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PROFESSOR: In the last lecture, we discussed discrete
time processing of continuous time signals.
And, as you know, the basis for that arises essentially
out of a sampling theorem.
Now in that context, and also in its own right, another
important sampling issue is the sampling of discrete time
signals, in other words, the sampling of a sequence.
One common context in which this arises, for example, is,
if we've converted from a continuous time signal to a
sequence, and we then carry out some additional filtering,
then there's the possibility that we can resample that
sequence, and as we'll see as we go through the discussion,
save something in the way of storage or whatever.
So discrete time sampling, as I indicated, has important
application in a context referred to here, namely
resampling after discrete time filtering.
And closely related to that, as we'll indicate in this
lecture, is the concept of using discrete time sampling
for what's referred to as sampling rate conversion.
And also closely associated with both of those ideas is a
set of ideas that I'll bring up in today's lecture,
referred to as decimation and interpolation of discrete time
signals or sequences.
Now the basic process for discrete time sampling is the
same as it is for continuous time sampling.
Namely, we can analyze it and set it up on the basis of
multiplying or modulating a discrete time signal by an
impulse train, the impulse train essentially, or pulse
train, pulling out sequence values at the times that we
want to sample.
So the basic block diagram for the sampling process is to
modulate or multiply the sequence that we want to
sample by an impulse train.
And here, the impulse train has impulses spaced by integer
multiples of capital N. This then becomes
the sampling period.
And the result of that modulation is then the sample
sequence x of p of n.
So if we just look at what a sequence and a sampled version
of that sequence might look like, what we have here is an
original sequence x of n.
And then we have the sampling impulse train, or sampling
sequence, and it's the modulation or product of these
two that gives us the sample sequence x of p of n.
And so, as you can see, multiplying this by this
essentially pulls out of the original sequence sample
values at the times that this pulse train is on.
And of course here, I've drawn this for the case where
capital N, the sampling period, is equal to 3.
Now the analysis of discrete time sampling is very similar
to the analysis of continuous time sampling.
And let's just quickly look through the
steps that are involved.
We're modulating or
multiplying in the time domain.
And what that corresponds to in the frequency domain is a
convolution.
And so the spectrum of the sampled sequence is the
periodic convolution of the spectrum of the sampling
sequence and the spectrum of the
sequence that we're sampling.
And since the sampling sequence is an impulse train,
as we know, the Fourier transform of an impulse train
is itself an impulse train.
And so this is the Fourier transform of
the sampling sequence.
And now, finally, the Fourier transform of the resulting
sample sequence, being the convolution of this with the
Fourier transform of the sequence that we're sampling,
gives us then a spectrum which consists of a sum of
replicated versions of the Fourier transform of the
sequence that we're sampling.
In other words, what we're doing, very much as we did in
continuous time, is, through the sampling process when we
look at it in the frequency domain, taking the spectrum of
the sequence there were sampling and shifting it and
then adding it in--
shifting it by integer multiples of
the sampling frequency.
In particular, looking back at this equation, what we
recognize is that this term, k times 2 pi over capital N, is
in fact an integer multiple of the sampling frequency.
And the same thing is true here.
This is k times omega sub s, where omega sub s, the
sampling frequency, is 2 pi divided by capital N.
All right, so now let's look at what this means pictorially
or graphically in the frequency domain.
And as you can imagine, since the analysis and algebra is
similar to what happens in continuous time, we would
expect the pictures to more or less be identical to what
we've seen previously for continuous time.
And indeed that's the case.
So here we have the spectrum of the signal
that's we're sampling.
This is its Fourier transform, with an assumed highest
frequency omega sub m, highest frequency over a 2pi range, or
over a range of pi, rather.
And now the spectrum of the sampling signal is what I show
below, which is an impulse train with impulses occurring
at integer multiples of the sampling frequency.
And then finally, the convolution of these two is
simply this one replicated at the
locations of these impulses.
And so that's finally what I show below.
