Spectral Optimization for Communication in the Presence of Crosstalk Rohit V

Spectral Optimization for Communication in the Presence of Crosstalk Rohit V. Gaikwad and Richard G. Baraniuk Department of Electrical and Computer Engineering Rice University, 6100 Main Street

Houston, Texas 77005

Email: rohitg, richb ¡ @rice.edu Web: www.dsp.rice.edu

Abstract— We present a general framework for designing op- NEXT NEXT timal transmit spectra for communication services dominated by Tx 1 Rx1 crosstalk — for example Digital Subscriber Lines (DSLs) and wireless LANs. Using the channel, noise, and interference transfer functions, we set up and solve an optimization problem to maximize Rx 2 Tx 2 the joint channel capacity. We employ joint signaling techniques and optimal power distribution to yield signiﬁcant gains in bit rates FEXT

(or performance margins). Furthermore, by design, the spectra are spectrally compatible with existing neighboring services. The Rx 3 Tx 3 framework is quite general; it does not depend on the exact choice of the modulation scheme, for example. It is also extremely simple and of low computational complexity. Fig. 1. NEXT and FEXT between neighboring lines in a binder. Tx’s Keywords—Digital subscriber line services (xDSL), multi-user in- are transmitters and Rx’s are receivers. terference, crosstalk, spectral compatibility, capacity, performance margins, power allocation, transmit spectra, joint signaling tech-

niques. ¦ kft long.

HDSL2 — High bit-rate DSL 2 — will provide a sym- ¨§§ I. INTRODUCTION metric bit–rate of £ Mbps over a single twisted pair

( £ kft long) without repeaters. The explosion in communications technologies has led Telephone lines are packed closely together into to significant resource-sharing among users. In particular, binders in a cable. Crosstalk (near-end (NEXT) and far- multiple users share the available bandwidth in many sys- end (FEXT)) results due to the proximity of the lines (see tems. Proximity of different paths or channels between Figure 1) and significantly limits achievable bit-rates [1]. users can lead to multi-user interference or crosstalk. For Complete cancellation or suppression of crosstalk is example, in digital subscriber lines (DSLs) in a telephone not always possible, since it is difficult and/or expensive. cable and users in a indoor wireless LAN, crosstalk can Rather, we employ crosstalk avoidance using orthogonal be significantly high and can limit achievable bit rates. signaling techniques to design optimal transmit spectra. Efficient usage of available bandwidth in the presence of We solve an optimization problem to maximize the joint crosstalk is thus key to improving capacity of such com- channel capacity given the channel, noise and crosstalk munication systems. characteristics. This is an information-theoretic result that Digital subscriber line (DSL) modems, the next gener- maximizes achievable bit rates in crosstalk-dominated en- ation of high-speed telephone line modems, exploit large

vironments. By design, we maintain spectral compatibil- ¢¥£ bandwidths ( ¢¤£ MHz) to yield high bit rates ( Mbps). ity with existing neighboring services. This problem was The various DSL services (xDSL in general) are catego- first solved in [2] for symmetric-bit-rate services facing rized according to the bit rates they deliver: self-NEXT (NEXT from same-service lines) and white ADSL — Asymmetric DSL — provides a high-speed (on additive Gaussian noise (AGN). Here, we solve the prob- the order of ¦ Mbps) downstream (from central office lem in presence of self-NEXT, self-FEXT, AGN, and in- to subscriber) channel and a low-speed (on the order of terference from other services. Optimization can also be ¦¨§ © kbps) upstream (from subscriber to the central office) done under an additional peak frequency-domain power channel over each twisted pair. constraint [3]. The techniques developed here are general VDSL — Very high bit-rate DSL — will provide a sym- and can be applied to any symmetric-bit-rate communi- metric or asymmetric high-bit-rate (on the order of ¨© cation channel with appropriate crosstalk characteristics. Mbps) channel over a single twisted pair less than to In this paper, we target symmetric-bit-rate DSL services, This work was supported by the National Science Foundation, grant e.g., HDSL2 [4]. no. MIP–9457438, Nortel Networks, and SBC Communications. Section II outlines the definitions and notation used. 0

10 self-NEXT transfer function [7]

. .=<

. . 1

0 3?B

#:9C#

576 8 8 @ −2 if 0

10 '(!$#&% (2)

2 © Hi,k H (f) otherwise @ C

−4 10 and self-FEXT transfer function [7] X i,k

2 .= H (f) .

