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`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 44. NO
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`IO, OCTOBER 1996
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`A Method to Reduce the Probability of Clipping in DMT-Based Transceivers
`Denis J. G. Mestdagh and Paul M. P. Spruyt
`
`Abstract-A new method allowing a reduction in the probabil-
`ity of clipping in discrete multitone (DMT)-based transceivers
`is described. The method does not use any kind of precod-
`ing and can easily be implemented within conventional DMT-
`transceivers. The main advantage of the proposed method is an
`improvement of system performance in terms of overall signal-
`to-noise ratios (SNR’s): with the simplest implementation option
`of the proposed method, up to about 8 dB improvement in the
`SNR as compared with previously reported brute force clipping
`methods can be achieved.
`
`I. INTRODUCTION
`HE discrete multitone (DMT) modulation technique is
`emerging as a very powerful technique for applications
`ranging from asymmetric digital subscriber line (ADSL), dig-
`ital audio broadcast (DAB) to interactive video on demand
`(IVOD) over CATV networks [l]-[3].
`A DMT signal is the sum of N independently quadrature
`amplitude modulated (QAM) signals each being carried over
`a distinct carrier frequency. The frequency separation of the
`N carriers is equal to 1/T where T is the time duration of a
`DMT symbol. The real part of the complex envelope of the
`generated DMT signal can be expressed as
`
`where r i denotes the QAM-phasor of carrier k (at frequency
`k / T ) of the m-th DMT symbol and (u)t is a rectangular
`transmit pulse of duration T .
`In a practical transceiver, the DMT symbol (1) is generated
`by means of an inverse fast Fourier transform (IFFT) on the
`complex phasors { r k ~ } ,
`IC E [0, N - 11 [4].
`Fig. 1 shows the instantaneous amplitude A(t) of two DMT
`symbols generated with two distinct sets of QAM-phasors
`and {~i}~,
`and N = 256. For both symbols, 16-
`QAM carrier modulation is assumed. A noticeable feature of
`the symbol in Fig. l(b) as compared with the one in Fig. l(a)
`is that it exhibits large amplitude spikes which arise when
`several frequency components add in-phase. These spikes may
`have a serious impact on the design complexity and feasibility
`of the transceiver’s analog front-end (Le., high resolution of
`D/A-A/D convertors and line drivers with a linear behavior
`over a large dynamical range). In addition, regulations can
`limit the peak envelope power or the probability of clipping
`[SI. The effect of amplitude clipping in DMT transceivers has
`Paper approved by S. Benedetto, the Editor for Signal Design, Modulation,
`and Detection of the IEEE Communications Society. Manuscript received
`March 31, 1995; revised September 11, 1995.
`The authors are with the Alcatel Corporate Research Center, Antwerp, B-
`20 18 Belgium.
`Publisher Item Identifier S 0090-6778(96)07387-4.
`
`been analyzed in the literature [6], [7] and methods based
`on encoding the input data in order to reduce the peak-to-
`average power ratio of the DMT signal have been proposed
`[SI, [9]. The coding methods, however, require an increase in
`data rate and hence a reduction of the energy per bit for the
`same transmit power, resulting in performance degradation in
`terms of information handling capacity of the communication
`system.
`In this letter, an alternative method is proposed. Since N is
`usually large (say N 2 128), A(t) can be accurately modeled
`as a Gaussian random process (central-limit theorem) with a
`zero mean and a variance o2 equal to the total signal power. Its
`probability density function (pdf), denoted as p(s), is given
`by 161
`
`Therefore, large amplitude spikes arise very rarely (thanks to
`statistical averaging) so that by applying a specific processing
`(no coding) only on DMT signals whose amplitudes exceed a
`one can obtain a DMT symbol stream
`given amplitude &lip,
`with almost no amplitudes exceeding Aclip.
`The paper is organized as follows. Section I1 presents the
`basics of the proposed method. In Section 111, the resulting
`improvement of system performance in terms of signal-to-
`noise ratio (SNR) is derived. Conclusions are reported in
`Section IV.
