# 2-Norm-Based Iterative Design of Filterbank Transceivers: A Control Perspective

Authors: Yang Shi; Tongwen Chen
Journal: Journal of Control Science and Engineering
 (2008)

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## Abstract

This paper considers design of filterbank-based transceivers. A composite error criterion is proposed
to capture all the three traditional distortions. Incorporating noise attenuation and filter bandlimiting
properties into this error criterion, an optimal design procedure is developed and applied to a transceiver
design example, yielding an FIR transceiver that has good frequency-selective properties and is close to
perfect reconstruction. As a least-squares solution is given in closed form in each iteration, the algorithm
is easy to implement.


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## Body

## 1. Introduction

Multirate systems, that is, digital systems with
signals of different sampling rates, have wide applications in control
[1], signal processing
[2], communications,
econometrics and numerical mathematics. There are several reasons for this
[3]: (1) in large
scale multivariable digital systems, often it is unrealistic, or sometimes
impossible, to sample all physical signals uniformly at one single rate; in
such situations, one is forced to use multirate sampling; (2) multirate systems
can often achieve objectives that cannot be achieved by single-rate systems.
The study of multirate systems goes back to late 1950s [4]. A renaissance of research
in multirate systems has occurred since 1980 in the control, signal processing,
and communications communities.(i)
In the control community, two directions of
research stand out: first, using multirate control to achieve what single rate
control cannot as well as the limitations of doing this, and second, the
optimal design of multirate controllers [1].(ii)
The driving force for studying multirate
systems in signal processing comes from the need of sampling rate conversion,
subband coding, and their ability to generate wavelets [2].(iii)
In the communications community, multirate
sampling is used for blind system identification and equalization.Intersymbol interference (ISI) is a common problem in
telecommunication systems, such as terrestrial television broadcasting, digital
data communication systems, and cellular mobile communication systems. Usually
the channel distortion results in ISI, which, if left uncompensated, causes
high error rates. The solution to the ISI problem is to design a receiver that
employs a means for compensating or reducing the ISI in the received signal.
The compensator for the ISI is the so-called equalizer. For these problems, the
filterbank approach [5–10] has gained practical interest recently.The multirate filterbank-based transceiver model was
proposed in [7] as a
unifying framework able to encompass existing modulations and equalization schemes.
Further in [9], the
multirate filterbank-based transceiver model, as shown in Figure 1, was studied
in detail. The filterbank-based transceiver introduces transmitter redundancy
using filterbank precoders and generalizes existing modulations including OFDM,
DMT, TDMA, and CDMA schemes encountered with single- and multiuser
communications [9].
Recently in [8],
based on the filterbank approach, the optimal channel equalization is studied:
an optimal receiver filterbank, which is a modified Kalman filter, is obtained.Figure 1
Multirate
discrete-time baseband equivalent transmitter/channel/receiver model.Motivated by the generality and importance of filterbank-based transceivers, this paper focuses on frequency-domain analysis and optimal design issues for filterbank-based transceivers. The contributions of this work are as follows.(i)
Contrary to a time-domain study [9], our work is developed
along the line of frequency-domain analysis. Frequency-domain models are
obtained using the blocking technique. Based on such models, an composite error
criterion is proposed to quantify the degree to perfect reconstruction. This
frequency-domain criterion captures the traditional distortions.(ii)
In the transceiver design, we apply and
incorporate the model matching methodology [11]: instead of designing for perfect reconstruction, we
design for close to perfect reconstruction with effective band separation. This
idea was explored in filterbanks [11] and transmultiplexers [12]. Under some mild condition,
one can always get arbitrarily close to perfect reconstruction by trading off
filter complexity and time delay in reconstruction.(iii)
Control of filter stopband energy is
incorporated in our design. This is important, since narrowband noise could
induce serious impairment due to poor stopbands of filters involved [13]: if the receiving filters
have poor stopband attenuation, all the neighboring bands would be affected
when there is a strong narrowband noise. The resulting ISI can seriously
degrade the system performance [14].The rest of the paper is organized as follows. In
Section2 the blocked model of the filterbank-based transceiver and the perfect
reconstruction are briefly reviewed. In Section 3 the transceiver system is
analyzed in several aspects based on the blocked model. Section 4 formulates
the optimal design problem as a least squares one and develops an iterative
design procedure for the filterbank-based transceiver. The proposed design
method is illustrated in detail with an example in Section 5. Finally, Section
6 offers some concluding remarks.We conclude this section by introducing some notation.
The signals are denoted by small letters, for example,r. r¯ (underlining denotes blocking) is the blocked
signal. r^¯(z) denotes the z-transform of r¯.
The systems are represented as time-domain operators, denoted by capital
letters, for example, T.
If a system T is linear time invariant (LTI), its transfer
function (matrix) is written as T^(z).
The notation ∥⋅∥2 stands for the 2-norm for transceiver
matrices.


