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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 11, DECEMBER 2008
Eigenfilter Approach to the Design of
One-Dimensional and Multidimensional
Two-Channel Linear-Phase FIR Perfect
Reconstruction Filter Banks
Bhushan D. Patil, Pushkar G. Patwardhan, and Vikram M. Gadre
Abstract—We present an eigenfilter-based approach for the design of two-channel linear-phase FIR perfect-reconstruction (PR)
filter banks. This approach can be used to design 1-D two-channel
filter banks, as well as multidimensional nonseparable two-channel
filter banks. Our method consists of first designing the low-pass
analysis filter. Given the low-pass analysis filter, the PR conditions
can be expressed as a set of linear constraints on the complementary-synthesis low-pass filter. We design the complementary-synthesis filter by using the eigenfilter design method with linear constraints. We show that, by an appropriate choice of the length of the
filters, we can ensure the existence of a solution to the constrained
eigenfilter design problem for the complementary-synthesis filter.
Thus, our approach gives an eigenfilter-based method of designing
the complementary filter, given a “predesigned” analysis filter, with
the filter lengths satisfying certain conditions. We present several
design examples to demonstrate the effectiveness of the method.
Index Terms—Eigen-filter design, least-square filter design, twochannel filter bank.
I. INTRODUCTION
O
NE-DIMENSIONAL (1-D) two-channel FIR perfect-reconstruction (PR) filter banks have been extensively
studied in the literature [1], [2]. In many applications, it is
desirable that the filters in the filter bank have linear phase.
The analysis and design of linear-phase PR FIR filter banks
has received a lot of attention in the literature [1]–[5]. Various
methods for the design of 1-D linear-phase two-channel PR
filter banks have been proposed. Design methods using lattice
structures in the polyphase domain have been presented in
[4]–[6]. These methods rely on a lattice-structure parameterization of the polyphase matrix, and the lattice parameters
are then chosen using some optimization method so that the
filters approximate the desired passband shape. However, the
objective function of the optimization is a nonlinear function
of the parameters, so the optimization becomes difficult with
increasing number of parameters (i.e., with increasing filter
lengths). Design approaches based on spectral factorization
are well known. In these approaches, a halfband product filter
Manuscript received December 30, 2007; revised March 20, 2008. First published May 20, 2008; current version published December 12, 2008. This paper
was recommended by Associate Editor Y.-P. Lin.
The authors are with the Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai 400 076, India (e-mail: bhushanp@ee.
iitb.ac.in).
Digital Object Identifier 10.1109/TCSI.2008.925818
is first designed and is then factored into two filters, yielding
the analysis and synthesis low-pass filters in the filter bank.
Various time-domain optimization approaches have been presented [8]–[10]. In these approaches, the filters in the filter
bank are designed by a “time-domain” optimization of the filter
coefficients (i.e., a direct optimization of the filter coefficients,
without any parameterization) after imposing the PR constraints
on the coefficients.
The design of multidimensional (MD) nonseparable twochannel filter banks is more complicated than the 1-D case
because of the lack of factorization theorems for general MD
polynomials. Thus, the spectral-factorization-based approaches,
which are commonly used for the design of 1-D two-channel
filter banks, can no longer be used for the MD case. The method
of transformations [13]–[15] is a commonly used approach for
the design of MD two-channel filter banks. This method divides
the MD filter-bank design problem to a set of two independent
problems: 1) that of designing a 1-D two-channel filter bank
and 2) to design an MD transformation kernel satisfying certain
constraints. The design of 2-D two-channel Quincunx filter
banks has been considered in [15]–[17], [28], and[29]. Design
of MD nonseparable filter banks using lattice structures has
been presented in [16] and [24]. However, unlike the 1-D case,
the lattice structures are not complete. Furthermore, due to the
considerations of the “shape” of the frequency passbands, the
optimization of the lattice parameters becomes increasingly
difficult with increasing number of filter coefficients.
