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532 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 40. NO. 3,
MARCH 1992
Approximation of FIR by IIR Digital Filters: An Algorithm Based
on Balanced Model Reduction
Bartlomiej Beliczynski, Member, IEEE, Izzet Kale, Member, IEEE,
and Gerald D. Cain, Senior Member, IEEE
Abstract-This paper presents an algorithm for the approxi-
mation of FIR filters by IIR filters. The algorithm is based on a
concept of balanced model reduction. The matrix inversions normally
associated with this procedure are notoriously error prone due to
ill conditioning of the special matrix forms re- quired. This
difficulty is circumvented here by directly for- mulating a reduced
state-space system description which is input/output equivalent to
the system that would more conven- tionally be obtained following
the explicit step of constructing an (interim) balanced
realization. Various examples of FIR by IIR filter approximations
are included.
1. INTRODUCTION T is fairly easy to assemble an armory of highly
effec- I tive design tools for linear-phase FIR filters and to
tra-
verse from these to other sorts of FIR designs. Whether it is
merely a matter of applying a window to an elemen- tary nonoptimal
FIR filter or of invoking an elaborate op- timization approach, FIR
design is readily amenable to assault from a variety of directions
and with finely grad- uated investment of algorithmic
sophistication. Serious filter design necessitates a reasonably
comprehensive pro- gramming suite embracing a range of design
options which can be deployed to advantage in the face of var-
iously challenging target specifications [ 1 3 .
The practioners landscape is less lush as regards IIR design.
The venerable Buttenvorth, Chebyshev, and el- liptic standbys can
often be satisfactorily employed, and a few optimization techniques
are on hand for effective use. But generally our view is that the
route to routine production of filter designs is more tedious with
IIR as opposed to FIR targets. In view of the great filter dimen-
sionality advantages associated with IIR filters and their
consequent appeal in resource-constrained implementa- tion, it
seems sensible to try to fashion a two-stage design process: quick
amval at an acceptable FIR design (which perhaps is an intentional
overkill of the specification) followed by conversion to an IIR
approximant which, though never an exact rendition of the FIR
design, still meets the filter specification. Having a design
environ- ment heavily slanted toward FIR work, we found this
prospect to be a powerful motivation.
Manuscript received April 20. 1990; revised December 20, 1990. B
. Beliczynaki was with the School of Electronic and
Manufacturing
Systems Engineering, Polytechnic of Central London, London WI M
8JS. U.K., on leave from the Warsaw University of Technology,
Warsaw. Po- land, while this work was being done.
I . Kale and G. D. Cain are with the School of Electronic and
Manufac- turing Systems Engineering, Polytechnic of Central London.
London W I M 8JS. U.K.
IEEE Log Number 9105668.
The problem of conversion from FIR filters to reduced order IIR
filters has surfaced from time to time in the sig- nal processing
literature for a number of years (e.g., [2]- [4]). IIR
approximation of linear-phase FIR filters has re- ceived particular
attention, especially in the context of de- lay equalization.
One of the many possible avenues to FIR filter approx- imation
is via balanced state-space realization. This method of dynamic
system reduction has been a big at- traction in certain reaches of
system theory for the past decade, with a lot of attention focusing
on state-space equation reduction [5]-[7]. The state-space
realization ought to be in the balanced form, that is, such that
the state-space realization has controllability and observabil- ity
grammians equal and diagonal.
In filter design and approximation, the concept of bal- anced
realization seems to have been rarely used [8], [9], although one
especially pertinent application of the con- cept may be found in
[9]. A usual recipe for the reduction of a transfer function order
is the following: convert the transfer function into state-space
form, calculate control- lability and observability grammians, take
the square roots of the singular values of their product (resulting
in the Hankel singular values), determine necessary similarity
transformation and balanced system realization, inspect the Hankel
singular values and decide upon the reduced system order, and
finally convert the reduced order state- space form into a transfer
function if required. The orig- inal and reduced order system can
then be compared against various criteria and a new order of
approximation attempted if necessary. The primary principle used
for system reduction lies in the elimination of a subsystem
associated with the smallest Hankel singular values.
Several of todays commercial software packages like PRO-MATLAB,
CTRL-C, and CC can be used to under- take the various steps needed
in the model reduction al- gorithm. However, a computational
problem may arise, especially when the original system is of
relatively high order.
In this paper we use a balanced realization technique to convert
single-input/single-output FIR filters to corre- sponding IIR
filter approximations, simultaneously (and hopefully substantially)
reducing system order. The start- ing point of our algorithm is a
canonical controllable form of an FIR filter. By using the Hankel
matrix factorization we derive a simple formula for similarity
transformation which leads to a balanced state-space realization.
How-
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BELICZYNSKI er al.: APPROXIMATION OF FIR BY IIR DIGITAL FILTERS
533
ever, in order to avoid calculation of a possibly ill-con-
ditioned balanced realization, we use a reduced state-space form of
the system that is input/output equivalent to the balanced system,
but which is obtained without inverting matrices.
