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B. The Smirnov Class Ni
The reader is referred to [5] for a full account of the Smirnov
class N+ and the Nevanlinna class N. We note that the proper
inclusions HP c N+ c N are valid.
The following theorem is a generalization of Theorem 1; the
proof of Theorem 3 ,is found in [5].
Theorem 3: Every r X m matrix function H(z) of class N+
with rank H( ej”) = m a.e. can be expressedin the form
(20)
H(z) = %(z)Hi(z)
where H,(z) is inner, and H,(z) is any column outer function
whose boundary value H,( ej”) satisfies the same magnitudesquared specification as H( ej”). Moreover, the factorization (20)
is unique up to multiplication by a constant unitary matrix.
By using Theorem 3 and the Parseval identity, we obtain the
following theorem, which is a generalization of Theorem 2.
Theorem 4: Let H(z) be an r X m matrix function of class
N+ with rank H(e@) = m a.e. Then the three properties l), 2),
and 3) presented in Theorem 2 are also equivalent if every phrase
“class HZ ” is replaced by the phrase “class N+ ” in statements
2) and 3).
The proof of this theorem can be made in the same way as the
’ proof of Theorem 2.
III.
CONCLUSIONS
We have considered the linear discrete-time systems with matrix-valued
transfer functions of the Hardy class H2 and more
generally of the Smimov class N+, and obtained that the notion
of minimum phase is equivalent to that of minimum-energy delay
for these classes.This equivalence has been presented by means
of the inner-outer factorization of transfer functions and the
Parseval identity for L2 class functions.
REFERENCES
111 A. V. Oppenheim and R. W. Schafer, Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975.
121 E. A. Robinson, Random Wavelets and Cybernetic Systems. New York:
Hafner, 1962.
[31 Yu. A. Rozanov, “Spectral properties of multivariate stationary processes
and boundary properties of analytic matrices,“Theo~ Prob. Appl. (USSR),
vol. 5, pp. 362-376, 1960.
[41 Yu. A. Rozanov, Stationary Random Processes (A. Feinstain, Trawl.)
San Francisco: Holden-Day, 1967.
PI Y. Inouye, “Linear systems with transfer functions of bounded type:
Canonical factorization,” IEEE Trans. Circuits Syst., vol. CAS-33, pp.
581-589, June 1986.
161 P. L. Dtiren, Theory of HP Space. New York: Academic Press, 1970.
High-Speed Delayed Multipath Two-Dimensional
Digital Filtering Architecture
HON KEUNG KWAN,
KOTARO HIRANO,
SENIOR
SENIOR
MEMBER,
MEMBER,
IEEE, AND
IEEE
ON CIRCUITS
AND
SYSTEMS,
NO. 2, FEBRUARY
1985
I. IHTR~DUCTION
High-speed digital filtering is important in many practical
signal-processing applications. Generally speaking, high-speed
filtering can be achieved by using high-speed digital componenti
and/or by making use of concurrency in arithmetic computation.
The latter can be achieved by utilizing the pipeline processing
technique and/or parallel-processing technique.
Recently, a general decomposition theorem has been proposed
[l] for the expansion of a general multidimensional rational
function in terms of functions of one-dimensional (1-D) only.
Consequently, multidimensional digital filters can be implemented with great modularity and parallelism. Another method
to the realization of two-dimensional (2-D) finite-impulse response (FIR) and infinite-impulse response (IIR) digital filter:
can be devised [2] using “lower-upper (LU) triangular decomposition” of the matrix coefficients of their two-dimensional
polynomials. The method enjoys a number of attributes for VLSI
implementation including high parallelism, modularity, and regularity. Moreover, parallel and high throughput structures can also
be obtained [3] for the realization of 2-D digital filters UT;AL,
well-known transforms such as the discrete Fourier and
Walsh-Hadamard.
