A DESIGN METHOD OF TWO-DIMENSIONAL LINEAR PHASE FIR
FILTERS USING FRITZ JOHN’S THEOREM
Yasunori SUGITA
Nihon University
1 Aza-Nakakawara, Tokusada, Tamura-machi, Koriyama-shi, Fukushima, 963-8642 Japan
Naoyuki AIKAWA
Nihon University
1 Aza-Nakakawara, Tokusada, Tamura-machi, Koriyama-shi, Fukushima, 963-8642 Japan
Keywords:
successive projection, Frits John’s theorem, two-dimensional FIR filters.
Abstract:
This paper presents a design method of 2-dimensional (2-D) FIR filters by successive projection (SP) method
using multiple extreme frequency points based on Fritz John’s theorem. The proposed method enables an
update of coefficients using multiple extreme frequency points by Fritz John’s theorem. Moreover, we also
present two methods as how to choose the extreme frequency point for the update coefficients. As a result, the
solution converges less iteration number and computing time than the previous method.
1 INTRODUCTION
The SP method proposed by A. A. -Taleb et al. in
1984 (Taleb and Fahmy, 1984) has been applied to
the design problem of many filters, such as FIR fil-
ters with complex desired frequency response, IIR fil-
ters, a class of time-constrained FIR filters, FIR filters
which approximate the amplitude characteristics and
the step response simultaneously (Sugita and Aikawa,
2004). In the SP method which is an iterative approx-
imation method, only one extreme frequency point
at which the deviation from the given specification
is maximized is used in the update of the filter co-
efficients. Hence, this algorithm is extremely simple
since it only requires the search for the maximum of
the error function over a closed frequency region in
each iteration. However, because this method is used
only one extreme frequency point in the update of the
filter coefficients, this algorithm requires a large it-
eration number to satisfy the given specification. In
(Sugita and Aikawa, 2004), authors proposed a new
algorithm to reduce the iteration number for designing
1-dimensional (1-D) filter which satisfies the given
specification. This method uses multiple extreme fre-
quency points in order to update the filter coefficients
by Fritz John’s theorem. As a result, it is possible to
reduce computing time further than the conventional
SP method.
In this paper, we propose a design method of 2-
D FIR filters by SP method using multiple extreme
frequency points based on Fritz John’s theorem. The
proposed method is possible to reduce the iteration
number and computing time further than the con-
ventional SP method by using multiple extreme fre-
quency points for the updating coefficients. More-
over, we propose two selection methods of the mul-
tiple extreme frequency points for updating coeffi-
cients.
2 DESIGN FORMULATION AND
SOLUTON BY SP METHOD
The amplitude characteristics of 2-D linear phase FIR
filters can easily be shown (Taleb and Fahmy, 1984)
to have the form
H (ω
1
, ω
2
) =
N
X
i=1
a
i
φ
i
(ω
1
, ω
2
). (1)
Where the coefficients a
i
(i = 1, 2, · · · , N) are re-
lated to the impulse response samples of the filter, φ
i
are frequency dependent functions having a form de-
pending on the type of symmetries (e.g., half plane,
quadrantal, or octagonal) imposed on the amplitude
characteristics and N is an integer which is defined
by the filter mask size.
Then, the design problem considered here is to find
the coefficients a
i
satisfying





D (ω
1
, ω
2
) −
N
X
i=1
a
i
φ
i
(ω
1
, ω
2
)





≤ λ (ω
1
, ω
2
) . (2)
320
SUGITA Y. and AIKAWA N. (2005).
A DESIGN METHOD OF TWO-DIMENSIONAL LINEAR PHASE FIR FILTERS USING FRITZ JOHN’S THEOREM.
In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Signal Processing, Systems Modeling and
Control, pages 320-323
DOI: 10.5220/0001158703200323
Copyright
c
 SciTePress

