DMT Signals with Low Peak-to-Average Power Ratio
`
`Mohamed Zekri, Patrick Boets and Leo Van Biesen
`Free University of Brussels (VUB), Dept. ELEC
`Pleinlaan 2, B-IO50 Brussels, Belgium
`E-mail: mzekri, pboets, Ivhiesen@,vuh.ac. be
`
`2. The Proposed Method to reduce PAR
`
`The basic characteristic of DMT is that the data are
`modulated on N QAM subcamers X , (k = 0, 1, . . . , N - 1)
`each of equal bandwidth. This frequency multiplexing can
`easily be implemented by using an Inverse Fast Fourier
`Transform (IFFT) in the modulator that transforms the
`subcarrier vector X = [X, ,XI,. . . , X N - , ]’ , with
`Xi =X;-,.+* for
`i = 2 , ..., N/2 and
`the
`t denotes
`transpose and * the complex conjugate, into the time
`domain. This results in the discrete time representation
`x = [xo,x,, . .. , xN-l ]’ with
`j2Ek,
`1 N-I
`n=0,1, ..., N-1
`x, =-
`f i k=O
`X k e
`The sequence xis real and is often referred to as a
`DMT symbol. It’s well known that for large N, the output
`time vector distribution can be well approximated by a
`Gaussian distribution and has a relatively large PAR
`defined as:
`[xmax I *
`
`(1)
`
`0,
`and 0, are respectively the maximum value of
`where x,,,
`x, and the root-mean square value of x over all possible
`symbols.
`The real sequence given by Eq. (1) can also be
`obtained by using a complex IFFT of the modulated
`carriers which are organized in the form of the following
`complex sequence Y(n) :
`
`N
`Y ( n ) = { X ( n ) for O < n < -
`2
`N
`for - - < n < N - l
`2
`
`10
`
`(3)
`
`PAR(X) =
`
`(2)
`
`Abstract
`This contribution proposes a new family of methods to
`reduce
`the Peak-to-Average power Ratio (PAR) in
`Discrete Multi-Tone (DMlJ and Orthogonal Frequency
`Division Multiplexing (OFDM) systems. This method
`avoids an use of extra
`Inverse Fast Fourier
`Transformations (IFFTs) as was done in some previously
`is based on
`published
`techniques, but
`instead
`transformations in the time domain of the original
`sequence available at the output of the IFFT processor.
`The improved statistics of peak power in the optimized
`iransmit signal are demonstrated by simulation results.
`
`1. Introduction
`
`Multi-carrier modulation has recently being used for
`many applications such as high-speed voice band modems
`and digital subscriber line systems (xDSL). These systems
`use DMT over wire media and OFDM for wireless
`communication.
`The amplitude distribution of a DMT signal with
`random input data is approximately Gaussian for large
`number of carriers. Therefore the DMT signal wilI
`occasionally present very high peaks. These peaks require
`a high dynamic range of the ADC (analog to digital
`converter) and analog front end in absence of any clipping
`or peak reduction technique. This would result
`in
`inefficient amplifiers (with excessive power dissipation)
`and expensive transceivers.
`Recently, there has been a variety of proposed methods
`to reduce PAR [2]-[6], but none of these methods is able
`to achieve simultaneously a large reduction in PAR with
`low complexity, without performing an extra IFFT
`calculations, without loss in data rate, without increase in
`the transmitter average power and without decrease in the
`minimum distance or equivalently without loss in the
`noise margin
`This paper proposes a new technique based on [l],
`which can achieve all these goals.
`
`0-7695-0250-4/99 $10.00 0 1999 IEEE
`
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`Herein, X ( n ) is a complex value representing the n’th
`modulated carrier of the multi-camer symbol to be
`
`transmitted, E i s the number of camers, and n is an
`2
`integer index. The IFFT of the complex sequence Y(n) is
`another complex sequence y(n) denoted as:
`
`A n ) = &I+ A n )
`(4)
`Herein, U(.) and b(n) are two real sequences of length N
`and j = f i . The output y(n) of the IFFT is a complex
`time domain sequence of length N whose real part U(.)