And here I made one particular choice for
the sampling period.
This in particular corresponds to a sampling period which is
capital N equal to 3.
And so the sampling frequency, omega sub s, is
2pi divided by 3.
Now when we look at this, what we recognize is that we have
basically the same issue here as we had in continuous time,
in the sense that when these individual replications of the
Fourier transform, when the sampling frequency is chosen
high enough so that they don't overlap, then we see the
potential for being able to get one of them back.
On the other hand, when they do overlap then what we'll
have is aliasing, in particular, discrete time
aliasing, much as we had continuous time aliasing in
the continuous time case.
Well notice in this picture that what we have is we've
chosen this picture so that omega sub s minus omega sub m
is greater than omega sub m, or equivalently, so that omega
sub s is greater than 2 omega sub m.
And so with omega sub s greater than 2 omega sum m,
that corresponds to this picture.
Whereas, if that condition is violated then, in fact, the
picture that we would have is a picture that looks like.
And in this picture, the individual replications of the
Fourier transform of the original signal overlap.
And we can no longer recover the Fourier transform of the
original signal.
And this, just as it was in continuous time, is referred
to as aliasing.
Now let's look more closely at the situation in which there
is no aliasing.
So in that case, what we have is a Fourier transform for the
sampled signal, which is as I indicated here, and the
Fourier transform for the original signal, as I indicate
at the top.
And the question now is how do we recover this
one from this one.
Well, the way that we do that, just as we did in a continuous
time case, is by a low pass filtering.
In particular, processing in the time domain or in the
frequency domain, this with an ideal low pass filter has the
effect of extracting that part of the spectrum that in fact
we identify with the original signal that we began with.
So what we see, again, is that the process is
very much the same.
As long as there's no aliasing, we can recover the
original signal by ideal low pass filtering.
So the overall system is, just to reiterate, a system which
consists of modulating the original sequence with a pulse
train or impulse train.
And then that is going to be processed
with a low pass filter.
The spectrum of the original signal x of n
is what I show here.
The spectrum of the sampled signal, where I'm drawing the
picture on the assumption that the sampling period is 3, is
now what's indicated, where these are replicated, where
the original spectrum is replicated.
This is now processed through a filter which, for exact
reconstruction, is an ideal low pass filter.
And so we would multiply this spectrum by this one.
And the result, after doing that, will generate a
reconstructed spectrum which, in fact, is
identical to the original.
So the frequency domain picture is the same.
And what we would expect then is that the time domain
picture would be the same.
Well, let's in fact look at the time domain.
And in the time domain, what we have is an analysis more or
less identical to what we had in continuous time.
We of course have the same system.
And in the time domain, we are multiplying
by an impulse train.
Consequently, the sample sequence is an impulse train
whose values are samples of x of at integer multiples of
capital N. For the reconstruction, this is now
processed through an ideal low pass filter.
And that implements a
convolution in the time domain.
And so the reconstructed signal is the convolution of
the sample sequence and the filter impulse response.
And expressed another way, namely writing out the
convolution as a sum, we have this expression.
And so it says then that the reconstruction is carried out
by replacing the impulses here, these impulses, by
versions of the filter impulse response.
Well, if the filter is an ideal low pass filter, then
that corresponds in the time domain to sine nx over sine x
kind of function.
And that is the interpolation in between the samples to do
the reconstruction.
Also, as is discussed somewhat in the text, we can consider
other kinds of interpolation, for example discrete time zero
order hold or discrete time first order hold, just as we
had in continuous time.
And the issues and analysis for the discrete times zero
order hold and first order hold are very similar to what
they were in continuous time-- the zero order hold just
simply holding the value until the next sampling instant, and
the first order hold carrying out linear interpolation in
between the samples.
Now in this sampling process, if we look again at the wave
forms involved, or sequences involved, the process
consisted of taking a sequence and extracting from it
individual values.
And in between those values, we have sequence
values equal to 0.
