Turbo Multiuser Detection for Coded DMT VDSL Systems

Turbo Multiuser Detection for Coded DMT VDSL Systems∗ Huaiyu Dai, Student Member, IEEE, and H. Vincent Poor, Fellow, IEEE Department of Electrical Engineering, Princeton University Princeton, NJ 08540 Tel: (609)258-4634 Fax: (609)258-1560 Email: [email protected], [email protected]

ABSTRACT In recent years, iterative processing techniques with soft-in/soft-out (SISO) components have received considerable attention. Such techniques, based on the so-called turbo principle, are exemplified through turbo decoding, turbo equalization and turbo multiuser detection. In this paper, turbo multiuser detection is applied to a discrete multitone (DMT) very-high-rate digital subscriber line (VDSL) system to combat crosstalk signals and to obtain substantial coding gain. The proposed iterative DMT receiver is shown to achieve an overall 7.0 dB gain over the uncoded optimum receiver at a bit error rate of 10 −7 for a channel with severe intersymbol interference and additive white Gaussian noise and with one dominant crosstalk signal. Impulse noise is detrimental to the proposed scheme but can be overcome through erasure decoding techniques, as is shown by example.

Index Terms Coded DMT, Crosstalk, Gray coding, Impulse noise, Multiuser detection, Turbo decoding, VDSL.

∗ Manuscript received Jan. 5, 2001; revised May 23 and June 28, 2001. This research was supported by the National Science Foundation under Grant CCR-99-80590

I.

Introduction*

Digital Subscriber Line (DSL) technology provides transport of high-bit-rate digital information over telephone subscriber lines. Various DSL techniques (Basic Rate ISDN, HDSL, ADSL, and VDSL) involving sophisticated digital transmission schemes and extensive signal processing have recently become practical due to advances in microelectronics. The latest in DSL technology is very-high-rate DSL (VDSL), which provides tens of megabits per second to those customers who desire broadband entertainment or data services. At such high rates, signals on twisted pairs can be reliably transmitted at most to a few thousand feet. Thus, VDSL will primarily be used for loops fed from an optical network unit (ONU) or a central office (CO) to a customer premises, i.e., it addresses the socalled “last mile” problem. The modulation scheme for VDSL can either be multicarrier-based or single carrierbased, typically discrete multitone (DMT) and carrierless amplitude/phase modulation (CAP)/quadrature amplitude modulation (QAM). The duplexing methods can be either time-division duplexing (TDD) or frequency-division duplexing (FDD) [9], [20].

Intersymbol interference (ISI) is one of the major obstacles to high-data-rate, bandwidth-efficient communications. Multicarrier modulation (MCM), following Shannon’s optimum transmission suggestion, achieves the highest performance in channels with ISI. DMT is a particular form of MCM that has been found to be well suited for DSL application and is adopted in ANSI T1.413 ADSL standards. With this approach, a channel is divided into many independent ISI-free subchannels in the frequency domain, and power and bits are allocated adaptively according to the channel characteristics [8], [20]. The advantages of using DMT for VDSL include optimality for data transmission, adaptivity to changing environments and flexibility in bandwidth management.

Normally VDSL signals occupy the band 300 KHz to 30 MHz within the twisted-pair bandwidth, and are separated from POTS/ISDN signals by splitter devices. Noise on a phone line usually occurs because of imperfect balance of the twisted pair. There are many types of noises that couple through imperfect balance into the phone line, the most common of which are crosstalk noise, radio noise and impulse noise. While the radio noise problem can be solved or at least alleviated by restricting VDSL transmission within radio bands, crosstalk and impulse noise are two *

This paper was partly presented at the 2001 Conference on Information Sciences and Systems, held at The Johns Hopkins University, Baltimore, MD, Mar. 21-23, 2001.

1

principal sources of degradation in VDSL transmission systems. The traditional single-user detector (SUD) for such systems merges crosstalk into the background noise, which is assumed to be white and Gaussian. Actually, crosstalk is the result of the sum of several filtered discrete data signals. Its distribution deviates from Gaussian, and its power spectral density (PSD) is significantly greater than that of background additive white Gaussian noise (AWGN). Recent research has explored the nature of crosstalk signals and has shown the potential benefits of robust multiuser detection to jointly mitigate crosstalk and impulsive noise for contaminated VDSL signals [7], [10].