## 2. Blocked Models and Perfect Reconstruction

Figure1 shows the discrete-time multirate filterbank
model for the baseband communication system [9]. It consists of the transmitter filterbank F^m(z) (m=0,…,M−1), the receiver filterbank H^p(z) (p=0,…,P−1), and the communication channel modeled by
the transfer function C(z),
which is assumed to be causal and stable. The input serial data stream s(n) and its successively time-advanced versions
are downsampled by a factor M to get M parallel substreams s0(n),s1(n),…,sM−1(n) as shown in Figure 1. These substreams are
then upsampled by a factor P and processed by filters F^m(z);
the combined output u(n) is then transmitted over the channel C^(z),
which is corrupted at the output by an additive noise v(n),
assumed to be stationary and white. At the receiver end, the received signal y(n) and its successively shifted versions are then
downsampled by a factor P,
upsampled by a factor M,
and processed by filters H^p(z);
the combined output forms the reconstructed signal r(n).By blocking the transmitter, channel, and receiver,
respectively, we can obtain the blocked model
[6, 8]
for the multirate system in
Figure 1:(1)r^̲(z)=H^̲(z)C^̲(z)F^̲(z)s^̲(z)+H^̲(z)v^̲(z),where F^̲(z), C^̲(z), and H^̲(z) are the transfer matrices of the blocked
transmitter filterbank, the blocked channel, and the receiver filterbank,
respectively.The blocked general multirate system (in the absence
of noise) is LTI withT^̲(z)=H^̲(z)C^̲(z)F^̲(z).
The transceiver achieves perfect reconstruction if in Figure 1r is a delayed version of s,
that is, if there exists nonnegative integer d such that T=Td,
where Td is the time-delay system with transfer
function T^d(z)=z−d.
Blocking Td the same way as we blocked T,
the perfect reconstruction condition is equivalent to [8](2)H^̲(z)C^̲(z)F^̲(z)=z−d[0z−1IlIM−l0], where integers d and l satisfy k≥0 and 0≤l≤M−1,
and Il and IM−l are the l×l and (M−l)×(M−l) identity matrices, respectively. Moreover, if
this condition is satisfied, T^(z)=z−(kM+l).


## 3. Analysis

In this section, we study the distortion analysis, the
effect of noise, and the frequency selectivity.


