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[MUSIC PLAYING]
PROFESSOR: Last time, we began the discussion of the
z-transform.
As with the Laplace transform in continuous time, we
developed it as a generalization
of the Fourier transform.
The expression that we got for the z-transform is the sum
that I indicate here.
We also briefly talked about an inverse z-transform
integral and some other informal methods of computing
the inverse z-transform.
But we focused in particular on the relationship between
the z-transform and the Fourier transform, pointing
out first of all that the z-transform, when we choose
the magnitude of z equal to 1--
so the magnitude of z of the form e to the j omega--
just simply reduces to the Fourier
transform of the sequence.
Then, in addition, we explored the z-transform for z, a more
general complex number.
In the discrete time z-transform case, we expressed
that complex number in polar form as r e to the j omega,
and recognize that the z-transform expression in fact
corresponds to the Fourier transform of the sequence
exponentially weighted.
Because of the exponential weighting, the z-transform
converges for some values of r corresponding to some
exponential waiting, and perhaps not for others, and
that led to a notion which corresponded to the region of
convergence associated with the z-transform.
We talked some about properties of the region of
convergence, particularly in relation to
the pole-zero pattern.
Now, they z-transform has a number of important and useful
properties, just as the Laplace transform does.
As one part of this lecture, what we'll want to do is
exploit some of these properties in the context of
systems described by linear constant coefficient
difference equations.
These particular properties that play an important role in
that context are the properties
that I indicate here.
In particular, there is--
as with continuous time--
a linearity property that tells us that the z-transform
of a linear combination of sequences is the same linear
combination of the z-transforms, a shifting
property that indicates that the z-transform of x of n
shifted is the z transform of x of n multiplied by a factor
z to the minus n 0.
Then the convolution property for which the z-transform of a
convolution of sequences is the product of the associated
z-transforms.
With all of these properties, of course, there is again the
issue of what the associated region of convergence is in
comparison with the region of convergence of
the original sequences.
That is an issue that's addressed somewhat more in the
text and let's not go into it here.
With the convolution property, the convolution property as in
continuous time, of course, provides a mechanism for
dealing with linear time invariant systems.
In particular, in the time domain a linear time invariant
system is described through convolution--
namely, the output is the convolution of the input and
the impulse response.
Because of the convolution property associated with the
z-transform, the z-transform of the output is the
z-transform of the input times the z-transform
of the impulse response.
Again, very much the same as what we had in continuous time
and also what we had in the context of the discussion with
the Fourier transform.
In fact, because of the relationship between the
z-transform and the Fourier transform, the z-transform of
the impulse response evaluated on the unit circle--
in other words, for the magnitude of z equal to 1--
in fact corresponds to the frequency
response of the system.
More generally, when we talk about the z-transform of the
impulse response, we will refer to it as the system
function associated with the system.
Now, the convolution property and these other properties, as
I indicated, we will find useful in talking about
systems which are described by linear constant coefficient
difference equations, and in fact, we'll do that shortly.
But first what I'd like to do is to continue to focus on the
system function for linear time invariant systems, and
make a couple of comments that tie back to some things that
we said in the last lecture relating to the relationship
between the region of convergence of a system, or--
I'm sorry, the region of convergence of a z-transform--
and the issue of where that is in relation to the poles of
the z-transform.
In particular, we can draw some conclusions tying back to
that discussion about the pole locations of the system
function in relation to whether the system is stable
and whether the system is causal.
In particular, recall from one of the early lectures way back
when that stability for a system corresponded to the
statement that the impulse response
is absolutely summable.
Furthermore, when we talked about the Fourier transform,
the Fourier transform of a sequence converges if the
sequence is absolutely summable.
So in fact, the condition for stability of a system and the
condition for convergence of the Fourier transform of its
impulse response are the same condition, namely absolute
summability.
Now what does this mean?
What it means is that if the Fourier transform converges,
that means that the z-transform converges on the
unit circle.
Consequently, if the system is stable, then the system
function, the z-transform or the impulse response, must
also converge on the unit circle.
In other words, the impulse response must have a Fourier
transform that converges.
So for a stable system, then, the region of convergence of
the system function must include the unit
circle in the z-plane.
So we see how stability relates to the location of the
region of convergence, and we can also relate causality to
the region of convergence.
In particular, we know that if a system is causal, then the
impulse response is right-sided.
For a sequence that's right-sided, the region of
convergence of its z-transform must be outside
the outermost pole.
