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ADSL, VDSL, and Multicarrier Modulation . John A. C. Bingham Copyright # 2000 John Wiley & Sons, Inc. Print ISBN 0-471-29099-8 Electronic ISBN 0-471-20072-7

6

DFT-BASED MCM (MQASK, OFDM, DMT)

[Weinstein and Ebert, 1971] described the simplest way of performing the modulation shown generally in Figure 5.1. There is no ®ltering of the output of the constellation encoders, and the real and imaginary parts of each of the N_car words are used to quadrature amplitude shift key (QASK) the N_cartones. An example of the QASKing of tone 8 with three successive symbols is shown in Figure 6.1.

Figure 6.1 Three symbols of a 16-pt QASKedtone 8. 91
The multiple QASK operation is performed using an inverse discrete Fourier transform (IDFT). Important things to note about this transform are:

^* The input comprises (N_car À 1) complex numbers, which are quadrature modulated onto tones 1 to (N_car À 1), plus two real numbers, which are modulated onto dc and tone N_car.

^* It generates N ˆ 2N_carreal samples and is called an N-point IDFT. ^* It can be performed ef®ciently as an IF(ast)FT,¹as described in Appendix C.

Subsequent implementations of MQASK, with different names, were described in [Keasler and Bitzer, 1980], [Hirosaki, 1981], and [Fegreus, 1986], but the only two that have survived are orthogonal FDM (OFDM) for wireless use (see the specialized bibliography at the end of the references) and DMTfor DSL. I will use the name MQASK whenever the emphasis is on the rectangular nature of the envelope, and DMTwhen discussing overall systems.

The baseband pulse is a rectangle of duration T, and the DFTof keyed tone n, F(n,k), comprises terms with amplitudes proportional to sinc(nÀk) and sinc(n ‡ k), but the phases of these and how they combine depends on the phase of the keyed sinusoid. We are mainly interested in the PSD of the signal, so it is convenient to write

jF…n; k†j² ˆ sinc²…n À k†‡ sinc²…n ‡ k†…6:1†

The ( n ‡ k) term causes a slight asymmetry about k = n, but for the values of n used for xDSL (typically 57), it is insigni®cant. Figure 6.2 shows |F(n,k)|²for n ˆ 8 with k treated as a continuous variable. It also shows the powers averaged across each subchannel band centered at integer values of k and width Áf (ˆ 1/T ). That is,

…mˆ‡0:5
jF_smothed…n; k†j² ˆ jF…n; k ‡ m†j²dm …6:2†
mˆÀ0:5

These are more indicative of the sidelobe magnitudes than the alternating zeros and peaks. As would be expected, the sum of the left-hand sides over all k values is unity. It will be useful later to have a simple mathematical model for this smoothed spectrum:

j
F
smoothed

…
n
;
k

†j
² %¹
_{22…jk À njÀ 1†} forjk À nj52 …6:3† This is also shown in Figure 6.2 as superimposed `Â’s. ¹Strictly speaking, the D refers to the algorithm and the F to the implementation, but for simplicity we will use the F for both from now on.
GUARD PERIOD 93
Figure 6.2 Sidelobe attenuation of QASKed tone 8.

Receiver. The matched receiver for a QASK signal is a demodulator followed by a complex integrate and dump; these operations can be performed by an FFT.

6.1 GUARD PERIOD

If an MQASK signal is passed through a channel with a ®nite IR, the envelope of every toneÐeach ideally rectangularÐwill be different. It is, however, useful to consider a generic envelope as shown in Figure 6.3; this suggests two ways of using an extra samples to overcome the distortion caused by the ®nite IR.

Figure 6.3 DistortedQASK envelope.

1. Add the last samples to the beginning of the signal (a cyclic pre®x), and then delay collecting the N samples in the receiver until the transient response has ®nished.

2. Add the ®rst samples to the end of the signal (a cyclic suf®x), and then similarly delay collecting the samples. The only difference between this and method 1 is that the apparent phase of every tone is delayed by 2nv/N.

The most important property of all envelopesÐthat the trailing edge is the complement of the leading edgeÐsuggests a third way of overcoming distortion.

