ECE 598: The Speech Chain

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ECE 598 The Speech Chain

• Lecture 1 Dimensional Analysis Cosines
• Logs and Exponentials

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in todays lecture, you will learn

• About this course
• Goals of this course
• Teaching style homework, labs and exams
• The language of acoustics
• Dimensional analysis
• Cosines frequency, phase, and amplitude
• All of the calculus you will ever need to
  know---in one slide!
• Musical scales, decibels, and other logarithmic
  things

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the goals of this course

• Acoustics
• Loudness Pitch
• Noise Reverberation
• Frequency Response
• Speech
• What sounds can the mouth produce? Why?
• Language
• Information defined as the unpredictable part
  of an utterance

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pre-requisites

• I assume that you
• have never taken calculus
• know that ylog10x means x10y, but would
  appreciate some practice
• know what cos(x) looks like, but would
  appreciate some practice
• know Newtons second law (fma) but would
  appreciate some practice
• know the ideal gas law (PVnRT) but would
  appreciate some practice

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how will this course be taught?

• Lectures Mondays 1-4 PM, Siebel 1105
• 100-130 Quiz (every week)
• 130-145 Break
• 145-245 New material (slides)
• 245-300 Break
• 300-400 Sample problems (on the whiteboard)
• Office Hours
• Daily, 930-1100, 2011 Beckman
• Every Monday, after lecture, 4-5 PM in the Siebel
  cafe

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how will this course be taught?

• Labs
• Weekly, 2 hours times TBA
• Written homework
• Weekly first one is due Monday!
• Exams and Quizzes
• Weekly 30-minute quizzes
• One final exam
• Purpose of the weekly quizzes
• If many people have trouble with a concept, I
  want to know about it RIGHT AWAY so I can CHANGE
  THE LECTURE.

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Today Units, Cosines, and Logs

• Units
• In acoustics, many problems can be solved by
  paying careful attention to the way in which
  something is measured.
• Cosines
• Every measured signal can be expressed as the sum
  of shifted, scaled cosines. More on this amazing
  fact in lecture 4. For the time being, lets
  understand more about cosines.
• Logarithms
• Musical pitch log (frequency)
• Sound pressure level log (pressure)

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a warning

• Todays lecture (and Mondays lecture) will seem
  like pure math with no acoustics.
• Thats because, in order to speak the language
  of acoustics (specifically, the language in
  which Acoustic Phonetics is written), we need
  two lectures of math.
• If you feel lost or abandoned, please come to
  office hours.

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The Five Fundamental Units (MKS)

• Length 1 meter
• Mass 1 kilogram
• Time 1 second
• Temperature 1 degree Kelvin
• Angle 1 radian
• All other acoustic units can be expressed as
  combinations of these five fundamental units.
• Many problems can be solved by just figuring out
  the units used on both sides of an equation.

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Units The Metric System

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

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Units fun conversions

• 1 year 31.557 Ms (megaseconds)
• 1 day 86.4 ks (kiloseconds)
• 0 degrees K -273 degrees C
• 1 radian 57o
• 1 cycle 2p radians (once around the circle)

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Why radians?

• A one-radian slice of cake is a slice with the
  same length on all three sides (two straight
  sides and one curved side).
• Why does cycle 2p radians?
• Because circumference C2pr

r meters
angle 1 radian
r meters
r meters
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Dimensional Analysis

• A little girl sits on the outermost edge of a
  merry-go-round. The merry-go-round has a radius
  of 2m. It is rotating at 0.5 radians/second.
  How fast is the little girl traveling, in
  meters/second?
• (0.5 second) ? (2 radian)

radians
meters
meters
1 second
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Dimensional Analysis

• A merry-go-round is spinning at p/10
  radians/second (about 0.314 radians/second).
  How long before it spins 1 cycle?
• Goal seconds/cycle.

seconds
1 second
2(10)
2p radians
20

cycle
p/10 radians
1 cycle
1
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Dimensional Analysis Newtons Law

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

f ma
5
15 m/s
kg m
(1 kg)
5
5 N
3 s
s2

• 1 Newton (N) is just a shorthand for 1 kg m/s2

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Dimensional Analysis Newtons Law

• A pencil has mass of 0.1kg (100g). Gravity
  accelerates the pencil at 10 m/s2 toward the
  center of the Earth. What is the force of
  gravity on the pencil?