Alternatively, instead of transforming the transfer function of
a digital filter into another form with different coefficient value:
which facilitates high-speed realization as adopted in [l]-[3], the
original transfer function can be used directly with the same
coefficient values for realization. In fact, this is the approach
characterized by the delayed N-path structure advanced recently
in [4] and [5] for the realization of 1-D FIR. and IIR digital
filters. The resultant structure possesseshigh regularity, modularity, and parallelism. Moreover, the resultant throughput is N2
times that of the conventional direct-form’ realization. In this
contribution, the concept of the delayed N-path 1-D structure is
formulated into a delayed multipath structure for 2-D FIR and
IIR digital filters. This results in a significant improvement in the
throughput rate of 2-D digital filters [7]. These are presented in
Sections II and III. In Section IV, a comparison of the throughput and hardware requirements of the conventional direct-form
realizations with those of the delayed multipath direct-form
realizations of both FIR and IIR digital filters is given. Illustrative examples are also given in Section V.
II.
DELAYED MULTIPATH FIR DIGITAL FILTER
Consider the transfer function of a 2-D FIR digital filter
H(z,‘,z;‘)
= c
M2
c
u(m,n)z;mz;n.
m=O n=O
In general, (1) can be decomposed as
Nl-1
N2-1
H(Z;‘,z;‘)
Manuscript received January 21, 1986; revised July 8, 1986.
H. K. Kwan is with the Denartment of Electrical and Electronic Enaineering, University of Hong Kong, Pokfulam Road, Hong Kong.
K. Hirano is with the Department of Electronic Engineering, Kobe University, Rokkodai Nada, Kobe 651, Japan.
\
IEEE Log Number 8611463.
CAS-34,
Each of these transfer function polynomials can be realized efficiently ii
terms of a number of processors. Consequently, high-speed computation
can be obtained which can be applied to various digital-filtering applications. For a two-dimensional finite-impulse response digital filter of order.
Ml and M2, the maximum throughput can be (M2+ l)[(Ml + 1)/2]’
times that of the conventional direct-form realization using one processor.
Ml
Absrrucr -In this paper, a delayed multipath method for the realization
of two-dimensional nonrecursive and recursive digital filters is presented.
In this method, a two-dimensional transfer function polynomial is decomposed into a number of shorter transfer function polynomials in parallel.
VOL.
=
c
j - 0
Z;jL2
c
Z;‘Aij(Z;“‘,Z;‘)
(2)
i=O
‘In this paper, the conventional direct-form reahzation refers to the direct
form realization in the form of a tapped delayed line (i.e., [2, figs. 1.12 and
1.151 using one processor.
0098-4094/87/02OO-0190$01.0001987 IEEE
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191
1987
INPUT
INf
ow
Fig. 2. An Nl-path structure for A,,(z;~‘,
z;l) of Fig. 1.
Assuming that a 2-D input signal is sampled from the upper
left-hand corner and is proceeded from left to right, top to
bottom according to a normal raster scanning manner. As the
2-D transfer function is decomposed into N2Nl parallel transfer functions, Aii(zcN1, z;‘)(i = 0, 1,2;. ., Nl - 1; j =
0,1,2;. .) N2 - l), and each of these N2Nl transfer functions
involves LlL2 multiplications and additions (i.e., LlL2 coefficients) instead of (Ml + l)( M2 + 1) multiplications and additions
(i.e., (Ml + l)( M2 + 1) coefficients) for the original 2-D transfer
function, the throughput improvement is thus N2 Nl times. With
Fig. 1. A delayed direct-form 2-D FIR digital-filter structure suitable for
multipath realization.
where
u-1
A,(z;~‘,z;~)
= 1
k=O
L2-1
1
a(kNl+i,Z+
jL2)zckN’z;‘.
(3)
I=0
In (2) and (3), N2 represents the number of z;JLz (j =
0,1,2;. . , N2 - 1) branches from the input, L2 represents the
incremental power of the z;jL2 branches, Nl represents the
number of z;‘(i = 0,1,2;. . , Nl - 1) branches connected to the
output of each zs-jL2 branch, and Ll represents the number of
in Aij(z;N’,~;l).
