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[MUSIC PLAYING]
PROFESSOR: In discussing the
sampling theorem, we saw that
for a band limited signal,
which is sampled at a
frequency that is at least twice
the highest frequency,
we can implement exact
reconstruction of the original
signal by low pass filtering an
impulse train, whose areas
are identical to the
sample values.
Well essentially, this low
pass filtering operation
provides for us an interpolation
in between the
sampled values.
In other words, the output of a
low pass filter, in fact, is
a continuous curve, which fits
between the sampled values
some continuous function.
Now, I'm sure that many of you
are familiar with other kinds
of interpolation that we could
potentially provide in between
sampled values.
And in fact, in today's lecture
what I would like to
do is first of all developed
the interpretation of the
reconstruction as an
interpolation process and then
also see how this exact
interpolation, using a low
pass filter, relates to other
kinds of interpolation, such
as linear interpolation
that you may
already be familiar with.
Well to begin, let's again
review what the overall system
is for exact sampling
and reconstruction.
And so let me remind you that
the overall system for
sampling and desampling, or
reconstruction, is as I
indicate here.
The sampling process consists
of multiplying
by an impulse train.
And then the reconstruction
process corresponds to
processing that impulse train
with a low pass filter.
So if the spectrum of the
original signal is what I
indicate in this diagram, then
after sampling with an impulse
train, that spectrum
is replicated.
And this replicated spectrum
for reconstruction is then
processed through a
low pass filter.
And so, in fact, if this
frequency response is an ideal
low pass filter, as I indicate
on the diagram below, then
multiplying the spectrum of
the sample signal by this
extracts for us just the
portion of the spectrum
centered around the origin.
And what we're left with,
then, is the spectrum,
finally, of the reconstructed
signal, which for the case of
an ideal low pass filter is
exactly equal to the spectrum
of the original signal.
Now, that is the frequency
domain picture of the sampling
and reconstruction.
Let's also look at, basically,
the same process.
But let's examine it now
in the time domain.
Well in the time domain, what we
have is our original signal
multiplied by an
impulse train.
And this then is the sample
signal, or the impulse train
whose areas are equal to
the sample values.
And because of the fact that
this is an impulse train, in
fact, we can take this term
inside the summation.
And of course, what counts
about x of t in this
expression is just as values
at the sampling instance,
which are displaced in time by
capital T. And so what we can
equivalently write is the
expression for the impulse
train samples, or impulse train
of samples, as I've
indicated here.
Simply an impulse train, whose
areas are the sampled values.
Now, in the reconstruction we
process that impulse train
with a low pass filter.
That's the basic notion
of the reconstruction.
And so in the time domain, the
reconstructed signal is
related to the impulse train
of samples through a
convolution with the filter
impulse response.
And carrying out this
convolution, since this is
just a train of pulses, in
effect, what happens in this
convolution is that this
impulse response gets
reproduced at each of the
locations of the impulses in x
of p of t with the
appropriate area.
And finally, then, in the time
domain, the reconstructed
signal is simply a linear
combination of shifted
versions of the impulse response
with amplitudes,
which are the sample values.
And so this expression, in
fact then, is our basic
reconstruction expression
in the time domain.
Well in terms of a diagram, we
can think of the original
waveform as I've shown here.
And the red arrows denote the
sampled wave form, or the
train of impulses, whose
amplitudes are the sampled
values of the original
continuous time signal.
And then, I've shown here what
might be a typical impulse
response, particularly typical
in the case where we're
talking about reconstruction
with an ideal low pass filter.
Now, what happens in the
reconstruction is that the
convolution of these impulses
with this impulse response
means that in the
reconstruction, we superimpose
one of these impulse
responses--
whatever the filter impulse
response happens to be--
at each of these
time instance.
And in doing that, then
those are added up.
And that gives us the total
reconstructed signal.
Of course, for the case in which
the filter is an ideal
low pass filter, then what we
know is that in that case, the
impulse response is of the
form of a sync function.