3. Use a quiet period between symbols, and in the receiver add the last samples to the ®rst.

All three methods result in zero sensitivity to IR terms h_iwith i4v, and the same sensitivity to terms with i > v (see Section 6.2). The important differences are in the spectra of the transmitted signals and the effective transfer functions of the receivers:

1,2. Pre®x and suf®x. The frequency separation of the zeros of the output spectrum is N/ T (N ‡ ). Since the tone separation is 1/T, this means that the spectral zeros do not coincide with the tone frequencies.²The frequency separation of the zeros of the receiver transfer function, on the other hand, is 1/ T, so these zeros do coincide with the tone frequencies.

3. Quiet period. Because the transmitted pulse is of duration T, the frequency separation of the zeros of the output spectrum is 1/T. Conversely, because the receiver uses all (N ‡ )samples,the frequency separation of the zeros of the receiver transfer function is N/ T(N ‡ v).

These frequency separations become signi®cant when methods of shaping the output PSD (see Section 6.5) and canceling RFI (Sections 10.6.4 and 11.6) are considered.

The guard period wastes samples, so the data rate ef®ciency is
” ˆ¹ …6:4†_{1 ‡ =N}
How this translates into decibels of margin depends on the number of bits/s/Hz. For ADSL N ˆ 512 and ˆ 32, so ” % 0.94. Therefore, in round numbers:

² The orthogonality of the MQASK signals is sometimes “explained” by saying that the spectrum of each is zero at that other tone frequencies. This is clearly not the reason, because with a cyclic pre®x the zeros do not fall at the tone frequencies, yet orthogonality is achieved!

^* For a data rate of 6 ‡ Mbit/s using 1 MHz of bandwidth 0.35 bit/s per

Hertz are wasted, for a loss in margin of approximately 1 dB. ^* For a data rate of 1.5 ‡ Mbit/s on a long loop using 0.5 MHz of
bandwidth, the loss would be approximately 0.5 dB.

6.1.1 Length of the Guard Period

In some of the early writings on MCM it was argued that if the duration of the guard period exceeds the variation of the group delay across the band, all keyed tones will have “arrived” at the receiver by the end of the guard period, and orthogonality will be ensured. A simple counterexample to this is a channel that has a single pole; the group delay will have some ®nite maximum, but the IR will have in®nite duration, and orthogonality will not be preserved. Choice of the length of the guard-period requires a compromise between ef®ciency as de®ned by (6.4) and ease of designing the equalizer (see Sections 8.4.4 and 11.2).

6.2 EFFECTS OF CHANNEL DISTORTION

Because no practical channels have a ®nite IR, the ideal of an IR that is contained within the cyclic pre®x is never achieved in practice. The following analysis of the effects of a longer IR was ®rst described in [Jacobsen, 1996].

For all xDSL systems it is an acceptable approximation to limit the length of the theoretically in®nite IR to the symbol length (N samples). Therefore, let the causal IR of the channel³be de®ned as h_ifor 04i4NÀ1, and let a cyclic pre®x of terms be used. The equalization and timing recovery process described in Section 8.4.4 selects the block of ( ‡ 1) contiguous h terms that contains the maximum energy. That is, it ®nds the value of k for which the “windowed” energy (h² ‡ h² ‡ÁÁÁ‡ h² ) is maximized. In general, the best value of k_{‡1 ‡}
will not be zero (i.e., there will be both pre- and postcursors), but for the sake of clarity in this ®rst explanation we will assume that it is.

Let us also use a simple, speci®c set of parameters for the DMTsystem to be analyzed⁴: N ˆ 8 and ˆ 3 for a total symbol length of 11. If the mth symbol set before the cyclic pre®x is added is de®ned as [x_m;i] for i ˆ 1 to 8, then after transmission through the channel the samples of interest for the reception of the (m ‡ 1)th symbol are

yˆ‰xm;6; xm;7; xm;8; xm;1;FFF ; xm;8; xm‡1;6; xm‡1;7; xm‡1;8; xm‡1;1;FFF ; xm‡1;8Š

É h₀; h₁;FFF ; h₇Š…6:5† After stripping off the (now distorted) cyclic pre®x, the column vector [y_m‡1] for input to the DFTcan be written as the sum of two vectors:

³Including both converters, all ®lters, and the loop.
⁴Extrapolation to the general, or any other speci®c, case should be easy.

