Design of Multi Channel Near Perfect Reconstruction

Signal Processing 83 (2003) 1079–1091

www.elsevier.com/locate/sigpro

Design of multi-channel near-perfect-reconstructiontransmultiplexers using cosine-modulated %lter banks

Fernando Cruz-Rold,an∗, ,Angel M. Bravo-Santos, Pilar Mart,1n-Mart,1n,Roberto Jim,enez-Mart,1nez

Dept. de Teor�a de la Senal y Comunicaciones, Escuela Politecnica, Universidad de Alcala, 28871 Alcala de Henares, Spain

Received 16 November 2001

Abstract

The paper considers the design of nearly perfect reconstruction transmultiplexer systems based on cosine-modulated %lterbanks. Under the same scheme of cosine-modulation, several families of transmultiplexer systems are derived, the maindi8erence of which is the prototype %lter. We show several ways of obtaining this %lter and we also propose a new techniquefor designing high-quality prototype %lters. A comparative analysis is included to con%rm the validity of the theory.? 2003 Elsevier Science B.V. All rights reserved.

Keywords: Transmultiplexing; Filtering

1. Introduction

The theory and design of transmultiplexer (TMUX)systems are widely reported in the literature (see forexample Refs. [1–7,19,20,24,26,28,29,32,33,36–40].These systems are traditionally used for inter-conversion between the time-division multiplexingformat (TDM) and the frequency-division multi-plexing (FDM) format, and have been successfullyutilized to describe several popular communica-tion applications such as code division multipleaccess (CDMA), discrete multi-tone (DMT) or or-thogonal frequency-division multiplexing (OFDM)[1–3]. This topic is very important in digital sig-nal transmission systems, especially in multi-usercommunications, because the core element of theaforementioned systems can be viewed from the

∗ Corresponding author. Tel.: +34-91-885-66-93; fax: +34-91-885-66-99.

E-mail address: [email protected] (F. Cruz-Rold,an).

perspective of a synthesis (transmitting) and analysis(receiving) con%guration.A schematic diagram of an M -channel TMUX sys-

tem with a channel is shown in Fig. 1a. Our studyfocuses on a uniform-band structure in which all theincoming data signals are interpolated at the sam-pling rate M and the received signal is splitted andup sampled at the same integer factor (M). Withinthe di8erent families of TMUX, modulated %lterbank-based systems are emphasized because all thesub-channel %lters can be obtained from a single pro-totype %lter. So, the design process of the TMUXsystem comes down to the design of this prototype%lter. The most widely used modulated TMUX sys-tem is based on the discrete Fourier transform (DFT)%lter bank [19,20,36] because it has been proposedfor implementing several multi-carrier systems (seefor example Refs. [13–18,22,23]. However, it is wellknown that the selectiveness of DFT as a %lter bank israther limited. The sub-channel responses consist ofa main lobe which partly overlaps with immediately

0165-1684/03/$ - see front matter ? 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0165-1684(02)00508-X

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1080 F. Cruz-Roldan et al. / Signal Processing 83 (2003) 1079–1091

Fig. 1. (a) M -channel transmultiplexer system over a channel and (b) M -channel maximally decimated %lter bank.

adjacent channels and high side lobes that extendover a wide frequency band. The level of the %rstside lobe is only −13 dB. So, these systems do notprovide a good performance with regard to the e8ectof narrowband interference.Another very attractive way of designing modulated

TMUX systems consists of using cosine-modulated%lter banks. These transmultiplexers should be con-sidered as a serious candidate when choosing fu-ture coding methods for the following reasons. (A)High-selectivity and high-discrimination systemscan be easily designed. These characteristics allowtolerance of stronger narrowband interferences andisolate each channel from the rest. (B) The resultingsub-channel (transmitting and receiving) %lters arealso derived from a single real-coeJcient prototype%lter. (C) Fast algorithms for eJciently implement-ing the sub-carrier modulators in a parallel processingstructure can be derived in particular cases.In this paper, we deal with the design of near-

perfect-reconstruction TMUX systems based oncosine-modulated %lter banks and two novel contri-butions are presented. First, we propose the design ofdi8erent near-perfect reconstruction TMUX systemsbased on the same scheme of cosine-modulation. Themain di8erence between these systems resides in the

prototype %lter characteristics. Secondly, we showseveral ways of obtaining this prototype %lter with-out using highly non-linear optimization algorithms,and we also propose a new technique for designinghigh-quality prototype %lters.The rest of this article is organized as follows: In