So what we're doing in this case is retaining the same
number of sequence values and simply setting some number of
them equal to 0.
Well, let's say, for example, that we want
to carry out sampling.
And what we're talking about is a sequence.
And let's say this sequence is stored in a computer memory.
As you can imagine, the notion of sampling it and actually
replacing some of the values by zero is somewhat
inefficient.
Namely, it doesn't make sense to think of storing in the
memory a lot of zeros, when in fact those are zeros that we
can always put back in.
We know exactly what the values are.
And if we know what the sampling rate was in discrete
time, then we would know when and how to put
the zeros back in.
So actually, in discrete time sampling, what we've talked
about so far is really only one part or
one step in the process.
Basically, the other step is to take those zeros and just
throw them away because we could put them in any time we
want to and really only retain, for example in our
computer memory or list of sequence values or whatever,
only retain the non-zero values.
So that process and the resulting sequence that we end
up with is associated with a concept called decimation.
What I mean by decimation is very simple.
What we're doing is, instead of working with this sequence,
we're going to work with this sequence.
Namely, we'll toss out the zeros in between here and
collapse the sequence down only to the sequence values
that are associated with the original x of n.
Now, in talking about a decimated sequence, we could
of course do that directly from this step down to here,
although again in the analysis it will be somewhat more
convenient to carry that out by thinking, at least
analytically, in terms of a 2-step process--
one being a sampling process, then the other being a
decimation.
But basically, this is a decimated version of that.
Now for the grammatical purists out there, the word
decimation of course means taking every tenth one.
The implication is not that we're always sampling with a
period of 10.
The idea of decimating is to pick out every nth sample and
end up with a collapsed sequence.
Let's now look at a little bit of the analysis and understand
what the consequence is in the frequency domain.
In particular what we want to develop is how the Fourier
transform of the decimated sequence is related to the
Fourier transform of the original sequence or the
sample sequence.
So let's look at this in the frequency domain.
So what we have is a decimated sequence, which consists of
pulling out every capital Nth value of x of n.
And of course that's the same as we can either decimate x of
n or we can decimate the sample signal.
Now in going through this analysis, I'll kind of go
through it quickly because again there's the issue of
some slight mental gymnastics.
And if you're anything like I am, it's usually best to kind
of try to absorb that by yourself quietly, rather than
having somebody throw it at you.
Let me say, though, that the steps that I'm following here
are slightly different than the steps that
I use in the text.
It's a slightly different way of going through the analysis.
I guess you could say for one thing that if we've gone
through it twice, and it comes out the same, well of course
it has to be right.
Well anyway, here we have then the relationship between the
decimated sequence, the original sequence, and the
sampled sequence.
And we know of course that the Fourier transform of the
sample sequence is just simply this summation.
And now kind of the idea in the analysis is that we can
collapse this summation by recognizing that this term is
only non-zero at every nth value.
And so if we do that, essentially making a
substitution of variables with n equal to small m times
capital N, we can turn this into a summation on m.
And that's what I've done here.
And we've just simply used the fact that we can collapse the
sum because of the fact that all but every nth value is
equal to zero.
So this then is the Fourier transform all
of the sampled signal.
And now if we look at the Fourier transform of the
decimated signal, that Fourier transform, of course, is this
summation on the decimated sequence.
Well, what we want to look at is the correspondence between
this equation and the one above it.
So we want to compare this equation to this one.
And recognizing that this decimated sequence is just
simply related to the sample sequence this way, these two
become equal under a substitution of variables.
In particular, notice that if we replace in this equation
omega by omega times capital N, then these two equations
become equal.
So the consequence of that, then, what it all boils down
to and says, is that the relationship between the
Fourier transform of the decimated sequence and the
Fourier transform of the sampled sequence is simply a
frequency scaling corresponding to dividing the
frequency axis by capital N.
So that's essentially what happens.
That's really all that's involved in
the decimation process.