### 3.1. Distortion Analysis

Perfect reconstruction synthesis filterbanks at the
transmitter and analysis filterbanks at the receiver allow perfect recovery of
communication symbols, but the challenges arise with ISI-inducing channels and
noise, either of which destroying the perfect reconstruction property. Many
practical transceivers do not achieve perfect reconstruction but get close to
perfect reconstruction. In order to measure the degree of closeness to perfect
reconstruction, we will introduce three traditional quantities to measure
sources of distortions: aliasing distortion, magnitude and phase distortions
[2]. These distortion
measures are based on the blocked model T^̲(z).Lemma 1  (see [15]).
An LPTV (linear periodic
time variant) systemG with period t can be uniquely decomposed
into(3)G=Gti+Gtvsatisfying the two
properties
(i)
Gti is the optimal LTI approximation of G in the sense that it minimizes ∥G^̲(z)−Q^̲(z)∥2 over the class of LTI Q's (G̲ denotes the blocked system LtGLt−1,
similarly for Q̲.
Here Lt is the lifting operator, and Lt−1 the inverse lifting operator [1]).(ii)
∥G^̲(z)∥22=t∥G^ti(z)∥22+∥G^tv̲(z)∥22.Back to our transceiver problem, the system froms to r is LPTV with period M;
decompose this into Gti+Gtv,
where Gti is the LTI component and Gtv the time-varying component. Thus we have(4)T^̲(z)=G^ti̲(z)+G^tv̲(z).Aliasing distortion in the system is defined
by(5)AD=∥G^tv̲(z)∥2Even if AD is zero, and then the LPTV system reduces to
an LTI systemGti,
it may still have errors in magnitude and phase compared with the ideal time
delay z−d;
define the following quantities:(6)MD=[12π∫02π(|G^ti(ejω)|−1)2dω]1/2,PD={12π∫02πsin2[∠G^ti(ejω)+d×ω]dω}1/2.Note that MD and PD are defined across all frequencies: (MD)2 is the energy of the magnitude distortion and PD the energy of sine of the phase distortion ϕ(ω)=∠G^ti(ejω)+dω.
It is worth noting that there are two reasons why we apply sine to characterize
the phase distortion PD: (1) if ϕ(ω) is within ±(π/2),
which is usually the case, sin2(ϕ(ω)) is a good indicator of the size of ϕ(ω);
(2) a connection with the 2-norm-based distortion measure
J, to be introduced, could be conveniently
established.Next we propose a composite distortion measure which
captures all the three types of distortions and is relatively easy to use in
design. The new distortion measure is the 2-norm of the blocked error transfer
matrix:(7)J=∥T^̲(z)−T^d̲(z)∥2.Such a measure is appropriate
because in the next theorem we establish connections between J and the three types of distortions discussed
earlier.Lemma 2  (see [16]).
LetG be a stable LTI system. Comparing G^(z) with the time delay z−d,
one has the following inequalities:
(8)|G^(ejω)−e−jdω|≥||G^(ejω)|−1|,|G^(ejω)−e−jdω|2≥sin2[∠G^(ejω)+dω].Theorem 1.
AD and MD relate toJ via (9)AD2+M(MD)2≤J2,whereas AD and PD relate to J via(10)AD2+M(PD)2≤J2.Proof.
From (4) and Lemma 1 we get(11)J2=∥T^̲(z)−T^d̲(z)∥22=∥G^ti̲(z)+G^tv̲(z)−T^d̲(z)∥22=M∥G^ti(z)−z−d∥22+∥G^tv̲(z)∥22.Note (5) to get(12)J2=AD2+M∥G^ti(z)−z−d∥22.Now by definition,(13)∥G^ti(z)−z−d∥22=12π∫02π|G^ti(ejω)−e−jdω|2dω.Invoke Lemma 2 to get
(14)|G^ti(ejω)−e−jdω|≥||G^ti(ejω)|−1|,|G^ti(ejω)−e−jdω|2≥sin2[∠G^ti(ejω)+dω].Combining these two
equalities with (13) and noting the definitions of MD and PD in (6), one has
(15)∥G^ti(z)−z−d∥2≥MD,∥G^ti(z)−z−d∥2≥PD.The proof is complete by
noting (12) and the above two inequalities.Then it is clear from Theorem1 that all distortions (AD, MD,
and PD) are bounded above by J.
Therefore, it makes sense to minimize J in transceiver design because this
suboptimizes the three distortions simultaneously.


### 3.2. Noise Suppression

In Figure2, q(n) represents the noise effect at the receiver
end. If we want to minimize the root-mean-square value of q,
it is equivalent to minimize the 2-norm of the blocked transfer matrix from v̲(n) (standard white noise) to q̲(n).
Therefore, the objective function for attenuating the output noise effect can
be(16)JN=∥H^̲(z)∥2.Figure 2
A model
of the noise effect at the receiver end.


### 3.3. Frequency Selectivity of Filters

Filters with frequency selectivity are of particular
importance in communications systems. In addition, frequency responses of
designed filters would be deteriorated in the iterative design procedure, if no
constraints are imposed on the filters. In order to obtain better bandlimiting property,
we minimize the stopband energy of filters involved. To see this, take the
objective function(s) forHi in the receiver filterbank as an example; we
have(17)JHi=∫ΩHi|H^i(w)|2dw,i=0,1,…,P−1,where ΩHi defines the stopband frequency interval(s) for Hi.
Similarly, JFi can be computed.In this paper, we will consider only FIR filters. The
frequency response of a realN-tap FIR filter Hi is given by(18)H^i(ω)=∑n=0N−1hi(n)e−jnω=hiTϕi(ω),where(19)hiT=[hi(0)hi(1)hi(2)…hi(N−1)],ϕiH(ω)=[1ejωej2ω…ej(N−1)ω].(The superscript H indicates the complex conjugate transpose.)
The objective function in (17) can then be written as(20)JHi=hiTQihi,where the fixed N×N matrix Qi is defined by(21)Qi=∫ΩHiϕi(w)ϕiH(w)dw.The elements qmn for Qi can be calculated easily if ΩHi is given; for example, assume the filters Hi have the passband [ωpi1,ωpi2] over [0,π],
then(22)Qi=∫0ωpi1ϕi(w)ϕiH(w)dw+∫ωpi2πϕi(w)ϕiH(w)dw,and thus the elements qmn for Qi are(23)qmn={sin(wpi1(m−n))−sin(wpi2(m−n))(m−n),m≠n,π+wpi1−wpi2,m=n.