`
`11. THE PROPOSED METHOD
`The basic idea behind the proposed method can be described
`as follows. Assume that the maximum amplitude of the clipped
`DMT signal, ACllp. is chosen so that the probability of
`amplitude clipping is lower than a specified value. In the DMT
`transmitter, the symbols generated by the IFFT are analyzed by
`a peak detector which provides an indication of the presence
`or absence of amplitude clipping. According to this indication,
`two distinct actions are taken:
`Case a: If the amplitude of the DMT symbol never exceeds
`Acllp, then the symbol is sent to the transmitter front-end
`without any change.
`Case b: If the generated DMT symbol has at least one
`sample whose amplitude exceeds Acllp, then it is not passed
`directly to the transmitter front-end. Instead. the phasor of
`each QAM-modulated carrier is changed by means of a fixed
`phasor-transformation and a new DMT symbol is generated by
`the IFFT. By careful selection of the phasor-transformation,
`the probability of clipping this new symbol (second pass) will
`be very low. (The resulting overall clipping probability will
`be calculated later on.)
`
`0090-6778/96$05.00 0 1996 IEEE
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`0
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`T
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`Fig. 1. Instantaneous amplitude A(t) of two DMT symbols generated with two distinct sets of phasors { r i } ~ and { & } 2 .
`
`and = 256.
`
`The receiver at the far-end is informed about the application
`(or not) of the phasor-transformation at the transmitter, and
`applies the inverse transformation (Case b) or not (Case
`a) after demapping the QAM-modulated carriers. This extra
`information requires only one bit per DMT symbol. This
`bit could be provided by modulation of the pilot tone that
`otherwise carries no information and that is permanently used
`to maintain synchronization. Forward error correction coding
`andor duplication of this information over another or several
`tones can be envisaged to improve the reliability of this data
`recovery.
`Many phasor-transformations can be used. An easy-to-
`implement fixed random phasor transformation (known at the
`receiver) will be considered in what follows. Several other
`(more involved) transformations can be used as well without
`affecting the main results presented here.
`The overall probability of clipping with the "two-pass"
`method described above can readily be obtained using (2).
`The probability that a given sample in the DMT symbol has
`an absolute amplitude larger than AcliP (Aclip > 0) is simply
`
`given by
`
`where erf(t) is the error function defined by
`
`Assuming 2N independent samples per DMT symbol, the
`probability that the symbol must be clipped after the first pass
`(Le., at least one sample has an absolute amplitude larger than
`Aclip) is given by
`PClip/l = 1 - (1 - P y .
`(4)
`The validity of (4) has been confirmed with great accuracy
`(better than 1 %) by extensive computer simulations.
`We assume that due to the random phasor-transform (with
`large N ) , the probability that the symbol must be clipped
`after the second-pass, Pclip,2, is equal to Pclip/l. This is
`particularly the case if the transformation is a random bijection
`
`
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`IEEE TRANSACTlONS ON COMMUNICATIONS, VOL. 44, NO. IO, OCTOBER 1996
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`1 E+@
`1 E-01
`1 E-02
`1 E-03
`1 E-04
`.-
`0.
`‘p 1E-05
`5 1E-06
`L
`1 E47
`1 E-08
`1 E-09
`1 E-1 0
`1E-11
`2.5
`
`3
`
`3.5
`
`4
`
`4.5
`
`5
`
`Fig. 2. Probability of clipping the transmitted DMT signal as a function of
`
`r,
`for p = 1,2,nnd3
`
`on the set of constellation points. This means that the sets of
`constellation points belonging to the sub-ensemble of clipped
`symbols are equiprobably transformed into the whole ensem-
`ble of DMT symbols. As the decision is taken after the first
`pass, only clipped symbols are submitted to a second pass, so
`that the overall clipping probability is given by
`
`2
`(5)
`PClip/Total = rClip/l
`‘ pClip/2 = pClip/l.
`Again, extensive computer simulations have been carried
`out to confirm the validity of (5).
`Defining the dimensionless parameter p = Aclip/a and
`making use of (3), (4), and (5), we obtain
`
`we will restrict ourselves to the cases p = 0,1, and 2 .
`Extension of the present analysis to the case where p > 2 is
`straightforward.
