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[MUSIC PLAYING]
PROFESSOR: We concluded the
last lecture with the
statement of the sampling
theorem.
And just as a quick reminder,
the sampling theorem said that
if we have a continuous-time
signal and we have equally
spaced samples of that signal,
sampled at a sampling period,
which I indicate is capital
T and if x of t is
band-limited-- in other words,
the Fourier transform is zero
outside some band where omega
sub m is the highest
frequency-- then under the
condition that the sampling
frequency, which is 2 pi divided
by the period, is
greater than twice the
highest frequency.
The original signal is uniquely
recoverable from the
set of samples.
And the sampling theorem
essentially was derived by
observing or using the notion
that sampling could be done by
multiplication or modulation
with an impulse train.
And the sampling theorem
developed by examining the
consequence of the modulation
property in the context of the
Fourier transform.
In particular, if we have
our signal x of t and if
multiplied by an impulse train
to give us a sampled signal--
another impulse train whose
values or areas are samples of
the original time function, as
I indicate here-- then in
fact, if we examine this
equation or equivalently,
bringing x of t inside this sum,
if we examine either of
these equations in the frequency
domain, the Fourier
transform of x of p of t is the
convolution of the Fourier
transform of the original
signal and the Fourier
transform of the
impulse train.
Now the impulse train is
a periodic signal.
It's Fourier transform.
Therefore, as we talked about
with Fourier transforms is
itself an impulse train.
And when we do this convolution,
then using the
fact that the Fourier transform,
the impulse train
is an impulse train.
The result of this convolution,
then tells us
that the Fourier transform of
the sample signal or the
impulse train, which represents
the samples, is a
sum of frequency-shifted
replications of the Fourier
transform of the original
signal.
So mathematically, that's
the relationship.
It essentially says that after
sampling or modulation with an
impulse train, the resulting
spectrum is the original
spectrum added to itself,
shifted by integer multiples
of the sampling frequency.
Well, let's see that
as we did last
time in terms of pictures.
And again, to remind you of the
basic picture involved, if
we have an original signal with
a spectrum as I indicated
here-- where it's band-limited
with the highest frequency
omega sub m-- and if the time
function is sampled so that in
the frequency domain we convolve
this spectrum with
the spectrum shown below, which
is the spectrum of the
impulse train, the convolution
of these two is then the
Fourier transform or spectrum
of the sample time function.
And so that's what we
end up with here.
And then as you recall, to
recover the original time
function from this-- as long as
these individual triangles
don't overlap--to recover it
just simply involves passing
the impulse train through a
low-pass filter, in effect
extracting just one of
these replications of
the original spectrum.
So the overall system then for
doing the sampling and then
the reconstruction of the
original signal from the
samples, consists of multiplying
the original time
function by an impulse train.
And that gives us then
the sampled signal.
The Fourier transform I show
here of the original signal
and after modulation with the
impulse train, the resulting
spectrum that we have is that
replicated around integer
multiples of the sampling
frequency.
And then finally, to recover
the original signal or to
generate a reconstructed signal,
we then multiply this
in the frequency domain by the
frequency response of an ideal
low-pass filter.
And what that accomplishes for
us then is recovering the
original signal.
Now in this picture, an
important point that I raised
last time, relates to the
fact that in doing the
reconstruction--well we've
assumed-- is that in
replicating these individual
versions of the original
signal, those replications
don't overlap and so by
passing this through a low-pass
filter in fact, we
can recover the original
signal.
Well, what that requires is that
this frequency, omega sub
m, be less than this
frequency.
And this frequency is omega
sub s minus omega sub m.
And so what we require is that
the frequency omega sub m be
less than omega sub s
minus omega sub m.
Or equivalently, what we require
is that the sampling
frequency be greater than twice
the highest frequency in
the original signal.
Now, if in fact that condition
is violated, then we end up
with a very important effect.
And that effect is referred
to as aliasing.
In particular, if we look back
at our original example--we
are here-- we were able to
recover our original spectrum
by low-pass filtering.
If in fact the sampling
frequency is not high enough
to avoid aliasing, then what
happens in that case is that
the individual replications of
the Fourier transform of the
original signal overlap and what
we end up with is some
distortion.