`will be selected for transmission by the real part selector
`REAL. It is further remarked that:
`x(n) = 2 4 1 ) = 2 Reb(n)]
`( 5 )
`represents the real IFFT of the multi-camer
`if&)
`symbol X ( n ) , i.e. the IFFT of the sub-camer complex
`conjugate vector built up from X ( n ) .
`When the real part of a sample U(.) of the complex
`time domain multi-camer symbol y(n) exceeds a certain
`threshold value, the time domain multi-camer symbol is
`subjected to a transformation. The threshold condition is
`verified by the observation device OBS which thereupon
`controls the transformation unit TRANSF to apply the
`transformation on y(n). Observing the time domain
`multi-camer symbol y(n) and transforming it is executed
`iteratively. The proposed transformation unit TRANSF is
`able to apply seven different transformations on the
`complex time domain multi-carrier symbol y(n) . These
`seven transformations will be discussed further. If none
`of the seven transformations produces a transformed time
`domain multi-camer symbol whose respective real part
`t h ) , t,(n), f 3 ( d t , ( d f s ( A f6(n)Or t,(n) consist
`of samples with an amplitude below the threshold value,
`a clip event occurs. This may cause harmonics and out-
`of-band radiation on the communication line. Moreover,
`the clipping will result into an increase of the binary error
`rate.
`In a first step, the transformation unit TRANSF
`decomposes the time domain multi-camer symbol y(n) in
`four partitions each containing information with respect
`to special subsets of carriers of the multi-camer symbol
`Y(n). Suppose thereto that the frequency domain multi-
`carrier symbol Y(n) would be decomposed into two
`partitions, the first partition Ye carrying the even
`numbered camers and the second partition Yocarrying
`the odd numbered camers. These partitions can
`mathematically be expressed as follows:
`
`Ye(n)=i(l+ejm).Y(n) for n = O , l , . . , , N - l
`
`(6)
`
`and
`
`whereby Y(n) = Ye (n)+ Y, (n) .
`Ye and Yo now can each be decomposed into two further
`partitions:
`
`j z n )
`Yo(n)=4(l+e
`.Ye(.)
`
`for n=0,1, ..., N-1
`
`(8)
`
`whereby Ye (e) = Yo (n)+ Y, ( n ) , and
`
`I
`
`jXLt
`
`~ , ( n ) = - ] + e 2 .q,(n) for n=0,1, ..., N - I ( I ~ )
`2
`whereby Y , (n) = Y, (n)+ Y3 (n).
`Using the Fourier transform properties, it can be
`verified that the time domain sequences corresponding to
`the previous frequency domain sequences Y e , Y, , Yo,
`Y, , Y, and Y, are given by:
`y,(n)= a,(n)+jb,(n), with
`
`(12)
`
`ae(n)=!-(a(n)+u(n+:))
`
`for n=O,l, ..., N-1, and
`
`2
`Uu(n)=L(~(fl)-u(n+:)’
`
`for n=O,l, ..., N - 1 , and
`
`and:
`
`yo (n) = a, (n)+ jb, (n), with
`
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`converter and converted into a transformed frequency
`domain multi-camer symbol, respectively T, (n) , T,(n),
`T3(n), T,(n), T,(n), T,(n) or T,(n), by the fast fourier
`transfomer FFT. The pilot carrier is monitored by the
`tone demodulator PILOT DMOD, and
`pilot
`the
`information demodulated from this pilot carrier is used to
`control the inverse transformation unit INV TRANSF so
`that the exact inverse transformation is applied to the
`transformed frequency domain multi-carrier symbol
`(n), i = 1,. . . ,7 . Since the inverse transformation is
`executed in the frequency domain, the transformations
`applied by the transformation unit TRANSF in the
`transmitter need to have a simple frequency domain
`equivalent
`to be usable
`in practice. When
`the
`transformation consists of a rotation proportional to the
`camer index and a fixed rotation equal for all carriers of
`partition, this condition is obviously satisfied.