The algorithm we derive is applied to investigate sev- eral
examples of FIR-to-IIR filter conversions. When using a design
technique of this nature the designer may employ two somewhat
unconventional visual aids for guidance. These are the Hankel
singular value plots and the 3D magnitude and phase plots. The
third dimension in these "waterfall" plots is the reduced order of
the IIR approximation of the prototype FIR. Both these aids are, in
our opinion, very important for a qualitative assess- ment of
design effectiveness, and we have made use of them in the examples
included in this paper.
The paper is organized as follows: in Section I1 prin- cipal
facts about balanced model reduction are reviewed; the algorithm
for FIR-to-IIR filter conversion is derived and discussed in
Section 111, various examples are in- cluded and treated in Section
IV, and finally conclusions are drawn in Section V.
11. BALANCED REALIZATION AND MODEL REDUCTION Important
highlights from balanced realization and
model reduction theories are presented in this section. These
facts wil! be used in subsequent sections to prove properties and
correctness of our FIR-to-IIR approxima- tion algorithm.
Let ( A , B , C ) be matrices forming a minimal realization of a
stable z-domain transfer function F(z) of order n; that is,
F(z) = C(ZZ - A ) - ' B . (1) If one selects any nonsingular
matrix T and calculates a new system of matrices ( A , B , C) such
that
2 = T-IAT, B = T - I B , C = CT ( 2 )
_ - -
- -
then - C(zZ - A ) - 9 = F(z). ( 3 )
there are two special matrices P and Q defined for a state-
space realization of ( A , B , C ) . These are solutions of the
following Lyapunov equations:
APA' + B B ~ = P
A'QA + C'C = Q .
(4) and
( 5 )
The matrices P and Q are known as the controllability and
observability grammians, respectively. Furthermore, P and Q are
positive-definite matrices if the system is sta- ble (i.e., Re ( A
i ( A ) ) < 0 for every i), and if and only if pairs ( A , B )
and ( A , C ) are completely controllable and completely
observable, respectively. For proof see, for example, [lo].
If transformation T is applied to the system ( A , B , C )
with grammians P and Q it will then lead to a new system - ( A ,
B , C ) according to (2) and to new grammians and Q , which are
solutions of the following equations:
_ - _
A j q ' + BE' = p
;i'QA + CTC = e. (6)
and
(7)
If (2) is substituted into (6) and (7) then one obtains -
(8) - P = T - I P ( T ~ ) - I
Q = TTQT. (9)
The grammians P and Q depend strongly on the state- space
coordinates, but the eigenvalues of their product
AI (PQ)
are invariant under state-space transformations, and are hence
input/output invariants.
We required all singular values of matrix A to be neg- ative,
i.e., Re (A,(A)) < 0. Hence, for every i, the Han- kel singular
values of F(z) are defined as
a,(&z)) !2 {A l (PQ)}1 '2 . (10)
And by convention these values are ordered in a descend- ing
fashion:
01 (F(z)) 2 01 + I (W). The maximum singular value of (T,
defines the Hankel norm:
( 1 1) II F(Z)IlH Li max 01 (F(z)). I
A state-space realization such that
P = Q = C = diag ( a l , ( T ~ , . * , a,) (12) is referred to
as a balanced realization [ 5 ] .
If the system ( A , B , C ) is stable and furthermore it is
controllable and observable, then the P and Q matrices are
positive-definite and can be factorized as
P = R R ~ (13)
Q = S'S. (14)
SPS' = sRRTsT (15)
SPST = U C 2 U T (16)
And then the matrix
is a symmetric matrix which can be factorized as follows:
where
UU' = I
and I is the unit matrix.
formation: Having S, U , and C, one defines this following
trans-
T = s - 1 u C ' / 2 . (18)
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Substituting (2) into (8) and (9) and applying (14), (16), and
(1 7) we have
p = C-' /QJTSpSqJ(C- l /2)T = C-'PUTUC2UTUC-' /2 =
and - Q = C'/2uT(S-I)TsTss-IuC1/2 =
So the transformation T applied to the ( A , B , C) system in
accordance with (2) will result in a balanced system.
Next let us assume that ( A h , B,,, cb) is a balanced sys- tem,
and decompose C into two parts:
where
C, = diag (al , . . , ak) (20)
and
C, = diag (uk+,, * * * , on). (21)
According to the partition, the state-space matrices can now be
represented as
IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL. 40. NO 3. MARCH
1992
(22) and the two subsystems are
( A , ,, B , , C,)-the truncated system
(A22, B,, C2)-the rejected system.
(23)
(24)
If system ( A b , Bb, C,) is asymptotically stable and bal-
anced, then (23) and (24) are also found to be asymptot- ically
stable and balanced.
A balanced realization is unique up to an arbitrary state- space
transformation Th such that
TbC = ETb and TiTb = I. (25)
See [lo] for proof.
balanced realization of F(z) to the first k states then If Fk(z)
is a transfer function obtained by truncating the
IlF(2) - FdZ)IIH 5 2 tr (E,). (26) The result (26) is presented
in [ 101 and [ 1 11, albeit in the s domain.
An interesting fact, stressed in 1121, is that the Hankel norm
lies between two other conventional norms: those of the sum of
squares and Chebyshev norms. A Hankel matrix can be used in the
calculation of the Hankel sin- gular values.
For the system ( A , B , C ) the Markov parameters Hk are
defined as
(27) Hk = C A - I B fork = 1, 2, 3, * . e .
