An Analysis of Linear Crosstalk Cancellation and Pre-compensation Techniques which Achieve Near Capacity in DSL Channels

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An Analysis of Linear Crosstalk Cancellation and

Pre-compensation Techniques which Achieve Near

Capacity in DSL Channels

Raphael Cendrillon and Marc Moonen

Department of Electrical Engineering (ESAT), Katholieke Universiteit Leuven, Belgium

{raphael.cendrillon,marc.moonen}@esat.kuleuven.ac.be

Abstract

This work was carried out in the frame of IUAP P5/22, Dynamical Systems and Control: Computation, Identification and Modeling and P5/11, Mobile

multime-dia communication systems and networks; the Concerted Research Action GOA-MEFISTO-666, Mathematical Engineering for Information and Communication Systems Technology; FWO Project G.0196.02, Design of efficient communication techniques for wireless time-dispersive multi-user MIMO systems and was

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I. INTRODUCTION

Linear crosstalk cancellation/pre-compensation schemes are preferable to non-linear schemes due to • Lower complexity

• No decision feedback errors

Easier performance analysis. More predictable performance. In decision feedback schemes feedback occurs before decoding. This leads to a high error rate.

• Linear schemes more amenable to partial implementations. See [1] for details. II. SYSTEMMODEL

A. Upstream

yk = Hkxk+ zk where Hkis the N × N channel matrix.

B. Downstream

yk = HTkxk+ zk (1)

C. Diagonal Dominance

When the line terminals (LT) are co-located the upstream channel matrix Hkis column-wise diagonal dominant in that

|hn,n_k | À |hn,m_k | , ∀m 6= n

Recall that the LTs are the modems at the central office (CO) and the NTs are the modems at the customer premises (CP). We quantify the degree of column-wise diagonal dominance using the parameter αk

|hn,m_k | ≤ |hn,n_k | tan αk, ∀m 6= n (2)

According to the semi-empirical model employed in the ETSI standards[2] αkis modeled such that

tan αk = Kf extfk √

l

where Kf ext= 10−45/20, fkis the frequency at tone k in MHz and l is the length of the crosstalk channel in kilometers.

Theorem 1: Let Hk be a column-wise diagonal dominant matrix which satisfies (2). Denote the QR decomposition of Hk

Hkqr= QkRk (3) The following bound holds

|r_kn,n| ≥ |hn,n_k |p1 − 4(N − 1)α2 ₍₄₎

where hn,m_k , [Hk]n,mand r n,m

k , [Rk]n,m.

Proof: See [3].

III. UPSTREAMCOMMUNICATION- LINEARZEROFORCINGCROSSTALKCANCELLER

A. Linear Zero Forcing Canceller

We use a linear filter to estimate the transmitted symbols

b

xk= Wkyk where Wk , H−1_k .

B. Performance Bounds

Theorem 2: Regardless of receiver structure the rate of user n on tone k is upper bounded

cnk,opt ≤ log2

³

1 +£1 + (N − 1) tan2α¤Sk|hn,n_k |2σ−2_k Γ−1 ´

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Proof: See Appendix I.

Theorem 3: With the linear ZF crosstalk canceller the rate of user n on tone k is lower bounded

cn k,zf≥ log2 ³ 1 +£1 − 4(N − 1)α2¤_Sk_|hn,n k | 2 σ−2 k Γ−1 ´ (6)

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IV. DOWNSTREAMCOMMUNICATION- LINEARDIAGONALIZINGCROSSTALKPRE-COMPENSATOR

A. Linear Zero Forcing Pre-compensator

We apply a linear pre-filter to pre-compensate for the effects of crosstalk

xk= Pkxk

where Pk ,_β1_zfH−Tk . The factor βzf ensures that the pre-filter Pkdoes not increase the power of any of the TXs βzf , max n ° °£_H−T k ¤ row n ° ° ' max n 1 |hn,n_k |

The result is that

yk = 1 βzfxk+ zk ' min n |h n,n k | xk+ zk

where the approximation on line 2 is based on the column-wise diagonal dominance of Hk.

So application of the ZF pre-compensator causes all of the users to see the channel of the worst user in the system. If the users have different line lengths this leads to very poor performance.

B. Linear Diagonalizing Pre-compensator

Again we apply a linear pre-filter to pre-compensate for the effects of crosstalk

xk= Pkxk however now Pk, 1 βdiagH −T k diag {Hk} and βdiag, max n ° °£_H−T k diag {Hk} ¤ row n ° ° ₍₇₎

The effect of this pre-filter is to diagonalize the channel.

yk =

1

βdiagdiag {Hk} xk+ zk

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' diag {Hk} xk+ zk

The approximation on line 2 is based on the column-wise diagonal dominance of Hkwhich causes βdiag' 1.