Whrere D (ω
1
, ω
2
) is the desired amplitude charac-
teristics and λ(ω
1
, ω
2
) is the positive maximum al-
lowable deviation from the desired amplitude charac-
teristics.
As a result, the iterative algorithm of A. A. -Taleb
et al. is
a
n+1
i
=a
n
i
+



e
n
p


−λ(ω
1
, ω
2
)
p

sign(e
n
p
)
N
P
m=1
φ
2
m
(ω
1
, ω
2
)
p
φ
i
(ω
1
, ω
2
)
p
,
(3)
where
e
n+1
p
= D (ω
1
, ω
2
)
p
−
N
X
i=1
a
n+1
i
φ
i
(ω
1
, ω
2
)
p
. (4)
The detail of this algorithm has been presented in
(Taleb and Fahmy, 1984).
It is clear from (3) that this algorithm only requires
a search for the maximum error function over a closed
frequency region. However, enormous amount of iter-
ation numbers are necessary for research the solution
which satisfies the given specification, because only
one extreme frequency point at which the deviation
from the given specification is maximized is used in
the update of the filter coefficients.
3 NEW ALGORITHM
In this section, the update method of coefficients us-
ing multiple extreme frequency points based on Fritz
John’s theorem is described.
[ Fritz John’s Theorem ]
Let X be a nonempty open set in n-dimensional
Euclidean space E
n
, and let f : E
n
→ E
1
, g
i
: E
n
→
E
1
for i = 1, ..., m. Consider Problem P to
Minimize f(x)
subject to g
i
(x) ≤ 0 for i = 1, ..., m
x ∈ X
Let
¯
x be a feasible solution, and let I =
{i : g
i
(
¯
x) = 0}. Furthermore, suppose that g
i
for
i /∈ I is continuous at
¯
x, that f
i
and g
i
for i ∈ I
are differentiable at
¯
x. If
¯
x locally solves Problem P ,
then there exist scalars u
0
, u
i
for i ∈ I such that
u
0
∇f (
¯
x) +
X
i∈I
u
i
∇g
i
(
¯
x) = 0 (5a)
u
0
, u
i
≥ 0 for i ∈ I (5b)
(u
0
, u
I
) 6= (0, 0) (5c)
where u
I
is the vector whose components are u
i
for
i ∈ I. Furthermore, if g
i
for i /∈ I is also dif-
ferentiable at
¯
x, then the Fritz John conditions can
be written in the following equivalent form where
u = (u
1
, ..., u
m
)
t
.
u
0
∇f (
¯
x) +
m
X
i=1
u
i
∇g
i
(
¯
x) = 0 (5d)
u
i
g
i
(
¯
x) = 0 for i = 1, ..., m (5e)
u
0
, u
i
≥ 0 for i = 1, ..., m (5f)
(u
0
, u) 6= (0, 0) (5g)

A proof of this theorem can be found in (Bazaraa
and Shetty, 1979).
We consider the optimization problem as following
in order to use Fritz John’s theorem for design prob-
lem of section 2. :
Minimize f

a
n+1

=
N
X
i=1

a
n+1
i
− a
n
i

2
(6)
Subject to
g
1

a
n+1

=


e
n+1
1


− λ(ω
1
, ω
2
)
1
≤ 0
g
2

a
n+1

=


e
n+1
2


− λ(ω
1
, ω
2
)
2
≤ 0
.
.
.
g
l

a
n+1

=


e
n+1
l


− λ(ω
1
, ω
2
)
l
≤ 0
.
.
.
g
m

a
n+1

=


e
n+1
m


− λ(ω
1
, ω
2
)
m
≤ 0.
(7)
Where the error function e
n+1
l
in lth condition
g
l

a
n+1

is
e
n+1
l
= D (ω
1
, ω
2
)
l
−
N
P
i=1
a
n+1
i
φ
i
(ω
1
, ω
2
)
l
(8)
from (4). For simplicity, we put λ
l
with λ(ω
1
, ω
2
)
l
.
If a
n+1
is an locally optimal solution of this prob-
lem, with Fritz John’s theorem, we get
u
0
∇f

