Discrete Multitone ModulationUsing Wavelet
ÆNSC 88-2213-E-002-080 878188731 !"#$%&'($)*+ email: smp@cc.ee.ntu.edu.tw 1 ¡¢£ (DMT) Æ , Æ !"#
$ADSL channel capacity 8-9 dB%&.'(
Æ )*+,,-./DFTDMT.0DFT DMT1234'5 6.789:;,<= >?@ABC,DFTDMTD12 .0 >EC,12F DMTGHDFT DMTI J. KFL", (DMT),12FDMT.
Abstract. The DMT (discrete multitone
modu-lation) techniquehas beenwidely applied to data
transmissionoverfading channelsoftwistedpairs.
IthasbeenshownthattheDMTsystemwithideal
lterscan achievewithin 8to9dBof thechannel
capacityofADSL.TheDFTbasedDMTsystemis
proposed asapracticalDMTimplementation but
its optimality is never asserted. In this report we
will show that the DFT based DMT systems are
asymptoticallyoptimalalthoughtheyarenot
opti-malfor nitenumberofchannels. TheDFTbased
DMT system andthe DMT systemwith ideal
l-ters achievethesamebound. However,fora
mod-estnumberofchannelstheoptimaltransceivercan
provide substantial gainoverthe DFT based
sys-tem.
Keywords: transceiver,discretemultitone
modu-lation (DMT),optimalDMT.
2 ¤¥¦§¨
Recently there has been great interest in
apply-ingthediscretemultitonemodulation(DMT)
tech-niqueto high speeddatatransmissionoverfading
channels such as ADSL and HDSL [1][2]. Fig. 1
shows an M-channel DMT system over a fading
channelC(z)withadditivenoisee(n). Thechannel
isdividedintoM subchannelsusing the
transmit-ting lters F
k
(z) andreceiving ltersH
k
(z). The
inputis parsedand coded asmodulation
symbol-s, e.g.,QAM(quadrature amplitudemodulation).
Withjudiciouspowerandbitallocation,DMTcan
provide signi cant gain over fading channels. In
[3], Kaletshowsthat theDMT system with ideal
lterscanachievewithin 8to 9dB ofthechannel
capacityofADSL.
InthewidelyusedDFTbasedDMTsystem,the
transmittingand receiving lters are DFT lters.
For a given probability of error and transmission
power,bitscanbeallocatedamongthesubchannels
to achieve maximum total bit rate R
b;max . Very
highspeeddatatransmissioncanbeachievedusing
DFTbasedDMTsystematarelativelylowcost[1].
This technique is currently playing an important
roleinhighspeedmodemsforADSLandHDSL.
IntheDMTsystemthebitrateR
b;max
depend-s on the choice of the transmitting and receiving
lters. The useof moregeneral orthogonal
trans-mitting lters instead of DFT lters is proposed
in [4]. From the view point of multidimentional
signal constellations it is shown that, for AWGN
fading channels the optimal transmitting and
re-ceiving ltersareeigenvectorsassociatedwiththe
channel. Howeverin ADSLorHDSLapplications,
thechannelnoiseisoftenthecoloredNEXT noise
duetocrosstalk[3]. Forfadingchannelswith
gen-eralcolorednoisesource,theoptimaltransceiveris
derivedin [5]. Theoptimal transceiver
decompos-esthechannelintoeigenchannelsbyincorporating
thechannelfrequencyresponseandthenoisepower
spectrum.
practical DMT implementation but its optimality
is not asserted. In this report we will show that
the DFT based DMT systems are asymptotically
optimal. TheperformanceoftheDFTbasedDMT
systemsbecomesclosetothatofoptimalDMT
sys-tems when the channel number M is suÆciently
large. Furthermore theasymptoticperformanceof
these twosystemsisthesameasthatoftheDMT
systemwithideal ltersin[3]. AlthoughtheDFT
basedDMTsystemisasymptoticoptimal,the
op-timaltransceiverprovidessigni cantgainoverthe
DFT based systemfor a modestnumber of
chan-nels. AnexamplewithNEXT noisesourcewill be
givento demonstratethis.
3 ©ª¦«¬
ConsiderthesystemmodelofanM-channelDMT
transceiverovera fading channel C(z) with
addi-tivenoisee(n)inFig.1. SupposethechannelC(z)
isanFIR lterwithorderL,whichisareasonable
assumptionafterchannelequalization. Inpractice,
to cancel ISI (inter-symbolinterference) some
de-gree ofredundancy isintroduced andthe
interpo-lation ratioN >M. UsuallywehaveN=M+L.
Thelengthofthetransmittingandreceiving lters
isalsoN.
Polyphase representation.
TheDMT systemcan beredrawnasin Fig.2using polyphasedecompo-sition [2]. The transmitter G is an N M
con-stant matrix; the kth column of G contains the
coeÆcients of the transmitting lter F
k
(z). The
receiverS is an MN constant matrix; the kth
rowof S contains the coeÆcientsof thereceiving
lterH
k
(z). ThematrixC(z)isanNN pseudo
circulantmatrix[6] withthe rstcolumn givenby
c 0 c 1 c L 0 0 T where fc n g L n=0
is the channel impulse response.
