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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 55, NO. 8, AUGUST 2008
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On Parallelepiped-Shaped Passbands for
Multidimensional Nonseparable th-Band
Low-Pass Filters
Qutubuddin Saifee, Pushkar G. Patwardhan, and Vikram M. Gadre
Abstract—The “shape” of the desired frequency passband is
an important consideration in the design of nonseparable multi-D multirate systems. For
-D
dimensional ( -D) filters in
th-band filters, the passband shape should be chosen such that
th-band constraint is satisfied. The most commonly used
the
-D th-band low-pass filters is the
shape of the passband for
T . In this
so-called symmetric parallelepiped (SPD)
paper, we consider the more general parallelepiped passband
T , and derive conditions on such that the th-band
constraint is satisfied. This result gives some flexibility in designing
-D th-band filters with parallelepiped shapes other than the
1 . We present design examples of
commonly used case of
2-D th-band filters to illustrate this flexibility in the choice of .
MM
SPD( L )
M
M
M
SPD( M )
L
M
L=M
Index Terms—Multidimensional filters,
Mth-band filters.
L
Fig. 1. 2-D
I. INTRODUCTION
Notation:
OLD-FACED lower case
letters are used to represent
are used
vectors, and bold-faced upper case letters
denotes the transpose of
denotes the
for matrices.
inverse of , and
denotes the inverse of
.
denotes the absolute value of the determinant of .
,
the symmetric parallelepiped (SPD) of matrix , is defined as
.
is defined as the set of integer
the set
. All the matrices are of size
vectors of the form
, where is the number of dimensions. For the 2-D case,
.
which is extensively used for the design examples
A 1-D th-band filter ( is an integer in the 1-D case) is defined as satisfying the following constraint on the filter impulse
response [1]:
Mth-band impulse response for M = [ 22
defined as satisfying the following constraint on the
pulse response:
B
for
constant
for
(1a)
(1b)
It has been shown in [2] that for the impulse response of an
ideal 1-D low-pass filter to satisfy the th-band constraint of
(1), the passband cutoff frequency must be an integer multiple
. Various methods for the design of 1-D th-band
of
filters have been proposed [4], [6]–[10], [15], [16].
Now consider the multidimensional ( -D) case. An -D
th-band filter, where
is a nonsingular integer matrix, is
Manuscript received October 1, 2007; revised December 12, 2007. First published April 30, 2008; last published August 13, 2008 (projected). This paper
was recommended by Associate Editor Y.-P. Lin.
The authors are with the Department of Electrical Engineering, Indian Institute of Technology, Mumbai 40076, India.
Digital Object Identifier 10.1109/TCSII.2008.921785
1
01 ].
for
constant
As an example, a 2-D
-D im-
(2a)
(2b)
for
th-band filter, for
,
is shown in Fig. 1. The “zero” samples are denoted by a “0,”
whereas nonzero samples are denoted by “X.”
In the frequency domain, the -D th-band constraint is
manifested as
constant
(3)
th-band filters, design of 2-D
Unlike the case of 1-D
th-band filters with general nondiagonal
has been addressed to a much lesser extent. Chen and Vaidyanathan [11]
present a method of designing nonseparable -D filters from
1-D filters. It is also shown in [11] that this method can be used
to design -D
th-band filters from 1-D
th-band
filters. In [12], the design of 2-D th-band eigenfilters having
various symmetries is presented.
For -D th-band filters, the most commonly used passband is the parallelepiped
. As an example,
with
is shown in Fig. 2(a). The
problem of considering other passbands for -D th-band
filters has been inadequately addressed in the literature. Motivated by the result on the constraints on the ideal passbands
for 1-D th-band filters derived in [2], our aim in this paper
is to derive analogous results on the constraints on ideal parallelepiped-shaped passbands for
-D
th-band filters. In
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SAIFEE et al.: PARALLELEPIPED-SHAPED PASSBANDS FOR
-D
TH-BAND FILTERS
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Thus, we have
for all integers
(5)
With this,
Let
, where
we have from (5)
Fig. 2. For Example-1 (a) SPD(
M
), (b) SPD(L ).
for all integers
particular, we consider the parallelepiped passband
,
and derive conditions on such that the th-band constraint is
satisfied. We note that, in general, the matrix has noninteger
(rational) elements. We also present design examples of 2-D
th-band filters, with different choices for , to demonstrate
the flexibility in choosing the passband.
