ECE 598: The Speech ECE 598: The Speech ChainChain

Lecture 1: Dimensional Lecture 1: Dimensional Analysis; Cosines;Analysis; Cosines;

Logs and ExponentialsLogs and Exponentials

in today’s lecture, you will in today’s lecture, you will learnlearn

About this courseAbout this course Goals of this courseGoals of this course Teaching style; homework, labs and examsTeaching style; homework, labs and exams

The language of acousticsThe language of acoustics Dimensional analysisDimensional analysis Cosines: frequency, phase, and amplitudeCosines: frequency, phase, and amplitude All of the calculus you will ever need to know---All of the calculus you will ever need to know---

in one slide!in one slide! Musical scales, decibels, and other logarithmic Musical scales, decibels, and other logarithmic

thingsthings

the goals of this coursethe goals of this course AcousticsAcoustics

Loudness; PitchLoudness; Pitch Noise; ReverberationNoise; Reverberation Frequency ResponseFrequency Response

SpeechSpeech What sounds can the mouth produce? Why?What sounds can the mouth produce? Why?

LanguageLanguage ““Information” defined as the unpredictable part Information” defined as the unpredictable part

of an utterance of an utterance

pre-requisitespre-requisites I assume that you:I assume that you:

1.1. … … have never taken calculushave never taken calculus

2.2. … … know that y=logknow that y=log1010x means x=10x means x=10yy, , but would appreciate some practicebut would appreciate some practice

3.3. … … know what cos(x) looks like, but know what cos(x) looks like, but would appreciate some practicewould appreciate some practice

4.4. … … know Newton’s second law (f=ma) know Newton’s second law (f=ma) but would appreciate some practicebut would appreciate some practice

5.5. … … know the ideal gas law (PV=nRT) but know the ideal gas law (PV=nRT) but would appreciate some practicewould appreciate some practice

how will this course be how will this course be taught?taught?

Lectures: Mondays 1-4 PM, Siebel 1105Lectures: Mondays 1-4 PM, Siebel 1105 1:00-1:30: Quiz (every week)1:00-1:30: Quiz (every week) 1:30-1:45: Break1:30-1:45: Break 1:45-2:45: New material (slides)1:45-2:45: New material (slides) 2:45-3:00: Break2:45-3:00: Break 3:00-4:00: Sample problems (on the 3:00-4:00: Sample problems (on the

whiteboard)whiteboard) Office HoursOffice Hours

Daily, 9:30-11:00, 2011 BeckmanDaily, 9:30-11:00, 2011 Beckman Every Monday, after lecture, 4-5 PM in the Every Monday, after lecture, 4-5 PM in the

Siebel cafeSiebel cafe

how will this course be how will this course be taught?taught?

LabsLabs Weekly, 2 hours; times TBAWeekly, 2 hours; times TBA

Written homeworkWritten homework Weekly; first one is due Monday!Weekly; first one is due Monday!

Exams and QuizzesExams and Quizzes Weekly 30-minute quizzesWeekly 30-minute quizzes One final examOne final exam

Purpose of the weekly quizzes:Purpose of the weekly quizzes: If many people have trouble with a concept, I If many people have trouble with a concept, I

want to know about it RIGHT AWAY so I can want to know about it RIGHT AWAY so I can CHANGE THE LECTURE.CHANGE THE LECTURE.

Today: Units, Cosines, and Today: Units, Cosines, and LogsLogs

UnitsUnits In acoustics, many problems can be solved In acoustics, many problems can be solved

by paying careful attention to the way in by paying careful attention to the way in which something is measured.which something is measured.

CosinesCosines Every measured signal can be expressed as Every measured signal can be expressed as

the sum of shifted, scaled cosines. More on the sum of shifted, scaled cosines. More on this amazing fact in lecture 4. For the time this amazing fact in lecture 4. For the time being, let’s understand more about cosines.being, let’s understand more about cosines.

LogarithmsLogarithms Musical pitch = log (frequency)Musical pitch = log (frequency) Sound pressure level = log (pressure)Sound pressure level = log (pressure)

a warninga warning Today’s lecture (and Monday’s lecture) Today’s lecture (and Monday’s lecture)

will seem like pure math with no will seem like pure math with no acoustics.acoustics.

