IEEE
TRANSACTIONS ON
SIGNAL
PROCESSING,
VOL.
43,
NO.
3,
MARCH
1995
649
A
New Class
of
Two-Channel Biorthogonal
Filter Banks and Wavelet Bases
See-May Phoong,
Student Member,
IEEE,
Chai
W.
Kim,
P. P. Vaidyanathan,
Fellow,
IEEE,
and Rashid Ansari
Abstract-We propose a novel framework for a new class of
two-channel biorthogonal filter banks. The framework covers two
useful subclasses:
i) causal stable IIR filter banks
ii) linear phase FIR filter banks.
There exists a
very
effcient structurally perfect reconstruction
implementation for such a class. Filter banks of high frequency
selectivity can be achieved by using the proposed framework
with low complexity. The properties of such a class are discussed
in detail. The design of the analysis/synthesis systems reduces
to the design of a single transfer function. Very simple design
methods are given both for FIR and IIR cases. Zeros of arbitrary
multiplicity at aliasing frequency can be easily imposed, for the
purpose of generating wavelets with regularity property. In the
IIR case, two new classes of IIR maximally fit filters different
from Butterworth filters are introduced. The filter coefficients
are given in closed form. The wavelet bases corresponding to
the biorthogonal systems are generated. We also provide a novel
mapping of the proposed
1-D
framework into
2-D.
The mapping
preserves the following:
i) perfect reconstruction
ii) stability in the IIR case
iii) linear phase in the FIR case
iv) zeros at aliasing frequency
v) frequency characteristic of the filters.
I.
INTRODUCTION
IG. l(a) shows a two-channel maximally decimated filter
F
bank, and Fig. l(b) shows the well-known polyphase form
for this system. The applications of such multirate systems are
well-known [I]-[7]. If for all input
~(n),
the output of the
system
2(n)
=
CX(~
-
no)
for
some nonzero constant
c
and
integer
no,
the system is called a perfect reconstruction (PR)
system. In the maximally decimated case, PR is equivalent to
biorthogonality [5].
A
number of PR or nearly PR systems
have been reported before.
In
this paper we develop several
new results for two-channel biorthogonal filter banks based
on
a useful class of polyphase matrices.
A.
Previous Work
In
FIR filter banks, all the four filters
Ho,
HI,
Fo,
and
Fl,
are FIR filters while in the case of IIR filter banks. some or all
Manuscript received October
8,
1993; revised August
18,
1994. This work
was supported by Office of Naval Research under Grant NOO 014-93-0231 and
funds from Tektronix, Inc. The associate editor coordinating the review
of
this
paper and approving it for publication was Prof. Roberto
H.
Bamberger.
S.-M.
Phoong and P. P. Vaidyanathan are with the Department of Electrical
Engineering, California Institute of Technology, Pasadena, CA, 91 125 USA.
C.
W.
Kim is with the Department of Electrical Engineering, University of
Pennsylyania, Philadelphia, PA, 19104 USA.
R.
Ansari is with Bellcore MRE, Morristown,
NJ,
07960 USA.
IEEE
Log
Number 9408226.
-
analysis bank
-
synthesis bank
m
I
R(z)l
I
I
ml
polyphase matrices
A\
\WJ
Fig.
1.
by using the polyphase representation.
(a) Two-channel analysislsynthesis filter bank (b) redrawing
of
(a)
of these filters are IIR filters. The earliest good designs for the
IIR case were such that the analysis bank was paraunitary and
the polyphase components of
Ho(z)
and
HI(%)
were allpass
(see [7], and p. 201
of
[2]). Even though all the
IIR
filters
are causal stable, the reconstructed signal suffers from phase
distortion. IIR PR filter banks typically have noncausal stable
filters
or
causal unstable filters [8]-[ 101. Recently the authors
in
[
111 proposed a IIR PR technique providing causal stable
solutions, but
no satisfactory design method was given.
In
earlier design
of
2-D filter banks, separable filters have
been considered because
of
their advantage of low complexity.
However nonseparable filters offer more freedom in the design
and hence in general will give better performance. Recently,
some results
on
the nonseparable filter banks have emerged
[
121-[ 141. However, few design techniques are available for
nonseparable PR filter banks. In [12], a design method based
on
space domain approach is given.