Section 2, we show the TMUX con%guration and wedevelop the input–output relationships. In this section,we also show di8erent ways of measuring the pro-duced TMUX system’s performance and we proposesome parameters to evaluate the cross-talk and theamplitude distortion as well as the spread–time andthe spread–frequency of the sub-channel %lters ob-tained. In Section 3, several methods for obtaining theprototype %lter are presented; we also state the mod-ulation by which the transmitting and the receivingsub-channel %lters are derived from the prototype %l-ter. The validity of the theory is illustrated in Section4 with several examples and, %nally, Section 5 sum-marizes our conclusions.

2. The transmultiplexer con�guration and errormeasures

The TMUX system con%guration consists of Mtransmitting %lters Fk(z) and M receiving %lters

F. Cruz-Roldan et al. / Signal Processing 83 (2003) 1079–1091 1081

Hk(z); 06 k6M − 1 (Fig. 1a). In this paper, wewill consider an ideal environment in which thereis an ideal and noise-free channel (C(z) = 1 ande[n] = 0 ∀ n). In this case, the output X i(z) of the ithbranch due to all the inputs X‘(z) can be written as

X i(zM ) =1M

M−1∑k=0

Hi(zW kM )M−1∑‘=0

F‘(zW kM )X‘(zM );

(1)

whereWM=e−j2�=M . The corresponding transfer func-tion between output signal X i(zM ) and input signalXr(zM ) is obtained as

Ti;r(zM ) =M−1∑k=0

Hi(zW kM )Fr(zWkM ): (2)

The terms Ti;r(zM ); i �= r, describe the cross-talk be-tween di8erent channels. In the case of cross-talk-freeTMUX systems, these functions are null. However,when cross-talk exists—there is signal leakage fromone channel to another—this error can be quanti%ed[40] by means of the inter-carrier (inter-channel) in-terference (ICI) as

ICI = max!; i

M−1∑r=0; i �=r

|Ti;r(ej!)|2 : (3)

The total cross-talk error for the ith channel is de%nedas [24]

ei =∫ �=Mo

M−1∑r=0r �=i

|Ti;r(ej!M )|2 d!; (4)

and the maximum cross-talk error is

emax = max06i6M−1

ei: (5)

On the other hand, the functions Ti; i(zM ) relate theoutput signal X i(zM ) to the input signal Xi(zM ). Forperfect reconstruction (PR) systems, these functionsare simple delays of the form z−pM ; p∈Z+, and theoverall distortion transfer function T0(z), given by

T0(z) =1M

M−1∑k=0

Fk(z)Hk(z) (6)

can be expressed as T0(z)=cz−n0 ; c∈R and n0 ∈Z+.However, if the TMUX system presents amplitude

distortion, we measure the peak di8erence on thisfunction as

�pp = (|T0(ej!)|max − |T0(ej!)|min); !∈ [0; �]:(7)

Even if both amplitude distortion and cross-talkerror are zero, the ith channel may still have errorscompared with the ideal time delay. We can evaluatethese errors by measuring the inter-symbol interfer-ence (ISI), de%ned as

ISI = maxi

(∑n

(ti; i[n]− �[n− nd])2); (8)

where ti; i[n] is the impulse response of the ith channel,�[n] is the unit impulse, and nd is a proper delay [40].

Recently, Chen et al. have proposed a new classof transmultiplexer designed with the %lter responsesspread in both time and frequency domains [6]. TheTMUX systems proposed in this work are not opti-mized in this sense. However, in order to evaluatethese characteristics, we use two parameters to mea-sure the spread of the %lter responses in time as wellas in the frequency domain. The time–spread of a %l-ter response fi[n]; 06 n6N − 1, characterizes thetime location property [21] and is de%ned as

�2ni =1Ei

∑n

(n− Pni)2|fi[n]|2; (9)

where Ei and Pni, are, respectively, the energy and thetime centre of the function fi[n]:

Ei =∑n

|fi[n]|2; Pni =1Ei

∑n

n|fi[n]|2: (10)

The frequency–spread [21] of a %lter response fi[n]is given by

�2!i =1

2�Ei

∫ �−�

(!− P!i)2|Fi(!)|2 d!; (11)

where

Fi(!) =N−1∑n=0

fi[n]e−j!n; (12)

P!i =1

2�Ei

∫ �−�! |Fi(!)|2 d!: (13)


Time and frequency domain energy concentration canbe used in the case of basis design for communicationsor to obtain robust systems against narrow-band orshort-time interferences.