So application of the diagonalizing pcompensator causes the users to see their own direct channels crosstalk perfectly re-moved. Column-wise diagonal dominance of Hk means that there is little to gain by exploiting the crosstalk channels for trans-mission. That is, the attenuations of the crosstalk channels are huge compared to those of the direct channels. The result is that perfect crosstalk cancellation is effectively the best that MIMO transmission can achieve. This in turn implies the near-optimality of the linear diagonalizing pre-compensator.

C. Performance Bounds

Theorem 4: Regardless of transmitter structure the rate of user n on tone k is upper bounded by

cnk,opt ≤ log2

³

1 +£1 + (N − 1) tan2α¤Sk|hn,n_k |2σ−2_k Γ−1 ´

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Proof: See Appendix III.

Theorem 5: With the linear diagonalizing crosstalk pre-compensator the rate of user n on tone k is lower bounded by

cnk,diag≥ log2 Ã 1 + ¡ 1 − 4(N − 1)α2¢2 1 + (N − 1) tan2_αSk|h n,n k | 2 σ_k−2Γ−1 ! (10)

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V. BOUNDS ONCAPACITYLOSS WITHLINEARSCHEMES

Corollary 1: Using the linear cancellation techniques described in Sec. III capacity loss can be tightly bounded by

cnk,opt− cnk,linear≤ log2

Ã 1 +£1 + (N − 1) tan2_α¤_Sk_|hn,n k | 2 σ_k−2Γ−1 1 + [1 − 4(N − 1)α2_{] S} k|hn,nk | 2 σ−2_k Γ−1 ! (11)

Proof: Follows directly from (5), (6).

Corollary 2: Using the linear cancellation techniques described, capacity loss can be loosely bounded by

cn

k,opt− cnk,linear ≤ log2

µ

1 + (N − 1) tan2_α

1 − 4(N − 1)α2

¶

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Proof: Follows from (13) and the observation that when a ≥ b, a_b ≥ 1+a_1+b.

Corollary 3: Using the linear pre-compensation techniques described in Sec. IV capacity loss can be tightly bounded by

cn

k,opt− cnk,linear≤ log2

Ã 1 + (N − 1) tan2_{α +}£_{1 + (N − 1) tan}2_α¤2_Sk_|hn,n k | 2 σ−2_k Γ−1 1 + (N − 1) tan2_{α + (1 − 4(N − 1)α}2₎2_Sk_|hn,n k | 2 σ−2 k Γ−1 ! (13)

Proof: Follows directly from (9) and (10).

Corollary 4: Using the linear pre-compensation techniques described, capacity loss can be loosely bounded by

cnk,opt− cnk,linear ≤ 2 log2

µ

1 + (N − 1) tan2_α

1 − 4(N − 1)α2

¶

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Proof: Follows from (13) and the observation that when a ≥ b, a_b ≥ 1+a 1+b.

VI. SIMULATIONS

A. Scenario

1 x 300 m and 1 x 1200 m lines using ETSI semi-empirical models for direct and crosstalk channel transfer functions[2]. LTs co-located. B. Upstream C. Downstream VII. CONCLUSIONS APPENDIXI PROOF OFTHEOREM2 Since conditioning never decreases mutual information

cnk,opt = I(xnk; yk)

≤ I(xnk; yk| x1k, . . . , xn−1k , xn+1k , . . . , xNk)

where I(x;y) denotes the mutual information between x and y. The term on the second line is simply the capacity of user n when the TXs (NTs) of all other users are disabled. This is essentially the matched filter bound (MFB) of user n since it is the capacity that user n achieves when all other user’s are disabled and a maximum-ratio combiner (MRC) is applied to the received signals at the CO.

With all other user’s TXs disabled

yk= hnkxk+ zk where hn_k , [Hk]_{col n}. Applying a MRC gives the following estimate

ˆ xnk = (hn k) H khn kk 2yk = xnk + (hn k) H khn kk 2zk The resulting SNR is SN Rnk,opt = Skσk−2khnkk2 ≤ Skσ_k−2|hn,n_k |2£1 + (N − 1) tan2_α¤

where we use (2) to define the bound. The capacity of user n with the optimal RX structure is

cn k,opt= log2 ¡ 1 + SN Rn k,optΓ−1 ¢

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APPENDIXII PROOF OFTHEOREM3 We will use the following Lemma in our proof.