So for causality, the region of convergence of the system
function must be outside the outermost pole.
For stability, the region of convergence must include the
unit circle, and we can also then draw from that the
conclusion that if we have a system that's causal unstable,
then all poles must be inside the unit circle because of the
fact that the poles must--
because of the fact that the region of convergence must be
outside the outermost pole and has to also
include the unit circle.
So for example, if we had let's say a system with a
system function as I indicate here with the algebraic
expression for the system function being the expression
that I indicate here with the pole at z equals a third and
another pole at z equals 2 and a zero at the origin.
If, in fact, the system was causal, corresponding to an
impulse response that's right-sided, this then would
be the region of convergence of the system function.
Alternatively, if I knew that the system was stable, then I
know that the region of convergence must include the
unit circle, and so this would be then the region of
convergence.
And what you might now want to ask yourselves is if instead
the region of convergence for the system function is this,
then is the system causal?
That's the first question.
The second question is is the system stable?
Remembering that for causality, the region of
convergence must be outside the outermost pole, and for
stability it must include the unit circle.
Now what I'd like to do is look at the properties of the
z-transform, and in particular exploit these properties in
the context of systems that are described by linear
constant coefficient difference equations.
The three basic properties that play a key role in that
discussion are the linearity property, the shifting
property, and the convolution property.
These, then, are the properties
that I want to exploit.
So let's do that by first looking at a first order
difference equation.
In the case of a first order difference equation, which
I've written as I indicate here--
no terms on the right hand side, but we could have terms
of course, in general--
y of n minus ay of n minus 1 equals x of n.
We can use the linearity property, so that if we take
the z-transform of both sides of this expression, that will
then be the z-transform of this term plus the z-transform
of this term.
So using those properties and together with the shifting
property, the property that tells us that the z-transform
of y of n minus 1 is z to the minus 1 times the z-transform
of y of n, we then convert the difference equation to an
algebraic expression.
And we can solve this algebraic expression for the
z-transform of the output in terms of the
z-transform of the input.
Now what we know from the convolution property is that
for a system, the z-transform of the output is the system
function times the z-transform of the input.
So this factor that we have, then, must correspond to the
system function, or equivalently the z-transform
of the impulse response of the system.
In fact, then, if we have this z-transform, we could figure
out what the impulse response of the system is by computing
or determining what the inverse z-transform, except
for the fact that expression is an algebraic expression and
doesn't yet totally specify the z-transform because we
don't yet know what the region of convergence is.
How do we get the region of convergence?
Well, we have the same issue here as we had with the
Laplace transform--
namely, the point that the difference equation tells us,
in essence, what the algebraic expression is for the system
function, but doesn't specify the region of convergence.
That is specified by either explicitly, because one way or
another we know what the impulse response is, or
implicitly, because we know certain properties of the
system, such as causality and or stability.
If I, let's say, imposed on this system, in addition to
the difference equation, the condition of causality, then
what that requires is that the impulse response be
right-sided or the region of convergence be outside the
outermost pole.
So, for this example that would require, then, that the
region of convergence correspond to the magnitude of
z greater than the magnitude of a.
What you might think about is whether I would get the same
condition if I required instead that
the system be stable.
Furthermore, you could think about the question of whether
I could specify or impose on this system that it be both
stable and causal.
The real issue-- and let me just kind of point to it--
is that the answer to those questions relate to whether
the magnitude of a is less than 1 or the magnitude of a
is greater than 1.
If the magnitude of a is less than 1 and I specify
causality, that will also mean that the system is stable.
In any case, given this region of convergence, then the
impulse response is the inverse transform of that
z-transform, which is a to the n times u of n.
Now let's look at a second order equation, and there's a
very similar strategy.
For the second order equation, the one that I've picked is of
this particular form, and I've written the coefficients
parametrically as I indicate here for a specific reason,
which we'll see shortly.
Again, I can apply the z-transform to this
expression, and I've skipped an algebraic step or two here.
When I do this, then, again I use the linearity property and
the shifting property, and I end up with this algebraic
expression.
If I now solve that to express y of z in terms of x of z and
a function of z, then this is what I get
for the system function.
So this is now a second order system function, and we'll
have two zeroes at the origin and it will have two poles.
Again, there's the question of what we assume about the
region of convergence--
I haven't specified that yet.
But if we, let's say, assume that the system is causal,
which I will tend to do, then that means that the region of
convergence is outside the outermost pole.
Now, where are the poles?