The rest of the paper is organized as follows. Section II
-D imderives the conditions on , such that the ideal
pulse response corresponding to the parallelepiped passband
satisfies the th-band constraint. We then consider
a class of matrices for which satisfies these conditions. We
also consider 2-D examples to illustrate some of the passbands
matrices. In Section III, we
possible using this class of
th-band FIR filters using
present design examples of 2-D
the eigenfilter design method. This demonstrates that good 2-D
th-band filters with parallelepiped passbands as derived in
Section II can be designed.
II. CONSTRAINTS ON SO THAT
“ALLOWED” PASSBAND FOR A -D
Consider an ideal zero-phase
passband
, i.e.,
TH-BAND
IS AN
FILTER
-D low-pass filter with the
if
otherwise
(4a)
(4b)
The -D impulse response for this ideal filter can be obtained
by taking the -D inverse Fourier transform [3], [11] of (4)
But
implies
, where
By expressing the above equation in terms of , we have
.
Now imposing the
we have
,
th-band constraint (2) on the above
for
Note that
is an integer vector.
(6)
Equation (6) implies that at least one of the ’s should be a
nonzero integer, for all integer
This can be obtained if the matrix is such that
has at
least one row with all integer values.
. The frequencyAnother constraint is imposed on
domain th-band constraint of (3), when applied to the ideal
, implies that
times the hyperpassband
should be an integer multiple of the hyvolume of
pervolume of
, where is the
Identity matrix.
Thus, we have
(7)
where is a nonzero positive integer.
can be interpreted as the “overlap factor” by which the region
covers the hypercube
.
We note that this “overlap factor”
also exists in
the case of the ideal passband for a 1-D
th-band
filter [2], where the cutoff frequency is constrained as
. The “overlap factor”
is the ratio of
the
and
. In the 1-D case, this is the ratio
. Thus, (7) says that the “overlap factor” should
be a nonzero integer.
The admissibility condition on the ideal frequency supports
of the filters in a -D nonseparable filter bank [13], [14] is
different from the concept of the allowed overlap that is described above. The admissibility condition for the 2-D nonseparable filter banks in [13], [14] requires that there be no overlap
in the shifted versions of the ideal parallelogram supports.
To summarize, we have the following result,
Proposition-1: Consider the ideal parallelepiped passband
for an ideal -D zero-phase filter. The matrix
has, in general, noninteger (rational) elements. The ideal impulse response corresponding to this passband will satisfy the
-D th-band constraint if is chosen such that the following
are true.
1)
has at least one integer row.
2)
, for some nonzero positive integer .
We now consider a class of matrices, resulting in parallelepiped passbands
, which can be considered as
a generalization of the 1-D passbands of [2]. Consider the following choice for :
, where is a matrix satisfying the following.
a)
has at least one integer row.
b)
is a nonzero positive integer .
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 55, NO. 8, AUGUST 2008
Fig. 4. The eight unique regions of Fig. 3 completely fill the region
SPD(I) = [0; ) .
Fig. 3. Shifts of SPD(L ) by f2 M
ample-1.
k , with k 2 N (M )g for Ex-
It is evident that this choice of
satisfies the conditions
of Proposition-1, and thus the ideal impulse response corresponding to the passband
satisfies the th-band
constraint.
Note that for the 1-D case,
, and
. Thus,
for the 1-D case, this leads to the result of [2] that the cutoff
frequency of the ideal passband should be an integer multiple
of
.
We now present specific examples for the 2-D case that will
illustrate the different possible passbands, with
chosen as
above.
Example-1: We consider
and
with
.
is shown in Fig. 2(a), and
(b) shows
.
The overlap factor for this case is
. We
now illustrate that the shifts of
by
, with
, completely cover the region
with an
overlap factor of 2, i.e., the region
is covered two
times by the shifted copies of
. In this case, the set
Fig. 3 shows the region
and its shifts by
, for
. The region
is shown dotted. In Fig. 3, we note the following points.
a) We have shown each
as a concatenation of
four-shifted parallelogram (FPD) regions
,
where
is the region
.
