[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.
