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[MUSIC PLAYING]
The last lecture we consider some basic structures
for infinite impulse response and actually also
for finite impulse response systems.
In this lecture, I would like to focus specifically
on finite impulse response systems and indicate, to show,
that for that class of systems, there
are some structures that specifically
exploit properties of FIR, or finite impulse
response, systems.
First of all, let's consider the general form
of a finite impulse response system.
One of the characteristics of a finite impulse response system,
of course, is the fact that the impulse response has
only a finite number of nonzero samples,
or equivalently the system function involves a sum only
over a finite range.
So the system function in general for a finite impulse
response system is of this form.
And the difference equation for this system
is that the form of a sum from k will 0 to capital N minus 1
of h of k, the unit sample response, x of n minus k.
We've stressed the fact, of course,
in a variety of the earlier lectures,
that the difference equation for a finite impulse response
system, in fact, corresponds to the convolution
sum for finite impulse response systems,
or the coefficients in the difference equation
are, in fact, the values of the unit sample response.
Now, for a finite impulse response system
we can, of course, talk about the same types of structures
that we talked about in a previous lecture.
For example, we can generate a direct form structure
from the difference equation, or equivalently
from the z transform.
And we see that the direct form structure corresponds
to generating a weighted combination of delayed inputs
where the weighting coefficients are the values in the unit
sample response h of n.
So for a general direct form structure,
we have x of n, x of n minus 1 at the output of this delay, x
of n minus 2, et cetera, x of n minus 1 multiplied by h of 1,
x of n multiplied by h of 0, down through the chain.
The results added up at the output
to produce the overall output y of n.
So this is what would correspond to applying
the notion of a direct form structure
to a finite impulse response system.
Likewise, we could talk about a transposed form
of this network.
We could also talk about a cascade form realization
of a finite impulse response filter,
simply by factoring this into a product,
not a poles and zeros, because, of course,
there aren't poles for a finite impulse response system,
but as a cascade of zeros, either
implemented as first or second order sections.
Something to give some thought to incidentally
is whether for a finite impulse response system
you can generate a parallel form structure
like the parallel form structure that we talked about last time.
The answer to that, in fact, is no.
And it's tied to the fact that an FIR
system doesn't have any poles.
Although as we'll see later on in this lecture,
there is a structure that is of the form of a parallel form
for implementing FIR systems.
So the first point then is that we
can apply many of the ideas of the previous lecture
to FIR systems.
But in addition, there are some properties of some FIR systems
that lead to some specific network structures that
in general can't be used for implementing IIR,
or recursive, systems.
The first of these that I'd like to talk about
is directed specifically toward the implementation
of linear phase FIR systems.
Now recall from several lectures ago--
and we've again raised this point a number of times--
that one of the differences between IIR and FIR systems
is that because of the fact that the impulse response of an FIR
system is a finite length, those systems
can in fact be designed to have exactly linear phase,
whereas the corresponding statement of course
is not true for the case of infinite impulse response
systems.
In particular, a linear phase finite impulse response system
is one for which the values in the unit sample response
satisfy a certain symmetry condition, namely the symmetry
condition as I've indicated it here,
h of n equal to h of capital N minus 1 minus little n,
where capital N is the length of the finite length impulse
response.
For example, a finite, a linear phase FIR impulse response
as one example might look, as I've indicated here,
where I've drawn it for capital N equal to 9.
And this symmetry property says that h of 0,
if I substitute n equals 0, is equal to h of capital N
minus 1, or h of 8.
And we can see that that's the symmetry that we have.
And then likewise, the linear phase property
says that as I go through the impulse response
from both ends, I see identical values
so that these values are identical.
Then these values are identical.
These, and, of course, this one is identical to itself.
And that then is the general symmetry of a linear phase FIR
impulse response.