3. Transmultiplexers design based oncosine-modulated �lter banks

Several authors have proved the mathematicalequivalence of TMUX systems and the %lter banks[24,38,39]. Therefore, analysis and synthesis %lterof an M -channel maximally decimated %lter bank(Fig. 1b) can %rst be designed and then the resultsapplied directly to obtain the corresponding TMUXsystems.In this work, we consider a special group of nearly

perfect TMUX systems based on cosine-modulation.These TMUX systems can be useful in many sig-nal processing and communications applications,because of their very eJcient implementation andbecause all analysis and synthesis %lters can be de-rived from a single low pass prototype %lter (Fig.2). The prototype %lter p[n] must have a narrowtransition bandwidth and high stop-band attenuation.In this way, the separation between channels is al-most absolute and cross-talk present in the outputsignals can be reduced. We consider pseudo-QMFbanks because they are an alternative to PR systems

Fig. 2. (a) Typical ideal magnitude response of the prototype %lter P(z) and (b) typical ideal magnitude responses of the transmittingFk (z) and receiving Hk (z) sub-channel %lters.

for avoiding the highly non-linear optimization nec-essary to obtain the %lter coeJcients in a PR sys-tem. The most popular pseudo-QMF banks are thefollowing:

(a) Conventional pseudo-QMF design [34,31]. Theprototype %lter is obtained by optimization usingan objective function, as shown in the followingsubsection.

(b) Spectral factorization approach to pseudo-QMFdesign [25,12], where the prototype %lter is aspectral factor of a 2M th band %lter.

(c) Near-perfect-reconstruction pseudo-QMF banks[30]. These %lter banks are a hybrid of conven-tional pseudo-QMF design—the same scheme ofcosine-modulation is used—and the spectral fac-torization approach—the prototype %lter is a spec-tral factor of a 2M th band %lter.

All the aforementioned pseudo-QMF banks can beincluded in the same family of modulated TMUX sys-tems, because their schemes of cosine modulation canbe obtained from the scheme of modulation proposedin Section 3.2.

3.1. Prototype 2lter design

Various methods exist which are based on di8erentobjective functions for optimizing the prototype %ltercoeJcients. The most commonly used or recently


proposed are:

1. The conventional technique can be found in [9].The objective function is based on simultaneouslyminimizing the stop-band energy of the prototype%lter and the overall amplitude distortion in the%lter bank (�¿ 0; 0¡"¡ 1):

#=min!

{"∫ �=M0

(|P(ej!)|2

+|P(ej(!−�=M))|2 − 1)2 d!

+(1− ")∫ ��+�=(2M)

|P(ej!)|2 d!}: (14)

2. A technique to obtain the exact spectral factors of2M th band %lters can be found in [30]. The methodlends itself to eJcient systems, because the onlyreconstruction error in the %lter bank is the alias-ing error, and it is comparable to the prototype %l-ter stop-band attenuation. However, designing theoptimized %lter for the modulation requires a greatcomputational e8ort.

3. The method proposed by Creusere and Mitra(CMT) uses the Parks–McClellan technique [8].The prototype %lter length, relative error weightingand stop-band edge are %xed before the optimiza-tion procedure is started, while the pass-band edgeis adjusted to minimize

#=max!

{|P(ej!)|2 + |P(ej(!−�=M))|2 − 1};

!∈ (0; �=M): (15)

4. Another designing technique is the Kaiser Win-dow Approach (KWA) proposed by Lin andVaidyanathan [27]. Let p[n] be a %lter designed as

p[n] = pi[n]w[n]; (16)

where w[n] is an (N+1)-length Kaiser window and

pi[n] =sin((n− N=2)!c)�(n− N=2) ; n∈Z: (17)

Next, G(ej!) is de%ned as G(ej!) = |P(ej!)|2.The design process of the prototype %lter is re-duced to the optimization of the ideal %lter cuto8

frequency !c in order to minimize the objectivefunction given by

#= maxn;n �=0

|g[2Mn]|: (18)

This condition ensures that p[n] is approximatelya spectral factor of a 2M th band %lter.

