- Email: [email protected]
A calculation of the capacity of a twisted-wire pair
SlGNAL
PROCESSING ELSEVIER
Signal Processing 41 (1995)391-393
A calculation of the capacity of a twisted-wire pair Lev Goldfeld, Dov Wulich* Department of Electrical & Computer Engineering, Ben-Gurion Universiiy of the Negev, Beer-Sheva. POB 653, Israel
Received 21 January 1994;revised 20 May 1994
Abstract An exact value of the capacity of a twisted-pair channel dominated by near-end cross-talk (NEXT) and additive white Gaussian noise (AWGN) is determined. The results obtained are compared to the upper bounds found recently (Kaiet and Shamai, 1990). Zusammenfassung Ein exakter Wert fiir die Kapazitlt eines Twisted-Pair-Kanals wird bestimmt fiir den Fall, da13dieser dominiert wird durch Nahnebensprechen (NEXT) und additives we&s GauBrauschen (AWGN). Die Ergebnisse werden verglichen mit den Obergrenzen, die kiirzlich (Kalet und Shamai, 1990) gefunden wurden. Rbumi! Une valeur exacte de la capacitt d’un canal ti paire torsadQ domini: par une interfkrence croisee proche (near-end cross-talk, NEXT) et un bruit blanc gaussien est dtterminte. Les rksultats obtenus sont cornpa& g des bornes supkrieures rkemment trouvks dans (Kalet et Shamai, 1990). Keywords:
Twisted-wire pair; Capacity
1. Introduction
High speed digital subscriber loop (HDSL) transmission is based on existing twisted-pair loop channels. In recent publications [l, 21 the capacity of such channels has been discussed. In Cl], the capacity of a twisted-wire channel dominated by near-end cross-talk (NEXT) is analyzed and the exact solution for NEXT-only channel is given. When, in addition to NEXT, additive white
Gaussian noise (AWGN) is also present in the channel, only the bounds on capacity are found. Our aim is to go one step further and to calculate the exact value of the capacity of the NEXT + AWGN channel. The analytical considerations will be based on the results obtained in [l].
2. General solution for C,,,, +AWGN Our work is based on the results obtained
*Corresponding author. Tel.: 912-l-461537. Fax: 972-7276338.E-mail: [email protected]. 0165-1684/95/$9.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0165-1684(94)00110-3
in [l]; therefore,
IIll.
let us recall the main results from
392
L. Goldfeld, D. Wulich / Signal Processing
The capacity for a NEXT-dominated pair loop with AWGN is given by
twisted-
Substituting (7) into (6) yields C=
C NEXT+AWGN m = sup
41 (1995) 391-393
s 0 log2 [ l+
IK(f)12K(f) lH,(f)]‘Ps(f) + No/2
1
df, (1)
where IH,(f)12 is the attenuation transfer function approximated in [l] by exp( -a$), lH,.(f)12 is the cross-talk transfer function approximated by /3f3j2, Ps(f) is the two-sided power spectral density
2
2 Cl
Nf)l,
-
(9)
0
where R(f) = (IH,(f)12)/(lHc(fd)(2). Substituting (4), (5) and (9) into (3) gives the optimal value of the psd of s(t): &Jf)
= +@(.M)
for_%_&,
(IO)
where
2CJw) - 11 ‘(“)
=
[14fL(f)12 + IK(f)121 + WW)14
(psd) of the transmitted signal s(t), and No/2 is the double-sided noise spectral density. The “sup” operation is carried out over all Ps(f) satisfying the average power constraint 2
cc Mf)df<
Ps.
s0
(2)
Using classic calculus of variation techniques the following result was obtained [l]:
ps,Jf) =
-
b
+ Jb2 2a
Substituting (10) into (8) we have fA No
s0
@(f,fd)df=
WW)12
+
Ps.
(11)
Eq. (11) represents a parametric equation from whichf, can be found analytically or numerically. Havingf,, the capacity can be found according to
forfG_L
(3)
=
s[ fd
0
a = I~x(f)12CI~x(f)12
+ IJ-W)l”)“”
c NEXT+AWGN 4ac
where
b= +
+ 41~,(f)12R(f)CI~,(f)12
IK(f)121,
+ IKU-N21,
(4)
log2
l
+
IKm2@cM4) IH,(f)pqj&)
+
1
1 dJ
(12)
In [l] only upper bounds were found for (12).