And now, again, let's look at that pictorially
and see what it means.
So what we want to look at, now that we've looked in the
time domain in this particular view graph, we now want to
look in the frequency domain.
And in the frequency domain, we have, again, the Fourier
transform of the original sequence and we have the
Fourier transform of the sampled sequence.
And now the Fourier transform of the decimated sequence is
simply this spectrum with a linear frequency scaling.
And in particular, it simply corresponds to multiplying
this frequency axis by capital N. And notice that this
frequency now, 2 pi over capital N, that frequency ends
up getting rescaled to a frequency of 2 pi.
So in fact now, in the rescaling, it's that this
point in the decimation gets rescaled to this point.
And correspondingly, of course, this whole spectrum
broadens out.
Now we can also look at that in the context of
the original spectrum.
And you can see that the relationship between the
original spectrum and the spectrum of the decimated
signal corresponds to simply linearly scaling this.
But it's important also to keep in mind that that
analysis, that particular relationship, assumes that
we've avoided aliasing.
The relationship between the spectrum of the decimated
signal and the spectrum of the sample signal is true whether
or not we have aliasing.
But being able to clearly associate it with just simply
scaling of this spectrum of the original signal assumes
that the spectrum of the original signal, the shape of
it, is preserved when we generate the sample signal.
Well, when might discrete time sampling, and for that matter,
decimation, be used?
Well, I indicated one context in which it might be useful at
the beginning of this lecture.
And let me now focus in on that a little more
specifically.
In particular, suppose that we have gone through a process in
which the continuous time signal has been converted to a
discrete time signal.
And we then carry out some additional
discrete time filtering.
So we have a situation where we've gone through a
continuous to discrete time conversion.
And after that conversion, we carry out some
discrete time filtering.
And in particular, in going through this part of the
process, we choose the sampling rate for going from
the continuous time signal to the sequence so that we don't
violate the sampling theorem.
Well let's suppose, then, that this is the spectrum of the
continuous time signal.
Below it, we have the spectrum of the output of the
continuous to discrete time conversion.
And I've chosen the sampling frequency to be just high
enough so that I avoid aliasing.
Well that then is the lowest sampling frequency I can pick.
But now, if we go through some additional low pass filtering,
then let's see what happens.
If I now low pass filter the sequence x of n, then in
effect, I'm multiplying the sequence
spectrum by this filter.
And so the result of that, the product of the filter
frequency response and the Fourier transform of x of n
would have a shape somewhat like I indicate below.
Now notice that in this spectrum, although in the
input to the filter this entire band was filled up, in
the output of the filter, there is a band that in fact
has zero energy in it.
So what I can consider doing is taking the output sequence
from the filter and in fact resampling it, in other words
sampling it, which would be more or less associated with a
different sampling rate for the continuous
time signals involved.
So I could now go through a process which is commonly
referred to as down sampling that is lowering
the sampling rate.
When we do that, of course, what's going to happen is that
in fact this spectral energy will now fill
out more of the band.
And for example, if this was a third, then in fact if I down
sampled by a factor of three, then I would fill up the
entire band with this energy.
But since I've done some additional low pass filtering,
as I indicate here, there's no problem with aliasing.
If I had, let's say, down sampled by a factor of three
and I'm now taking that signal and converting it back to a
continuous time signal, then of course the way I can do
that is by simply running my output clock for the discrete
to continuous time converter.
I can run my output clock at a third the rate
of the input clock.
And that, in effect, takes care of the
bookkeeping for me.
So here we have now the notion of sampling a sequence, and
very closely tied in with that, the notion of decimating
a sequence, and related to both of those, the notion of
down sampling, that is changing the sampling rates so
that, if we were trying this in with continuous time
signals, we've essentially changed our clock rate.
And we might also want to, and it's important to, consider
the opposite of that.
So now a question is what's the opposite of decimation.
Suppose that we had a sequence and we decimate it.