Coding is a common way to reduce the gap in channel capacity experienced by uncoded systems. A concatenated coding scheme consisting of an inner trellis code (a 4-D Wei’s code) and an outer Reed-Solomon (RS) code was proposed for ADSL DMT systems to provide a 5 dB coding gain at bit error rate (BER) 10-7 without bandwidth expansion [26], [27]. There are two problems with this approach. First, since the constellation size varies from tone to tone, a time-varying trellis-coded modulation (TCM) encoder is required. Second, further improvement is very difficult from a practical implementation perspective because of the complexity of Viterbi decoding for multidimensional TCM. More recently, powerful turbo coding has been proposed for DMT systems [2], [13], [16], [22]. One typical turbo code is used to code across all subchannels. The coded bits are then interleaved and allocated to various tones for quadrature amplitude modulation. Thus, a single standard binary decoder can be employed at the receiver and further improvement in turbo coding is easily incorporated. A coding gain of 6.0 dB for bandwidth efficiency of 2 bits/s/Hz and 4.1 dB for 3 bits/s/Hz was reported for a channel with severe ISI at BER 10-5 [16].

In recent years, iterative processing techniques with soft-in/soft-out (SISO) components have received considerable attention. The basic idea is to break up optimum joint signal processing, e.g. concatenated decoding, joint equalization and decoding, or joint decoding and multiuser detection (MUD), which are typically very complex and require large amounts of memory, into separate components, iterating between them with the exchange of probabilities or “soft” information. This approach typically results in almost no loss of information. This so-called turbo principle is exemplified through turbo decoding [15], turbo equalization [11] and turbo multiuser detection [18], [25]. An iterative decoding technique, called soft cancellation, was used to mitigate the effect of home-LANs on uncoded VDSL systems in [6]. In this paper, we consider the application of turbo multiuser detection in a coded DMT VDSL system to combat crosstalk and to obtain substantial coding gain. We also consider the effects of

2

impulse noise, which has been found to greatly impact the performance of our proposed receiver, and an erasure decoding technique is proposed as a remedy.

The paper is organized as follows. In Section II a signal model for the DMT VDSL communication system is described, together with the iterative receiver structure for demodulation (with multiuser detection) and channel decoding. In Section III, we describe MUD-based schemes for DMT VDSL signal detection, while in Section IV details of the turbo decoding process are given. Simulation results are given in Section V, and Section VI concludes the paper.

II.

System Description

We consider a convolutionally encoded DMT system with crosstalk as shown in Fig. 1. The information bits d are first encoded into coded bits b with a standard binary convolutional encoder with code rate R. A code-bit interleaver is used to decorrelate the noise on the coded bits at the input of the channel decoder. The interleaved bits are optimally allocated to N subchannels and mapped to QAM signals of various constellation sizes. Then the conjugate-symmetric vector of length N = 2 N is transformed using the inverse fast Fourier transform (IFFT) to get a real time-domain vector. After parallel-to-serial and digital-to-analog conversion, the DMT VDSL signal x (t ) is transmitted into the channel, where it is corrupted by additive coupled crosstalk signals and background noise. At the receiver end, after analog-to-digital and serial-to-parallel conversion, the received signal r (t ) is transformed back to the frequency domain using an FFT, where it can be written as M

Yi = H i ⋅ X i + ∑ Fi ,m ⋅ Ci ,m + Ei , i = 1,..., N , m =2

(1)

where for the ith subchannel, H i is the channel gain, X i is the transmitted (complex) DMT symbol, C i ,m is the mth crosstalk signal, m = 2 ,

, M , Fi ,m is the corresponding crosstalk coupling function, and Ei is the

background noise. In this paper, we will primarily assume that the noise Ei in (1) is Gaussian in order to focus on the issue of crosstalk reduction. However, in Section V, we will consider a non-Gaussian model briefly in order to treat the effects of impulse noise. Output values of the FFT are fed into the demodulator and decoder for further

3

processing. Note that when the VDSL signal and crosstalk signals are asynchronous (almost always the case), the crosstalk coupling functions can vary in time. However, our model considers only one DMT block at a time so we omit the time index for simplicity. In practice, the crosstalk coupling functions can be estimated for each block to allow the application of our models.