Well, in fact, to stress once again that the phase is indeed
linear, we can consider generating a unit sample
response or impulse response h1 of n, which corresponds
to this unit sample response shifted
so that it's centered around the origin, in which case h of n
is equal to h1 of n plus capital N minus 1.
Or in this case, h of n is h1 of n shifted to the right by 4.
And consequently, the frequency response
for this system, or the Fourier transform of the unit sample
response is the Fourier transform of the unit sample
response of this system multiplied by e to the j omega
capital N minus 1 over 2.
That is just due to the fact that these two unit sample
responses are related to each other by a linear shift.
Now, this unit sample response is even.
That is it's even about n equals 0.
And consequently, its Fourier transform
is a real function of omega.
So this is real.
And consequently, the phase associated
with the causal system h of n is just simply this linear phase
term, e to the j omega N minus 1 over 2.
So this is an argument, again, that
stresses the fact that if the unit sample
response has this symmetry, then the phase of the system
is linear.
Now, how can we exploit this in the implementation
of finite impulse response systems?
Well, let's look, first of all, at the general expression
for the z transform of the system,
h of z given by a sum of h of n z to the minus n.
And essentially the idea in exploiting this symmetry
for the implementation of FIR systems
is to take advantage of the fact that because roughly half
of the values in h of n are identical,
we should be able to implement the system with essentially
only half the number of multiplies
that would be required in a more general FIR system.
Well, to show how this works, let's
work through this for the case where
we assume that capital N, the number
of non-zero points in the unit sample response, is even.
In the previous example, we'd considered capital N odd.
Here, in working through this let's
assume that capital N is even.
And a similar derivation would apply
and has gone through in the text for capital N odd.
OK, well, we can express the transfer function h of z
as a sum from n equals 0 to capital N over 2 minus 1
of h of n z to the minus n--
those are the first half points, the first half of the non-zero
points--
plus the sum over the second half
of the non-zero points from capital N over 2 to capital
N minus 1.
And basically where we want to get to
is to exploit the fact that these points are numerically
equal to these, except that the index runs
in the other direction.
Well to develop that formally, let's manipulate this sum
by using a substitution of variables.
In particular, let's make the substitution of variables
that r is equal to capital N minus 1 minus little n,
or equivalently that little n is equal to capital N minus 1
minus r.
If we do that, then changing this
to a summation on r, where we have that little n
is equal to capital N over 2, little n equal to capital N
over 2, r is equal to capital N over 2 minus 1,
as I've indicated here.
And when little n is equal to capital N minus 1,
little r is equal to 0.
So this gets converted to a sum from r equals 0
to capital N over 2 minus 1.
h of what was n, which is now capital N minus 1
minus r substituted in here.
And then capital N minus 1 minus r again substituted
for little n.
Basically, what we've done is reduce this
to a sum over the same limits as we have here
and with the indices running in the opposite direction.
And now, we can recognize that because of the symmetry
property, because of the fact that we're
talking about a linear phase system, h of capital
N minus 1 minus r is equal to h of r.
That's the symmetry property that
corresponds to stating that the system is linear phase.
All right, well, so here we have a sum from n
equals 0 to N over 2 minus 1.
Some things involving h of n.
Here we have a sum from r equals 0 to N over 2 minus 1,
some things involving h of r.
This index of summation, of course,
is just a dummy variable.
And we can change it back to n if we want to, which we do.
So rewriting these two sum, in particular rewriting
this one in terms now of a summation on n,
this becomes the sum from n equals 0
to capital N over 2 minus 1, h of n z to the minus capital N
minus 1 minus little n.
This sum, of course, stays the same.
And now, because we have h of n appearing
in both of these sums, we can combine this together
into a single summation, from 0 to N over 2 minus 1 of h of n,
the values in the unit sample response,
times z to the minus n plus z to the minus capital
N minus 1 minus little n.
Notice now that we only have in this summation
half the terms that we had originally.
