Sopa de cebada

The discussion above provides all the components needed to do basic capacity calculations for DSL systems. The characteristics of twisted-pair wires as a communication medium are addressed in Chapters 2 and 3. The frequency-dependent attenuation of the transmitted signal can be computed based on the loop models given in Chapter 2. Given an ABCD matrix for any loop, its insertion loss can be computed (as in Equation 2.63) according to

H_iloss( f ) = ZS( f ) + ZL( f )

ZS( f ) (C( f )ZL( f ) + D( f )) + A( f )ZL( f ) + B( f ), (4.78) where A( f ), B( f ), C( f ), and D( f ) are the frequency elements of the ABCD matrix (see Section 2.3.7), and Z_S( f ) and ZL( f ) are the source and termination impedances at either end of the loop. The PSD of the received signal is then given by

S_p( f ) = Sx( f ) |Hiloss( f )|², (4.79) where S_x( f ) is the PSD of the transmitted signal.

As discussed in detail in Chapter 3, crosstalk is the primary noise source in DSL systems.

For the purpose of estimating the performance of a DSL system, the crosstalk PSD needs to be computed, in most cases by using one of the models presented in Section 3.6. Both the crosstalk noise and the background noise are modelled as additive Gaussian noise.

Subsection 4.6.3 describes basic data rate calculations for PAM, QAM, and CAP systems in the presence of additive Gaussian noise (for example, crosstalk and background noise).

Subsection 4.6.4 describes the corresponding basic data rate calculations for DMT systems.

By combining all the above information, it is possible to do basic data rate calculations for DSL systems.

The following example demonstrates basic data rate calculations for PAM, QAM, CAP, and DMT systems. In this example, the channel is a loop consisting of a 1 km long 0.4 mm wire with worst-case crosstalk noise according to the ETSI ADSL FB disturber model (see [ETSI TS 101 388]). The data rates are calculated for four hypothetical DSL systems that are identical except for the line code used. All four hypothetical DSL systems use the same transmit PSD with flat−40 dBm/Hz transmit power from 4 kHz up to 1.1 MHz. The loop insertion loss is computed according to Equation 4.78 and using the PE04 wire parameters from Table A.1 of [ETSI TS 101 388]. The received signal PSD is computed according to Equation 4.79 and is plotted in Figure 4.9. The crosstalk PSD profile is taken from Tables 14 and 15 of [ETSI TS 101 388], and the NEXT and FEXT noise are computed as described in Chapter 3. The received NEXT and FEXT noise PSDs are also shown in Figure 4.9. The

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FIGURE 4.9

Signal and noise PSD for example rate calculations.

signal and (combined) noise profiles in Figure 4.9 are used to calculate SNR values to use in the data rate calculations.

The PAM data rate can be calculated according to Equation 4.64 using the SNR derived from the signal and noise profiles in Figure 4.9. Because the signal bandwidth in this example is 1.1 MHz, it is appropriate to use a symbol rate of 2.2 MHz. The QAM and CAP data rates are computed according to Equation 4.40 assuming carrier frequency fc= 550 kHz and symbol rate 1.1 MHz. The DMT data rate is computed according to Equation 4.76 assuming 255 carriers with spacing off = 4.3125 kHz and a symbol rate of 4 kHz.

Table 4.2 gives the calculated data rate for all the systems. For comparison, the table also contains the Shannon channel capacity for the loop. Comparison of the data rates for the different systems shows that they all have very similar performance. This is not surprising, considering that the data rate formulas for all these systems are very similar in nature. The PAM modulation and QAM/CAP modulation have exactly the same (theoretical) perfor-mance. When bit error rate is considered, PAM and QAM/CAP (square constellations) have the same Shannon gap and therefore the same performance. The slightly lower data rate for

TABLE 4.2

Example Calculation of Transmission Rates

Calculated Modulation Γ[dB] fs[kHz] fc[kHz] Rate [kbps]

Shannon 0 N/A N/A 15260

PAM 9.95 2200 N/A 11635

QAM/CAP 9.95 1100 550 11635

DMT 9.95 4 (×255) k×4.3125 10827

DMT is because the calculations account for the 7.8 percent overhead due to cyclic-prefix (see Section 7.3) by assuming a symbol rate of 4 kHz but sub-carrier spacing of 4.3125 kHz.

Practical PAM and QAM/CAP systems require some excess bandwidth (see Chapter 6), which hasn’t been included in the calculations. The excess bandwidth would reduce the data rates from the values given here.

When all the different implementation issues are considered, there can be slightly larger variations in the calculated performance, but even then there is relatively little difference in performance among the PAM, QAM, CAP, and DMT line codes. The main differences among the four line codes are various practical implementation issues, where each line code has its strengths and its weaknesses.

References

[ANSI T1.413-2001] ANSI T1.413-2001, Spectrum Management for Loop Transmission Systems, American National Standard, 2001.

[ETSI TS 101 388] ETSI TS 101 388 V1.3.1 (2002–05), Asymmetric Digital Subscriber Line (ADSL) — European specific requirements, ETSI Technical Specification, 2002.

[ETSI TS 101 524] ETSI TS 101 524 V1.2.1 (2003–03), Symmetric Single Pair High Bitrate Digital Subscriber Line (SDSL), ETSI Technical Specification, 2003.

[Forney 1998] G.D. Forney Jr. and G. Ungerb ¨oeck, Modulation and coding for linear Gaussian channels, IEEE Transactions on Information Theory, Vol. 44, Issue 6, pp. 2384–2415, Oct. 1998.

[ITU-T G.991.2] ITU-T G.991.2 (02/2001), Single-pair high-speed digital subscriber line (SHDSL) transceivers, ITU-T Recommendation, 2001.

[Lee 1994] E.A. Lee and D.G. Messerschmitt, Digital Communication, 2nd ed., Boston: Kluwer, 1994.

[Papoulis 2002] A. Papoulis and S.U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed., New York: McGraw-Hill, 2002.

[Richards 1973] D.L. Richards, Telecommunication by Speech: The Transmission Performance of Telephone Networks. London: Butterworth, 1973.

[Shannon 1948] C.E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J., Vol. 27, pp.

379–423 and 623–656, July and Oct. 1948.

[Shannon 1949] C.E. Shannon, Communication in the presence of noise, Proc. IRE, Vol. 37, pp. 10–21, 1949.

[Shannon 1959] C.E. Shannon, Probability of error for optimal codes in a Gaussian channel, Bell Syst. Tech.

J., Vol. 38, pp. 611–656, May 1959.

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