Figure 2 shows the turbo structure for iterative demodulation and decoding. It consists of two stages: a soft metric calculator (the demodulation stage) and a SISO channel decoder (the decoding stage). The two stages are separated by an interleaver and a de-interleaver. The crosstalk signals are first detected via a multiuser detection technique, discussed in Section III. Then a channel log-likelihood ratio (LLR) for the kth bit carried by the ith subchannel symbol is calculated as follows:

Λ 1 (bk ,i ) = log

P(bk ,i = 1 | {r (t )}) P(bk ,i = −1 | {r (t )})

,

(2)

where {r(t)} is the received waveform as shown in Fig. 1(c). Using Bayes’ formula and discarding the common term p ({r (t )}) , (2) can be written as

Λ 1 (bk ,i ) = log

p({r (t )} | bk ,i = 1) P (bk ,i = 1) + log , p({r (t )} | bk ,i = −1) P(bk ,i = −1)

λ1 ( bk ,i )

(3)

λ2 p ( bk ,i )

where the second term λ 2p ( b k ,i ) represents the a priori LLR delivered from the decoding stage in the previous iteration. For the first iteration, this term is set to zero if we assume equally likely coded bits. The first term λ1 (bk ,i ) , denoting the extrinsic information obtained from the demodulation stage about the bit bk ,i , is then de-interleaved and sent to the channel decoder as its a priori information. Similarly, the SISO channel decoder computes the a posteriori LLR of each code bit and then excludes the influence of a priori knowledge to get extrinsic information from the decoding stage about the bit b j as follows:

λ2 (b j ) = Λ 2 (b j ) − λ1p (b j ) = log

P(b j = 1 | decoding) P(b j = −1 | decoding)

− λ1p (b j ) ,

(4)

where b j is the de-interleaved version of bk ,i , alternatively the coded bits before the interleaver in Fig. 1(a). The above factorization is derived in Section IV. Again, this extrinsic information is interleaved and fed back to the

4

demodulation stage as a priori knowledge for the next iteration. At the last iteration, the SISO decoder also computes the a posteriori LLRs for information bits, which are used to make final decisions. More details on this turbo decoding process will be given in Section IV.

III.

Mitigation of Crosstalk via Multiuser Detection

As we mentioned in Section I, it is possible to apply multiuser detection to jointly detect the VDSL signal and the crosstalk signals and thereby to greatly improve the system performance. According to the system model given in Fig. 1, the optimal maximum likelihood multiuser detector (ML-MUD) for Gaussian noise is one that estimates the VDSL input and crosstalker inputs in unison so as to minimize the distance between the channel output received signal and all the possible discrete waveform outcomes. Although it is possible that the crosstalk signals are incorrectly estimated, the probability of erroneous selection of the desired VDSL signal will be lower for such a detector than if the crosstalk signals are merely absorbed into the background Gaussian noise for detector design. We would expect a greater improvement in performance using multiuser detection when the difference between the power spectral density level of the crosstalk signals and that of background noise is larger. There are two types of crosstalk, the so-called near-end crosstalk (NEXT) caused by signals traveling in the opposite direction as the signal of interest, and far-end crosstalk (FEXT) caused by signals traveling in the same direction as the signal of interest. Generally speaking, crosstalk strength increases with frequency: NEXT with f 1.5 and FEXT with f 2 . Fortunately, FEXT experiences the same line attenuation as the desired signal; but unfortunately, NEXT does not. For VDSL systems, high-frequency NEXT is the most detrimental type of crosstalk, and consequently is also the most promising for reduction via MUD. A typical background noise level in VDSL transmission is –140dbm, while the typical NEXT is –90 ~ –110dbm; thus we can expect substantial gain from multiuser detection relative to traditional single user detection in this situation. Moreover, in DMT VDSL subchannels where there are substantially stronger crosstalk signals (typically in the high frequency bands on long loops), the so-called "near-far" problem of wireless code-division multiple-access (CDMA) systems, single-user detection (SUD) will fail to work properly while optimal MUD will essentially achieve the single-user lower bound. Note that the crosstalk signals in DSL transmission are of various types and cannot be represented under a uniform framework, to the best of the authors’ knowledge. In our application of MUD to signal detection in DSL systems, we deal mainly with NEXT of other

5

types (in contrast to the self NEXT coming from the phone lines carrying the same VDSL service) for the following reason. FEXT experiences the same line attenuation as the desired signal while NEXT does not, which makes NEXT the most detrimental type of interference, especially at high frequencies. Self NEXT can be largely alleviated by duplexing methods that separate the upstream and downstream data in time or frequency. Therefore, the other-type NEXT provides the best opportunity for performance gain. Nevertheless, although we consider other-type NEXT, multiuser detection is a valid technique for mitigation of crosstalk of all types, albeit modifications of the techniques proposed here may be necessary for each specific situation.

