IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 8, AUGUST 2007
517
M
Design of 2-D th Band Lowpass
FIR Eigenfilters With Symmetries
Pushkar G. Patwardhan and Vikram M. Gadre
Abstract—In this letter, we extend the eigenfilter filter design
th band lowpass filmethod for the design of two-dimensional
ters, with the filter impulse response having quadrantal or diagonal
symmetry. We show that imposing the
th band and symmetry
constraints puts restrictions on the possible choices of the matrix
. We identify sufficient conditions on
, and show a class of
matrices satisfying those conditions, so that th band lowpass filters with quadrantal or diagonal symmetry can be designed.
Index Terms—Eigenfilters, multidimensional finite impulse reth band filters.
sponse (FIR) filter, two-dimensional (2-D)
Throughout this letter we use
to denote the specific Quincunx
.
sampling matrix
We now briefly review the eigenfilter method for filter design
as it applies to the design of lowpass 2-D FIR filters (we refer to
[2] and [5] for details). The objective is to minimize a quadratic
measure of the passband and stopband error function, which can
be formulated as
(1)
where
I. INTRODUCTION
HE eigenfilter method [1] has been shown to be an effective technique for designing discrete time filters. Various generalizations of this method have been proposed (refer
to [2] and [3] for a good review of the eigenfilter design method
and its various extensions). Eigenfilter design of one-dimensional (1-D) th Band filters has been proposed in [1], [4]. The
eigenfilter method has been extended to design two-dimensional
(2-D) filters with the filters having quadrantal symmetry [5], [6].
In this letter, our aim is to extend the eigenfilter method of [5]
to design 2-D th band lowpass finite impulse response (FIR)
filters having quadrantal or diagonal symmetries (in the general
2-D case, is a nonsingular 2 2 integer matrix). We note that
the earlier references for 2-D eigenfilter design do not impose
both the constraints of th band and symmetry. We observe
that imposing the th band constraint and the quadrantal or
diagonal symmetry constraint puts a restriction on the possible
choices for . We analyze this restriction on , and present
sufficient conditions on
so that quadrantally or diagonally
symmetric th band filters can be designed.
The rest of the letter is organized as follows. In the rest of
this section we briefly review the eigenfilter method from [5].
In Section II, we formulate the eigenfilter method for design of
2-D th band FIR lowpass quadrantally symmetric filters, and
present design examples. In Section III, we consider the case of
diagonally symmetric th band lowpass FIR filters.
Notation: Boldfaced lower-case letters are used to represent
vectors, and bold-faced upper case letters are used for matrices.
denotes the transpose of . The lattice generated by a matrix , denoted as LAT
, is defined as the set of vectors
such that
for integer vectors . SPD
, the symmetric parallelepiped of is defined as the set
,
1,1 .
T
Here,
and
is the desired frequency
response at a selected reference frequency point
in the passband.
is the actual frequency response of the
2-D filter
is the passband error.
is the stopband error. ,
are the weighting constants.
The error function can be written in the form
,
where is a real, symmetric and positive definite matrix, and
is a real vector whose elements are related to the 2-D filter
impulse response
. By Rayleigh principle, the eigenvector
associated with the smallest eigenvalue of matrix minimizes
the total error .
In the sections below, we formulate the error function for the
design of th band filters with filters having quadrantal symmetry (Section II) and diagonal symmetry (Section III).
II. DESIGN OF QUADRANTALLY SYMMETRIC
TH BAND FIR FILTERS
Our goal is to design 2-D th band FIR filters, with the filter
impulse response having quadrantal symmetry. A 2-D filter with
impulse response
is th band if [2]
(2)
is a nonsingular integer matrix. (Note that, in general,
where
need not be diagonal).
And
is said to have quadrantal symmetry, with center of
symmetry at the origin, if [7], [8]
(3)
Manuscript received July 5, 2006; revised December 10, 2006. The associate
editor coordinating the review of this manuscript and approving it for publication was Dr. Malcolm D. Macleod.
The authors are with the Department of Electrical Engineering, Indian
Institute of Technology, Bombay 400 076, India (e-mail: [email protected]; [email protected]).
Digital Object Identifier 10.1109/LSP.2007.891322
where
and
. (Note that
). An example of a quadrantally symmetric
is shown in Fig. 1(a). In this letter, we only consider the above
case of quadrantal symmetry, where the center of symmetry is
1070-9908/$25.00 © 2007 IEEE
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on December 1, 2008 at 00:28 from IEEE Xplore. Restrictions apply.