## 3.1. Distortion Analysis

Perfect reconstruction synthesis filterbanks at the
transmitter and analysis filterbanks at the receiver allow perfect recovery of
communication symbols, but the challenges arise with ISI-inducing channels and
noise, either of which destroying the perfect reconstruction property. Many
practical transceivers do not achieve perfect reconstruction but get close to
perfect reconstruction. In order to measure the degree of closeness to perfect
reconstruction, we will introduce three traditional quantities to measure
sources of distortions: aliasing distortion, magnitude and phase distortions
[2]. These distortion
measures are based on the blocked model T^̲(z).Lemma 1  (see [15]).
An LPTV (linear periodic
time variant) systemG with period t can be uniquely decomposed
into(3)G=Gti+Gtvsatisfying the two
properties
(i)
Gti is the optimal LTI approximation of G in the sense that it minimizes ∥G^̲(z)−Q^̲(z)∥2 over the class of LTI Q's (G̲ denotes the blocked system LtGLt−1,
similarly for Q̲.
Here Lt is the lifting operator, and Lt−1 the inverse lifting operator [1]).(ii)
∥G^̲(z)∥22=t∥G^ti(z)∥22+∥G^tv̲(z)∥22.Back to our transceiver problem, the system froms to r is LPTV with period M;
decompose this into Gti+Gtv,
where Gti is the LTI component and Gtv the time-varying component. Thus we have(4)T^̲(z)=G^ti̲(z)+G^tv̲(z).Aliasing distortion in the system is defined
by(5)AD=∥G^tv̲(z)∥2Even if AD is zero, and then the LPTV system reduces to
an LTI systemGti,
it may still have errors in magnitude and phase compared with the ideal time
delay z−d;
define the following quantities:(6)MD=[12π∫02π(|G^ti(ejω)|−1)2dω]1/2,PD={12π∫02πsin2[∠G^ti(ejω)+d×ω]dω}1/2.Note that MD and PD are defined across all frequencies: (MD)2 is the energy of the magnitude distortion and PD the energy of sine of the phase distortion ϕ(ω)=∠G^ti(ejω)+dω.
It is worth noting that there are two reasons why we apply sine to characterize
the phase distortion PD: (1) if ϕ(ω) is within ±(π/2),
which is usually the case, sin2(ϕ(ω)) is a good indicator of the size of ϕ(ω);
(2) a connection with the 2-norm-based distortion measure
J, to be introduced, could be conveniently
established.Next we propose a composite distortion measure which
captures all the three types of distortions and is relatively easy to use in
design. The new distortion measure is the 2-norm of the blocked error transfer
matrix:(7)J=∥T^̲(z)−T^d̲(z)∥2.Such a measure is appropriate
because in the next theorem we establish connections between J and the three types of distortions discussed
earlier.Lemma 2  (see [16]).
LetG be a stable LTI system. Comparing G^(z) with the time delay z−d,
one has the following inequalities:
(8)|G^(ejω)−e−jdω|≥||G^(ejω)|−1|,|G^(ejω)−e−jdω|2≥sin2[∠G^(ejω)+dω].Theorem 1.
AD and MD relate toJ via (9)AD2+M(MD)2≤J2,whereas AD and PD relate to J via(10)AD2+M(PD)2≤J2.Proof.
From (4) and Lemma 1 we get(11)J2=∥T^̲(z)−T^d̲(z)∥22=∥G^ti̲(z)+G^tv̲(z)−T^d̲(z)∥22=M∥G^ti(z)−z−d∥22+∥G^tv̲(z)∥22.Note (5) to get(12)J2=AD2+M∥G^ti(z)−z−d∥22.Now by definition,(13)∥G^ti(z)−z−d∥22=12π∫02π|G^ti(ejω)−e−jdω|2dω.Invoke Lemma 2 to get
(14)|G^ti(ejω)−e−jdω|≥||G^ti(ejω)|−1|,|G^ti(ejω)−e−jdω|2≥sin2[∠G^ti(ejω)+dω].Combining these two
equalities with (13) and noting the definitions of MD and PD in (6), one has
(15)∥G^ti(z)−z−d∥2≥MD,∥G^ti(z)−z−d∥2≥PD.The proof is complete by
noting (12) and the above two inequalities.Then it is clear from Theorem1 that all distortions (AD, MD,
and PD) are bounded above by J.
Therefore, it makes sense to minimize J in transceiver design because this
suboptimizes the three distortions simultaneously.