That’s because, in order to speak the That’s because, in order to speak the “language of acoustics” (specifically, the “language of acoustics” (specifically, the language in which language in which Acoustic PhoneticsAcoustic Phonetics is is written), we need two lectures of math.written), we need two lectures of math.

If you feel lost or abandoned, please If you feel lost or abandoned, please come to office hours.come to office hours.

The Five Fundamental Units The Five Fundamental Units (MKS)(MKS)

1.1. Length: 1 meterLength: 1 meter2.2. Mass: 1 kilogramMass: 1 kilogram3.3. Time: 1 secondTime: 1 second4.4. Temperature: 1 degree KelvinTemperature: 1 degree Kelvin5.5. Angle: 1 radian Angle: 1 radian All other acoustic units can be expressed All other acoustic units can be expressed

as combinations of these five fundamental as combinations of these five fundamental units.units.

Many problems can be solved by just Many problems can be solved by just figuring out the units used on both sides figuring out the units used on both sides of an equation.of an equation.

Units: The Metric SystemUnits: The Metric System 1m (meter) = 1m (meter) =

0.000001 Mm (megameter) =0.000001 Mm (megameter) = 0.001 km (kilometer) =0.001 km (kilometer) = 0.01 hm (hectometer) =0.01 hm (hectometer) = 0.1 Dm (decameter) =0.1 Dm (decameter) = 10 dm (decimeters) =10 dm (decimeters) = 100 cm (centimeters) =100 cm (centimeters) = 1000 mm (millimeters) =1000 mm (millimeters) = 1,000,000 1,000,000 m (micrometers)m (micrometers)

Units: fun conversionsUnits: fun conversions 1 year = 31.557 Ms (megaseconds)1 year = 31.557 Ms (megaseconds) 1 day = 86.4 ks (kiloseconds)1 day = 86.4 ks (kiloseconds) 0 degrees K = -273 degrees C0 degrees K = -273 degrees C 1 radian = 571 radian = 57oo

1 cycle = 21 cycle = 2 radians (once around radians (once around the circle)the circle)

Why radians?Why radians? A one-radian slice of cake is a slice with A one-radian slice of cake is a slice with

the same length on all three sides (two the same length on all three sides (two straight sides and one curved side).straight sides and one curved side).

Why does cycle = 2Why does cycle = 2 radians? radians? Because circumference C=2Because circumference C=2rr

r meters

r metersr meters

angle =1 radian

Dimensional AnalysisDimensional Analysis A little girl sits on the outermost A little girl sits on the outermost

edge of a merry-go-round. The edge of a merry-go-round. The merry-go-round has a radius of 2m. merry-go-round has a radius of 2m. It is rotating at 0.5 radians/second. It is rotating at 0.5 radians/second. How fast is the little girl traveling, in How fast is the little girl traveling, in meters/second?meters/second?

(0.5 (0.5 secondsecond) ● (2 ) ● (2 radianradian) =) =

radiansradians metersmeters metersmeters1 1 secondsecond

Dimensional AnalysisDimensional Analysis A merry-go-round is spinning at A merry-go-round is spinning at

radians/second (about 0.314 radians/second (about 0.314 radians/second). How long before radians/second). How long before it spins 1 cycle?it spins 1 cycle?

Goal: seconds/cycle.Goal: seconds/cycle.

secondsseconds

cyclecycle== 1 second1 second

/10 radians/10 radians

22 radians radians

1 cycle1 cycle

2(10)2(10)2020

11

Dimensional Analysis: Newton’s Dimensional Analysis: Newton’s LawLaw

A girl releases a 1kg ball in strong A girl releases a 1kg ball in strong winds. 3 seconds later, the ball is winds. 3 seconds later, the ball is already traveling 15 m/s. What is already traveling 15 m/s. What is the force of wind on the ball?the force of wind on the ball?

f = maf = ma

= (1 kg) = (1 kg) 15 m/s15 m/s

3 s3 s

55

= 5= 5kg mkg m

ss22= 5 N= 5 N

1 “Newton” (N) is just a shorthand 1 “Newton” (N) is just a shorthand for 1 kg m/sfor 1 kg m/s22