But generally, we may want to
consider other kinds of
impulse responses.
And so in fact, the
interpolating impulse response
may have and will have, as this
discussion goes along,
some different shapes.
Now what I'd like to do is
illustrate, or demonstrate,
this process of effectively
doing the interpolation by
replacing each of the impulses
by an appropriate
interpolating impulse response
and adding these up.
And I'd like to do this
with a computer
movie that we generated.
And what you'll see in the
computer movie is,
essentially, an original
wave form, which is
a continuous curve.
And then below that in the movie
is a train of samples.
And then below that will be
the reconstructed signal.
And the reconstruction will be
carried out by showing the
location of the impulse response
as it moves along in
the wave form.
And then the reconstructed curve
is simply the summation
of those as that impulse
response moves along.
So what you'll see then is an
impulse response like this--
for the particular case of an
ideal low pass filter for the
reconstruction--
placed successively at the
locations of these impulses.
And that is the convolution
process.
And below that then will be
the summation of these.
And the summation of those
will then be the
reconstructed signal.
So let's take a look at, first
of all that reconstruction
where the impulse response
corresponds to the impulse
response of an ideal
low pass filter.
Shown here, first, is the
continuous time signal, which
we want to sample and then
reconstruct using band limited
interpolation, or equivalently,
ideal low pass
filtering on the
set of samples.
So the first step then
is to sample this
continuous time signal.
And we see here now the
set of samples.
And superimposed on the samples
are the original
continuous time signal to focus
on the fact that those
are samples of the top curve.
Let's now remove the continuous
time envelope of
the samples.
And it's this set of samples
that we then want to use for
the reconstruction.
The reconstruction process,
interpreted as interpolation,
consists of replacing
each sample with a
sine x over x function.
And so let's first consider
the sample at t equals 0.
And here is the interpolating
sine x over x function
associated with that sample.
Now, the more general process
then is to place a sine x over
x function at the time location
of each sample and
superimpose those.
Let's begin that process at the
left-hand set of samples.
And in the bottom curve, we'll
build up the reconstruction as
those sine x over x functions
are added together.
So we begin with the
left-hand sample.
And we see there the sine x over
x function on the bottom
curve is the first step
in the reconstruction.
We now have the sine x over x
function associated with the
second sample.
Let's add that in.
Now we move on to the
third sample.
And that sine x over x
function is added in.
Continuing on, the next sample
generates a sine x over x
function, which is superimposed
on the result
that we've accumulated so far.
And now let's just speed
up the process.
We'll move on to the
fifth sample.
Add that in.
The sixth sample, add that in.
And continue on through
the set of samples.
And keep in mind the fact that,
basically, what we're
doing explicitly here is the
convolution of the impulse
train with a sine x
over x function.
And because the set of samples
that we started with were
samples of an exactly band
limited function, what we are
reconstructing exactly is the
original continuous time
signal that we have
on the top trace.
OK, so that then kind of gives
you the picture of doing
interpolation by replacing
the impulses by
a continuous curve.
And that's the way we're fitting
a continuous curve to
the original impulse train.
And let me stress that this
reconstruction process--
by putting the impulses
through a filter--
follows this relationship
whether or not this impulse
response, in fact, corresponds
to an ideal low pass filter.
What this expression always says
is that reconstructing
this way corresponds to
replacing the impulses by a
shifted impulse response with
an amplitude that is an
amplitude corresponding
to the sample value.
Now the kind of reconstruction
that we've just talked about,
and the ideal reconstruction,
is often referred to as band
limited interpolation because
we're interpolating in between
the samples by making the
assumption that the signal is
band limited and using the
impulse response for an ideal
low pass filter, which has a cut
off frequency consistent
with the assumed bandwidth
for the signal.
So if we look here, for example,
at the impulse train,
then in the demonstration that
you just saw, we built up the
reconstructed curve by replacing
each of these
impulses with the
sync function.
