Lecture 24, Butterworth Filters - Signals and Systems Video Lecture - Electronics and Communication Engineering (ECE) | Best Video for Electronics and Communication Engineering (ECE)

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ALAN V. OPPENHEIM: Last time,
we introduced the notion of
mapping continuous time filters
to discrete time
filters, and we developed
impulse invariance as one
useful technique for carrying
out that type of mapping.
What I'd like to do in this
lecture is illustrate impulse
invariance as a design procedure
in the context of
one specific class of filters,
namely Butterworth filters.
And then following that, we'll
proceed on to discuss another
very important and useful
mapping or design procedure
referred to as the bilinear
transformation.
So to begin, let me just discuss
briefly what the class
of Butterworth filters is.
And specifically, the
Butterworth filters are
defined through their frequency
response or transfer function.
And I'm using capital B to
denote a Butterworth filter.
And by definition, the magnitude
squared of the
frequency response of a
Butterworth filter is given by
this expression.
And for example, if N were equal
to 1, then this would
simply correspond to the
magnitude squared of the
frequency response for a first
order differential equation.
Now if you look at the frequency
response, B of omega
for this class of filters--
I've illustrated that below--
and what we see is that the
frequency response starts at
unity, because that's the way
that it's normalized, and it
has a monotonic characteristic
in the pass band
and in the stop band.
At a frequency equal to the
parameter omega sub c, up
here, which is referred to as
the cutoff frequency, the
Butterworth filter frequency
response always goes through
the same point, namely 0.707.
And as the order of the
Butterworth filter, capital N,
increases, the transition from
the pass band to the stop band
becomes sharper and sharper.
So for higher order filters,
then, the frequency response
is flatter in the pass band and
drops off more quickly,
and attenuates more
in the stop band.
Now in designing Butterworth
filters, what we want to look
at is the location of the poles
of the system function.
And we can infer those from
our definition of what the
frequency response for the
Butterworth filter is.
In particular, we have this
expression for the magnitude
squared of the frequency
response.
And we recognize that, of
course, as B of j omega times
B of minus j omega, that's what
the magnitude squared
will be equal to.
And in order to convert this
to an expression describing
the system function, or in
terms of the more general
Laplace transform variable s,
what we recognize is that j
omega, in the more general
setting, simply corresponds to
the Laplace transform
variable s.
So this product, in fact, is
the Laplace transform for s
equal to j omega.
More generally, then, this is
the result of evaluating B of
s times B of minus s at
s equals j omega.
Consequently, comparing this
expression with this statement
leads us to the conclusion that
the transfer function B
of s of the Butterworth filter,
times B of minus s, is
given by the expression that
I indicate here, simply
replacing j omega by s.
Now what we want to look at
are the poles of B of s.
That's what we'll want
to get as we design
a Butterworth filter.
And we can recognize the poles
of this product simply by
looking at the roots of the
denominative polynomial.
And those roots are--
just taking account of
this factor 2N--
those roots are at j omega
sub c times the 2N
roots of minus 1.
Those roots, in fact, all
lie on a circle, and the
consequence of that is that the
poles of this expression
are on a circle.
The circle is of radius omega
sub c, and the poles are
distributed around the circle.
So here I've illustrated the
poles of B of s times B of
minus s, for this specific
case where capital
N is equal to 3.
So there are a total of six
poles around this circle.
And then for this specific case,
the poles are spaced by
60 degrees.
Now we have B of s times
B of minus s.
To get the system function for
the Butterworth filter, we'd
like to get B of s, and
the question now is
how do we get that?
Well, the thing to recognize is
that wherever this factor
has a root, this factor
has to have a root at
the negative location.
So in fact, when we look at
these poles, we compare them
with this, for example,
associated with B of s, this
associated with B of minus s.
And likewise, we compare these
two together, likewise we
compare these two together.
And so we can extract B of s
from this product, simply by
taking one pole out of
each of those pairs.
Now a question, of course, is
out of each pair, which one do
we associate with B of s, and
which do we associate with B
of minus s?
And the answer drops out fairly
simply if we recognize
that if we want to design
filters that are stable, then
B of s, the transfer function
that we're designing, must
have all its poles in the left
half of the s plane.