7 6 76 7
6
7 6 76 7
6

3 2 32 3
7 6 76 7 6
²y_m‡1;1 h₀00 00 h₃ h₂ h₁ x_m‡1;1⁶y_{m‡1;27 6}h₁ h₀00 0 h₄ h₃ h₂₇₆x_m‡1;27
6 7 6 76 7
⁶y_m‡1;3^{7 6}h₂ h₁ h₀00 h₅ h₄ h₃⁷⁶x_m‡1;3⁷
6 7 6 76 7
⁶y_{m‡1;47 ˆ 6}h₃ h₂ h₁ h₀ 0 h₆ h₅ h₄₇₆x_m‡1;47
6 7 6 76 7
⁶y_m‡1;5^{7 6}h₄ h₃ h₂ h₁ h₀ h₇ h₆ h₅⁷⁶x_m‡1;5⁷
6 7 6 76 7
⁶y_m‡1;67 6h₅ h₄ h₃ h₂ h₁ h₀ h₇ h₆76x_m‡1;67
6 7 6 76 7
4y_m‡1;75 4h₆ h₅ h₄ h₃ h₂ h₁ h₀ h₇54x_m‡1;75 y_m‡1;8 h₇ h₆ h₅ h₄ h₃ h₂ h₁ h₀ x_m‡ 32
76 7 6
^{1;8 …6:6†2}000 0 h₇ h₆ h₅ h₄ x_m;1^{3 6}000 0 0 h₇ h₆ h₅76x_m;27
6 76 7
⁶000 0 0 0 h₇ h₆⁷⁶x_m;3⁷
6 76 7
⁶000 0 0 0 0 h^{76 7}
7
76x_m;47_‡6
6 76 7
⁶000 0 0 0 0 0⁷⁶x_m;5⁷
6 76 7
⁶000 0 0 0 0 0⁷⁶x_m;67
6 76 7
4000 0 0 0 0 0 54x_m;75 000 0 0 0 0 0 x_m;8
! 7 6 76

The second vector clearly represents intersymbol interference (the effect of x_mon y_m‡1), but for easiest understanding of the detection method, the ®rst vector, which represents the effects of the “present” symbol, should be split into two:

y_m‡1 ˆ Hx_m‡1 ‡ H₀x_m‡1 ‡ H₁x_m …6:7†

where ²0 h₇ h₆ h₅ h₄00 0^3
600 h₇ h₆ h₅00 0⁷

3
7
6
²h₀ h₇ h₆ h₅ h₄ h₃ h₂ h₁
⁶h₁ h₀ h₇ h₆ h₅ h₄ h₃ h₂7
6 7
⁶h₂ h₁ h₀ h₇ h₆ h₅ h₄ h₃⁷
6 7
7
⁶H ˆ^{6 7
6h3 h2 h1 h0 h7 h6 h5 h47 …6:8†6}h₄ h₃ h₂ h₁ h₀ h₇ h₆ h₅⁷
6 7
⁶h₅ h₄ h₃ h₂ h₁ h₀ h₇ h₆7
6 7
4h₆ h₅ h₄ h₃ h₂ h₁ h₀ h₇5
h₇ h₆ h₅ h₄ h₃ h₂ h₁ h₀
7
6
6 7
6 7
⁶00 0 h₇ h₆00 0⁷
6 7
⁶00 0 0 h₇00 0⁷
H
0
ˆÀ
6 7
^{6 7 …6:9†6}00 0 0 0 0 0 0⁷
6 7
⁶00 0 0 0 0 0 0⁷
6 7
6 7
400 0 0 0 0 0 0 5
00 0 0 0 0 0 0

H is a circulant matrix, and the ®rst term of (6.7) would be the only one if the input x were cyclic or if the cyclic pre®x were long enough to span the IR. The subscripted H’s represent distortion: H₀ de®nes the interference of the “present” symbol with itself (i.e., intrasymbol/interchannel interference), and H₁ de®nes the interference from the “previous” symbol (i.e., intersymbol/ interchannel interference). It can be seen that each contains h_ionly with i>v; that is, no distortion results from IR terms within the range of the cyclic pre®x. Also H₀and H₁contain the same distribution of h_iterms, albeit in different places.