And the sum of those built up
the reconstructed curve.
Well, there are lots of other
kinds of interpolation that
are perhaps maybe not as exact
but often easier to implement.
And what I'd like to
do is focus our
attention on two of these.
The first that I want to mention
is what's referred to
as the zero order hold, where
in effect, we do the
interpolation in between these
sample values by simply
holding the sample value until
the next sampling instant.
And the reconstruction that we
end up, in that case, will
look something like this.
It's a staircase, or box car,
kind of function where we've
simply held the sample value
until the next sampling
instant and then replaced by
that value, held it until the
next sampling instant,
et cetera.
Now that's one kind
of interpolation.
Another kind of very common
interpolation is what's
referred to as linear
interpolation, where we simply
fit a straight line between
the sampled values.
And in that case, the type of
reconstruction that we would
get would look something like I
indicate here, where we take
a sample value, and the
following sample value, and
simply fit an interpolated curve
between them, which is a
straight line.
Now interestingly, in fact, both
the zero order hold and
the linear interpolation, which
is often referred to as
a first order hold, can also
be either implemented or
interpreted, both implemented
and interpreted, in the
context of the equation that
we just developed.
In particular, the processing
of the impulse train of
samples by a linear time
invariant filter.
Specifically, if we consider
a system where the impulse
response is a rectangular
function, then in fact, if we
processed the train of samples
through a filter with this
impulse response, exactly the
reconstruction that we would
get is what I've shown here.
Alternatively, if we chose an
impulse response which was a
triangular impulse response,
then what in effect happens is
that each of these impulses
activates this triangle.
And when we add up those
triangles at successive
locations, in fact, what we
generate is this linear
interpolation.
So what this says, in fact, is
that either a zero order hold,
which holds the value, or
linear interpolation can
likewise be interpreted as a
process of convulving the
impulse train of samples
with an appropriate
filter impulse response.
Well, what I'd like to do is
demonstrate, as we did with
the band limited interpolation
or the sync interpolation as
it's sometimes called--
interpolating with
a sine x over x--
let me now show the process.
First of all, where we have
a zero order hold as
corresponding to this
impulse response.
In which case, we'll see
basically the same process as
we saw in the computer generated
movie previously.
But now, rather than a sync
function replacing each of
these impulses, we'll have
a rectangular function.
That will generate then our
approximation, which is a zero
order hold.
And following that, we'll do
exactly the same thing with
the same wave form, using
a first order hold or a
triangular impulse response.
In which case, what we'll see
again is that as the triangle
moves along here, and we build
up the running sum or the
convolution, then we'll, in
fact, fit the original curve
with a linear curve.
So now let's again look at that,
remembering that at the
top we'll see the original
continuous curve, exactly the
one that we had before.
Below it, the set of samples
together with the impulse
response moving along.
And then finally below that,
the accumulation of those
impulse responses, or
equivalently the convolution,
or equivalently the
reconstruction.
So we have the same continuous
time signal that we use
previously with band limited
interpolation.
And in this case now, we want to
sample and then interpolate
first with a zero order
hold and then with
a first order hold.
So the first step then
is to sample the
continuous time signal.
And we show here the set
of samples, once again,
superimposed on which we have
the continuous time signal,
which of course is exactly
the same curve as
we have in the top.
Well, let's remove that envelope
so that we focus
attention on the samples that
we're using to interpolate.
And the interpolation process
consists of replacing each
sample by a rectangular signal,
whose amplitude is
equal to the sample size.
So let's put one, first of all,
at t equals 0 associated
with that sample.
And that then would be the
interpolating rectangle
associated with the sample
at t equals 0.
Now to build up the
interpolation, what we'll have
is one of those at each
sample time, and
those are added together.
We'll start that process,
as we did before, at the
left-hand end of the set of
samples and build the
interpolating signal
on the bottom.
So with the left-hand sample,
we have first the rectangle
associated with that.
