos(t-t-) )

dx/dt = –dx/dt = –sin(sin(t-t-))

dd22x/dtx/dt22 = – = –2 2 cos(cos(t–t–))

Spring-Mass System:Spring-Mass System:

dd22x/dtx/dt22 = –(k/m) x(t) = –(k/m) x(t)

––22 cos( cos(t–t–) = – (k/m) cos() = – (k/m) cos(t-t-))

It only works at the “natural frequency:”It only works at the “natural frequency:”

0 0 √(k/m)√(k/m)

Linearity means that you can multiply by “A” here, if you want to, and the same “A” will appear here.

The Other Function We The Other Function We KnowKnow

First and Second Derivatives of an Exponential:First and Second Derivatives of an Exponential:

x(t) = ex(t) = ett

dx/dt = dx/dt = eett

dd22x/dtx/dt22 = = 2 2 eett

Could x(t)= Could x(t)= eett also solve the spring-mass also solve the spring-mass

system?system?

dd22x/dtx/dt22 = –(k/m) x(t) = –(k/m) x(t)

22 e ett = – (k/m) e = – (k/m) ett

It only works if:It only works if:

√√(-k/m)(-k/m)

Imaginary NumbersImaginary Numbers Definition (the part that you memorize):Definition (the part that you memorize):

jj√-1√-1

Linearity:Linearity:

2j2j√-4 √-4 √4 √4 √-1√-1

3j3j√-9 √-9 √9 √9 √-1√-1

Solution to the spring-mass system:Solution to the spring-mass system:

√√(-k/m) (-k/m) √-1 √-1 √(k/m) √(k/m) jj

x(t) = ex(t) = ett = e = ejjtt0

Complex ExponentialsComplex Exponentials Definition (another bit to memorize):Definition (another bit to memorize):

x(t) = ex(t) = ej(j(tt) ) = cos(= cos(tt) + j sin() + j sin(tt))

Then the “real part” of x(t) is defined to be:Then the “real part” of x(t) is defined to be:

Re{x(t)} = cos(Re{x(t)} = cos(tt))

So x(t)=eSo x(t)=ejjtt and x(t)=cos( and x(t)=cos(t) are actually exactly the same solution! t) are actually exactly the same solution!

The “imaginary part,” Im{x(t)}=sin(The “imaginary part,” Im{x(t)}=sin(t), doesn’t change the solution.t), doesn’t change the solution.

If you like, visualize x(t) as a movement in two dimensions:If you like, visualize x(t) as a movement in two dimensions:

Re(x(t)) is movement in the horizontal directionRe(x(t)) is movement in the horizontal direction

Im(x(t)) is movement in Buckaroo Banzai’s mysterious 8Im(x(t)) is movement in Buckaroo Banzai’s mysterious 8thth

dimension.dimension.

All we really care about is the movement in the horizontal All we really care about is the movement in the horizontal

direction; the movement in the Buckaroo Banzai direction is a direction; the movement in the Buckaroo Banzai direction is a

convenient fiction that just happens to make the math work out.convenient fiction that just happens to make the math work out.

How do you Plot a Complex How do you Plot a Complex Exponential?Exponential?

Answer: you can’t!Answer: you can’t!

What you CAN do: plot either the real part or the imaginary What you CAN do: plot either the real part or the imaginary

partpart

x(t) = ex(t) = ej2tj2t

Re{x(t)} = cos(Re{x(t)} = cos(t)t)

Im{x(t)} = sin(Im{x(t)} = sin(t)t)

-4

-3-2

-10

1

23

4

0.00 0.79 1.57 2.36 3.14 3.93 4.71 5.50 6.28

3cos(2t)

3cos(2t-pi/2)

Special NumbersSpecial Numbers

eejj = cos(= cos() + j sin() + j sin())