Thinking about it as a 2-step process, that would correspond
to first multiplying by an impulse train, where there are
bunch of zeros in there, and then choosing, throwing away
the zeros and keeping only the values that are non-zero,
because the zeros we can always recreate.
Well, in fact, the inverse process is very specifically a
process of recreating the zeros and then doing the
desampling.
So in the opposite operation, what we would do is undo the
decimation step.
And that would consist of converting the decimated
sequence back to an impulse train and then processing that
impulse train by an ideal low pass filter to do the
interpolation or reconstruction, filling in the
values which, in this impulse train, are equal to zero.
So we now have the two steps.
We take the decimated sequence and we expand it
out, putting in zeros.
And then we desample that by processing it through a low
pass filter.
So just kind of looking at sequences again, what we have
is an original sequence, the sequence x of n.
And then the sample sequence is simply a sequence which
alternates, in this particular case, those
sequence values was zero.
Here what we're assuming is that the sampling period is 2.
And so every other value here is equal to zero.
The decimated sequence then is this sequence, collapsed as I
show in the sequence above.
And so it's, in effect, time compressing the sample
sequence or the original sequence so that we throw out
the sequence values which were equal to zero
in the sample sequence.
Now in recovering the original sequence from the decimated
sequence, we can think of a 2-step process.
Namely, we spread this out alternating with zeros, and
again, keeping in mind that this is drawn for the case
where capital N is 2.
And then finally, we interpolate between the
non-zero values here by going through a low pass filter to
reconstruct the original sequence.
And that's what we show finally on the bottom curve.
So that's what we would see in the time domain.
Let's look at what we would see in the frequency domain.
In the frequency domain, we have to begin with the
sequence on the bottom, or the spectrum on the bottom, which
would correspond to the original spectrum.
Then, through the sampling process, that is periodically
replicated.
Again, this is drawn on the assumption that the sampling
frequency is pi or the sampling period is equal to 2.
And so this is now replicated.
And then, in going from this to the spectrum of the
decimated sequence, we would rescale the frequency axis so
that the frequency pi now gets rescaled in the spectrum for
the decimated sequence to a frequency which is 2 pi.
And so this now is the spectrum of
the decimated sequence.
If we now want to reconvert to the original sequence we would
first intersperse in the time domain with zeros,
corresponding to compressing in the frequency domain.
This would then be low pass filtered.
And the low pass filtering would consist of throwing away
this replication, accounting for a factor which is the
factor capital N, and extracting the portion of the
spectrum which is associated with the spectrum of the
original signal which we began with.
So once again, we have decimation and interpolation.
And the decimation can be thought of as a time
compression that corresponds to a frequency expansion then.
And the interpolation process is then just the reverse.
Now there are lots of situations in which decimation
and interpolation and discrete time sampling are useful.
And one context that I just want to quickly draw your
attention to is the use of decimation and interpolation
in what is commonly referred to as sampling rate
conversion.
What the basic issue and sampling rate conversion is is
that, in some situations, and a very common one is digital
audio, a continuous time signal is sampled.
And those sampled values are stored or whatever.
And kind of the notion is that, perhaps when that is
played back, it's played back through a different system.
And the different system has a different assumed sampling
frequency or sampling period.
So that's kind of the issue and the idea.
We have, let's say, a continuous time signal which
we've converted to a sequence through a sampling process
using an assumed sampling period of T1.
And these sequence values may then, for example, be put into
digital storage.
In the case of a digital audio system, it may, for example,
go onto a digital record.
And it might be the output of this that we want to recreate.
Or we might in fact follow that with some additional
processing, whatever that additional processing is.
And I'll kind of put a question mark in there because
we don't know exactly what that might be.
And then, in any case, the result of that is going to be
converted back to a continuous time signal.
But it might be converted through a system that has a
different assumed sampling period.
And so a very common issue, and it comes up as I indicated
particularly in digital audio, a very common issue is to be
able to convert from one assumed sampling period, T1,
our sampling frequency, to another
assumed sampling period.