That's shown now on
the bottom curve.
We now have an interpolating
rectangle with a second sample
that gets added into
the bottom curve.
Similarly, an interpolating
rectangle with the zero order
hold with the third sample.
We add that into the
bottom curve.
And as we proceed, we're
building a staircase
approximation.
On to the next sample, that gets
added in as we see there.
And now let's speed
up the process.
And we'll see the staircase
approximation building up.
And notice in this case, as in
the previous case, that what
we're basically watching
dynamically is the convolution
of the impulse train of samples
with the impulse
response of the interpolating
filter, which in this
particular case is just
a rectangular pulse.
And so this staircase
approximation that we're
generating is the zero order
hold interpolation between the
samples of the band limited
signal, which is at the top.
Now let's do the same thing
with a first order hold.
So in this case, we want to
interpolate using a triangular
impulse response rather then
the sine x over x, or
rectangular impulse responses
that we showed previously.
So first, let's say with the
sample at t equals 0, we would
replace that with a triangular
interpolating function.
And more generally, each impulse
or sample is replaced
with a triangular interpolating
function of a
height equal to the
sample type.
And these are superimposed
to generate the linear
interpolation.
We'll begin this process with
the leftmost sample.
And we'll build the
superposition below in the
bottom curve.
So here is the interpolating
triangle for
the leftmost sample.
And now it's reproduced below.
With the second sample, we
have an interpolating
triangle, which is added
into the bottom curve.
And now on to the
third sample.
And again, that interpolating
triangle will be added on to
the curve that we've
developed so far.
And now onto the next sample.
We add that in.
Then we'll speed
up the process.
And as we proceed through, we
are building, basically, a
linear interpolation in between
the sample points,
essentially corresponding to--
if one wants to think
of it this way--
connecting the dots.
And what you're watching, once
again, is essentially the
convolution process convulving
the impulse train with the
impulse response of the
interpolating filter.
And what we're generating,
then, is a linear
approximation to the band
limited continuous time curve
at the top.
OK, so what we have then is
several other kinds of
interpolation, which fit within
the same context as
exact band limited
interpolation.
One being interpolation in the
time domain with an impulse
response, which is
a rectangle.
The second being interpolation
in the time domain with an
impulse response, which
is a triangle.
And in fact, it's interesting
to also look at the
relationship between that and
band limited interpolation.
Look at it, specifically,
in the frequency domain.
Well, in the frequency domain,
what we know, of course, is
that for exact interpolation,
what we want as our
interpolating filter is an
ideal low pass filter.
Now keep in mind, by the way,
that an ideal low pass filter
is an abstraction, as I've
stressed several
times in the past.
An ideal low pass filter is a
non-causal filter and, in
fact, infinite extent, which is
one of the reasons why in
any case we would use some
approximation to it.
But here, what we have is the
exact interpolating filter.
And that corresponds to an
ideal low pass filter.
If, instead, we carried out the
interpolating using the
zero order hold, the zero order
hold has a rectangular
impulse response.
And that means in the frequency
domain, its
frequency response is of the
form of a sync function, or
sine x over x.
And so this, in fact, when we're
doing the reconstruction
with a zero order hold, is the
associated frequency response.
Now notice that it does some
approximate low pass filtering.
But of course, it permits
significant energy outside the
past band of the filter.
Well, instead of the zero order
hold, if we used the
first order hold corresponding
to the triangular impulse
response, in that case then in
the frequency domain, the
associated frequency response
would be the Fourier transform
of the triangle.
And the Fourier transform of a
triangle is a sine squared x
over x squared kind
of function.
And so in that case, what we
would have for the frequency
response, associated with the
first order hold, is a
frequency response
as I show here.
And the fact that there's
somewhat more attenuation
outside the past band of the
ideal filter is what suggests,
in fact, that the first order
hold, or linear interpolation,
gives us a somewhat smoother
approximation to the original
signal than the zero
order hold does.
