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Volume 2007, Article ID 61396, 12 pagesdoi:10.1155/2007/61396 Research Article Design of Nonuniform Filter Bank Transceivers for Frequency Selective Channels Han-Ting Chiang, 1 See-May P

Volume 2007, Article ID 61396, 12 pages

doi:10.1155/2007/61396

Research Article

Design of Nonuniform Filter Bank Transceivers

for Frequency Selective Channels

Han-Ting Chiang, 1 See-May Phoong, 1 and Yuan-Pei Lin 2

1 Department of Electrical Engineering, Graduate Institute of Communication Engineering, National Taiwan University,

Taipei 10617, Taiwan

2 Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu 300, Taiwan

Received 14 January 2006; Revised 16 July 2006; Accepted 13 August 2006

Recommended by Soontorn Oraintara

In recent years, there has been considerable interest in the theory and design of filter bank transceivers due to their superior fre-quency response In many applications, it is desired to have transceivers that can support multiple services with different incoming data rates and different quality-of-service requirements To meet these requirements, we can either do resource allocation or design transceivers with a nonuniform bandwidth partition In this paper, we propose a method for the design of nonuniform filter bank transceivers for frequency selective channels Both frequency response and signal-to-interference ratio (SIR) can be incorporated

in the transceiver design Moreover, the technique can be extended to the case of nonuniform filter bank transceivers with rational sampling factors Simulation results show that nonuniform filter bank transceivers with good filter responses as well as high SIR can be obtained by the proposed design method

Copyright © 2007 Hindawi Publishing Corporation All rights reserved

The orthogonal frequency division multiplexing (OFDM)

system has enjoyed great success in many wideband

com-munication systems due to its ability to combat

intersym-bol interference (ISI) [1] It is known that the transmitting

and receiving filters of the OFDM transceiver have poor

fre-quency responses As a result, many subchannels will be

af-fected when there is narrowband interference, and the

per-formance degrades significantly [2] Many techniques have

been proposed to solve this problem

One of the solutions is the filter bank technique In recent

years, there has been considerable interest in the application

of filter banks to the design of transceivers with good

fre-quency characteristics [2 10] Many of these previous studies

[3 6] have focused on the design of filter bank transceivers

(or transmultiplexers) under the assumption that the

trans-mission channel is an ideal channel that does not create ISI

When the channel is a frequency selective channel, these

fil-ter bank transceivers suffer from severe ISI effect [7,8], and

post processing technique is needed at the receiver for

chan-nel equalization [4] Recently the authors in [10] studied

the filter bank transceiver for frequency selective channels

The transmitting and receiving filters are optimized for SIR

(signal-to-interference ratio) maximization Like OFDM sys-tems, simple one-tap equalizers can be employed at the re-ceiver for channel equalization It has been demonstrated that filter bank transceivers with high SIR and good fre-quency responses can be obtained [10]

In many applications, it is desired to have transceivers that can support multiple services [11, 12] Different ser-vices might have different incoming data rates and different quality-of-service requirements One solution to this prob-lem is to judiciously allocating the resources to meet the re-quirements, see, for example, [11] Another solution is to use a nonuniform filter bank transceiver The theory and de-sign of nonuniform filter banks have been studied by a num-ber of researchers [13–18] These results are extended to the design of transceivers and transmultiplexers with nonuni-form band separation in [12,19] In [12], the authors pro-posed a general building block for the design of nonuniform filter bank transmultiplexers Near perfect reconstruction transmultiplexers with good frequency property can be ob-tained by the proposed method therein In [19], a design of nonuniform transmultiplexers using semi-infinite program-ming was proposed The proposed algorithm was efficient and good results were achieved However these nonuniform transceiver designs do not consider the channel effect When

x0 (n)

x1 (n)

x M 1(n)

N0

.

N1

F0 (z)

F1 (z)

F M 1(z)

C(z)

v(n)

z l0

.

.

H0 (z)

H1 (z)

H M 1(z)

N0

.