Dimensional Analysis: Newton’s Dimensional Analysis: Newton’s LawLaw

A pencil has mass of 0.1kg (100g). A pencil has mass of 0.1kg (100g). Gravity accelerates the pencil at 10 Gravity accelerates the pencil at 10 m/sm/s22 toward the center of the Earth. toward the center of the Earth. What is the force of gravity on the What is the force of gravity on the pencil?pencil?f = maf = ma= (0.1 kg)(10 m/s= (0.1 kg)(10 m/s22) ) = 1 kg m/s= 1 kg m/s22 = 1N = 1N

Useful Composite Units in Useful Composite Units in AcousticsAcoustics

Frequency: 1 Hz = 1 cycle/second (Hertz)Frequency: 1 Hz = 1 cycle/second (Hertz) Audible sound is roughly 10 kHz > f > 10 HzAudible sound is roughly 10 kHz > f > 10 Hz

Density: 1 kg/mDensity: 1 kg/m33 = 1 mg/(cm) = 1 mg/(cm)33

Air is about 1.29 mg/(cm)Air is about 1.29 mg/(cm)33

Force: 1 N = 1 kg m/sForce: 1 N = 1 kg m/s22 (Newtons) (Newtons) The force of gravity on a pencilThe force of gravity on a pencil

Pressure: 1 Pa = 1 N/mPressure: 1 Pa = 1 N/m22 (Pascals) (Pascals) Air pressure at sea level is about 100,000 PaAir pressure at sea level is about 100,000 Pa Audible pressures are usually 20Audible pressures are usually 20Pa < p(t) < 20PaPa < p(t) < 20Pa

Energy: 1 J = 1 N m (Joules) Energy: 1 J = 1 N m (Joules) Burning 1kcal of sugar yields 4184 JBurning 1kcal of sugar yields 4184 J

Power: 1 W = 1 J/s = 1 N m/s (Watts)Power: 1 W = 1 J/s = 1 N m/s (Watts) The heat given off by one human ~ 100WThe heat given off by one human ~ 100W

Review, Topic #1: Dimensional Review, Topic #1: Dimensional AnalysisAnalysis

There are five fundamental units: length (m), There are five fundamental units: length (m), mass (kg), time (s), temperature (deg K), and mass (kg), time (s), temperature (deg K), and angle (radians)angle (radians)

Every other unit can be expressed in terms of Every other unit can be expressed in terms of these fundamental unitsthese fundamental units

Units can be divided and canceled just like Units can be divided and canceled just like anything else in an equationanything else in an equation

Carefully evaluating the units is called Carefully evaluating the units is called “dimensional analysis.”“dimensional analysis.”

Many important problems in acoustics have been Many important problems in acoustics have been solved using only dimensional analysis.solved using only dimensional analysis.

More about cosinesMore about cosines Every time you see a Every time you see a

formula involving formula involving cosines, please cosines, please remember: remember: There is a circle hiding There is a circle hiding

behind it.behind it. Here is the circle. Here is the circle.

It is centered at (0,0). It is centered at (0,0). It has a radius of r. It has a radius of r. It has a point, It has a point,

somewhere on its edge, somewhere on its edge, called (x,y).called (x,y).

is the angle between is the angle between (x,y) and the positive x (x,y) and the positive x axis.axis.

x

yr

This point is called(x,y)

sohcahtoasohcahtoa sinsin = = = =

coscos= = = =

tantan = = = =x

yr

oppositeopposite

hypotenusehypotenuse

adjacentadjacent

hypotenusehypotenuse

oppositeopposite

adjacentadjacent

yy

rr

xx

rr

yyxx

Circles and TrianglesCircles and Triangles x=r cosx=r cos y=r siny=r sin y/x = tany/x = tan = = xx22+y+y22=r=r22, so…, so…

coscos22 +sin +sin22 =1=1 x

yr

This point is called(x,y) sinsin

coscos

If Daughter is on a Merry-Go-If Daughter is on a Merry-Go-Round, Daddy’s Walk is a Round, Daddy’s Walk is a

CosineCosine Imagine a girl going Imagine a girl going

around a merry-go-round around a merry-go-round with a radius of A meters. with a radius of A meters. As she moves, her daddy As she moves, her daddy walks back and forth in walks back and forth in order to stay level with order to stay level with her. His position is x(t) = A her. His position is x(t) = A cos(cos(t). t).

is called the “angular is called the “angular velocity,” in velocity,” in radians/second.radians/second.