`We consider an additive white Gaussian noise (AWGN)
`channel. Different noise sources will contribute to the overall
`SNR: a) at the transmitter: the clipping noise (only for p =
`1 or 2 ) and the quantization noise of the D/A convertor, and
`b) at the receiver; the AWGN and the quantization noise of
`the A/D convertor.
`We will assume a resolution of b b for the D/A and A/D
`convertors, and a quantization noise for p = 0 that is a factor
`01 lower than the AWGN. Therefore, without clipping (p = 0),
`the quantization noise of the D/A and A/D convertors and the
`AWGN are, respectively, given by
`
`In general, for a p-pass method, the overall probability of
`= P&ip/l, which tends rapidly
`clipping is given by P ~ l
`i ~ / ~ ~ ~ ~ l
`
`to zero as p increases.
`In practice, the number of passes ( p ) is limited by the speed
`of the IFFT at the transmitter. It is noteworthy, however, that
`e.g. the 2-pass method does not necessarily require a factor of
`2 increase of the (1)FFT speed. Indeed, since the probability
`of clipping after the first pass is low, say less than
`the
`speed of the IFFT should only be increased by a few percent
`provided that a DMT symbol buffer is used to absorb the
`delay incurred by the second pass.
`Note also that for p > 2, the method requires the transmis-
`sion of log, p b to inform the far-end receiver about the number
`of passes that have been applied to generate the DMT symbol.
`In practice, this additional control information is negligible
`compared to the data rate.
`Fig. 2 represents the clipping probability for the 1-pass, 2-
`pass, and 3-pass methods as a function of p. For example, with
`p = 4 the 2-pass and the 3-pass methods reduce the probability
`of clipping down to
`and less than 3.10W5, respectively.
`
`111. PERFORMANCE IMPROVEMENTS
`
`The system performance can be expressed in terms of the
`SNR for p = 0 (Le., no clipping) and q5 2 1. In this section,
`
`and where A,,,
`is the maximum amplitude of the DMT
`symbols. A,,,
`can be expressed as a function of the signal
`power cz and the crest factor v: A,,, = v . 0. The crest factor
`is fixed by N and the QAM constellation size according to [6]
`
`where L2 is the QAM constellation size (e.g., L = 4 for
`16-QAM).
`When clipping is applied 0, = 1 or 2 ), the quantization
`noise becomes
`
`The noise due to clipping has been derived in [6] for p = 1
`and is rewritten here for convenience
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`1237
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`50
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`45
`
`a @ e
`w 5 35
`
`30
`
`2.5
`
`3
`
`3.5
`
`4
`
`4.5
`
`5
`
`CI
`
`(a)
`
`9
`8
`7
`6
`= 5 e
`( 3 4
`3
`2
`1
`0
`2.5
`
`3
`
`3.5
`
`4
`
`4.5
`
`5
`
`I.r
`
`(b)
`(a) The SNR's (in dB) for IJ = 0 (no clipping), 1) = 1 dnd 1) = 2 d\ a function of hi for II = 2 , b = 1 2 and (b) the gain G (in dB) as
`Flg. 3.
`for u = 2 , b = 1 2 and b = 14
`a function of
`
`(1 1)
`
`where erfc(t) = 1 - erf(t).
`In order to calculate the clipping noise for p = 2, we
`first need to determine the pdf of A(t) for IAl >AcliI1, and
`then follow the same reasoning as in [6]. This pdf is readily
`obtained as
`
`(12)
`~ ' ( 2 ) pclip/l 'p(x) for 1x1 >Aclip
`where p(x) and Pclipll are given by (2) and (4), respectively.
`Therefore, the total power of the clipped portion of the DMT
`symbols for p = 2 is given by
`
`Making use of Amax/Aclip = v / p , the overall noise for
`p = 0, 1 and 2 can be expressed as
`N,,o = 2 . QO + AWGN = (2 + a ) . Q0
`
`(14.a)
`
`(The factor 2 in the right-hand side of (14) is due to
`the quantization noise of the D/A at the transmitter and the
`quantization noise of the A/D at the receiver.)