N1

N M 1



x0 (n)



x1 (n)



x M 1(n)

Figure 1: A nonuniform filter bank transceiver with integer sampling factors

the transmission channel is frequency selective, an additional

equalizer is needed at the receiver

In this paper, we consider the design of nonuniform

transceiver for frequency selective channels Both the cases

of integer and rational sampling factors are considered As

the effect of channel is taken into consideration at the time

the filter bank is optimized, simple one-tap equalizers can be

used at the receiver for channel equalization Unlike the

uni-form case, the equivalent system from the transmitter input

to the receiver output is no longer LTI and ISI-free condition

needs to be derived Furthermore we will show that like the

uniform case [10], SIR can be formulated as a Rayleigh-Ritz

ratio of filter coefficients The optimal filters that maximize

the SIR can be obtained from an eigenvector of a positive

def-inite matrix Moreover, an iterative algorithm that can

incor-porate the frequency response is proposed for SIR

maximiza-tion Simulation results show that we can obtain nonuniform

transceivers with very high SIR (around 50 dB) and good

fre-quency response (stopband attenuation around 40 dB)

This paper is organized as follows InSection 2, we study

nonuniform filter bank transceivers with integer sampling

factors The ISI-free condition is derived and the SIR is

for-mulated as a Rayleigh-Ritz ratio of transmitting and

receiv-ing filters Then SIR-optimized transmittreceiv-ing and receivreceiv-ing

filters are given Moreover, the design method can be

ex-tended to the case of unknown frequency selective

chan-nels In Section 3, an iterative algorithm is proposed to

al-ternatingly optimize the transmitting and receiving filters for

SIR maximization We will show how to incorporate the

fre-quency response into the objective function The results are

extended to the case of rational sampling factor inSection 4

InSection 5, simulation examples are given to demonstrate

the usefulness of the proposed method A conclusion is given

inSection 6

Notation

The N-fold downsampled and upsampled versions of x(n)

are respectively denoted by [x(n)] ↓ Nand [x(n)] ↑ Nin the time

domain, and by [X(z)] ↓ N and [X(z)] ↑ Nin thez domain The

convolution of two sequencesx(n) and y(n) is represented

byx(n) ∗ y(n).

WITH INTEGER SAMPLING FACTORS

Figure 1 shows a nonuniform filter bank transceiver The downsampling and upsampling ratios N i are integers and they can be different for different i A larger Ni indicates a lower data rate and also implies that a smaller bandwidth is allocated to theith subband For a filter bank transceiver, the

integersN isatisfyM −1

i =0 1/N i ≤1, which is a necessary condi-tion for recovering the input signalsx i(n) When the

equal-ity M −1

i =0 1/N i = 1 holds, the transceiver is said to be crit-ically sampled The transmission channel is modeled as an

Lth-order LTI channel with transfer function

C(z) =

L



l =0

The additive noise is denoted byv(n) Because our

formu-lation is based on the signal-to-interference ratio, the chan-nel noise does not affect the transceiver design Therefore in Sections2,3, and4, we setv(n) = 0 For convenience, an advance operatorz l0is added at the receiver to account for the system delay caused by channelC(z) In practice, this

ad-vance element can be replaced by an appropriate delay In this paper, we consider only FIR filter banks The transmit-ting and receiving filters are, respectively,

F i(z) =

N fi



n =0

f i(n)z − n, H i(z) =

N hi



n =0

h i(n)z n (2)

The orders of these filtersN f i andN h ican be larger thanN i For notational simplicity, we use the noncausal expression for the receiving filters Causal filters can be obtained easily

by adding sufficient delays In addition, we assume that the input signalsx i(n) are uncorrelated, zero mean, wide sense

stationary (WSS), and white random processes with the same varianceEx That is,

E

x i(n)

=0, E

x i(n)x ∗ j(m)

=Ex δ(i − j)δ(n − m).

(3) This assumption is usually satisfied by properly interleaving the input data

2.1 ISI-free condition

The filter bank transceiver shown inFigure 1is said to be

ISI-free if in the absence of noise, for all possible input signals

x i(n), the outputs are



x i(n) = G i x i(n), (4) for some constantG i In this case, a zero-forcing solution can

be obtained by cascading a simple one-tap equalizer

Express-ing the output signal at the jth subband in the z domain, we

have



X j(z) =

M−1

i =0



X i



z N i

F i(z)z l0C(z)H j(z)

↓ N j

= X j(z)

F j(z)z l0C(z)H j(z)

↓ N j

+

M−1

i =0

i = j



X i



z N i

F i(z)z l0C(z)H j(z)

↓ N j

(5)