5. Recently, other methods for designing prototype%lters have been proposed in [35]. The statementof the problems are: given '; M and N , %nd thecoeJcients of P(z) to minimize their stop-bandenergy

E2 =∫ �(1+')�=(2M)

|P(ej!)|2 d!; (19a)

or to minimize

E∞ = max!∈[!s;�]

|P(ej!)|; (19b)

subject to

1− �16 |T0(ej!)|6 1 + �1

for !∈ [0; �] (20)

and

|T‘(ej!)|6 �2 for !∈ [0; �]; (21)

where 16 ‘6 (M − 1) and T‘(z) = (1=M)∑M−1k=0 Fk(z)Hk(zW

‘M ). EJcient prototype %lters

can be obtained using these techniques. However,the optimization problem involves the T‘(z) func-tions and the computational cost increases consid-erably when the number of channels M increasestoo.

In this subsection, we propose a new designtechnique that completes that proposed in [10].This technique is based on the fact that the 3 dBpoint of the prototype %lter amplitude responsemust be located at approximately ! = �=2M . Thiscondition must be satis%ed in order to guaranteethat:

(a) P2(z) is approximately a 2M th band linear-phaseFIR %lter;

(b) The frequency response of the prototype%lter satis%es approximately the power


complementary property, i.e.,

|P(ej!)|2

+|P(ej(�=M−!))|2 ≈ Mc; 06!6�=M;

|P(ej!)|2

+|P(ej(−�=M−!))|2 ≈ Mc; −�=M6!60;(22)

where c∈R+.

The proposed optimization problem is formulateddepending on the technique used to obtain the proto-type %lter. If we use the Parks–McClellan algorithm,the following optimization problem is considered: Theprototype %lter order N , relative error weighting, andthe stop-band edge are %xed before the optimizationprocedure is started, while the pass-band edge is ad-justed to minimize

#= ‖P(ej�=2M )| − 1=√2|: (23)

When we use the window technique to design theprototype %lter, the statement of the problem is formu-lated in the following way [10]: Before the procedureis started, the (N + 1)-length prototype %lter is ob-tained as expression (16) established, where w[n] isan (N +1)-length window and pi[n] is the shifted im-pulse response of an ideal low-pass %lter (Eq. (17)).Then, the ideal %lter cuto8 frequency !c is adjusted tominimize Eq. (23).In this way, and as the results obtained show,

it is possible to reduce amplitude distortion andcross-talk errors introduced into the TMUXsystem.

3.2. Transmitting and receiving 2lters

In this subsection, we put forward the modulationby which the transmitting and receiving sub-channel%lters are obtained. We also show how conventionalpseudo-QMF banks can be derived from the schemeof cosine-modulation proposed.Let p[n] be the prototype %lter (with real coeJ-

cients) designed by means of the method describedin the preceding subsection. We de%ne sk [n] signals

[25,12,11], 06 k6M − 1; 06 n6N − 1, as

sk [n] = 2p[n] cos((k +

12

)�Mn+ #k

): (24)

Terms #k are essential in order to ensure the reduc-tion of amplitude distortion and the cancellation ofthe most signi%cant aliasing terms (related with thecross-talk error in the TMUX systems [38,39,24].The transmitting sub-channel %lters fk [n] are ob-

tained by alternating sk [n] and its ‘Uipped’ versionssk [N − 1− n] as

fk [n] =

{sk [N − 1− n]; k even;sk [n]; k odd:

(25)

In order to guarantee that the TMUX system willalso be free from phase distortion, the receiv-ing sub-channel %lters are chosen according tohk [n] = fk [N − 1− n].Amplitude distortion is reduced when initial phase

factor #0 and %nal phase factor #M−1 verify

#0 =�4

(±(2r + 1)− 1

M(N − 1)

); (26)