Lemma 1: Define the permutation matrix Π = [IN]rows 1,...,n−1, n+1,...,N, n. If the channel matrix Hksatisfies (2) then

Hk , ΠHkΠT also satisfies (2).

Proof: Follows by inspection.

Define

Hk , ΠHkΠT (15) and the corresponding QR decomposition

Hkqr= QkRk (16) Using Theorem 1 and Lemma 1

|rn,n_k | ≤ ¯ ¯ ¯hn,nk ¯ ¯ ¯p1 − 4(N − 1)α2 ₍₁₇₎ Now ˆ xk = H−1k yk = xk+ H−1k zk and ˆ xn k = xnk+ £ H−1 k ¤ row nzk Hence SN Rnk,zf = Skσk−2 ° °£_H−1 k ¤ row n ° °−2 = Skσ_k−2 ° ° ° h ΠH−1k ΠT i row n ° ° °−2 = Skσ_k−2 ° ° ° h H−1k ΠT i row N ° ° °−2 = Skσ−2 k ° ° ° h H−1_k i row N ° ° °−2 (18) From (16) H−1_k = R−1k Q H k . Hence h H−1k i row N = h R−1k i row NQ H k = 1 rn,n_k £ Qk ¤H col N where hn,m_k ,£Hk ¤ n,mand r n,m k , £ Rk ¤ n,m. This implies ° ° ° h H−1k i row N ° ° °2 = |rn,n_k |−2 ≤ ¯ ¯ ¯hn,nk ¯ ¯ ¯−2¡1 − 4(N − 1)α2¢ = ¯ ¯ ¯hN,N_k ¯ ¯ ¯−2¡1 − 4(N − 1)α2¢

where we use (17) and (15) in the second and third line respectively. Combining this with (18) yields the following SNR bound

SN Rnk,zf ≥ Skσ−2k |h

n,n k |

2¡

1 − 4(N − 1)α2¢

The capacity of user n is with the ZF canceller is

cn k,zf = log2 ¡ 1 + SN Rn k,zfΓ−1 ¢ which leads to (6).

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APPENDIXIII PROOF OFTHEOREM4 We write the following obvious bound

cn

k,opt = I(xnk; ynk)

≤ I(xk; ynk)

The term on the second line is simply the capacity of user n when all TXs (LTs) are used to communicate to the RX (NT) of user n. This is essentially the matched filter bound (MFB) of user n since it is the capacity that user n achieves when we apply a matched filter prior to transmission at the CO.

From (1) yn k = £ HT k ¤ row nxk+ zk = (hnk)Txk+ zk The resulting SNR is SN Rn k,opt = Skσk−2khnkk2 ≤ Skσ_k−2|hn,n_k |2£1 + (N − 1) tan2_α¤

where we use (2) to define the bound. The capacity of user n is cnk,opt= log2 ¡ 1 + SN Rnk,optΓ−1 ¢ which leads to (9). APPENDIXIV PROOF OFTHEOREM5 Lemma 2: Define f (α) , s 4(N − 1)α2_{+ (N − 1) tan}2_α 1 − 4(N − 1)α2 then |rn,m_k | |r_km,m| ≤ f (α), ∀n 6= m (19)

Proof: First observe that

k[Rk]_{col m}k = k[Hk]_{col m}k , ∀m This implies |r_kn,m|2 = k[H]_{col m}k2₂− |rm,m_k |2− X i6=n,m ¯ ¯ ¯ri,mk ¯ ¯ ¯2 ≤ |hm,m_k |2+X i6=m ¯ ¯ ¯hi,mk ¯ ¯ ¯2− |rm,m_k |2 ≤ |hm,m_k |2£1 + (N − 1) tan2_α¤_{− |r}m,m k | 2 ≤ |r_km,m|2 · 1 + (N − 1) tan2_α 1 − 4(N − 1)α2 − 1 ¸ = |r_km,m|2 · 4(N − 1)α2_{+ (N − 1) tan}2_α 1 − 4(N − 1)α2 ¸ , ∀n 6= m

Examining (8) we see that the SNR with the diagonalizing pre-compensator is

SN Rn k,diag= 1 β2 diag Skσ−2 k |hn,nk | 2 (20)

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From (3) HT_k = RT kQTk hence H−T k = ¡ QT k ¢H R−T k Hence £ H−T_k diag {Hk}¤_{row n} = h¡QT k ¢Hi row nR −T k diag {Hk} = h¡QTk ¢Hi row nB = X m (qknm)∗bn where B , R−T_k diag {Hk}, bn, [B]_{row n}and qnmk , [Qk]_n,m. This implies that