6.2.1 Total Distortion: Signal/Total Distortion Ratio
If the average energy of each transmit sample, x_m;iis normalized to unity, the total signal energy of the (m ‡ 1)th symbol is
X jHj² ˆ N h² …6:10†
iˆ0
and the total distortion energyÐ contributed equally by the mth and (m +1)th symbolsÐcan be seen from (6.9) and (6.6), respectively, to be
jH₀j²‡jH₁j² ˆ 2…h24 ‡ 2h25 ‡ 3h26 ‡ 4h27†…6:11†
Therefore, since N ˆ 2N_car, the signal/total distortion ratio (STDR)^5,6is
X
STDR ˆ N
^{,X 2} …6:12†_car h² …i À 3†h
iˆ0 iˆ4
⁵The STDR is a single wideband measure of distortion; in Section 6.2.3 we consider the SDRs on the individual subchannels.
⁶ The signal and distortion can be considered as passing through the “window” and being splattered on the “wall,” respectively.
which can be generalized to
STDR ˆ N _N ,_N
^Xh^{2 X} …i À †h² …6:13†_car
iˆ0 iˆ‡1

If we wish to compare this STDR to that for a single-carrier system ( N_car ˆ 1), we should set the length of the DFE equal to , so that both systems will be immune to IRs shorter than ( ‡ 1). Then

,
STDR
^{X 2 X 2} …6:14†_MCM;ˆ0 ˆ N_car h …i À †h
iˆ0 iˆ‡1
but
STDR ^{, X 2} …6:15†_SCM ˆ h20 h
iˆ‡1

For N large, STDR_MCM) STDR_SCM. This should make the equalization task much easier, but as we shall see, there are many factors that must be taken into account. It is interesting (and perhaps counterintuitive) that the performance of an MCM system, in which all the terms of the IR contribute to the signal energy, is only as good as that of an DF-equalized SCM system, in which only the ®rst term contributes!

6.2.2 Case of Both Post- and Precursors

Having established the ramp weighting of the energy for an IR with only postcursors, it is easy to generalize to the case where there are IR terms both before and after the selected window. If, in our previous example, the set [h₂;FFF ; h₅]⁷were chosen, the total distortion would be given by

jwallj² ˆ 2:…2h20 ‡ h21 ‡ h26 ‡ 2h27†…6:16†

and the speci®c- and general-case denominators of (6.12) and (6.13) would be changed appropriately. This ramp weighting of the walls is shown informally in Figure 6.4.

6.2.3 Distortion on Individual Subchannels: SDR(j)

The wideband STDR is not, however, the complete measure of the effects of distortion in an MCM system, because the total distortion will be distributed unequally among the subchannels, and the effects of that distributed distortion

⁷Being careful not to use negative subscripts for h, which would imply that the IR was noncausal. Figure 6.4 Weighting of window and walls.

will depend on the SNRs of the subchannels. For example, an SDR of 20 dB on a subchannel that carries only 2 bits would be fairly inconsequential; on a subchannel carrying 12 bits it would be disastrous.

Calculation of SDR( f ) is a tedious process; it involves:

1. Convolving the QASK signal (including the cyclic pre®x) for every used tone with the IR (equalized if appropriate) of the channel
2. Discarding the cyclic pre®x portion and FFTing the remaining N samples to generate the response to each tone and the contribution of that keyed tone to the total distortion
3. Accumulating the contributions for all used tones

Undoubtedly, several DMTsystem designers have done this, but detailed results have remained proprietary. A reportable result is that for the ADSL upstream channel, the distortion resulting from the loop, the transformers, and the POTSprotecting high-pass ®lters is such that at the low end of the band (typically, tones 7 to 11, as de®ned in Section 8.1.2), SDR < SNR, and there is a signi®cant loss of capacity.

Un®nished Business. The long path from the parameters of an equalizer to the effects on the overall channel capacity makes optimizing the design of an equalizer nearly impossible. This path might be shortened and/or smoothed, or a frequency-domain equalizer (see Section 11.2.2) might be used to link cause and effect more closely.

6.3 THE SIDELOBE PROBLEM

As can be seen from Figure 6.2, the sidelobes of an MQASK modulator or demodulator fall off slowly. This has several consequences, which we consider in turn.

