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DENNIS FREEMAN: Everybody would
like to believe that they just
simply sort of think about
the subject a little bit,
and it will come to them.
It's kind of like the
osmosis theory of learning.
But in fact, the way you
get good at something
is practicing.
It's the same for music.
It's the same for sports.
It's the same for academics.
So the way you get
good in this subject
is to work on the homework.
We'll give two
kinds of homework.
There will be conventional
kinds of problems followed
by engineering design problems.
The conventional
kinds of problems
are intended to be
the kinds of problems
that have simple, unambiguous,
easily checkable answers--
so well-defined problem,
single, unambiguous answer,
the kinds of problems that
you might be expecting
to see on an exam, for example.
But then we'll also have
engineering design problems.
Engineering design
problems are intended
to be a little more
ambitious, a little more
fun, a little harder, perhaps,
perhaps not completely
well-specified, sort of more
like the kind of a problem
that your boss might give
you after you've graduated.
So part of the exercise
will be figuring out
exactly how to convince
somebody that you've
got the right answer.
So those kinds of
problems are often--
it will often be the case
that numerics will help you
in making an argument.
So it often will be
the case that we'll
ask you to plot something.
Or maybe you just decide that a
plot is the most effective way
to communicate your result.
We are completely ambivalent
about what programming
language that you use.
Since 6.01 and 6.02
are prerequisites,
we assume you all know Python.
So all the examples that
I give in lecture handouts
or in homework solutions
will be in Python.
But if you'd rather use
Matlab, I just don't care.
OK, we're completely ambivalent
about what programming language
you use.
But it will be important
that in some of the problems
you'll find that useful
to be able to generate
a numerical kind of solution.
These are very different
kinds of problems.
And to help you
with both of them,
we have a different
idea about each.
So to help you with the
conventional problems, the kind
that have precise,
easy-to-check answers,
we've developed for the first
time ever over the summer
a tutor-type
environment for 6.003.
So this was intended to be
like the tutor environment
that we use in 6.01.
It's one of the more
popular aspects of 6.01,
the fact that you
can put in an answer
and hit the button
that says Check.
OK, so we're going
to have one of those.
So the idea is going to be that
in the case that we ask you
a simple question that
has a precise answer,
we'll use a tutor environment
to let you check your answer
to see if you got
the right answer.
In the case of the
engineering design problems,
that won't be the case.
It will not be the case
that it's very easy at all
to think about a computer
program that would check you.
So the alternative is
going to be that we will
have extended office hours.
So we're going to have something
that I think of as open
or block office hours.
Monday and Tuesday
afternoon and early evening,
we've reserved the
basement of Stata, 32-044,
for use of this class,
the idea being that that's
a nice space you
can just come there
to work any time you want to
hopefully because it's nice,
and hopefully because you all
know that there will be
other 6.003 people there,
you'll just decide that's
a good place to work.
And that's convenient because
in these engineering design
problems, if you run
into something that's
unclear to you, we
don't really want
you to spend an hour pondering
what was being asked.
So we want to make it
easy for you to get help.
And you can get help from
peers, from other students who
are there, but
we'll also make sure
that there is at least one
6.003 staff member there all
the time.
OK, so the homeworks
will be due on Wednesday.
That's the strategic value
of Mondays and Tuesdays.
So you can use that time
to get ready for turning
in the homework on Wednesday.
Questions or comments?
OK, we're required
to say something
about what we mean by
a collaboration policy.
So in this course, we would
like you to talk to each other
and help each other figure
out what's going on.
At the same time, we'd
like to reward you
for being on the honest side,
by which I'm going to mean,
if you sign your name to
something, you actually did it.
So we'd like to believe that
the homework that you turn in
under your name, you did.
You're perfectly welcome
to talk to each other,
to talk to the staff,
to talk to friends who
took this course previously.
Get all the help
you like in trying
to understand the concepts.
But when it comes down to
writing up your homework
and turning it in, we expect
that what you signed your name
to is something you did.
If you got especially
large amount of help
at the conceptual level, we'd
like you to acknowledge that.
You don't need to acknowledge
that the TAs helped you.
We expect that.
But if you collaborated
with somebody
to understand what
the problem was,
we'd like you to tell us that.
Just write at the top,
collaborated on figuring out
the concepts with so-and-so.
But we'd like you to have
written your homework.
