[MUSIC PLAYING]
PROFESSOR: Last time, we began the discussion of
discreet-time processing of continuous-time signals.
And, as a reminder, let me review the basic notion.
The idea was that we convert from a continuous-time signal
to a sequence through an operation which I represent as
a continuous to discrete time converter.
And then that sequence is used as the input to an appropriate
discreet-time system.
And after appropriate discreet-time processing, that
sequence is converted back to a continuous-time signal
through an operation which I label as a discrete to
continuous time converter.
Now, in the lecture last time, we carried out some analysis
which related for us the spectra in the first step of
this operation.
Namely in the transformation from a continuous-time time
signal to a sequence.
And let me, by the way, draw your attention to the fact
that, in the real world, this operation is essentially
implemented by what you would typically label as an analog
to digital converter, if in fact the discreet-time
processing is being done digitally.
Now, it's important to emphasize that it's not
exactly what an analog to digital converter does, but,
in some sense at least, you should think of this mapping
from continuous-time to discreet-time in very much the
same way that one would think of an
analog to digital converter.
And the mapping back then corresponds, in some sense, to
what would happen with a digital to analog converter.
Well, let me review what is involved in the mapping from
the continuous-time signal to the sequence.
And, let me stress again that, this operation is basically--
in the continuous to discreet-time conversion--
a two-step process.
In the first part of the process, the continuous-time
signal is modulated with impulse train, where the
period of the impulse train is capital T. And so we have a
continuous-time time impulse train signal which captures
the samples of the original continuous-time signal.
That impulse train is then put through an operation which,
essentially, re-labels the samples so that the sample
values, the impulse areas, are re-labeled as sequence values.
And the result of that conversion is then the
sequence x of n.
So the overall process, then, is a sampling process,
followed by what is simply, in this box,
a re-labeling process.
And, although as I indicated just a minute ago, that is
essentially what an analog to digital converter does.
An analog to digital converter doesn't necessarily carry it
out in those two steps, but particularly, in terms of
carrying through an analysis, thinking of it as a two-step
process is particularly convenient.
Now we talked last time about what this mapping from
continuous-time to discreet-time means, both in
the time domain, and terms of the spectra.
And in particular, in the time domain we begin with the
continuous-time signal, which is then sampled with an
impulse train and converted to a sequence by simply
generating a sequence whose values are the
areas of the impulses.
And I stress the fact that what this corresponds to,
essentially, is a normalization of the time
axis, essentially, by dividing the time axis by capital T.
In the frequency domain, then, we had the spectrum of the
original signal, which, because of the sampling
process, is replicated at integer multiples of the
sampling frequency omega sub s, or 2 pi over capital T. And
then, in converting the impulses to a sequence, we are
essentially normalizing the frequency axis, so that the
frequency 2 pi over capital T gets re-labeled as 2 pi.
And the resulting discreet-time spectrum looks
like I indicate here.
Which really is nothing more than a frequency scaling
corresponding to the associated time scaling.
So the mapping from the impulse train spectrum to the
discreet-time spectrum corresponds to a mapping
specified by capital omega equal to small omega times
capital T.
And equivalently, it's the frequency 2 pi over capital T,
which is, of course, the sampling frequency which gets
normalized to the frequency 2 pi.
And so, in the frequency domain, there is a frequency
normalization associated with the fact that corresponding to
this spectrum is a time sequence or discreet-time
sequence, as I showed previously, and the
discreet-time sequence is related to the original
continuous-time signal through a time normalization.
In other words, these sequence values are simply samples of
the continuous-time signal with the time axis
renormalized.
Now, what we want to consider--
this is the conversion from continuous-time to
discreet-time--
what we want to consider now is the overall system which
implements not just the conversion, but filtering, and
then coming back out of the conversion back to
continuous-time.
So let's look at the overall system.
And the overall system, of course, as I've stressed
several times in the past, consists of first, the
sampling process, conversion to an impulse train, and the
impulse train converted to a sequence.
That sequence is then processed through our
discreet-time filter.
And after the discreet-time time processing, the result of
that is converted back to an impulse train.
So this resulting process sequence is then converted
back to an impulse train.
And then, finally, we carry out the desampling process by
simply using a low-pass filter with a cutoff associated with
the sampling frequency that we used.
Now, typically in a system like that-- which implements
discreet-time processing of continuous-time time signals--
we need to ensure in one way or another that the bandwidth
of the input is sufficiently limited, so
that we avoid aliasing.
One way to do that is to force it one way or another, or
simply know that our signal satisfies the bandwidth
constraint.