Daughter’s movement= circle of radius A

Daddy’s movementx(t) = A cos(t)

0

0.7854

1.5708

2.35619

3.14159

3.92699

4.71239

5.49779

6.28318

-2 -1 0 1 2x(t)

t

Daddy moves backand forthrather like apendulum…

Frequency = 1/PeriodFrequency = 1/PeriodAngular Velocity = 1/(Time Angular Velocity = 1/(Time

Constant)Constant)

x(t) = cos(t)x(t) = cos(t) Period: Period:

T = 2T = 2 ~ 6.28 seconds/cycle ~ 6.28 seconds/cycle Frequency: Frequency:

f = 1/T = 1/2f = 1/T = 1/2 cycles/second cycles/second Angular Velocity: Angular Velocity:

= 1 radian/second= 1 radian/second Time Constant: Time Constant:

= 1 second/radian= 1 second/radian

-1.5

-1

-0.5

0

0.5

1

1.5

0.00 0.79 1.57 2.36 3.14 3.93 4.71 5.50 6.28

-4-3

-2-10

12

34

0.00 0.79 1.57 2.36 3.14 3.93 4.71 5.50 6.28

x(t) = 3cos(2t)x(t) = 3cos(2t) Period: Period:

T = T = ~ 3.14 seconds/cycle~ 3.14 seconds/cycle Frequency: Frequency:

f=1/T = 1/f=1/T = 1/ cycles/second cycles/second Angular Velocity: Angular Velocity:

= 2 radians/second= 2 radians/second Time Constant: Time Constant:

= 0.5 seconds/radian= 0.5 seconds/radian

tt

x(t)x(t)

Root Mean Square (RMS)Root Mean Square (RMS)

t

x(t) = A cos(x(t) = A cos(t)t) A is called the “amplitude” or A is called the “amplitude” or

“maximum amplitude” of x(t). “maximum amplitude” of x(t). A has the same units as x(t).A has the same units as x(t).

The average value of a cosine is The average value of a cosine is always E[x(t)]=0.always E[x(t)]=0.

-4

-3

-2

-1

0

1

2

3

4

0.00 0.79 1.57 2.36 3.14 3.93 4.71 5.50 6.28

x(t)

t

A more useful “average amplitude” is the “root mean A more useful “average amplitude” is the “root mean square,” defined as the square root of the mean of the square,” defined as the square root of the mean of the square of x(t),square of x(t),

xxRMSRMS = √E[x = √E[x22(t)](t)] For a cosine, xRMS turns out to beFor a cosine, xRMS turns out to be

xxRMSRMS = A/√2 ≈ 0.707 A = A/√2 ≈ 0.707 A

Review, Topic #2: CosinesReview, Topic #2: Cosines

If a little girl is going around a merry-go-If a little girl is going around a merry-go-round, and her daddy is following in round, and her daddy is following in front of her, then his movement is front of her, then his movement is x(t)=Acos(x(t)=Acos(t-t-)) We’ll come back to the We’ll come back to the part in a minute part in a minute

= 2= 2f is called the angular velocityf is called the angular velocity A is called the amplitudeA is called the amplitude xxRMSRMS ≈ 0.707A is called the RMS ≈ 0.707A is called the RMS

amplitudeamplitude

Time DelayTime Delayor: what if the merry-go-round starts or: what if the merry-go-round starts

later?later?

-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(2(t-1))

3cos(2t)t

x(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(2(t+1))

3cos(2t)

Time AdvanceTime Advanceor: what if the merry-go-round starts or: what if the merry-go-round starts

earlierearlier??

t

x(t)

General Form of a Time General Form of a Time ShiftShift

-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(2(t-1))

3cos(2t)

To delay by To delay by seconds, write seconds, write

x(x(tt) = A cos( ) = A cos( (t - (t - ) ) )) To advance by To advance by seconds, write seconds, write x(x(tt) = A cos( ) = A cos( (t + (t + ) ) ))

t

x(t)

Phase ShiftPhase Shift

-4

-3-2

-1

0

12

3

4

0.00 0.79 1.57 2.36 3.14 3.93 4.71 5.50 6.28

3cos(2t-2)

3cos(2t)

““Phase Shift” and “Time Shift” are different words for Phase Shift” and “Time Shift” are different words for exactly the same thing:exactly the same thing:

x(t) = A cos( x(t) = A cos( (t-(t-) ) ) = A cos( ) = A cos( tt – – ), ), == What are the units of What are the units of ??