So for example, we would
like you not to say,
I copied so-and-so's is
homework word for word
even though that's an honest
statement, that's not something
that we're actually looking for.
OK, is that clear?
We want you to work
together, but we
would like you to write
up your own solutions.
OK, we have firm deadlines.
The homeworks will always
be due on Wednesday.
You'll be able to slip
on one without penalty.
So if you turn in one
late, it won't change--
it won't have any effect
on your grade whatever.
If you do twice, that
could have an effect.
Unless you're excused by
a dean or an instructor
or a medical person,
that submission
will count half what it
would normally count.
I should have mentioned when
I was talking about homework,
homework 1 is already posted.
It will be due next Wednesday.
The online submission part
is almost ready to be posted
and will be posted
later this afternoon.
So right now, you can
see all the problems.
But the online will become
available this afternoon.
OK, here is 003 at a glance.
It's not too unlike what
you might have expected.
Just a couple of things to
point out, the order of coverage
is not the same as
the order of coverage
in Oppenheim and Willsky.
The reason for doing that
is that having taught
this class for 15
years, I just think
there's some subjects that
are easier starting points
than others.
So I'm doing what I
think is the easiest
entry to this
material with the idea
that after you've got the
easy stuff under your belt,
it's easier to move on to
the more difficult material.
So the order is not
precisely the same
as that in Oppenheim
and Willsky.
But there is a map on the
website that gives you
a week-by-week breakdown of what
part of Oppenheim and Willsky
are we working on right now.
So Oppenheim and Willsky
is the recommended text.
But it's not quite
in the regular order
of Oppenheim and Willsky.
See We will eventually cover
exactly the same material that
is in--
Oppenheim and Willsky was
written for this class.
We'll cover the
same material, just
not quite in the same order.
Another thing to
notice is exams.
We'll have three midterm
exams and a final.
The midterm exams are all
evening exams, 7:30 to 9:30
on Wednesday evening.
The idea is that
there's intended
to be a gentle ramp
through the exams.
Exam 1 is worth 10%.
Exam 2 is worth 15%.
Exam 3 is worth 20%.
The final exam is worth 40%
so the idea is that if you
have a mismatch on the way--
if you don't quite
understand the way
we're going to be
asking questions
and so forth, there's
not a big penalty
for screwing up the first exam.
That's the idea.
So there's intended
to be a gentle ramp
in how much of the exams count.
OK, and finally,
I'd like feedback
on what you think is happening.
So I would like to ask
for a few volunteers--
few is four or fewer--
who would be interested
to meet with me
once a week just to
tell me what you think.
Ideally, those people
would be kind of outgoing
and tell me things that
other people think too.
But if you're
completely introverted
and want to only tell me what
you think, that's fine too.
So this is an
opportunity for you
to tell the staff,
to give us feedback
on how things are going.
We're going too slow.
We're going too fast.
We're going too boring.
We're intensely too
interesting, that kind of stuff.
And it's an opportunity for you
to understand our perspective.
Well, the reason we
did that was blah.
And you can tell us, yeah,
but that's not important.
So it's an opportunity
for you to convey to us
how you think things are
going and how you think
things should be different.
It's also an opportunity for
you to learn about teaching.
So if you think about
teaching as a career,
it's a good thing to do--
completely voluntary,
has absolutely no effect
on your grade.
If you think you
might be interested,
it's probably going to meet on--
we'll probably meet
on Thursday afternoon.
But that's negotiable since
it's only four people.
And if you are interested,
please send me an email.
And the first four people
who ask, they're there.
OK?
That's it on administration.
Are there things
you'd like to know
about course
administration before we
go on to technical things?
Wonderful.
OK, this course
is about systems.
This is not the first course
you've had about systems.
You've had lots of
courses in systems.
Every physics course you've
ever taken is about systems.
What's different
about this course
is the way we think
about systems.
We think about systems
in a particular way that
turns out to be extraordinarily
powerful, useful,
shows up all the time.
And that's what the
focus of the course is.
That
The abstraction that
we're going to use
is we think about a
system as a thing that
has an input and an output.
Given the input and the system,
you can compute the output.
Systems have one
input and one output.
It's a very special
way of thinking
about a system that turns out
to be surprisingly powerful.
It's so powerful that that's
really the thing that we
will focus on in this class.
So it's the thing that I will
call the 6.003 abstraction.
That's going to be the
theme in everything we do,
that representation.