`The associated SNR's are obtained using (3), (4), (7), (1 l),
`(13), and (14) and their closed-form expressions are given by
`
`SNR,,o
`
`3.2''
`(2 + a ) . v 2
`
`SNR,=I ={ [?(:)'+Q]. ($)
`
`(15.a)
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`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 44, NO. IO, OCTOBER I996
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`(15.b)
`
`does not increase the actual transmission data rate. Significant
`improvements in SNR of about 3 dB up to 8 dB as compared
`to the case or brute force clipping can be achieved. This
`can be used at profit to reduce the required resolution of the
`D/A-AD convertors, to decrease the maximum amplitude of
`the transmitted signal, or to provide an extra signal-to-noise
`margin of the communication system.
`
`(15.c)
`
`The SNR’s (in dB) for p = 0, 1, and 2 derived from (15)
`are plotted in Fig. 3(a) as a function of p for a = 2, b = 12
`b and v = 25.9 [(9) with L = 4 and N = 2561. It is seen
`that, as compared with the p = 0 case, an improvement in
`SNR is obtained provided that p >_ 3.7 for p = I and p 2 3.4
`for p = 2. This improvement in SNR can be used at profit to
`reduce the required resolution of the D/A-A/D convertors as
`discussed in [6].
`The system performance improvement provided by the
`proposed method ( p = 2) in comparison with p = 1 can be
`characterized by the gain G in SNR: G = SNRp,2/SNRp,1.
`Fig. 3(b) shows the gain G (in dB) as a function of p for
`a = 2, b = 12 and b = 14. It is seen that G is bell-shaped and
`that its maximum value increases with b. The maximum gain
`is 3.4 dB for b = 12 and is as high as 7.9 dB for b = 14.
`Notice that for p = 2 the relevant values of G are only
`those associated with the values of p that satisfy the condition
`SNRP,2 2 SNR,,o. Fortunately, for a given b, the value of
`p that provides the maximum gain G is very close to the value
`of p that satisfies the condition SNRp,2 = SI?IR,,o.
`
`IV. CONCLUSION
`A method to decrease the probability of clipping DMT
`symbols by several orders of magnitude has been described.
`The method does not use any kind of pre-coding and hence,
`
`ACKNOWLEDGMENT
`The authors are very grateful to J.-F. Van Kerckhove for
`his computer simulations that have confirmed the validity of
`(4) and (5). They also wish to thank the reviewers for their
`detailed and valuable comments.
`
`REFERENCES
`
`[l] J. A. C. Bingham, “Multicarrier modulation for data transmission: an
`idea whose time has come,” IEEE Commun. Mag., pp. 5-14, May 1990.
`[2] R. Lassalle et al., “Principles of modulation and channel coding for
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`pp. 168-190, 1987.
`[3] T. de Cousanon et al., “OFDM for digital TV broadcasting,” Signal
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`[4] A. Ruiz et al., “Discrete multiple tone modulation with Coset coding
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`[SI Draft American National Standard for Telecommunications-Network
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`161 D. J. G. Mestdagh et al., “Effect of amplitude clipping in DMT-ADSL
`transceivers,” Elecrron. Lett., vol. 29, no. 15, pp. 1354-1355, July 1993.
`See also, D. J. G. Mestdagh et ul., “Analysis of clipping effect in DMT-
`based ADSL systems,” in Pruc. IEEE Int. Conf Commun. ICC’94. Mav
`1-5, 1994, pp: 293-300.
`171 Richard Gross el al., “SNR and soectral orooerties for a cliooed DMT
`. _
`ADSL signal,” in Proc. IEEE In; con$ ‘Cok“., 1 ~ ~ 2 9 4 : May 1-5,
`1994, pp. 843-847.
`[XI A. E. Jones et al., “Block coding scheme for reduction of peak to
`mean envelope power ratio of multicarrier transmission schemes,” IEE
`Electron. Lett., vol. 30, no. 25, pp. 2098-2099, Dec. 1994. See also,
`T. A. Wilkinson and A. E. Jones, “Minimization of the peak to mean
`envelope power ratio of multicarrier transmission schemes by block
`coding,” IEEE 45th Veh. Technol. Con$, Chicago, 1995, pp. 825-829.
`[9] S. J. Shepherd ez nl., “Simple coding scheme to reduce the peak factor
`in QPSK multicarrier modulation,” Electron. Lett., vol. 31, no.14, pp.
`1131-1132, July 1995.
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