MoreZl -kN1 terms (k=0,1,2,.+.,Ll-1)
over, we have
for i =1,2
Mi+l=NiXLi
l<N2<M2+1
(4
(5)
and
tical parallel Aij(zcN1, z;‘) sections as shown in Fig. 2, the
overall throughput is thus increased to N2N12 times as compared to that of the conventional direct-form structure using one
processor. The corresponding number of processorsrequired for
such a delayed multipath structure is N2Nl’.
Assuming Sl X S2 is the size of the input data to be processed,
the memory requirements of each processor for storing coefficients and data (i.e., present and past input data) are, respectively, LlL2 and [(( L2 - l)Sl+ (Ll- 1) Nl + l)/Nl],
(this is
because the whole set of input data (see(3)) which have just been
used in the present iteration shall be used again for the future
iterations. For [Xl,, see Table I). Besides, the shared memory
requirement for the delayed path is (N2- 1) L2Sl+ Nl. In each
processor for an Nl-path structure, the past input data to be
stored (but not the coefficients) have to be updated at intervals of
NlT, seconds by storing the new incoming input data and
decarding the last unused input data.
III.
IIR DIGITALFILTER
DELAYEDMULTIPATH
Consider the transfer function of a 2-D IIR digital filter
1~ Nl;
1~ Ll.
(6)
Ml
As seen from (2) and (3), the transfer function of a 2-D FIR
digital filter is decomposedinto N2 Nl parallel transfer functions
with each transfer function represented by a polynomial
Aij(ZCN’, z;‘) in zi-N1 and z~;’ together with a multiplication
factor of zLiz;jL2. Fig. 1 gives a representation of (2). In
practice, each of the N2Nl transfer functions Alj(zcN1, z;‘)
can be realized in terms of Nl identical parallel sections as
shown in Fig. 2. In Fig. 2, the sampled input signal is connected
sequentially by the input switch to the input of each of the Nl
identical parallel sections in a round robin fashion at intervals of
TI (where z;’ = e-j”‘q) seconds. The output of each section is
collected by the output switch which is in synchronization with
the input switch. By incorporating Fig. 2 into Fig. 1, at any
discrete time instant, the overall structure will give us a true
realization of H( z; ‘, z; ‘) as given in (1).
E,
H(z;‘,z;l)=
1+
M2
“~04w;“z;”
M1
M2
c
.
(7)
c b(m,n)Z;mZ;n
m=O
n=O
m=n#O
In general, (7) can be decomposedas
H( z;‘, z;‘) =
i+
N2-1
Nl-1
&“,
i=O
C
j=O
z;jL2
K-1
c
i=O
(8)
Z;iBii(z;N’,z;l)
_ ~
192
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1987
Fig. 3. A delayed direct-form 2-D IIR digital-filter structure suitable for
multipath realization.
TABLE I
THROUGHO~ANDHARDWAREREQUIREMEN-ISOFTHECONVENTIONALDIRECT-FORMAND
THEDELA.AYEDMUL~PATHDIRECT-FORM~-DFIRDIGITAL-FILTERSTRUCTURES
structure
Throughput
Improvement
Conventional
Direct-Form
Delayed
Multipath
Direct-Form
No. of
PlWXSSOt3
Required
1
N212
1
Memory Requirement
of Each Processor
Coefficient
Data
storage
storuge
(Ml+l)(M2+1)
N2N12
LlL2
m2Sl+Ml+l
[((L2-1)Sl
+(Ll-1)NI
+ %‘Nll,
[Xl, = X if X is an integer; otherwise, [Xl, = integer part of X+ 1.
and
where
Aii(z;““,z;l)
=
i?,,(z;N’,z;l)
Ll-1
L2-1
c
c
k=O
I-0
Ll-1
E-1
= kxo
,Fo
l<N1;
a(kNl+i,I+jL2)~;~~~z;’
(9)
b(kNl+i,I+jL2)z;kNlz;’
i=j=k=[+O
(10)
Mi+l=NiXLi
-
Mi+l=xxz
for i =1,2
(11)
for i =1,2
l<N2<M2+1
(13)
l<N2<M2+1
l<Nl;
l<Ll
(14)
(15)
l<E.