Fig. 5. For Example-2: (a) SPD( M
) and (b) SPD( L ).
Fig. 6. (a) Shifts of SPD( L ) by f2 M k , with k 2 N (M )g for Example-2, (b) The twelve regions of (a) completely fill the region SPD( I) =
[0; ) .
b) We have marked each shifted
region such that
we give the same marking to two regions if they completely overlap. Also, we have used a “ ” sign for the
second marking of the same region i.e., the region marked
as “
” completely overlaps with the region marked as
“A.”
c) We have also done “modulo” operations
on some marked regions, to show that after the
” operation the region actually com“modulopletely overlaps with another region, thus getting the same
marking. By “modulo” operation we mean
moving the region by the shift-vector
,
where p and q are integers.
d) Thus, from Fig. 3 we see that there are a total of sixteen
marked regions, out of which eight regions are “unique”
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SAIFEE et al.: PARALLELEPIPED-SHAPED PASSBANDS FOR
-D
TH-BAND FILTERS
Fig. 7. Frequency response of the
Mth-band filter of Design Example 1.
Fig. 8. Frequency response of the
Mth-band filter of Design Example 2.
(i.e.,markedwithoutthe“ ”sign).Andtheremainingeight
regions (marked with the “ ” sign) overlap with the one of
the “unique” eight regions, as shown by the markings.
e) Thus, it can be seen that the “overlap factor” is 2.
Finally, it now remains to be shown that the union
of “modulo” of the unique eight regions is
.
Fig. 4 shows that all the parts of the unique eight regions which lie outside the
region come inside the
-region by a modulooperation, such that the
-region is fully covered by the eight unique regions.
The parts of the regions which undergo a modulooperation are shown dotted.
789
, and
Example-2: We consider
with
.
is shown in
Fig. 5(a), and 5(b) shows
.
Fig. 6(a) shows the shifts of
by
, with
. Here we have marked the regions by numbers.
In this example, the set
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 55, NO. 8, AUGUST 2008
As can be seen from Fig. 6(a), in this case there are no overlapping regions. This is consistent with the fact that, in this case,
. So, there is no overlap.
the overlap factor
is covered
Fig. 6(b) shows that the region
completely by the shifted copies of
.
Thus, as can be seen from the above examples, the choice of
as above
does result in parallelepiped passdifferent from the commonly used case of
bands
. Also, depending on the overlap factor , the hypervolume of the passband
can be greater than (actually, an integer multiple) that of the commonly used passband
.
III. DESIGN EXAMPLES FOR 2-D
TH-BAND
conditions on such that the th-band constraint is satisfied.
We presented design examples for 2-D
th-band filters to
th-band filters can be designed by
demonstrate that good
, with appropriate conchoosing the passband as
straints on . This gives some flexibility in designing
-D
th-band filters with parallelepiped passbands
,
with choices of
other than the commonly used case of
.
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[13] Y. Lin and P. P. Vaidyanathan, “Theory and Design of Two-parallelogram filter banks,” IEEE Trans. Signal Process., vol. 44, no. 11, pp.
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M
FILTERS
In this section, we present design examples of 2-D th-band
filters with the passbands
, where is chosen as
, as discussed in Section II. These design examples
demonstrate that good th-band filters can be designed using
the passbands chosen according to Proposition-1. For the design
examples, we consider a zero-phase 2-D FIR filter:
N
Note that, for an th-band zero-phase filter, some of the filter
coefficients will be zero. We use the eigenfilter design method
[4], [5] for all the designs. For imposing the th-band constraint on the filter, we use the technique discussed in [4], [5].
Design Example 1: Consider the matrices as in Example-1
in Section II.
, and
. We
choose
. The passband is chosen as
,
and the stopband is chosen as
.
Fig. 7 shows the frequency response of the designed th-band
filter.
Design Example 2: For this design example, we consider the
matrices as in Example-2 in Section II.
and
. Again, as in design example 1, we
. Fig. 8 shows the frequency response of
choose
,
the designed th-band filter with the passband as
and the stopband as
.
IV. CONCLUSION
In this paper, we addressed the problem of choosing parallelepiped-shaped passbands for -D th-band filters, other
than the commonly used
. We considered the
more general parallelepiped passband
, and derived
N
M
M
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M
M