So in fact, out of each of these
pairs we would associate
the left half plane pole with
B of s, and so the transfer
function for the Butterworth
filter for this particular
case, where the this designates
the parameter omega
sub c and capital N is 3,
namely a third order
Butterworth filter, is this
set of pole locations.
Given those, of course, we can
figure out, simply through
Algebraic means, what the
transfer function B of s is.
All right, so that's what
Butterworth filters are.
And now what I'd like to do is
talk about the design of a
digital Butterworth filter using
the design technique
that we introduced last time,
namely impulse invariance.
And the context in which I will
phrase the design is the
context of mapping a continuous
time signal to a
discrete time signal, carrying
out filtering using the
discrete time filter that we're
designing, and then
mapping back.
So we're talking about now a
discrete time filter that we
want to design through impulse
invariance from continuous
time Butterworth filters, and
we're going to get our design
specifications in the context of
having considered discrete
time processing of continuous
time signals, where we will
map from a continuous time
signal to a sequence, carry
out the filtering with the
discrete time filter that
we're going to design.
And then we will take the
resulting filtered output and
map it back to the continuous
time signal.
But this discrete time filter is
the one that we're talking
about designing.
And for a choice of parameter,
there's a sampling frequency,
of course, involved
in this process.
And the value that I'll pick for
the sampling frequency is
10 kilohertz.
OK, all fairly straightforward
so far.
And so since we have a sampling
rate of 10 kilohertz,
we want to first look at our
specifications on the desired
continuous time filter and then
map those to appropriate
specifications on the discrete
time filter.
And what I'll pick for the
desired specifications on the
continuous time filter is that
at 1 kilohertz, I will ask
that the continuous time
frequency response be down by
no more than 1 db in comparison
with its value at
omega equals 0.
So that, in effect, specifies
the behavior in the pass band,
or the specifications
on the pass band.
And for the stop band, I'll
specify that the filter is
down by 15 db by the time we've
gotten to 1.5 kilohertz.
So we have, essentially, the
beginning of the transition
band-- the end of the pass
band--at 1 kilohertz, and the
beginning of the stop band
at 1.5 kilohertz.
And since we're talking about
designing a Butterworth
filter, we know that the
Butterworth filter is
monotonic in the pass band and
stop band, and so we'll have a
filter specification, something
as I show here.
This represents the allowable
pass band tolerance.
This is the allowable
stop band tolerance.
And if I can draw this without
getting myself into trouble,
essentially we're looking for
a filter, then, that always
stays between the specified
boundaries here.
Now what we have to figure out
is what the corresponding
specifications are for
the digital filter.
And the strategy, let me
emphasize, is that we have a
situation where we're doing
discrete time processing of
continuous time signals,
and we have a set of
specifications associated
with that.
That imposes specifications on
our discrete time filter, and
then we want to design the
discrete time filter using
impulse invariance, and that's
the discrete time filter that
we'll use in the
overall system.
All right, now we want
specifications on the discrete
time filter, and we want the
continuous overall equivalent
system to meet certain
specifications at certain
frequencies related to
continuous time frequencies.
Recall that when we sample
a continuous time signal,
there's a very specific mapping
from the continuous
time frequency axis to the
discrete time frequency axis.
In particular, the sampling
frequency gets mapped to 2 pi.
Well, that means that our other
critical frequencies get
mapped in proportion to that.
So 1 kilohertz, which is
a 1/10 of the sampling
frequency, would then convert
to a discrete time
frequency of 0.2 pi.
And 1.5 kilohertz will convert
to a discrete time
frequency of 0.3 pi.
So what this says is that for
the discrete time filter, we
would like the same behavior,
but at frequencies--
or the same specifications, but
at frequencies normalized
to the discrete time
frequency axis.
That means that we want the
discrete time frequency
response magnitude to be greater
than or equal to minus
1 db at 2/10 pi--
corresponding to the 1 kilohertz
in continuous time--
and for the beginning of the
stop band, that would occur at
0.3 pi, at which point we
want this less than or
equal to minus 15.
So those are our discrete
time specifications.
And we now want to design the
discrete time filter using
impulse invariance.
Now in impulse invariance, as
you recall, it corresponds to
generating an impulse response
which is a sampled version of
the continuous time impulse
response, and there is a
temptation naturally to think
all of this parameter, capital
T, as necessarily identical to
the sampling in the system in
which the filter is
going to be used.