(( radians/second) ( radians/second) ( seconds) = seconds) = radians radians

t

x(t)

0.00

0.79

1.57

2.36

3.14

3.93

4.71

5.50

6.28

-2 -1 0 1 2

Phase Shift: ExamplePhase Shift: Example

Suppose that the little Suppose that the little girl’s starting position is at girl’s starting position is at an angle of -an angle of -/2 radians /2 radians (relative to the “x axis”)(relative to the “x axis”)

Then daddy’s position isThen daddy’s position is

x(t)=A cos(x(t)=A cos(t - t - /2) /2)

Suppose the little girlstarts here…

…so daddy starts here…

x(t)

t

…and his position is a cosine, delayed by /2 radians.

Four Particularly Interesting Four Particularly Interesting Phase Shifts: Phase Shifts: /2, /2, , 3, 3/2, and /2, and

22

-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)

-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)

-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-3pi/2)

-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-2pi)

cos(t-/2) = sin(t) !!! cos(t-) = -cos(t) !!!

cos(t-3/2) = -sin(t) !!! cos(t-2) = cos(t) !!!

Review, Topic #3: Phase Review, Topic #3: Phase ShiftShift

““Phase shift” = “Time shift” Phase shift” = “Time shift” ““Time shift:” in secondsTime shift:” in seconds ““Phase shift:” (radians/second)●(seconds) = Phase shift:” (radians/second)●(seconds) =

radiansradians Four particularly interesting phase shifts:Four particularly interesting phase shifts:

Shift by Shift by /2 radians (1/4 cycle): cosine turns into /2 radians (1/4 cycle): cosine turns into sinesine

Shift by Shift by radians (1/2 cycle): cosine turns into radians (1/2 cycle): cosine turns into negative cosinenegative cosine

Shift by 3Shift by 3/2 radians (3/4 cycle): cosine turns /2 radians (3/4 cycle): cosine turns into negative sineinto negative sine

Shift by 2Shift by 2 radians (1 full cycle): what radians (1 full cycle): what happened?happened?

A meters

Girl on the Merry-Go-Round: Girl on the Merry-Go-Round: How Fast is She Moving?How Fast is She Moving?

radiansradians metersmeters metersmeters= = A A secondsecond secondsecond ● ● A A radianradian

radians/second meters/second

Girl on the Merry-go-Round: Girl on the Merry-go-Round: How Fast is How Fast is DaddyDaddy Moving? Moving?

Fastest: Daddy moves Fastest: Daddy moves as fast as Daughter as fast as Daughter when she’s passing in when she’s passing in front of him:front of him:

x(t) = 0x(t) = 0 v(t) = v(t) = ±± AA Slowest: Daddy stops Slowest: Daddy stops

and turns around and turns around whenwhen

x(t) = x(t) = ± ± AA v(t) = 0 m/sv(t) = 0 m/s

At all times:At all times: x(t) = A cosx(t) = A costt v(t) = - v(t) = - A sinA sintt

0

0.7854

1.5708

2.35619

3.14159

3.92699

4.71239

5.49779

6.28318

-2 -1 0 1 2x(t)

t

0.00

0.79

1.57

2.36

3.14

3.93

4.71

5.50

6.28

-2 -1 0 1 2

t

v(t)

What is What is Differential Differential CalculusCalculus??

““dx/dt” is calculus notation for “the rate at which dx/dt” is calculus notation for “the rate at which

x(t) is changing.”x(t) is changing.” If x(t) is position, then v(t)=dx/dt is velocity.If x(t) is position, then v(t)=dx/dt is velocity. If v(t) is velocity, then a(t)=dv/dt is acceleration.If v(t) is velocity, then a(t)=dv/dt is acceleration. If J(t) is energy, then W(t)=dJ/dt is power.If J(t) is energy, then W(t)=dJ/dt is power.

You do not need differential calculus in order to You do not need differential calculus in order to

take ECE 598. You only need to memorize two x(t) take ECE 598. You only need to memorize two x(t)

dx/dt pairs; the rest of the course will proceed dx/dt pairs; the rest of the course will proceed

just fine using only these two pairs. They are given just fine using only these two pairs. They are given

on the next slide.on the next slide.