The Markov parameters represent the impulse response sequence of
the system ( A , B , C). The Hankel matrix is hence defined as
follows:
r HI H2 1
and can be written as
H = WOWc
where WO = [CTATCT . * . (AT)kCT, . . * I T (30)
(31) w, = [ B AB . A ~ B . -1. A further basic fact from
realization theory is that
(32) rank (Hj = degree of F(z)
which is known as Kronecker's theorem [13], [14]. Since the
solution of (4) can be written as
(33)
P = wcw; (34)
Q = WTWo. (35)
P = B B ~ + A B B ~ A ~ + . . - + A ~ B B ~ ( A ~ ) ~ + . e
then
and similarly one can show that
The Hankel singular values of F(z) are the singular values of H
and the following holds true [lo]:
[The ith singular value of HI2 = h,(PQ) = uf(F(z)).
(36)
111. FIR TO IJR CONVERSION ALGORITHM The method presented in
Section I1 is a general method
for balanced model reduction, which can be applied to any
filter, be it FIR or IIR. The resulting filter will always be an
IIR. In many practical applications FIR filters must be of quite
high order; in the process of converting from FIR to IIR this fact
leads to intensive computations. These suffer from very long
calculation times and numerical in- accuracies, even if very high
precision arithmetic is em- ployed. Moreover, balanced realization
calculations are often ill conditioned. To try to circumvent these
difficul- ties, we employ a unique property of the FIR filter (it
can be structurally represented in a very special form) and re-
duce the system order without calculating explicitly its balanced
realization.
We start with an N-coefficient FIR filter (an nth order system,
where n = N - l j written in the following trans- fer function
form:
F(2) = CO + q 2 - I + . . + C,,z-'l = CO + F,(z ) . (37)
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535 BELICZYNSKI er al.: APPROXIMATION O F FIR BY IIR DIGITAL
FILTERS
The filter F(z) may be represented as a set of difference
equations
where
A =
x ( k + 1) = h ( k ) + Bu(k) y ( k ) = Cx(k) + Du(k)
, B =
0 0 - * . 0 0
1 0 - . . 0 0 . . .
-0 0 * * * 1 0
0
0
C = [CI ~2 ~3 * * * c,] , D = CO. (40) Notice that filter F, ( z
) is representable by the A , B, C ma- trices alone.
Now for the ( A , B , C) system having a finite impulse
response, the rows and columns of zeros are omitted and the
following finite Hankel matrix is used:
The matrices WO and W, defined by (30) and (31) have their
dimensions also limited
WO, w, E anxn (42) where an ,I is the set of real matrices. W,
and W,. are usually called observability and controllability
matrices, respectively. The H matrix (41) is a symmetric matrix, so
it can be factorized in this manner:
H = VAL (43)
W T = I (44)
where
with A being a diagonal matrix and I a unit matrix. Notice that
H is not necessarily a positive-definite matrix.
Corollary: For the system ( A , B, C) where the A , B , C
matrices are given by (40), the controllability matrix, and the
controllability grammian are unit matrices, i.e.,
w, = I (45) P = I (46)
and the state transformation
T = V l A l - / 2 (47)
will lead to a balanced realization of the system where I/ and A
are defined by (43) and (44) and I * I denotes the absolute value
of the matrix elements.
Proof: By simply using (40) and (31) along with (42), one
obtains (45). Substituting (45) into (34) results in formula
(46).
Now according to (29), (45), (35), and (43)
H = W,WC = W , = VAV
and
so also
PQ = VA2V = V 1 A ) 2 V T .
If one uses the state transformation (47) then, by employ- ing
(8) and (9), one obtains
7 = e = l A J . H Having achieved a balanced realization one can
reduce
the system by means of (23). However, a balanced real- ization
calculated via transformation (47) might be ill conditioned,
particularly if the diagonal elements of A vary dramatically in
magnitude. The theorem which fol- lows is then helpful.
7heorem: If the Hankel matrix H of an nth order FIR filter ( A ,
B , C) (where A , B , C are given by (40)) is fac- torized as (43),
then a kth order reduced balanced system is input/output equivalent
to the system
(4, 4 3 C,) (48) where
Ak = VrAVk
Bk = V:B
c, = cv, and Vk is a (n X k ) rectangular matrix obtained from
the following partitioning:
v = [V, Vn- ,] . Proof: If I A I is partitioned as follows:
then, by using (47), one can determine balanced realiza-
tion
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536 IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL. 40. NO. 3.
MARCH 1992
The balanced reduced-order system defined by ( 2 3 ) then has
the transfer function
Gk(z) = CVk I Ak 1 - / * ( z l - I Ak I I2 VrAVk 1 Ak 1 - * ) -
I * 1 Ak 1 I2 V:B
= CI/,IA~)-~/*(ZIA~~~/*)A~)-~*
= C V ~ ( Z I - VlAVk)- VrB.
- I I I * v , T A vk 1 1 - I 2 ) - 1 I W:B
Remark: Because of the very special forms of matrices A and B ,
(40), one can prove that
Ak = V ( 2 : n , l : / ~ ) ~ V ( l : n - 1, 1:k)
Bk = V(1, I : k ) T
C, = CV(l :n , 1 : k ) (49) where V(i : j , k : m) denotes an
extraction of matrix Vs rows from i t o j and its columns from k to
m . Now, at last, a complete algorithm can be presented.