(16)
A 2-D IIR digital filter represented by (8) can be realized in a
delayed direct-form structure shown in Fig. 3. In a manner
similar to the FIR case,-each of the transfer functions
Aij(zcN’, z;‘) and Bi,(z;N’,l)
of Fig. 3 can be realized,
respectively, using Nl and Nl identical parallel sections as
shown in Figs. 2 and 4. Consequently, the overall throughput rate
is N2N12 or --2
N2Nl (whichever is smaller) times faster than the
conventional direct-form II realization (using one processor). The
number of processors required for such a delayed multipath
structure is N2N12 •k--2
N2Nl . Assuming Sl X S2 is the size of
the input data to be processed,the memory requirements of each
processor for storing coefficients and data (present and past
input data) are, respectively, LlL2 and [((--L2- 1)Sl + (Ll1) Nl + l)/Nl]l_for
the- nonrecursive path or Ll L2 and [((n 1)Sl + (n - 1)Nl + l)/Nl], for the recursive path (seeTable II).
IEEE TRANSACTIONS
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1587
OUTPUT
INPUT
Fig. 4. A x-path
structure for B,j(r-z,
2;‘) of Fig. 3
TABLE II
THROUGHPUT
AND HARDWARE
THE DELAYED
MULTIPATH
structure
REQUIREMENTS
DIRECT-FORM
Throughput
Impr0vement
Conventional
Direct-Form II
1
No. of
Processors
Required
1
OF THE CONVENTIONAL
DIRECT-FORM
II AND
2-D IIR DIGITAL FILTER STRUCTURES
Coefficient
storage
(Ml+l)(M2+1)+
(Ml +l)(M2
Delayed
Multipath
Direct-Form
N2N12
--2 0*
N2 Nl
whichever
is smaller
N2N12
-- + 2
N2 Nl
Memory Requirement
of Each Processor
Data
storage
M2Sl+Ml+l
fl)-1
LlL2
0*
OX-
M2Sl+ Ml fl
whichever is
larger
[((L2-1)Sl
+(Ll-1)Nl
Ll L2
+ l)/Nll,
[((EO! 1)Sl
+(Ll -1)Nl
+l)/Nl II
[Xl, = X if X is an integer; otherwise, [Xl, = integer part of X + 1.
Besides, the shared memory
for the delayed path is
- requirement
(N2 - 1) L2Sl+ Nl or (N2 - l)L2Sl+ Nl (whichever is larger).
In each processor, the input data to be stored has to be updated
(in a manner similar to the FIR case)- at intervals of NW,
seconds for the nonrecursive path and NlT, seconds for the
recursive path.
IV.
COMPARISON
OF THROUGHPUT
AND
sets of values for Nl and Ll; they are: (a) Nl= 3, Ll= 5 and
(b) Nl = 5, Ll = 3. Altogether, there are eight possible combinations of N2, L2, Nl, and Ll; hence, there are eight possible
realizations. Table III gives a summary of the throughput and
hardware requirements of these eight realizations. For illustration, the delayed multipath structure for N2 = 3, L2 = 5 and
Nl = 3, L2 = 5 is shown in Fig. 5.
HARDWARE
REQUIREMENTS
Tables I and II summarize the throughput and hardware
requirements of the conventional direct-form structures and the
delayed multipath direct-form structures of 2-D FIR and IIR
digital filters.
V. REALIZATION EXAMPLES
In this section, we shall illustrate the delayed multipath realization procedures of one 2-D FIR and one 2-D IIR digital filter
examples. In general, the procedures apply to all other cases.
A. 2-D FIR DigitaI Filter
A 14 X 14th-order 2-D FIR digital filter is chosen for illustration. From (l), we have
II(z;l,z;‘) = fJ 5 t2(m,n)z;mz;n
m-o n-o
whereM1+1=15andM2+1=15.From(4)and(5),thereexist
four possible sets of values for N2 and L2; they are: (a)
N2 =l, L2 =15; (b) N2 = 3, L2 = 5; (c) N2 = 5, L2 = 3; and
(d) N2 =15, L2 =l. From (4) and (6), there exist two possible
B. 2-D IIR Digital Filter
A 14 x 14th-order 2-D IIR digital filter is chosen for illustration. From (7), we have
f,
ff(z;‘,z;‘)
=
n~04-Y
14
~)Z7%”
14
1+ c c b(m,n)z;mz;”
m=On=O
m-n20
where M1+1=15, M2+1=15, M1+1=15, and M2+1=15.