As you can see, if we try to
pass this through a low-pass
filter to recover the original
signal, in fact we won't
recover the original signal
since these individual
replications have overlapped.
And this is the case where omega
sub s minus omega sub m
is less than omega sub s.
In other words, the sampling
frequency is not greater in
this case than twice the
highest frequency.
So what happens here then is
that in effect, higher
frequencies get folded down
into lower frequencies.
What would come out of the
low-pass filter is the
reflection of some higher
frequencies into lower
frequencies.
As I suggested a minute
ago, that effect is
referred to as aliasing.
And in order to both understand
that term better
and to understand in fact the
effect better, it's useful to
examine this a little more
closely for the specific
example of a sinusoidal
signal.
So let's concentrate on that.
And what we want to look at is
the effect of aliasing when
our input signal is a
sinusoidal signal.
Now to do that, what I want
to show shortly is a
computer-generated movie
that we've made.
And let's first walk through a
few frames of it to give you--
first of all, to set up our
notation and to suggest what
it is that we're trying
to demonstrate.
Well, what we have is
an input signal-- is
a sinusoidal signal.
And the spectrum or Fourier
transform of that is an
impulse in the frequency
domain at the
frequency of the sinusoid.
We then have samples of that and
when we sample that-- and
for this particular example,
it's sampled at 10 kilohertz--
this spectrum is then replicated
at multiples of the
sampling frequency.
And I haven't shown negative
frequencies here, but the
contribution due to the negative
frequency is at 10
kilohertz minus the
input sinusoid.
We then carry out a
reconstruction with an ideal
low-pass filter.
And the ideal low-pass filter
is set at half the sampling
frequency or 5 kilohertz.
So what we have then is the
input signal x of t and the
impulse train x of p of t.
And then the reconstructed
signal is the output from the
low-pass filter which I
denote as x of r of t.
Now as the input frequency x of
t increases, this impulse
moves up in frequency,
but this impulse
moves down in frequency.
And so let's just look at a
few frames as the input
frequency increases.
So we have here a case where the
input frequency has moved
up close to 5 kilohertz.
As we continue further, these
two impulses will cross and
what we'll end up with, as
I indicated, is aliasing.
So here now is a case where
we have aliasing.
The replication of the negative
frequency has crossed
into the passband of the filter
and the reconstructed
sinusoid will now be the
frequency associated with this
impulse rather than the
frequency associated with the
original sinusoid.
And to dramatize that even
further, here is the example
where now the input frequency
has moved up close to 10
kilohertz, but what comes out
of the low-pass filter is a
much lower frequency.
And in fact, you can see that
here is the reconstructed
sinusoid, whereas here we
have the input sinusoid.
Well, now what I'm going to want
to do is demonstrate this
as I indicated with a
computer-generated movie.
And what we'll see is the effect
of reconstructing from
the samples using a low-pass
filter for an input which
changes in frequency and with a
sampling rate of 10 kilohertz.
And what we'll see in the first
part of this movie is
the input x of t and the
reconstructed signal x of r of
t without explicitly showing
the samples.
And then, at a later point,
we'll also show this and
indicate that in fact the
samples of those two are
equal, even though they
themselves are not.
So at the top, we'll have the
input sinusoid without showing
the samples.
And its Fourier transform is
an impulse in the frequency
domain as we've indicated.
And if we sample it, that
impulse then gets replicated.
And so its samples, in
particular, will have a
Fourier transform not only with
an impulse at the input
sinusoidal frequency,
but also at 10
kilohertz minus that frequency.
Now for the reconstruction, we
passed the samples through an
ideal low-pass filter.
I picked the cutoff frequency of
the low-pass filter at half
the sampling frequency,
namely 5 kilohertz.
And here, what we see is that
the output reconstructed
signal in fact matches in
frequency the input signal.
Now as we change the input
frequency, the reconstructed
sinusoid is identical until we
get to an input frequency,
which exceeds half the
sampling frequency.
At that point we have aliasing
and while the input frequency
is increasing, the output
frequency in fact is
decreasing because that's what's
inside the passband of
the filter.
Now let's sweep it back.
And as the input frequency
decreases, the output
frequency increases until
there's no aliasing and now
the output reconstructed signal
is equal to the input.
So we've sampled a signal and
then reconstructed the signal
from the samples.