Well, let me just kind of indicate that if--
and you can verify this
algebraically at your leisure--
that if the cosine theta term is less than 1, then the roots
of this polynomial will be complex.
And in fact, the poles are at r e to the
plus or minus j theta.
So for cosine theta less than 1, then the poles are complex,
and the complex poles at an angle theta and with a
distance from the origin equal to the parameter r.
In fact, let's just look at that.
What I show here is the pole zero pattern.
Assuming that r is less than 1, and that cosine theta is
less than 1, and we have a complex pole pair shown here--
now if we assume that the system was causal, that means
that the region of convergence is outside these poles.
That would then include the unit circle, which means that
the system is also stable.
In fact, as long as the reach of convergence includes the
unit circle, we can also talk about the frequency response
of the system--
namely, we can evaluate the system
function on the unit circle.
We, in fact, evaluated the Fourier transform associated
with this pole zero pattern last time.
Recall that the frequency response, then, is one that
has a resonant character with the resonant peak being
roughly in the vicinity of the angle of the pole location.
As the parameter r varies-- let's say, as r gets smaller--
this peak tends to broaden.
As r gets closer to 1, the resonance
tends to get sharper.
This is now a look at the z-transform, and we see very
strong parallels to the Laplace transform.
In fact, throughout the course, I've tried to
emphasize--
and it just naturally happens--
that there are very strong relationships and parallels
between continuous time and discrete time.
In fact, at one point we specifically mapped from
continuous time to discrete time when we talked about
discrete time processing of continuous time signals.
What I'd like to do now is turn our attention to another
very important reason for mapping from continuous time
to discrete time, and in the process of doing this, what
we'll need to do is exploit fairly heavily the insight,
intuition, and procedures that we've developed for the
Laplace transform and the z-transform.
Specifically, what I would like to begin is a discussion
relating to mapping continuous time filters to discrete time
filters, or continuous time system functions to discrete
time system functions.
Now, why would we want to do that?
Well, there are at least several reasons for wanting to
map continuous time filters to discrete time filters.
One, of course, is the fact that in some situations what
we're interested in doing is processing continuous time
signals with discrete time systems--
or, said another way, simulate continuous time systems with
discrete time systems.
So it would be natural in a setting like that to think of
mapping the desired continuous time filter to a
discrete time filter.
So that's one very important context.
There's another very important context in which this is done,
and that is in the context or for the purpose of exploiting
established design procedures for continuous time filters.
The point is the following.
We may or may not be processing a sample continuous
time signal with our discrete time filter-- it may just be
discrete time signals that we're working with.
But in that situation, still, we need to design the
appropriate discrete time filter.
Historically, there is a very rich history associated with
design of continuous time filters.
In many cases, it's possible and very worthwhile and
efficient to take those designs and map them to
discrete time designs to use them as discrete time filters.
So, another very important reason for talking about the
kinds of mappings that we will be going into is to simply
take advantage of what has been done historically in the
continuous time case.
Now, if we want to map continuous time filters to
discrete time filters, then in continuous time, we're talking
about a system function and an associated
differential equation.
In discrete time, there is the corresponding system function
and the corresponding difference equation.
Basically what we want to do is generate from a continuous
time system in some way a discrete time system that
meets an associated set of desired specifications.
Now, there are certain constraints that it's
reasonable and important to impose on whatever kinds of
mappings we use.
Obviously, we want a mapping that will take our continuous
time system function and map it to a
discrete time system function.
Correspondingly in the time domain, there is a continuous
time impulse response that maps to the associated
discrete time impulse response.
These are more or less natural.
The two that are important and sometimes easy to lose sight
of are the two that I indicate here.
In particular, if we are mapping a continuous time
filter with, let's say, a desired or desirable frequency
response to a discrete time filter and we would like to
preserve the good qualities of that frequency response as we
look at the discrete time frequency response, then it's
important what happens in the s-plane for the continuous
time filter along the j omega axis relate in a nice way to
what happens in the z-plane around the unit circle,
because it's this over here that represents the frequency
response in continuous time and this contour over here
that represents the frequency response in discrete time.
So that's an important property.
We want to kind of the j omega axis to
map to the unit circle.
Another more or less natural condition to impose is a
condition that if we are assured in some way that our
continuous time filter is stable, then we would like to
concentrate on design procedures that more or less
preserve that and will give us stable digital filters.
So these are kind of reasonable conditions to
impose on the procedure.