Now this is a fairly subtle,
complicated, tongue twisting
issue, but the bottom line on
it, the essential point, is
that the parameter capital
T that we use in impulse
invariant design is a totally
different, unrelated, and in
fact, as it turns out, arbitrary
parameter, which is
not necessarily pegged to
the sampling frequency.
And I think it would be
difficult for me to totally
clarify that during
the lecture.
It's discussed more in the
book, and certainly, you
should take plenty of time
to reflect on it.
All right, but let's now look,
then, at where we are in our
design procedure, and
specifically, what it is that
we need to do in order
to design the
digital Butterworth filter.
Now we have a set of
specifications that we've
generated relating, essentially,
to how we want
the pass band of the digital
filter and the stop band of
the digital filter to behave.
Of course, since this isn't an
ideal filter, it has some
transition from pass band to
stop band, and as we discussed
last time, there is aliasing,
which we need to at
least be aware of.
We specified certain frequencies
along this axis
which are easily converted by
relating to the two axes
through this mapping, are easily
related back to the
continuous time frequency
axis, as we have here.
And in particular, now if we
were to simply pick that
parameter in the impulse
invariant design, capital T,
as equal to unity, and I
indicated just a minute ago
that we can pick it arbitrarily,
if I pick it as
unity, then the procedure would
consist of designing the
continuous time Butterworth
filter with meeting or
exceeding the appropriate
specifications and then going
through the impulse invariant
procedure.
All right, so let's
do that then.
We want the discrete time
impulse response to be the
continuous time impulse response
sampled, and for
convenience, I'm going to
pick this parameter
capital T equal to 1.
That means that the frequency
normalization between the
discrete time frequency axis
and the continuous time
frequency axis in fact is--
those axes are scaled
identically, because capital T
is equal to 1.
And so now we want the analog,
or continuous time,
specifications.
And so what we need to
do then is design
a Butterworth filter.
I'm using capital B here again
to denote the frequency
response of the Butterworth
filter.
The Butterworth filter to have
a magnitude which is greater
than or equal to minus 1 db
prior to the frequency 0.2 pi,
and less than or equal
to minus 15 at a
frequency beyond 0.3 pi.
And so now what we need to do
is determine capital N and
omega sub c in order to
meet or exceed those
specifications.
Now if you go through the
associated algebra in doing
that, let's say that you decide
that you want to pick
capital N and omega sub c
to exactly meet those
inequalities at the two
frequencies, 0.2 pi and 0.3
pi, what you'll find after going
through the algebra is
that they're exactly met if
capital N is 5.88 and omega
sub c is 0.7047.
And this obviously isn't
satisfactory as parameters for
the Butterworth filter.
Why is that?
The reason is that capital N
isn't an integer, and in the
class of Butterworth filters,
to generate a rational
transfer function,
capital N, this
parameter, must be an integer.
So since it can't be 5.88, the
natural thing to do is to move
it up to the next closest
integer, namely 6.
And that means that we'll end
up with a filter that does
even better than the
specifications.
On the other hand, there's
something kind of underneath
the surface that is inherent
in the impulse invariant
design procedure, namely the
fact that there will always be
some aliasing.
So one strategy, and a natural
one, often, in impulse
invariant design, is to choose
N as the next highest integer
as I've done, and then choose
the parameter omega sub c so
that the pass band
specifications are exactly
met, and the stop band
specifications are then
slightly exceeded, and
that will leave
some margin for aliasing.
All right, now continuing this
example, then, how would we
complete the design?
Well, we know what our two
parameters, capital N and
omega sub c are.
That means that we can
determine B of s
times B of minus s.
Those poles are located on a
circle in the complex s plane.
And the poles on that circle
are paired, some being
associated with B of s and
some with B of minus s.
And in particular, to determine
B of s, we would
simply take the poles on the
portion of the circle that's
in the left half
of the s plane.
Now what that gives us is a
Butterworth continuous time
filter, which we are then
mapping through impulse
invariance with capital
T equal to 1 to a
discrete time filter.
That discrete time filter to be
used in a system which has
an associated sampling
frequency which is 10 kilohertz.
So to get the discrete time
filter, then, we first
determine B of s,
as we just did--
or at least I indicated
how to do it.