And keep in mind, that given a
set of samples, there are lots
of continuous curves
that we can thread
through the set of samples.
The one that we picked, of
course, is the one consistent
with the assumption about
the signal bandwidth.
In particular, we've
reconstructed the signal whose
spectrum falls within the
passband of the filter.
Now what I'd like to show is
the same reconstruction and
input as I showed before, but
now let's look at the samples
and what we'll see is that when
there's aliasing, even
though the output-- the
reconstructed signal-- is not
identical to the input.
In fact it's consistent with
the input samples that is
sampling the reconstructed
signal.
It gives a set of samples
identical to the samples of
the input and it's just that the
interpolation in between
those samples is an
interpolation consistent with
the assumed bandwidth of
the input based on
the sampling theorem.
So let's now look at that with
the samples also shown along
with the sinusoid.
So at the top, we have the input
sinusoid together with
its samples.
The bottom trace is the Fourier
transform of the
sampled waveform.
The middle trace is the
reconstructed sinusoid
together with its samples.
And notice, of course, that the
samples of the input or
reconstructed signal
are identical.
And also the input sinusoidal
frequency and the output
sinusoidal frequency
are identical.
And we now increase the
frequency at the input.
The reconstructed sinusoid
tracks the input in frequency
and, of course, the samples
of the two are identical.
The interpolation in between
the samples is identical
because of the fact that the
input frequency is still less
than half the sampling
frequency.
And so, as long as the input is
frequency is less than half
the sampling frequency, not
only will the samples be
identical, but also the
reconstructed continuous
waveform will match the
input waveform.
Now when we get to half the
sampling frequency, we're just
on the verge of aliasing.
This isn't aliasing quite yet,
but any increase in the input
frequency will now generate
aliasing.
We now have aliasing, the output
frequency is lower than
the input frequency,
but notice that
the samples are identical.
Now the low-pass filter is
interpolating in between those
samples with a sinusoid that
falls within the passband of
the low-pass filter, which no
longer matches the frequency
of the input sinusoid.
But the important point is
that even when we have
aliasing, the samples of the
reconstructed waveform are
identical to the samples of
the original waveform.
And notice that as the input
frequency increases, in fact
the interpolated output, the
reconstructed output has
decreased in frequency.
Now as the input frequency
begins to get closer to 10
kilohertz-- in fact your eye
tends to also interpolate
between the samples with a
frequency that is lower than
the input frequency.
And that's particularly
evident here.
Notice that the input samples
in fact look like they would
be associated with a much lower
frequency sinusoid, than
in fact was the sinusoid
that generated them.
The lower-frequency sinusoid
in fact corresponds to the
reconstructed one.
Now as we sweep back down, the
aliasing eventually disappears
and the output sinusoid
tracks the
input sinusoid in frequency.
So we've seen the effect of
aliasing for sinusoidal
signals in terms of waveforms.
Now let's hear how it sounds.
Now what we have for this
demonstration is an oscillator
and a sampler.
And the output of the sampler
goes into a low-pass filter.
So the input from the oscillator
goes into the
sampler and the output of
the sampler goes into
the low-pass filter.
The sampler frequency
is 10 kilohertz.
And so the low-pass filter has
a cutoff frequency as I
indicate here, of 5 kilohertz.
And what we'll listen to is the
reconstructed output as
the oscillator input
frequency varies.
And recall that what should
happen is that when the
oscillator input frequency gets
past half the sampling
frequency, we should
hear aliasing.
So we'll start the oscillator
at 2 kilohertz.
[OSCILLATOR SOUND
IN BACKGROUND]
PROFESSOR: And keep in mind that
what you see on the dial
is the input frequency,
what you hear
is the output frequency.
As long as the input frequency
is less than half the sampling
frequency-- in other words, 5
kilohertz -- the reconstructed
signal sounds identical
to the input.
Now at 5 kilohertz, we're
right on the verge of
aliasing, and when we increase
the input frequency past 5
kilohertz, the reconstructed
frequency
in fact will decrease.
So as we move, for example, from
5 kilohertz up to let's
say, 6 kilohertz.
6 kilohertz in fact gets
aliased down to, what?
It gets aliased down
to 4 kilohertz.
So 6 kilohertz at the input is
4 kilohertz at the output.
