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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.