#M−1 =�4

(±(2r + 1)−

(2− 1M

)(N − 1)

);

(27)

where r ∈Z [11]. With regard to aliasing error(cross-talk error in the TMUX system), the mostsigni%cant terms can be cancelled when cosine mod-ulation phase factors #k+1; 06 k6 (M − 2), satisfy

#k+1 =±(2m+ 1)�2− (k + 1)(N − 1)

�M

− #k;(28)

where m is any integer [12,11].Note that the proposed scheme of modulation, with

the appropriate phase factors #k , provides good %lterbanks or TMUX systems when we use a prototype%lter designed by means of the method described inthe preceding subsection or a spectral factor of a 2M thband %lter designed as proposed in [25,30].When the resulting prototype %lter has a linear-

phase response, we can obtain a conventionalpseudo-QMF bank from the proposed %lter bank byusing the following way. One possible choice for#k; 06 k6 (M − 1), which simultaneously satis%es


conditions (26)–(28) is

#k =�4

(1− (2k + 1)(N − 1)

1M

): (29)

Substituting Eq. (29) into Eq. (24), we get

sk [n] = 2p[n] cos

( (k +

12

)

× �M

(n− N − 1

2

)+�4

): (30)

The impulse responses of the receiving and the trans-mitting sub-channel %lters can be expressed as

hk [n] =

2p[n] cos((k +

12

)

× �M

(n− N − 1

2

)+�4

); k even;

2p[N − 1− n] cos((k +

12

)

× �M

(N − 1

2− n)+�4

); k odd;

(31)

fk [n] = hk [N − 1− n]: (32)

Using the relation p[n] = p[N − 1 − n], valid forType 1 or 2 linear-phase %lters—with a symmetricimpulse response—and the fact cos(−x)=cos(x), wecan rewrite expressions (31) and (32) as

hk [n] = 2p[n] cos

((2k + 1)

�2M

×(n− N − 1

2

)+ (−1)k

�4

); (33)

fk [n] = 2p[n] cos

((2k + 1)

�2M

×(n− N − 1

2

)− (−1)k

�4

); (34)

which correspond with conventional pseudo-QMFbanks.

4. Simulation results

In this section, the performance of the proposedTMUX systems is illustrated by several simulationexperiments. The simulated systems are 8-channel,16-channel, and 32-channel TMUX systems obtainedwith prototype %lters designed using the proposedtechnique and with the CMT [8] and KWA [27] meth-ods. We only consider %lters obtained using thesemethods because these techniques present objectivefunctions that require little computational e8ort to ob-tain the prototype %lter coeJcients, and the resultingpseudo-QMF banks work reasonably well. The resultsare presented and compared in Tables 1–3. Fig. 3shows the magnitude response plot for di8erent proto-type %lters, and Fig. 4 shows the magnitude responseof their resulting transmitting sub-channel %lters.In our experimental simulation we observe that the

systems obtained from the KWA technique have al-most the same performance as those designed from theproposed Kaiser window-based method. However, ifwe use the CMT technique to obtain the prototype %l-ter, the results are very di8erent from those obtainedwith the proposed Parks–McClellan-based method.We also %nd that if we use the window-based

technique, the convergence of the algorithm isalways ensured even if a high or low relation(prototype filter order)=(number of channels),i.e., (N − 1)=M , is needed. This good behaviour isnot ensured for the proposed Parks–McClellan basedtechnique.As regards amplitude distortion, note that a low

quote of this error in the %lter banks—measured by the�pp parameter—implies a low quote of inter-symbolinterference in the corresponding TMUX system. Inour design examples, the best results have always beenobtained by using the proposed technique with theBlackman window. Fig. 5 shows the magnitude re-sponses of the overall distortion transfer function forthe two best 8-channel, 16-channel, and 32-channelTMUX systems.On the other hand, we have included the parameterEa in Tables 1–3 in order to see in an experimentalway the relationship between the aliasing in the %lterbank and the cross-talk and the inter-channel interfer-ence in the TMUX system. Ea is the worst aliasingerror peak [36], and it is de%ned as the higher value ofthe aliasing function Tal(ej!) = (

∑M−1‘=1 |T‘(ej!)|2)1=2.