In this paper, we use a time-domain formulation of the PR
problem and present an eigenfilter approach for the design
of two-channel linear-phase PR filter banks. We use the term
“time-domain formulation” in the 1-D as well as the MD case to
refer to a formulation which directly uses the filter coefficients
and does not use any other parameterization. This approach
can be used for the design of 1-D, as well as nonseparable
MD, two-channel filter banks. Although the eigenfilter method
and its extensions have been effectively used for the design of
1-D and MD filters [11], [12], [20], [21], [25], the eigenfilter
method has not been applied to the problem of filter-bank
design. Our method consists of first designing the low-pass
analysis filter. Given the low-pass analysis filter, the PR conditions can be expressed as a set of linear constraints on the
complementary-synthesis low-pass filter. We then design the
complementary-synthesis filter by using the eigenfilter design
method with linear constraints. We show that, by appropriately
1549-8328/$25.00 © 2008 IEEE
PATIL et al.: EIGENFILTER APPROACH TO THE DESIGN OF 1-D AND MD FIR PR FILTER BANKS
choosing the lengths of the filters, we can ensure the existence
of a solution to the constrained eigenfilter design problem for
the complementary-synthesis filter. Thus, our approach gives
an eigenfilter-based method of designing the complementary
filter, given a “predesigned” analysis filter, with the filter
lengths satisfying certain conditions.
The paper is organized as follows. Section II gives a brief
review of the eigenfilter method as it applies to the design of
FIR filters. In Section III, we consider the case of 1-D twochannel filter banks. In Section III-A, we present a formulation
of the 1-D two-channel FIR PR filter-bank design problem and
cast it in an eigenfilter-design framework in Section III-B. We
present design examples of 1-D filter banks in Section III-C.
In Section IV, we present the extension of the formulation for
the case of MD two-channel nonseparable filter banks, and in
Section IV-A, we present the detailed formulation for the case
of the 2-D nonseparable Quincunx filter banks and present design examples.
Notation: Boldfaced lowercase letters
are used to represent vectors, and bold-faced uppercase letters
are used for
denotes the transpose of .
denotes the inmatrices.
denotes the determinant of the matrix .
verse of .
denotes the absolute value of the scalar .
The following notation is required for Section IV which discusses the case of MD filter banks.
vector raised to a
vector power gives a
A
scalar defined as follows:
, where
and
.A
vector raised to a matrix power , where is a
matrix, is defined as follows:
is a vector whose th entry is
, where ,
, are the columns of the
denotes the set of all
integer vectors. The
matrix .
lattice generated by a
nonsingular integer matrix
is denoted by
and is defined as the set of all
vectors of the form
, where
.
is the
, with
.
set of integer vectors of the form
II. REVIEW OF THE EIGENFILTER DESIGN METHOD [11], [12]
In this section, we briefly review the eigenfilter design
method, as it applies to the design of zero-phase FIR filters
[11], [12]. We will first review the 1-D case and then review
the extensions of the method to the MD case. We then review
the technique of imposing linear constraints on the eigenfilter
design method (we refer to [11] and [12] for more details on
the eigenfilter design method and its various extensions for the
design of 1-D and MD filters).
1) Eigenfilter Method for the Design of 1-D FIR Filters: In
the eigenfilter design method, the objective is to formulate the
, where
error function to be minimized in the form
is a real, symmetric, and positive-definite matrix, and is
a real vector. The goal is to find a vector which minimizes
. For the design of 1-D FIR filters, the elements of
the vector are related in some manner to the filter impulse response
. The constraint
is imposed to avoid trivial
should be chosen to properly
solutions. The error measure
reflect the deviation of the passband and the stopband from the
. Once such an error
ideal values of the desired response
measure is chosen, by the Rayleigh principle [11], [12], [26],
3543
the eigenvector associated with the smallest eigenvalue of the
matrix minimizes the error .
, where
Consider a 1-D zero-phase low-pass FIR filter
. We note that, within a delay factor, this is
. Since
an odd-length linear-phase filter, with length
is zero-phase, we have
. With this, the frequency response of
takes the form
Defining the vectors
(1)
can be written as
.
The frequency response of the 1-D low-pass filter should apgiven by
proximate a desired response
where
and
are the passband and stopband cutoff frequencies, respectively.
With this desired response, the stopband error can be defined
as
can be written in the form
, where
is a real, symmetric, and positive-definite matrix.
The passband error can be expressed as
, where
is the deviation of the response from the zero frequency response, which is given as
.
Thus,
can be written in the form
, where
is a real, symmetric, and positive-definite matrix.
The total error to be minimized is
(2)
is a real, symmetric, and positive-definite matrix, as required
for the eigenfilter formulation.
associated with the smallest
With this, the eigenvector
eigenvalue of matrix minimizes the error . can then be
used to obtain the filter coefficients using (1).