Although, a fairly typical thing to do in addition to
this sampling process is to include what is referred to an
anti-aliasing filter.
In other words, this is a filter that would band-limit
the input at at least half the sampling frequency, so that we
are guaranteed, then, that there is no aliasing that's
carried out in this process.
And it's important to stress that, in this kind of
processing--
discreet-time processing of continuous-time signals--
except in certain special situations, it's very
important to avoid aliasing because we're going to want to
do a reconstruction after we do the sampling and
processing.
OK, now, this is the sequence of steps in the time domain.
Let's examine what happens as a consequence of this in the
frequency domain.
Well, let's choose some type of simple
representative spectrum.
And, of course, what's important about it is that the
spectrum we choose is band-limited, or that there's
an anti-aliasing filter.
And it's not the shape, of course, that is critical.
And as we work our way through the system, this is the
continuous-time spectrum.
After sampling, that spectrum is replicated at multiples of
the sampling frequency--
integer multiples of the sampling frequency--
and so there would be another one over here, and another one
over here, et cetera.
And then, in converting to a discreet-time sequence, there
is the associated frequency normalization, so that the
sampling frequency gets normalized to a
frequency of 2 pi.
OK, now, at that point, where we are in the system is at
this point, so that we've converted to a sequence.
We now want to carry out some filtering, and then, after
that filtering, convert back to a continuous-time signal.
All right, so, here we are at the spectrum associated with
the sequence.
And now, the processing that we're carrying out is linear
time and variant filtering in the discreet-time domain.
And what that corresponds to, then, is multiplying this
spectrum by the filter frequency response.
And I've chosen a particular shape.
And again, it's not the shape that's important to the
discussion, but the fact, for example, that it has a
particular cutoff frequency, which we will track as we work
through this.
And so now, the spectrum of y of n, the output of the
digital filter, is the product of this spectrum, and the
Fourier transform, or frequency response, of the
digital filter.
Now, in working our way through, we're going to take
the output of the filter and undo the two-step process.
So we now want to take that sequence, convert it to an
impulse train, and then take that impulse train and
desample through a low-pass filter.
So, here we are now at the output of the digital filter.
We then convert that to an impulse train.
Well that's really undoing the original time normalization.
And so, what that means, is that we are undoing the
frequency normalization.
In particular, we're dividing the frequency axis by capital
T. Whereas, this point in y of omega was 2 pi, now it's 2 pi
over capital T. What that means is that, equivalently,
we're multiplying this spectrum by the frequency
response of the digital filter, but now
linearly-scaled in frequency, so that what was a cutoff
frequency of omega sub c is now cutoff frequency of omega
sub c, divided by capital T.
So now, the next step in the process is the reconstructing
low-pass filter.
And what that extracts is simply the portion of this
periodic spectrum around the origin.
And so finally, then, the spectrum of the output of the
overall system will be the spectrum of the input
multiplied by a frequency response, which is the digital
filter frequency response frequency scaled by dividing
that digital filter frequency axis by capital T.
OK, now, what we can ask is, we've got this processing--
we've converted to discreet-time, and we've gone
back to continuous-time, and one can ask now what
equivalent, overall continuous-time system does
that correspond to?
In other words, if we--
that, of course, is a continuous-time system, it's a
continuous-time input and continuous-time time output--
and the overall system, then, would be one that would give
us exactly the same output spectrum as we're getting.
Well, what is that?
What we have is an output spectrum, which is the product
of the input spectrum and the digital filter frequency
characteristic frequency-scaled.
And so, in fact, the resulting continuous-time time filter is
simply the digital filter with an
appropriate frequency scaling.
In other words, with the frequency axis divided by
capital T. So said another way, if we show here the
frequency response of the original digital filter, then
the corresponding continuous-time filter would
be this, frequency-scaled.
And then, because of the associated low-pass filtering
and the reconstruction, we would select out just one of
these periods-- in particular, the portion around the origin.
And the essential consequence of that is that the
corresponding continuous-time filter,
then, is given by this.
And these two are related simply by a linear scaling of
the frequency axis.
And note that, where the digital filter has a cutoff
frequency of omega sub c, the continuous-time filter has a
cutoff frequency of omega sub c divided by capital T.
So that's the linear frequency scaling.
And, by the way, plant away for now-- and we'll return to
this point later--
the observation that even if the digital filter frequency
response is fixed, which we would assume it is, by
changing the sampling frequency, in fact, what we're
able to do is affect a linear scaling all of the equivalent
continuous-time filter.