1 = cos(1 = cos() + j sin() + j sin() = e) = ej0j0

j = cos(j = cos() + j sin() + j sin() = e) = ejj/2/2

1 = cos(1 = cos() + j sin() + j sin() = e) = ejj

(e(ejj/2/2))22 = e = ejj

(j)(j)22 = = 11

Linearity AgainLinearity Again ““Real Part” and “Imaginary Part” are linear Real Part” and “Imaginary Part” are linear

operators:operators:

x(t) = A f(t) + B g(t)x(t) = A f(t) + B g(t)

Re{x(t)} = A Re{f(t)} + B Re{g(t)}Re{x(t)} = A Re{f(t)} + B Re{g(t)}

Im{x(t)} = A Im{f(t)} + B Im{g(t)}Im{x(t)} = A Im{f(t)} + B Im{g(t)}

ExampleExample

x(t) = 2 ex(t) = 2 ejjt t + 3 e+ 3 ej(j(tt) )

Re{x(t)} = 2cos(Re{x(t)} = 2cos(t) + 3cos(t) + 3cos(tt) )

Im{x(t)} = 2sin(Im{x(t)} = 2sin(t) + 3sin(t) + 3sin(tt) )

Amplitude, Phase, and Amplitude, Phase, and Frequency of Cosines vs. Frequency of Cosines vs.

ExponentialsExponentials Exponential:Exponential:

x(t) = Aex(t) = Aej(j(tt)) = A e = A ejj eejjtt

Cosine:Cosine:

Re{x(t)} = A cos(Re{x(t)} = A cos(tt))

The Life Story of a CosineThe Life Story of a Cosine Vocal fold motion is a cosine at (say) Vocal fold motion is a cosine at (say) 400400, with amplitude of , with amplitude of

A=0.001m:A=0.001m:

x(t) = 0.001 + 0.001 ex(t) = 0.001 + 0.001 ej400j400tt m m

Notice this looks like an exponential, but it’s “really” a cosine. Notice this looks like an exponential, but it’s “really” a cosine.

Re{x(t)} = 0.001 + 0.001cos(400Re{x(t)} = 0.001 + 0.001cos(400t)t)

Air puffs come through when the vocal folds are open, with a Air puffs come through when the vocal folds are open, with a

maximum rate of 0.001 mmaximum rate of 0.001 m33/second: /second:

uuGlottisGlottis(t)=0.0005+0.0005e(t)=0.0005+0.0005ej400j400tt m m33/s/s

The same air puffs reach the lips 0.5ms later: The same air puffs reach the lips 0.5ms later:

uuLipsLips(t)=0.0005+0.0005e(t)=0.0005+0.0005ej400j400(t(t0.0005) 0.0005) =0.0005 + 0.0005 e=0.0005 + 0.0005 e0.2j0.2j eej400j400tt liter/s liter/s

Air pressure at the lips is the derivative of u(t), times 0.003 kg/s:Air pressure at the lips is the derivative of u(t), times 0.003 kg/s:

ppLipsLips(t)= 0 + 0.0003j(t)= 0 + 0.0003j e ejj eej400j400tt = 0.0003 = 0.0003 e ejj eej400j400tt Pascals Pascals

Acoustic wave reaches the listener’s ear 2ms later:Acoustic wave reaches the listener’s ear 2ms later:

ppEarEar(t)= 0.0003(t)= 0.0003 e e0.3j0.3j eej400j400tt0.02) 0.02) = 0.0003= 0.0003 e ej7.7j7.7 eej400j400tt

Look Closer:Look Closer:

x(t) = 0.001 ex(t) = 0.001 ej400j400tt

uuGlottisGlottis(t) = 0.0005 e(t) = 0.0005 ej400j400tt

uuLipsLips(t) = 0.0005 e(t) = 0.0005 e0.2j0.2j eej400j400tt

ppLipsLips(t) = 0.0003(t) = 0.0003 e ejj eej400j400tt

ppEarEar(t) = 0.0003(t) = 0.0003 e ej7.7j7.7 eej400j400tt

Amplitude, frequency, phase. Which Amplitude, frequency, phase. Which

one stays the same?one stays the same?

