2.3 Passband Transmission

   


2.3 Passband Transmission

Passband line codes have no energy at or near DC. The reason for their use in DSLs is because DSLs normally are transformer coupled (for purposes of isolation) from transceiver equipment, and thus the transformers pass no DC or low frequencies. This section reviews four of the most popular passband transmission line codes.

2.3.1 Quadrature Amplitude Modulation (QAM)

Quadrature amplitude modulation ( QAM ) is a N = 2-dimensional modulation method. The two basis functions are (for transmission at time 0):

graphics/02equ26.gif


where (t) is a baseband modulation function like sinc or a square-root raised cosine pulse shape. The multiplication of the pulse shape by sine and cosine moves energy away from baseband to avoid the DC notch of the twisted-pair transformer coupling. QAM pulses suffer severely from line attenuation in DSLs and compensation is expensive. Some proprietary DSL systems with QAM. QAM is often used in voiceband modem transmission where line-characteristics are considerably more mild over the small 3 “4 kHz bandwidth, allowing DC to be avoided and a receiver to be implemented with tolerable complexity. (See Section 2.4 on equalization.)

For successive transmission, QAM is implemented according to

graphics/02equ27.gif


in which one notes the sinusoidal functions are not offset by kT on the k th symbol period. Because of the presence of the sinusoidal functions and the potential arbitrary choice of a carrier frequency with respect to the symbol rate, QAM functions do not appear the same within each symbol period. That is, QAM basis functions are not usually periodic at the symbol rate, x n ( t ) x n ( t + kT ), even if the same message is repeatedly transmitted. However, the baseband pulse is repeated every symbol period. This aperiodicity is not typically of concern, but the use of periodic functions can allow minor simplification in implementation in some cases with the so-called "CAP" methods of the next subsection.

2.3.2 Carrierless Amplitude/Phase Modulation (CAP)

Carrierless amplitude/phase modulation (CAP) was proposed by Werner, who notes that the carrier modulation in QAM is superfluous because the basic modulation is two-dimensional and a judicious choice of two DC-free basis functions can sometimes simplify the transmitter implementaton [10]. The potentially high receiver complexity of QAM is still apparent with CAP (see Section 8.4.2 of [1] on Equalization).

CAP basis functions appear the same within each symbol period:

graphics/02equ26.gif


Successive transmission is implemented with

graphics/02equ29.gif


This form is a more natural extension of one-dimensional successive transmission and the basis-function concept. There is no carrier frequency, and f c is simply a parameter that indicates the center of the passband used for transmission, whence the use of the term "carrierless," while nevertheless the amplitude and phase of x(t) are modulated with the two-dimensional basis set. CAP transmission systems are not standardized for use in DSL, but have been used by a few vendors in proprietary implementations [9] and appear in HDSL reports (see Chapter 4).

CAP and QAM are fundamentally equivalent in performance on any given channel given the same receiver complexity ”only the implementations differ , and only slightly. For other quadrature modulation schemes, see [1].

2.3.3 Constellations for QAM/CAP

The easiest constellations for QAM are the QAM square constellations, which are essentially the same as two PAM dimensions treated as a single two-dimensional quantity. Indeed, square QAM and PAM are essentially equivalent when appropriate normalization to the number of dimensions occurs, graphics/02inl11.gif . Figure 2.8 shows SQ QAM constellations for even and odd numbers of bits per two-dimensional symbol. The expressions for probability of error are then

graphics/02equ30.gif


Figure 2.8. QAM/CAP square constellations.

graphics/02fig08.jpg

Another popular series of constellations for b 5 are the cross constellations that use the general constellation shown in Figure 2.9 and have relationship to probability of symbol error

graphics/02equ31.gif


Figure 2.9. General QAM cross for b > 4 and odd.

graphics/02fig09.gif

2.3.4 Complex Baseband Equivalents

Figure 2.10 shows a passband transmission system's energy location. The designer only cares about the region of bandwidth used and not the entire transmission band .

Figure 2.10. Passband channel input and output spectra.

graphics/02fig10.gif

Typically, these systems are two-dimensional and many designers prefer to describe the transmission system with scalar complex (rather than real) functions and analysis. There are two types of complex representations in common use: baseband equivalents that are most useful in analyzing QAM systems and analytic equivalents that are most useful in analyzing CAP systems. We first define the complex data symbol x b,k = x 1, k + jx 2, k .

A QAM waveform is determined by taking the real part of the complex waveform

graphics/02equ32.gif


where x a (t) is the analytic equivalent of the QAM signal. The baseband equivalent is just

graphics/02equ33.gif


The baseband equivalent does not explicitly appear to depend on the carrier frequency and essentially amounts to shifting the passband signal down to baseband. That is,

graphics/02equ60.gif


The baseband equivalent output of a noiseless linear channel with transfer function H(f) can be found as

graphics/02equ34.gif


Defining P(f) = H(f - f c ) F (f) as the pulse response of the channel, and scaling the channel output (signal and noise) down by a factor of graphics/02inl12.gif (to eliminate the extra square-root 2 factors in the normalized basis functions of QAM and the artificial doubling of noise inherent in the complex representation) produces the convenient complex baseband channel model

graphics/02equ36.gif


where x b,k are complex symbols at the channel input and n b (t) is complex baseband equivalent white noise with power spectral density 2 s 2 or equivalently s 2 per real dimension. This baseband equivalent channel has exactly the same form as with successive transmission with real baseband signals such as PAM, except that all quantities are complex. Thus, PAM is a special case. This representation allows a consistent single theory of equalization/modulation as will be followed by subsequent sections of this text and generally throughout the DSL industry.

CAP systems can also be modeled by a complex equivalent system above except that the channel output is still passband and is the analytic signal

graphics/02equ36.gif


where graphics/02inl16.gif and the noise is statistically again WGN with power spectral density s 2 per real dimension.

In either the QAM or CAP case, a complex equivalent channel has been generated and the effects of carrier and/or center frequencies can be subsequently ignored in the analysis, which can then concentrate on detecting x b,k from the complex channel output.


   
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DSL Advances
DSL Advances
ISBN: 0130938106
EAN: 2147483647
Year: 2002
Pages: 154

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