`In the implementation of the presented technique,
`some specific phasor transformations are used for the
`inverse transformations. Such phasor transformations are
`defined by a phasor transformation vectorZ(n). In
`general this phasor
`transformation vector may be
`expressed as:
`Z(n)= A.eJpn with n = 0,1,. , . , N - 1
`(18)
`Herein, Ais a complex constant with unit magnitude in
`order to not decrease the minimum distance of the
`27r
`constellation, and 9, is any multiple of -proportional
`N
`to the index n, and can be enlarged by a fixed angle q 0 .
`In other words, pn can be written as:
`Pn =PO +"N
`2nM
`(19)
`With M being an arbitrary integer and po being a
`constant angle. Generally, since the time domain multi-
`carrier symbol y(n) is divided into four partitions y o (n) ,
`y , (n), y 2 (n) and y 3 (n) , four transformations can be
`defined: Z , (n) , Z , (n) , Z 3 (n) and Z, (n) . The multi-
`carrier symbol obtained at the output of the camer
`transformer FFT then can be expressed as:
`+ 2, (n) . q (n) +
`q n ) = 2, (n) . I&)
`Z h ) * y2 (4 +zh) *
`r, (4
`
`(20)
`
`and the time domain multi-camer symbol transmitted is
`given by:
`
`ti (n) = 2 Re[lFFT(Ti (n))]
`(21)
`In the example described below, only two phasor
`transformations Z, (n)
`and Z 2 (n) are used to perform
`the transformations:
`
`and:
`y, (n) = a, (n)+ jb, (n) , with
`
`b, (n)= '(. 2
`(n)-aa( n +:))
`
`for n = 0,1, .. . , N -1
`
`and:
`
`y 2 (n) = a2 (n)+ jb2 (n), with
`
`a 3 n
`
`( ) =- ;( u,(n)-b, ( n+- 7)) for n=0,1, ..., N-I
`b3(n)=- ( b,(n)+u, ( n+-
`
`for n=0,1, ..., N - 1
`
`d
`
`2
`
`364
`
`2
`4
`first
`transformation unit TRANSF
`the
`Thus,
`decomposes the time domain multi-camer symbol y(n)
`in 4 partitions y o ( n ) , y,(n), y 2 ( n ) and y 3 ( n ) .
`Afterwards, the real and imaginary parts of these
`partitions will be used to constitute a transformed time
`domain multi-camer symbol whose real part is denoted
` t 2 ( 4 3 t h ) , t 4 ( 4 7 f s ( 4 d n ) or 4).
`by d
`More specifically, the camers of a partition must be
`rotated over an angle which is an integer multiple of
`2K -radians,
`the integer multiple being proportional to the
`N
`camer index n, and will be rotated over a fixed angle
`equal for all camers of that partition before the partitions
`are re-combined into the transformed time domain multi-
`camer symbol with real parts ti (n), i = 1,. . . ,7 .
`One of the transformed real time domain symbols
`ti (n), i = 1,. . . ,7 is selected for transmission. It will be
`parallel to serial converted and transferred to the multi-
`camer
`receiver. An
`indication of
`the
`applied
`transformation is modulated on the pilot carrier by the
`pilot tone modulator PILOT MOD and added to the
`multi-camer symbol before transmission towards the
`receiver. Obviously, the applied transformation may be
`reported in another way to the multi-carrier receiver too.
`In the receiver, the transformed time domain multi-camer
`symbol is again parallelised by the serial to parallel
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`+ I- j
`I
`Z l ( n ) = e - for n even
`f i
`2
`j > 1 + j
`zl(n)=e - for n odd
`f i
`I
`and:
`
`I
`
`the
`by
`applied
`transformations
`seven
`The
`transformation unit TRANSF on the basis of the phasor
`transformations 2, (n) and 2, (n) , and corresponding
`inverse
`transformations
`applied by
`the
`inverse
`transformation unit INV TRANSF are listed hereafter.