Now, if we move up even further,
7 kilohertz at the
input gets aliased down to 3
kilohertz at the output.
So that, then is an audio
demonstration of aliasing.
So to summarize, if we sample a
signal and then reconstruct
from the samples using a
low-pass filter, as long as
the sampling frequency is
greater than twice the highest
frequency in the signal we
reconstruct exactly.
If on the other hand, the
sampling frequency is too low,
less than twice the highest
frequency, then we get aliasing.
In other words, higher
frequencies get folded or
reflected down into lower
frequencies as they come
through the low-pass filter.
Now, one of the common
applications of the whole
concept of sampling is the use
of sampling to convert a
continuous-time signal into a
discrete-time signal to carry
out what's often referred to as
discrete-time processing of
continuous-time signals.
And this in fact is something
that we'll be talking about in
a fair amount of detail,
beginning
with the next lecture.
But let me indicate that for
that kind of processing,
essentially what happens, is
that we begin with the
continuous-time signal and
convert it to a discrete-time
signal, carry out the
discrete-time processing, and
then convert back to
continuous-time.
And the conversion from a
continuous-time signal to a
discrete-time signal in fact,
is done by exploiting
sampling, specifically by
sampling the continuous-time
signal with an impulse train
and then converting the
impulse train into a sequence
in a matter that I'll talk
about in more detail
next time.
Now in doing that-- of course,
as you can imagine-- it's
important since we want an
accurate representation of the
original continuous-time signal,
to choose the sampling
frequency, to very carefully
avoid aliasing.
And so in fact, in that context
and in many other
contexts, aliasing is something
that we're very
eager to avoid.
However, it's also important
to understand that aliasing
isn't all bad.
And there are some very specific
contexts in which
aliasing is very useful and
very heavily exploited.
One example of a very useful
context of aliasing is when
you want to look at things that
happen at frequencies
that you can't look at, for
one reason or another.
And sampling and aliasing is
used to map those into lower
frequencies.
One very common example of
that is the use of the
stroboscope which was invented
by Dr. Harold Edgerton at MIT.
And sometime earlier, in fact
we had the opportunity to
visit Dr. Edgerton's laboratory
at MIT and see some
examples of this.
So I'd like to-- as a conclusion
to this lecture--
take you on a visit to the
strobe lab at MIT.
In the lecture-- in discussing
aliasing-- we've stressed the
fact that in most situations,
it's something that
we'd like to avoid.
However, right now we're at MIT,
in Strobe Alley as it's
called, on the way to visit
the laboratory of my MIT
colleague, Professor Harold
Edgerton, where in fact
aliasing is an everyday
occurrence.
Basically, the idea is the
following-- that if in fact
you want to make measurements
at frequencies that, for one
reason or another, you can't
measure, then sampling and,
consequently, aliasing can
be used to bring those
frequencies down into
a frequency range
that you can measure.
Well, Professor Edgerton alias
Doc Edgerton invented the
stroboscope for exactly
that reason.
And, kind of, the idea
is the following.
The eye, essentially, is a
low-pass filter and so there
are things that happen at
frequencies above which your
eye can track.
And by sampling with light
pulses, sampling in time, what
in effect you're able to do is
sample in such a way that
higher frequencies get aliased
down to lower frequencies so
that, in fact, your eye
can track them.
So let's take a look inside
the lab and in fact see an
illustration of this strobe
and some of its effects.
Let me introduce you to my MIT
colleague, Doc Edgerton.
Also by the way, this is a great
place for kids of all
ages and so my daughter,
Justine, insisted on coming
along to also help out.
Doc, maybe we could begin with
you just saying a little bit
about what the strobe is and
what some of the history is?
DR. HAROLD EDGERTON: Sure, it's
a very simple application
of intermittent light.
And this is a xenon lamp that
flashes in a controlled rate
depending on this knob which
Justine's going to turn.
And we're going to look at
a motor that's driving an
unbalanced weight to set
up some [INAUDIBLE]
oscillations in the spring.
I'll turn on the motor.
I'll turn on the strobe.
[STROBOSCOPE SOUND
IN BACKGROUND]
DR. HAROLD EDGERTON: Just
get the right range.
All right, Justine, turn that
now, until it stops.
See that, Justine, the frequency
is that the light,
which corresponds to the
frequency of the motor.