## 3.2. Noise Suppression

In Figure2, q(n) represents the noise effect at the receiver
end. If we want to minimize the root-mean-square value of q,
it is equivalent to minimize the 2-norm of the blocked transfer matrix from v̲(n) (standard white noise) to q̲(n).
Therefore, the objective function for attenuating the output noise effect can
be(16)JN=∥H^̲(z)∥2.Figure 2
A model
of the noise effect at the receiver end.


## 3.3. Frequency Selectivity of Filters

Filters with frequency selectivity are of particular
importance in communications systems. In addition, frequency responses of
designed filters would be deteriorated in the iterative design procedure, if no
constraints are imposed on the filters. In order to obtain better bandlimiting property,
we minimize the stopband energy of filters involved. To see this, take the
objective function(s) forHi in the receiver filterbank as an example; we
have(17)JHi=∫ΩHi|H^i(w)|2dw,i=0,1,…,P−1,where ΩHi defines the stopband frequency interval(s) for Hi.
Similarly, JFi can be computed.In this paper, we will consider only FIR filters. The
frequency response of a realN-tap FIR filter Hi is given by(18)H^i(ω)=∑n=0N−1hi(n)e−jnω=hiTϕi(ω),where(19)hiT=[hi(0)hi(1)hi(2)…hi(N−1)],ϕiH(ω)=[1ejωej2ω…ej(N−1)ω].(The superscript H indicates the complex conjugate transpose.)
The objective function in (17) can then be written as(20)JHi=hiTQihi,where the fixed N×N matrix Qi is defined by(21)Qi=∫ΩHiϕi(w)ϕiH(w)dw.The elements qmn for Qi can be calculated easily if ΩHi is given; for example, assume the filters Hi have the passband [ωpi1,ωpi2] over [0,π],
then(22)Qi=∫0ωpi1ϕi(w)ϕiH(w)dw+∫ωpi2πϕi(w)ϕiH(w)dw,and thus the elements qmn for Qi are(23)qmn={sin(wpi1(m−n))−sin(wpi2(m−n))(m−n),m≠n,π+wpi1−wpi2,m=n.