Let us consider the detection problem for the data model given in (1) for the case of Gaussian ambient noise. The traditional single user detector demodulates QAM symbols tone-by-tone independently. On the other hand, joint maximum-likelihood detection of both VDSL and crosstalk signals selects a set of N inputs {X i } and the crosstalk

{

}

sequences C (ml ) = C1,m ( l ) , C 2,m ( l ) ,..., C N ,m ( l ) , m = 2 ,

, M , to satisfy

N

M

} i =1

m=2

X i = arg{ min ( l ) ∑ Yi − H i ⋅ X i − ∑ Fi ,m ⋅ Ci ,m { X i },{Ci ,m

2 (l )

},

i = 1,..., N ,

(5)

where the minimization is taken over the DMT signal alphabet and all possible crosstalk sequences

[

]

C m = C (ml ) , l = 1,..., C m , m = 2,

C m , m = 2,

, M , that can occur within the VDSL symbol period of interest. The size

, M , of the set of all possible crosstalk sequences can be large but is always finite when all the

crosstalkers are digital signals or are derived from digital signals.

Just as its counterpart in wireless CDMA does, the maximum-likelihood multiuser detector achieves optimum performance but suffers from very high complexity. A full search in the input domain requires approximately N |C||M| squared-error computations, where N is the number of subchannels, | C |=

M

∏ | C m | is the number of

m=2

possible crosstalk sequences, and |M| is the average size of the transmitted alphabet. In practice N and especially |C| can be very large, introducing prohibitive computational complexity. The large number of possible crosstalk sequences also means an exponentially greater number of states, making dynamic programming inappropriate. Therefore, we need to consider a simplified receiver structure that maintains satisfactory performance while

6

requiring far less computational complexity. As we mentioned before, the crosstalk signals in DSL transmission are of various types and cannot be represented under a uniform framework. The type we examine here is the dominant near-end QAM-like crosstalk (e. g. [5]).

One lower-complexity approach to MUD is to employ a linear multiuser detection technique, such as decorrelating (zero forcing) or MMSE multiuser detection. However, unlike CDMA or space-division multiple-access (SDMA) where linear detection has been effective, there is no identifying signature such as the spreading code for CDMA or the steering vector for SDMA, to aid linear detection in VDSL. Moreover, the desired signals and crosstalk signals are often of different data format. An alternative approach that is better suited to this situation is to employ interference cancellation (IC) [21], i.e., to attempt excision of the crosstalk from the received signal before applying traditional DMT VDSL signal detection. We adopt this approach here.

The interference cancellation approach is based on a natural idea: if decisions have been made about an interfering user’s information bits, the interfering signal can be reconstructed and subtracted at the receiver. Of course, this will achieve perfect interference elimination if the decisions were correct; however, with incorrect decisions things could be worse than without the canceller. Nevertheless, in VDSL applications, incorrect detection of the crosstalk sequence may not be as bad as one might think. Dominant NEXTs from other communication systems [4], [5], [6] usually are detected in the time domain and a DMT symbol interval will contain more than one crosstalk bit. Often distortion effects in the frequency (resp. time) domain will diffuse in the time (resp. frequency) domain, in which further gain can be obtained by making hard decisions in the high-SNR scenario. These phenomena make IC a good choice in this application.

Our interference cancellation multiuser detection (IC-MUD) scheme is described as follows: 1) A hard decision is made on the VDSL signal in the frequency domain. This decision can be obtained through the received signal Y = (Y1 , values

2

, Y N ) T directly (for the first iteration) or through soft LLR

(see (4)) from a SISO decoder.

ˆ = (X , 2) The DMT symbols X 1

, X N ) T are reconstructed based on these detected bits.