In
[13], a subclass
of
2-D paraunitary systems (which can
be
represented as
a cascade of I-D paraunitary systems
of
degree one) is
3053-587)</95$04.OO
0
1995 IEEE
650
IEEE
TRANSACIIONS ON SIGNAL PROCESSING,
VOL.
43,
NO.
3,
MARCH
1995
considered. However,
in
both of the polyphase approaches
above, the optimization in the designs involves a large number
of nonlinear constraints. Thus other approaches, such as 1-
D to 2-D mapping, have been considered [14]-[19]. In [14],
even though PR property is preserved by the mapping, the
frequency responses of the filters will change. In [15] and
[16], a mapping of
1-D filter banks to 2-D filter banks
is given. The authors apply the technique on a 1-D two-
channel orthogonal IIR system to achieve a 2-D IIR filter
bank. The resulting systems have either phase distortion
or
stability problem.
In
[17], the authors employ McClellan’s
transformation on the
1-D maximally flat FIR halfband filters
to obtain a 2-D biorthogonal filter bank. However, because
of
the lack of factorization theorems in the 2-D case, one
of the lowpass filters is constrained to have all its zeros at
the aliasing frequency. In addition, there is no simple way
to ensure the frequency selectivity of all the filters. In [18],
the authors introduce a mapping which can be viewed as the
generalization McClellan’s transformation. 2-D two-channel
PR
systems with good frequency selectivity can be obtained
by judiciously designing the mapping. However, the mapping
works for the FIR case only and the resulting filters usually
have a large number of coefficients.
Some of the results in this paper were reported in the earlier
conference papers [20]-[22]. For the 1-D case [20], both of
the
linear phase FIR and causal stable IIR solutions for PR
filter banks similar to those proposed in this paper were given.
For the 2-D quincunx case [21], the authors constructed a I-D
to 2-D mapping (which is the same as the mapping given in
Section
V
in this paper) that preserved many of the desired
properties. However many of the properties of the 1-D and 2-
D biorthogonal systems were not addressed in [20], [21], for
example, the problem of imposition of zeros at the aliasing
frequency which is important for the purpose of generating
smooth wavelet basis functions.
B.
The New Idea and Its Merits
In this paper, we constrain the polyphase matrix
E(z)
such
that
[det
E(z)]
is a delay. Furthermore, we consider
E(z)
and
R(z)
to
be
either i) both causal stable IIR or ii) both FIR.
In each case, the following properties can
be
simultaneously
satisfied:
1) Perfect reconstruction is preserved structurally and the
structural complexity is very low.
2) All analysis and synthesis filters are designed by con-
trolling a single transfer function
P(z) [allpass in the
IIR case, and Type 2 (i.e., odd order symmetric linear
phase FIR) in the FIR case].
So
the design procedure is
very simple. It is very easy to design
P(z)
so
that all
filters have
good
responses (lowpass or highpass as the
case may
be).
3)
In
the IIR case, all the analysis and synthesis filters are
causal and stable.
4)
In some applications such as image coding, the linear
phase property of the analysis and synthesis filters is
desired. In the FIR case, the filters are exact linear-phase.
In the IIR case, we can force the phase response of the
filters to
be
nearly linear in the passband, as we shall
explain and demonstrate.
5)
The lowpass analysis filter
Ho(z)
can be forced to have
arbitrary number of zeros at
w
=
T.
Furthermore the
lowpass synthesis filter
Fo(z)
is guaranteed to have the
same number of zeros at
T
as
Ho(z).
In both of the
IIR and FIR cases, we give closed form expression for
the filter coefficients that provide maximum number of
zeros at
T.
A new class of biorthogonal wavelet basis functions can be
generated from the above filter bank. The regularity property
can be directly controlled by imposing multiple zeros at
T
as desired. In the IIR case, since all filters are causal (in
addition to being stable), the basis functions
are
all causal.
In the FIR case, the linear phase property ensures symmetry
of the wavelets, while at the same time providing a simple
control on regularity (because the number of zeros at
T
is
trivially controlled).
I)
A
1-D
to
2-0
Mapping:
Furthermore, we also provide a
novel mapping of the proposed 1-D filter banks into the 2-
D quincunx case, preserving all the desirable properties. In
particular, there is the following:
1)
The perfect reconstruction property is preserved.