. . 1

3?D

8 #:9;#8 −6 576

10 if 0A@

!$#&%

) (3) © otherwise F 2 i,k H (f) −8 F

# E

Here 8 are the center frequencies (see Figure 2) of the

FHG £ I EJ¡ @ bins with index @ . We consider real signals −10 Channel

Magnitude squared frequency response

10 x x self−NEXT with symmetric frequency responses. Thus, we denote self−FEXT quantities only over the non-negative frequency region.

−12

K:L !$#&% 10 ' 0 100 200 300 400 500 Similarly, DS-NEXT is denoted by , DS- fk

Frequency (kHz)

K:L !$#&% MON¨!/#&% FEXT by ) , and AGN by . We sum the DS-NEXT, DS-FEXT, and AGN to get the total Gaussian Fig. 2. Magnitude-squared transfer function of the channel (CSA loop

noise as

6), self-NEXT interferers, and self-FEXT interferers.

MJ!/#&%QP MON!$#&%SRTK:L !$#&%URTK:L !/#&% ) ' (4) Details on obtaining optimal transmit spectra are pre-

sented in Section III. We discuss simulation results in DSL modems transmit in two directions on the same line V Section IV and present conclusions in Section V. via a § – line hybrid circuit. We denote the upstream

and downstream transmit power spectral densities (PSDs)

L[Z !$#&% \]W^!$#&% by LXWY!/#&% and , respectively. Similarly, and

II. DEFINITIONS AND NOTATION F \]Z !/#&% denote the PSDs in frequency bin . An Equal PSD

There are two types of crosstalk (see Figure 1): (EQPSD) signaling scheme in frequency bin F is one for

1 1

\`Z !$#&%ba © # Near-end crosstalk (NEXT): Interference between neigh- which \W_!$#&% for all in the bin. (that is,

both upstream and downstream transmissions occupy the

. .=<

boring lines that arises when signals are transmitted in op- >

#:9C# 8 posite directions. If the neighboring lines carry the same band 0 in the same way). A Frequency Divi-

sion Signaling (FDS) scheme in frequency bin F is one for 1

type of service, then the interference is called self-NEXT; 1

vice versa (that is, both transmissions occupy orthogonal

. .=<

Far-end crosstalk (FEXT): Interference between neigh- >

#:9;# 8 boring lines that arises when signals are transmitted in the frequency bands within 0 ). same direction. If the neighboring lines carry the same III. OPTIMAL TRANSMIT SPECTRA type of service, then the interference is called self-FEXT; otherwise, it is called as different-service (DS) FEXT. We consider a full-duplex symmetric-bit-rate communication channel where self-NEXT dominates and self- The term self-interference refers to the combined self- FEXT is small (see Figure 2). This is the case of interest NEXT and self-FEXT. Channel noise is modeled as AGN. for HDSL2. However, self-FEXT factors into our design DS NEXT and DS FEXT can also be modeled as AGN for in a signiﬁcant way. This is a new, non-trivial extension capacity purposes [5]. of the work of [2].

Figure 2 illustrates the channel, self-NEXT, and self- W

Our goal is to maximize the upstream capacity ( d ) '(!$#&% FEXT transfer functions, denoted by "!$#&% , , and

and the downstream capacity ( deZ ) given an average to-

)*!$#&% , respectively. We assume that the channel can be

tal power constraint of fhgXikj and the equal capacity con-

characterized as a linear time-invariant system. We divide 1 deZ straint dlW .

the transmission bandwidth + of the channel into K nar- Consider the case of two neighboring lines carrying the

row frequency bins; each of width , Hz and assume that

d V same service. Line £ upstream capacity is and line the channel, noise and the crosstalk characteristics vary

downstream capacity is deZ . Under the Gaussian channel slowly enough with frequency that they can be approxi- assumption, we can write these capacities (in bps) as

mated as constant over each bin. monqp

We use the following notation for the channel transfer 1

d 0[£`R

r stvu]w r=xtvu]wzy|{

}~ -

function of line [6] 6

. .

