第 17 讲：插值
本讲内容：将由样本重建信号理解为插值过程；带限插值；近似插值，包括零阶保持和一阶保持（线性插值）；通过图像说明采样与插值；采样在计算机信号处理中的应用。
授课教师：艾伦·V. 奥本海姆教授
英文字幕稿
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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.