From (ll), (13), and (15), there exist eight possible sets of values
for N2, L2, Nl, and Ll; they are: (a) N2 =l, L2 =15, Nl = 3,
and Ll = 5; (b) N2 =l, L2 =15, Nl = 5, and Ll = 3; (c) N2 =
15, L2 =l, Nl= 3, and Ll=5; (d) N2=15, L2=1, Nl=5,
and Ll= 3; (e) N2 = 3, L2 = 5, Nl= 3, and Ll= 5; (f) N2 = 3,
L2=5,
Nl=5,
and Ll=3; (g) N2=5, L2=3, Nl=3, and
Ll = 5; and (h) N2 = 5, L2 = 3, Nl= 5, and Ll= 3. Similarly,
from (12), (14), and (16), there exist eight* such identical sets of
‘Under the conditions that Mi =z for i =1,2, if optimal throughput (see
Table II) is desirable, Ni and Li should be, respectively, equal to Ni and z for
i=1,2.
194
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SYSTEMS,
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2, FEBRUARY
1981
J+
I
Azzki’,
AND
ZZ’ I
%G”T
Fig. 5. The delayed multipath direct-from structure of a 14x 14th-order 2-D
FIR digital filter (N2 = Nl = 3, L2 = Ll = 5).
TABLE III
THROUGHPUT
AND HARDWARE
REQUIREMENTS
OF THE CONVENTIONAL
DIRECT-FORM
AND THE DELAYED MULTIPATH
DIRECT-FORM
OF A
14 x ~~TH-ORDER
2-D FIR DIGITAL-FILTER
STRUClVRE
FOR PROCESSING INPUT DATA OF 256 X 256
structure
Throughput
Improvement
Conventional
No. of
Processors
Required
Memory Requirement
of Each Processor
Coefficient
Data
storage
storage
1
1
225
3599
9
9
15
1199
25
25
45
719
21
21
25
346
75
75
15
207
Direct-Form
Delayed
Multipath
DirectForm
N2=1
N1=3
N2=1
N1=5
N2=3
N1=3
N2=3
Nl=5
N2=5
N1=3
N2=5
N1=5
N2=15
N1=3
N2=15
N1=5
L2=15
L1=5
L2=15
L1=3
L2=5
L1=5
L2=5
L1=3
L2=3
L1=5
45
45
15
175
L2=3
L1=3
L2=1
125
125
9
105
L1=5
135
135
5
5
L2=1
L1=3
375
315
3
3
--VI. CONCLIJDI&G REMARKS
values for N2, L2, Nl, and a. The throughput and hardware
requirements of these eight realizations are summarized in Table
In this paper, a delayed multipath method for the realization of
IV.- For illustration, the delayed
multipath structure for N2 2-D FIR and IIR digital filters has been presented. The method ’
=N2=3, L2=n=5,
Nl=N1=3,
and Ll=L1=5
is shown is flexible in the sensethat the rate of throughput improvement
in Fig. 6.
and the hardware complexity depend on the choices of values for
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1987
OUTPUT
Fig. 6. The delayed multipath direct-form structure of a 14 X 14th-order 2-D
II&digitalfilter(N2=~=Nl=~=l,L2=L2=
Ll=Ll=5).