7

3) The desired signal is subtracted and the known crosstalk coupling function is applied to get a

~ ~ frequency domain estimate of the entire crosstalk sequence C m = (C1,m ,

~ , C N ,m ) T via

~ −1 ˆ ), ˆ − ∑ Fi C C m = Fm (Y − H X i

(6)

i 0 , 0 ≤ ε ≤ 1 , and κ ≥ 1 . Here, the

(40)

(0,σ 2 ) term represents the nominal background noise (Gaussian

with zero mean and variance σ 2 ), and the N (0, κσ 2 ) term represents an impulse component (Gaussian with zero mean and variance κσ 2 ), with ε representing the probability that impulses occur in a given subchannel. It is

18

assumed that noise samples in disjoint frequency bins are independent. In our simulation we choose parameters

ε = 0.01 and κ = 100 , which means the impulse spike is 20 dB higher than the background noise floor with occurrence probability of 1% per frequency bin. Again we do not include crosstalk signals and MAP is used in the SISO decoder. Figure 13 shows that the performance of the proposed receiver is greatly degraded with impulse noise. The use of erasure decoding can remedy this. In the demodulation stage, for those bits associated with impulse-contaminated symbols, instead of calculating soft metrics for them, the a priori information is used as a substitute, i.e., λ1 (bk ,i ) = λ2p (b j ) , where b j is the de-interleaved version of bk ,i . (For the first iteration, these are set to zeros.) In DMT systems, the erasure positions where impulse spikes appear, can possibly be detected in advance through pilot tones. In Fig. 14, we see that the performance of the proposed receiver experiences almost no performance loss with impulse noise with the aid of erasure decoding. The reader is referred to [10] for an alternative approach for combating impulse noise when crosstalk signals are present.

VI.

Conclusions

In this paper, a new coded DMT VDSL receiver structure using turbo multiuser detection has been proposed and has been shown to achieve an overall 7.0 dB gain over the uncoded optimum (maximum likelihood) receiver at BER

10 −7 for a channel with severe ISI, AWGN, and one dominant crosstalk signal. The traditional single-user detection scheme produces extremely poor and totally unacceptable performance in our settings. Without multiuser detection, the decoding process turns out to be useless. The effect of impulse noise is detrimental to the proposed scheme but can be overcome through an erasure decoding technique. For our proposed scheme, we also see that natural coding is better than Gray coding.

In this paper, we have assumed knowledge of the line transfer function and crosstalk coupling functions. In reality, however, channel identification is needed, and the effects of channel estimation error should be taken into consideration. This issue is of interest for further study. The problem of detecting impulse spike positions (for erasure decoding purposes) also deserves further study.

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References

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[14] J. Hagenauer and P. Hoeher, “A Viterbi algorithm with soft-decision outputs and its applications,” Proc. IEEE Global Telecommunications Conf., pp. 47. 1.1-47. 1.7, 1989 [15] J. Hagenauer et al., “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inform. Theory, vol. 42, no. 2, pp. 429-445, Mar. 1996. [16] J. P. Lauer and J. M. Cioffi, ‘Turbo coding for discrete multitone transmission,” Proc. IEEE Global Telecommunications Conf., pp. 3256-3260, 1998. [17] P. Luukkanen and P. Zhang, “Comparison of optimum and sub-optimum turbo decoding schemes in 3rd generation cdma200 mobile system,” Proc. IEEE Wireless Communications and Networking Conf., pp. 437441, 1999 [18] H. V. Poor, “Turbo multiuser detection: A primer,” J. Commun. Networks, vol. 3, no. 3, Sept. 2001. [19] P. Roberson et al., “A comparison of optimal and sub-optimal MAP decoding algorithms operating in the log domain,” Proc. IEEE International Conf. on Communications, pp. 1009-1013, 1995 [20] T. Starr, J. M. Cioffi and P. J. Siverman, Understanding Digital Subscriber Line Technology, Prentice-Hall: Upper Saddle River, NJ, 1999. [21] S. Verdú, Multiuser Detection, Cambridge University Press: Cambridge, UK, 1998. [22] A. J. Viterbi et al., “A pragmatic approach to trellis-coded modulation,” IEEE Commun. Mag., vol. 27, no. 7, pp. 11-19, July 1989 [23] A. J. Viterbi, “An intuitive justification and simplified implementation of the MAP decoder for convolutional codes,” IEEE J. Select. Areas Commun., vol. 16, no. 2, pp. 260-264, Feb. 1998 [24] X. Wang and H. V. Poor, "Robust multiuser detection in non-Gaussian channels,” IEEE Trans. Commun., vol. 47, no. 2, pp. 289-305, Feb. 1999. [25] X. Wang and H. V. Poor, “ Iterative (turbo) soft interference cancellation and decoding for coded CDMA”, IEEE Trans. Commun., vol. 47, no. 7, pp. 1046-1061, July 1999. [26] T. N. Zogakis et al., “Analysis of a concatenated coding scheme for a discrete multitone modulation system,” Proc. 1994 IEEE Military Communications Conf., pp.433-437, 1994 [27] T. N. Zogakis et al., “A coded and shaped discrete multitone system,” IEEE Trans. Commun., vol. 43, no. 12, pp. 2941-2949, Dec. 1995.