From the above equation, we see that in general the system

from the inputx i(n) to the output xj(n) is not LTI unless

N j is a factor of N i This is very different from the case of

uniform filter bank transceivers, in which allN i = N Let g i, j

be the greatest common divisor (gcd) ofN iandN j Define

two coprime integersp i, j = N i /g i, j and p j,i = N j /g i, j Then

we can write



X j(z) = X j(z)

F j(z)z l0C(z)H j(z)

↓ N j

+

M−1

i =0

i = j

X i



z p i, j

F i(z)z l0C(z)H j(z)

↓ g i, j ↓ p j,i (6)

Define

T i, j(z) =F i(z)z l0C(z)H j(z)

↓ g i, j

=

L



l =0

c(l)

F i(z)H j(z)z l0− l

↓ g i, j

(7)

for 0≤ i, j ≤ M −1 As the input signalsx i(n) are arbitrary,

one can show (see the appendix for a proof) that the ISI-free

conditionXi(z) = G i X i(z) is satisfied if and only if

T i, j(z) =

⎧

⎨

⎩

G i, j = i,

For convenience of discussion, we express [F i(z)H j(z)z l0− l]↓ g i, j

in terms of the two sequencesα i,l(n) and β i, j,l(n) as



F i(z)H j(z)z l0− l

↓ g i, j =

⎧

⎪

⎪

⎪

⎪

α i,l(0) +

n

=0

α i,l(n)z − n, i = j,



n

(9)

for 0 ≤ i, j ≤ M −1, and 0 ≤ l ≤ L Note that since F i(z)

andH j(z) are of finite length, α i,l(n) and β i, j,l(n) have finite

nonzero terms only Using the above definition, we can write thejth output signalx j(n) as



x j(n) =

L

l =0

α j,l(0)c(l)



x j(n)

+

L



l =0

c(l)

α j,l(n) − α j,l(0)δ(n)

∗ x j(n)

+

M−1

i =0

i = j

L

l =0

c(l)β i, j,l(n) ∗x i(n)

↑ p i, j



↓ p j,i

.

(10)

The first, second, and third terms on the right-hand side of the above expression are the desired signal, the intraband ISI and the cross-band ISI, respectively To get an ISI-free transceiver, we need to find the transmitting filtersF k(z) and

receiving filtersH k(z) so that the second and third terms are

equal to zero The general solution to this problem is still unknown In the following, we will show how to reduce the

effect of ISI by finding a solution that maximizes the signal-to-interference ratio (SIR)

2.2 Matrix formulations of α i,l(n) and β i, j,l(n)

In this section, we will formulate the sequencesα i,l(n) and

for the optimization of the transceivers Recall from (9) that

α i,l(n) and β i, j,l(n) are obtained from the convolution of f k(n)

andh k(n) Let us define the following vectors:

α i(n) =

⎡

⎢

⎢

⎢

⎣

α i,0(n)

α i,1(n)

α i,L(n)

⎤

⎥

⎥

⎥

⎦

, β i, j(n) =

⎡

⎢

⎢

⎢

⎣

β i, j,0(n)

β i, j,1(n)

β i, j,L(n)

⎤

⎥

⎥

⎥

⎦

,

hi =

⎡

⎢

⎢

⎢

⎣

h i(0)

h i(1)

h i



N h i



⎤

⎥

⎥

⎥

⎦

, fi =

⎡

⎢

⎢

⎢

⎣

f i(0)

f i(1)

f i



N f i



⎤

⎥

⎥

⎥

⎦

.

(11)

Then from (9), it is not difficult to verify that the vectors

α i(n) and β i, j(n) can respectively be expressed as

α i(n) =Ai(n)h i,

β (n) =Bi, j(n)h j, (12)

where the matrices Ai(n) and B i, j(n) are respectively given by

Ai(n)

=

⎡

⎢

⎢

⎢

⎣

f i



nN i+l0



f i



nN i+l0+1

· · · f i



nN i+l0+N h i



f i



nN i+l0−1f i



nN i+l0−1+1· · · f i



nN i+l0−1+ N h i



f i



nN i+l0− L

f i



nN i+l0− L+1

· · · f i



nN i+l0− L+N h i



⎤

⎥

⎥

⎥

⎦

,

Bi, j(n)

=

⎡

⎢

⎢

⎢

⎣

f i



ng i, j+l0



f i



ng i, j+l0+1

· · · f i



ng i, j+l0+N h j



f i



ng i, j+l0−1f i



ng i, j+l0−1+1

· · · f i



ng i, j+l0−1+N h j



f i



ng i, j+l0− L

f i



ng i, j+l0− L+1

· · · f i



ng i, j+l0− L+N h j



⎤

⎥

⎥

⎥

⎦

.