. .=

"!$#&% LXWY!/#&%

. . 02143

576 8 #:9;#8

if 0A@

. . . .

@ 0 0 (5)

"!/#&% (1)

MJ!/#&%SR 'O!/#&% L !$#&%UR ()*!/#&% L !$#&%`S

Z W otherwise @

§ u §

and s (f) sd (f) ¦

α = 1 ¦ α = 1 «

« 2a 2a

1 mknp ¦

α = 0.8 ¦ α = 0.8

£R d

r stvu]w r=xItvu]w {

}

. 0

. α = 0.5 1−α = 0.5 1−α = 0.5 α = 0.5

¬ ¬ LZ!/#&%

"!$#&% a a

. 0 . . 0

. (6)

MJ!/#&%UR '(!$#&% L !$#&%hR )*!$#&% L !/#&%]S Z

W 1−α = 0.2 1−α = 0.2 1−α = 0 1−α = 0

0 L[Z!/#&%

The supremum is taken over all possible L[W_!/#&% and 0

¨ ¨ ©

satisfying ª

0 W 2 W f 0 W 2 W f

W Z

L !$#&%Q© L !$#&%Q©&# @

@ Fig. 3. Upstream and downstream transmit spectra in a single frequency

²´³ ®¯¶µ:³ bin ( ®¯b°]± EQPSD signaling and FDS signaling). and the average power constraints for the two directions

(for different values of · ) are illustrated in Figure 3. The

< <

W Z

V L !$#&% # f V L !$#&% # f

gXikj gXikj

@ ¸`½ º$»&¼

(7) PSDs ¸]¹^º$»&¼ and are “symmetrical” or power com-

yT{ yT{

} }

plementary to each other for different values of · . This

W Z d We can solve for the capacities d and using ensures that the upstream and downstream capacities are

“water–ﬁlling” if we impose the restriction of EQPSD, 1

equal ( ¾I¹O¿¾½ ). L[Z !/#&%S# that is LXW_!$#&% . However, this gives low ca- Next, we show that given this setup, the optimal signal-

pacities. Therefore, we employ FDS ( L[WY!/#&% orthogonal ing strategy uses only FDS or EQPSD in each bin.

to L[Z!$#&% ) in spectral regions where self-NEXT is large enough to limit our capacity and EQPSD in the remain- A. Optimal Spectrum: One frequency bin ing spectrum. This gives much improved performance. If we deﬁne the achievable rate as

To ease our analysis, we divide the channel into E bins

ÀzÁ

¹ ½

º/»&¼ÂÃ¸ º/»&¼o¼[¿

of equal bandwidth (see Figure 2) and continue our de- º$¸ ÄÅ

sign and analysis on the single frequency bin F assuming

¸]¹_º$»&¼oÑ »_Â

the bin frequency responses (1)–(3). For ease of notation, Ò (11)

Æ ÇÈÉ ÊÌËÎÍÐÏ

¸ º$»&¼Ó ¸ º$»&¼ÔbÕ×Ö

½ ¹ Ï

in this section set Ï

1 1

1 then

D D

B B

P P P

56 8 576 8 56 8

@ @

@ in (1)–(3) (8)

¹ ¹ ½

¾ ¿ ØÌÙ¨Ú º$¸ º$»&¼ÂÃ¸ º/»&¼o¼Ùàqá

ÆÛ ÜÝ&Þ×ÝSß

MP MJ!$# % F 8

and let denote the total noise PSD in bin . ÀcÁ

½ ½ ¹

¾ ¿ ØÌÙÚ º/¸ º/»&¼Â¸ º$»&¼k¼â ÜIÝ^ÞÝ&ß

ÆIÛ (12)