All the Calculus You’ll Ever All the Calculus You’ll Ever Need to Know, on One SlideNeed to Know, on One Slide

x(t)x(t) dx/dtdx/dt

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

x(t) = A ex(t) = A ett dx/dt = dx/dt = A eA ett

A, A, , , , and , and can be replaced by any expression, no can be replaced by any expression, no

matter how horrible and complicated, as long as the matter how horrible and complicated, as long as the

expression doesn’t contain any t.expression doesn’t contain any t.

Magical eMagical ett: dx/dt=x(t): dx/dt=x(t)

The function x(t)=eThe function x(t)=ett is is called the “my function” called the “my function” of calculus,* because…of calculus,* because…

x(t)=ex(t)=ett is the only is the only function with the function with the following propertyfollowing property

x(t)=ex(t)=ett dx/dt = dx/dt = x(t)x(t)

The property dx/dt=x(t) The property dx/dt=x(t) only works for the only works for the constant e=2.718282.constant e=2.718282.

0

5

10

15

20

25

0 1 2 3

* “my function” = “eigenfunction” in German. German is not a pre-requisite for ECE 598.* “my function” = “eigenfunction” in German. German is not a pre-requisite for ECE 598.

0

5

10

15

20

25

0 1 2 3

x(t) = ex(t) = ett

dx/dt = edx/dt = ett

t

t

x(t) (in meters)

dx/dt (in m/s)

Magical e Moves FasterMagical e Moves Faster

One way to make eOne way to make ett move move faster is to multiply it by a faster is to multiply it by a constant. The derivative constant. The derivative then gets multiplied by then gets multiplied by the same constant:the same constant:

x(t)=Aex(t)=Aett dx/dt = Ae dx/dt = Aett

The magical derivative The magical derivative property still holds:property still holds:

dx/dt=x(t)dx/dt=x(t)

0

5

10

15

20

25

0 1 2 3

0

5

10

15

20

25

0 1 2 3

x(t) = 2ex(t) = 2ett

dx/dt = 2edx/dt = 2ett

t

t

x(t) (in meters)

dx/dt (in m/s)

0

200

400

600

800

1000

0 1 2 3

Magical e Moves Magical e Moves MuchMuch FasterFaster

We can make eWe can make ett move move muchmuch faster by squaring faster by squaring it:it:

x(t)= (ex(t)= (ett))2 2 = e= e2t2t

dx/dt = 2edx/dt = 2e2t2t

In general, for any In general, for any constants A and constants A and ,,

x(t) = A ex(t) = A ett

dx/dt = dx/dt = A eA ett

0

100

200

300

400

500

0 1 2 3

x(t) = ex(t) = e2t2t

dx/dt = 2edx/dt = 2e2t2t

t

t

x(t) (in meters)

dx/dt (in m/s)

0

500

1000

1500

2000

2500

0 1 2 3

What About x(t)=10What About x(t)=10tt?? Define the “natural Define the “natural

logarithm” as follows:logarithm” as follows:y=ln(x)y=ln(x) if and only if x=e if and only if x=ey y

For example,For example,10=e10=eyy if and only if if and only if

y = ln(10) ≈ 2.3y = ln(10) ≈ 2.3

So x(t) = 10So x(t) = 10tt = (e = (eln(10)ln(10)))tt ≈ ≈ ee2.3t2.3t is a special case of is a special case of x(t)=ex(t)=ett, for the constant , for the constant =ln(10) ≈ 2.3, and its =ln(10) ≈ 2.3, and its derivative isderivative is

dx/dt dx/dt = ln(10) e= ln(10) eln(10)tln(10)t

= ln(10) 10= ln(10) 10tt

0

200

400

600

800

1000

1200

0 1 2 3

x(t) = 10x(t) = 10tt = e = e(ln10)t(ln10)t

dx/dt = ln(10) 10dx/dt = ln(10) 10tt

t

t

x(t) (in meters)

dx/dt (in m/s)

Review, Topic #4: CalculusReview, Topic #4: Calculus

x(t)x(t) dx/dtdx/dt

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

x(t) = A ex(t) = A ett dx/dt = dx/dt = A eA ett

Topic #5: SoundTopic #5: SoundCosine Changes in Air PressureCosine Changes in Air Pressure

When you hear a sound, it is because When you hear a sound, it is because the air pressure at your ear drum is the air pressure at your ear drum is changing in small rapid fluctuationschanging in small rapid fluctuations