Phasor NotationPhasor Notation

xx = 0.001 = 0.001

uuGlottisGlottis = 0.0005 = 0.0005

uuLipsLips = 0.0005 e = 0.0005 e0.2j0.2j

ppLipsLips = 0.0003 = 0.0003 e ejj

ppEarEar = 0.0003 = 0.0003 e ej7.7j7.7

Phasor notation: write down only the Phasor notation: write down only the

amplitude and phase, not the amplitude and phase, not the

frequency.frequency.

Definition of Phasor NotationDefinition of Phasor Notation A phasor specifies the amplitude and A phasor specifies the amplitude and

phase of a cosine, but not its frequency.phase of a cosine, but not its frequency.

xx = A e = A ejj

To get x(t) back, you look back through To get x(t) back, you look back through

the problem definition in order to find the problem definition in order to find , ,

then writethen write

x(t) = Re{ x(t) = Re{ xx e ejjtt } }

(Written in boldface if possible)

ExampleExample

Phasor Phasor

xx, , uu, , ppComplex Complex

x(t), u(t), p(t)x(t), u(t), p(t)Real Real

x(t), u(t), p(t)x(t), u(t), p(t)

Vocal Fold x(t)Vocal Fold x(t) 1010 10 e10 ej400j400tt 10cos(40010cos(400t)t)

Vocal Fold u(t)Vocal Fold u(t) 55 5 e5 ej400j400tt 5 cos(4005 cos(400t)t)

Lips u(t)Lips u(t) 5 e5 e0.2j0.2j 5 e5 e0.2j0.2j eej400j400tt 5 cos(4005 cos(400tt0.20.2

Lips p(t)Lips p(t) 33 e ejj 33 e ejj eej400j400tt 33cos(400cos(400t+t+

Ear p(t)Ear p(t) 33 e ej7.7j7.7 33 e ej7.7j7.7 eej400j400tt 33cos(400cos(400tt7.77.7

The Main Purpose of Phasors: The Main Purpose of Phasors: They Turn Derivatives into They Turn Derivatives into

MultiplicationMultiplication x(t) = Re{ x(t) = Re{ xx e ejjt t }}

dx/dt = Re{ jdx/dt = Re{ jxx e ejjtt} }

dd22x/dtx/dt22 = Re{ (j = Re{ (j))22xx e ejjtt } = Re{ } = Re{ 22xx e ejjtt } }

The Main Purpose of Phasors: The Main Purpose of Phasors: They Turn Derivatives into They Turn Derivatives into

MultiplicationMultiplication In regular notation:In regular notation:

v(t) = dx/dtv(t) = dx/dt

x(t) = A cos(x(t) = A cos(tt) ) v(t) = v(t) = A sin(A sin(tt))

In phasor notation:In phasor notation:

vv = = jjxx

xx = Ae = Aejj v v = = jjAeAe-j-j

In regular notation:In regular notation:

dd22x/dtx/dt22 = = (k/m) x(t)(k/m) x(t)

In phasor notation:In phasor notation:

22 xx = = (k/m) (k/m) xx

Solution: Solution: =√(k/m) !!!=√(k/m) !!!

SummarySummary Linearity means you can: Linearity means you can:

add two solutions, or add two solutions, or scale by a constant.scale by a constant.

x(t)=ex(t)=ett could solve the spring-mass system, but only if could solve the spring-mass system, but only if =√(-k/m)=√(-k/m) jj22 = = 1 1 eejj = cos( = cos() + j sin() + j sin()) We don’t really need the imaginary part; Re{eWe don’t really need the imaginary part; Re{ejjtt} = cos(} = cos(t)t) We don’t really need the frequency part either; it is never changed We don’t really need the frequency part either; it is never changed

by any linear operation (e.g., scaling, time shift, derivative)by any linear operation (e.g., scaling, time shift, derivative) Phasor notation encodes just the amplitude and phase of a cosine: Phasor notation encodes just the amplitude and phase of a cosine:

x x = Ae= Aejj

To get back to the time domain: To get back to the time domain:

x(t) = Re{ x(t) = Re{ xxeejjt t }}