TABLE Iv
THROUGHPUT
AND HARDWARE
REQ~HWMENTS
OF THE CONVENTIONAL
DIRECT-FORM
II AND THE DELAYED MULTIPATH
DIRECT-FORM
OF A
14 X 14111-0~~~~
2-D IIR DIGITAL-FILTER
STRUCTURE
FOR PROCESSING INPUT DATA OF 256 x 256
structure
Throughput
Improvement
No. of
PRXt?SSWS
Required
Conventional
Direct-Form II
NZ-z-1
Nl=N1=3
N2=N2=1
Nl=x=5
N2-x=3
Nl=N1=3
Delayed
Multipath
DirectForm
.N2=E=3
L2===15
Ll=L1=5
LZ=‘i;l=15
Ll=E=3
LZ=E=5
Ll= L1=5
L2===5
Memory Requirement
of Each Processor
Coefficient
Data
storage
StoraRe
1
1
449
3599
9
18
75
1199
25
50
45
719
27
54
25
346
Nl=N1=5
Ll=L1=3
75
150
15
207
N2=N2=5
L2=L2=3
Ll= L1=5
L2= L2=3
45
90
15
175
125
250
9
105
135
270
5
5
375
750
3
3
Nl=N1=3
N2===5
Nl-N1-5
N2=??i=15
Nl=‘iJr=3
N2=%?=15
Nl=x=5
Ll--=3
L2=L2=1
Ll=L1=5
L2=
L2 =l
Ll=L1=3
196
IEEE TRANSACTlONS
N2, L2, Nl,
_---
Ll (and N2, L2, Nl, Ll). One can always choose
such parameters according to the throughput improvement that is
required and the cost of the hardware that one can afford (see
Tables I and II). As seen from (4)-(6) and (ll)-(16),
there are
--restrictions on the values of N2, L2, Nl, Ll, N2, L2, Nl, and z
to be chosen. In practice, these restrictions can easily be avoided,
if necessary,by assuming that one or more of Ml, M2, Ml, and
M2 are raised by the addition of some zero coefficients such that
restrictions fixed by (4)-(6) and (ll)-(16) are observed. In this
paper, we have considered only the direct-form transfer functions
as given in (1) and (7). In fact, if a 2-D FIR or IIR transfer
function is given in terms of parallel or cascade sections, the
method presented can also
-- be applied to individual section
provided its Ml, M2 (and Ml, M2) are large enough such that
(4)-(6) or (ll)-(16) are observed.
Further work on 2-D and 3-D delayed multipath realization
methods can be found in [8] and [9]. The finite word-length
roundoff error analysis of the 2-D delayed multipath structure
and its comparisons with other implementation structures are
topics that are currently under investigation.
REFERENCES
111 A. N. Venetsanopoulos and B. G. Mertzios, “Decomposition of multidi-
mensional filters,” IEEE Tram. Circuits Syst., vol. CAS-30, pp. 915-917,
Dec. 1983.
C.
L. Nikias. A. P. Chrvsafis. and A. N. Venetsanoooulos. “The LU
PI
decomposition theorem and its implications to the r&lization of twodimensional digital filters,” IEEE Trans. Acourt., Speech, Signal Process.,
vol. ASSP-33, pp. 694-711, June 1985.
[31 J. K. Pitas and A. N. Venetsanopoulos, “Two-dimensional realization of
digital filters by transform decomposition,” IEEE Trans. Circuits Syst.,
vol. CAS-32, pp. 1029-1040, Oct. 1985.
[41 K. K. Dhar and K. Hirano, “A digital filter design algorithm suitable for
multi-DSP implementation,” in Workshop Dig. of IEEE Int. Workshop on
Digital Signal Process., 1985, pp. 2g-l-2g-7.
[51 K. Hayashi, K. K. Dhar, K. Sugahara, and K. Hirano, “Design of
high-speed digital filters suitable for multi-DSP implementation,” IEEE
Tram. Circuits Sysf., vol. CAS-33, pp. 202-217, Feb. 1986.
161 V. Cappelhni, A. G. Constantinides, and P. Emiliani, Digital Filters and
their Applications.
London, New York, San Francisco: Academic Press,
1978, ch. 1, pp. 49-50.
[71 H. K. Kwan and K. Hirano, “High speed delayed multipath 2-D digital
filter structures,” in Proc. IEEE Int. Conf. on Acourt., Speech, Signal
Process., vol. 2, Apr. 1986, pp. 1029-1032.