Example: pure toneExample: pure tone

p(t) = Pp(t) = P00 + A cos( + A cos(t)t)

PP00 = Atmospheric pressure ≈ 10 = Atmospheric pressure ≈ 1055 Pa Pa

A = Sound pressure, in PascalsA = Sound pressure, in Pascals

Musical TonesMusical Tones

p(t) = Pp(t) = P00 + A cos( + A cos(t)t)

Pitch is related to Pitch is related to =2=2f. For example:f. For example: Middle A: f = 440 HzMiddle A: f = 440 Hz E#, a musical fifth above middle A: f = E#, a musical fifth above middle A: f =

660Hz660Hz A, a musical fourth above the E#: f = 880HzA, a musical fourth above the E#: f = 880Hz A, an octave above that: f = 1760 HzA, an octave above that: f = 1760 Hz

Pythagoras’ Other RulePythagoras’ Other Rule

Musical intervals are (nearly) defined by Musical intervals are (nearly) defined by integer frequency ratios:integer frequency ratios: ff22/f/f11=2: pitch(f=2: pitch(f22)=pitch(f)=pitch(f11) + one octave) + one octave ff22/f/f11≈3/2: pitch(f≈3/2: pitch(f22)=pitch(f)=pitch(f11) + musical fifth) + musical fifth ff22/f/f1 1 ≈≈ 4/3: pitch(f4/3: pitch(f22)=pitch(f)=pitch(f11) + musical ) + musical

fourthfourth ff22/f/f1 1 ≈≈ 5/4: pitch(f5/4: pitch(f22)=pitch(f)=pitch(f11) + major third) + major third ff22/f/f1 1 ≈≈ 6/5: pitch(f6/5: pitch(f22)=pitch(f)=pitch(f11) + minor third) + minor third ff22/f/f1 1 ≈≈ 9/8: pitch(f9/8: pitch(f22)=pitch(f)=pitch(f11) + one tone) + one tone ff22/f/f1 1 ≈≈ 18/17: pitch(f18/17: pitch(f22)=pitch(f)=pitch(f11) + semitone) + semitone

Pythagoras’ Other RulePythagoras’ Other Rule

Entertaining historical footnote: The followers of Entertaining historical footnote: The followers of Pythagoras attributed cosmic significance to the Pythagoras attributed cosmic significance to the integer ratios of consonant string lengths. They integer ratios of consonant string lengths. They developed a numerological system in which:developed a numerological system in which: 1 point = 1 point1 point = 1 point 2 points = 1 line2 points = 1 line 3 points = 1 plane3 points = 1 plane 4 points = 1 solid4 points = 1 solid The number pyramid 10=4+3+2+1 is a symbol of the The number pyramid 10=4+3+2+1 is a symbol of the

unity of creationunity of creation A musical fifth (a 3/2 ratio) is the harmony achieved by a A musical fifth (a 3/2 ratio) is the harmony achieved by a

plane intersecting a lineplane intersecting a line

Logarithms Turn Division Into Logarithms Turn Division Into SubtractionSubtraction

pitch(fpitch(f22)=pitch(f)=pitch(f11) + one octave:) + one octave:

ff22/f/f11=2=2

ln(f2/f1) = ln(2)ln(f2/f1) = ln(2)

ln(f2) – ln(f1) = ln(2)ln(f2) – ln(f1) = ln(2)

Number of octaves = Number of octaves = ln(fln(f22) – ln(f) – ln(f11))

ln(2)ln(2)

There are 12 Semitones in an There are 12 Semitones in an OctaveOctave

pitch(fpitch(f22=2f=2f11) = pitch(f) = pitch(f11) + 12 semitones:) + 12 semitones:

Number of semitones = (ln(fNumber of semitones = (ln(f22)-)-ln(fln(f11))))

1212

ln(2)ln(2)

The Semitone ScaleThe Semitone Scale

Let’s choose a global “reference note.” For Let’s choose a global “reference note.” For example, we could choose Aexample, we could choose A00, the lowest , the lowest note on a piano. Anote on a piano. A00 rings at a frequency of rings at a frequency of f=22.5Hz, or f=22.5Hz, or =2=2f=55f=55 radians/second. radians/second.