In other words, there are capital N
over 2 different coefficients that we're
using, whereas in the direct form implementation,
there were capital N coefficients that we
were using.
And here, we're exploiting the symmetry in the coefficients.
Well, let's see what this implies
in terms of a filter structure.
Here, I've simply rewritten, again,
the general form for the transfer function assuming
that the system is linear phase, as we've just previously worked
out.
This is just simply a rewrite of the previous expression.
And let's look at this, for example,
I had assumed that that capital N was even.
So let's choose as an example capital
N equal to 8, in which case this expression becomes a sum from n
equals 0 to 3, h of n, z to the minus n plus z to the minus--
capital N minus 1 is 7, so minus 7 minus n.
Well, how many delays are involved in implementing
this system function?
Apparently, there are seven delays involved whether or not
the system is linear phase.
And that shouldn't be surprising because the length
of the impulse response is still 8.
So seven delays in general are required,
but only four coefficients are required.
And to see what structure is implied by this,
we recognize that h of z is constructed
by summing a system which corresponds to a delay.
Looking at this expression for n equal 0, no delay.
n equal 0 substituted in gives us a transfer function of 1.
Plus a delay with n equal 0 substituted in of 7.
The output of those two delays added together and multiplied
by h of 0, and consequently, if we consider a chain of delays
as I've drawn here, we have the output
before we go through any delays added
to the output from the seventh delay--
that's looking at this expression for n equal 0--
and the result of that summation multiplied by h of 0
to form the first term in this sum.
To form the next term in the sum for n equals 1,
we have the output of one delay plus the output
from the 7th minus 1th delay.
That is the 6th delay.
So we have this output, the output from the first delay,
and this output, the output from the 6th delay.
And now that is multiplied by h of 1 coefficient, which
is h of 1, to form the next term in the sum.
And if we go through the remaining terms in the sum,
we have the output from the 2nd delay and the output
from the 5th delay added together.
That multiplied by h of 2.
The output from the 3rd delay and the output
from the 4th delay added together
and that multiplied by h of 3.
That forms the individual terms in this sum.
And now we want finally to add those together.
So we can simply add those together
at their outputs to form the output y of n.
So here is a structure which implements
a finite impulse response system of order 8, or of length 8.
It involves seven delays, which of course
the direct form also involves.
But the important aspect of it is
that it involves only half the number of multiplies
that the direct form would normally require.
Taking advantage of the fact that it's a linear phase system
then, we're able to implement the system
with half the number of multiplies
than would be required in general.
All right, so this is one structure.
I stress again that it's not the only structure for implementing
a linear phase system.
Clearly, we can generate another structure, linear phase
structure, by simply applying the transposition theorem
to this structure.
And we would end up with a transposed configuration.
And an example of that is shown in the text.
I won't bother going through it.
But the important aspect here is that we were able to exploit--
in the case a finite impulse response systems
we're able to exploit the linear phase characteristics
of the system, if we're talking about a system that
is linear phase.
The second structure that is peculiar to find an impulse
response systems and in some instances has some advantages
is a structure that is often referred to as a frequency
sampling structure.
The reason for that being that the coefficients
in the structure correspond to samples of the frequency
response of the system.
And I'll indicate in a few minutes
why that might be an advantage.
But first let's look at what the specific structure
is that's involved.
A frequency sampling structure is
derived by essentially implementing the system,
or inquiry as to what the implementation would be,
if we specify the system in terms, rather than of the unit
sample response, of the discrete Fourier transform coefficients.
We recall, and I hope that this is second nature to you by now,
in general we have a unit sample response.
This is a finite impulse response system.
So this is a finite length sequence.
So we can describe it in terms of it's discrete Fourier
transform.
And the discrete Fourier transform coefficients
are referred to as h of k.
Or we can describe it as we can describe
almost all sequences in terms of its z transform,
or a system in terms of its system function.
The z transform is the sum of h of n times
e to the minus, n as it always is.