!/#&% F £

Let \ denote the PSD in bin of line upstream di-

F V

rection and \]Z!$#&% denote the PSD in bin of line down-

¹ ½

º/»&¼ ¸ º$»&¼

stream direction (for capacity purposes we will consider Due to the power complementarity of ¸ and ,

¾I¹ ¾½

the bin F demodulated to baseband). Denote the corre- the channel capacities and are equal. There-

¾¹

W IZ

sponding capacities of bin F by and . fore, we will only consider the upstream capacity ex- ÀzÁ

We desire a signaling scheme that includes FDS, pression. Further, we will use the shorthand for ÀcÁ

EQPSD, and all combinations in between in each bin. º/¸¹_º$»&¼Â¸`½ º$»&¼k¼ in the remainder of this section.

Therefore, we divide bin F in half and set Substituting the PSDs (9) and (10) into (11) and using

(12), we obtain

< <

if 0@

Êèé[ê

\ !/#&%

!o£"9 % # ,

if 0 (9)

ßì^Þí

¾ ¿¥ã ä ØÌÙ¨Ú

ÆÛ ÜIÝ^ÞÝ&ß¢å

ÇÈÉÊçæÍÐÏ

Êèé[î

Êkèé[ï

Å ð

otherwise, Å

Ï?ë Ï

ßì^Þí

ÊèéXê

< < >

!£*9 % © #

0A@

if ßì^Þí

1 (13)

> <

ÊÌæ

ÏñÇÈÉ ÍÐÏ

Êkè ï

Êkè î

\ !$#&%

Å Å

ð*ò >

# ,

if 0 (10)

Ï?ë Ï

Êè ó¤¿

Let ÅÌô denote the SNR in the bin. Then, we can ,

Here f¢¡ is the average power over the bandwidth in rewrite (13) as

< <

F ©q £

bin and . The factor controls the power

1 1

·¢óeÑ

W Z

©£ \ !$#&% \ !/#&%

¿ ØÌÙÚ ã äöõ

distribution in the bin. When , ¾

ÆIÛ ÜIÝ^ÞÝ&ß

ÇÈÉ ÍÐÏ

º ·ø¼ólÓ ·¢óeÔbÕ

&#?G¤ © ,¤¥ £ \]WY!/#&%

Í*÷ Ï

(EQPSD signaling); when , ÍÐÏ

º ·ø¼óeÑ

and \`Z!$#&% are disjoint (FDS signaling). These two ex- Í*÷

â (14)

ÏñÇÈÉÊ ËùÍÐÏ

º ·ø¼óeÔúÕøû

treme transmit spectra along with other possible spectra ·¢ólÓ

ÍÐÏ Ï Í*÷

Note from (12) and (14) that the expression after the üÌý¨þ another optimization technique (optimization in the pres-

¡ ÿ¡ in (14) is the achievable rate ÿ . Differentiating the ence of self-interference) [10].

expression in (14) with respect to yields 3. Loop between 1 and 2 until convergence is reached.

¢ We have found a simple test condition that closely ap-

1 £ £

0 0

proximates the optimal switch-over bin [8]. This

D D

B B

!/V 9£]%¥¤V! 9 %UR ! 9 %

approximation reduces the above algorithm to a single,

9"J!o£ÐR %ù¥§¦ @ (15) computationally simple step of optimal power distribu-

tion.

¦¢©* GC!7© £¥ with @ . The resulting spectra may not have contiguous power Setting this derivative to zero gives us the single sta-

1 allocation over frequency. However, we present optimal

1 ways of grouping bins in [8] to yield contiguous upstream

G !7©£ £¥ ©£

tonic in the interval @ . If the value and downstream spectra. corresponds to a maximum, then it is optimal to perform

1 It is key to note that the optimal transmit spectra do not ©£ EQPSD signaling in this bin. If the value corre- dictate any speciﬁc modulation scheme, but rather sim- sponds to a minimum, then the maximum is achieved by

1 ply describe how a modulation scheme should optimally £ the value , meaning it is optimal to perform FDS distribute its power over frequency. Thus, optimal spec-

signaling in this bin. No other values of are an optimal tra can be used with a number of different modulation option. We can write test conditions to determine the sig- schemes, including, but not limited to, DMT, CAP, QAM, naling nature (FDS or EQPSD) in a given bin by solving PAM, etc.