2) In the IIR case, all the analysis and synthesis filters
remain causal and stable. In the FIR case the linear phase
property is preserved.
3)
Even though the filter bank is nonseparable, the com-
plexity is that
of
a separable filter bank, growing linearly
with the filter order.
4) The frequency response supports for the filters are the
diamond and diamond-complement as desired for the
quincunx case
[
151,
[2]. Moreover the filter frequency re-
sponses are ensured to be
good
simply by designing the
1-D filter having a good frequency response. Any desired
specifications can be met by designing a 1-D transfer
function
p(
z)
appropriately as we shall demonstrate.
5)
If the 1-D lowpass filter
H&)
has
k
zeros at
T,
then the
resulting 2-D lowpass filter will have its ith-order total
derivative equal to zero at
(T,
T),
for
i
=
0,1,.
. .
,
IC
-
1.
See Section
V
for details.
We also provide a design example to show that the mapping
can be easily applied to any dilation matrix (i.e., decimation
matrix) with determinant 2.
2)
Relation to Other Results in the Literature:
All the de-
signs proposed in this paper are based on a single class of
polyphase matrices to
be
described in Section
V.
However,
some of the filter banks reported by other researchers are
related to our work. In [23], the authors derive a class of
biorthogonal linear phase FIR filter bank which turns out to
be
a special case of our two-channel framework. In the IIR
maximally flat halfband case, our solution is different from
the traditional IIR Butterworth design and has approximately
linear-phase in the passband. In the FIR maximally flat half-
band case, the solution agrees with the classical FIR maximally
flat design [24]. However, our construction is different from
those in [25] and
[6]
since the analysis filters are factors of
maximally flat halfband filters in [25] and [6], whereas our
PHOONG
er
al.:
NEW
CLASS
OF
TWO-CHANNEL BIORTHOGONAL
FILTER
BANKS
AND
WAVELET
BASES
65
1
analysis filters are themselves maximally flat halfband. The
2-D mapping proposed earlier in [15] and
[I61 is different
from ours because it is known that the earlier mapping will
not preserve the
PR
property in general.
C.
Outline
of
the Paper and Notations
Our
presentation will go as follows: In the next section,
we will derive a framework for the two-channel biorthogonal
filter banks. Some properties of such class will be described
in detail. In Section
111,
we will discuss both the
IIR
and FIR
filter banks which are covered in the proposed framework.
In Section
IV,
wavelet basis functions generated from the
proposed filter banks will be presented and imposition of zeros
at aliasing frequency will
be
considered. Two new classes
of
IIR
maximally flat solution are given in closed form. In
Section
V,
we will first introduce a novel 2-D mapping for the
quincunx case. Some properties of the mapping are discussed.
Then both the
IIR
and
FIR
cases are considered. Furthermore
numerical examples will be provided throughout the discussion
to demonstrate the idea.
1)
Notations and Definitions:
Capital boldfaced letters are
used to denote matrices.
I
represents the identity matrix.
The determinant of the matrix
A
is denoted by
[det
A].
Superscript2-D is used to represent the 2-D function obtained
by applying the mapping, for example,
E2-D(z~,
21)
is ob-
tained by applying the 2-D mapping to
E(z).
The z-transform
of
h(n)
is represented by
H(z).
The relation between the filters
{Hk(z),
Fk(z)}
and the polyphase matrices E(z) and
R(z)
can be described as follows:
Hk(Z)
=
Ek,0(Z2)
+
z-1Ek,l(z2),
Fk(2)
=
z-lRo,k(z2)
+
Rl,k(z2)
and
where
Ei,j(z)
and
Ri,j(z)
are, respectively, the zjth elements
of the matrices
E(z) and R(z).
A
filter
Hk(z)
is halfband if
either one of its polyphase components
Ek,o(z),
Ek,l(z)
is a
delay.
11.
A
FRAMEWORK
FOR
1-D
BIORTHOCONAL FILTER BANKS
Consider Fig.
1,
where a two-channel system is shown. In
general,
R(z)
=
E-l(z) for perfect reconstruction. It is not
easy to constrain
[det
E(
z)]
to
be
minimum phase for stability
of
R(z); therefore, let us make it a delay. An example is
Then, we get the following expressions for the analysis filters:
H1(z)
=
-a(z2)Ho(z)
+
Z-2"-1.