And the best way to
get across, I think,
how that's different from
other representations
that you're already
quite familiar with
is to just look at an example.
So here's a system that I'm
sure you all know all about.
Right?
This is not the
first time you've
seen this kind of a system.
So you all know how to do this.
Free-body diagrams,
F equals ma--
there's an enormous
number of ways
that you can analyze
such a system.
We're going to take a
very special approach
where we think about
the system as having
an input and an output.
OK, that's a little
bit arbitrary.
We'll say a little more in
a minute about how arbitrary
it is.
But the idea is going
to be that this system,
the mass-spring system,
the anvil with a spring,
there's an input, and
there's an output.
So for example, for
some particular purpose,
I might want to think
that the input is
the position of my hand.
One reason I might want
to think about that is it
is under my control.
So it might be
reasonable to think
about my hand being the input.
So for example, my
hand can do this.
Right?
So I might think about
the position of my hand
being the input.
So if I did that,
then what I could do
is characterize the input
by some waveform, a signal.
In order to use the
abstraction, I also
have to say what the output is.
I might be interested,
for example,
in knowing the
position of the mass.
Then the output would be the
displacement of the mass.
That assignment of input and
output is somewhat arbitrary.
I get to choose that to
make the problem that I'm
interested in as
easy as possible.
So for example, if I'm thinking
about this mass and spring
system here, using
my hand as the input,
that seems kind of
natural because that's
the thing I control.
Thinking about the position
of the mass as the output,
that seems reasonable
because that's
the thing you can
take a picture of.
It's the thing you can observe
from where you are now.
But it's not unique.
I might have wanted to
think about the force
that I'm putting into
it as in the input.
I might have wanted to
think about the output being
the acceleration of the mass.
Maybe there's some limit
to how much acceleration
some mass can take.
So you get to choose the
input and the output.
But you have to choose
an input and an output.
That's part of the abstraction.
And that's part of the power.
We'll see later that part of
the power of doing it this way
is that you draw attention
to the things of interest
and push into the background the
parts that are not of interest,
at least not now.
This kind of an abstraction
is an extremely versatile.
Here's a completely
different kind of problem.
Imagine water
flowing into a tank.
The tank is leaky,
so it flows out.
There's a second tank.
OK, what on earth could this be?
This could be--
maybe this is the--
it's appropriate today.
It's raining, or it was raining.
It was pouring.
Rain When I came
in, it was pouring.
Maybe this is nature
delivering rain.
This is the Woburn
reservoir from which
we get water in Boston.
Maybe this is the Fresh
Pond reservoir from which
we get water in Cambridge.
Maybe this is the
rate at which water
goes from Woburn to Fresh Pond.
Maybe this is the rate at
which water leaves Fresh Pond
and goes into consumers' houses.
The point is that
there's some abstraction.
There's some
physical-- actually,
I didn't say that right.
There's a physical thing--
rain, Woburn, Woburn reservoir,
Fresh Pond reservoir,
consumers' houses.
There's some physical thing.
There's some laws of
physics that dictate things.
But we ignore-- or ignore
is not quite the right way
to think about it.
We think about those
rules of physics
as rules that govern the
input-output relationship.
So we think about then
the entire system.
Rather than thinking about it
as reservoirs and rain and water
and flow and all
that sort of thing,
we think about it as
there's a signal in, rain,
and there's a signal
out, water usage.
And we take all the
details of the system
and bury it in this box.
It's an abstraction.
It's a way to suppress some
details to highlight others.
OK, and we use this in a
huge variety of situations.
We can think about a third
example here, our cell phone.
Cell phone is an enormously
complicated system.
But for the purpose of
understanding the input-output
characteristics, if what I
really want to know is how good
is the quality of the audio--
not that that's even a little
bit important for cell phones.
Right?
Cell phones are important for
texting and taking pictures
and have nothing
to do with voice.
But in the old days, cell
phones had to do with voice.
And I made this slide back
when that was still true.
So I apologize it's out of date.
So here, the important thing
of the cell phone system
was the sound in
and the sound out.
And the idea was to represent
the cell phone system
by the transformation between
how the sound comes in
and how the sound comes out.
OK, so we do this for an
enormous number of reasons.
One is that it's
widely applicable.
You can do this kind of a
characterization for systems
from electrical systems,
mechanical systems,
optical systems, acoustic
systems, biological systems,
financial systems,
all over the place.