And so, in fact, just to compare
these two, we can see
that here is the ideal filter.
Here is the zero order hold,
corresponding to generating a
box car kind of reconstruction.
And here is the first order
hold, corresponding to a
linear interpolation.
Now in fact, in many sampling
systems, in any sampling
system really, we need to use
some approximation to the low
pass filter.
And very often, in fact, what
is done in many sampling
systems, is to first use just
the zero order hold, and then
follow the zero order
hold with some
additional low pass filtering.
Well, to illustrate some of
these ideas and the notion of
doing a reconstruction with a
zero order hold or first order
hold and then in fact adding
to that some additional low
pass filtering, what I'd like
to do is demonstrate, or
illustrate, sampling and
interpolation in the context
of some images.
An image, of course, is a
two-dimensional signal.
The independent variables
are spatial
variables not time variables.
And of course, we can sample
in both of the spatial
dimensions, both in x and y.
And what I've chosen as a
possibly appropriate choice
for an image is, again,
our friend and
colleague J.B.J. Fourier.
So let's begin with the original
image, which we then
want to sample and
reconstruct.
And the sampling is done by
effectively multiplying by a
pulse both horizontally
and vertically.
The sample picture is then
the next one that I show.
And as you can see, this
corresponds, in effect, to
extracting small brightness
elements out of
the original image.
In fact, let's look in
a little closer.
And what you can see,
essentially, is that what we
have, of course, are not
impulses spatially but small
spatial pillars that implement
the sampling for us.
OK, now going back to the
original sample picture, we
know that a picture can be
reconstructed by low pass
filtering from the samples.
And in fact, we can do that
optically in this case by
simply defocusing the camera.
And when we do that, what
happens is that we smear out
the picture, or effectively
convulve the impulses with the
point spread function of
the optical system.
And this then is not too
bad a reconstruction.
So that's an approximate
reconstruction.
And focusing back now
what we have again
is the sample picture.
Now these images are, in fact,
taken off a computer display.
And a common procedure in
computer generated or
displayed images is in fact the
use of a zero order hold.
And if the sampling rate is
high enough, then that
actually works reasonably
well.
So now let's look at the result
of applying a zero
order hold to the sample image
that I just showed.
The zero order hold corresponds
to replacing the
impulses by rectangles.
And you can see that what that
generates is a mosaic effect,
as you would expect.
And in fact, let's go in a
little closer and emphasize
the mosaic effect.
You can see that, essentially,
where there were impulses
previously, there are now
rectangles with those
brightness values.
A very common procedure with
computer generated images is
to first do a zero order hold,
as we've done here, and then
follow that with some additional
low pass filtering.
And fact, we can do that low
pass filtering now again by
defocusing the camera.
And you can begin to see that
with the zero order hold plus
the low pass filtering, the
reconstruction is not that bad.
Well, let's go back to
the full image with
the zero order hold.
And again, now the effect of
low pass filtering will be
somewhat better.
And let's defocus again here.
And you can begin to see that
this is a reasonable
reconstruction.
With the mosaic, in fact, with
this back in focus, you can
apply your own low pass
filtering to it either by
squinting, or if you have the
right or wrong kind of
eyeglasses, either taking them
off or putting them on.
Now, in addition to the zero
order hold, we can, of course,
apply a first order hold.
And that would correspond to
replacing the impulses,
instead of with rectangles as
we have here, replacing them
with triangles.
And so now let's take a look at
the result of a first order
hold applied to the
original samples.
And you can see now that the
reconstruction is somewhat
smoother because of the fact
that we're using an impulse
response that's somewhat
smoother or a corresponding
frequency response that
has a sharper cut off.
I emphasize again that this is
a somewhat low pass filtered
version of the original because
we have under sampled
somewhat spatially to bring out
the point that I want to
illustrate.
OK, to emphasize these effects
even more, what I'd like to do
is go through, basically,
the same sequence again.