(4)
A.
Obtaining Ideal Responses with
(4)
First notice that the filter
Ho(z)
can be made an ideal
lowpass filter if
P(z)
has the following magnitude and phase
responses:
IP(ej2w)l
=
1,
vw
(54
(5b)
(-2N
+
l)w,
for
w
E
[0,~/2];
2N
+
1)w
AZ
x,
for
w
E
(T/~,T].
Lp(ej2w)
=
{
(-
From
(4),
we see that in the high-frequency region,
Hl(ej")
has unity gain since
IHo(ej")l
=
0.
The function
a(.)
does not affect
Ho(z)
and can be freely chosen to shape
the response of
Hl(z).
It should be chosen such that in the
low frequency region,
a(z2)H0(z)
cancels with
z-'"-l.
For exact magnitude cancellation,
la(ej")l
must
be
unity.
Since
Ho(z)
is linear phase, it is necessary that
a(.)
has
linear phase in the low-frequency region. Comparing these
two requirements and the conditions in
(3,
we realize that
p(z)
is a suitable candidate for
a(.).
Indeed, if
N'
=
2N-
1,
Hl(z)
is an ideal highpass filter.
In
this case, we have an
ideal filter bank, and the polyphase matrix
E(z)
in Fig. l(b) is
(
0.5
0)
(z;"
z-2N+1
P(z>
)
E(z)
=
-0.5P(~)
1
With this, we get the following expressions for the analysis
filters, which we will repeatedly use in this paper.
The perfect reconstruction can
be
achieved by choosing
R(z)
in Fig. l(b) to be
With this, we obtain
-
0.5p2(z)
Ho(z)
=
z-2N
+
z-lp(z2)
(2)
=
(
0.5zpNP(z)
but
Hl(z)
=
z-('"+l),
which is a delay. Thus, even though
Ho(z)
can be designed to be a good lowpass filter (as we
will show),
Hl(z)
is allpass and this is not useful for subband
Ho(z)
by taking the polyphase matrix to be
The corresponding synthesis filters can be verified to have
the following form:
coding applications. We can modify
HI
(2)
without affecting
Fo(z)
=
-H1(-z), Fl(Z)
=
Ho(-z).
(9)
This choice of synthesis filters in
(9)
ensures that
{Fo(z),
Fl(z)}
is a lowpass/highpass pair
if
{Ho(z),
Hl(z)}
is a
lowpass/highpass pair. From
(6)
and
(8),
we have the imple-
mentation of the filter bank shown in Fig. 2. The structure is
similar to a ladder network structure
[26].
(3)
0.5Kz)
=
(
-O!?'t(z)
-0.5a(z)P(z)
+
z-"
652
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1995
Ln-tr;lf-
-2N+1
-
E(z)
Fig.
2.
Implementation
of
the proposed biorthogonal filter bank.
Remarks:
Of course, the
n(z)
in
(3)
can be taken as
functions different from
p(z),
as in the case of
[20], [21],
[23].
This will provide more freedom in the design. However,
by taking them to be the same, the biorthogonal systems can
have some additional useful properties. Therefore, we will only
consider the case when
a(.)
=
p(z).
B.
Two
Useful Approximations
of
(5)
The ideal choice of
p(z)
as in (5) requires infinite com-
plexity. Therefore, we have to design
p(z)
to approximate the
conditions in (5). However the approximation will not change
the perfect reconstruction property because
E(z)
in
(6)
and
R(z) in
(8)
satisfy R(z)E(z)
=
0.5~-~~+~1,
regardless of the
choice of
P(z).
Fig.
2
shows that the frequency responses of all
the analysis and synthesis filters depend on one single function
p(
z)
only. The frequency selectivity of all four filters depends
on how well
p(z)
approximates conditions (5). This makes the
design procedure simple.
In the next section, we will provide
two simple but useful approximations which correspond to
the following two cases:
Stable IIR case:
Here,
p(z)
is chosen to
be
a causal
stable allpass function
so
that (Sa) is met exactly. We
design the phase response of the allpass filter
so
that (5b)
is approximately satisfied. This leads to a biorthogonal
system with
causal stable
IIR analysis and synthesis
filters.