(5) 3. Example of capacity calculation
and 1 is the Lagrangian constant determined from (2). fd is the maximal value off for which c < 0, or equivalently for which
IKm12 = y 1 is determined from the equation 2 I” %,,(f)df s
= Ps.
In our work we calculate CNExT+ AwoN for 24gauge line (without bridge taps) for ]Hc(f)12 = exp( --a& lHJf)l” = j?f312 and for the following parameters: c1= k(l/l,); k = 1.158, lo = 18OOOft and for B = 10V9 (see Cl]). The CNEXT + AWGN is numerically calculated for I= 600 ft and for I = 18 000 ft. The results are shown in Fig. 1 together with the upper bounds as obtained in [l, Fig. 63. We see that the upper bound is very close except for the small transition region between NEXT and AWGN domination.
393
L. Goldfeld, D. Wulich / Signal Processing 41 (199.5) 391-393
r
I
I
I 60 P/No
I
I
I
70 [dB-kHz]
Fig. 1. Curves of CNEXT+AWGN as a function of P/N, for 1= 600 and 18000 ft.
4. Conclusion The exact value obtained for the capacity of the NEXT + AWGN twisted-wire channel is very close, at least in two examples, to the upper bound found in [l]. The differences between the upper bound and the exact value are in the transition region between AWGN and NEXT. The capacities obtained are much higher than the bit rates available in practice over twisted-wire channels.
Acknowledgements We would like to thank Dr. I. Kalet for his help.
References [l] I. Kalet and S. Shamai (Shitz), “On the capacity of a twisted-wire pair: Gaussian model”, IEEE Trans. Comm., Vol. 38, No. 3, March 1990, pp. 379-383. [Z] S. Shamai (Shitz), “On the capacity of a twisted-wire pair: Peak-power constraint”, IEEE Trans. Comm., Vol. 38, No. 3, March 1990, pp. 368-378.
PROCESSING ELSEVIER
Signal Processing 41 (1995)391-393
A calculation of the capacity of a twisted-wire pair Lev Goldfeld, Dov Wulich* Department of Electrical & Computer Engineering, Ben-Gurion Universiiy of the Negev, Beer-Sheva. POB 653, Israel
Received 21 January 1994;revised 20 May 1994
Abstract An exact value of the capacity of a twisted-pair channel dominated by near-end cross-talk (NEXT) and additive white Gaussian noise (AWGN) is determined. The results obtained are compared to the upper bounds found recently (Kaiet and Shamai, 1990). Zusammenfassung Ein exakter Wert fiir die Kapazitlt eines Twisted-Pair-Kanals wird bestimmt fiir den Fall, da13dieser dominiert wird durch Nahnebensprechen (NEXT) und additives we&s GauBrauschen (AWGN). Die Ergebnisse werden verglichen mit den Obergrenzen, die kiirzlich (Kalet und Shamai, 1990) gefunden wurden. Rbumi! Une valeur exacte de la capacitt d’un canal ti paire torsadQ domini: par une interfkrence croisee proche (near-end cross-talk, NEXT) et un bruit blanc gaussien est dtterminte. Les rksultats obtenus sont cornpa& g des bornes supkrieures rkemment trouvks dans (Kalet et Shamai, 1990). Keywords:
Twisted-wire pair; Capacity
1. Introduction
High speed digital subscriber loop (HDSL) transmission is based on existing twisted-pair loop channels. In recent publications [l, 21 the capacity of such channels has been discussed. In Cl], the capacity of a twisted-wire channel dominated by near-end cross-talk (NEXT) is analyzed and the exact solution for NEXT-only channel is given. When, in addition to NEXT, additive white
Gaussian noise (AWGN) is also present in the channel, only the bounds on capacity are found. Our aim is to go one step further and to calculate the exact value of the capacity of the NEXT + AWGN channel. The analytical considerations will be based on the results obtained in [l].