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 11, DECEMBER 2008
the PR condition for the filter bank can be written as [1]–[5]
(3)
Fig. 1. One-dimensional two-channel filter bank.
2) Eigenfilter Method for the Design of MD FIR Filters:
The eigenfilter method can be extended to the case of MD
-dimensional
zero-phase FIR filters [12]. Consider an
, where
is an
zero-phase low-pass FIR filter
vector. Since
is zero-phase, we have
.
takes the form
With this, the frequency response of
, where
is the
set of integer vectors corresponding to the indexes of the
. Now,
independent coefficients of the zero-phase filter
by imposing some ordering on the independent coefficients of
, we can form a vector . With this,
can be written as
, where
is a vector consisting of elements
of the form
. With this “vectorization,” the design
can
of the -dimensional zero-phase low-pass FIR filter
be formulated as an eigenfilter design problem, similar to the
formulation done earlier for the 1-D case.
3) Imposing Linear Constraints on the FIR Eigenfilters:
, where is a matrix
Linear constraints of the form
is a vector having constant
having constant elements and
elements, can be imposed on the FIR eigenfilter design [11],
[12]. Note that, here, is the “coefficient vector” corresponding
can be
to the filter to be designed. The constraint
expressed in the following form (we refer to [12] for details):
, where
, where
is the
reference frequency which we choose to be the zero frequency
for the case of low-pass filters. The linear constraints
can be imposed as follows:
if and only if lies in
the null space of . Any such can be expressed as
,
where is a rectangular unitary matrix whose columns form an
and is any arbitrary
orthonormal basis for the null space of
vector. Using this, it can be shown that, after imposing linear
constraints, the error function to be minimized can be written as
. Therefore, the optimal is the eigenvector
. Once we
corresponding to the smallest eigenvalue of
.
find the optimal , the optimal is given by
For the PR condition in (3), any constant value instead of two
could be used. We use the constant value of two in this paper.
Defining the “product filter”
as
,
should be a halfband filter, i.e., the coeffi(3) states that
cients of
corresponding to the terms with even powers,
other than the origin, should be zero. This can be written as
(4)
where
is the inverse -transform of
is the convolution of
and
.
(5)
Let
be of length
and
be of length, i.e.,
(6)
Using (6) in (5), we have
(7)
and
are zero-phase, it follows that
Note that, since
is also zero-phase. Thus, we have only considered positive
is
, i.e.,
values for in (7). The length of
, for
.
But since
is also zero-phase,
.
and
can be
We would like to note that the filters
and
in the zeromade causal by having delay factors
and
, respectively. This results in a delay
phase
factor
in
, which retains PR with a delay factor
in the right-hand side of (3). However, in this paper, we will
assume zero-phase filters for convenience.
Using (7) in (4) gives
III. DESIGN OF 1-D TWO-CHANNEL LINEAR-PHASE FIR PR
FILTER BANKS
In this section, we present the design of 1-D linear-phase PR
filter banks using the eigenfilter approach.
A. Problem Formulation
A two-channel 1-D filter bank is shown in Fig. 1.
and
are zeroIn this paper, we will assume that
phase, i.e.,
and
. We note that,
within a delay factor, this corresponds to odd-length filters [3].
With the following choice of the two high-pass filters:
(8)
where
denotes the highest integer less than .
Thus, to design a PR two-channel filter bank, we need to deand
, such that (8) is satisfied.
sign zero-phase
PATIL et al.: EIGENFILTER APPROACH TO THE DESIGN OF 1-D AND MD FIR PR FILTER BANKS
B. Design Method
We observe that, given
, (8) gives a set of
linear
unknown independent
equations for the
coefficients of
, for
.
The set of linear (8) can be written in matrix notation as
, where is the
vector containing the “variables”
, for
of the linear equations,
is the
matrix containing the “constant coefficients” of the
is an
vector,
linear equations, and
whose first entry is two, and all other entries are zeros.
The elements of the matrix are given by
(9)
As an example, for
and
, the matrix is shown
as the equation at the bottom of the page. As discussed in
Section II, for the eigenfilter design formulation, the concan be written in the form
, where
straint
are the same as . Thus, the set of
the dimensions of
linear equations in
unknowns,
, will have a nontrivial solution if the rank of the
.
matrix ,
,
Since
, i.e.,
a nontrivial solution can be guaranteed if
.