518
IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 8, AUGUST 2007
As an example, an
having quadrantal symmetry and the
th band property, with
, is shown in Fig. 1(c)
(The zero values in are marked as “0,” whereas other alphabets
denote non-zero values).
hn
M
hn
hn
Fig. 1. (a) Quadrantally symmetric [ ], (b) diagonally symmetric [ ], (c)
quadrantally symmetric [ ] with th band constraint, and (d) diagonally symmetric [ ] with th band constraint.
hn
M
at the origin. This leads to zero-phase quadrantally symmetric
filters.
For designing th band quadrantally symmetric filter, we
need to impose the th band constraint of (2) and the quadrantal symmetry constraint (3) on the filter impulse response
. Using (2) in (3) we have
A. Formulation of the Error Function for the Eigenfilter
Design Method
We now formulate the error function for the eigenfilter
method. First, consider a 2-D FIR quadrantally symmetric
filter, with center of symmetry at the origin
and
(we will consider the th band constraint later). The frequency
response of the quadrantally symmetric filter can be written as
[5]
(4)
From (4), we observe that we require that the four matrices
,
,
,
generate the same lattice. We now mention a well known fact [2].
is the same as LAT
, where
is a
Fact-1: LAT
unimodular integer matrix, i.e., the lattice generated remains the
same if the lattice generator matrix
is postmultiplied by a
unimodular integer matrix.
and
generate
Using the above fact, it is obvious that
the same lattice. Further, let
be such that it satisfies the following:
and
where
and
Define the column vectors
as
..
.
(5)
where and are unimodular integer matrices.
When
satisfies (5), we have from Fact-1 that
and
also generate the same lattice as .
Summarizing this, we have the following.
Proposition-1: If
is a nonsingular integer matrix satisfying (5), then it is possible to design a quadrantally symmetric
2-D th band filter.
Consider the class of matrices
, where
0
is a diagonal integer matrix, and
0
is the Quincunx sampling matrix [2]. With this, we have
and
..
.
..
.
..
.
..
.
..
.
.
Here, we have used the facts that
a)
and commute since they both are diagonal;
b)
Similarly we have
of matrices
we can have
is chosen as
matrix
With this, the frequency response of the filter can be written
as
and the passband and stopband errors
can be written as
and
where
. Thus, the class
satisfies (5), and hence from Proposition 1
th band quadrantally symmetric filters when
. We note that the hexagonal sampling
can be written as
, and thus belongs
to the above class. Also, it is obvious that the Quincunx matrix
belongs to the above class (with
). Thus, this class does
include some commonly used sampling matrices.
(6)
Thus, the total error is
where
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on December 1, 2008 at 00:28 from IEEE Xplore. Restrictions apply.
(7a)
(7b)
PATWARDHAN AND GADRE: DESIGN OF 2-D
Fig. 2.
M
SP D(M
) for (a) M =
2
1
TH BAND LOWPASS FIR EIGENFILTERS
2
01
, (b) M =
2
2
1
01
.
Now, to impose the th band constraint, we extend the 1-D
technique proposed in [1] for our 2-D case: when we impose the
th band condition, some elements of the vector are forced
to zero. Thus, we can form a new vector by removing the zero
elements from , and a new matrix by removing the corresponding rows and columns of . Based on this eigenformulation, the vector which minimizes the error
is the
eigenvector corresponding to the smallest eigenvalue of .
Also, we note that the passband and stopband should be
chosen to be consistent with the
th band constraint. A
possible choice for the passband, which we use in the design
[2], where SPD(.) is the
examples in this letter, is SPD
symmetric parallelepiped as defined in the notation in Section I.
Fig. 2(a) shows SPD
for
, and
for
Fig. 2(b) shows SPD
. Note that
Fig. 2(a) has quadrantal symmetry, and Fig. 2(b) has diagonal
symmetry.
Design Example 1: In this example we design a th band
quadrantally symmetric filter using the above formulation, with
. The passband is chosen as SPD
Fig. 3. Plot of quadrantally symmetric M th band filter designed in example 1.
(2) in (8) we have
Using similar arguments as in Section II, we have:
Proposition-2: Let
be a nonsingular integer matrix satisfying the following:
and
where and are unimodular integer matrices.
With
satisfying (9), it is possible to design a diagonally
symmetric 2-D th band filter.
Consider the class of matrices
, where and are
as in Section II. With this, we have
Similarly we have
matrices
we can have
chosen as
matrix
TH BAND FILTERS
We now consider the design of th band FIR filters, with
the filter impulse response having diagonal symmetry.
is
said to have diagonal symmetry with center of symmetry at the
origin, if [7], [8]
(8)
where
and
(9)
, and
the stopband is chosen as
SPD
. We
choose
18,
0.5, and the reference passband frequency is taken to be the origin, i.e.
0 0 . The
plot of the frequency response magnitude is shown in Fig. 3.
III. DESIGN OF DIAGONALLY SYMMETRIC
519
.