The Algorithm Given co, c I , * , c,, the FIR filter
coefficients: 1) Create the Hankel matrix, H - (41). 2 ) Decompose
the H matrix to obtain V and A - (43). 3 ) Display the Hankel
singular values (i.e., the ele-
ments of 1 A I ) , and choose a desired order of approxi-
mation.
4) Calculate A k , Bk, Ck matrices - (49) 5) Convert the system
state-space representation (Ak,
Bk, Ck, D), where D = c,,, into the transfer function form.
IV . FIR-TO-IIR REDUCTION EXAMPLES We have scrutinized many
cases of FIR filters, with the
number of coefficients N ranging from 4 to more than 256 ,
successfully converting and reducing to an appropriate IIR form.
Since it is more convenient for us to dimension IIR results in
terms of the order n of the z-domain numerator and denominator
polynomials, we relinquish the more conventional (for FIR) N
notation and declare our original filters size in terms of n , so
that before-and-after size comparisons can be facilitated. In this
section we restrict ourselves to presenting six primary examples,
which clearly demonstrate this model reduction technique, its
strengths and areas where its use can be especially advan- tageous.
In these examples several conditions of phase (minimum phase,
linear phase, and maximum phase) for the original FIR filter,
having identical magnitude re- sponse, are investigated.
The results are presented in a variety of formats, high-
lighting the important aspects of our findings in a manner we
believe aids visualization. One of the most important graphs
included in all our findings is a plot of the Hankel singular
values, which serves as a guideline for order se- lection of the
reduced IIR realization. A common feature seen on all our Hankel
singular values plots is an arrow,
which serves as an indicator to the approximant choice
subsequently employed. Apart from the Hankel singular values plot,
waterfall plots of the transfer function mag- nitude and phase of a
selection of reduced-order IIR filters are presented for comparison
purposes. In addition a feather plot, which is a plot of both
magnitude and phase error together, for each frequency, is also
included. One final element presented in the cluster of graphical
representations of both the original FIR and reduced IIR filter, is
the pole/zero pattern (PZP).
A. Example 1 In this example, we look at a narrow stopband,
band-
reject filter, with monotonic passband characteristics, having
filter order n = 88 in the original prototype FIR. We investigate
and comment upon its reducibility as well as identifying and
stating some underlying factors present in the reduction
mechanism.
Fig. 1 is a sample plot for the reduction of the n = 88 linear
phase FIR to an n = 45 IIR. Here phase is linear only in the
passband of the IIR version. Before we discuss the results
presented in Fig. 1, it is necessary to clearly define and explain
what each segment of Fig. 1 repre- sents. Starting with Fig. l(a),
we have a plot of the Han- kel singular values versus the desired
IIR order. The Han- kel singular value plot, when closely examined,
readily reveals that the sharper the falloff to zero, the more
effec- tively and accurately the prototype FIR can be approxi-
mated. Thus, when the Hankel value becomes negligible, there is no
need to engage in increasing the IIR approxi- mation order. This
happens at about the location of the arrow (i = 45), which sets the
conditions leading to Figs. l(b) to (f). Fig. l(c) is a composite
plot which shows the prototype FIR filters magnitude response
represented by small dots, overlaid by the reduced IIR filters
magnitude response, represented by the ellipse-like symbols,
plotted as absolute gain against normalized frequency. This plot
reveals where the FIR specification is not being met by the IIR
approximation.
Similarly, Fig. I(d) employs the same symbology as in Fig. l(c),
only now exhibiting the integrity of the phase approximation
(calibrated in radians). Fig. l(b) is a handy form of vector error
plot, a feather plot showing mag- nitude error combined with the
corresponding phase er- ror, at each frequency displayed. The point
of view adopted is that the original FIR prototype given is the
ref- erence datum (dismissing consideration of the degree to which
that FIR might itself have been an inadequate ap- proximation to
some idealized filter specification). Thus, goodness is tantamount
to original FIR transfer func- tion closeness. Notice that the
vertical axis is a mea- sure of the absolute error and has been
scaled so as to magnify the quite modest error profile prevailing
for this particular Hankel value choice.
Figs. l(e) and (f) give the pole/zero patterns (PZPs) of the
prototype FIR and the approximating IIR, respec- tively. The PZPs
furnish invaluable graphical perspec-
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BELlCZYNSKl er a/: APPROXIMATION OF FIR BY IIR DIGITAL FILTERS
537
0 50 100 index i
(a)
FIR ooo IIR 4 7
- 4 0.25 0.5
normalized frequency (C)
T-not to scale
( e ) FIR
2.8
4) x 1.4 E t i 0
J
3 -1.4 4
0 0.25 0.5 normalized frequency
(b)
1 , ~ , ... FIRi,o;o IIR , 0.8
0.4 t :: 1 0 0.25 0.5 normalized frequency
(d)
(fi IIR Fig. 1 . (a) Hankel singular values versus the desired
IIR order. (b) Error (IIR relative to FIR). An n = 88 linear-phase
band-reject FIR prototype (shown by ellipses in (c) and (d)) and
its n = 45 IIR approximant (shown by dots in ( c ) and (d)) . The
corresponding z-plane pole/zero patterns are given in (e) and ( f )
.
tive for understanding why the reduction mechanism works and how
much further one can reduce.