Linear phase FIR case:
To satisfy the condition (Sb),
p(z)
can
be
chosen as a Type
2
linear phase function
[2]
(filter with a symmetric impulse response of even
length). The magnitude response of
p(z)
is optimized
to be as close to unity as possible
so
that (Sa) is well
approximated. This leads to a
linear phase
biorthogonal
system.
C.
Additional Properties
of
the Filter Banks Designed as Above
In Section
I,
we have outlined some properties. Properties
14
mentioned at the beginning of Section I-B
are
clear from
the above discussion and Property 5 will be discussed in the
Section IV. In addition to these
five
properties, we have the
following:
1)
Double haljband property:
In all the previous construc-
tions of two-channel PR filter banks,
Ho(z)Fo(z)
is
a
halfband filter, where
Ho(z)
is not necessary halfband
-2N+1
$F&
-1
z
but a factor of a halfband filter. However in our con-
struction above, one can verify that not only the product
Ho(z)Fo(z)
but also the filter
Ho(z)
is halfband.
Poles
of
filters:
In the IIR case, notice from Fig.
2
that there is no feedback loop in both the analysis and
synthesis ends in the ladder network. Therefore, the
filters have the same poles as those of
p(z2)
and stability
depends solely on
the
allpass function
/?(z).
Moreover
in the IIR case if the allpass filter
p(z)
is implemented
by using the robust lattice structure
[2],
the filter
bank
is stable even when it is realized with finite wordlength.
Robustness to round
of
noise:
The ladder structure
shown in Fig.
2
is similar to the structure considered
in
[26].
By using the same reasoning in
[26],
it can be
verified that the round
off
noise in the analysis end is
compensated by that in the synthesis end. Combining
this with the structurally PR property, we conclude that
the implementation in Fig.
2
preserves PR even when
all the coefficients
are
quantized to a finite precision and
all the intermediate results are rounded
off.
However, if
the subband signals are quantized (which is usually the
case), this property is lost.
Zeros
of
the filters:
We can verify that
FO
(z)
and
HI
(2)
in
(9)
and
(7)
can, respectively,
be
rewritten
as
FO(2)
=
(2fZN+1
-
p(z2))Ho(z),
HI(Z)
=
(22-2N+1
+
p(z2))F1(2).
(10)
These factorizations give the filter bank an interesting
structure shown in Fig.
3.
From
(lo),
it is clear that if
p(
z)
is FIR, the zeros of
HO
(2)
are also zeros of
Fo
(2).
Even when
P(z)
is an irreducible IIR transfer function,
this is true since
Ho(z)
is in the form of
(7)
and the
zeros of denominator of
p(z2)
cannot cancel the zeros
of
Ho(z).
Moreover, if
Ip(ej")l
<
2,
both
Fo(e3"')
and
&(e3")
have the same
set
of zeros on the unit circle.
The same is true for the pair of
Hl(z)
and
Fl(z).
In
particular, if
Ho(z)
has
T
zeros at
z
=
-1,
this implies
that
Fo(z)
has no fewer than
T
zeros at the same point.
This property is important in the generation of wavelets
since for biorthogonal wavelets, we need both of the
analysis and the synthesis wavelets to
be
regular. By
increasing the number of zeros of
Ho(z)
at
z
=
-
I,
our construction ensures that
Fo(z)
has at least the same
number
of
zeros at
z
=
-
1.
This is the property which
I
PHOONG
et
al.:
NEW
CLASS
OF
TWO-CHANNEL
BIORTHOGONAL
FILTER
BANKS
AND WAVELET
BASES
653
Fig.
3.
Redrawing
of
Fig.
2,
where
Ho(:)
=
O.~(Z-~'~
+
-'-l
j(z2))
and
FI(:)
=
HO(-:).
does not appear in the previously existing constructions
of biorthogonal filter banks.