f ma
(0.1 kg)(10 m/s2)
1 kg m/s2 1N
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Useful Composite Units in Acoustics

• Frequency 1 Hz 1 cycle/second (Hertz)
• Audible sound is roughly 10 kHz gt f gt 10 Hz
• Density 1 kg/m3 1 mg/(cm)3
• Air is about 1.29 mg/(cm)3
• Force 1 N 1 kg m/s2
  (Newtons)
• The force of gravity on a pencil
• Pressure 1 Pa 1 N/m2
  (Pascals)
• Air pressure at sea level is about 100,000 Pa
• Audible pressures are usually 20mPa lt p(t) lt 20Pa
• Energy 1 J 1 N m
  (Joules)
• Burning 1kcal of sugar yields 4184 J
• Power 1 W 1 J/s 1 N m/s (Watts)
• The heat given off by one human 100W

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Review, Topic 1 Dimensional Analysis

• There are five fundamental units length (m),
  mass (kg), time (s), temperature (deg K), and
  angle (radians)
• Every other unit can be expressed in terms of
  these fundamental units
• Units can be divided and canceled just like
  anything else in an equation
• Carefully evaluating the units is called
  dimensional analysis.
• Many important problems in acoustics have been
  solved using only dimensional analysis.

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More about cosines

• Every time you see a formula involving cosines,
  please remember
• There is a circle hiding behind it.
• Here is the circle.
• It is centered at (0,0).
• It has a radius of r.
• It has a point, somewhere on its edge, called
  (x,y).
• f is the angle between (x,y) and the positive x
  axis.

This point is called (x,y)
r
y
f
x
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sohcahtoa
opposite
y

• sinf
• cosf
• tanf

hypotenuse
r
adjacent
x
hypotenuse
r
r
y
opposite
y
f
x
adjacent
x
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Circles and Triangles

• xr cosf
• yr sinf
• y/x tanf
• x2y2r2, so
• cos2f sin2f 1

This point is called (x,y)
sinf
cosf
r
y
f
p/2
x
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If Daughter is on a Merry-Go-Round, Daddys Walk
is a Cosine

• Imagine a girl going around a merry-go-round
  with a radius of A meters. As she moves, her
  daddy walks back and forth in order to stay level
  with her. His position is x(t) A cos(wt).
• w is called the angular velocity, in
  radians/second.

Daughters movement circle of radius A
Daddys movement x(t) A cos(wt)
t
Daddy moves back and forth rather like a pendulum
x(t)
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Frequency 1/PeriodAngular Velocity 1/(Time
Constant)
x(t)
x(t)
t
t

• x(t) cos(t)
• Period
• T 2p 6.28 seconds/cycle
• Frequency
• f 1/T 1/2p cycles/second
• Angular Velocity
• w 1 radian/second
• Time Constant
• 1/w 1 second/radian

• x(t) 3cos(2t)
• Period
• T p 3.14 seconds/cycle
• Frequency
• f1/T 1/p cycles/second
• Angular Velocity
• w 2 radians/second
• Time Constant
• 1/w 0.5 seconds/radian

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Root Mean Square (RMS)

• x(t) A cos(wt)
• A is called the amplitude or maximum
  amplitude of x(t).
• A has the same units as x(t).
• The average value of a cosine is always Ex(t)0.

x(t)
t
t

• A more useful average amplitude is the root
  mean square, defined as the square root of the
  mean of the square of x(t),
• xRMS vEx2(t)
• For a cosine, xRMS turns out to be
• xRMS A/v2 0.707
  A

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Review, Topic 2 Cosines

• If a little girl is going around a
  merry-go-round, and her daddy is following in
  front of her, then his movement is
  x(t)Acos(wt-f)
• Well come back to the f part in a minute
• w 2pf is called the angular velocity
• A is called the amplitude
• xRMS 0.707A is called the RMS amplitude

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Time Delayor what if the merry-go-round starts
later?
x(t)
t
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Time Advanceor what if the merry-go-round
starts earlier?
x(t)
t
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General Form of a Time Shift
x(t)
t

• To delay by t seconds, write
• x(t) A cos( w
  (t - t) )
• To advance by t seconds, write
• x(t) A cos( w
  (t t) )