6.3.1 Noise Smearing and Resultant Enhancement

The windowing process in the receiver spreads the input noise in any one subchannel over many. Consider two subchannels, m₁and m₂. Some of the noise in m₁will appear in the DFToutput of m₂, and vice versa. If the noise in m₁is

Figure 6.5 Receivedsignal on 9 kft of 26 AWG with HDSL andT1 NEXT.

much higher than that in m₂, the sidelobes of the higher noise may contribute signi®cantly to the lower noise, and in extreme cases, even become the dominant contributor. If the input SNR in m₁and m₂is the same, this noise smearing will have little effect on the total capacity, but if the SNR in m₂is higher than that in m₁, m₂will lose more in capacity than m₁will gain.

Figure 6.5 shows the level of the received signal on 9 kft of 26 AWG. It also shows the NEXTfrom 20 HDSL interferers (one of the test situations de®ned in T1.413) and from 4 T1 interferers in an adjacent binder-group (very severe interference on that length of loop). Figure 6.6 shows the SNRs for HDSL NEXT at the input to the receiver and at the output of the DFTwhen all the noise powers have been smeared by the F_smoothedde®ned in (6.3); it can be seen that the noise from about 380 kHz upward (the region where the SNR is highest) has been increased by an average of about 5 dB. The noise-smearing effect would be moderately serious.

In practice, the front end of the receiver may contain an equalizer of some sort, which would amplify the signal at higher frequencies. Since it will also amplify the noise there, it will eliminate the harmful effects of smearing the lowfrequency noise into the higher frequencies. In most cases, however, smearing of high-frequency noise into the lower frequencies is, as we shall see in Section 6.3.2, much more serious.

Figure 6.6 SNRs with HOSL NEXT: without equalization.
6.3.2 Noise Enhancement from Linear Equalization

The most basic linear equalizer for any modem receiver is a linear two-port that equalizes the amplitude and phase of the channel over some used band so that the impulse response of the tandem connection of channel and two-port is an impulse. The zero-forcing equalizer has been discarded for many highperformance single-carrier systems because of its noise enhancement. A method that minimizes the sum of the residual distortion and noise can do a better job, but for channels with severe amplitude distortion the loss of capacity due to noise enhancement is still typically too great to be tolerable. The decisionfeedback equalizer (DFE) is a much better solution, and is well established and understood (see, e.g., [Honig and Messerschmitt,1984]).

For MCM the situation is very different, and the following argument has often been used to try to show that a linear equalizer would be adequate.

^* The capacity of the full channel is the sum of the capacities of the subchannels.
^* The capacity of each subchannel depends only on the SNR of that subchannel.
^* This SNR is not changed by linear equalization because signal and noise are ampli®ed or attenuated equally.
^* Therefore, the capacities of all the subchannels and of the full channel are not changed.

This argument is wrong because, although the equalization itself is linear, the overall detection process is not. In Section 6.3.1 we saw that because of the slow decay of the sidelobes of a rectangularly windowed DFT, colored input noise may result in noise ampli®cation at some frequencies (often, those that have the highest input SNR), and loss of capacity.

If the coloring of the noise is due only to crosstalk transfer functions, the effect is, as we saw in Section 6.3.1, only moderate, but if already colored noise is further colored by equalization the effect may be serious. Figure 6.5 shows the received ADSL downstream signal and noise levels before equalization, for an extreme example of 9 kft of 26 AWG with 4 T1s in an adjacent binder⁸; Figure 6.7 shows the SNRs after equalization at both the input and output of the DFT. It can be seen that the smearing would greatly reduce the SNR at low frequencies.

Figure 6.7 SNRs with TI NEXT: with equalization. ⁸Signal and noise are monotonically decreasing and increasing, respectively, with frequency, which results in the greatest coloring of the noise input to the FFT.
Depending on whether FDD or EC were used, this would probably result in a loss of 20 to 30% of capacity.

A crude predictor of the amount of noise enhancement is the sum of the equalization and the noise coloration (both in decibels) across the full band. A simple requirement would be that the extra noise that is spread into the lowest bin should be less than the noise that is already there; that is, it less than doubles the noise there and reduces the capacity by less than one bit. To satisfy this across a band of approximately 250 subcarriers

ÁdB_noise ‡ ÁdB_eq < 45 dB …6:17†
where ÁdB_noiseand ÁdB_eqare the increases in noise PSD and equalizer gain from one end of the band to the other.

NOTE: This calculation must be performed across the full band even if the upper subbands are unusable because of low SNR. Unless ®ltered out before the FFT, the noise up there will still be spread into the lower subbands.