2. General solution for C,,,, +AWGN Our work is based on the results obtained
*Corresponding author. Tel.: 912-l-461537. Fax: 972-7276338.E-mail: [email protected]. 0165-1684/95/$9.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0165-1684(94)00110-3
in [l]; therefore,
IIll.
let us recall the main results from
392
L. Goldfeld, D. Wulich / Signal Processing
The capacity for a NEXT-dominated pair loop with AWGN is given by
twisted-
Substituting (7) into (6) yields C=
C NEXT+AWGN m = sup
41 (1995) 391-393
s 0 log2 [ l+
IK(f)12K(f) lH,(f)]‘Ps(f) + No/2
1
df, (1)
where IH,(f)12 is the attenuation transfer function approximated in [l] by exp( -a$), lH,.(f)12 is the cross-talk transfer function approximated by /3f3j2, Ps(f) is the two-sided power spectral density
2
2 Cl
Nf)l,
-
(9)
0
where R(f) = (IH,(f)12)/(lHc(fd)(2). Substituting (4), (5) and (9) into (3) gives the optimal value of the psd of s(t): &Jf)
= +@(.M)
for_%_&,
(IO)
where
2CJw) - 11 ‘(“)
=
[14fL(f)12 + IK(f)121 + WW)14
(psd) of the transmitted signal s(t), and No/2 is the double-sided noise spectral density. The “sup” operation is carried out over all Ps(f) satisfying the average power constraint 2
cc Mf)df<
Ps.
s0
(2)
Using classic calculus of variation techniques the following result was obtained [l]:
ps,Jf) =
-
b
+ Jb2 2a
Substituting (10) into (8) we have fA No
s0
@(f,fd)df=
WW)12
+
Ps.
(11)
Eq. (11) represents a parametric equation from whichf, can be found analytically or numerically. Havingf,, the capacity can be found according to
forfG_L
(3)
=
s[ fd
0
a = I~x(f)12CI~x(f)12
+ IJ-W)l”)“”
c NEXT+AWGN 4ac
where
b= +
+ 41~,(f)12R(f)CI~,(f)12
IK(f)121,
+ IKU-N21,
(4)
log2
l
+
IKm2@cM4) IH,(f)pqj&)
+
1
1 dJ
(12)
In [l] only upper bounds were found for (12).
(5) 3. Example of capacity calculation
and 1 is the Lagrangian constant determined from (2). fd is the maximal value off for which c < 0, or equivalently for which
IKm12 = y 1 is determined from the equation 2 I” %,,(f)df s
= Ps.
In our work we calculate CNExT+ AwoN for 24gauge line (without bridge taps) for ]Hc(f)12 = exp( --a& lHJf)l” = j?f312 and for the following parameters: c1= k(l/l,); k = 1.158, lo = 18OOOft and for B = 10V9 (see Cl]). The CNEXT + AWGN is numerically calculated for I= 600 ft and for I = 18 000 ft. The results are shown in Fig. 1 together with the upper bounds as obtained in [l, Fig. 63. We see that the upper bound is very close except for the small transition region between NEXT and AWGN domination.
393
L. Goldfeld, D. Wulich / Signal Processing 41 (199.5) 391-393
r
I
I
I 60 P/No
I
I
I
70 [dB-kHz]
Fig. 1. Curves of CNEXT+AWGN as a function of P/N, for 1= 600 and 18000 ft.
4. Conclusion The exact value obtained for the capacity of the NEXT + AWGN twisted-wire channel is very close, at least in two examples, to the upper bound found in [l]. The differences between the upper bound and the exact value are in the transition region between AWGN and NEXT. The capacities obtained are much higher than the bit rates available in practice over twisted-wire channels.
Acknowledgements We would like to thank Dr. I. Kalet for his help.
References [l] I. Kalet and S. Shamai (Shitz), “On the capacity of a twisted-wire pair: Gaussian model”, IEEE Trans. Comm., Vol. 38, No. 3, March 1990, pp. 379-383. [Z] S. Shamai (Shitz), “On the capacity of a twisted-wire pair: Peak-power constraint”, IEEE Trans. Comm., Vol. 38, No. 3, March 1990, pp. 368-378.