Starting the examination of our results from Fig. l(e), we
observe that all the zeros of the prototype FIR filter are in what
we refer to as conjugal quartet form (i.e., in conjugate and having
another conjugate pair at the re- ciprocal of the radius applying
to the first pair). This con- struct is a well-known feature of
linear-phase FIR filters and prevails everywhere except where zeros
fall on the real axis at reciprocal radii or on the unit circle. If
we now examine the PZP of the (reduced) IIR version with n = 45 it
becomes clear that our resulting filters PZP takes on the form of
an almost all-pass construct. A slight an- gular rotation offset
prevents all-pass action and instead gives us a good band-reject
filtering characteristic with linear phase in the passband. This
suggests that the poles of the prototype FIR, once released from
the origin in the z plane, find themselves propelled on top of
those zeros inside the unit circle. This of course results in
pole/zero cancellation. Since half the zeros of the prototype FIR
were inside the unit circle, half of the poles freed from the
origin are expended in annihilating these zeros, leav-
ing the other half available for use in realizing the almost
all-pass behavior that characterized the FIR prototype.
This observation ties in with the information conveyed by the
Hankel singular value plot of Fig. l(a) which sug- gests that any
order reduction below n = 45 (approxi- mately the number of
minimum-phase zeros the prototype FIR possesses) would result in a
bad approximation. This further implies that poles and zeros which
do not naturally cancel one another (and hence have a role to play
in shap- ing th: desired filter response) are forcibly ignored. At
the other extreme, if IIR approximation orders in excess of 47 are
chosen, redundant poles and zeros (placed on top of one another)
are brought in, and serve no useful pur- pose apart from canceling
each other out. We might as well not have this excess baggage.
The point about allowing n, the IIR order, to be greater than
that required for the onset of negligible Hankel sin- gular values
(inevitably resulting in redundant poles and zeros) is demonstrated
in the second part of this example. If the magnitude and phase
approximation errors dis- played in Fig. l(b) are examined in
detail, we find that for n = 45, the error is less than 0.01 in the
passband, and less than 0.03 in the stopband. This is a fairly
decent approximation which suggests that, if linear phase in the
passband is required, there is no computational burden improvement
by realizing the filter with an IIR construct, since an IIR of
order 45 would have 45 numerator and 45 denominator coefficients,
resulting in a slightly more ex- pensive computational burden than
the original 89 coef- ficients of the FIR prototype.
If, on the other hand, passband phase may be assigned with
greater freedom, a quick glance at the prototype FIRS PZP leads to
a realization that, if a minimum-phase situation were to be chosen,
a massive saving ought to be possible. Fig. 2 displays the results
for this minimum- phase situation. Except for Figs. 2(e) and (f),
which are the counterparts of Fig. l ( f ) , but with a reduction
to n = 5 and n = 30, respectively, it is apparent that a tenfold
reduction improvement with respect to the linear-phase situation is
achieved, and that roughly comparable ap- proximation fidelity
occurs. It is worth noting that the magnitude errors in the n = 5
case are less than 0.016.
The explanation of this, as our intuition suggested, lies in the
much greater scope available for simultaneous polel zero
cancellation, and concomittant stability, when all the zeros are to
be found inside the unit circle. In the interest of conserving
space in Fig. 2 we have not shown the PZP of the minimum-phase
prototype (easily seen by expung- ing the zeros outside the unit
circle in Fig. l(e) and amending those inside to double-order
zeros). The case of n = 30 has a PZP (see Fig. 2(f)) which clearly
demon- strates the wasteful redundancy the Hankel singular val- ues
plot dictate, in the form of poles on top of zeros. Ob- viously
Fig. 2(e) has been contaminated by superfluous pole-zero pairs very
nearly canceling each other.
Fig. 3 is a plot of the Hankel singular values for the
maximum-phase case. The Hankel singular values clearly indicate
that there is no scope for reduction due to the fact
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538
,5 .._ F I R , ooo IIR I a I
$ 0 o.:bo;q -0.5
ca -1 0- -
0 0 2 5 0 5 normalized frequency
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1992
I
2 0.8
E o
--0.8
-1.6- 0 0.25 0.5
normalized frequency (b)
, ~ , ... FIR; y2; IIR , 0.8
0.25 0.5 normalized frequency
(d)
Fig. 2. (a) Hankel singular values versus the desired IIR order.
(b) Error (IIR relative to FIR). An n = 88 minimum-phase
band-reject FIR prototype and IIR approximants (?I = 5 in (b)-(e)
and n = 30 ( f ) ) . The :-plane pole/ zero patterns for these
approximants are given in (e) and ( f ) .
I 0 50 100
index i Fig. 3 . The Hankel singular values for the
maximum-phase case. An n =
88 maximum-phase band-reject FIR prototype.
that all the zeros are outside the unit circle and we have a
wide passband where phase control is required. So one is better off
using the prototype FIR in these circumstances.