That means that this
kind of an approach
is very powerful because
it's so widely applicable.
It also means that, for example,
every engineering discipline
everywhere has some
course like this.
It's just too
powerful not to do it.
So regardless of what
department you were in,
in the School of
Engineering, for example,
there is something like this.
Another reason
this is interesting
is that it provides a layer
of abstraction that lets
you focus on certain things.
So for example, let's expand
on the cell phone network.
Imagine that sound
comes into a cell phone.
Well, the job of the phone
is to communicate to a tower.
And that communication is via
some electromagnetic signal.
Then the job of the tower is to
communicate with another tower.
That happens in
all kinds of ways.
It could be a fiber optic link.
It could be a satellite link.
It could be electromagnetism.
It could be lots of
different things.
There's a lot of different
technologies for getting towers
to talk to towers.
Then towers talk
back to cell phones
via electromagnetic
signals just like these.
And the cell phone eventually
generates an acoustical output.
The idea is that by thinking
about the signals and systems
approach, we have abstracted
away everything other
than the flow of information.
It makes it easy to
concentrate, to follow
the flow of information
through a complex system.
Because we have
pushed every element
into the same framework--
every element has an
input and an output--
regardless of the
underlying substrate,
the analysis is similar.
That means that
components that are
characterized using
this abstraction
are easily combined.
So we refer to this as--
we say that these systems
are combinational.
Their combinational in
the same sense that Python
was combinational in 6.01.
If you represent things
by Python functions,
it's very easy to combine the
functions to have a bigger
function whose purpose and
details can be understood
without knowing what was inside
every individual function.
Similarly here, by knowing
the input behavior--
the input-output
behavior of the cell
phone, the input-output
behavior of the tower,
the input-output
behavior of this tower,
it's easy to compose then.
And in fact, we'll spend a
fair amount of time thinking
through the rules
of combination.
So that's kind of an overview of
the most important thing we're
going to talk about in this
class, which is the 6.003
abstraction, the
idea of representing
a system by the
way it transforms
an input into an output.
So with that kind of
overview, what I want to do
next is say a little more
about what are signals
and what are systems.
It's the signals and
systems abstraction.
We're going to need to know
some more details about what is
a signal and what is a system.
So basically, a signal is
just a mathematical function.
In all the examples I
talked about so far,
and in many of the
examples that we'll
talk about throughout
the term, the function
is a function of time.
And the signal can have
many different dimensions.
So for example, the
mass-spring system
evolved as a function of time.
The tank system, the
leaky tank system
evolved as a function of time.
The cell phone system
had acoustic signals
that were functions of time.
Time was the same in each case.
But the dependent variable
can be a variety of things.
Here it was position.
So here it was flow rates,
meter cubed per second of water.
Here it was perhaps pascals,
some unit of pressure
to characterize the
acoustic waveform.
So the point is that signals
are generally functions.
We'll see that just like
functions in mathematics,
there's a lot more to functions
than dependent variables
and independent variables.
And in fact, that's
going to be a key feature
of our analysis of signals.
But that's to come.
So for the time being, the
simplest and complete model
for what a signal is is
it's just a function.
The function doesn't have to
be a one-dimensional function.
In fact, a lot of the
interesting applications
of signals and
systems is to look
at multi-dimensioned functions.
So for example, my
research is in hearing.
I'm interested to study
how do the cells respond
to sound so that we can
understand how broken ears work
differently.
When you have a hearing
deficit, like I do,
when you have a
hearing deficit, what
is the manifestation of that
at a signal-processing level?
How do the cells
respond differently
to people who have impaired
hearing from people
who have normal hearing?
In order to study
that problem, we
take video pictures of the
cells at large magnifications
and watch them wiggle
when sounds hit them.
So that's a
picture-processing example.
So the signals, the independent
variable is not just time.
It's a picture.
So it might have x and y.
In fact, the pictures we
take are three dimensional.
So it has x, y, and z.
And in fact, they're four
dimensional because they
change with time.
We apply this kind
of a technique
on 4D signals-- x, y, z, time.
But it's still the property.
It still has a property
that the signals of interest
are functions.
We will be especially
interested in this class
in two distinct
representations of signals.
We will call them CT
signals and DT signals.
CT is continuous time.
DT is discrete time.
We're very interested in
that because we're engineers.
A lot of physics lives
in continuous time.