(13)

The dimensions of the matrices Ai(z) and B i, j(n) are,

respec-tively, (L + 1) ×(N h i+ 1) and (L + 1) ×(N h j + 1) Notice

thatg i, j = N iwheni = j Similarly, we can also express the

vectorsα i(n) and β i, j(n), respectively, in terms of the

trans-mitting filter fias

α i(n) = Ai(n)f i, β i, j(n) = Bi, j(n)f i, (14)

for some matricesAi(n) andBi, j(n) The matricesAi(n) and



Bi, j(n) consist of the transmitting filter coe fficients h j(n) and

they are very similar to Ai(n) and B i, j(n), respectively.

2.3 SIR-optimized receiving filters

In this section, we will design the receiving filters so that

the SIR is maximized for a fixed set of transmitting filters

As the jth receiving filter a ffects only the jth output signal



x j(n), the receiving filters can be designed separately; the jth

receiving filterF j(z) is optimized so that the SIR of the jth

output signalx j(n) is maximized Recall from (10) that the

output of the jth subband xj(n) consists of three

compo-nents, namely, the desired signal, the intraband interference,

and the cross-band interference As the input signals x i(n)

satisfy the uncorrelated and white property in (3), the

de-sired signal power and intraband interference power at the

jth output are given by

Psig(j) =Ex







L



l =0

α j,l(0)c(l)





2

,

Pintra(j) =Ex



n, n =0







L



l =0

α j,l(n)c(l)





2

, (15)

whereExis the power of the input signal defined in (3) The

computation of the cross-band interference power is more

complicated because the sequence [x j(n)] ↑ p i, j is not a WSS

process From multirate theory [20], we know that [x j(n)] ↑ p

is cyclo wide sense stationary with periodp i, j, or CWSS(p i, j) Letting u(n) = [x j(n)] ↑ p i, j, then its autocorrelation coeffi-cients satisfyE[u(n)u ∗(n − k)] = E[u(n+ p i, j)u ∗(n+ p i, j − k)].

Sincep i, jandp j,iare coprime, the quantity

L

l =0

c(l)β i, j,l(n) ∗x i(n)

↑ p i, j



↓ p j,i

(16)

is also CWSS(p i, j) [20] From (10), we see that the cross-band interference consists of (M −1) CWSS sequences with period p i, j for i = 0, , j −1,j + 1, , M −1 Let

P j be the least common multiple of the integers { p0,j, ,

p j −1,j,p j+1, j, , p M −1,j } Then the cross-band interference is

a CWSS(P j) random process We can compute the average cross-band interference power over one periodP j and it is given by

Pcross(j) =Ex



i,n

i = j

1

p i, j







L



l =0

β i, j,l(n)c(l)





2

Next we will express the three quantitiesPsig(j), Pintra(j), and

Pcross(j) in terms of the receiving filter coe fficients h j(n) To

do this, let us define the (L + 1) ×1 vector

c= c(0) c(1) · · · c(L) T (18) Then from (12), we can write

Ex







L



l =0

α j,l(0)c(l)





2

=ExcTAj(0)hj2

=Exh† jA† j(0)c∗cTAj(0)hj

(19)

Similarly, using the expressions ofα i(n) and β i, j(n) in (12),

we can also write the intraband and cross-band interference

powers in a quadratic form of hj In summary, the three pow-ers are given by

Psig(j) =h† jQsig,jhj, Pintra(j) =h† jQintra,jhj,

Pcross(j) =h† jQcross,jhj,

(20)

where the matrices Qsig,j, Qintra,j, and Qcross,jare, respectively, given by

Qsig,j =ExA† j(0)c∗cTAj(0),

Qintra,j =Ex



n, n =0

A† j(n)c ∗cTAj(n),

Qcross,j =Ex



i,n

i = j

1

p i, jB† i, j(n)c ∗cTBi, j(n).