Fig. 3. Magnitude response of di8erent prototype %lter: (a) 128-length %lter (proposed technique (PT)—Kaiser window, 2 = 10:06);(b) 128-length %lter (KWA, 2 = 10:06); (c) 256-length %lter (PT—Parks–McClellan); (d) 256-length %lter (CMT); (e) 384-length %lter(PT—Blackman window); and (f) 384-length %lter (PT—Hamming window).


Fig. 4. |Fk (ej!)| for the transmultiplexers obtained with the following prototype %lters: (a) 128-length %lter (PT—Kaiser window, 2=10:06);(b) 128-length %lter (KWA, 2 = 10:06); (c) 256-length %lter (PT—Parks–McClellan); (d) 256-length %lter (CMT); (e) 384-length %lter(PT—Blackman window); and (f) 384-length %lter (PT—Hamming window).


Fig. 5. |T0(ej!)| (periodic �=M) for the transmultiplexers obtained with the following prototype %lters: (a) 128-length %lter (PT—Blackmanwindow); (b) 128-length %ller (PT—Parks–McClellan); (c) 256-length %lter (PT—Blackman window); (d) 256-length %lter (CMT);(e) 384-length %lter (PT—Blackman window); and (f) 384-length %lter (CMT).


Table 1Results obtained on 8-channel transmultiplexers (sub-channel %lters of 128-length). PT is proposed technique

Prototype %lter ICI (dB) emax �pp ISI (dB) �2ni �2!i �2ni · �2!i Ea(dB)

PT—Blackman window −96:9995 2:39E− 10 15E− 4 −66:7478 31.59 14:2E− 3 0.4485 −96:76PT—Hammimg window −50:1592 9:29E− 6 11E− 3 −44:1974 41.07 13:7E− 3 0.563 −50:16PT—Kaiser window −100:473 1:07E− 10 37E− 4 −56:3508 29.22 14:4E− 3 0.421 −100:47PT—Parks–McClellan −126:23 2:75E− 13 24E− 4 −57:8897 27.62 14:6E− 3 0.4033 −126:23KWA −100:322 1:11E− 10 37E− 4 −58:8473 29.23 14:4E− 3 0.4209 −100:32CMT −97:8582 1:72E− 10 24E− 4 −64:6197 29.1 14:4E− 4 0.419 −97:86

Table 2Results obtained on 16-channel transmultiplexers (sub-channel %lters of 256-length)



Table 3Results obtained on 32-channel transmultiplexers (sub-channel %lters of 384-length)



The measurements carried out in the %lter bank andthe dual TMUX systems demonstrates that the bestbehaviour with regard to the aliasing error in the %lterbanks does not imply the lower inter-channel interfer-ence or cross-talk errors in the corresponding TMUXsystems (as example, see Table 3).

5. Concluding remarks

In this paper, we have presented a method for ob-taining nearly perfect-reconstruction TMUX systemsbased on cosine modulated %lter banks. The pro-posed scheme of modulation group’s di8erent TMUX

systems based on pseudo-QMF cosine modulated %lterbanks, including the conventional pseudo-QMF basedbanks. All the transmitting and receiving sub-channel%lters are obtained from a single prototype %lter. Todesign this %lter, we proposed a new technique andshowed that it achieves similar performance to thedesigns obtained by the methods of [8,27]. Moreover,we noted in our experimental simulation that theconvergence of the proposed algorithm—using thewindow technique—is ensured.In order to illustrate the validity of the proposed

TMUX systems, we have presented several examplesshowing that good results can be obtained. The result-ing systems present low levels of amplitude distor-


tion and cross-talk errors, as well as a good spread intime and frequency domains. To measure these errorand characteristics, we have proposed several param-eters and we have explored the performance of sev-eral TMUX systems. The simulation results showedthat those TMUX systems designed with the proposedtechnique for the prototype %lter design work reason-ably well.

Acknowledgements

The authors wish to thank H. G,omez-Moreno ofthe University of Alcal,a for his valuable and con-structive suggestions. This work was supported inpart by CAM Grant 07T/0025/2001 and CICYT GrantTIC2001-0751-C04-02.

References

[1] A.N. Akansu, P. Duhamel, X. Lin, M. de Courville,Orthogonal transmultiplexers in communication, IEEE Trans.Signal Process. 46 (4) (April 1998) 979–995.