OK, well, this is pretty much the process and the analysis,
but to highlight a number of the issues and emphasize these
points, what I'd like to do is illustrate some of this with a
videotape demonstration that, in fact, was made originally
as part of another course-- a course devoted entirely to
digital signal processing, which essentially is
discreet-time time processing, whether or not it's related to
continuous-time signals.
And what I'd like to now focus on are some of the details of
that demonstration.
In the demonstration, the specific impulse response that
is used for the digital filter, or discreet-time
filter, is the one that I show here.
And the associated frequency response is the frequency
response of a discreet-time, low-pass filter,
as I indicate below.
And the cutoff frequency of that filter--
as I indicate, the filter was designed as a discreet-time
filter with a cutoff frequency of pi over 5.
And let me just draw your attention to the fact that pi
over 5 is also a 10th of 2 pi.
And so in fact the digital or discreet-time filter cutoff
frequency is a 10th of 2 pi.
And as I'll stress again shortly, remember that, in the
frequency normalization or unnormalization, 2 pi
represents, in effect, the sampling frequency.
And so the consequence of that is that the cutoff frequency
really, is going to be associated with a 10th of the
sampling frequency.
But, for now, keep in mind that it's just
simply a 10th of 2 pi.
Now, the equivalent continuous-time system, in
terms of the impulse response, is, of course, a band-limited
interpolation of the impulse response associated with the
discreet-time filter.
And in the frequency domain, the frequency response is
correspondingly a time-scaled, frequency scaled version of
the frequency response.
So, in fact, in the frequency domain and in the time domain
related to the continuous-time signal, the associated impulse
response is what I indicate here--
a band-limited interpolation of the discreet-time impulse
response and time scale, in fact.
And the frequency response--
following the discussion that we've
previously gone through--
is a frequency-scaled version of the one associated with the
digital filter.
Well, the first thing that I'll want to look at is the
impulse response.
And when we do, let me just indicate that in the actual
implementation things are slightly different than they
are associated with the ideal analysis.
In particular, in converting from a discreet-time sequence
to the continuous-time signal, whereas this way of looking at
it is convenient in the context of the analysis, in
fact, the way it's done is using a more or less standard
digital to analog converter.
And what a digital to analog converter does, as I indicated
in the previous lecture, is to convert the sequence not to an
impulse train but, in fact, to go directly through a
zero-order hold.
And so, usually what comes out of a digital to analog
converter is a staircase type of signal associated with a
zero-order hold.
And then, the result of that is low-pass filtered to do the
reconstruction.
So what we'll want to look at, then, is that reconstruction,
first with just an impulse input.
And so, what we'll see after the low-pass filter, for the
impulse response, is a smooth curve like this.
But also, as part of the demonstration, what I'll do,
just to show the zero-order hold, is to take the low-pass
filter out temporarily and then put it back in.
So first, let's just look at the filter impulse response.
What we see here is the impulse response of the
overall system.
And we observe, for one thing, that it's a symmetrical
impulse response.
In other words, corresponds to a linear phase filter.
We could also look at the impulse response before the
desampling low-pass filter-- let's take out the desampling
low-pass filter slowly--
and what we observe is, basically, the output of the
digital to analog converter.
Which, of course, is a staircase, or boxcar,
function, not an impulse train.
In the real world, the output of a D to A converter,
generally, is a boxcar type of function.
We can put the desampling filter back in now and notice
that the effect of the desampling filter is,
basically, to smooth out the rough edges in the boxcar
output from the D to A converter.
OK, so, that's the impulse response of the system.
Now, what I'd like to show is the frequency
response of the system.
And to measure the frequency response, of course, what we
can do is put a sine wave into the system and look at the
sinusoidal output.
So, in particular now, what will happen is that, with the
system, we will put in a continuous-time sinusoid,
which is sampled, converted to a sequence.
The sampled continuous-time sinusoid is a
discreet-time sinusoid.
That goes through the digital filter and gets attenuated, or
amplified appropriately.
And then the output of that is converted back-- and that's,
again, a sinusoidal output--
that gets converted back to a continuous-time sinusoid.
Theoretically, as I indicate here-- but again, as we just
saw, really, represented by a zero-order hold, followed by a
low-pass filter.
So, that's the overall operation, with one
modification from the diagram that we have here.
In this particular diagram I've included and
anti-aliasing filter.
In fact, in the demonstration there is no
anti-aliasing filter.
And so, in fact, the input is a sinusoidal input which is
not band-limited by virtue of an anti-aliasing filter.