And the unit sample response can be
expressed in terms of the discrete Fourier
transform coefficients in terms of the sum of H
of k W sub N to the minus n k.
We have the 1/N out in front.
And again, this term R sub capital N then
to just simply truncate h of n after capital N points.
That is to extract one period from the periodic function
that this sum generates.
Now, we can substitute h of n expressed
in terms of its discrete Fourier transform into this expression.
The R sub N of n will disappear because we're only
summing from n equal 0 to capital N minus 1.
And if we substitute this in here
and interchange the order in which we evaluate the sums,
then the resulting expression for the z
transform for the system function
is the sum from K equal 0 to N minus 1 of the DFT values.
And this term and this term are involved in the sum on n.
And they result in the sum from 0 to capital N minus 1 of z
to the minus 1, W capital N to the minus k.
That raised to the n-th power.
Well, this is the kind of summation
that we've been carrying out again and again, particularly
in talking about finite length sequences.
And hopefully, this is another kind of summation
that by now you carry around in your hip pocket.
It's the sum from 0 to capital N minus 1
of something of the form alpha to the n,
where the alpha happens to be that.
And that sums up to 1 minus z to the minus capital N times--
let's see, 1 minus z to the minus capital N
and actually W sub capital N to the minus k capital N.
That is it sums up to this alpha raised
to the capital N-th power divided by 1 minus W sub N
to the minus k times z to the minus 1.
Well, what does W sub capital N to any integer multiple
of capital N?
That's just simply equal to unity.
So this term disappears.
And we end up simply with 1 minus z
to the minus n divided by this.
All right, so combining this together,
we can then express h of z as 1 minus z to the minus capital N,
we have this 1/N, which we have to account for somehow,
and then finally a sum from 0 to N minus 1 of H of k,
divided by this factor.
And so this is an expression for the z transform of the system,
or the transfer function of the system, where
the coefficients in that expression
are the discrete Fourier transform values H of k.
The discrete Fourier transform values,
as we've stressed several times, in fact
are samples on the unit circle of the z transform.
So these coefficients, the H of k's are
in effect samples of the frequency
response of the system.
In other words, they are frequency samples.
And consequently, the structure that this implies
is usually referred to as a frequency sampling structure.
Well, let's look at what the specific structure is.
We can view this as a cascade of the system that implements
this term in cascade with the system that
implements this summation.
This term is simply implemented by a gain
of unity in the forward direction,
and then a gain of minus z to the minus capital N--
I happen to have the arrow here in the reverse direction, which
is wrong.
This is as an arrow, again, in the forward direction.
That just simply implements the system, 1 minus z
to the minus capital N.
And that I guess makes enough of a mess out of it,
so that hopefully you can at least see what
direction that arrow goes in.
This then implements that piece.
To implement this sum, well, this sum
implies a parallel combination of systems,
each system implementing one of these pieces.
So if we look at this, say, for k equal 0,
we have a coefficient of 0 as I've indicated here and a pole.
And where is the pole?
Well, the pole for k equal 0 is at W sub N to the minus 0.
For k equal to 1, it's at W sub N to the minus 1
and on through.
So the system that implements this sum is then
a parallel combination of first order poles,
where the coefficient in the feedback path
is W sub N to the minus k.
And the gain in the forward path,
which can be placed either before or after the pole,
are the frequency samples H of 0, H of 1,
down through H of n minus 1.
And then, of course, we have the gain
of 1/N, which has to go someplace.
And for now, let's just leave it at the end.
I stress also that these, of course,
are complex poles, because W sub N to an integer power
is in general complex.
We can, of course, redraw this structure
by combining complex conjugate poles together.
And the structure that results is indicated in the text.
I won't bother to do that here.
But there are a couple of points about this structure that I'd
like to draw your attention to.
First of all, we started with a finite impulse response system.
Obviously, we've ended up with a finite impulse response system.