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Phase Shift
x(t)
t

• Phase Shift and Time Shift are different
  words for exactly the same thing
• x(t) A cos( w(t-t) ) A cos( wt
  f ), fwt
• What are the units of f?
• (w radians/second) (t seconds)
  f radians

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Phase Shift Example

• Suppose that the little girls starting position
  is at an angle of -p/2 radians (relative to the
  x axis)
• Then daddys position is
• x(t)A cos(wt - p/2)

Suppose the little girl starts here
so daddy starts here
t
and his position is a cosine, delayed by p/2
radians.
x(t)
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Four Particularly Interesting Phase Shifts p/2,
p, 3p/2, and 2p

cos(wt-p/2) sin(wt) !!!
cos(wt-p) -cos(wt) !!!
cos(wt-3p/2) -sin(wt) !!!
cos(wt-2p) cos(wt) !!!
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Review, Topic 3 Phase Shift

• Phase shift Time shift
• Time shift in seconds
• Phase shift (radians/second)?(seconds)
  radians
• Four particularly interesting phase shifts
• Shift by p/2 radians (1/4 cycle) cosine turns
  into sine
• Shift by p radians (1/2 cycle) cosine turns into
  negative cosine
• Shift by 3p/2 radians (3/4 cycle) cosine turns
  into negative sine
• Shift by 2p radians (1 full cycle) what happened?

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Girl on the Merry-Go-Round How Fast is She
Moving?
radians
meters
meters
wA second
w second
? A radian
w radians/second
wA meters/second
A meters
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Girl on the Merry-go-Round How Fast is Daddy
Moving?

• Fastest Daddy moves as fast as Daughter when
  shes passing in front of him
• x(t) 0
• v(t) wA
• Slowest Daddy stops and turns around when
• x(t) A
• v(t) 0 m/s
• At all times
• x(t) A coswt
• v(t) - wA sinwt

t
t
x(t)
v(t)
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What is Differential Calculus?

• dx/dt is calculus notation for the rate at
  which x(t) is changing.
• 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 J(t) is energy, then W(t)dJ/dt is power.
• You do not need differential calculus in order to
  take ECE 598. You only need to memorize two x(t)
  dx/dt pairs the rest of the course will
  proceed just fine using only these two pairs.
  They are given on the next slide.

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All the Calculus Youll Ever Need to Know, on One
Slide
A, w, f, and a can be replaced by any expression,
no matter how horrible and complicated, as long
as the expression doesnt contain any t.
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Magical et dx/dtx(t)
x(t) (in meters)

• The function x(t)et is called the my function
  of calculus, because
• x(t)et is the only function with the following
  property
• x(t)et dx/dt x(t)
• The property dx/dtx(t) only works for the
  constant e2.718282.

t
x(t) et
dx/dt (in m/s)
t
dx/dt et
my function eigenfunction in German.
German is not a pre-requisite for ECE 598.
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Magical e Moves Faster
x(t) (in meters)

• One way to make et move faster is to multiply it
  by a constant. The derivative then gets
  multiplied by the same constant
• x(t)Aet dx/dt Aet
• The magical derivative property still holds
• dx/dtx(t)

t
x(t) 2et
dx/dt (in m/s)
t
dx/dt 2et
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Magical e Moves Much Faster
x(t) (in meters)

• We can make et move much faster by squaring it
• x(t) (et)2 e2t
• dx/dt 2e2t
• In general, for any constants A and a,
• x(t) A eat
• dx/dt aA eat

t
x(t) e2t
dx/dt (in m/s)
t
dx/dt 2e2t
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What About x(t)10t?

• Define the natural logarithm as follows
• yln(x) if and only if xey
• For example,
• 10ey if and only if
• y ln(10) 2.3
• So x(t) 10t (eln(10))t e2.3t is a special
  case of x(t)eat, for the constant aln(10)
  2.3, and its derivative is
• dx/dt
• ln(10) eln(10)t
• ln(10) 10t

x(t) (in meters)
t
x(t) 10t e(ln10)t
dx/dt (in m/s)
t
dx/dt ln(10) 10t
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Review, Topic 4 Calculus
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Topic 5 SoundCosine Changes in Air Pressure