OK, the
mass-spring-dashpot system,
the signals were
functions of time.
Time was a continuously
varying independent variable.
So the leaky tank,
the cell phones, those
were all systems whose
signals evolve in time.
So they're all continuous.
You can start with a
second and break it in half
and get a half a second and a
half, and a half, and a half.
And there's no limit to how
many halves you can take.
By contrast, a lot of the
systems that we will look at
are things that evolve
in discrete time.
What's your bank account?
Well it only gets
computed once a day.
So it doesn't make sense to
talk about your bank account
at 9 AM and 2 PM.
Right?
The bank updates your
account once a day.
So it's something that
happens in discrete time.
It's something that happens
in all computational systems.
So computational
systems generally
operate on signals that are
functions of discrete time.
What was the state at time zero?
What was the state at time one?
What was the state at time two?
So the system is something that
eats a discrete time signal
and generates a discrete
time output signal.
And a unique part
of this class will
be converting between
the two representations.
Because as engineers,
we often want
to build something that
operates in the physical world--
masses and springs
and reservoirs
and that kind of stuff--
and do the processing
computationally.
Ever since the advent
of digital electronics,
it's just much easier.
Digital one, it's
just much easier
to process signals in
the digital domain.
So we will often
be interested in,
how do you represent
a signal who
naturally lives in
the physical world,
and is therefore a part--
whose signals are continuous
in nature, continuous time,
how do you convert it into
a discrete representation
so you can crunch
it on a computer?
And in fact, how do you go back?
So for example, if
we were thinking
about processing
audio signals, we
might want to think
about how we would
take a signal of
continuous time and turn it
into a discrete
time representation.
That's precisely what
we do when we want
to record something in MP3.
MP3 represents a sound.
A sound is something
that I think
of as a signal in
continuous time.
It's pascals as a
function of seconds.
But we want to represent
it by a sequence of numbers
because it's a sequence of
numbers that's easy to store,
to communicate, et cetera.
We do the same sort
of thing with images.
The images of the ear that I
talked about in my research
are things that I think about
happening at continuous space.
Every time we look at a
smaller and smaller dimension,
we get a different
answer because the image
has no quantization in space.
But when I crunch
it, I don't want
to work in something
in continuous space.
I want to work in a sampled
version like a JPEG version.
Similarly we want
to convert back.
I mean it would be useless
for the case of the cell phone
to convert it into a
discrete representation
and then not be able to hear it.
So we'll think about
the inverse process,
which is reconstruction.
If you had a discrete
representation of a signal,
how would you turn it into
a continuous representation?
And there's lots of ways people
do that here's a way that we
call zero-order hold
where you convert
every sample into a
corresponding voltage
and just hold that voltage until
the next sample comes along.
This is the representation
that is most commonly
used in things like MP3.
By contrast, we might
do something cleverer.
We might literally
extrapolate between samples.
That is commonly done
in picture processing.
The reason we use the
two different schemes is
that psychophysically,
the things that
are important to your
eyes and the things that
are important to your
ears are different.
So you would hear
the errors more
if you did the
linear interpolation.
And you would see
the errors more
if you did the
other interpolation.
And we'll do lots of examples
later on in the course
so you see why that's true.
OK, too much talking,
not enough thinking--
so I've told you a
lot about signals.
Now, I'd like you to
think about some signals.
And so a common thing I'm
going to do in this class
is ask you to think
about a question,
talk to your neighbor, come
to a consensus, and then vote.
So that's what we're
going to do here.
I'm going to play some sounds.
You're going to try to
figure out what signals were
represented by those sounds.
But I would like you to have an
opinion before we communicate
the answer because you
can prove that people
get more out of it when
there's something at stake.
So I would like you to
work with a partner.
So in order to work with a
partner, everybody stand up.
Introduce yourself
to your neighbor.
Figure out a good partner.
Figure out where they live.
[CLASSROOM CHATTER]
OK, so what I'm going
to do now is play
a computer-generated sound.
This is a computer-generated
speech sound
made by Bob Donovan who did
his PhD thesis figuring out
how to do
computer-generated speech.
And then I'm going to play four
transformations of that sound.
And your task is
to identify what's
the mathematical representation
of the transformations
that I'm telling you.
OK, so first, the original.
First is the signal
that Bob Donovan made.
COMPUTER-GENERATED
SPEECH: I must apologize
for speaking [INAUDIBLE].