). An example of a diagonally sym(Note that
metric
is shown in Fig. 1(b).
We only consider the above case of diagonal symmetry, where
the center of symmetry is at the origin, which leads to zero-phase
diagonally symmetric filters.
For designing th band diagonally symmetric filter, we need
to impose the th band constraint of (2) and the diagonal symmetry constraint (8) on the filter impulse response
. Using
. Thus, the class of
satisfies (9), and hence from Proposition-2
th band diagonally symmetric filters when
is
. As an example, the hexagonal sampling
can be written as
, and thus belongs
to the above class. Also, the Quincunx matrix belongs to the
).
above class (with
As an example, an
having diagonal symmetry and the
th band property, with
, is shown in Fig. 1(d)
(The zero values in are marked as “0,” whereas other alphabets
denote non-zero values).
A. Formulation of the Error Function for the Eigenfilter
Design Method
We now formlate the error function for the eigenfilter
method. First, consider a 2-D FIR diagonally symmetric filter,
with center of symmetry at the origin (we will consider the
th band constraint later)
and
(Note that for diagonal symmetry we need
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on December 1, 2008 at 00:28 from IEEE Xplore. Restrictions apply.
.)
520
IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 8, AUGUST 2007
Fig. 4. Plot of diagonally symmetric
M th band filter designed in example 2.
Fig. 5. Plot of diagonally symmetric
The frequency response of the filter can be written as
M th band filter designed in example 3.
. We
the stopband is chosen as
choose
12,
0.5, and the reference passband frequency is taken to be the origin, i.e.,
0 0 . The
plot of the frequency response magnitude is shown in Fig. 4.
. Again,
Design Example 3: In this example we take
, and the stopband is
the passband is chosen as SPD
chosen as
SPD
.
12,
0.5, and
0 0 as in Design Example 1. The
filter plot is shown in Fig. 5.
where
IV. CONCLUSION
Define the column vectors
and
We formulated the eigenfilter design method for the design
of 2-D th band lowpass FIR filters, with the filters having
quadrantal or diagonal symmetry. We observed that imposing
the th band and symmetry constraint puts a restriction on the
possible choices of . With a diagonal integer matrix and
is
the Quincunx sampling matrix , we showed that when
chosen as
we can design th band quadrantally
symmetric filters, and when
is chosen as
we can
design th band diagonally symmetric filters.
as
..
.
..
.
..
.
..
.
REFERENCES
..
.
..
.
M
With these definitions of and
, the total error is as in (6)
and (7). To impose the th band constraint, we follow a similar
approach as done for the quadrantal symmetry case in Section II,
to form the new vector and matrix
from and
.
Design Example 2: In this example we design a th band
diagonally symmetric filter using the above formulation, with
. The passband is chosen as SPD
[1] P. P. Vaidyanathan and T. Q. Nguyen, “Eigenfilters: a new approach
to least-squares FIR filter design and applications including Nyquist
filters,” IEEE Trans. Circuits Syst., vol. CS-34, no. 1, pp. 11–23, Jan
1987.
[2] P. P. Vaidyanathan, Multirate Systems and Filter Banks. Englewood
Cliffs, NJ: Prentice Hall, 1993.
[3] A. Tkacenko, P. P. Vaidyanathan, and T. Q. Nguyen, “On the eigenfilter
design method and its applications: a tutorial,” IEEE Trans. Circuits
Syst. II, vol. 50, no. 9, pp. 497–517, Sep. 2003.
[4] Y. Wisutmethangoon and T. Q. Nguyen, “A method for design of
th band filters,” IEEE Trans. Signal Processing, vol. 47, no. 6, pp.
1669–1678, Jun. 1999.
[5] S. C. Pei and J. J. Shyu, “2-D FIR eigenfilters: a least-squares approach,” IEEE Trans. Circuits Syst., vol. 37, no. 1, pp. 24–34, Jan 1990.
[6] ——, “A unified approach to the design of quadrantally symmetric linear-phase two-dimensional FIR digital filters by eigenfilter
approach,” IEEE Trans. Signal Processing, vol. 42, no. 10, pp.
2886–2890, Oct 1994.
[7] H. C. Reddy, I. Khoo, and P. K. Rajan, “2-D symmetry: theory and
filter design applications,” IEEE Circuits Syst. Mag., vol. 3, no. 3, pp.
4–33, 2003.
[8] M. N. S. Swamy and P. K. Rajan, “Symmetry in two-dimensional filters and its application,” in Multidimensional Systems: Techniques and
Applications, S. G. Tzafestas, Ed. New York: Mercel Dekker, 1986,
pp. 401–468.
, and
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on December 1, 2008 at 00:28 from IEEE Xplore. Restrictions apply.