2.2 Baseband Transmission

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Baseband codes are distinguished from passband codes (See Section 2.3) in that baseband codes can transmit energy at DC ( f = 0), while passband codes transmit at a frequency spectrum translated away from DC. Baseband line codes in DSLs have N = 1 and appear in the earliest DSLs: Examples are T1, ISDN, and HDSL.

2.2.1 The 2B1Q Line Code (ISDN and HDSL)

The 2B1Q baseband line code is heavily used in early DSLs. The 2B1Q line code ideally uses a basis function ^[2]

^[2] The sinc function is

which leads to the independence of successively transmitted data symbols when sampled at multiples of the symbol period. Thus, in the modulation of Section 2.1, N = 1, b = 2, M = 4, and the encoder is memoryless. On an ideal channel with no distortion, a corresponding matched filter in the receiver also maintains the independence of successive symbol transmissions. In practice, the sinc function cannot be exactly implemented, and transmission through the bandlimited channel distorts this basis function anyway. Thus, compromise pulse masks that attempt to predistort the transmit basis function to incur minimum distortion have been incorporated into the ISDN and HDSL standards documents [2,3] and are further detailed in [1].

The signal constellation for 2B1Q appears is basically ± 1, ± 3 (see [1]). The number of bits in a group is b = 2, and each group of bits maps into one of the four data symbol values. A one-dimensional value for x could be computed as x = 2 m - 3, m = 0,1,2,3. The name "2B1Q" derives from the mneumonic "2 bits per 1 quartenary" symbol. Because the transformer/hybrid coupling to the transmission line does not pass DC, baseband line codes like 2B1Q undergo severe distortion, and a straightforward implementation of a symbol-by-symbol detector will likely not perform acceptably. The receiver must compensate for the transformer and line distortion using the DFE of Section 2.5.

For purposes of analysis, the following function characterizes the one-sided output power-spectral density of the transmit shaping filters where the use of a rectangular basis function (with additional gain ) was assumed, along with a unity-DC-gain n th-order Butterworth filter,

where and R = 135 . For ISDN: n = 2 V _p = 2.5 V, 1/ T = f = f _{3 db} = 80 kHz, so the data rate is 160 kbps; and for HDSL: n = 4 V _p = 2.7 V, 1/ T = f = 392kHz, f _{3 db} = 196 kHz, so the data rate is 784 kbps (per line).

The probability of error, given an AWGN channel, is

with a straightforward encoder mapping. The gap for 2B1Q is 9.8 dB at P _e = 10 ^-7 . ISDN and HDSL also mandate an additional 6 dB of margin, leaving ISDN and HDSL at least 16 dB below the best theoretical performance levels. Additional loss of several dB occurs with ISI (see Section 2.5) on most channels. Thus, uncoded 2B1Q is not a high-performance line code, but clearly the transmitter is fairly simple to implement. The receiver for improved performance is more complex than necessary, leading in part to 2B1Q's later reduced use in DSL.

2.2.2 Pulse Amplitude Modulation (PAM)

2B1Q generalizes into what is known as pulse amplitude modulation (PAM), but has the same basis function as 2B1Q modulation, and simply has M = 2 ^b equally spaced levels symmetrically placed about zero, b = 1, ..., . The number of dimensions remains N = 1, and PAM also uses a memoryless encoder, x = 2 m - ( M - 1), m = 0, ..., M - 1 or a code like gray code in 2B1Q. Approximately a 6 dB increase in transmit power is necessary for the extra constellation points associated with each additional bit in a PAM constellation if the performance is to remain the same (i.e., P _e is constant). Eight-level PAM is sometimes called "3B1O" and is used in single-line HDSL or HDSL-2 standards ^[3] (see Chapters 5 and 6). In general for PAM,

^[3] The original HDSL makes use of 2 twisted pair (or 3 in Europe) while HDSL-2 uses only 1 twisted pair.

where T = 1 /f is the symbol period, R is the line impedance (typically 100 to 135 Ohms), and V _p = (M - 1 )d _min / 2 is the peak voltage. The power spectral density of the transmitted signal similarly generalizes to

with and f _{3 dB} equal to the cut-off frequency of a n th order (Butterworth) transmit filter. The data rate is log ₂ ( M )/ T bps. The probabilty of error is

assuming again an input bit encoding that leads to only 1 bit error for nearest neighbor errors. For more details on different baseband transmission methods like Manchester, differential encoding, J-ISDN, BnZs, HDB, and 4B5B, see [1].