It's only, of course, band-limited appropriately if
we choose the sinusoidal frequency that way.
So, there is no anti-aliasing filter,
and this is the system.
And one consequence of that is that, in fact, if we sweep the
input sinusoid only up to half the sampling frequency, we'll
see no aliasing.
But if we let it sweep past that, we're
going to get aliasing.
Now, in the demonstration, the sampling rate that's picked
for this part of the demonstration is a 20
kilohertz sampling rate.
That means, based on the sampling theorem, that as long
as the input frequency is below 10
kilohertz we get no aliasing.
When the input frequency goes beyond 10 kilohertz , that
higher frequency is going to get aliased down
into a lower frequency.
A consequence of that, then, is that as we go through the
processing, and we demonstrate the frequency response of the
system, what we'll see in the output is no aliasing when the
input is below 10 kilohertz.
As the input sweeps past 10 kilohertz--
when we let it, which we will eventually in the
demonstration--
then, in fact, that frequency, as it finally shows up here,
will begin to be aliased down into a lower frequency.
Another way of thinking about that is that, when we watch
the frequency response of the system, as we look at the
digital filter frequency response, what we're sweeping
as we go from 0 up to 10 kilohertz in the input
frequency is this portion of the frequency response.
As we sweep from 10 kilohertz out to 20 kilohertz, what
we'll see is this portion of the frequency response.
In other words, we'll see it periodically replicated.
Or, if we look at the corresponding continuous-time
frequency response, what it means, really, is that
sweeping from 0 to 10 kilohertz is
moving up this way.
And then sweeping from 10 kilohertz to 20 kilohertz on
the input, really because of the aliasing, reflects itself
in the digital filter by looking back toward lower
frequencies.
And so the continuous-time filter sweeps back down from
10 kilohertz back to 0.
OK, so, that's what we'll see, and we'll see it in several
different ways as explained in the demonstration.
So now let's look at the frequency
response of the filter.
Now what we'd like to illustrate is the frequency
response of the equivalent continuous-time filter.
And we can do that by sweeping the filter
with sinusoidal input.
So, what we'll see in this demonstration is, on the upper
trace, the input sinusoid, on the lower trace, the output
sinusoid, using a 20 kilohertz sampling rate, and a sweep
from 0 to 10 kilohertz.
In other words, a sweep from 0 to, effectively, pi, in terms
of the digital filter.
So what we'll observe as the input frequency increases, is
that the output sinusoid will have, essentially, constant
amplitude up to the cutoff frequency of the filter, and
then approximately zero amplitude past.
So let's now sweep the filter frequency response.
And there is the filter cutoff frequency.
Now, we can also observe the filter frequency response in
several other ways.
One way in which we can observe it is by looking,
also, at the amplitude of the output sinusoid as a function
of frequency, rather than as a function of time.
And so we'll observe that on the left-hand scope.
While on the right-hand scope, we'll have the same trace the
we just saw, namely two traces--
the upper trace is the inputs sinusoid, the lower trace is
the output sinusoid.
And, in addition to observing the frequency response, let's
also listen to the output sinusoid and observe the
attenuation in the output as we go from the filter passband
to the filter stopband.
Again, a 20 kilohertz sampling rate and a sweep range from 0
to 10 kilohertz.
Now, of course, we're in the filter stopband.
Now, if we increase the sweep range from 10 kilohertz the 20
kilohertz, so that the sweep range is equal to the sampling
frequency, in essence, that corresponds to sweeping out
the digital filter from 0 to 2 pi.
And, in that case, we'll begin to see some of the periodicity
in the digital filter frequency response.
So let's do that now with a 20 kilohertz sampling rate and a
sweep range of 0 to 20 kilohertz.
Now as we come near 2 pi, we get back the past-band.
And, finally, back to a 0 to 10 kilohertz sweep, so that
we're again sweeping only from 0 to pi with regard to the
digital filter.
Now, an important observation is that, with the digital or
discreet-time filter cutoff frequency fixed as I've
indicated here--
and I remind you that what the cutoff frequency is,
is a 10th of 2 pi--
with that cutoff frequency fixed, because of the
normalization that we get as we come back to a
continuous-time filter, in fact, what we have is a cutoff
frequency that is dependent on the sampling frequency or on
the sampling period.
And, more specifically, since the discreet-time, or digital,
filter or has a cutoff frequency which is a 10th of 2
pi, the normalization, as you recall, is that 2 pi, in
discreet-time frequency, corresponds to omega sub s,
the sampling frequency, in terms of
continuous-time frequency.