5) Ripple sizes
of
thefilters:
Since
Ho(z)
is a halfband filter
and
Ho(z)
+
Fl(z)
=
z-~~, we have the following
relationship between the passband ripple
6,
and the
stopband ripple
6,:
Moreover, by using (10) and the fact that
p(z2)
z
-z-2N+1 in the high frequency region, we get
~,(Fo)
=
~&(Ho),
&(HI)
=
3&(F1) (12)
(2010g3
x
9.5
dB). This property ensures that by
designing
Ho(z)
to have sufficiently high stopband
attenuation, we can ensure that all the other three filters
will also have good frequency selectivity.
6)
Complexity:
From Fig. 2, it is
very
clear that the analysis
and synthesis banks have the same complexity. Assume
that
,f?(z)
has order
N.
For the
IIR
case, by using the one
multiplier lattice structure for allpass function
[2],
we
need approximately
2N
multiplications,
6N
additions,
and
5N
delays. For the
FIR
case, by exploiting the
symmetry, we need approximately
N
multiplications,
2N additions and
3SN
delays. All the operations are at
a lower rate. Therefore, the analysis (or synthesis) bank
requires
N
and
0.5N
multiplications per input sample
for the
IIR
and FIR case, respectively.
7)
Near linear phase
in
the
IIR
cases:
From (7), since in
the passband the magnitude response
of
Ho(z)
is ap-
proximately one, the transfer function
,f?(z2)
x
z-2N+1.
Therefore,
Ho(z)
has approximately linear phase in the
passband. Similar argument is true for
Hl(z).
111.
DESIGN
PROCEDURES
FOR
THE
TWO
CLASSES OF
BIORTHOGONAL
FILTER
BANKS
In this section, we will discuss the two cases of the
approximations of
(5)
given in the last section. Simple design
procedures will
be
given for both cases.
A. Causal Stable IIR Biorthogonal Filter Banks
stable real allpass function
In this section,
/3(z)
in
(6)-(9)
is taken to be the causal
where
UN~,~
=
1
and
aNl,k
are real. In this case,
Ho(z)
is
a sum of a delay and an allpass function. See
(7).
It is an
IIR
halfband filter and has been studied by some researchers
[27], [28].
Ho(z)
can
be
made lowpass with large stopband
attenuation and small passband ripples by designing the phase
response of the allpass function to approximate
(5b)
[29].
I)
Choice
of
NI:
From the monotone decreasing phase
property [2] of a causal stable allpass function, we know
that the phase of
AN^(^^)
spans a range of 4N1x when
w
spans a range of
2x,
but from (5b),
,0(z2)
spans a range of
4Nx or 4(N
-
1)x.
To
make the range spanned by both
of the functions equal, we set
NI
=
N
or
N-
1
and this
results in two classes of causal stable
IIR
filter banks. Since
the derivation and properties of both of the classes are very
similar, in the rest of the paper, we consider only the case
NI
=
N
(we will point out at those places where the second
class has a different property). With this choice, the analysis
filters can
be
written as
~~(2)
=
-AN(z~)H~(~)
+
z-4N+1.
(14)
The relationship between the synthesis and analysis filters is
the same as
(9).
2)
Additional Properties
of
the Above
IIR
Filter Banks:
1)
Preservation
of
zero ut aliasing frequency:
Substituting
z
=
-
1 into the expression of
Ho(z)
in (14), we find
that
Ho(
z)
always has a zero at
z
=
-
1, independent of
the coefficients
aN,k.
In particular, the zero is preserved
even when all
UN,k
are quantized coarsely. This means
that one zero at
z
=
-
1 is structurally imposed. This is
important in the generation of wavelet bases since one
zero at
z
=
-
1 is a necessary condition for the existence
of the wavelet functions 161. Note also that Hl(z) will
always have a structurally imposed zero at
z
=
1.
2)
Low
sensitivity:
Since there exists low sensitivity lattice
structure for allpass function [2], the filters have low
passband sensitivity. Since the halfband property of
Ho(z)
is structurally imposed,
it
has low stopband
sensitivity as well.
3)
Bump
in
the transition
bund: Substituting
w
=
x/2
into
the expression for Hl(e3") and
Fo(eJ")
and using the
fact that
AN(-^)
=
(-l)N,
we find that (Hl(eJ")J
=
IFo(eJ")I
=
at
w
=
x/2,
independent of the
allpass function
AN
(2).
This means that
JH1(
e3")
1
and
IFo(eJ")I
always have a bump
of
approximately
4
dB