But the system appears to have some poles in it.
That is its a recursive structure,
and it has in fact capital N poles
involved in the implementation, although presumably
the overall system function has only zeros.
Well, how can that be?
Very simply because if you look at this term,
this term has capital N zeros equally distributed
around the unit circle in angle.
And in fact, a zero is occurring exactly
at the location where each of these poles occur.
So that, in fact, this piece in cascade with this
cancels out this feedback loop, or this pole simply
cancels out one of the zeros from this term.
This one cancels out another zero and on down
through the chain.
So in the overall system function, there are no poles.
There are only zeros.
And what happens in effect is that these poles
are used to cancel out different zeros appearing
in this first path.
That suggests one of the problems,
in fact, with the frequency sampling structure
because it requires poles, as I've drawn it here actually
on the unit circle--
in an actual implementation obviously
that would be slightly of the unit circle--
but it involves implementing poles
to cancel out the zeros appearing in this term 1 minus
z to the minus n.
And that has some drawbacks from a number of points of view.
In fact, it has some drawbacks--
we won't analyze this, but it turns out
to be true that due to finite register
length effects there are some drawbacks.
But there are also some advantages
to this kind of structure.
And the advantages in some situations
outweigh the disadvantage, the main disadvantage,
of canceling out zeros with poles or poles with zeroes.
One of the first advantages is that the coefficients
that are involved involve frequency samples.
Now, in many situations what we would like to implement
is a system in which perhaps a large number of the frequency
samples happen to be zero.
Imagine for example, implementing
a very narrow band, low pass or bandpass, filter
and specifying the filter in such a way
that the frequency samples in-band are non-zero
and the frequency samples out of band are equal to zero.
In that case, most of the coefficients
are coefficients which are zero coefficients.
And consequently, there will be only a few of these paths
that we have to implement, so that in that situation
where perhaps a large number of frequency samples are zero--
in fact, the number of multiplications involved
in implementing the system, and in fact, the number of delays
involved in implementing the system,
might be significantly reduced.
Another advantage to a frequency sampling structure
is the fact that it is basically a parallel form structure.
And that can be exploited in terms of implementation
of a large number of filters, or in terms of the implementation
of a filter bank.
Let me state more specifically what I mean by that.
In this system function, this piece and this piece,
except for the H of k's, are tied only to the fact
that the unit sample response of the system
is of length capital N. So that if I implement
a large number of systems whose impulse response is of length
capital N, the systems will look identical from the input
up to the point where I do the multiplication by the frequency
samples.
Consequently, if I were to implement a large number of
filters simultaneously.
I can use in common for all of those systems
the part of this structure from the input
to the output of the poles.
And then simply draw off those outputs,
multiply by the required frequency samples,
sum the outputs, and that implements the filter.
Consequently, I'm able to share among all
of the filters in the filter bank
these delays, these delays, and the multiplications by powers
of W. And the only additional coefficients
and the only additional pieces that come in that
differ between the filters and the filter bank
are the coefficients that I multiply these outputs by.
I need to multiply by the coefficients
and then form the linear combination.
So that then are the two primary advantages to frequency
sampling structures.
And in fact, frequency sampling structures
have found application primarily for those two reasons.
Well, there are lots of other structures
that one can talk about, both for a finite impulse
response an infinite impulse response systems.
And as I stressed at the end of the last lecture,
the discussion of structures in general
and the kinds of structures that can be generated is very vast.
In fact, it seems that new structures are generated faster
than they can be evaluated.
And I would like to terminate the discussion
of specific structures with just the ones that I've shown you
in the previous lecture and the ones
that I've shown for finite impulse response systems
in this lecture.
To conclude that discussion of digital networks
and digital filter structures, I'd
like to focus finally on just a few
of the issues, one primarily, involved
in the comparison of filter structures.
Now we've talked about different structures.