(15):

0 0

D D

£ 1

9CV! 9 % V¨f¢¡

@ Figures 4 and 5 illustrate the optimal transmit spectra 0

0 (16)

D D

Mñ, 9 9;

for HDSL2 on CSA loop ¦ in the presence of HDSL2 V¨§

interferers, and a combination of V§ T1 NEXT and 0

0 1

D D

£ 1

9CV! 9 %

V¨f¢¡ spectra to separate in frequency giving rise to an FDS re-

0 (17)

D D

B gion. At lower frequencies the upstream and downstream

Mñ, 9 9;

spectra share the same spectrum using EQPSD signaling.

0 Note: In each bin the optimal spectra exclusively employ The switch-over frequency represents the transi-

1 tion from EQPSD to FDS signaling. As an added bonus,

©£ £ EQPSD or FDS signaling; that is, or only. FDS scheme is a special case of the more general orthogonal no echo cancellation is required in the large FDS region. signaling concept. However, of all orthogonal signaling Bridged taps are short segments of twisted pairs that at- schemes, FDS signaling gives the best results in terms of tach to data-carrying twisted pairs between the subscriber spectral compatibility under an average power constraint and the central ofﬁce. Bridged taps are terminated at the and hence is used here (see proof in [8]). other end with some characteristic impedance. They lead to a non-monotonic channel transfer function (with nulls B. Optimal Spectra: All frequency bins in it) of the data-carrying twisted pair. Our technique ap- plies even to such channels and we can derive optimal The above analysis dealt with only a single frequency and near-optimal spectra for them. Figure 6 illustrates

bin centered around frequency #8 (see Figure 2). To ob-

tain the complete optimal spectra, we apply the test con- Simulation Details:

Bit rate ﬁxed at Mbps. Total average input power in each direc-

ditions in (16) and (17) to each frequency bin in @ and

o§ ')( tion "!$# %& dBm. use a variation of water-ﬁlling for the power distribution. Different service interference models obtained from Annex B of T1.413- A simple iterative algorithm yields the complete optimal 1995 (from [6], the ADSL standard), with exceptions as in [11]. Self- transmit spectra [8]: NEXT interference modeled as a 2-piece Unger model [7]. Margins calculated according to [12].

1. Estimate which bins employ EQPSD signaling and OPTIS transmit spectra obtained by tracking dBm/Hz below the OP- FDS using (16) and (17). We have shown that in our case TIS PSD masks [13]. OPTIS performance margin ﬁgures from [13].

AGN of *+-,/. dBm/Hz added to the interference.

(low self-FEXT) we get an EQPSD region to the left of DMT modulation scheme:

a switch-over bin and FDS region to the right of it Sampling frequency 021%34... kHz.

56%7 89%:.

0 Bin width kHz. Number of bins .

[8]. Estimate the switch-over bin . 4.§;< Start frequency = kHz. Bit error rate = .

2. Perform optimal power distribution within the EQPSD

§ ( SNR gap = dB. No cyclic preﬁx. No limitation on maximum number region using water-ﬁlling [9] and in the FDS region using of bits per tone. Upstream Upstream −35 −35

−40 −40

−45 −45

−50 −50

Amplitude (dBm/Hz) Amplitude (dBm/Hz)

0 100 200 300 400 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Downstream Downstream −35 −35

−40 −40

−45 −45

−50 −50

Amplitude (dBm/Hz) Amplitude (dBm/Hz)

0 100 200 300 400 500 0 100 200 300 400 500 ME2F Frequency (kHz) ME2F Frequency (kHz)

Fig. 4. Optimal upstream and downstream transmit spectra for HDSL2

on CSA loop = with self-NEXT and self-FEXT interferers. Fig. 5. Optimal upstream and downstream transmit spectra for HDSL2