What I'd like to do in the remainder of this lecture is
look at two common procedures for mapping continuous time
filters to discrete time filters.
The first one that I want to talk about is one that, in
fact, is very frequently used, and also one that, as we'll
see for a variety of reasons, is highly undesirable.
The second is one that is also frequently used, and as we'll
see is, in certain situations, very desirable.
The first one that I want to talk about is the more or less
intuitive simple procedure of mapping a differential
equation to a difference equation by simply replacing
derivatives by differences.
The idea is that a derivative is more or less a difference,
and there's some dummy parameter capital T that I've
thrown in here, which I won't focus too much on.
But in any case, this seems to have some plausibility.
If we take the differential equation and do this with all
the derivatives, both in terms of y of t and x of t, what
we'll end up with is a difference equation.
Now what we can use are the properties of the Laplace
transform and the z-transform to see what this means in
terms of a mapping--
in particular, using the differentiation property for
Laplace transforms.
In the Laplace transform domain, we would have this.
Using the properties for the z-transform, the z-transform
of this expression would be this.
So, in effect, what it says is that every place in the system
function or in the differential equation that we
would be multiplying by s when Laplace transformed.
In the difference equation, we would be
multiplying by this factor.
In fact, what this means is that the mapping from
continuous time to discrete time corresponds to taking the
system function and replacing s wherever we see it by 1
minus z to the minus 1 over capital T. So if we have a
system function in continuous time and we map it to a
discrete time system function this way by replacing
derivatives by differences, then that corresponds to
replacing s by 1 minus z to the minus 1 over capital T.
Now we'll see shortly what this mapping actually means
more specifically in relating the s-plane to the z-plane.
Let me just quickly, because I want to refer to this, also
point to another procedure very much like backward
differences which corresponds to replacing derivatives not
by the backward differences that I just showed, but by
forward differences.
In that case, then, the mapping corresponds to
replacing s by z to the minus 1 over capital T. It looks
very similar to the previous case.
So there the relationship between these system functions
is what I indicate here.
Let's just take a look at what those mappings correspond to
when we look at this specifically in the s-plane
and in the z-plane.
What I show here is the s-plane, and of course it's
things on the left half of the s-plane, poles on the left
half of the s-plane that would guarantee stability.
It's the j omega axis the tells us about the frequency
response, and in the z-plane it's the unit circle that
tells us about the frequency response.
Things inside the unit circle, or poles inside the unit
circle, that guarantee stability.
Now the mapping from s-plane to the z-plane corresponding
to replacing derivatives by backward differences in fact
can be shown to correspond to mapping the j omega axis not
to the unit circle, but to the little circle that I show
here, which is inside the unit circle.
The left half of the s-plane maps to the
inside of that circle.
What does that mean?
That means that if we have a really good frequency response
characteristic along this contour in the s-plane, we'll
see that same frequency response
along this little circle.
That's not the one that we want, though--
we would like to see that same frequency response around the
unit circle.
To emphasize this point even more-- suppose, for example,
that we had a pair of poles in our continuous time system
function that looked like this.
Then, where they're likely to end up in the z-plane is
inside the unit circle, of course.
But if the poles here are close to the j omega axis,
that means that these poles will be close to this circle,
but in fact might be very far away from the unit circle.
What would happen, then, is that if we saw in the
continuous time filter a very sharp resonance, the discrete
time filter in fact might very well have that resonance
broadened considerably because the poles are so far away from
the unit circle.
Now, one plus with this method, and it's about the
only one, is the fact that the left half of the s-plane maps
inside the unit circle-- in fact, inside a circle inside
the unit circle, and so stability is always
guaranteed.
Let me just quickly mention, and you'll have a chance to
look at this a little more carefully in the video course
manual, that for forward differences instead of
backward differences, this contour in the s-plane maps to
a line in the z-plane, which is a line tangent to the unit
circle, and in fact is the line that I showed there.
So not only are forward differences equally bad in
terms of the issue of whether they map from the j omega axis
to the unit circle, but they have a further difficulty
associated with them, which is the difficulty they may not
and generally don't guarantee stability.
Now, that's one method, and one, as I indicated, that's
often used partly because it seems
so intuitively plausible.
What you can see is that by understanding carefully the
issues and the techniques of Laplace and z-transforms, you
can begin to see what some of the difficulties with those
methods are.
The next method that I'd like to talk about is a method
that, in fact, is very commonly used.