## 4. Problem Formulation and Design

In view of the new distortion measureJ discussed in the preceding section, we wish to
design transmitter and receiver subsystems to minimize J.
Thus our optimal filterbank-based transceiver design problem using FIR
subsystems can be stated as follows: given the FIR channel and desired
reconstruction time delay d,
design FIR transmitter and receiver subsystems of some given lengths to
minimize J subject to some constraint on JN in (16) and the stopband energy constraints on JHi in (20).In order to incorporate both the noise attenuation and
filter bandlimiting constraints, such an optimal design problem can be recast
by including penalties onJN, JHi,
and JFi.
Because both F^̲(z) and H^̲(z) are designable, this optimization problem is
in general nonlinear and difficult to solve. Thus we take the following
iterative design procedure which turns out to be very effective in the design
example to follow.Step 1.
Design transmitter subsystems to
satisfy desired frequency limiting properties (without considering
reconstruction performance); these are used to initiate the iteration.Step 2.
Fixing the transmitter subsystems,
design FIR receiver subsystems by minimizing the following objective function
(using the blocked models)(24)minH^̲(z)(∥H^̲(z)C^̲(z)F^̲(z)−T^d̲(z)∥22+αN∥H^̲(z)∥22+∑i=1PαHiJHi):=minH^̲(z)(J1).Step 3.
Fixing the receiver subsystems
just designed, now redesign FIR transmitter subsystems by minimizing the
following objective function (using the blocked models)(25)minF^̲(z)(∥H^̲(z)C^̲(z)F^̲(z)−T^d̲(z)∥22+∑j=1MαFjJFj):=minF^̲(z)(J2).Step 4.
Repeat Steps2 and 3 until the
corresponding objective function is sufficiently small.We note that the idea of iteratively designing
transmitter and receiver filters was used effectively in transmultiplexers
design in [12]. The
advantage of this procedure is evident: by fixing either F^̲(z) or H^̲(z) in Steps 2 and 3, the optimization problems
become mathematically tractable; in fact, they are finite-dimensional, convex
optimization with a quadratic cost function, whose global optimal solution can
be always computed. Even analytical solutions can be obtained.For example, looking at the optimal design problem in
Step2, we define(26)P^(z)=H^̲(z)C^̲(z)F^̲(z)−T^d̲(z).This system is FIR and hence can
be represented by its finitely many coefficient matrices Pi.
By Parseval's equality,(27)J12=∥P^(z)∥22=[∑itrace(PiPi′)].Since F^̲(z) and T^d̲(z) are given and thus P(z) depends on H^̲(z) in an affine manner, it follows that Pi relates to the coefficients of H^̲(z) (to be designed) too in an affine manner. It
is obvious that both JN and JHi are of quadratic forms, therefore we can
rewrite the quantity in (24) in the following way:(28)J12=(MFx−b)′(MFx−b)+x′MNx+x′MHx.Here x is a column vector containing all the
parameters in H^̲(z), to be designed, b is a column vector depending on only T^d̲(z), MF is a matrix depending on F^̲(z) and the way x is formed, MN depends on αN and H^̲(z),
and finally MH depends on αHi and Hi(z).
The matrices M, MN, MH, and b can be computed and are independent of the
design parameters (x). Now the optimal design problem in Step 2
becomes a least squares problem:(29)minx[(MFx−b)′(MFx−b)+x′(MN+MH)x].If MF′MF+MN+MH is invertible, the optimal solution can be
obtained to be(30)xopt=(MF′MF+MN+MH)−1MF′b.From here we can recover the
optimal receiver subsystems. The optimal design problem in Step 3 can be solved
similarly.


## 5. Design Example

The filterbank-based transceiver withM=3 and P=8 is designed. The channel to be used in this
example is(31)C^(z)=1−0.3z−1+0.5z−2−0.4z−3+0.1z−4−0.02z−5+0.3z−6−0.1z−7.The transmitter and receiver filters involved in
design are all FIR and causal with a fixed order of 12. The magnitude Bode
plots of the initial transmitter filters are given in Figure3, and the
reconstruction time delay is taken as d=8.
The constants αN, αHi,
and αFj,
reflecting relative weightings among multiple objectives, are tuned in the
design process. In our design, these are taken to be αN=0.02, αHi=0.03(i=0,1,…,7), αFj=0.01(j=0,1,2).Figure 3
The magnitude Bode plots for the initial
transmitter filtersF0 (solid), F1 (dash-dot), and F2 (dotted): dB versus ω/2π.Next we apply the iterative design procedure to the
example. In the first iteration,J1=5.806,
and the resulting αN∥H^̲(z)∥=0.02∥H^̲(z)∥22=1.553.
After some iterations, the value of the objective function gradually decreases
as the number of iterations increases, and finally converges to the value J1=0.156, 0.02∥H^̲(z)∥22=0.0872,
and finally J=0.048.
The above comparison on J1 clearly illustrates the improvement obtained
by using the proposed iterative algorithm. The designed transmitter and
receiver filters are shown in Figures 4 and 5, respectively.Figure 4
The magnitude Bode plots for the designed
transmitter filtersF0 (solid), F1 (dash-dot), and F2 (dotted): dB versus ω/2π.Figure 5
The magnitude Bode plots for the designed
receiver filtersH0–H7:
dB versus ω/2π.


## 6. Conclusion

In this paper,
we investigated the problem of optimal design of filterbank-based transceivers.
We proposed quantities to measure various distortions. We also introduced a
composite distortion (J) that captures all distortions. Finally, by
incorporating two important practical issues, the noise suppression and filter
bandlimiting property, we developed an iterative design procedure based on
minimizing the objective function and successfully applied this procedure to
design of a filterbank-based transceiver. At each iteration, the least squares
solution should be found, thus the final transmitter and receiver filters can
be obtained with relative ease.



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*Source: 143085-2008-04-06.xml*