• When you hear a sound, it is because the air
  pressure at your ear drum is changing in small
  rapid fluctuations
• Example pure tone
• p(t) P0 A cos(wt)
• P0 Atmospheric pressure 105 Pa
• A Sound pressure, in Pascals

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Musical Tones

• p(t) P0 A cos(wt)
• Pitch is related to w2pf. For example
• Middle A f 440 Hz
• E, a musical fifth above middle A f 660Hz
• A, a musical fourth above the E f 880Hz
• A, an octave above that f 1760 Hz

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Pythagoras Other Rule

• Musical intervals are (nearly) defined by integer
  frequency ratios
• f2/f12 pitch(f2)pitch(f1) one octave
• f2/f13/2 pitch(f2)pitch(f1) musical fifth
• f2/f1 4/3 pitch(f2)pitch(f1) musical
  fourth
• f2/f1 5/4 pitch(f2)pitch(f1) major third
• f2/f1 6/5 pitch(f2)pitch(f1) minor third
• f2/f1 9/8 pitch(f2)pitch(f1) one tone
• f2/f1 18/17 pitch(f2)pitch(f1) semitone

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Pythagoras Other Rule

• Entertaining historical footnote The followers
  of Pythagoras attributed cosmic significance to
  the integer ratios of consonant string lengths.
  They developed a numerological system in which
• 1 point 1 point
• 2 points 1 line
• 3 points 1 plane
• 4 points 1 solid
• The number pyramid 104321 is a symbol of the
  unity of creation
• A musical fifth (a 3/2 ratio) is the harmony
  achieved by a plane intersecting a line

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Logarithms Turn Division Into Subtraction

• pitch(f2)pitch(f1) one octave
• f2/f12
• ln(f2/f1) ln(2)
• ln(f2) ln(f1) ln(2)
• Number of octaves

ln(f2) ln(f1)
ln(2)
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There are 12 Semitones in an Octave

• pitch(f22f1) pitch(f1) 12 semitones
• Number of semitones (ln(f2)-ln(f1))

12
ln(2)
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The Semitone Scale

• Lets choose a global reference note. For
  example, we could choose A0, the lowest note on a
  piano. A0 rings at a frequency of f22.5Hz, or
  w2pf55p radians/second.
• Any other note, p(t)cos(2pft), is N semitones
  above A1
• N .

12 (ln(f/22.5))
ln(2)
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Example a Full-Tone Scale
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Loudness

• p(t) P0 A cos(wt)
• Perceived pitch is mostly determined by w.
• Perceived loudness is mostly determined by the
  amplitude, A.

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What Loudness Differences Can People Hear?

• As of 1900, the answer was
• If x1(t)A1cos(wt), and x2(t)A2cos(wt), most
  people can hear the difference in loudness if and
  only if
• log10(A2/A1) gt 1/20
• 20 log10(A2/A1) gt 1

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decibel measurement of relative amplitudes and
powers

• If two signal powers differ by a factor of
  10L/10, they are L decibels apart (L dB)
• A22/A1210L/10
• log10(A22/A12) L/10
• If two signal powers differ by a factor of
  10L/10, then their amplitudes differ by a factor
  of 10L/20
• log10( (A2/A1)2 ) L/10
• 2 log10( A2/A1 ) L/10
• Thus the ratio of two signals, in decibels, is
  given by either
• L 20 log10( A2/A1 ), or
• L 10 log10( A22/A12 )

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Sound Pressure Level

• For a pure tone p(t)Acos(wt), sound pressure
  level can be defined to be
• L 20 log10 (PRMS/PREF)
• PRMS root mean square pressure. For a pure
  tone, recall that this is PRMS A/v2
• PREF, the reference pressure, has been defined to
  be 0.00002 Pa (almost, but not quite, the softest
  audible sound).
• SPL is written XX dB SPL.

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Examples of Sound Pressure Level
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Review, Topic 5 Sound
12

• Number of semitones (ln(f2/f1))
• Ratio of amplitudes, in dB 20 log10(A2/A1)
• Sound Pressure Level 20 log10(PRMS/PREF)

ln(2)
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Summary What you have learned today

• Dimensional analysis
• Frequency, Period, Angular velocity
• Time shift, Phase shift
• All the calculus youll ever need to know
• Exponentials (and faster exponentials)
• Logarithms (loge and log10)
• Semitone scale for musical pitch
• Sound pressure level decibels