In Fig. 4 we return to more productive territory: linear- phase
and minimum-phase conditions. Presented in Fig. 4 are waterfall
displays of the magnitude and (unwrapped) phase responses of the
linear-phase and the minimum- phase cases for a range of IIR
approximation orders. For Figs. 4(a) and (c), n varies from 48 down
to 43. A much more economical scene is portrayed in Figs. 4(b) and
(d), where n starts at 8 and decreases to only 3 . The first
(i.e.,
leftmost) filter response in all these waterfall plots is al-
ways that of the FIR prototype. Such 3D representation of the
approximation results serves as a very valuable form of visual aid
in conveying the sensitivity of the process of IIR approximation in
one plot.
B. Example 2 In this example we examine the extent of
reducibility,
as well as the dominant factors influencing reduction, for a
narrow-band bandpass filter having n = 88. As in the previous
example, let us look at the linear-phase condi- tion first. Our
starting point in the analysis of the results in this example is
the Hankel singular values plot as shown in Fig. 5(a), which
suggests the possibility of about a fourfold reduction, with
minimal errors. As seen from Figs. 5(c) and (d) and further
confirmed by Fig. 5(b), an IIR approximation with n = 12 results in
an approxima- tion which tracks the prototype FIR fairly closely (a
max- imum absolute passband magnitude error of some 0.01).
The results obtained above confirm a characteristic we found in
the many examples we tried: narrow passband filters, be they
low-pass, high-pass, bandpass, bandstop, or multiband, requiring
linear phase in their passbands, lend themselves favorably to
reduction because of the rapid falloff exhibited by the Hankel
singular values, re- sulting from the inherent narrow-band
simultaneous mag- nitude and phase constraint imposed. Through
experi- mentation with narrow passband filters we have further
observed that in the remaining two cases of phase, i.e., minimum
and maximum, the margin for order reduction in the reduced IIR
filter (for comparable error profiles with respect to the
linear-phase case) is fairly small. For the bandpass filter we are
dealing with in this example, re- duction by order 4 (by recasting
the FIR prototype as min- imum phase rather than linear phase) or
increase by order 4 (by designing maximum-phase rather than
linear-phase) is the extent to which phase selection can perturb
the or- der reduction. Since this is only a possible 5 % improve-
mentldegradation swing it can be said that the overriding influence
in the reduction mechanism in narrow passband cases is no longer
the phase, but rather the bandwidth of the passband. This is all
because phase is insignificant in the stopband. It should however
be made very clear that the minimum-phase case consistently
resulted in better re- duction than the other two conditions. In
Figs. 6(a) and (b) the Hankel singular value plots of the minimum
and maximum phase conditions for the n = 88 FIR narrow passband
bandpass filter are presented in turn. These two Hankel singular
value plots look very similar to the one that can be obtained for
the linear-phase case. The main difference in the three comes in
the decay rate to negli- gible Hankel values. Figs. 6(c)-(f) give
waterfall mag- nitude and phase plots of the minimum and maximum
phase cases respectively, each for a spread of eleven cred- ible
JIR filter approximation orders, having n ranging from 4 to 14 for
the minimum-phase case and 9 to 19 for the maximum phase.
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BELICZYNSKI er al.: APPROXIMATION OF FIR BY IIR DIGITAL FILTERS
53Y
Q A? (C) (d).
Fig. 4. Variations of IIR approximation quality starting with n
= 88 band-reject FIR prototypes (linear phase in (a) and (c) and
minimum phase in (b) and (d))
0 50 100 index i
(a)
FIR ooo IIR 4/
0 0.25 0.5 normalized frequency
(C)
-1.2 - 0 0.25 0.5
normalized frequency (b)
, ~ . . . FIR ,!ooo IIR 0.8
00
however, in an FIR design environment this task is easy. In this
example we investigate one such multiband filter. Fig. 7(a) shows
the Hankel singular values for an n = 135 minimum-phase (minimum
phase was chosen to re- sult in the best reduction) multiband FIR
filter; inspection of the Hankel singular values immediately
suggests that about a sixfold reduction should be possible. Figs.
7(b)-(d) are results for an IIR approximation of order n = 20.
Absolute magnitude errors across all frequencies for this choice of
IIR order lie below 0.09. The above filter's reducibility was
further explored for a variety of bandwidths, with identical filter
passband characteristics, and the results obtained substantiated
our findings stated in example 2. Figs. 7(e) and (f) are waterfall
plots of the IIR phase and magnitude approximations for n ranging
from 12 to 23. It is apparent from the waterfall plots in Figs.
7(e) and (f) that the choice of any approximant or- der above n =
20 does not appreciably improve the ap- proximation.
D. Example 4 Next we investigate an n = 127 linear-phase
wide-band
low-pass filter, and examine its reducibility in the context
Fig. 5 . (a) Hankel singular values plot. (b) Error (IIR relative
to FIR). An n = 88 linear-phase bandpass FIR prototype and its n =
12 IIR approxi- mant. shown in (c) and (d).
of findings we have established in the previous examples. The
situation so far as reducibility goes in this case is, as expected,
not very favorable since we have a wide band over which phase
linearity is required. Confirmation of this is clearly set out by
the Hankel singular values plot (Fig. 8(a)). A sample reduction
case of n = 51 creates
C. Example 3 Techniques for the design of multiband filters with
ar-
bitrary band locations and weightings are tedious and not well
established in traditional IIR design approaches;
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540 IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL. 40. NO. 3.