One of the advantages of a guard period (quiet period or cyclic pre®x) is, as we shall see in Section 8.4.3, that it is not necessary to completely equalize an input signal. An algorithm for designing a “partial” equalizer should strive to minimize this noise enhancement effect.

6.3.3 Reducing Noise Enhancement Noise enhancement can be reduced in three ways:

1. Increasing the size of the DFT (i.e., reducing Á f ). Doubling the size of the DFT(halving Á f ) would double the sidelobe number at any frequency and thereby reduce the magnitude of all sidelobes by approximately 6 dB. Crude calculations suggest, however, that for a typical ADSL system with as much as 50 dB variation of attenuation across the band a DFTsize of at least 4096 would be needed to bring the loss of capacity in the lower subbands due to noise enhancement down to an acceptable level. This would be impractical from both memory size and latency considerations.

2. Using a guard period. The guard period was ®rst used commercially in Telebit’s Trailblazer voiceband modem. The FFT size was 2048, and the attenuation distortion across a voiceband is usually less than 15 dB. Noise enhancement was not serious, and the only purpose of the guard period was to simplify the equalizer; shortening the impulse response to ( ‡ 1) samples obviously required fewer taps than shortening it to 1. For DSL the situation is different. ADSL latency requirements limit the symbol duration to about 250 ms, and therefore the DFTsize to 512; as we have seen, equalization to an impulse (with a linear equalizer of in®nite complexity!) would seriously degrade performance. A guard period allows for a very wide choice of shortened impulse responses⁹(SIRs), and in doing so, greatly simpli®es the design of the equalizer. Just as important, it may also reduce the noise enhancement to a tolerable level. The characteristics of the SIR and the design of an equalizer to produce this are discussed in Section 11.2.1, but a little pre-motivation may be useful. Figure 6.8 shows the amplitude responses of the equalizers may be useful. Figure 6.8 shows the amplitude responses of the equalizers sample [(1 ‡ 0.8D)(1 ‡ 1.4D ‡ 0.7D²)(1 ‡ 0.6D²)]. It can be seen that the amplitude variation across the passbandÐand thence the potential for noise enhancementÐcan be reduced signi®cantly by even a short SIR.

3. Using a demodulation method that attenuates the sidelobes much more rapidly. Three such methodsÐSMCM, frequency-domain partial response, and DWMTÐare discussed in Chapter 7, but very little has been published about the equalization of such signals.

Figure 6.8 Response of equalizers of 9 kft of 26 AWG to generate two SIRs.

⁹ The term desired impulse response was used in much of the literature on DFEs (e.g., [Honig and Messershmitt, 1984]), but it is misleading because it implies that the SIR can be de®ned a priori, which is rarely the case.

Figure 6.9 Sidelobes at the edge of an MQASK passband.
6.3.4 Band Limiting

Figure 6.9 shows the cumulative effect in the stopband of the sidelobes of a set of MQASK signals in a passband. A simple model for this that will be useful when designing ®lters (see Section 8.3.4) is

jF
N
^{X 1} …6:18†_stopband…k†j² ˆ_22m2
mˆk

where N is the total number of modulated tones in the passband, k is the number of tones removed from the band edge, and jF_passbandj²is normalized to unity. It can be seen that turning off tones provides only a mild band limiting of an MQASK signal that will usually have to be augmented by a ®lter (see Section 8.3.4).

6.4 REDUCING THE SIDELOBES: SHAPED CYCLIC PREFIX

Many ways of reducing the sidelobes have been proposed; in this section we describe some simple modi®cations of DMT(see also [Weinstein and Ebert,1971], [Bingham, 1995], and [Spruyt et al., 1996]); Chapter 7 describes some different MCM systems. A shaped cyclic pre®x is between the two extremes of unshaped cyclic pre®x and a guard period discussed in Section 6.1. The samples of the cyclic pre®x are weighted by w(i), i ˆ1to , and the last samples of the pulse are weighted by a complementary [1 w(i)]. Typically, the pulse is symmetrical; that is, [1Àw(i)] ˆ w( ‡ 1Ài), and the most common shape is sine-squared (also known as raised cosine); that is, w(i) ˆ sin²(i/2).