a
n+1

+
m
X
l=1
u
l
∇g
l

a
n+1

= 0 (9a)
(u
0
, u) 6= (0, 0) . (9b)
Substituting (6) and (7) in (9), we get
a
n+1
i
= a
n
i
+
1
2u
0
m
X
l=1
u
l
φ
i
(ω
1
, ω
2
)
l
. (10)
Moreover, substituting (10) in (7), and considering
equal to zero give Lagrange multipliers :









u
1
/u
0
u
2
/u
0
.
.
.
u
l
/u
0
.
.
.
u
m
/u
0









= G
−1
∗









2 (e
n
1
− λ
1
)
2 (e
n
2
− λ
2
)
.
.
.
2 (e
n
l
− λ
l
)
.
.
.
2 (e
n
m
− λ
m
)









.
(11)
A DESIGN METHOD OF TWO-DIMENSIONAL LINEAR PHASE FIR FILTERS USING FRITZ JOHN’S THEOREM
321

−1
−0.5
0
0.5
1
−1
0
1
0
0.1
0.2
0.3
0.4
ω
1
ω
2
Amplitude Error
Figure 1: The amplitude error characteristics of 2-D FIR
filters
Where
G =









G
1,1
G
1,2
· · · G
1,k
· · · G
1,m
G
2,1
G
2,2
· · · G
2,k
· · · G
2,m
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
G
l,1
G
l,2
· · · G
l,k
· · · G
l,m
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
G
m,1
G
m,2
· · · G
m,k
· · · G
m,m









(12)
and (l, k )-th component G
l,k
is given by
G
l,k
=
N
P
i=1
φ
i
(ω
1
, ω
2
)
l
· φ
i
(ω
1
, ω
2
)
k
.
(13)
Let’s ∆
l,k
be (l, k)-th cofactor of matrix G. Then,
the iterative algorithm using Fritz John’s theorem is
described as
a
n+1
i
=a
n
i
+
1
|G|
m
P
l=1
m
P
k=1
(|e
n
k
| − λ
k
)sign(e
n
k
)∆
l,k
φ
i
(ω
1
, ω
2
)
l
(14)
by substituting (11) in (10). Where m is the num-
ber of extreme frequency points used for the updating
coefficients and must not exceed the number N of un-
known coefficients.
By the way, because it is well known that the Haar
condition (Cheney, 1966) consists in designing 1-D
filter, the extreme frequency points exist at least N +1
point, where N is the number of unknown coefficient.
However, the amplitude error characteristic of 2-D fil-
ter is very complicated as shown in Fig. 1. Especially,
it is clear from Fig. 1 that the detection of the extreme
frequency points in band-edge is very difficult. In ad-
dition, it is known that N-dimensional linear space of
approximation function (1) does not generally satisfy
the Haar condition (Kamp and Thiran, 1975). There-
fore, because the extreme frequency points more than
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
ω
1
Amplitude Error
Figure 2: The amplitude error characteristics on ω
2
= 0
axis of 2-dimensional FIR filter in Fig. 1
the number of unknown coefficients are found, there
is a possibility that the proposed algorithm does not
converge. Hence, we propose two methods of finding
the multiple extreme frequency points less than the
number of unknown coefficients as follows.
(I) : The maximum extreme frequency point in
each approximation band is used.
(II) : The maximum extreme frequency point in
all approximation bands and the extreme frequency
points in an arbitrary cross section are used.
In method (I), since the number of approximation
band of a 2-D linear phase Lowpass or Highpass FIR
filter is two, the number of the extreme frequency
points is always less than the number N of unknown
coefficient.
In method (II), the amplitude error characteristic of
2-D filter in an arbitrary cross section behave like one
of 1-D filter, as shown in Fig. 2. Accordingly, it is
clear that the number of extreme frequency points on
this axis is always less than the number of unknown
coefficients.
4 RESULT
In this section, we show some examples of 2-D fil-
ter designed by the proposed method and the conven-
tional SP method (Taleb and Fahmy, 1984). All al-
gorithm used for the design examples are corded in
Visual C
++
6.0 and run on a Pentium 4/3.06GHz PC.
In proposed method, the simulation was carried out
for both of above method (I) and (II). In all of the ex-
amples, an initial coefficients vector a
0
is taken equal
to zero.
First, we will design a circularly symmetric low-
pass filter of specification as following.
ICINCO 2005 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL
322