X1

x1

21

X2 d

b Convolutional Encoder

Interleaver

QAM MOD

. . .

x2

IFFT N = 2N

XN

. . .

P/S & D/A

x(t)

xN

(a)

n(t ) DMT VDSL signal

x (t )

r (t ) VDSL Line Channel H

c2 ( t )

Crosstalk Coupling Function F 2

. . .

Crosstalk signals

. . .

cM (t )

Crosstalk Coupling Function F M

(b)

r (t )

A/D & S/P

y1

Y1

y2

Y2

. . .

FFT

N = 2N

yN

. . .

demod. & decoder

Detected data

YN (c)

Fig. 1 VDSL DMT System Configuration: (a) Transmitter; (b) Channel; (c) Receiver

22

Λ1 (bk ,i ) +

λ1 (bk ,i ) De-INT

− Soft Metric Calc. INT

SISO DEC

λ2 (b j ) − + Λ 2 (b j )

Fig. 2 Turbo structure for iterative demodulation and decoding

Bit allocation

bits per dimension

3

2

1

0

50

100 150 Subchannel i

200

250

Fig. 3 Bit allocation for DMT subchannels

23

10

10

Bit Error Rate

10

10

10

10

10

10

ML-MAP

0

-1

-2

-3

-4

-5

1st iteration 2nd iteration 3rd iteration 4th iteration 5th iteration

-6

-7

1

2

3

4 5 Geometric SNR (dB)

6

7

8

Fig. 4 Performance of the ML-MAP turbo multiuser receiver 10

10

Bit Error Rate

10

10

10

10

10

10

IC-MAP

0

-1

-2

-3

-4

-5


-6

-7

1

2

3


6

7

8

Fig. 5 Performance of the IC-MAP turbo multiuser receiver

24

10

10

Bit Error Rate

10

10

10

10

10

10

ML-SOVA

0

-1

-2

-3

-4

-5


-6

-7

1

2

3


6

7

8

Fig. 6 Performance of the ML-SOVA turbo multiuser receiver

10

10

Bit Error Rate

10

10

10

10

10

10

IC-SOVA

0

-1

-2

-3

-4

-5


-6

-7

1

2

3


6

7

8

Fig. 7 Performance of the IC-SOVA turbo multiuser receiver

25

10

10

Bit Error Rate

10

10

10

10

10

10

Performance Comparison

0

-1

-2

-3

-4

SUD IC-MUD ML-MUD IC-MUD+VA ML-MUD+VA ML-MUD+MAP IC-MUD+SOVA SUD+MAP

-5

-6

-7

0

1

2

3

4

5

6 7 8 9 10 Geometric SNR (dB)

11

12

13

14

15

Fig. 8 Performance comparison of various DMT VDSL receivers

10

10

Bit Error Rate

10

10

10

10

10

Gray Coding

-1


-2

-3

-4

-5

-6

-7

1

2

3


6

7

8

Fig. 9 Performance of the iterative DMT receiver with Gray coding

26

10

10

Bit Error Rate

10

10

10

10

10

10

Natural Coding

0


-1

-2

-3

-4

-5

-6

-7

1

2

3


6

7

8

Fig. 10 Performance of the iterative DMT receiver with natural coding

00

01

11

10

Fig. 11 Gray coding for 4-PAM

00

01

10

11

Fig. 12 Natural coding for 4-PAM

27

10

10

Bit Error Rate

10

10

10

10

10

10

Impulse Noise

0

-1

-2

-3

-4

-5


-6

-7

1

2

3


6

7

8

Fig. 13 Performance of the iterative DMT receiver with impulse noise 10

10

Bit Error Rate

10

10

10

10

10

10

Impulse Noise with Erasure Decoding

0

-1

-2

-3

-4

-5


-6

-7

1

2

3


6

7

8

Fig. 14 Performance of the iterative DMT receiver with erasure decoding with impulse noise

28