`I. According to the first transformation, the real and
`imaginary parts of the four partitions y o ( n ) , y , ( n ) ,
`y 2 (n) and y 3 (n) are combined by the transformation unit
`TRANSF into the following transformed time domain
`multi-carrier symbol:
`
`sr
`2 U 2 ( n+- Y +
`
`tl(n)=2(ao(n)+a3(n))+
`
`If the corresponding transformed frequency domain
`multi-camer symbol,
`i.e.
`the multi-carrier symbol
`obtained at the output of the fourier transformer FFT in
`the receiver RX when
`t l ( n ) is supplied thereto, is
`denoted by TI (n) , the original frequency domain multi-
`camer symbol can be recovered from Tl(n) by the
`transformation unit INV TRANSF
`inverse
`if
`the
`following inverse transformation is applied:
`
`I,, (n) for n = 0,- N
`
`2
`Tl (n) for n E (0 mod 4,3 mod 4)and
`N
`O I n I - - l
`2
`for n E (2 mod4)and 2 5 n 5 -- N 1(25)
`x(n)={*
`2, (n)
`2
`T (4
`N
` n~ (1mod4)and 1 I n I - - 1
`L f o r
`2 2 (n)
`2
`CO
`
`N
`2
`f o r - c n < N
`
`Herein,
`the notation
`(kmod2)stands
`for
`the set
`(k, k + 2, k + 21, k + 32 ,... } , e.g. (0 mod 4) stands for the set
`(0,4,8,12 ,... } .
`11. According to the second transformation, the four
`partitions yo (n), y , (n) , y 2 (n) and y 3 (n) are combined
`by the transformation unit TRANSF into the following
`transformed time domain multi-camer symbol:
`
`If the corresponding transformed frequency domain
`multi-camer symbol is denoted by T2 (n), the original
`frequency domain multi-carrier symbol can be recovered
`from T,(n) by the inverse transformation unit if the
`following inverse transformation is applied:
`N
`‘2T2 (n) for n = 0,-
`2
`T, (n) for n E (0 mod 4,l mod 4)and
`N
`x(n)={m
`O I n < - - l
`2
`2, (4
`for n~ {2mod4)and2In<--l N
`2
`- T2 (.)
`N
`2 2 (4
`for n E (3 mod4}and3 I n I --1
`2
`10
`N
`
`f o r - - < n < N 2
`four
`the
`third
`transformation,
`the
`In
`111.
`partitions y o (n) , yl (n) , y 2 (n) and y 3 (n) are combined by
`the transformation unit TRANSF into the following
`transformed time domain multi-carrier symbol:
`
`(27)
`
`If the corresponding transformed frequency symbol is
`denoted by T3 (n) , the original frequency domain multi-
`carrier symbol can be recovered fromT3(n) by the
`inverse transformation unit if the following inverse
`transformation is applied:
`
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`

`c
`(0) for n = 0, -
`N
`2 ~ ,
`2
`* (4
`~ , ( n ) for n even
`
`N
`X ( n ) = { L for n~{lmod4}and1In<--l
`2
`Z , ( n )
`N
`22 (4
`T3(n) for n e {3mod4}and35nS--l
`2
`10
`
`N
`for - < n < N
`2
`IV. According to the fourth transformation, the four
`partitions yo (n), y , (n), y 2 (n)and y 3 (n) are combined
`into the following transformed time domain multi-camer
`symbol:
`
`(29)
`
`c
`N
`2 ~ , ( n ) for n = 0,-
`2
`T, (n) for n E (0 mod 4,l mod 4)and
`N
`O<n<--l
`2
`n E (2 mod4)and 2 I n I -- N 1 (33)
`2
`N
`n~{3mod4}and3In5--1
`2
`
`x(n)={*for
`z, (4
`T5 (4
`-for
`2, tn)
`10
`N
`for--<ncN
`2
`VI. In the sixth transformation, the four partitions
`yo ( n ) , yl ( 0 ) , y 2 (n)and y 3 (n) are combined into the
`following transformed time domain multi-camer symbol:
`
`At the receiver, the original frequency domain multi-
`carrier symbol can be recovered from the corresponding