But in this case, what we'll
do is double the sample
spacing both horizontal
and vertically.
This of course, means that
we'll be even more highly
under sampled than in the ones
I previously showed.
And so the result of the
reconstructions with some low
pass filtering will be a much
more low pass filtered image.
So we now have the
sampled picture.
But I've now under sampled
considerably more.
And you can see the effect
of the sampling.
And if we now apply a zero order
hold to this picture, we
will again get a mosaic.
And let's look at that.
And that mosaic, of
course, looks even
blockier than the original.
And again, it emphasizes the
fact that the zero order hold
simply corresponds to filling
in squares, or replacing the
impulses, by squares, with the
corresponding brightness values.
Finally, if we, instead of a
zero order hold, use a first
order hold, corresponding to two
dimensional triangles in
place of these original
blocks.
What we get is the next image.
And that, again, is a smoother
reconstruction consistent with
the fact that the triangles
are smoother than the
rectangles.
Again, I emphasize that this
looks so highly low pass
filtered because of the fact
that we've under sampled so
severely to essentially
emphasize the effect.
As I mentioned, the images that
we just looked at were
taken from a computer, although
of course the
original images were continuous
time images or more
specifically, continuous
space.
That is the independent
variable
is a spatial variable.
Now, computer processing of
signals, pictures, speech, or
whatever the signals are is
very important and useful
because it offers a lot
of flexibility.
And in fact, the kinds of things
that I showed with
these pictures would have been
very hard to do without, in
fact, doing computer
processing.
Well, in computer processing
of any kind of signal,
basically what's required is
that we do the processing in
the context of discrete time
signals and discrete time
processing because of the
fact that a computer
is run off a clock.
And essentially, things happen
in the computer as a sequence
of numbers and as a sequence
of events.
Well, it turns out that the
sampling theorem, in fact, as
I've indicated previously,
provides us with a very nice
mechanism for converting our
continuous time signals into
discrete time signals.
For example, for computer
processing or, in fact, if
it's not a computer for some
other kind of discrete time or
perhaps digital processing.
Well, the basic idea, as I've
indicated previously, is to
carry out discrete time
processing of continuous time
signals by first converting the
continuous time signal to
a discrete time signal, carry
out the appropriate discrete
time processing of the discrete
time signal, and then
after we're done with that
processing, converting from
the discrete time sequence
back to a continuous time
signal, corresponding to the
output that we have here.
Well in the remainder of this
lecture, what I'd like to
analyze is the first step in
that process, namely the
conversion from a continuous
time signal to a discrete time
signal and understand how the
two relate both in the time
domain and in the frequency
domain.
And in the next lecture, we'll
be analyzing and demonstrating
the overall system, including
some intermediate processing.
So the first step in the process
is the conversion from
a continuous time signal to
a discrete time signal.
And that can be thought of as
a process that involves two
steps, although in practical
terms it may not be
implemented specifically
as these two steps.
The two steps are to first
convert from the continuous
time, or continuous time
continuous signal, to an
impulse train through a sampling
process and then to
convert that impulse train to
a discrete time sequence.
And the discrete time sequence
x of n is simply then a
sequence of values which are the
samples of the continuous
time signal.
And as we'll see as we walk
through this, basically the
step of going from the impulse
train to the sequence
corresponds principally to a
relabeling step where we pick
off the impulse values and use
those as the sequence values
for the discrete time signal.
So what I'd like to do as a
first step in understanding
this process is to analyze
it in particular with our
attention focused on trying
to understand what the
relationship is in the frequency
domain between the
discrete time Fourier transform
of the sequence,
discrete time signal, and the
continuous time Fourier
transform of the original
unsampled, and then the
sampled signal.
So let's go through that.
And in particular, what we have
is a process where the
continuous time signal is,
of course, modulated or
multiplied by an
impulse train.
And that gives us,
then, another
continuous time signal.
We're still in the continuous
time domain.
It gives us another continuous
time signal, which is an
impulse train.