And it's a little less or a
little more, when you lean to
go forward to backwards.
PROFESSOR: Doc, maybe
we could turn this
strobe off for minute.
And let me point out, by the
way, the fact that when we're
looking at this without the
strobe on, what we're seeing
essentially are frequencies
that our eye can't track.
So we can't see the motor
turning and we can't really
see other than with a blur.
We can't see the movement
of the spring.
And so I guess, your point is
that when we put the strobe
on, we're essentially
sampling this.
And now we brought this down to
a frequency that our eye is
able to track.
In fact, I guess if we turn the
incandescent light off,
what we'll be able to really
bring out are the alias
frequencies.
So now, what we're looking
at in fact are the alias
frequencies.
The spring, of course, is moving
a lot faster than we
see it, isn't that right?
DR. HAROLD EDGERTON:
Yes, it's going
approximately 30 times a second.
The motor is going far from
30 times a second.
I will speed this up while I hit
the next mode, where I get
a figure 8 out of this thing.
You want to see that now?
PROFESSOR: Yeah, great.
DR. HAROLD EDGERTON:
[INAUDIBLE]
[MACHINE NOISE GETS LOUDER]
PROFESSOR: So what we'll be
seeing now is essentially a
second harmonic, is it?
DR. HAROLD EDGERTON: Yes, that's
the second harmonic.
PROFESSOR: Justine, you think
you could make that spring
dance around a little bit by
changing the strobe frequency?
DR. HAROLD EDGERTON: Yeah, you
need to go around that way.
You go around this way.
PROFESSOR: Hey, that's
really neat.
Let's turn the lights
back on if we can.
DR. HAROLD EDGERTON: Tomorrow
[INAUDIBLE]
it's periodic, it has
to be periodic.
PROFESSOR: And what's
interesting now, if we look at
this in a-- let's see, can you
flip this strobe off again?
DR. HAROLD EDGERTON: Sure.
DR. HAROLD EDGERTON: Notice,
Justine, when we look at the
spring now, all that we
can see is a blur.
And you really can't see--
because your eye can't track
it, you can't see
things happening
spatially in frequency.
You said, by the way, that
this was originally
demonstrated at the
World's Fair.
DR. HAROLD EDGERTON: This
particular instrument was made
the World's Fair in Chicago--
not the last one, but the one
before that.
PROFESSOR: Wow.
DR. HAROLD EDGERTON: It was
a--you see it all scratched up
because it's a--
the [INAUDIBLE]
use this thing is to
break the springs.
Because of the uses, you try to
find the parts that fail.
PROFESSOR: I see.
You put them under stress
and fatigue and--
DR. HAROLD EDGERTON: If I run
this for half an hour and so,
the spring will break.
And they work on automobiles,
they run them
until something vibrates.
Then they find out what
the part is and what
frequency it is.
PROFESSOR: Well let's--by the
way, I bet you run this for a
lot more than half an
hour in this state.
DR. HAROLD EDGERTON: Oh, yeah,
we've broken many, many
springs in this thing--
and it's continuous.
We experiment, try
new things on it.
PROFESSOR: Maybe we could look
at a couple of other things.
How about the fan?
Maybe--
DR. HAROLD EDGERTON: Sure, I'll
plug this fan and this is
a classic experiment
for the strobe.
[MACHINE NOISE STOPS]
DR. HAROLD EDGERTON:
That's a good idea.
Get that thing off.
Makes too much noise.
PROFESSOR: Guess, we move
that over here.
DR. HAROLD EDGERTON: This is
just an ordinary electric fan,
but it has a mark on one
blade, so that you can
identify it.
We'll plug it in, get
it up to speed.
PROFESSOR: This looks like a fan
that was also demonstrated
in the World's Fair,
a few years ago.
DR. HAROLD EDGERTON: Yeah,
could've been.
There was a movie Quicker'n
a Wink had this
thing in there and--
PROFESSOR: With this very fan?
DR. HAROLD EDGERTON:
Well, one like it.
It was loaned to MGM.
And Pete Smith, he said he
wanted me to throw out a
custard pie into it.
I said, no, I'm a serious
scientist.
So he says, let's compromise
on the egg.
So we dropped an egg into it and
you would see a high-speed
movie of the egg dropping.