We would then expand that out
in a partial fraction
expansion, and then apply the
impulse invariant procedure,
which consists of mapping the
poles in the s plane to poles
in the z plane at locations e
to the s of k, capital T,
capital T is equal to 1.
And retaining the
residue, or the
coefficient, in the expansion.
And this, then, will give us the
transfer function for the
discrete time filter.
So if in fact we did that, then
the resulting frequency
response that we would get is
what I've indicated here.
And I indicated first on a
magnitude scale, linear
magnitude scale, and second
on a logarithmic scale.
And as we had originally
specified, the digital filter
is supposed to be greater than
or equal to minus 1 db at
point 2 pi, and less than
or equal to minus
15 db at 0.3 pi.
And in fact, this slightly
exceeds those specifications
since we had purposely allowed
some margin in the stop band.
Now this is an illustration
of the impulse invariant
procedure, and it has a number
of very nice properties, one
of which is that it takes
a continuous time filter
frequency response and it
converts it to a discrete time
frequency response, which in the
absence of aliasing, looks
identical, except for a linear
frequency scale change.
And in fact, if we picked
capital T equal to 1, then
there is no scale change.
It has the major disadvantage
that there is always aliasing,
and for some problems, for
example, if the filter that
we're trying to design is not
band limited or low pass, then
the aliasing will naturally
become intolerable.
Well, there's another design
procedure which I now want to
introduce, which totally
avoids the problems of
aliasing, but, obviously,
then has its own costs
associated with it.
And that's procedure is referred
to as the bilinear
transformation.
The bilinear transformation,
which I won't try to derive
here in any detail, is a mapping
of continuous time
filters to discrete time
filters, corresponding to
taking the Laplace transform
variable s in the continuous
time filter and replacing it
by what is referred to as a
bilinear function of z.
And so if I substitute this in
here, that will give me the
discrete time frequency
response.
Again in this procedure, there's
a parameter capital T,
which again is totally
irrelevant given the approach
that we're taking, and which
we will generally tend to
normalize out to unity.
And let me just say quickly and
in passing that although
we won't go through this, the
notion that the bilinear
transformation can be tied to
the concept of taking the
differential equation for the
continuous time filter,
converting it to an integral
equation by integrating enough
times on both sides, and then
converting that to a
difference equation by
approximating the integrals
with the trapezoidal rule.
And that, in effect, will
correspond to mapping the
continuous time filter to a
discrete time filter with the
bilinear transformation.
Well, we'll just focus on the
properties of the mapping.
And in particular, if we were
to substitute into the
expression z equal to e to the
j omega corresponding to the
unit circle, we would find
that the unit circle in
discrete time corresponds to
mapping the j omega axis in
continuous time, which is
exactly what we want.
Now the mapping between the
continuous time frequency and
the discrete time frequency is
a nonlinear mapping, which is
given by the algebraic
expression
that I indicate here.
And if we plot this mapping,
what we have is this curve.
And what this corresponds to,
then, is a mapping of the j
omega axis, or continuous
time frequency, to
discrete time frequency.
And if we think more generally
of the mapping represented by
the bilinear transformation in
the context of the s-plane and
the z-plane, it corresponds to
mapping the j omega axis in
the s-plane to once around the
unit circle in the z-plane.
And you could also convince
yourself that the left half of
the s-plane maps to the inside
of the unit circle.
And so that means that stable
continuous time filters will
always map to stable discrete
time filters, which is exactly
what we desire.
Now notice in this that there's
no issue of aliasing.
What's happened is that we've
replaced s by a function of z,
and it corresponds to mapping
the s-plane to the z-plane.
In fact, the whole j omega axis
has mapped to once around
the unit circle, which obviously
requires some type
of nonlinear mapping, because,
look, the j omega axis is
infinitely long.
The unit circle has
a finite radius.
And so essentially, what has to
happen is if you think of
walking along the continuous
time frequency axis, and
simultaneously walking around
the unit circle, if you walk
at a constant rate around the
unit circle and you're
simultaneously walking up the
j omega axis, if you want to
get around to pi by the time,
here, you've gotten to
infinity along the continuous
time frequency axis, you
better start walking faster,
and faster, and faster,
because you've got an infinite
distance to cover while you
just go over a finite
distance here.