TheconditionforzeroISIbecomesSC(z)G=I.
DFT based DMT systems.
IntheDFTbasedDMT system, the transmitting and receiving lters areM 2: The polyphase representation of the DMT
system.
DFT lters. Redundancy takestheform ofcyclic
pre x. De ne Was theMM DFTmatrixwith
[W ] mn = 1= p Me j2mn=M , for 0 m;n < M.
Onecanverifythat DFTbased DMTsystem has
transmitterandreceivergivenby
G=[WW 1 ] y ; S= 1 [0W ]; (1) where W 1
is asubmatrix ofW that contains the
rst L columns of W and is the diagonal
ma-trixdiag(C 0 ;C 1 ;;C M 1 )withfC k g M 1 k =0
denot-ingtheM pointDFTofc
n .
Zero ISI DMT systems.
Using singularvalue de-composition,wecandecomposeC0 as, C 0 =[U 0 U 1 | {z } U ] 0 NM V T =U 0 V T ; (2)
whereUandVareNNandMMunitary
ma-trices. ThecolumnvectorsofUandVare
respec-tively the eigenvectorsof C
0 C T 0 and C T 0 C 0 . The
matrixisdiagonalandthediagonalelements
k
arethesingularvaluesofC
0 .
Considerthecasethat thetransmitter is a
uni-tarytransformationfollowedbypaddingofLzeros,
inparticular G= G 0 0 ; (3) where G 0
is an arbitraryMM unitary matrix.
ForzeroISI,wecanchoose
S=G T 0 V 1 U T 0 : (4)
When thetransmitter is chosenasG 0 =V , the receiver is S= 1 U T 0
. This becomes the DMT
systemdevelopedin [4].
TransmissionPower
ForagivenaveragebitrateR
b
,thedesignofthe
transmitterandreceivera ectstherequired
trans-mission power. Let R
N
be the N N
autocor-relation matrix of the channel noiseprocess e(n).
TheM1outputnoisevectorof thereceiverhas
autocorrelationfunctiongivenby
b R=SR N S T :
Letthenumberofbitsallocatedtothek-th
chan-nel be b
k
, then the average bit rate is R
b = 1 N P M 1 k =0 b k
: The actual bit rate is 1
T R
b
, where
T is the sampling period of the system. Let
P(R
b ;P
e
;M) be the transmission power required
for the M channel transceiverto achieve an
aver-agebitrateofR
b
andprobabilityoferrorP
e . With
optimal bit allocation, thetransmission powerfor
the giventransceiverisminimized and is equalto
[5] P(R b ;P e ;M)=c2 2R b N =M ( M 1 k =0 [SR N S T ] k k ) 1=M ; (5)
wheretheconstantcdependsonthegiven
probabil-ityofsymbolerrorP
e
andthemodulationscheme.
In theDFT based DMTsystem, thereceiveris
S= 1
[0W ] asgivenin (1). Inthiscasewecan
verifythattransmissionpoweris
P DFT (R b ;P e ;M) = c2 2R b N =M M 1 k =0 [WR M W y ] k k 1=M det( y ) 1=M :
From (5) wesee that the transmission power can
befurther minimized byoptimizingthe
transceiv-er. Using the optimal transceiver, the minimum
transmissionpoweris[5] P opt (R b ;P e ;M)=c2 2RbN =M det (U T 0 R N U 0 ) 1=M det( 2 ) 1=M :
Asymptotic Performance In the following, we
will show that the DFT based DMT systems are
asymptotically optimal although they are not
op-timal for nite number of channels. For a given
error probability and bit rate, we will show that
thepowerrequiredinDFTbasedDMTsystem
ap-proaches that of theoptimal system for large M.
Inparticular, lim M!1 P opt (R b ;P e ;M) = lim M!1 P DFT (R b ;P e ;M) = c2 2Rb exp Z ln S ee (e j! ) jC(e j! )j 2 d! 2 : (6)
Note thatthis is thesame bound achievedby the
DMTsystemwithideal ltersasderivedin[3]. The
proofcanbedonein twosteps.
Step1: Usingthedistributionofeigenvaluesfor
Toeplitzmatrices[7], weareabletoshowthat
lim M!1 det( 2 ) 1=M = lim M!1 det( y ) 1=M =exp Z lnjC(e j! )j 2 d! 2 ; (7) where C(e j!
) is theFouriertransformof c
n . The
proofoftheaboveequationcanbefoundin [5].
Step2: Usingpropertiesofpositivede nite
ma-trices,wecanshowthat
lim M!1 det(U T 0 R N U 0 ) 1=M = exp Z lnS ee (e j! ) d! 2 : (8)
Ontheotherhand,propertiesofToeplitzmatrices
giveus [8], lim M!1 M 1 k =0 [WR M W y ] k k 1=M = exp Z lnS ee (e j! ) d! 2 : (9)
Withtheequalitiesin(7)-(9),wecanestablish(6).