), ), ),

EQPSD signaling takes place to the left of switch-over frequency on CSA loop = with self-NEXT, self-FEXT, and T1 NEXT >@?BADC and FDS to the right. (Note that the placement of the FDS interferers. EQPSD signaling takes place to the left of switch-over regions is nonunique. Here we choose a grouping such that upstream frequency >@?BADC and FDS to the right. and downstream transmissions use equal transmit powers and result in equal performance margins. Other groupings are possible; see [8] for details.) Upstream −35

TABLE I −40

Uncoded performance margins (in dB) for HDSL2 on CSA loop = : −45 OPTIS vs. Optimal. OPTIS ﬁgures were obtained from [13]. Diff = Difference between worst-case optimal and worst-case OPTIS. −50

Amplitude (dBm/Hz)

0 100 200 300 400 500 Crosstalk OPTIS Optimal Frequency (kHz) source Up Dn Up Dn Diff Downstream −35 49 HDSL 2.7 12.2 18.5 18.5 15.8 25 T1 19.9 17.5 21.3 21.3 3.8 −40 39 self 2.1 9.0 18.3 18.3 16.2 −45 24 self+24 T1 4.3 1.7 5.4 5.4 3.7 −50

Amplitude (dBm/Hz)

0 100 200 300 400 500 ME2F Frequency (kHz)

the optimal transmit spectra for HDSL2 on CSA loop § B (with bridged taps) in the presence of HDSL2 inter-

ferers. Again, we get a distinct EQPSD region at low Fig. 6. Optimal upstream and downstream transmit spectra for HDSL2

frequencies and a FDS region at high frequencies. The on CSA loop , with self-NEXT and self-FEXT interferers.

EQPSD signaling takes place to the left of switch-over frequency >

non-monotonicity of the channel leads to a similar effect ?EADC on the power distribution (see Figure 6). and FDS to the right. Note that the optimal transmit spectra vary significantly with the interference combination. ers were considered; for HDSL2, “self” comprises both self-NEXT and self-FEXT. Equal performance margins B. Performance margins were obtained for upstream and downstream transmis- The amount of noise (in dB) a channel can sustain sions. We can clearly see that the optimal scheme out- while maintaining a fixed bit rate and bit error rate is performs OPTIS with large gains in all the cases. known as the noise margin or performance margin [14]. Table I lists the performance margins of the optimal trans- C. Spectral compatibility mit spectra versus those obtained using the OPTIS fixed Whenever we optimize the bit rate of any xDSL ser- transmit spectra (ANSI T1E1.4 committee’s standard for vice on a given line we need to make sure that this service