No, not with the strobe, but
with this [INAUDIBLE]
PROFESSOR: That was with the
high-speed photography.
DR. HAROLD EDGERTON: High-speed
movies, yeah.
PROFESSOR: So again, I guess,
without the strobe, when we
look at it, what we're looking
at are frequencies that are
much higher than the
eye can follow.
And now, with the strobe on,
you can see both the alias
frequency and you can also see
the original frequency because
we had the incandescent
light on.
Let's turn down the background
light again.
And then, really all that we're
able to see are the
aliasing frequencies.
And I guess when we see more
than one mark, that means that
we're actually running it at--
DR. HAROLD EDGERTON: Four times
the speed of the fan.
PROFESSOR: --four times the
speed the fan, yeah.
DR. HAROLD EDGERTON: You see
a little variation in the--
PROFESSOR: Oh, yeah.
Right.
DR. HAROLD EDGERTON: It's
because the blades aren't
exactly the same.
PROFESSOR: Actually, this
gives me a chance to
illustrate another important
point related to the lecture.
Let's see, can we bring it down
to a frequency so that we
only get one mark?
DR. HAROLD EDGERTON: Sure.
You may miss this because
it's too lowered to just
one blade there now.
PROFESSOR: So the way we have
it now, we've essentially
aliased the fan's speed down
so that it's just a little
higher than DC.
And now, I'm right at DC.
And now, if I go down just a
little further, in fact it
looks like the fan is
turning backwards.
And if you think of this in the
context of aliasing, it's
like the two impulses in
the frequency domain
have crossed over.
And what you get in effect, if
you analyze it mathematically,
is you get a phase reversal.
And it wasn't until I first
understood about aliasing, by
the way, Doc, that I understood
why when I went to
Western movies, every once in
a while you'd see the wagon
wheels turning backwards.
Then there's the wagon wheels
of the Western movie going
backwards, I guess.
And, Justine, why don't
you see if you can--
DR. HAROLD EDGERTON: Too
much flicker there.
Why don't you bring it up
so you get two marks.
PROFESSOR: See if you can bring
the frequency up so that
you get two marks.
DR. HAROLD EDGERTON: You
turn that, Justine.
Grab right ahold of that and
give it a big twist.
You went past it.
They're not regular there now.
Here we are.
Now hold it right there.
Put your finger on there,
hold the dial.
It's flashing twice per
revolution now, Al.
PROFESSOR: I guess another thing
that this demonstrates
is something that I've heard a
long time ago, which is that
you should never use a power saw
with a fluorescent light
because the fluorescent light
gives you a little bit of a
strobe effect and you could
actually convince yourself
that that's standing still and
make the mistake of trying to
put your finger between
the blades.
DR. HAROLD EDGERTON: You want to
stick your finger in there?
PROFESSOR: No, I don't think
I want to try it.
How about you, Justine?
What do you think?
Is that standing still
or is that moving?
DR. HAROLD EDGERTON: She
knows it's going.
We won't let her get
close to that fan.
PROFESSOR: Actually if we turn
the lights back on again, what
that will let us see once again
is that we can see both
the alias frequencies when we
do that and we can also see
the higher frequencies because
of the incandescent lighting.
Maybe what we can do now
is take a look at
some other fun things.
And one I guess I'm curious
about is the disk that you
have over there.
Doc, maybe you can tell
us what we have here?
DR. HAROLD EDGERTON: Sure, Al.
This is a disk to show how you
can get motion pictures out of
a series of still pictures.
This circle is repeated
12 times.
The white dot goes from the
outer part of it on this side
to the inner part
on the other.
If I flash one time per
revolution on this, you'll see
it exactly as it is.
But if I skip one picture each
time, then you get the
relative motion of this ball.
Well, the object is to show the
ball rotating either this
way or that way depending on
whether the strobe was going
faster or slower
than the other.
This way motion pictures were
developed hundred years ago,
long before photography.
They drew pictures of
people in different
poses, animated pictures.
Like to see it run?
PROFESSOR: Yeah, great.
It's actually, the title, kind
of, is "Aliasing Can Be Fun."
DR. HAROLD EDGERTON:
That's right.
Let me get it up to speed.
On the way up, you get a lot
of different, sort of,
patterns as it goes through.