One of the things that obviously is true
is that from a strictly mathematical point of view,
all of these structures, different ways of implementing
the same transfer function, obviously
implement the same transfer function.
That is algebraically or theoretically,
all of the structures from an input output point of view
are indistinguishable.
In other words, they generate the same transfer function.
On the other hand, there are a variety
of issues when one considers the practical implementation
of digital filters that have to be considered in choosing
different filter structures.
One of them I suggested last time to be the number of delays
involved.
And we saw the difference between non-canonic and canonic
form structures.
Delays mean registers.
And registers mean hardware.
In talking about linear phase filters,
we saw that if we exploit the linear phase property, then
we can reduce the number of multiplications
involved by half.
And multiplications, again, mean hardware or speed or something
of that sort in terms of the implementation.
Another common characteristic that
can be focused on in comparing structures
is what you could refer to as modularity.
It's particularly important in hardware implementation
of digital filters, where perhaps you
would like your structure to have stages in it that
are identical, so that the implications in terms
of the hardware are that a section or sections can
be multiplexed and therefore used over and over again.
And another issue, which is the one
that I'd like to direct our attention to finally
are the effect of finite register
length on the characteristics of the system.
And these characteristics, or the finite register length
effects, in fact, are very heavily dependent on the filter
structures that are used.
There are two kinds of finite register length effects.
One of them has to do with the fact
that arithmetic in a digital filter,
specifically in a recursive digital filter,
has to be done, of course, with finite register length,
and it has to be truncated.
In other words, multiplications, the result of B bits times B
bits results in 2B bits.
If this goes on indefinitely, obviously the register length
grows indefinitely.
And so the results of arithmetic have
to be truncated or rounded.
And this generates some noise or essentially
noise in the filter output.
And the second thing is the fact that coefficients
have to be represented in finite register length.
That is coefficients or parameters
have to be truncated.
What I would like to do in closing the lecture
is just briefly give you a glimpse
into some of the issues of the coefficient, or parameter,
quantization and not delve any in any more detail
into the relatively complex issue of finite register length
effects.
But let's focus just a little bit, just a little,
on the effects of parameter quantization.
In particular, we have, of course, the general form
for a system function as a ratio of polynomials in z
to the minus 1, where the denominator polynomial,
as we've been writing it, is of the form
1 minus the sum of a sub k times z to the minus k.
Well, let's imagine that what we would like to do
or that it's reasonable to tie the investigation of H of z
in terms of parameter errors to the accuracy, with which
the poles of the transfer function can be placed.
We can express this polynomial, the denominator polynomial,
in factored form, where, of course, each of the factors
involves the poles of the system explicitly.
And now as we talk about implementing this system,
the coefficients in the denominator polynomial,
and in the numerator polynomial also,
have errors due to the fact that they have to be represented
in a finite register length.
So we can represent that as the coefficient that we actually
arrive at is the coefficient that we really
want plus some error.
And consequently, the pole location,
the location of each of the poles, the i-th pole,
for example, is where we want it plus some deviation.
Now, we can examine what this deviation
is by essentially assuming that the variation
in these parameters is small.
And expressing this error in terms
of the partial derivatives of the pole locations expressed
in terms of the coefficients a sub k.
In other words, we can express the movement
of the pole in terms of the movement of the coefficients
as a sum of the partial derivative
of the pole, the i-th pole-- that's
the one we're talking about here-- with respect
to the k-th coefficient times the error
in the k-th coefficient.
Well, this can be tracked through--
and this is done in the text.
I won't go through the algebra--
but basically, the result is that this partial derivative
can be expressed in this form where the important thing
to focus on is that what's involved in the denominator
is the product of the difference in the pole locations.
Let's assume, for example, that all of the poles
are clustered inside a circle of radius epsilon.
In other words, the largest distance between the poles
is epsilon.
What that says then is that when we substitute that
in for this expression, the absolute value
of the change in the pole location with respect
to a change in the coefficient has in the denominator epsilon
raised to the n-th power.