181 H. K. Kwan arid K. Hirano, “High speed 2-D FIR digital filtering using
similarity transformation,” in Proc. IEEE Int. Symp. on Circuits Syst.,
May 1986, pp. 523-526.
[91 H. K. Kwan and K. Hirano, “Hiah speed delayed multipath 3-D FIR
digital filter structures,” in Proc. IEEE Int. Conf. on Circkts Syst., May
1986, pp. 26-27.
ON CIRCUITS
AND
SYSTEMS,
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CAS-34,
NO. 2, FEBRUARY
1987
by an optimization procedure reported earlier. The number of multiplications per output sample required to implement these filters is shown to be
small.
I.
INTRODUCTION
Elliptically symmetric two-dimensional (2-D) filters are useful
in many situations. For example, there may be compelling reasons to use different sampling rates in the two spatial directions.
In such cases, the design of filters with a circularly symmetric
response reduces to the design of filters with elliptic symmetry.
One of the most powerful techniques for designing 2-D FIR
filters is the transformation of a one-dimensional (1-D) FIR filter
using the McClellan transformation [l]. Various techniques [l]-[6]
have been proposed to determine the coefficients of the McClellan transformation to approximate a given passband boundary of
a 2-D filter with piecewise-constant frequency response. Recently, a simple analytic technique to choose the coefficients of
the first-order McClellan transformation so as to map the cutoff
frequency of the 1-D filter onto a circular contour in a 2-D
frequency plane with a high degreeof accuracy was proposed [7].
Here, we extend this technique to 2-D filters with elliptic symmetry and obtain some interesting results. From the computations
of several examples, it is found, for a first-order transformation
and for a specific error criterion, that the results obtained by the
analytic technique are nearly identical to those obtained by using
the optimization procedure of [2]. The number of multiplications
required to implement these filters is shown to be quite small.
II. THE TECHNIQUE
The passband boundary of a low-pass elliptically symmetric
filter is described by
bI/d+(~2/~2J2
=l
(1)
where wi and wz are the frequency variables of the 2-D filters.
The variables tilr and a2r are the passband edge frequencies on
the or-axis and the ~,-a&, respectively. (We shall assume
ozc ’ q,; the case w2c< wlc is exactly analogous.)The first-order
McClellan transformation is given by
c0sw~F’(o,,w,)
= t& + tie cos q + $1 cos 02 + q1 cos cd1cos cd*. (2)
When expressed in sine functions, (2) reduces to
sin2 ( o/2) P F( w1, t.d2)= too+ t,, sir? ( w1/2) + t,, sin2( w2/2)
+ t,, sin2( @i/2) sin2( oz /2)
Design of Elliptically Symmetric Two-Dimensional
FIR Filters Using the McClellan Transformation
M. SUDHAKARA
SATYA i\r. Hz&%&
REDDY
SENIOR
AND
MEMBER,
IEEE
Abstract-An
analytic technique for choosing the coefficients of the
first-order McClellan transformation for the design of elliptically symmetric two-dimensional FIR filters is described. It is found that the results
obtained by this analytic technique are nearly identical to those obtained
(3)
where w is the frequency variable of the 1-D filters, in rad/s. We
choose the coefficients fP4 using the following considerations.
The point o = 0 maps onto (O,O),and the point w,, the cutoff
frequency of the 1-D protot e filter, maps onto the points
(wIc,O), (0, wzc), and (olc/ F2,w,,./fi)
in the (w,,w,)-plane.
This gives &=O,
tia=a/b,,
fci=a/b,,
and t,,=a[b,(b,/h)c,c21/b2cIc2,where
a P sin’ ( 0,/2)
bi P sin2(oJ2),
(44
i=1,2
(4b)
and
cjPsin2(wi,/2fi),
Manuscript received February 25, 1986; revised June 19, 1986.
The authors are with the Department of Electrical Engineering, Indian
Institute of Technology, New Delhi 110016 India.
IEEE Log Number 8611464.
i=1,2.
(44
Finally, we choose a, i.e., wc, such that
009%4094/87/0200-0196$01.00 01987 IEEE
O<F(w,,w,)
<1
(5)