Any other note, p(t)=cos(2Any other note, p(t)=cos(2ft), is N ft), is N semitones above Asemitones above A11::

N = .N = .12 (ln(f/22.5))12 (ln(f/22.5))

ln(2)ln(2)

Example: a Full-Tone ScaleExample: a Full-Tone Scale

LoudnessLoudness

p(t) = Pp(t) = P00 + A cos( + A cos(t)t)

Perceived pitch is mostly determined Perceived pitch is mostly determined by by

Perceived loudness is mostly Perceived loudness is mostly determined by the amplitude, A.determined by the amplitude, A.

What Loudness Differences What Loudness Differences Can People Hear?Can People Hear?

As of 1900, the answer was: As of 1900, the answer was:

If xIf x11(t)=A(t)=A11cos(cos(t), and xt), and x22(t)=A(t)=A22cos(cos(t), t), most people can hear the difference in most people can hear the difference in loudness if and only ifloudness if and only if

loglog1010(A(A22/A/A11) > 1/20) > 1/20

20 log10(A20 log10(A22/A/A11) > 1) > 1

decibel measurement of decibel measurement of relative amplitudes and powersrelative amplitudes and powers If two signal powers differ by a factor of 10If two signal powers differ by a factor of 10L/10L/10, they , they

are L decibels apart (L dB):are L decibels apart (L dB):

AA2222/A/A11

22=10=10L/10 L/10

loglog1010(A(A2222/A/A11

22) = L/10) = L/10 If two signal powers differ by a factor of 10If two signal powers differ by a factor of 10L/10L/10, then , then

their amplitudes differ by a factor of 10their amplitudes differ by a factor of 10L/20L/20::

loglog1010( (A( (A22/A/A11))2 2 ) = L/10) = L/10

2 log2 log1010( A( A22/A/A11 ) = L/10) = L/10

Thus the ratio of two signals, in decibels, is given by Thus the ratio of two signals, in decibels, is given by eithereither

L = 20 logL = 20 log1010( A( A22/A/A11 ), or), or

L = 10 logL = 10 log1010( A( A2222/A/A11

2 2 ))

Sound Pressure LevelSound Pressure Level

For a pure tone p(t)=Acos(For a pure tone p(t)=Acos(t), sound t), sound pressure level can be defined to bepressure level can be defined to be

L = 20 log10 (PL = 20 log10 (PRMSRMS/P/PREFREF) ) PPRMSRMS = root mean square pressure. For a = root mean square pressure. For a

pure tone, recall that this is Ppure tone, recall that this is PRMSRMS = A/√2 = A/√2 PPREFREF, the reference pressure, has been , the reference pressure, has been

defined to be 0.00002 Pa (almost, but defined to be 0.00002 Pa (almost, but not quite, the softest audible sound).not quite, the softest audible sound).

SPL is written “XX dB SPL.”SPL is written “XX dB SPL.”

Examples of Sound Pressure Examples of Sound Pressure LevelLevel

SPLSPL DescriptionDescription

0 dB SPL0 dB SPL Threshold of HearingThreshold of Hearing

40 dB SPL40 dB SPL Whispering; Office Fan NoiseWhispering; Office Fan Noise

65 dB SPL65 dB SPL ConversationConversation

80 dB SPL80 dB SPL YellingYelling

120 dB SPL120 dB SPL Sensation ThresholdSensation Threshold

140 dB SPL140 dB SPL Pain TresholdPain Treshold

Review, Topic #5: SoundReview, Topic #5: Sound

Number of semitones = (ln(fNumber of semitones = (ln(f22/f/f11))))

Ratio of amplitudes, in dB = 20 Ratio of amplitudes, in dB = 20 loglog1010(A(A22/A/A11))

Sound Pressure Level = 20 Sound Pressure Level = 20 loglog1010(P(PRMSRMS/P/PREFREF))

1212

ln(2)ln(2)

Summary: What you have Summary: What you have learned todaylearned today

Dimensional analysisDimensional analysis Frequency, Period, Angular velocityFrequency, Period, Angular velocity Time shift, Phase shiftTime shift, Phase shift

All the calculus you’ll ever need to All the calculus you’ll ever need to knowknow

Exponentials (and faster exponentials)Exponentials (and faster exponentials) Logarithms (logLogarithms (logee and log and log1010)) Semitone scale for musical pitchSemitone scale for musical pitch Sound pressure level; decibelsSound pressure level; decibels