MARCH 1992
0-
Fig. 6 . Variation of approximation quality for n = 88 bandpass
FIR pro- totypes (minimum-phase in (a), (c), and (e) and maximum
phase in (b). (d), and (0).
Figs. 8(b) to (d). This choice of n resulted in moderately small
magnitude errors.
This example reinforces our earlier observations, that demand
for phase linearity over a wide band in conjunc- tion with
simultaneous magnitude constraints will result in increased system
order n, in order to satisfy both these requirements.
E. Example 5 Finally, we present results for a bandwidth
situation
which is a contrast to that considered in example 4, namely,
very narrow band and low pass. As example 4, however, the phase is
linear. In formulating this case we reduced the bandwidth of the
low-pass filter of example 4 by a factor of 35, which resulted in
the Hankel singular values of this narrow-band low-pass filter (see
Fig. 9(a)) exhibiting a rolldown to negligible values at a much re-
duced (by a factor of 10 in comparison to the wide-band case)
system order index i.
The implication of this in terms of required IIR approx- imant
order is that a much lower n value than that seen in example 4
should suffice. Indeed, satisfactory results for an approximant
choice of n = 6 are presented in Figs. 9(b) to (d).
v - 0 50 100 150
index i (a)
4 , ... FIR. pooIIR ,
0 0.25 0.5 normalized frequency
(C)
-1 6 I 0 0.25 0.5
normalized frequency (b)
... FIR, ooo IIR 1.2 m
o * 1 P 0 0.25 0.5
normalized frequency
(e) Fig. 7. An n = 135 minimum-phase multiband FIR prototype and
its n = 20 IIR approximant ((a)-(f)). Variations of IIR
approximation quality are indicated in (e)-(f).
0.8 1
v 0 50 100 150
index i (a)
... FIR, ooo IIR 4
. . . ..- m
2
a $ 0
-2
- 4 0 0 25 0 5
normallzed frequency
- 0 a / 0 0 2 5 0 5
normalized frequency (b)
FIR, ooolIR 1 2
M 0~~~ 0 6 0 4
0 2
0 0 0 25 0 5
normalized frequency
(4 Fig. 8. (a) Hankel singular values plot. (b) Error (IIR
relative to FIR). An n = 127 linear-phase wide-band low-pass FIR
prototype and its n = 51 IIR approximant, see (c) and (d).
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BELICZYNSKI er U / . : APPROXIMATION OF FIR BY I I R DIGITAL
FILTERS 54 I
0 50 100 150 index i
(a)
... FIR, ooo IIR 47
. 0 ...... ........ ............... ....
.........._..__.._..
i -2 - 4
0 25 0 5 normalized frequency
(C)
-0 4- 0 0.25 0.5
normalized frequency (b)
... FIR, ooo IIR
0 . 2 b , 1 0 0 0.25 0.5
normalized frequency (d)
Fig. 9. (a) Hankel singular values plot. (b) Error (IIR relative
to FIR). An n = 127 linear-phase narrow-band low-pass FIR prototype
and its n = 6 IIR approximant, see (c) and (d).
V . CONCLUSIONS The method of balanced model reduction has been
used
to convert an FIR filter to an IIR filter. Because of the
special form of FIR filters, the general algorithm for bal- anced
model reduction was simplified. The canonical controllable
state-space form of an FIR filter, the corol- lary, and the theorem
we have presented, give the basis for the efficient reduction
algorithm. This algorithm takes into account both magnitude and
phase simultaneously.
The Hankel singular values have been successfully de- termined
from the filters Hankel matrix, not via gram- mians. Balanced
realization is not directly calculated at all, but exactly the same
results are obtained as in model reduction, via a transformation
which does not employ any matrix inversion. This procedure gives a
robust al- gorithm for conversion from FIR to IIR. We have imple-
mented our algorithm through the use of the Matlab pack- age,
processing several dozen different FIR filters with orders up to
256 or so. In some cases (mostly high di- mensional) we had
numerical problems. For filters with 500 or more coefficients we
have found the accuracy of Matlab generally not sufficient to
implement the algo- rithm we have described.
A key role in the selection of order of reduction is played by
the Hankel singular values plot. This plot read- ily supports a
qualitative judgement of how far it is pos- sible to reduce the
filter order, guiding the user to the threshold where higher orders
fail to secure significant improvements in approximation
quality.
Several practical observations we have made are as fol-
lows.
1) Narrow passband filters can be reduced more effec- tively
than wider passband filters of the same phase type.
2) For various filters having the same magnitude re- sponse, the
best reduction is achieved if the original filter is strictly a
minimum-phase filter. The maximum-phase condition resulted in
significantly poorer order reduction. (However, in one case not
explicitly reported above, the poorest reduction ever encountered
has been in a comb filter, where all the zeros of the FIR are
placed in an equispaced manner from dc to Nyquist around the unit
circle. In such cases sensible reduction is not possible.)
3) The technique we presented is very appealing for almost
linear-phase IIR filter design. Efficient model reduction is
possible due to the fact that the prototype FIR linear-phase filter
has linear phase in the whole range of frequencies, but its reduced
IIR form has its phase non- linearity harmlessly concentrated in
stopbands, retaining linear phase where it matters-in the
passbands.