,
We would like to note that, when
there exists a unique solution, i.e., the constraints completely
determine the unknowns, and there are no “free variables” to
optimize. Thus, we choose the values of and so that
and so that we can use the eigenfilter
design method to optimize the “free variables.”
Thus, this suggests the following procedure for the design of
and
.
the filters
of length
. We note that
1) Design zero-phase
cannot be arbitrary. As shown in [4], a necessary and
to be an analysis filter of a
sufficient condition for
PR filter-bank pair is that its two polyphase components
). The
be coprime (except for possible zeros at
techniques that we use in this paper to ensure that
is valid (i.e., satisfies the earlier condition) are as follows.
a) Design
using the unconstrained eigenfilter
method and, then, verify explicitly by factorization
that its polyphase components do not have common
3545
zeros. This technique works well, because, for a
general 1-D filter, it is “almost always” true that
its polyphase components are coprime. In fact, this
holds for the filter design examples that we show in
Section III-C as follows.
as the analysis filter of a “known” PR
b) Choose
filter-bank pair. For example, in one of the design
as the analexamples as follows, we choose
ysis filter of the Daub97 biorthogonal filter bank [2].
is valid “by design.”
Therefore, in such cases,
as a halfband
c) Another simple way is to design
filter. Since one of the polyphase components is a conis
stant (or a monomial, in general), a halfband
always valid.
.
2) Choose a such that
3) Design zero-phase
of length
by imposing the
, where is as defined in (9),
linear constraints
and
This design can be done using the constrained eigenfilter
method as described in Section II. As discussed earlier,
with chosen as in 2) in the list, a solution is guaranteed
to exist.
In the earlier procedure, it is very easy to increase the length of
the filters. We would also like to note that any FIR filter design
, the only requirement being
method can be used to design
should be valid, as discussed earlier. In fact, the earlier
that
, given
procedure can design the “complementary filter”
, within certain constraints on the lengths
any valid filter
of the two filters.
We would like to note that our proposed design method im, whereas the design
poses constraints on the design of
is relatively mildly constrained (it needs to be valid).
of
Thus, to achieve a certain frequency-response criteria (for example, stopband attenuation), the required filter length of
will be larger than that of
(with the same frequency-response criteria).
C. Design Examples
1) Design Example-1: In this example, for
, we use the
nine-tap analysis low-pass filter from the Daub97 filter bank
, we design the complementary-synthesis
[2]. Given this
filter
with
. In the eigenfilter error formulation
,
, and
. The freof (2), we use
and
are shown in
quency-response plots of
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 11, DECEMBER 2008
Fig. 2. Plot of
H (!) for Example-1.
Fig. 3. Plot of
G (!) for Example-1.
Fig. 4. Plot of
H (!) for Example-2.
Fig. 5. Plot of
G (!) for Example-2.
TABLE I
COEFFICIENTS OF [ ]OF DESIGN EXAMPLE 1
g n
Figs. 2 and 3, respectively. The coefficients of
are shown
in Table I.
2) Design Example-2: For this example, we use the unconwith
strained eigenfilter method for the design of
, and after obtaining the coefficients of
, we explicitly
verify that it is valid. We then design
with
. Again,
,
, and
. The frequency-response
we use
and
are shown in
plots of the filters
and
Figs. 4 and 5, respectively. The coefficients of
are shown in Tables II and III, respectively.
3) Design Example-3: For this example, we design a halfusing the eigenfilter design method with
.
band
, every even indexed coefficient
Note that, for the halfband
for the design
except the origin is zero. We then choose
. The frequency-response plots of the filters
and
of
are shown in Figs. 6 and 7, respectively.
4) Design Example-4: One of the advantages of using the
eigenfilter method is the ease with which certain time- and frequency-domain constraints can be incorporated in the design
[12]. In this example, we demonstrate that this flexibility of
the eigenfilter method can be effectively used in the filter-bank
PATIL et al.: EIGENFILTER APPROACH TO THE DESIGN OF 1-D AND MD FIR PR FILTER BANKS
3547
TABLE II
COEFFICIENTS OFh [n]OF DESIGN EXAMPLE 2
TABLE III
COEFFICIENTS OFg [n]OF DESIGN EXAMPLE 2
design. To demonstrate this, we impose the constraints correin both of the filters
sponding to having zeros at
and
. For both the filters, we impose a third-order zero at
. We design
by using the eigenfilter method after
,
imposing the zero constraints. Moreover, in the design of
the zero constraints are added to the “PR constraints.” Adding
the zero constraints increased the number of constraints, and
, i.e., the value of ,
so the minimum required length of
and
increases accordingly. For this example, we use
for the design of
and
, respectively. As
for the earlier examples, we use
,
,
. The frequency-response plot of the filters
and
and
are shown in Figs. 8 and 9, respectively. The
coefficients of
and
are shown in Tables IV and V,
respectively.