(21)

Asx i(n) and x j(n) are uncorrelated for i = j, the total ISI

power at the jth output is Pisi(j) = Pintra(j) + Pcross(j) Thus

the SIR of thejth output is given by

γ j = Psig(j)

Pisi(j) =h

†

jQsig,jhj

h† jQisi,jhj

where Qisi,j =Qintra,j+ Qcross,j Notice that both Qsig,j and

Qisi,jare positive semidefinite matrices Furthermore, except

for some very rare cases, the matrix Qisi,jis positive definite

From the above expression, we see that the SIR is expressed as

a Rayleigh-Ritz ratio of hj The optimal unit-norm vector hj

that maximizesγ jis well known [21] Let Q1isi,/2 jbe the

posi-tive definite matrix such that Qisi,j =Q1isi,/2 jQ1isi,/2 j The optimal

hjis given by

hj,opt =Q−isi,1/2 j arg max

v=0

v†Q−isi,1/2 jQsig,jQ−isi,1/2 j v

The optimal vector v is the eigenvector corresponding

to the largest eigenvalue of the positive definite matrix

Q−isi,1/2 j Qsig,jQ−isi,1/2 j

2.4 SIR-optimized transmitting filters

In this section, we consider the SIR optimization of the

trans-mitting filtersf i(n) given a fixed set of the receiving filters As

theith transmitting filter f i(n) a ffects only the ith input

sig-nalx i(n), we can consider the SIR due to the ith transmitted

signalx i(n) Consider the transmission scenario when only

theith subband is transmitting, that is, x j(n) =0 for j = i.

Then from (10), the outputs of the receiver are given by



x i(n) =

L

l =0

α i,l(0)c(l)



x i(n)

+

L



l =0

c(l)

α i,l(n) − α i,l(0)δ(n)

∗ x i(n),



x j(n) =

L

l =0

c(l)β i, j,l(n) ∗x i(n)

↑ p i, j



↓ p j,i

, fori = j.

(24) Note that the first and second terms on the right-hand side

of (24) are respectively the desired signal and the intraband

interference due to theith transmitted signal x i(n) On the

other hand,x j(n) represents the cross-band interferences due

tox i(n) By following a procedure similar to that in the

pre-vious section, we can compute the signal power and

interfer-ence powers and express the SIR as a Rayleigh-Ritz ratio as

follows:



γ i =f

†

iQsig,ifi

fi †Qisi,ifi, (25)

where the matrices Qsig,iandQisi,iare positive semidefinite

matrices that have a form very similar to Qsig,i and Qisi,i,

respectively Hence the optimal unit-norm fithat maximizes

the SIR can be obtained by solving the above Rayleigh-Ritz

ratio

2.5 SIR optimized for unknown channels

In many applications, the exact channel impulse response may not be available, and we may have only the statistics

of the transmission channels The above design method can easily be modified to obtain transceivers that are optimized for unknown channels Assume that the vector containing

the channel impulse response, c, is zero-mean with

autocor-relation matrix

Rc = E

cc†

In this case, the exact channel impulse response is not known From previous discussions, we know that the sig-nal power and interference powers at the output of the jth

subband are respectively given by (20) and (21) When the channel is not known, we can compute the average signal power and interference powers by taking the expectation with respect to the channel impulse response c(l) It is not

difficult to verify that the average SIR can also be expressed

as a Rayleigh-Ritz ratio of the filter coefficients hi Similarly, given the receiving filters, we can modify

the optimization of transmitting filters fi for the case of unknown channels by using the average SIR In many situ-ations, we do not know the statistics of the channel In this case, it is often assumed that the channel impulse responses are independent identical distribution, that is, i.i.d channels The autocorrelation matrix of the channel impulse response

becomes Rc = σ2

cI.