While relatively harmless at low speeds, 2B1Q, and more generally PAM, should be avoided in DSL at symbol rates exceeding 500 kHz. Wideband PAM unnecessarily creates unacceptably high and unyielding crosstalk into other DSLs.

2.2.3 Alternate Mark Inversion (AMI)

Alternate mark inversion (AMI) is less efficient than PAM and used in early T1 transmission. The modulation basis function is dependent on past data symbols in that a 0 bit is always transmitted as a zero level, but that the polarity of successive 1's alternate, hence the name. One recalls that early data communciations engineers called "1" a "mark." Thus, AMI is an example of sequential encoding. The basis function can contain DC because the alternating-polarity symbol sequence has no DC component, thus the sinc basis function can again be used. However, the distance between the signal level representing a zero, 0 Volts, and 1 (either plus or minus a nonzero level) is 3 dB less for the same transmit power as binary PAM, so

This 3 dB performance loss simplifies receiver processing. An AMI transmitter is shown in Figure 2.6. Differential encoding of binary PAM signals ("1" causes change, "0" causes no change) provides signal output levels of ½ and - ½. The difference between successive outputs (+1, 0, or -1) is then input to the basis function shown to complete modulation. The receiver maps

Figure 2.6. Encoder/ modulator for AMI.

The transmit power spectral density of AMI is shaped as

assuming 6th-order Butterworth transmit filtering and no transformer high pass. A high pass filter that represents a transformer is sometimes modeled by an additional factor of where f _t is the 3dB cut-off frequency of the high pass).

ANSI T1 transmission uses AMI with the basis function shown in Figure 2.7.

Figure 2.7. T1.403 "T1" pulse shape (basis function after 1-D operation in T1 alternate mark inversion (AMI), 1 = 648 ns).

2.2.4 Successive Transmission

Data transmission consists of successive transmission of independent messages over a transmission channel. The subscript of n denoting dimensionality is usually dropped for one dimensional transmission and a time-index subscript of k is instead used to denote the message transmitted at time t = kT, or equivalently the k th symbol. Thus, x _k corresponds to one of M possible symbols at time k. The corresponding modulated signal then becomes

a sum of translated single-message waveforms. The symbols x _k may be independent or may be generated by a sequential encoder. A receiver design could sample the output of matched filter (- t ) on an AWGN channel at time instants t = kT. The design of the modulation filter ( t ) may not be such that successive translations by one symbol period are orthogonal to one another. However, this property is desirable and the function is called a Nyquist pulse when it satisfies the orthogonality constraint

Nyquist pulses exhibit no overlap or intersymbol interference between successive symbols at the matched-filter output in the receiver. The Nyquist criterion can be expressed in many equivalent forms, but one often encountered is that the combined transmit-filter/matched-filter shape q ( t ) = ( t )* (- t ) should have samples q ( kT ) = d _k or the "aliased" spectrum should be flat,

The function is a Nyquist pulse, but also one that "rings" significantly with time, because the amplitude of the sinc function at non-sampling instants decays only linearly with time away from the maximum at time t = 0. This leaves a transmission system designed with sinc pulses highly susceptible to small timing-phase errors in the sampling clock of the receiver. There are many Nyquist pulses, but perhaps the best known are the raised-cosine pulses, which satisfy the Nyquist criterion and have their ringing decay with 1 /t ³ instead of 1 /t. These pulses are characterized by a parameter a that specifies the fraction of excess bandwidth, which is the bandwidth in excess of the minimum 1 /T necessary to satisfy the Nyquist criterion with sinc pulses. The raised cosine pulses have response

The actual pulse shapes for transmit filter and receiver filter are square-root raised-cosine pulses and have transform and time-domain impulse response:

This pulse convolved with itself (corresponding to the combination of a transmitter and receiver matched filter) is a raised-cosine pulse. The time and frequency response of a square-root raised cosine transmit filter appear in [1].

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