Table 1: Comparison of the number of the iteration
Proposed Method
Conventional SP (I) (II)
filter mask M deviation λ Iteration Time Iteration Time Iteration Time
number m : sec. number m : sec. number m : sec.
5 × 5 100 0.268 4401 1 : 02 2657 0 : 37 1989 0 : 28
7 × 7 100 0.126 2438 0 : 56 1866 0 : 43 938 0 : 21
9 × 9 100 0.118 792 0 : 27 481 0 : 16 203 0 : 10
25 × 25 200 0.030 588 7 : 49 347 5 : 47 198 3 : 34
filter mask 5 × 5
D(ω)=

1 · · · 0 ≤
p
ω
2
1
+ ω
2
2
≤ 0.4π
0 · · · 0.6π ≤
p
ω
2
1
+ ω
2
2
≤ π
The amplitude characteristics of the filter obtained
is presented in Fig.3. The maximum amplitude error
of the resulting filter is 0.268 in passband and 0.268
in stopband. These results are equal to A. A. Taleb et
al.’s results (Taleb and Fahmy, 1984). In evaluating
the design errors, the frequency ω
1
and ω
2
are sam-
pled at the size π/100. In our proposed algorithm,
the design took 1989 iterations and 28 seconds to sat-
isfy the given specification. If this filter is designed
by conventional SP method, the iteration number of
4401 iterations and computing time of 62 seconds are
required. Clearly, the proposed method has provided
a very significant savings.
Table 1 shows a comparison of the algorithm due
to conventional SP method (Taleb and Fahmy, 1984)
and the proposed method in the design of the other
examples. It is proven from table 1 that the pro-
posed method is much faster than the conventional
SP method in all of the examples. Moreover, the it-
eration number and computing time necessary for re-
search the solution decrease by increasing of extreme
frequency points used for the updating coefficients.
Therefore, the number of the iteration can be dras-
tically reduced by increasing the number of extreme
frequency points also in the case of a 2-D filter.
5 CONCLUSION
In this paper, we proposed a design method of 2-
D FIR filter by SP method using multiple extreme
frequency points based on Fritz John’s theorem. In
case of the design of 2-D FIR filter, the determina-
tion method of the extreme frequency point for the up-
dating coefficients becomes a problem. We proposed
two determination methods for multiple extreme fre-
quency points used for the updating coefficients. As
a result, the proposed method is possible to reduce
the number of the iteration and the computing time
−1
−0.5
0
0.5
1
−1
0
1
0
0.5
1
1.5
ω
1
ω
2
Amplitude
Figure 3: The amplitude characteristics of filter mask(5×5)
necessary for research the solution which satisfies the
given specification, further than the conventional SP
method. We confirmed that the proposed method con-
verged in many examples.
REFERENCES
Bazaraa, M. S. and Shetty, C. M. (1979). Nonlinear pro-
gramming. John Wiley & Sons, Inc.
Cheney, E. W. (1966). Introduction to Approximation The-
orey. McGraw-Hill, New York.
Kamp, Y. and Thiran, J. P. (1975). Chebyshev approxima-
tion for two-dimensional non-recursive digital filters.
IEEE Trans. Circuits Syst., CAS-22(3):208–218.
Sugita, Y. and Aikawa, N. (2004). Designing filters by suc-
cessive projection using multiple extreme frequency
points based on fritz john’s theorem. IEICE trans. on
fundamentals of elec., comm. and computer sciences,
E87-A(8):2029–2036.
Taleb, A. A. and Fahmy, M. M. (1984). Design of fir tow-
dimensional digital filters by successive projections.
IEEE Trans. Circuits and Systems, CAS-31(9):801–
805.
A DESIGN METHOD OF TWO-DIMENSIONAL LINEAR PHASE FIR FILTERS USING FRITZ JOHN’S THEOREM
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