WITH FREQUENCY CRITERIA

From the previous discussions, we know that when the trans-mitting filters are given, we can obtain optimum receiving fil-ters so that SIR is maximized Conversely, given the receiving filters we can design the transmitting filters that maximize the SIR One can therefore alternatingly optimize the receiv-ing and transmittreceiv-ing filters so that SIR is maximized Because

in each iteration, the solution obtained in the previous iter-ation is also a candidate, the SIR cannot decrease1when the number of iterations increases As we will see in the numeri-cal examples, the increase in SIR is substantial as the number

of iterations increases However because no constraint is ap-plied on the filters, their frequency responses will often de-grade significantly as the number of iterations increases To solve this problem, we can incorporate the filter stopband en-ergy in the optimization Let us consider the design of the

receiving filters hj The stopband energy of thejth receiving

filterH j(z) is given by

Pstop(j) = 1

2π



 h, j

H j

e jω2

where h, jis the stopband region ofH j(z) Define the

vec-tor eN(z) = [1 z · · · z N]T Then the weighted stopband

1 In general, it is not guaranteed that the SIR is monotonically increasing.

x0 (n)

x1 (n)

x M 1(n)

N0

.

N1

F0 (z)

F1 (z)

F M 1(z)

M0

.

M1

M M 1

C(z)

v(n)

z l0

.

.

M0

M1

H0 (z)

H1 (z)

H M 1(z)

N0

N1

N M 1



x0 (n)



x1 (n)



x M 1(n)

Figure 2: Nonuniform filter bank transceiver with rational sampling factors

energy can be expressed as

Pstop(j) =h† jQstop,jhj, (28)

where the matrix Qstop,jis given by

Qstop,j = 1

2π



 h, j

eN h j



e jω

e† N h j

e jω

The new objective function that incorporates the frequency

response is

†

jQsig,jhj

h† j

Qisi,j+c h, jQstop,j



hj

wherec h, j ≥0 is a weight that adjusts the relative importance

of the frequency responses Whenc h, j =0, the new objective

functionη jreduces to the SIR expressionγ j in (22) and no

frequency criteria are applied One can see thatη j is also a

Rayleigh-Ritz ratio of hj We can choose hj to be the

unit-norm vector that maximizes this ratio Similarly, one can

in-corporate the stopband energy into the optimization of the

transmitting filters f i(n) One will get a new objective

func-tion



†

iQsig,ifi

fi † Qisi,i+c f ,iQstop,i]fi, (31)

where fi †Qstop,ifiis the term corresponding to the stopband

energy of the filter f i(n) The optimal f iis the unit-norm

vec-tor that maximizesηi

Note that in the new objective function, the passband

re-sponses of the filters are not included For unit-norm filters,

when the stopband energy is small, the passband energy will

be close to one In transceiver designs, nearly zero ISI

prop-erty can be guaranteed by a high SIR and the flatness of

pass-band response is not needed

The iterative algorithm for transceiver optimization is

summarized as follows

(1) Select a set of the receiving filters H(0)

i (z) with good

frequency responses

Fork ≥1, repeat the following steps

(2) Given the receiving filtersH(k −1)

i (z), optimize F(k)

j (z) so

thatηjis maximized for 0≤ j ≤ M −1

(3) Given the transmitting filter F(k)

j (z), optimize H(k)

i (z)

so thatη iis maximized for 0≤ i ≤ M −1

(4) Stop if the SIR is higher than the desired value or if it reaches the maximum number of iterations; otherwise,

k = k + 1 and return to step (2).

WITH RATIONAL SAMPLING FACTORS

In this section, we generalize the design method to the case of rational sampling factors We will first employ the technique

in [15] to convert the transceiver with rational sampling tors into an equivalent transceiver with integer sampling fac-tor Then the optimization method developed in the previ-ous sections can be adopted The block diagram of a nonuni-form filter bank transceiver with rational sampling factors is shown inFigure 2 At the transmitter, the input signalx i(n)

goes through an N i-fold expander and anM i-fold decima-tor The bandwidth of theith subband is proportional to the

ratio M i /N i Without loss of generality, we assume that the integersM iandN iare coprime If they are not coprime, then

it is known [20] that theith subband can be replaced with

an equivalent system with coprimeM i andN i , and such an equivalent system will have a lower complexity Furthermore,

to ensure symbol recovery, we assume

M−1

i =0

M i

Let us decompose thekth transmitting and receiving

fil-ters using the polyphase representation as

H k(z) =

Mk −1

=0

z E 

z M k

,

F k(z) =

Mk −1

=0

z − R 

z M k

.

(33)

Note that no coefficient of H k(z) or F k(z) appears in more

x k(n) N k F k(z) M k

x k,0(n)

.

x k,1(n)

.