The consequence is that this cutoff frequency, in
fact, is 1/10 of--
not 2 pi now because of the normalization--
it's 1/10 of the sampling frequency.
So, consequently, as we change the sampling frequency, what
will happen is that, even with the discreet-time filter
cutoff fixed, the cutoff frequency of the equivalent
continuous-time filter will change.
Now, that's what I want to demonstrate.
But let me again stress and ask you to keep in mind that
this demonstration is done without an
anti-aliasing filter in.
And we are going to be changing the sampling
frequency and, so keep in mind that, as we look at this, as
the input frequency sweeps past half the sampling
frequency--
whatever sampling frequency we happen to be looking at--
then, because of the fact that there's no anti-aliasing
filter we'll get aliasing.
In other words, the frequency and the digital filter, or
discreet-time filter, as we sweep the input frequency up,
will move up in frequency until we get past half the
sampling frequency and then essentially will move back
down in frequency.
Consequently, what we'll get, then--
or what we'll see-- are periodic replications of the
frequency response when we swept past half
the sampling frequency.
All right, so now, let's look at the same digital filter,
but the frequency response, as we change
the sampling frequency.
Now, what we would like to demonstrate is the effect of
changing the sampling frequency.
And we know that the effective filter cutoff frequency is
tied to the sampling frequency and, for this particular
filter, corresponds to a 10th of the sampling frequency.
Consequently, if we double the sampling frequency, we should
double the effective filter passband width, or double the
filter cutoff frequency.
And, so, let's do that now.
Again a 0 to 10 kilohertz sweep range, but a 40
kilohertz sampling frequency.
And we should observe that the filter cutoff frequency has
now doubled out to four kilohertz.
Now, let's begin to decrease the filter sampling frequency.
So from 40, let's change the sampling
frequency to 20 kilohertz.
We should see the cutoff frequency cut in half.
Now, we can go even further.
We can cut the sampling
frequency down to 10 kilohertz.
And remember that the sweep range is 0 to 10 kilohertz.
So now we'll be sweeping from 0 to 2 pi.
So as we get close to 2 pi, we'll see the passband again.
And, now, let's cut down the sampling frequency even
further, to 5 kilohertz.
Here we are at 2 pi.
And then at 4pi.
All right, so, that illustrates the effect of
changing the sampling frequency.
Now let's conclude this demonstration of the effect of
the sampling frequency on the filter cutoff frequency by
carrying out some filtering on some live audio.
What we'll watch, in this case, is the output audio
waveform as a function of time on the single tray scope, and
also we'll listen to the output.
We'll begin it with a 40 kilohertz sampling rate, then
reduce that to 20 kilohertz, 10
kilohertz, and then 5 kilohertz.
And in each of those cases, the effective filter cutoff
frequency, then, is cut in half from 4 kilohertz, to 2
kilohertz, to 1 kilohertz, and then to 500 cycles.
So let's begin with a 40 kilohertz sampling frequency,
or an effective filter cutoff frequency of 4 kilohertz.
Now, let's reduce that a 20 kilohertz sampling frequency,
or a 2 kilohertz filter.
Then a 10 kilohertz sampling frequency.
And, finally, a 5 kilohertz sampling frequency
corresponding to a 500 cycle equivalent analog filter.
Alright, now, let's finally conclude by returning to a
little higher quality ragtime by changing the sampling
frequency back to 40 kilohertz.
Alright, well, hopefully what you've seen in the
demonstration and in this lecture gives you a sense and
a feeling for the analysis and the use of discreet-time
filters for processing continuous-time signals.
And as you may be aware, and as I've tried to indicate
previously in the past, this, in fact, is one very
important-- but not the only-- but one very important context
in which discreet-time filtering is used.
And this, in fact, is an area that is developing rapidly
because of the fact that microprocessors, digital
technology, computers, et cetera afford considerable
flexibility in carrying out digital processing of signals.
And when digital processing is used, that naturally
corresponds to implementing the processing and analyzing
it in discreet-time.
Now, in the next lecture we'll be
continuing on another aspect--
developing another aspect--
of sampling.
And, in particular, what we'll be talking about is sampling
of discreet-time signals.
As I'll indicate there, one of the contexts in which
discreet-time sampling, in fact, plays an important role
is in the context in which we are processing continuous-time
signals using discreet-time processing.
Where, in fact, one step that we might want to take-- in
addition to the steps so we've talked about here--
is an additional sampling process following whatever
kinds of filtering that we do.
Well, that's a discussion and a topic that we'll be going
into in the next lecture.
Thank you.