IV. EXTENSIONS TO THE MD CASE
We now extend the approach to the case of MD two-channel
linear-phase filter banks. An -dimensional two-channel filter
bank is shown in Fig. 10.
is the sampling matrix, which is a
nonsinHere,
. Denoting the integer
gular integer matrix with
Fig. 6. Plot of H
(
!) for Example-3.
Fig. 7. Plot of G
(
! ) for Example-3.
vectors in
as
and , we can assume without loss of
(the
zero vector). Let the high-pass
generality that
filters be chosen as [15], [22]
(10)
,
where
are the columns of
, and the symbol denotes the ele.
mentwise product
With the choice of the high-pass filters as in (10), the PR
condition is [15]
(11)
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 11, DECEMBER 2008
TABLE V
COEFFICIENTS OF [ ]OF DESIGN EXAMPLE 4
g n
Fig. 8. Plot of
H (!) for Example-4.
Fig. 10. MD two-channel filter bank.
Note that
is the MD convolution of
and
(13)
Thus, from (12), we have the PR conditions as
(14)
Fig. 9. Plot of
Now, given
, (14) imposes a set of linear constraints on
. By choosing the number of coefficients of
appropriately, we can ensure that a solution to (13) exists. Thus, this
problem can be cast into the framework of the eigenfilter design method with linear constraint, in a manner similar to that
done for the 1-D case in Section III. We now present the specific formulation and design examples for the case of 2-D Quincunx filter banks. A similar formulation can also be done for
the design of 3-D two-channel face-centered orthorhombic filter
banks.
G (!) for Example-4.
TABLE IV
h [n]OF DESIGN EXAMPLE 4
COEFFICIENTS OF
By defining the product filter
condition of (11) can be written as [15]
for
, the PR
denotes the set of all
where
is the MD inverse -transform of
Equation (12) states that
on the lattice
.
Consider the 2-D Quincunx filter bank, whose sampling matrix is
for
where
and
A. Design of 2-D Quincunx Filter Banks
(12)
integer vectors and
[23].
on all the nonzero points
. Assume that the low-pass analysis and
and
are zero-phase. Note that, for
synthesis filters
the 2-D case,
.
Assume that the filters have a square region of support, as
defined as follows:
(15)
PATIL et al.: EIGENFILTER APPROACH TO THE DESIGN OF 1-D AND MD FIR PR FILTER BANKS
Therefore, in this case, the product filter is given as follows:
3549
and
. Then, we have
(16)
is zero-phase,
Since
and a set of independent coefficients of
,
are
(17)
independent coefficients of
Thus, there are
.
Using this, we can rewrite (16) in terms of the independent
as follows:
coefficients of
where
denotes the largest integer less than .
From (18) and (20), we now have a set of
equations in
unknowns (which are the independent coefficients
of
).
Thus, by arranging the independent coefficients of
in a vector , (20) can be written as
(21)
is a matrix with
rows and
where
columns, which is obtained from (20), and is a vector formed
from the “unknowns” (coefficients of
) of size
.
To ensure the existence of nontrivial solutions for (21), we
require that the number of rows of are less than the number
of columns
(22)
(18)
Since
and
are both zero-phase,
is also zero-phase, i.e.,
. Thus, the independent set of equations are obtained by using the following
values for ,
in (18):
Now, from (12), we require that
(19)
, where
,
The condition
,
can also be written as follows:
are integers and
Thus, we have (20), shown at the bottom of this page.
Let
denote the number of locations
,
where
, i.e.,
denotes the number
of locations
in the set
is as given earlier.
where
Thus, the overall procedure to design Quincunx filter banks
is as follows.
a) Design a 2-D FIR filter
,
,
,
with a diamond-shaped passband. Again, like the 1-D
case, we note that
cannot be arbitrary. It has
been shown in [27] that a necessary and sufficient condition for an MD filter
to be an analysis filter
of a PR filter-bank pair is that its two polyphase components with respect to the subsampling matrix
be coprime. The techniques b) and c) used for the 1-D case (see
Section III-B) can also be used for the 2-D case (and for
the MD case in general) to ensure that
is valid (i.e.,
satisfies the earlier condition). We would like to note that
the method a) of Section III-B, which works well for the
1-D case, is in general difficult for the 2-D case. In the design examples as follows, we present examples using both
the techniques b) and c) to design
.
b) Choose such that (22) is satisfied.
c) Use the eigenfilter design method with the formulation
presented in this section to obtain the filter
.