When it eventually reaches its
speed, which is about 1,100
per minute, you'll
see it stop.
PROFESSOR: And the background
blur, basically at the high
frequencies that the eye can't
follow and then, kind of,
superimposed on that again, we
can see the frequencies that
are aliased down.
And that's what the
eye can follow.
DR. HAROLD EDGERTON: Right now,
we have one flash per
revolution, so you can see the
part of the disk that's
illuminated with the strobe
exactly as if it
was standing still.
Now if I increase the frequency,
so they skip one
circle, then you get
the illusion
that, that dot is moving.
PROFESSOR: In fact, let's really
enhance the revolution,
let's turn the incandescent
lights off again.
And now, now what we see
really are the alias
frequencies.
What do you think
of this Justine?
JUSTINE: Neat.
DR. HAROLD EDGERTON: It
looks like magic.
I still have great joy in
watching this thing, though
it's so simple.
PROFESSOR: Now, while we're
watching this, something also
I might point out for the
lecture--for the course-- is
that actually there really are
two sampling frequencies that
we're seeing.
One is the strobe, which is the
strobe that you're running.
The other is the inherent frame
rate for the TV, that's
running at 30 frames a second.
And that's one of the reasons,
by the way, that people
watching this on the video
course are in fact seeing a
flicker or a beating or
modulation between the two
unsynchronized frame rates.
DR. HAROLD EDGERTON: I'll run
the frequencies of the strobe
up, so we get two of
them in there.
You keep watching,
we had all these
other interesting patterns.
There's two now.
And I'll make the two bounce
on each other.
You get all these patterns
for free.
You design a disk to show one
thing and then when you run
it, you find all the
other patterns.
PROFESSOR: I think it would be
a terrific homework problem
for the video course, to have
them all sit down and analyze
all the frequencies that
they're seeing and what
they're being aliased to.
What do you think of that?
DR. HAROLD EDGERTON:
That's a good idea.
As a teacher, I love
to give quizzes.
Find out whether the students
are listening.
PROFESSOR: I think that'll chase
a few people away from
the course, that's
what I like--
DR. HAROLD EDGERTON: No, it
attracts them because you get
involved in these optical
things, there's no limit on
what you can do.
PROFESSOR: Let's bring the
incandescent lights back up
again, just to remind everybody
that in back of all
these are some frequencies that
are a lot higher than the
ones that we begin to get the
impression that we're watching.
DR. HAROLD EDGERTON: It's just
a motor running at constant
speed with a pattern on it.
PROFESSOR: Doc, I have to say
that there aren't many people
I know that have as much fun
in their work as you do.
DR. HAROLD EDGERTON: Well,
I'm a lucky man.
PROFESSOR: Well, what I'd like
to do now, maybe, is take a
look at one last experiment,
if you could.
DR. HAROLD EDGERTON: Sure.
PROFESSOR: And what I'd like to
do is go take a look at, I
guess, what sometimes is called
the--well, not the
water drop experiment-- what's
the name of the--
DR. HAROLD EDGERTON: You mean
the Double Piddler Hydraulic
Happening Machine?
PROFESSOR: That's the one
I was thinking of.
Let's take a look over there.
DR. HAROLD EDGERTON: Come
on, Justine, let's go
and turn on the water.
PROFESSOR: So, Doc, this is
the--what did you call it
DPHHM for Double Piddler
Hydraulic Happening Machine?
Got it.
DR. HAROLD EDGERTON: It looks
like a continuous
stream, but it's not.
It's a pump over there.
It's pumping 60 pulses
a second.
The water is coming
out in spurts.
PROFESSOR: So actually, again
it's the 60 pulses a second
your eye can't follow.
DR. HAROLD EDGERTON: Your eye's
no good at 60 a second.
PROFESSOR: Basically
looks like a blur.
DR. HAROLD EDGERTON: It is a
blur, a nice juicy blur.
Now we put the strobe on.
PROFESSOR: So again, I guess
we have this essentially
aliased down.
And again with the incandescent
light, you can
see both the high frequency
and the alias frequency.
And let's see, I guess that's
what the frequency close to DC
and we can adjust it so
that it's stopped.
DR. HAROLD EDGERTON: All right,
make the water go up.
PROFESSOR: And then we can
actually make it go up.