Proof of (8): Note that the matrix U T 0 R N U 0
istheMM leadingprinciplesubmatrixof P=
U T
R
N
U, where U is as de ned in (2). Let the
eigenvalues of P be ordered as 0 1 N 1
. Usingtheinterlacingpropertyofeigenvalues
for positivede nite matrices [9], it canbe shown
thatdet(U T 0 R N U 0
)isboundedbetweenthe
prod-uctof theM largesteigenvaluesand the product
oftheM smallesteigenvalues,i.e.,
0 1 M 1 det (U T 0 R N U 0 ) L L+1 N 1 :
SupposethepowerspectraldensityS
ee (e
j!
)ofthe
channel noise has minimum S
min
> 0 and
max-imum S
max
< 1. Then these eigenvalues are
bounded betweenS min andS max , inparticular, S min 0 1 N 1 S max :
It followsthat det(U T 0 R N U 0 ) N 1 k =L k = detP L 1 k =0 k detP L 0 detP S L min det(U T 0 R N U 0 ) M 1 k =0 k = detP N 1 k =M k detP L N 1 detP S L max
Combiningtheabovetwoequalities,wehave
detP S L max det(U T 0 R N U 0 ) detP S L min : (10)
Also observe that detP = detR
N
. The matrix
R
N
isToeplitzanditistheNN autocorrelation
matrixofS ee (e j! ). Itisknownthat [10] lim N!1 ( detR N ) 1=N =exp Z lnS ee (e j! ) d! 2 : LettingM goto1in (10),wearriveat(8). 44
NotethattheDMTsystemdevelopedin[4]does
notachievethisboundasymptotically. Toseethis,
letC(z)=1,thenthetransmitterandreceiverare
identitymatrices. Thecodinggainofthesystemin
[4]isoneregardlessofthenumberofchannels. On
the other hand,the coding gain corresponding to
theasymptoticboundin(6)isalwaysgreaterthan
oneifthechannel noiseisnotwhite.
Example.
Suppose the channel C(z) is an FIR lter of order 1 and C(z) = 1+0:5z1
: For the
sameprobabilityoferrorandsamebitrate,Fig.3
shows Popt(R b ;Pe;M) PD F T (Rb;Pe;M)
,theratioofpowerneededin
optimalsystemoverthepowerneededinthe
DFT-based system. We plot the ratio as afunction of
M fortwodi erentnoisesources,theAWGNand
NEXTnoisesource,whichiscoloredchannelnoise
due tocrosstalk[3].
FromFig. 3wesee that,for bothnoisesources
the ratio Popt(Rb;Pe;M) P D FT (R b ;P e ;M)
approaches unity as the
channel numberM increases. But for the NEXT
noisechannel,theratioapproachesunitaryonlyfor
verylargeM. Wecanseethatforamodestnumber
ofchanneltheoptimalsystemprovidessubstantial
gain.
Ư§
[1] P. S. Chow, J. C. Tu, and J. M. CioÆ,
\Performance Evaluation of a Multichannel
Transceiver System for ADSL and VHDSL
Services," IEEE J. Select. Areas Commun.,
Aug.1991.
[2] A.N.Akansu,et.al.,\OrthogonalT
ransmulti-plexersinCommunication: AReview,"IEEE
Trans.SP,April1998.
20
40
60
80
100
120
0.4
0.5
0.6
0.7
0.8
0.9
1
M
AWGN
NEXT noise source
M3: Theratioofthepowerneededintheoptimal
DMTsystemoverthepowerneededinDFTbased
system for the same probability of error and the
samebitrate.
[3] I. Kalet, \Multitone Modulation," in
A. N. Akansu and M. J. T. Smith, Eds.,
Subband and Wavelet Transforms: Design
andApplications, Boston,MA:Kluwer,1995.
[4] S. Kasturia, J. T. Aslanis, and J. M. CioÆ,
\VectorCodingforPartialResponse
Channel-s,"IEEETrans.Inform. Theory,July1990.
[5] Yuan-Pei Lin and See-May Phoong, \Perfect
Discrete Multitone Modulation withOptimal
Transceivers,"submittedtoIEEETrans.SP.
[6] P. P. Vaidyanathan, Multirate Systems and
Filter Banks,Prentice-Hall,1993.
[7] R. M. Gray, \On the Asymptotic
Eigenval-ue Distribution of Toeplitz Matrices," IEEE
Trans.InformationTheory, Nov.1972.
[8] A. Gersho and R. M. Gray, Vector
Quanti-zation and Signal Compression, Kluwer
Aca-demic Publishers,Boston,1991.
[9] R. A. Hornand C.R. Johnson,Matrix
Anal-ysis, CambridgeUniversityPress,1985.
[10] N. S. Jayant and P. Noll, Digital Coding of