`transformed frequency multi-camer symbol
`if
`the
`following inverse transformation is applied:
`‘2T4(n) for n =o,- N
`2
`T, (n) for n E (0 mod 4,3 mod 4)and
`
`The original DMT symbol X(n)can be recovered
`from the corresponding frequency symbol T6 (n) by
`applying the following transformation:
`c
`N
`2r6 (n) for n = 0,-
`2
`T6(fl) for n even
`
`N
`for n E (1 mod4)andl I n S --1
`2, (4
`2
`10
`N
`2
`V. In the fifth transformation, the four partitions
`yo (n), y , (n), y2 (n)and y 3 (n) are combined into the
`following transformed time domain multi-camer symbol:
`
`T (4
`N
`6 for n~{3mod4)and3InS--l
`4 (4
`2
`N
`to
`
`for - c n < N 2
`VII. The seventh transformation is obtained by a
`simple modification of the previous transformation, the
`corresponding time domain multi-carrier symbol is given
`by:
`+?\I
`t7 (n)= 2a, (n)+ &(- a, (n +3N)-b,
`8
`+ u3 (n + f ) + b3 ( n
`
`( n +?)
`
`(36)
`
`The original DMT symbol X(n)can be recovered
`from the corresponding frequency symbol T5 (n) by
`applying the following transformation:
`
`The original frequency domain DMT symbol X(n)can
`be recovered from the output of the fourier transformer
`FFT in the receiver, denoted T7 (n), corresponding to the
`input t7 (n)if the following transformation is applied:
`
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`

`N
`2T, (n) for n = 0,-
`2
`T7(n) for n even
`z, (4
`X(n)={-T,(")
`for n e {lmod4}and15n5--1 N
`2
`T (4
`N
`7 for n~{3mod4}and3In5--1
`z, (4
`2
`Lo
`
`N
`
`f o r - - < n < N 2
`
`3. Complexity and gain
`
`(37)
`
`%
`
`At the transmitter, one needs to compute a preliminary
`output to detect if the algorithm must be applied or not,
`this extra information is encoded into a pilot and send to
`the receiver. If the proposed algorithm was applied, the
`complexity at the transmitter measured in terms of extra
`number of additions and multiplications is limited by
`(5+3*3+2*3)N real additions for T, , T2 , T4 and T5 and
`(5+3*4+2*2)N real additions for T 3 , T, and T, and 2N
`real multiplications for each transformation. At the
`receiver, the original DMT symbol can be recovered by
`performing maximum
`complex multiplications at the
`output of FFT transformer.
`One can mention that the seven transformations
`described above are an example corresponding to the
`most complex case. There is still room to improve this
`technique by decreasing the implementation complexity
`as will be shown in the next section.
`Another remark is that the proposed algorithm is not
`restricted to ADSL systems. It can be used for all systems
`wherein a multi-camer modulation is applied with QAM-
`constellations of different size used in each camer.
`Figure 1 shows the clip probability versus the allowed
`PAR for an ADSL implementation using N=512 and 16-
`QAM in each camer. The curves denoted by R4 and R8
`correspond to Orckit method E71 when applying the
`original lFFTs and three or seven transformations
`respectively. Similarly, the curves denoted by S4 and S8
`correspond
`to applying
`the original
`IFFTs and
`transformations I, 11, I11 or the complete transformations
`respectively. The maximum gain in PAR introduced by
`the proposed technique is about 3.7 dB instead of 2.7 dB
`given in [7].