And in fact, we've gone
through this analysis
previously.
And what we have is this
multiplication or taking this
term inside the summation and
recognizing that the impulse
train is simply an impulse
train with areas of the
impulses, which are
the samples of the
continuous time function.
We can then carry out
the analysis in
the frequency domain.
Now in the time domain, we have
a multiplication process.
So in the frequency domain, we
have a convolution of the
Fourier transform of the
continuous time signal, the
original signal, and the Fourier
transform of the
impulse train, which is itself
an impulse train.
So in the frequency domain then,
the Fourier transform of
the sampled signal, which is
an impulse train, is the
convolution of the Fourier
transform of the sampling
function P of t and the Fourier
transform of the
sampled signal.
Since the sampling signal is a
periodic impulse train, its
Fourier transform is
an impulse train.
And consequently, carrying out
this convolution in effect
says that this Fourier
transform simply gets
replicated at each of the
locations of these impulses.
And finally, what we end up
with then is a Fourier
transform after the sampling
process, which is the original
Fourier transform of the
continuous signal but added to
itself shifted by integer
multiples of
the sampling frequency.
And so this is the basic
equation then that tells us in
the frequency domain what
happens through the first part
of this two step process.
Now I emphasize that it's
a two step process.
The first process is sampling,
where we're still essentially
in the continuous time world.
The next step is essentially a
relabeling process, where we
convert that impulse train
simply to a sequence.
So let's look at
the next step.
The next step is to take the
impulse train and convert it
through a process
to a sequence.
And the sequence values are
simply then samples of the
original continuous signal.
And so now we can
analyze this.
And what we want to relate is
the discrete time Fourier
transform of this and the
continuous time Fourier
transform of this, or in fact,
the continuous time Fourier
transform of x of C of T.
OK, we have the impulse train.
And it's Fourier transform we
can get by simply evaluating
the Fourier transform.
And since the Fourier
transform of this--
since this corresponds
to an impulse train--
the Fourier transform, by the
time we change some sums and
integrals, will then have this
impulse replaced by the
Fourier transform of the shifted
impulse, which is this
exponential factor.
So this expression is the
Fourier transform of the
impulse train, the continuous
time Fourier transform.
And alternatively, we can look
at the Fourier transform of
the sequence.
And this, of course,
is a discrete
time Fourier transform.
So we have the continuous time
Fourier transform of the
impulse train, we have the
discrete Fourier transform of
the sequence.
And now we want to look at
how those two relate.
Well, it pretty much falls out
of just comparing these two
summations.
In particular, this term and
this term are identical.
That's just a relabeling of what
the sequence values are.
And notice that when we compare
these exponential
factors, they're identical as
long as we associate capital
omega with little omega times
capital T. In other words, if
we were to replace here capital
omega by little omega
times capital T, and replace x
of n by x of c of nt, then
this expression would be
identical to this expression.
So in fact, these two are equal
with a relabeling, or
with a transformation, between
small omega and capital omega.
And so in fact, the relationship
that we have is
that the discrete time Fourier
transform of the sequence of
samples is equal to the
continuous time Fourier
transform of the impulse train
of samples where we associate
the continuous time frequency
variable and the discrete time
frequency variable through
a frequency scaling as I
indicate here.
Or said another way, the
discrete time spectrum is the
continuous time spectrum of the
samples with small omega
replaced by capital
omega divided by
capital T. All right.
So we have then this
two step process.
The first step is taking the
continuous time signal,
sampling it with an
impulse train.
In the frequency domain, that
corresponds to replicating the
Fourier transform
of the original
continuous time signal.
The second step is relabeling
that, in effect turning it
into a sequence.
And what that does in the
frequency domain is provide us
with a rescaling of the
frequency axis, or as we'll
see a frequency normalization,
which is associated with the
corresponding time
normalization in the time domain.
Well, let's look at those
statements a little more
specifically.
What I show here
is the original
continuous time signal.