Well, what does that mean?
What it means is that if the poles are clustered
within a radius within a circle of radius epsilon,
for any given pole the higher the order of the polynomial
that we use to implement the pole,
the greater the sensitivity of the pole location with respect
to the coefficient.
So this says that if we implement
the poles of the system, or when we implement
the poles of the system, it obviously
is better to implement the poles through a lower order
polynomial rather than through a higher order polynomial.
What does this mean in terms of the filter
structures we've been talking about?
It means basically that the direct form structures are
the most sensitive, that the cascade form structures are
certainly less sensitive than the direct form structure,
and the parallel form structure in terms
of the implementation of the poles
have the same type of hypersensitivity
as the cascade form and likewise are less
sensitive than the direct form.
Even in the cascade form structure,
though, there's some flexibility.
And there are some issues with regard to how
the poles are implemented.
And let me close the lecture by just giving you
some indication of what some of the-- in particular what two
of the possibilities might be.
Based on the discussion that we've just gone through,
the indication is that we would want to implement,
or it might be it advantageous from the point of view
of parameter quantization, to implement a system as a cascade
or parallel combination of first or second order sections.
And let's make them second order sections.
Let's look once again at the implementation
of a simple second order pole pair.
I've neglected the implementation of the zeros.
But clearly the point that I'm about to make
extends also for the implementation
of two poles and two zeros.
Here, I have implemented in what we
had referred to as direct form, a second order section which
we might use in a cascade form or parallel form
realization, which implements the complex pole
pair at radius r, an angle plus, and minus theta.
In the actual implementation of a cascade or parallel form
structure using a second order section of this form,
it is these coefficients that are
quantized to fit within a finite register length.
We see then that the quantization
of the coefficients for this particular network form
corresponds to quantizing the parameter r squared
and quantizing the parameter 2r cosine theta.
What this implies in the z-plane is that for, let's say,
a choice of register length for this particular case
equal to 3, for 3-bit register length, the pole
locations which can be implemented with that system
are restricted to fall at the intersection of the grid points
that I've indicated here due to the fact
that quantization of the parameter r cosine theta
implies that we must lie along lines equally spaced
along the horizontal axis.
Recall that r cosine theta is just the real part of the pole
location.
And quantization of the parameter r
squared corresponds to the requirement
that the poles must lie along circles,
which are spaced according to the quantization of r square.
So the realizable pole positions with a second order section
in the form that I indicated on the previous slide
are given by the constellation, or by the intersection
of the grid points, that we have shown here.
Another possible implementation for a second order pole pair
is the so-called coupled form implementation.
I won't justify here by inspection
that the transfer function is given by the same expression as
for the direct form structure that I showed previously,
but, in fact, it is.
And the poles in this case fall at,
again, are equal to e to the j theta and r
e to the minus j theta.
But now, the coefficients we see are r cosine theta and r sine
theta.
Quantization of r cosine theta implies, as it did previously,
that the poles must lie along lines equally spaced
along the horizontal axis.
Quantization of r sine theta implies
that the poles must line along lines equally spaced
along the vertical, or imaginary, axes.
And so for that case, the choice of realizable pole positions
are governed by the constellation of grid points
that I've indicated here.
So this is just simply, this last issue,
is just simply an introduction to the fact
that even given the notion that the cascade form structure is
better than the direct form structure,
there are additional issues involved in specifically
how to implement a pole pair, or a pole zero combination
in order to account for parameter quantization effects.
There are lots of remaining issues,
as you'll see in the text, related
to both the issues of digital filter structures
and the effects of parameter and arithmetic finite register
length effects, which we won't be going
into in this set of lectures.
But you should at least be aware be aware of their existence.
In the next several lectures, we'll
begin on a new topic, namely a discussion
of some of the issues involved in the design
of digital filters.
Thank you.
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