ACKNOWLEDGMENT The authors wish to thank A. Wood of the Centre
for
Microelectronic Systems Applications at the Polytechnic of
Central London for his help in providing the necessary interface
routines and the software installation in the preparation of this
paper.
REFERENCES
[ I ] T. W. Parks and C. S . Bums . Digital Filter Design. New
York: Wiley, 1987.
[2] J . B. Bednar, On the approximation of FIR by IIR digital
filters. IEEE Trans. Acoust.. Speech, Signal Processing, vol.
ASSP-31, no. 1, pp. 28-34, Feb. 1983.
[3] D. H. Friedman, On appproximating an FIR filter using
discrete orthonormal exponentials, IEEE Trans. Acoust., Speech,
Signal Processing, vol. ASSP-29, no, 4 , pp. 923-926, Aug.
1981.
[4] K. Hackelmann and R. Unbehauen, Approximation of the
frequency response of a FIR filter by an IIR filter. in Proc. Euro.
Con$ Cir- cuits Theory Design (Paris, France), Sept. 1-4. 1987, pp.
477-482.
[SI B. C. Moore, Principal component analysis in linear systems:
Con- trollability, observability, and model reduction. IEEE Trans.
Au- tomat. Contr., vol. AC-26, no. 1, Feb. 1981.
[6] L. M. Silverman and M. Bettayeb. Optimal approximation of
linear systems, in Proc. Joint Automat. Contr. Conf., Pap. FA8-A,
1980.
[7] L. Prenobo and L. M. Silverman, Model reduction via balanced
state space representation, IEEE Trans. Automat. Contr., vol.
AC-27, no. 2, pp. 382-387, Apr. 1982.
[8] S . Y . Kung, A new identification and model reduction
algorithm via singular value decomposition, in Proc. 12th IEEE
Asilomar Con$ Circuits Sysl. Comput. (Pacific Grove, CA), Nov.
1978.
[9] H. Kimura and Y. Honoki, Balanced approximation of digital
FIR filter with linear phase characteristics. in Proc. Inr. Symp.
Circuits Syst. (Kyoto, Japan), vol. I . June 5-7. 1985. pp.
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[IO] K. Glover, All optimal Hankel-norm approximation of linear
mul- tivariable systems and their L-error bounds, lnt. J . Contr..
vol. 3 9 , n o . 6 , p p . 1115-1117. 1984.
[ I I] D. F. Enns. Model reduction with balanced realizations:
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[I21 S . Y. Kung and D. W . Lin, Optimal Hankel-norm model
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Cam- bridge University Press, 1988.
[ 141 F. R . Gantmacher. Theory ofMotrices. Moscow: Nauka, 1966,
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542 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 40, NO. 3.
MARCH 1992
Bart lomiej Beliczynski (M91) was born in Kielce, Poland, in
1946. He received the M.S. and Ph.D. degrees in electronics and
electrical engi- neering, respectively, from the Warsaw Univer-
sity of Technology in 1968 and 1974, respec- tively.
Since 1969 he has been with the Department of Electrical
Engineering of the Warsaw University of Technology. In 1974/1975 he
spent one year with the Shizuoka University in Hamamatsu, Ja- pan,
as a research student and from 1987 to 1990
he was a Visiting Lecturer at the Polytechnic of Central London.
His re- search interests include computer-controlled and adaptive
systems, along with digital signal processing.
Izzet Kale (M88) was born on November 23, 1960 in Akincilar,
Cyprus. He received the B.Sc (Hon.) degree in electrical and
electronic engi- neering from the Polytechnic of Central London
(PCL), London, England, in 1983, and the M.Sc. degree in the design
and manufacture of mrcro- electronic systems from Edinburgh
University, Edinburgh, Scotland, in 1984.
Since 1984 he has been a member of the lectur- ing staff in the
School of Electronic and Manufac- turing Systems Engineering at
PCL, actively en-
gaging in research in the areas of digital signal processing and
VLSI and participating in the development and delivery of
continuing education pro- grams in these areas.
Mr. Kale is an Associate Member of IEE.
Gerald D. Cain (M66-S69-M7O-SM90) was born in Anniston, AL. He
received the B.S.E.E. degree from Auburn University, Auburn, AL, in
1963, and the M.S.E.E. degree from the Univer- sity of New Mexico,
Albuquerque, in 1965, and the Ph.D. degree in communication theory
from the University of Florida, Gainesville, in 1970.
He participated in the technical development program of Sandia
Laboratories, carrying out de- velopment work for laser radar and
test range tim- ing and control instrumentation. He ioined Tele-
-~-c
dyne Brown Engineering Company in 1965 and led a small team of
radar analysts engaged in system modeling and simulation. Since
1971 he has been with the Polytechnic of Central London (PCL),
London, England, teaching and coordinating joint research activity
in signal processing with several international partners.
Presently, he is Head of PCLs School of Electronic and
Manufacturing Systems Engineering and directs the Centre for
Microelectronic Systems Applications which brings together digital
sig- nal processing and VLSI. He has actively developed continuing
education programs in data communication and digital signal
processing topics and is currently involved in several
trans-European educational exchange pro- grams