[2] A.N. Akansu, M.J. Medley (Eds.), Wavelet, Subband andBlock Transforms in Communications and Multimedia,Kluwer Academic Publishers, Norwell, MA, 1999.

[3] A.N. Akansu, M.V. Tazebay, R.A. Haddad, A newlook at digital orthogonal transmultiplexers for CDMAcommunications, IEEE Trans. Signal Process. 45 (l) (January1997) 263–267.

[4] M.G. Bellanger, On computational complexity in digitaltransmultiplexer %lters, IEEE Trans. Comm. 30 (July 1982)1461–1465.

[5] M.G. Bellanger, J.L. Daguet, TDM-FDM transmultiplexer:digital polyphase and FFT, IEEE Trans. Comm. COMM-19(October 1971) 628–634.

[6] L. Chen, K.-P. Chan, T.Q. Nguyen, X.H. Dai, Algorithm forthe design of transmultiplexers with time-frequency spreadcriteria, Electron. Lett. 36 (17) (August 2000) 1499–1500.

[7] B.-S. Chen, L.-M. Chen, Optimal reconstruction in multiratetransmultiplexer systems under channel noise: Wienerseparation %ltering approach, Signal Process. 80 (4) (April2000) 637–657.

[8] C.D. Creusere, S.K. Mitra, A simple method for designinghigh-quality prototype %lters for M-band pseudo-QMF banks,IEEE Trans. Signal Process. 43 (4) (April 1995) 1005–1007.

[9] R.E. Crochiere, L.R. Rabiner, Multirate Signal Processing,Prentice-Hall, Englewood Cli8s, NJ, 1983.

[10] F. Cruz-Rold,an, P. Amo-L,opez, S. Maldonado-Basc,on, S.S.Lawson, An eJcient and simple method for designingprototype %lters for cosine-modulated pseudo-QMF banks,IEEE Signal Process. Lett. 9 (1) (January 2002) 29–31.

[11] F. Cruz-Rold,an, P. Amo-L,opez, P. Mart,1n-Mart,1n, F.L,opez-Ferreras, Alternating analysis and synthesis %lters:a new pseudo-QMF bank, Digital Signal Process. 11 (4)(October 2001) 329–345.

[12] F. Cruz-Rold,an, F. L,opez-Ferreras, P. Amo-L,opez, J.D. Os,esdel Campo, Arbitrary-length spectral factor applied to thedesign of pseudo-QMF cosine-modulated %lter banks, IEEETrans. Circuits Systems II: Analog Digital Signal Process. 48(3) (March 2001) 321–325.

[13] European Telecommunications Standards Institute, Integratedservices digital network (ISDN) basic rate access digitaltransmission system on metallic local lines, ETSI ETR 080,July 1993.

[14] European Telecommunications Standards Institute, Trans-mission and multiplexing (TM); asymmetric digital subscriberline (ADSL); requirements and performance, ETSI ETR 328ed 1, 1996.

[15] European Telecommunications Standards Institute, Radioequipment and systems, high performance radio local areanetwork (HIPERLAN) Type 1, ETS 300–652, October 1996.

[16] European Telecommunications Standards Institute, DigitalAudio broadcasting (DAB); DAB to mobile, portable and%xed receivers, Norme ETSI, document ETS 300 401, 1995–1997.

[17] European Telecommunications Standards Institute, Digitalbroadcasting systems for television, sound and data services;framing structure, channel coding and modulation for digitalterrestrial, satellite and cable television, Norme ETSI,documents ETS 300 744, ETS 300 421, ETS 300 429, 1997.

[18] European Telecommunications Standards Institute,HIPERLAN Type 2 functional speci%cation, Part 1—Physicallayer, DTS/BRAN0300003-1, June 1999.

[19] E. Feig, Practical aspects of DFT-based frequency divisionmultiplexing for data transmission, IEEE Trans. Comm. 38(7) (July 1990) 929–932.

[20] N.J. Fliege, DFT polyphase transmultiplexer %lter banks withe8ective reconstruction, in: Proceedings of the EUPSICO-92,Vol. I, Brussels (Belgium), August 1992, pp. 235–238.