But you see, I have no brain.
DENNIS FREEMAN:
And now, I'm going
to play a sequence of
four transformations--
one, two, three, four.
And the question
is going to be is
this the correct mathematical
representation for this sound.
The sound you just
heard was f of t.
Is the following sound,
which by definition is f1,
the same as f of 2t?
COMPUTER-GENERATED
SPEECH: [INAUDIBLE].
DENNIS FREEMAN: You're
allowed to talk.
[CLASSROOM CHATTER]
OK, remember that answer.
Now, is the following
sound minus f of t.
COMPUTER-GENERATED
SPEECH: [INAUDIBLE].
[CLASSROOM CHATTER]
DENNIS FREEMAN: OK,
remember that answer.
Is the following sound f of 2t?
COMPUTER-GENERATED
SPEECH: [INAUDIBLE].
[CLASSROOM CHATTER]
DENNIS FREEMAN: And finally,
is the following signal
one third of f of t?
COMPUTER-GENERATED
SPEECH: I must apologize
for speaking [INAUDIBLE].
But you see, I have no brain.
DENNIS FREEMAN: OK,
confer with your partner.
Figure out how many are true.
Because in 30 seconds, I'm going
to ask you to raise your hand.
[CLASSROOM CHATTER]
Question or?
AUDIENCE: Question,
f of t [INAUDIBLE]?
DENNIS FREEMAN: f of t
was the original signal.
AUDIENCE: I know,
but like [INAUDIBLE].
DENNIS FREEMAN: Ah,
pressure, pressure
in the acoustic waveform.
AUDIENCE: If you were
to have minus f of t,
your ear wouldn't [INAUDIBLE].
DENNIS FREEMAN: That's true.
OK, how many of those
statements are true?
Raise your hand with some number
of fingers, keeping in mind
that you're voting having
collaborated with your partner.
So it's not your
fault. You have a--
right, anyway.
OK, so how many?
Everybody raise your hands.
I want to see.
OK, it's about 90%--
no, less than that
about 85% correct.
OK, so first one,
was that f of 2t?
AUDIENCE: Yes.
DENNIS FREEMAN: Why
do you think that?
Everybody seems
to be saying yes.
It was fast.
It sounded faster.
So what's faster mean from
a signal point of view?
AUDIENCE: [INAUDIBLE].
DENNIS FREEMAN:
Change the time scale.
That's a very good way
of thinking about it.
So if we thought about--
if we made a simple
representation for f--
ignore what it really is.
Let's say it's that.
What would f1 look like in
order to make it sound faster?
AUDIENCE: [INAUDIBLE].
DENNIS FREEMAN: OK, we
have people doing this.
And we have people doing this.
And I'm not quite
sure what that means.
So could somebody be a
little more descriptive?
Hand gestures are fine.
Yes.
AUDIENCE: Squished.
DENNIS FREEMAN:
Squished, exactly right.
So what we want to have happen,
in order to sound faster,
we want to have it
come out faster.
So we would like
that to happen to it.
So it's all over in one
second if the original
was all over in two seconds.
Right?
That's what faster means.
So we would like
that to be true.
Is that what's going
on in that function?
Yeah.
OK, so that's right.
Secondly-- yes.
AUDIENCE: Is that the reason
also why it was higher pitched?
DENNIS FREEMAN: That also
makes it higher pitched.
In fact, we will talk
a lot about this later.
But pitch has to do with a
related kind of waveform.
If I thought about a waveform
that was a single tone,
that might look like this.
So if you were to play an oboe
and play just a C constantly,
you'd get some
periodic waveform.
If I did the same transformation
on that periodic waveform,
what would happen
to that waveform?
What would happen to this
if I did transformation one?
Squish, what's squish mean?
More higher frequency,
more cycles per second.
Yeah.
AUDIENCE: So how can you make
the same waveform go twice as
fast, but have the
same pitch [INAUDIBLE]?
DENNIS FREEMAN:
That's very hard.
In fact, I worked on that
as a research project.
So one of the
techniques that we used
to try to fix people's ears--
it turns out that
people who have
hearing disorders like mine
can more easily understand
male speech than female
speech just because
of the shift in frequencies.
Male speech is primarily
about an octave lower
on average than female speech.
So we tried to
convert every female
into a male speaker
and every male
into the Jolly Green Giant.
And it worked.