It's a very important, useful method, which kind of can be
motivated by thinking along the lines of mapping the
continuous time filter to a discrete time filter in such a
way that the shape of the impulse response is
preserved--
and, in fact, more specifically so that the
discrete time impulse response corresponds to samples of the
continuous time impulse response.
And this is a method that's referred to as impulse
invariance.
So what impulse invariance corresponds to is designing
the filter in such a way that the discrete time filter
impulse response is simply a sample version of the
continuous time filter impulse response with a sampling
period which I denote here as capital T. That will turn into
a slightly confusing parameter shortly, and perhaps carried
over into the next lecture.
Hopefully, we'll get that straighted out, though, within
those two lectures.
Remembering the issues of sampling, the discrete time
frequency response, then since the frequency responses the
Fourier transform of the impulse response is related to
the continuous time impulse response as I indicate here,
what this says is that it is the superposition of
replications of the continuous time frequency response,
linearly scaled in frequency and shifted and
added to each other.
It's the same old sort of shifting, adding, or aliasing
issue-- the same sampling issues that
we've addressed before.
This equation will help us understand what the frequency
response looks like.
But in terms of an analytical procedure for mapping the
continuous time system function to a discrete time
system function, we can see that and develop it in the
following way.
Let's consider the continuous time system function expanded
in a partial fraction expansion.
And just for convenience, I'm picking first order poles--
this can be generalized multiple order poles, and we
won't do that here.
The same basic strategy applies.
If we expand this in a partial fraction expansion, and we
look at the impulse response associated with this-- we know
how to take the inverse of Laplace transform of this,
where I'm just naturally assuming causality throughout
the discussion--
the continuous time impulse response, then, is the sum of
exponentials with these amplitudes and at these
complex exponential frequencies.
Now, impulse invariance corresponds to sampling this,
and so the discrete time impulse response is simply a
sampled version of this.
The A sub k, of course, carries down.
We have the exponential--
we're sampling at t equals n capital T, and so we've
replaced that here, and then the unit step to truncate
things for negative time.
Let's manipulate this further, and eventually what we want to
get is a relationship--
a mapping--
from the continuous time to the discrete time filter.
We have this step, and we can rewrite that now as I show
here, just simply taking this n outside, and we have e to
the s sub k capital T. Now this is of the form the sum of
terms like A sub k times is beta to the n.
We can compute the z-transform of this, and the z-transform
that we get I show here--
it's simply A sub k over 1 minus e to the s sub k capital
T z to the minus 1, simply carrying this term or this
parameter down.
So we started with a continuous time system
function, which was a sum of terms like A sub k over s
minus s sub k-- the poles were at s sub k.
We now have the discrete time filter in this form.
Consequently, then, this procedure of impulse
invariance corresponds to mapping the continuous time
filter to a discrete time filter by mapping the poles in
the continuous time filter.
According to this mapping, the continuous time filter pole at
s sub k gets mapped to a pole e to the s sub k capital T,
and the coefficients A sub k are preserved.
That, then, algebraically, is what the procedure of impulse
invariance corresponds to.
Let's look at how we interpret some of this in
the frequency domain.
In particular, we have the expression that tells us how
the discrete time frequency response is related to the
continuous time frequency response.
This is the expression that we had previously when we had
talked about issues of sampling.
So that means that we would form the discrete time
frequency response by taking the continuous time 1, scaling
it in frequency according to this parameter capital T, and
then adding replications of that together.
So if this is the continuous time frequency response, just
simply an ideal low-pass filter with a cutoff frequency
of omega sub c, then the frequency scaling operation
would keep the same basic shape but linearly scale the
frequency axis so that we now have omega sub c times T. Then
the discrete time frequency response would be a
superposition of these added together at multiples of 2 pi
in discrete time frequency.
So that's what we have here--
so this is the discrete time frequency response.
This looks very nice-- it looks like impulse invariance
will take the continuous time frequency response, just
simply linearly scale the frequency axis, and
incidentally periodically repeat it.
We know that for an ideal low-pass filter,
that looks just fine.
In fact, for a band-limited frequency response,
that looks just fine.
But we know also that any time that we're sampling--
and here we're sampling the impulse response--
we have an effect in the frequency domain or the
potential for an affect an effect that
we refer to as aliasing.
So in fact, if instead of the ideal low-pass filter we had
taken a filter that was an approximation to a low-pass
filter, then the corresponding frequency scale version would
look as I've shown here.
And now as we add these together, then what we will
have is some potential for distortion corresponding to
the fact that these replications overlap, and what
that will lead to is aliasing.