the HDSL2 service [13]) for CSA loop ¦ . For different does not significantly interfere with the existing neighbor- service interferers (HDSL and T1), only the NEXT pow- ing xDSL services. In other words, we need to check if TABLE II 3. Optimal spectra are not bound to any particular modu- Spectral-compatibility margins (in dB) for HDSL2 on CSA loop = : lation scheme. OPTIS vs. Optimal. OPTIS numbers were obtained from [11]. 4. There exist near-optimal transmit spectra that are very Crosstalk xDSL Optimal easy to compute, even for complicated loops such as those source service OPTIS Up Dn with bridged taps. 5. FDS regions require no echo cancellation. 49 HDSL HDSL 7.86 (OPTIS and Optimal 6. Transmit spectra can be adapted on-line to changes in not involved) the line conditions (e.g., temperature variations, etc.). 39 HDSL2 HDSL 7.84 13.28 20.84 7. Optimal spectra yield bounds on maximum achievable 49 HDSL2 HDSL 7.26 12.71 20.15 bit rates. Our scheme requires a priori knowledge of the characteristics of neighboring interfering services. These can ei- the service being optimized is spectrally compatible with ther be estimated online or analyzed in a worst-case man- existing services. Spectral compatibility is measured in ner for each DSL line. This information could also be terms of noise margins (called spectral compatibility mar- obtained from a central office database that specifies the gins) of neighboring services in the presence of the opti- type of services in each binder group in the telephone ca- mized service. ble. By design, the optimal transmit spectra achieve good REFERENCES spectral compatibility margins. Through optimal power distribution techniques, we distribute more power of the [1] I. Kalet, “The Multitone Channel,” IEEE Trans. Commun., vol. 37, Feb 1989. optimized xDSL service in the regions of low interference [2] A. Sendonaris, V. Veeravalli, and B. Aazhang, “Joint Signaling and less power in the regions of high interference. Thus, Strategies for Approaching the Capacity of Twisted Pair Chan- nels,” IEEE Trans. Commun., vol. 46, May 1998. we avoid the higher-power transmission frequencies of [3] R. Gaikwad and R. Baraniuk, “Optimal Transmit Spectra for neighboring lines and therefore simultaneously reduce the HDSL2 under a Peak Frequency–Domain Power Constraint,” effect of the leakage power from optimized xDSL into ANSI T1E1 Contribution, no. T1E1.4/98-188R2, 1998. [4] R. Gaikwad and R. Baraniuk, “Optimal Transmit Spectra for these neighboring lines. HDSL2,” ANSI T1E1 Contribution, no. T1E1.4/98-162R1, 1998. To illustrate, consider the spectral compatibility be- [5] K. Kerpez, “Near-End Crosstalk is almost Gaussian,” IEEE Trans. tween HDSL2 and HDSL. Table II lists the spectral com- Commun., vol. 41, Jan. 1993. [6] American National Standard for Telecommunications, “Network patibility margins of the optimal transmit spectra versus and Customer Installation Interfaces– Asymmetric Digital Sub- OPTIS [13] for CSA loop 6. We compare the perfor- scriber Line (ADSL) Metallic Interface,” ANSI T1E1 Standard, mance margins for HDSL in the presence of two types no. T1.413-1995, 1995. [7] K. Kerpez, “Full-duplex 2B1Q Single-pair HDSL Perfor- of interferers; other HDSL lines and HDSL2 lines. The mance and Spectral Compatibility,” ANSI T1E1 Contribution, column “OPTIS” lists the margins obtained using OP- no. T1E1.4/95-127, 1995. [8] R. Gaikwad and R. Baraniuk, “Spectral optimization and joint sig- TIS transmit spectra and the column “Optimal” lists the naling techniques for communication in the presence of crosstalk,” margins obtained using optimal transmit spectra (differ- Tech. Rep. 9806, Rice University, Electrical and Computer Engi- ent for each combination of interferers). We see that the neering Department, Houston, Texas, July 1998. [9] R. Gallager, Information Theory and Reliable Communication. optimal spectra have better spectral compatibility margins New York: Wiley, 1968. than OPTIS. Likewise, analysis for other xDSL services [10] J. Aslanis and J. Cioffi, “Achievable Information Rates on Digi- like T1 and ADSL yields similar results. tal Subscriber Loops: Limiting Information Rates with Crosstalk Noise,” ANSI T1E1 Contribution, vol. 40, Feb 1992. [11] G. Zimmerman, “Performance and Spectral Compatibility of OP- V. CONCLUSIONS TIS HDSL2,” ANSI T1E1 Contribution, no. T1E1.4/97-237, 1997. [12] G. Zimmerman, “Normative Text for Spectral Compatibility Eval- We solved an optimization problem to jointly maximize uations,” ANSI T1E1 Contribution, no. T1E1.4/97-180R1, 1997. [13] J. Girardeau, M. Rude, H. Takatori, and G. Zimmerman, “Updated capacity of DSLs in the presence of crosstalk. The re- OPTIS PSD Mask and Power Specification for HDSL2,” ANSI sulting optimal transmit spectra yield large performance T1E1 Contribution, no. T1E1.4/97-435, 1997. margin gains. We can trade these increased performance [14] M. Barton and M. Honig, “Optimization of Discrete Multitone to Maintain Spectrum Compatibility with Other Transmission Sys- margins for increased bit rates or decreased average trans- tems on Twisted Copper Pairs,” IEEE J. Select. Areas Commun., mission power. The spectra are by design spectrally com- vol. 13, pp. 1558–1563, Dec 1995. patible with existing services. The key advantages of this technique are: 1. Optimal transmit spectra yield significant performance margin gains as compared to fixed transmit spectra. 2. Optimal transmit spectra are inherently spectrally compatible with existing services.