The shaping can be done in many different ways, which differ in whether they provide the extra sidelobe attenuation in the transmit PSD or the receive transfer function, and also in the separation of the spectral zeros [Áf or N Á f/(N ‡ )]:

1. A cyclic pre®x is added and shaped as described above. Figure 6.10 shows one side of the transmit PSD with N ˆ 512 and ˆ 32: with a rectangular shaping as de®ned for ADSL in T1.413, and sine-squared shaping. As would be expected, shaping only 64 of the 544 samples has very little effect on the close-in sidelobes, but there is a very useful extra attenuation of the far-out sidelobes. The zeros of the PSD are separated by Áf (i.e., they fall on the tone frequencies). In the receiver, the ®rst samples, instead of being discarded as in the unshaped case, are added to the last samples before input to the DFT. The receive transfer function has the large sidelobes, and its zeros are separated by NÁf/(N ‡ ).

Figure 6.10 One sideband of QASK PSD with rectangular and shaped cyclic-pre®x envelopes.

2. A cyclic pre®x is added and the pulse envelope is transmitted unshaped. The transmit PSD has the large sidelobes, and its zeros are separated by NÁf/(N ‡ ). The shaping is applied in the receiver, and then, as in method 1, the ®rst samples are added to the last before input to the DFT. The receive transfer function has the attenuated sidelobes, and its zeros are separated by Áf.

3. As in method 1, a cyclic pre®x is added with its samples weighted by w(i), and a cyclic suf®x is also added, with its samples [i ˆ (N ‡ ‡ 1) to (N ‡ 2)] weighted by [1Àw(i)]. The transmit PSD has the attenuated sidelobes, and its zeros are separated by NÁf/(N ‡ ). Because the envelope pulse has been extended to (N ‡ 2) samples, successive symbols overlap. That should not matter, however, because in the receiver all the shaped samples are discarded. The transfer function has the large sidelobes, and its zeros are separated by Áf.

4. As a combination of methods 1 and 2, half-shaping [i.e., w(i) ˆ sin (i/2)] can be applied in both transmitter and receiver.
7 6
3
7 6
6.4.1 Sensitivity to Channel Distortion

The sensitivity to the h_i terms is the same for all four methods, so we will analyze only method 1. [Jacobsen, 1996] showed that for our example of N ˆ 8, ˆ 3, H is as given in (6.8), but if the shaping is symmetrtical, H₀and H₁as given in (6.9) and (6.6) must be modi®ed to

²0 h₇ h₆ h₅ h₄ w₃h₃ w₂h₂ w₁h₁⁶00 h₇ h₆ h₅ w₃h₄ w₂h₃ w₁h₂7
6 7
⁶00 0 h₇ h₆ w₃h₅ w₂h₄ w₁h₃⁷
6 7

⁶ 00 0 0 h₇ w₃h₆ w₂h₅ w₁h^7
47 …6:19†₀ˆÀ⁶H^{6 7
6}00 0 0 0 w₃h₇ w₂h₆ w₁h₅⁷

6 7
⁶00 0 0 0 0 w₂h₇ w₁h₆7
6 7
400 0 0 0 0 0 w₁h₇5 ₂00 0 0 0 0 0 0_{300 0 0 h7 w3h6 w2h5 w1h4}
⁶00 0 0 0 w₃h₇ w₂h₆ w₁h₅₇
6 7
⁶00 0 0 0 0 w₂h₇ w₁h₆⁷
6 7
⁶00 0 0 0 0 0 w₁h₇7 ^{H1 ˆ 6 7 …6:20†6}00000 000⁷
6 7
⁶0 h₇ h₆ h₅ h₄ w₃h₃ w₂h₂ w₁h₁7
6 7
400 h₇ h₆ h₅ w₃h₄ w₂h₃ w₁h₂5 00 0 h₇ h₆ w₃h₅ w₂h₄ w₁h₃
7 6
7 6
7 6
As in the unshaped case, the total distortion energy is contributed equally by the mth and (m ‡ 1)th symbols, and
jH₀j²‡jH₁j² ˆ 2‰w21h21‡…w21 ‡ w22†h22 ‡ Sh23‡…1 ‡ S†h24‡…2 ‡ S†h25ÁÁÁŠ …6:21† where
Sˆ…w21 ‡ w22 ‡ w23†
This can be generalized to
j
H
0
j
²‡jH₁j² ˆ 2‰w21h21‡…w21 ‡ w22†h22‡ÁÁÁ‡ Sh²‡…1 ‡ S†h²