Well, what all that says,
really, is that that's why, in
fact, we're taking
the entire--
or we're able to take the entire
omega axis and just map
it into an interval
of length pi.
All right, now that means that
there is a nonlinear
distortion of the frequency
axis if we were to take a
continuous time filter and
convert it to a discrete time
filter with the bilinear
transformation.
How do we account for that, or
for that matter, when can we
really use it?
And we can see how to both take
account of it, and what
its limitations are, by
recognizing the following.
Suppose that I want to design
a discrete time filter and
what is going to happen is that
it will be mapped to a
continuous time filter, or the
relationship between the two
frequency axes will be
given by this curve.
So let's suppose that the
continuous time frequency
response looks as I've shown
here, with a pass band cutoff
frequency and a stop band
cutoff frequency.
If this were mapped through the
bilinear transformation to
a discrete time filter, then
this cutoff frequency would
fall over here, which is related
through this curve,
and this cutoff frequency, the
stop band edge would be here.
Again, it's these frequencies
reflected through this curve.
So let's suppose now that what
I'd like to do is design a
discrete time filter, where the
discrete time filter has
certain frequency
specifications, which is what
our previous example has.
It has, let's say, a specified
stop band edge and a specified
pass band edge, and a specified
stop band edge.
The design procedure would then
correspond to mapping
those frequencies to the
corresponding continuous time
frequencies, designing the
continuous time filter to meet
the specifications based on
those critical frequencies,
then taking the continuous time
design and mapping it
back to a discrete time filter
through the bilinear
transformation.
All right, now what we want to
do is again map a Butterworth
filter, continuous time
Butterworth filter to a
digital filter in such a way
that the digital filter
approximately meets the
specifications that we had before.
And let me just remind
you of where we were.
The critical frequencies were
at 0.2 pi and 0.3 pi.
And what we had asked for is
that the frequency response be
down by no more than 1 db up
to 0.2 pi, and down by at
least 15 db at 0.3 pi.
So we want to design the same
discrete time filter as we did
before with impulse
invariance.
We now want to do it with the
bilinear transformation
applied to an appropriate
Butterworth design.
OK, well, let's see what kind
of specifications we have.
We know that the frequency, the
critical frequencies, are
mapped through this curve,
or this equation.
And I indicated again that this
parameter capital T is
arbitrary in the design
procedure.
That may seem confusing,
initially, but there's some
further discussion of it in the
book, and it's true and
important to sort out.
So we're going to pick capital
T equal to unity.
And that means then that the
corresponding critical
frequencies of the continuous
time filter are at twice the
tangent of 0.2 pi over 2,
and twice the tangent
of 0.3 pi over 2.
So that means then that the
specifications on our
continuous time filter are given
by this, essentially
pass band and stop band
edges warped through
that nonlinear curve--
this nonlinear curve.
And I'm using capital G here to
denote the system function
for the resulting continuous
time filter.
And so now we could think of
designing a Butterworth filter
that, let's say, exactly meets
these specifications.
Well, if you do that, what
you'll find is that you get
exact equality here if you pick
capital N equal to 5.3.
And again, we have the issue
that if we want to meet or
exceed the specifications, we
can't make the filter order
lower, we have to
make it higher.
And so we would make it, instead
of 5.3, we would make
the filter order equal to 6.
And now, again, we have several
options and trade offs.
Before, with impulse invariance,
we essentially
decided to meet the pass band
specifications and exceed the
stop band specifications
to provide
some margin for aliasing.
Here, we don't have any
aliasing, and we can trade
things off anyway we'd like.
The way that I've chosen to do
it is to exactly meet the stop
band cut off, and exceed the
pass band specifications.
And the result of doing that
is to choose a filter order
capital N equal to 6, and the
parameter omega sub c in the
Butterworth filter is
given by 0.76622.
All right, so now we have the
parameters for the continuous
time Butterworth filter, which
when mapped to the discrete
time filter through the bilinear
transformation, will
exceed our requirements in the
pass band and just meet the
stop band cutoff.
And so now we want to complete
the design procedure.
And given the parameters of the
Butterworth filter, we can
draw our appropriate circle in
the s-plane, which happens to
be the same circle as before,
but with a different radius.
And then associate poles on this
circle with B of s and B
of minus s.