And then below it is
the sampled signal.
And these two are signals in
the continuous time domain.
Now, what is the conversion
from this
impulse train to a sequence?
Well, it's simply taking these
impulse areas, or these sample
values, and relabeling them, in
effect as I show below, as
sequence values.
And essentially, I'm now
replacing the impulse by the
designation of a
sequence value.
That's one step.
But the other important step to
focus on is that whereas in
the impulse train, these
impulses are spaced by integer
multiples of the sampling
period capital T. In the
sequence, of course, because
of the way that we label
sequences, these are always
spaced by simply integer
multiples of one.
So in effect, you could say that
the step in going from
here to here corresponds to
normalizing out in the time
domain the sampling
period capital T.
To stress that another way, if
the sampling period were
doubled so that in this picture,
the spacing stretched
out by a factor of two.
Nevertheless, for the discrete
time signal, the spacing would
remain as one.
And essentially, it's the
envelope of those sequence
values that would then get
compressed in time.
So you can think of the step
in going from the impulse
train to the samples as,
essentially, a time
normalization.
Now let's look at this in
the frequency domain.
In the frequency domain, what
we have is the Fourier
transform of our original
continuous signal.
After sampling with an impulse
train, this spectrum retains
its shape but is replicated at
integer multiples of the
sampling frequency 2 pi over
capital T, as I indicate here.
Now, we know that a discrete
time spectrum must be periodic
in frequency with a
period of 2 pi.
Here, we have the periodicity.
But it's not periodic with
a period of 2 pi.
It's periodic with a period,
which is equal to
the sampling frequency.
However, in converting from the
samples to the sequence
values, we go through
another step.
What's the other step?
The other step is a time
normalization, where we take
the impulses, which are spaced
by the sampling period.
And we rescale that, essentially
in the time
domain, to a spacing
which is unity.
So we're dividing out in the
time domain by a factor, which
is equal to the sampling
period.
Well, dividing out in the time
domain by capital T would
correspond to multiplying in
the frequency domain the
frequency axis by capital T.
And indeed, what happens is
that in going from the impulse
train to the sequence values,
we now rescale this axis so
that, in fact, the axis gets
stretched by capital T. And
the frequency, which
corresponded to 2 pi over
capital T, now gets
renormalized to 2 pi.
So just looking at this again,
and perhaps with the overall
picture, in the time domain,
we've gone from a continuous
curve to samples, relabeled
those, and in effect
implemented a time
normalization.
Corresponding in the frequency
domain, we have replicated the
spectrum through the initial
sampling process and then
rescaled the frequency axis
so that, in fact, now this
periodicity corresponds to a
periodicity here, which is 2
pi, and here, which is the
sampling frequency.
So very often, in fact--
and we'll be
doing this next time--
when you think of continuous
time signals, which have been
converted to discrete time
signals, when you look at the
discrete time frequency axis,
the frequency 2 pi is
associated with the sampling
frequency as it was applied to
the original continuous
time signal.
Now as I indicated, what we'll
want to go on to from here is
an understanding of what
happens when we take a
continuous time signal, convert
it to a discrete time
signal as I've just gone
through, do some discrete time
processing with a linear time
invariant system, and then
carry that back into the
continuous time world.
That is a procedure that we'll
go through, and analyze, and
in fact, illustrate in some
detail next time.
In preparation for that, what I
would be eager to encourage
you to do using the study guide
and in reviewing this
lecture, is to begin the next
lecture with a careful and
thorough understanding
of the arguments that
I've just gone through.
In particular, understanding the
process that's involved in
going from a continuous time
signal through sampling to a
discrete time signal.
And what that means in the
frequency domain in terms of
taking the original spectrum,
replicating it because of the
sampling process, and then
rescaling that so that the
periodicity gets rescaled so
that it's periodic with a
period of 2 pi.
So we'll continue with that next
time, focusing now on the
subsequent steps in
the processing.
Thank you.