Now how do we do that?
Well in fact, we do that by using the ideas of decimation
and interpolation.
In particular, if we had, for example, a situation where we
wanted to convert from a sampling period, T1, to a
sampling period which was twice as long as that, then
essentially, we're going to take the sequence and process
it in a way that would, in effect, correspond to assuming
that we had sampled at half the original frequency.
Well how do we do that?
The way we do it is we take the sequence we have and we
just throw away every other value.
So in that case, we would then, for this sampling rate
conversion, down sample and decimate.
Or actually, we might not go through this step formally.
We might just simply decimate.
Now we might have an alternative situation where in
fact the new sampling period, or the sampling period of the
output, is half the sampling period of the input,
corresponding to an assumed sampling frequency, which is
twice as high.
And in that case, then., we would go through a process of
interpolation.
And in particular, we would up sample and interpolate by a
factor of 2 to one.
So in one case, we're simply throwing
away every other value.
In the other case, what we're going to do is take our
sequence, put in zeros, put it through a low pass filter to
interpolate.
Now life would be simple if everything happened in simple
integer amounts like that.
A more common situation is that we may have an assumed
output sampling period which is 3/2 of the
input sampling period.
And now the question is what are we going to do to convert
from this sampling period to this sampling period.
Well, in fact, the answer to that is to use a combination
of down sampling and up sampling, or up sampling and
down sampling, equivalently interpolation and decimation.
And for this particular case, in fact, what we would do is
to first take the data, up sample by a factor of 2, and
then down sample the result of that by a factor of 3.
And what that would give us is a sampling rate conversion,
overall, of 3/2, or a sampling period conversion of 3/2.
And more generally, what you could think of is how you
might do this if, in general, the relationship between the
input and output sampling periods was some rational
number p/q.
And so in fact, in many systems, in hardware systems
related to digital audio, very often the sampling rate
conversion, most typically the sampling rate conversion, is
done through a process of up sampling or interpolating and
then down sampling by some other amount.
Now what we've seen, what we talked about in a set of
lectures, is the concepts of sampling a signal.
And what we've seen is that the signal can be represented
by samples under certain conditions.
And the sampling that we've been talking about is sampling
in the time domain.
And we've done that for continuous time and we've done
it for discrete time.
Now we know that there is some type of duality both
continuous time and discrete time, some type of duality,
between the time domain and frequency domain.
And so, as you can imagine, we can also talk about sampling
in the frequency domain and expect that, more or less, the
kinds of properties and analysis will be similar to
those related to sampling in the time domain.
Well I want to talk just briefly about that and leave
the more detailed discussion to the text and
video course manual.
But let me indicate, for example, one context in which
frequency domain sampling is important.
Suppose that you have a signal and what you'd like to measure
is its Fourier transform, its spectrum.
Well of course, if you want to measure it or calculate it,
you can never do that exactly at every single frequency.
There are too many frequencies, namely, an
infinite number of them.
And so, in fact, all that you can really calculate or
measure is the Fourier transform at a set of sample
frequencies.
So essentially, if you are going to look at a spectrum,
continuous time or discrete time, you can only really look
at samples.
And a reasonable question to ask, then, is when does a set
of samples in fact tell you everything that there is to
know about the Fourier transform.
That, and the answer to that, is very closely related to the
concept of frequency domain sampling.
Well, frequency domain sampling, just to kind of
introduce the topic, corresponds and can be
analyzed in terms doing modulation in the frequency
domain, very much like the modulation that we carried out
in the time domain for time domain sampling.
And so we would multiply the Fourier transform of the
signal whose spectrum is to be sampled by an
impulse train in frequency.
And so shown below is what might be a representative
spectrum for the input signal.
And the spectrum, then for the signal associated with the
frequency domain sampling, consists of multiplying the
frequency domain by this impulse train.
Or correspondingly, the Fourier transform of the
resulting signal is an impulse train in frequency with an
envelope which is the original spectrum
that we were sampling.