So some things that we can say about impulse invariance is
that we have an algebraic procedure--
and I'll illustrate with another example shortly--
for taking a continuous time system function and mapping it
to a discrete time system function.
It has a very nice property in terms of mapping, the mapping
from the frequency axis in continuous time due to the
unit circle--
namely, to a first approximation.
As long as there's no aliasing, the mapping is just
simply a linear scaling of the frequency axis, although there
may be some aliasing.
That means, of course, that this method can only be used
if the frequency response that's being mapped, or if the
system that's being mapped, has a frequency response
that's approximately low-pass--
it has to be approximately band-limited.
Then what we have is some of potential distortion, which
comes about because of aliasing.
Also because of the mapping, the fact that poles at s sub k
get mapped to poles at e to the s sub k capital T, if the
analog or continuous time filter is stable--
meaning that the real part of s sub k is negative--
then the discrete time filter is guaranteed to be stable.
In other words, the magnitude of z sub k will be guaranteed
to be less than 1.
I'm assuming, of course, in that discussion that we are
always imposing causality on the systems.
To just look at the algebraic mapping a little more
carefully, let's take a simple example.
Here is an example of a system, a continuous time
system, where I simply have a resident pole pair with an
imaginary part along the imaginary axis of omega sub r
and a real part of minus alpha.
And so the associated system function then is just the
expression which incorporates the two poles, and I've put in
a scale factor of 2 omega sub r.
And now to design the discrete time filter using impulse
invariance, you would carry out a partial fraction
expansion of this, and that partial fraction expansion is
shown below.
We have a pole at minus alpha minus j omega r and at minus
alpha plus j omega r.
And to determine the discrete time filter based on impulse
invariance, we would map the poles and preserve the
coefficients a sub k, referred to as the residues.
And so the discrete time filter that we would end up
with as a system function, which I indicate here--
and we have, as I said, preserved the residue, and the
pole is now at a to the minus alpha T, e to the minus j
omega sub r T. That's one term, and the other term in
the sum has a pole at the complex conjugate location.
If we were to add these two factors together, then what we
would get is both poles and a 0 at the origin.
In fact, then, the pole is defined by its angle, and this
angle is e to the j omega sub r capital T, and by its
radius, and this radius is e to the minus alpha capital T.
Now we can look at the frequency response associated
with that, and let's just do that.
For the original continuous time frequency response, what
we have is simply a resonant character, as I've shown here.
And if we map this using impulse invariance, which we
just did, the resulting frequency response that we get
is the frequency response which I indicate.
We see that that's basically identical to the continuous
time frequency response, except for a linear scaling in
the frequency axis, if you just compare the dimensions
along which the frequency axis is shown except for one minor
issue, which is particularly highlighted when we look at
the frequency response at higher frequencies.
What's the reason why those two curves don't quite follow
each other at higher frequencies?
Well, the reason is aliasing.
In other words, what's happened is that in the
process so applying impulse invariance, the frequency
response of the original continuous time filter is
approximately preserved, except for some distortion,
that distortion corresponding to aliasing.
Well, just for comparison, let's look at what would
happen if we took the same example--
and we're not going to work it through here carefully.
We're not work it through it all, not even not carefully.
If we took the same example and mapped it to a discrete
time filter by replacing derivatives by backward
differences, what happens in that case is that we get a
frequency response that I indicate here.
Notice that the resonance in the original continuous time
filter is totally lost.
In fact, basically the character of the continuous
time frequency response is lost.
What's the reason?
Well, the reason goes back to the
picture that I drew before.
The continuous time filter had a pair of resident poles close
to the j omega axis.
When those get mapped using backward differences, they end
up close to this little circle that's inside the unit circle,
but in fact for this example, are far away
from the unit circle.
So far we have one useful technique for mapping
continuous time filters to discrete time filters.
In part to highlight some of the issues, I focused
attention also on some not so useful methods--
namely, mapping derivatives to forward or backward
differences.
Next time what I would like to do is look at impulse
invariance for another example--
namely, the design of a Butterworth filter, and I'll
talk more specifically about what Butterworth filters are
at the beginning of that lecture.
Then, in addition, what we'll introduce is another very
useful technique, which has some difficulties which
impulse invariance doesn't have, but avoids the principal
difficulty that impulse invariance does have--
namely, aliasing.
That method is referred to the bilinear transformation, which
we will define and utilize next time.
Thank you.