In particular, take
the ones in the
left half of the s-plane--
same ones we had before except
we have a different value for
omega sub c--
and those, then, represent
B of s.
So we can now determine
B of s.
And once we have the transfer
function B of s, we then map
that to a discrete time filter
by mapping through the
bilinear transformation.
And that then is the design
procedure which we would
follow this example.
All right, well, let's just see
what the result looks like
when we're done.
Here I show, again, on both a
linear magnitude scale and on
a logarithmic or DB scale, the
frequency response of the
resulting filter.
And recall that the
specifications were that at
0.2 pi, we should be down by no
more than 1 db, and clearly
we've exceeded that
considerably.
And at 0.3 pi, we should be down
by at least 15 db, and we
chose the design so that we
would exactly meet that edge.
So we've met this point and
exceeded this point, and this
is our design.
Now let's just compare this
with the impulse invariant
design that we saw a
few minutes ago.
The impulse invariant design
is what I indicate here.
The bilinear transformation
design is now
overlaid on top of it.
It's hard to see much difference
on a linear
magnitude scale, but let's look
on a logarithmic scale.
And this one that I'm lifting
is the bilinear
transformation, and this
is impulse invariance.
And notice, in fact, something
very interesting, which is
that the filter that we obtained
by mapping the
Butterworth filter to a digital
filter through the
bilinear transformation in fact
falls off in frequency
much more rapidly than the one
that we got through impulse
invariance.
The question is why?
Now one thought that might come
to mind is, well, impulse
invariance has aliasing, the
bilinear transformation
doesn't have aliasing,
that must be a
consequence of aliasing.
In fact, that's not
the reason.
Aliasing, as it turns out in
this particular design, was
relatively minimal in the
impulse invariant design.
The reason has to do with this
nonlinear mapping in the
bilinear transformation from the
continuous time frequency
to the discrete time
frequency.
Keep in mind that through that
mapping, as you start walking
around the unit circle and
moving up to j omega axis, as
you move up the j omega axis,
you have to move faster, and
faster, and faster,
and faster.
And what's in continuous time
frequency at infinity is what
you get to in the discrete time
frequency by the time you
get around to pi.
So in fact, what we're looking
at as we look out along this
frequency axis is we're seeing
higher, and higher, and higher
frequencies in the continuous
time filter.
By the time we get to pi, we
should in fact be in the
continuous time filter
equivalently off to infinity,
which sounds like a pretty
uncomfortable place to be.
OK, now this was a fairly rapid
trip through a number of
issues, in particular, some of
the issues associated with the
bilinear transformation, and
also this issue of how you
pick this parameter capital T,
and how it might be associated
with a sampling frequency if
you're doing discrete time
processing of continuous
time signals.
And we don't have time to
explore some of those issues
more fully in this lecture.
But I'd like to conclude by
making a couple of comments.
One comment is that the two
techniques that we've talked
about, impulse invariance and
the bilinear transformation,
are the two techniques that are
principally used when one
thinks of mapping continuous
time filters to discrete time
filters for whatever
application.
And I stress again that you may
want to do that mapping
whether or not the discrete
time filter is eventually
going to be used for processing
continuous time signals.
Now impulse invariance had the
very nice characteristic that
it corresponds to a linear
mapping between the two
frequency axes, except for the
issue of aliasing, and that's
a problem with it, and in
particular, limits its
usefulness to filter designs, or
for mapping continuous time
filters that are band limited.
On the other hand, we have the
bilinear transformation as a
design procedure, which totally
avoids aliasing, but
has the disadvantage or
difficulty that it represents
a nonlinear mapping from the
continuous time filter to the
discrete time filter.
Now this nonlinear distortion
is perfectly acceptable if
we're designing or attempting
to design filters that have
flat frequency characteristics.
It's not acceptable if, for
example, we had a linear
frequency characteristic that we
wanted to map to a discrete
time filter and end up with
a linear frequency
characteristic.
It won't come out to be linear
because of this nonlinear
mapping of the frequency axis.
Now there are also a number of
other design procedures, which
we won't go into, for designing
discrete time
filters, and among them are
a variety of techniques,
including, for example,
computer aided design
procedures.
And I invite you, if you're
interested and want to dig
into that in more detail and
more deeply, to explore that
topic by making reference to
various of the books listed in
the bibliography in the text.
Thank you.