Well, this of course is what we would do in
the frequency domain.
It's modulation by an impulse train.
What does this mean in the time domain?
Well, let's see.
Multiplication in the time domain is convolution in the
frequency domain.
Convolution in the frequency domain is multiplication--
I'm sorry.
Multiplication in the frequency domain, then, is
convolution in the time domain.
And in fact, the process in the time domain is a
convolution process.
Namely, the time domain signal is replicated at integer
amounts of a particular time associated with the spacing in
frequency under which we're doing the
frequency domain sampling.
So in fact, if we look at this in the time domain, the
resulting picture corresponds to an original signal whose
spectrum or Fourier transform we've sampled.
And a consequence of the sampling is that the
associated time domain signal is just like the original
signal, but periodically replicated, in time now, not
frequency, but in time, at integer multiples of 2 pi
divided by the spectral sampling interval omega 0.
And so this then is the time function associated with the
sample frequency function.
Now, that's not surprising because what we've done is
generated an impulse train and frequency
with a certain envelope.
We know that an impulse train in frequency is the Fourier
transform of a periodic time function.
And so in fact, we have a periodic time function.
We also know that the envelope of those impulses--
we know this from way back when we talked about Fourier
transforms--
the envelope, in fact, is the Fourier
transform of one period.
And so all of this, of course, fits together as it should in
a consistent way.
Now given that we have this periodic time function whose
Fourier transform is the samples in the frequency
domain, how do we get back the original time function?
Well, with time domain sampling, what we did was to
multiply in the frequency domain by a gate, or window,
to extract that part of the spectrum.
What we do here is exactly the same thing, namely multiply in
the time domain by a time window which extracts just one
period of this periodic signal, which would then give
us back the original signal that we started with.
Now also let's keep in mind, going back to this time
function and the relationship between them, then again,
there is the potential, if this time function is too long
in relation to 2 pi divided by omega 0, there's the potential
for these to overlap.
And so what this means is that, in fact, what we can end
up with, if the sample spacing and the frequency is not small
enough, what we can end up with is an overlap in the
replication in the time domain.
And what that corresponds to and what it's called is, in
fact, time aliasing.
So we can have time aliasing with frequency domain sampling
just as we can have frequency aliasing
with time domain sampling.
Finally, let me just indicate very quickly that, although
we're not going through this in any detail, the same basic
idea applies in discrete time.
Namely, if we have a discrete time signal and if the
discrete time signal is a finite length, if we sample
its Fourier transform, the time function associated with
those samples is a periodic replication.
And we can now extract, from this periodic signal, the
original signal by multiplying by an appropriate time window,
the product of that giving us the reconstructed time
function as I indicate below.
So we've now seen a little bit of the notion of frequency
domain sampling, as well as time domain sampling.
And let me stress that, although I haven't gone into
this in a lot of detail, it's important.
It's used very often.
It's naturally important to understand it.
But, in fact, there is so much duality between the time
domain and frequency domain, that a thorough understanding
of time domain sampling just naturally leads to a thorough
understanding of frequency domain sampling.
Now we've talked a lot about sampling.
And this now concludes our discussion of sampling.
I've stressed many times in the lectures associated with
this that sampling is a very important topic in the context
of our whole discussion, in part because it forms such an
important bridge between continuous time and discrete
time ideas.
And your picture now should kind of be a global one that
sees how continuous time and discrete time fit together,
not just analytically, but also practically.
Beginning in the next lecture, what I will introduce is the
Laplace transform and, beyond that, the Z transform.
And what those will correspond to are generalizations of the
Fourier transform.
So we now want to turn our attention back to some
analytical tools, in particular developing some
generalizations of the Fourier transform in both continuous
time and discrete time.
And what we'll see is that those generalizations provide
us with considerably enhanced flexibility in dealing with
and analyzing both signals and linear time invariant systems.
Thank you.