We now present some design examples to demonstrate this
method.
1) Design Example-5: For this example, we use the technique b) to obtain a valid analysis filter
, i.e., we
choose
to be the analysis filter of a “known” PR
Quincunx filter bank. We obtain
by using the method
(20)
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 11, DECEMBER 2008
Fig. 11. Plot of
H (z ; z ) for Design Example-5.
Fig. 13. Plot of
G (z ;z ) for Design Example-6.
Fig. 12. Plot of
G (z ; z ) for Design Example-5.
Fig. 14. Plot of
H (z ;z ) for Design Example-7.
Fig. 15. Plot of
G (z ;z ) for Design Example-7.
of McClellan transformations [23] on the analysis low-pass filter
of a 1-D PR filter bank. We use the following five-tap zero-phase
1-D analysis filter from the spline family of filter banks:
Moreover, we use the following 2-D kernel as the transformation
kernel:
With this, we have
. The plot of the filter
obtained is shown in Fig. 11.
For designing
, we use
, which satisfies (22).
The plot of the filter
obtained is shown in
Fig. 12.
2) Design Example-6: In this example, we use the same
as Design Example-5. For
, we use
. The plot of
obtained is shown in Fig. 13.
3) Design Example-7: In this example, we use a Quincunx
halfband filter as
. The region of support of
is
, with
. We then design
using the earlier formulation with
. The plots of
and
obtained are shown in Figs. 14 and
15, respectively.
PATIL et al.: EIGENFILTER APPROACH TO THE DESIGN OF 1-D AND MD FIR PR FILTER BANKS
V. CONCLUSION
In this paper, we presented an eigenfilter-based approach to
the design of two-channel linear-phase PR filter banks. This
method can be used to design 1-D as well as MD filter banks.
We independently designed the low-pass analysis filter. Given
the low-pass analysis filter, the PR conditions can be written as
a set of linear constraints on the synthesis filter coefficients. We
casted this problem of the design of the synthesis filter into an
eigenfilter design problem with linear constraints. We presented
detailed formulations of this method for the design of 1- and 2-D
filter banks and presented several design examples.
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M
Bhushan D. Patil received the B.E. degree in instrumentation engineering from Government College of
Engineering, Jalgaon, India, and the M.E. degree in
instrumentation engineering from Shri Guru Gobind
Singhji Institute of Engineering and Technology,
Nanded, Jalgaon. He is currently working toward
the Ph.D. degree in the field of communication
and signal processing at the Indian Institute of
Technology Bombay, Mumbai, India.
His research interests include wavelets and filter
banks and their applications.
Pushkar G. Patwardhan received the B.E. degree (with an award for achieving second rank in
the college) in electronics engineering from K. J.
Somaiya College of Engineering, University of
Mumbai, Mumbai, India, in 1995 and the M.Tech.
degree (with the Prof. G. N. Revankar Award for
achieving second rank in the Electrical Engineering
Department) in communications engineering and the
Ph.D. degree from the Indian Institute of Technology
Bombay, Mumbai, in 1997 and 2007, respectively.
He is currently with India Design Center, Tensilica
Inc., Pune, India. His primary research interests are in the area of multidimensional multirate systems, filter banks, and wavelets.
Vikram M. Gadre received the B.Tech. and Ph.D.
degrees in electrical engineering from the Indian Institute of Technology (IIT), New Delhi, India, in 1989
and 1994, respectively.
He is currently a Professor with the Department of
Electrical Engineering, IIT Bombay, Mumbai, India.
His research interests broadly include communication and signal processing, with emphasis on multiresolution approaches.
Dr. Gadre was the recipient of the President of
India Gold Medal from IIT Delhi in 1989 and the
Award for Excellence in Teaching from IIT Bombay in 1999 and 2004. He
was also the recipient of the SVC Aiya Memorial Award and the Prof. K.
Sreenivasan Medal from the Institution of Electronics and Telecommunication
Engineers for contribution to education in Electronics and Telecommunication.