° °£_H−T k diag {Hk} ¤ row n ° °2 ≤ X m |qnmk |2kbnk2 ≤ ³max i kbik 2´ Ã X m |qnm k |2 ! = max i kbik 2 (21) Define Gk , R−1_k and gnm_k , [Gk]n,m. Now

Hence, using Lemma 2

|gn,m_k | ≤ m−1_X i=n ¯ ¯ ¯r_ki,m ¯ ¯ ¯ |r_km,m| ¯ ¯ ¯gn,ik ¯ ¯ ¯ , ∀m > n ≤ f (α) m−1_X i=n ¯ ¯ ¯gkn,i ¯ ¯ ¯ , ∀m > n For example ¯ ¯ ¯g_kn,n+1 ¯ ¯ ¯ ≤ 1 |r_kn,n|f (α) ¯ ¯ ¯gkn,n+2 ¯ ¯ ¯ ≤ 1 |r_kn,n| ¡ f (α) + f (α)2¢ In general |g_kn,m| ≤ 1 |rn,n_k | m−n_X l=1 µ m − n − 1 l − 1 ¶ f (α)l Hence |gnmk |2≤ 1 |rmm k | 2   m−n_X l=1 µ m − n − 1 l − 1 ¶ Ã (N − 1)¡4α2_{+ tan}2_α¢ 1 − 4(N − 1)α2 !l/2  2 (22)

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To simplify this expression we ignore terms of order O(α3) or O(tan3_{α) and higher. Thus (22) can be approximated} |gnmk |2≤ 1 |rmm k | 2 (N − 1)¡4α2_{+ tan}2_α¢ 1 − 4(N − 1)α2 Hence |gmn k |2|hmmk |2 ≤ |hmm k | 2 |rmm k | 2 (N − 1)¡4α2_{+ tan}2_α¢ 1 − 4(N − 1)α2 ≤ (N − 1) ¡ 4α2_{+ tan}2_α¢ (1 − 4(N − 1)α2₎2 Hence kbik2 ≤ 1 1 − 4(N − 1)α2 Ã 1 + (N − 1) ¡ 4α2_{+ tan}2_α¢ 1 − 4(N − 1)α2 !

With (7) and (21) this implies

1 β2 diag ≥ ¡ 1 − 4(N − 1)α2¢2 1 + (N − 1) tan2_α

Examining (20) we see that the SNR can now be bounded

SN Rnk,diag≥ ¡ 1 − 4(N − 1)α2¢2 1 + (N − 1) tan2_αSkσ −2 k |h n,n k | 2

The capacity of user n is

cnk,diag= log2 ¡ 1 + SN Rnk,diagΓ−1 ¢ which leads to (10). REFERENCES

[1] R. Cendrillon, M. Moonen, et al., “Partial Crosstalk Cancellation Exploiting Line and Tone Selection in Upstream DMT-VDSL,” Submitted to EURASIP

Journal on Applied Signal Processing, Feb 2003, available at http://www.esat.kuleuven.ac.be/~rcedrill.

[2] Transmission and Multiplexing (TM); Access transmission systems on metallic access cables; VDSL; Functional Requirements, ETSI Std. TS 101 270-1/1, Rev. V.1.2.1, 1999.

[3] G. Ginis and J. Cioffi, “Vectored Transmission for Digital Subscriber Line Systems,” IEEE J. Select. Areas Commun., vol. 20, no. 5, pp. 1085–1104, June 2002.

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TABLE I SIMULATIONPARAMETERS Number of DMT Tones 4096 Tone Width 4.3125 kHz Symbol Rate 4 kHz Coding Gain 3 dB Noise Margin 6 dB

Symbol Error Probability < 10−7

Transmit PSD Flat -60 dBm/Hz

FDD Band Plan 998

Cable Type 0.5 mm (24-Gauge)

Source/Load Resistance 135 Ohm

Noise Model ETSI Type A

0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Data−rate Loss (%) Frequency (MHz) Loose Bound Tight Bound Simulation

Fig. 1. US Data Rate Loss vs. Frequency (300 m)

0 2 4 6 8 10 12 0 5 10 15 20 25 30 Data−rate Loss (%) Frequency (MHz) Loose Bound Tight Bound Simulation

Fig. 2. US Data Rate Loss vs. Frequency (1200 m)

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0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Data−rate Loss (%) Frequency (MHz) Loose Bound Tight Bound Simulation

Fig. 4. DS Data Rate Loss vs. Frequency (300 m)

0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 8 9 10 Data−rate Loss (%) Frequency (MHz) Loose Bound Tight Bound Simulation

Fig. 5. DS Data Rate Loss vs. Frequency (1200 m)