DR. HAROLD EDGERTON: Of course,
nobody believes that.
PROFESSOR: Yeah, in fact, let me
just again, to stress this
point to the class.
The idea here of the phase
reversal-- of course, you can
see it in the time domain-- you
just think about when the
flashes of light come.
But if you think of these
impulses that we have in the
frequency domain and we're
aliasing as we change the
sampling frequency, what happens
is that these impulses
cross over and what that means
is that we get a phase
reversal depending on which
phases are associated with
which side of DC so that's
kind of the idea
of the phase reversal.
Let's turn the--
DR. HAROLD EDGERTON: Well, we
tried to have Justine put her
finger in between
those two drops.
PROFESSOR: Yeah, let's turn the
incandescent light off first.
And--
DR. HAROLD EDGERTON: Take
one finger out now.
Put it right in between
those two drops.
PROFESSOR: Justine,you think
you can do that?
DR. HAROLD EDGERTON: Better
get on the other side.
Use your other hand, so
they can see with it.
You can--
PROFESSOR: Think you can get
your finger in there?
PROFESSOR: Whoop, there's water
there all the time.
PROFESSOR: Well I
don't know, Doc.
It seems to me if we-- can't
we just adjust this so that
the dots just go through
each other?
DR. HAROLD EDGERTON: Sure.
PROFESSOR: Now if the dots can
do it, Justine, how come you
can't get your finger
in there?
JUSTINE: I don't know.
PROFESSOR: Why don't you
try that once more?
I guess not.
DR. HAROLD EDGERTON: No,
that's one thing you
can't do with it.
PROFESSOR: Well let's bring the
lights back up and again,
just to stress the point, here
we are at DC, here we are at a
frequency that's just a little
above DC, and we can go back
down to DC and we can actually
get a phase reversal.
And I guess, if we do this long
enough, we can empty out
the whole ocean and
put it back in
wherever it comes from.
Isn't that right?
DR. HAROLD EDGERTON: And we
caution the students when they
run this, not to run
it too long--
PROFESSOR: That's right.
DR. HAROLD EDGERTON: We've
got the bucket here.
PROFESSOR: You have
to be careful--
DR. HAROLD EDGERTON: --and
it's been a while since
somebody believes me.
PROFESSOR: Well, I don't know
about them, but I guess I
believe you, Doc.
DR. HAROLD EDGERTON: I'll put
a little more pressure on so
we get little more interesting
patterns.
Little patterns or surface
tension that's pulling in
things together.
We have these machines, they're
all over the place.
They're a lot of fun.
PROFESSOR: Well, Doc, this
is really terrific.
I think that this whole idea of
using aliasing and strobes
and the kinds of things
that you do with
them are just fantastic.
And we really appreciate the
chance to come in here and see
the demonstration.
DR. HAROLD EDGERTON: Well,
that's the whole game.
We've [? been happy to use ?]
them for years and probably will
for many years to come.
PROFESSOR: So as I emphasized
at the beginning, in lots of
situations aliasing can, in
fact, be very useful.
Also what this demonstrates is
that particularly when you
have a colleague, like Doc
Edgerton, aliasing and for
that matter science, in
general, can be an
awful lot of fun.
DR. HAROLD EDGERTON: Thanks
for coming in.
PROFESSOR: Thanks a lot, Doc.
DR. HAROLD EDGERTON:
See you again.
PROFESSOR: And thank
you, Justine.
JUSTINE: You're welcome.
PROFESSOR: Well, I have to say
that visit was an awful lot of
fun for me and for Justine and
in fact, for the whole camera
crew that was there.
And hopefully, all of you at
some point will also have a
chance to visit at
Strobe Alley.
Well, hopefully what we've gone
through today gives you a
good feeling for the concepts
of sampling and aliasing and
both, why it might be useful
and why we might
want to avoid it.
In the next lecture, we'll
continue on the
discussion of sampling.
And in particular, what I'll
be talking about is the
interpretation of the
reconstruction process not in
the frequency domain, but in a
time domain and interpretation
specifically associated with the
concept of interpolating
between the samples.
We'll then proceed from there
to a discussion as I've
alluded to in several lectures
of what I've referred to as
discrete-time processing of
continuous-time signals, very
heavily exploiting the
concept and issues
associated with sampling.
Thank you.