`The simulation results show that a given maximum
`clipping probability can be satisfied whose power level is
`at least 1.5 dB below the clipping power level obtained
`by the technique described in [7] corresponding to the
`same maximum clipping probability.
`
`367
`
`1 oo
`
`.
`
`2.
`\
`\ '
`
`10
`
`11
`
`12
`
`13
`
`14
`
`15
`
`Figure 1. Clipping probability
`
`4. Reducing the implementation complexity
`
`In this section we propose some modifications to
`reduce the complexity of the previous algorithm. As can
`be observed, in each transfonnation described above,
`there are two scale factors 2 and f i , in order to
`eliminate the factor f i , two possible solutions can be
`proposed:
`is to choose a suitable phasor
`The first one
`transformation 2, ( n ) and Z2 (n) as for example:
`
`j--n
`(j.e
`
`for n odd
`
`r
`jEn
`j.e 2
`,=
`for n odd
`(eJT"
`In this case, the number of multiplications at the
`transmitter will be reduced by 50% at the expense of an
`increase of the number of additions by an amount 2N per
`transformation.
`The second one is to rotate all the four partitions
`Y,(n), Y,(n), Y2(n)and Y,(n)by a specific phasor
`transformation Zi (n) as defined in Eq. (18) and then a
`new frequency sequence can be given by:
`
`and:
`
`Z2(.)={
`
`for n even
`
`(39)
`
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`

`[2] Denis J. Mestdagh and Paul P. Spruyt, “A Method to reduce
`the probability of clipping in DMT-based transceivers”, IEEE
`trans. on Communications, October 1996, Vol. 44, No 10, pp.
`1234-1 238.
`[3] Stefan H. Muller and Johannes B. Huber, “A Novel Peak
`Power Reduction Scheme for OFDM”, Proc. of Int. Symposium
`on Personal,
`Indoor and Mobile Radio Communications
`PlMRC’97, Helsinki, Finland, Sep. 1997, pp. 1090-1094.
`
`[4] Jose Tellado and John M. Cioffi, “PAR reduction with
`minimal or zero bandwidth loss”, ANSI Document, T1E1.4
`Technical Subcommittee, no. 98-173,Jun 1998, pp. 1-12.
`
`[5] Jose Tellado and J. Cioffi, “Revisiting DMT’s PAR”, VDSL
`ETSI TM6:TD08, Antwerp, Belgium, April 20-24, 1998.
`
`[6] M. Zekri, L. Van Biesen and P. Spruyt, “A New Peak Power
`Reduction Scheme for Multicarrier Modulation”, Proc. Of the
`IASTED Int. Conf on Networks and Communication Systems,
`Pittsburgh, PA, USA, May 13- 16, 1998, pp. 53-57.
`
`[7] Rami Verbin, “Efficient algorithm for clip probability
`reduction”, T1 El .4 Committee contribution number 97-323,
`Minneapolis, September 2 1-25, 1997.
`
`The complexity of this solution is exactly the same as
`in the previous case with a difference that the entire
`symbol now is scaled by a factor fi instead of2.
`The simulation results show that the improvements in
`PAR reduction given by this solution are almost the same
`given in figure 1.
`
`5. Conclusion
`
`A new method for clip probability reduction is
`proposed by reducing the PAR of multi-camer signal.
`This method is based on the shift property of an FFT in
`the time domain. The new time domain signals can be
`easily computed at the transmitter and the receiver can
`efficiently recover the original frequency symbol from
`the received frequency symbol. A gain of more than 3.7
`dB can be achieved with low complexity and redundancy.
`
`6. References
`
`[I] M. Zekri and L. Van Biesen, “Super Algorithm for Cli
`$
`Probability Reduction of DMT Signals”, Proceedings of the 13
`International Conference on Information Networking (ICOIN-
`13), Cheju Island, Korea, January 27-29,1999, Vol. 11, pp. 8B-
`3.1-3.6.
`
`368
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