.

z b k,Mk 1 M k

x k,M k 1 (n)

N k z a k,Mk 1 R k,M k 1 (z)

(a)



x k,0(n)

z a k,1 E k,1(z) N k



x k,1(n)

.

.

.

.

z a k,Mk 1 E k,M k 1 (z) N k



x k,M k 1 (n)

M k z b k,Mk 1

(b)

Figure 3: (a) Equivalent circuit of thekth subband in the transmitting bank, (b) equivalent circuit of the kth subband in the receiving bank.

than oneE (z) or R (z) As M kandN kare coprime, we can

always find positive integersa and b such that aM k − bN k =

1 Leta k,1 andb k,1 be the smallest integers that satisfy this

condition Define

a k,l = la k,1, b k,l = lb k,1 (34)

Using the polyphase representation and the noble identities

[20], we can redraw thekth subbands of the transmitter and

receiver, respectively, as those shown in Figures3(a)and3(b)

Moreover, sinceM kandb k,1are coprime, we have2



b k,1



M k,

b k,2



M k, ,

b k,M k −1



M k

=1, 2, , M k −1

, (35) where [p] q represents p modulo q, which is a number

be-tween 0 and q −1 Thus, in Figure 3(a), the signal x k(n)

is split into its polyphase components { x k,0(n), x k,1(n), ,

x k,M k −1(n) } Similarly, { x k,0(n), xk,1(n), ,x k,M k −1(n) } in

Figure 3(b) are the polyphase components of the signal

2 See homework [ 20 , Problem 4.9].



x k(n) Using these results, we can redrawFigure 2asFigure 4 The transceiver inFigure 4has the same structure as that in

Figure 1 Since input signalsx i, j(n) are also uncorrelated, we

can apply the design method developed in previous sections

to obtain the optimalR (z) and E (z) The filters F k(z) and

H k(z) can be obtained from (33)

In this section, we provide two examples to show the perfor-mance of nonuniform filter bank transceivers designed by us-ing the proposed method It is emphasized that in transceiver designs, the nearly zero ISI property is guaranteed by a high SIR value and passband flatness is not needed We assume that the channel noisev(n) is AWGN in the following

exam-ples

Example 1 In this example, we design nonuniform filter

bank transceivers with integer sampling factors The num-ber of subbands isM =4 and the sampling factors are{ N0,

N1,N2,N3} = {2, 4, 8, 8} Four-tap channels are used here A

total of 100 randomly generated iid channels are employed in the simulation We assume that channel impulse responses are known All the transmitting and receiving filters are of

x0,0 (n) N0 R0,0(z) C(z)

v(n)

x M 1,1(n) N M 1 z a M 1,1 R M 1,1(z) z a M 1,1 E M 1,1(z) N M 1 xM 1,1(n)

x M 1,M M 1 1 (n) N M 1 z a M 1,MM 1 1 R M 1,M M 1 1 (z) z a M 1,MM 1 1 E M 1,M M 1 1 (z) N M 1 xM 1,M M 1 1 (n)

.

.

.

.

.

.

.

.

.

.

.

Figure 4: Equivalent circuit of the nonuniform filter bank transceiver with rational sampling factors inFigure 2

order 56 We consider the iterative algorithm for both cases

of with and without frequency criteria For the case with

fre-quency criteria (indicated as fc), the weights for the stopband

energy are chosen asc f ,0 = c f ,1 = c h,0 = c h,1 = 0.05, and

c f ,2 = c f ,3 = c h,2 = c h,3 = 0.4 We plot the SIR averaged

over the 100 random channels versus the number of

itera-tions and the results are shown inFigure 5 From the figure,

we see that the average SIR increases with the number of

it-erations When no frequency criteria are applied, the average

SIR increases by about 15 dB and it can be as high as 56 dB

after 400 iterations Even when the frequency criteria are

ap-plied, the average SIR increases by more than 8 dB Thus the

incorporation of frequency criteria results in a loss of SIR

by 7 dB To show the improvement in frequency response

when the frequency criteria are applied, we plot the

magni-tude responses of the transceiver optimized for one

partic-ular channel—Channel A after the 200th iteration The

im-pulse response of Channel A is given by

Channel A= 0.2218 −0 475 0.3906 0.2845 (36)