[21] R.A. Haddad, A.N. Akansu, A. Benyassine, Time-frequencylocalization in M-band %lter banks and wavelets, J. Opt. Eng.32 (7) (July 1993) 1411–1429.

[22] IEEE 802.11, Wireless LAN medium access control (MAC)and physical layer (PHY) speci%cations, November 1997.

[23] IEEE 802.11, Draft supplement to standard for telecommuni-cations and information exchange between systems—LAN/MAN speci%c requirements—Part 11: Wireless MACand PHY speci%cations: high-speed physical layer in the5 GHz band, P802.11a/D6.0, May 1999.

[24] R.D. Koilpillai, T.Q. Nguyen, P.P. Vaidyanathan, Someresults in the theory of crosstalk-free transmultiplexers, IEEETrans. Signal Process. 39 (10) (October 1991) 2174–2183.

[25] R.D. Koilpillai, P.P. Vaidyanathan, A spectral factorizationapproach to pseudo-QMF design, IEEE Trans. Signal Process.41 (1) (January 1993) 82–92.

[26] C.-W. Lin, B.-S. Chen, State space model and noise %lteringdesign in transmultiplexer systems, Signal Process. 43 (1)(April 1995) 65–78.


[27] Y.-P. Lin, P.P. Vaidyanathan, A Kaiser window approach forthe design of prototype %lters of cosine modulated %lterbanks,IEEE Signal Process. Lett. 5 (June 1998) 132–134.

[28] T. Liu, T. Chen, Design of multichannel nonuniformtransmultiplexers using general building blocks, IEEE Trans.Signal Process. 49 (l) (January 2001) 91–99.

[29] S.K. Mitra, C.D. Creusere, H. Babic, Perfect transmultiplexersusing IIR %lter banks, in: Proceedings of the EUPSICO-92,Vol. I, Brussels (Belgium), August 1992, pp. 223–226.

[30] T.Q. Nguyen, Near-perfect-reconstruction pseudo-QMF %lterbanks, IEEE Trans. Signal Process. 42 (1) (January 1994)65–76.

[31] H.J. Nussbaumer, Pseudo-QMF %lter bank, IBM Tech.Disclosure Bull. 24 (November 1981) 3081–3087.

[32] R.P. Ramachandran, P. Kabal, Bandwidth eJcienttransmultiplexers, Part 1: synthesis, IEEE Trans. SignalProcess. 40 (1) (January 1992) 70–84.

[33] R.P. Ramachandran, P. Kabal, Bandwidth eJcient trans-multiplexers, Part 2: subband complements and performanceaspects, IEEE Trans. Signal Process. 40 (5) (May 1992)1108–1121.

[34] J.H. Rothweiler, Polyphase quadrature %lters—a new subbandcoding technique, in: Proceedings of the IEEE InternationalConference on Acoustics, Speech, and Signal Processing,Boston, MA, 1983, pp. 1280–1283.

[35] T. SaramXaki, R. Bregovic, An eJcient approach for designingnearly perfect-reconstruction cosine-modulated and modi%edDFT %lter banks, in: Proceedings of the IEEE InternationalConference on Acoustics, Speech, and Signal Processing, SaltLake City, Utah (USA), May 2001.

[36] P.P. Vaidyanathan, Multirate Systems and Filter Banks,Prentice-Hall, Englewood Cli8s, NJ, 1993.

[37] P.P. Vaidyanathan, Filter banks in digital communications,IEEE Circuits Systems Mag. 1 (2) (Second Quarter 2001)4–25.

[38] M. Vetterli, A theory of multirate %lter banks, IEEE Trans.Acoust. Speech Signal Process. ASSP-35 (3) (March 1987)356–372.

[39] M. Vetterli, Perfect transmultiplexer, in: Proceedings of theIEEE International Conference on Acoustics, Speech andSignal Processing, April 1990, pp.1723–1726.

[40] A. Viholainen, J. Alhava, J. Helenius, J. Rinne, M. Renfors,Equalization in %lter bank based multicarrier systems,in: Proceedings of the IEEE International Conference onElectronics, Circuits and Systems, Pafos, Cyprus, September1999, pp. 1467–1470.

Documents

Design of Multi Channel Near Perfect Reconstruction