Males sounded like
the Jolly Green Giant.
And females sounded like males.
And it didn't help
hearing at all.
And that's primarily
because it's a hard problem.
But we will say more about that
as we go on through the course.
That is a hard problem
because you end up
having to do other
than just squish.
OK, second one, is this is
the right transformation?
No, why not?
AUDIENCE: [INAUDIBLE]
so most of the speech
is measured in the
frequency domain.
But if you just flip
over the amplitude,
it shouldn't really
make a difference
DENNIS FREEMAN: If you
flipped over the amplitude--
so flip over the
amplitude, you're
saying that didn't do it.
What was the transformation?
If it wasn't flip over
the amplitude, what is it?
Yeah.
AUDIENCE: f of minus t.
DENNIS FREEMAN: f of minus t--
it was flipped this way.
Right, flipping this way
is largely inaudible.
If you had a pressure waveform
versus a rarefaction waveform,
positive pressures versus
negative pressures,
it's very hard to tell.
Not impossible, if you're one
of those audio types like me,
you can probably
hear the difference.
But it's very hard.
Hearing this, not hard.
OK, how about this one?
Obviously wrong because
this one was right.
OK, how about this one?
Yeah.
So the idea was that there
were two that were correct.
So the 85% of you or so
who got two, presumably
you got the right two.
Right?
But I won't ask.
OK, here's an image
processing question.
Think about this
picture of Stata.
I know it's hard to think
about pictures of Stata.
That's OK.
Think about this
picture is Stata.
I have indexed x and y.
And I've written that
as f, a signal that
depends on two independent
variables, x and y.
And I've got three
transformations--
the f1 transformation,
the f2, and the f3.
The question is, is this
the right mathematical
representation for transforming
that picture into this one?
Is this the one for that one?
And is this a
representation for this one?
Is this the representation
for that one?
Take 30 seconds,
talk to your partner,
and figure out how many of those
transformations are correct.
[CLASSROOM CHATTER]
OK, so how many of the
transformations are correct?
Raise your hand.
Everybody raise your hand.
That's a zero or a three?
OK, just checking.
OK, about 85% correct again.
OK, probably a different 85%.
So how do you think
about this problem?
So there's a variety of ways
you could think about it.
Let me show you a
way that focuses
on an idea called
mapping because I
think it's a very powerful way.
So if you think about
a map, how does the t
variable in the
first problem map
to the t variable in
the second problem?
So a way you can
think about this
is if this is true for
all x and y, it's true
for a particular x.
So let's ask is it
true for x equals 0.
So how does the point x equals
0 map from one image to another?
So if you substituted x
equals 0, then f1 of 0--
so if you look at
this transformation,
the claim would be that f1
of 0 is the same as f of 0.
Is that true?
So f1 of 0 bisects Stata.
f of 0 bisects Stata.
So yeah, that's the right thing.
Everybody see that?
Is the statement true
for x equals 250?
Well, f1 of 250--
so f1 of 250 is through
the right-hand side
of Stata sort of where the steel
and the bricks come together.
Is that the same as
substituting in here, if x is
250, then f of 500--
f of 500 is off the screen.
So is that transformation hold
at the point x equals 250?
No.
OK, it can't be right.
OK?
One bad sample
proves that it can't
be the general transformation.
OK, everybody see that?
Similarly, for the
second transformation,
let's just try some points.
If we try the point x equals 0,
this says that f2 at location 0
should be f at
location minus 250.
Is that correct?
So f of 0, well, that's the
left-hand side of Stata.
And so f2 of 0, that's the
left-hand side of Stata. f
of 2 times 0 minus 250
is f of 250 minus 250,
that's also the left of Stata.
OK, so by using
that reasoning, you
can go through and see
if certain points map
to the corresponding place
where they were supposed to go.
That's the idea of mapping.
And in this particular case,
if we traced two points,
and the map 2x minus
250 is a linear function
of x, so two points
determine a straight line.
So after you've determined
two points, you're done.
Seem right?
So similarly, you can figure
out that the particular points
didn't work here.
And so the answer is that only
one of those transformations
was right.
OK, so I spent a fair
amount of time on signals.
In the last minute,
oh, minute, no,
I'm not going to
do that am I. OK,
I spent a fair amount of
time talking about signals.
Next time, we'll talk about
similar ways of thinking
about simple systems.
So have a good weekend.