400 350 300 250 200 150 100 50 0

The number of iterations 40

42 44 46 48 50 52 54 56 58

100 random channels

100 random channels frequency criteria

Figure 5: SIR versus the number of iterations

0.8

0.6

0.4

0.2

0

Normalized frequencyω/π

80

70

60

50

40

30

20

10

0

Figure 6: Magnitude responses of the transmitting filters (no

fre-quency criteria)

1

0.8

0.6

0.4

0.2

0

Normalized frequencyω/π

80

70

60

50

40

30

20

10

0

Figure 7: Magnitude responses of the receiving filters (no frequency

criteria)

The results are shown in Figures6,7,8, and9 Comparing the

results in Figures6and7with those in Figures8and9, we can

see that the incorporation of the frequency criteria improves

the frequency characteristics of the transceiver significantly

The tradeoff is a loss in SIR of around 7 dB

Example 2 In this example, we design two-band

nonuni-form filter bank transceivers with rational sampling factors,

whereN0= N1=5,M0=2, andM1=3 A total of 100 iid

channels with 4 taps are randomly generated The filter

or-ders areN f0 = N h0 = 58 andN f1 = N h1 = 87 The

trans-mitting filtersF0(z) and F1(z) are, respectively, initialized as

good lowpass and highpass filters with a passband bandwidth

of 2π/5 We consider 3 cases: (i) optimization without

fre-1

0.8

0.6

0.4

0.2

0

Normalized frequencyω/π

80 70 60 50 40 30 20 10 0

Figure 8: Magnitude responses of the transmitting filters (with fre-quency criteria)

1

0.8

0.6

0.4

0.2

0

Normalized frequencyω/π

80 70 60 50 40 30 20 10 0

Figure 9: Magnitude responses of the transmitting filters (with fre-quency criteria)

quency criteria (indicated byc =0); (ii) optimization with frequency criteria and the weights on the stopband energy arec f ,0 = c f ,1 = c h,0 = c h,1 = c =0.1 (indicated by c =0.1);

(iii) optimization with frequency criteria and the weights on the stopband energy arec f ,0 = c f ,1 = c h,0 = c h,1 = c =10 (in-dicated byc =10) The SIR averaged over 100 random chan-nels versus the number of iterations are given inFigure 10for the three different values of c From the figure, we see that the SIR is smaller when we impose frequency criteria The heav-ier the frequency criteria, the lower the SIR Comparing the cases ofc =10 andc =0, the loss of SIR (after 200 iterations)

is around 6 dB Even with the frequency weighting ofc =10, the SIR can be as high as 47 dB, a value that is good enough for many applications To demonstrate the effect of adding frequency criteria, we plot the filter magnitude responses for

200 150

100 50

0

The number of iterations 42

44

46

48

50

52

54

c =0

c =0.1

c =10

Figure 10: SIR versus the number of iterations

1

0.8

0.6

0.4

0.2

0

Normalized frequencyω/π

110

100

90

80

70

60

50

40

30

20

10

0

c =0

c =0.1

c =10 Initial

Figure 11: Magnitude response ofF0(z).

one particular channel—Channel B after 200 iterations The

impulse response of Channel B is

Channel B= −0 44270 −0 42492 0.39377 0.34971

(37) The magnitude responses of the initial filters are given in

Figures11and12 The results are shown in Figures11,12,

13, and14(for the purpose of comparison, we also plot the

initial | F i(e jω)| in the same figure) From the figure, it is

clear that without any frequency weighting, the magnitude

responses degrade significantly after 200 iterations and the

1

0.8

0.6

0.4

0.2

0

Normalized responseω/π

110 100 90 80 70 60 50 40 30 20 10 0

c =0

c =0.1

c =10 Initial

Figure 12: Magnitude response ofF1(z).

1

0.8

0.6

0.4

0.2

0

Normalized frequencyω/π

90 80 70 60 50 40 30 20 10 0

c =0

c =0.1

c =10

Figure 13: Magnitude response ofH0(z).

frequency weighting can greatly enhance the selectivity of fil-ters

In this paper, we propose a method for designing nonuni-form filter bank transceivers for frequency selective channels

By expressing the SIR as Rayleigh-Ritz ratios of transmitting and receiving filters respectively, we can iteratively optimize the filters so that SIR is maximized Moreover, a new cost function that incorporates the filter frequency response is in-troduced Iterative optimization algorithm based on the new