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
5
FUNDAMENTALS OF MULTICARRIER MODULATION
5.1 BLOCK DIAGRAM
A very simpli®ed block diagram of an MCM transmitter is shown in Figure 5.1; the receiverÐat least at the present level of detailÐis the mirror image of the transmitter. The input to the S/P converter is a sequence of symbols of B bits each; the output for each symbol is Ncargroups of b(n) bits each. That is,
X
B ˆ b…n†…5:1†
n4Ncar
Some of the b(n) may be zero, but that need not concern us yet. The groups of b(n) are then constellation-encoded, perhaps ®ltered, and then modulated onto Ncarsubcarriers; the methods of encoding and modulation are considered in Chapters 6 and 7.
The output of the modulator for the mth block is given by
X
y…mT ‡ t†ˆ realfp…m; n; t†exp…j2n…t=T†g for 0 < t4T …5:2†
n4Ncar
Figure 5.1 Simpli®ed block diagram of an MCM transmitter. 79
where p(m,n,t) is the baseband pulse resulting from any ®ltering of the output of the constellation encoder. [Zervos and Kalet, 1989] and [Kalet,1989] showed that if the integration of (4.4) is replaced by a summation over some large but ®nite set of subcarriers, each with bandwidth f, then (4.4) applies also to multicarrier modulation. That is,
X
R ˆ fB ˆ f b…n†…5:3†
n4Ncar
where ()
b
…
n
†ˆ
log
2
1
‡
3SNR…n†
…mar=cg†‰QÀ1…Pe=4†Š2 …5:4†
Equations (5.4) can be simpli®ed by combining the terms that are not functions of n into one variable:
ˆmar ‰QÀ1…†Š2 …5:5†3cg
so that
b…n†ˆ log21 ‡SNR…n† …5:6†
For an ADSL system with BER ˆ 10À7, mar ˆ 4 (6 dB), and cg ˆ 2.51 (4 dB), the “gap”1 % 14:0.
Equations (5.3) and (5.6) are a little more restrictive than (4.4) in that the range for n must be narrowed so that the argument of the log is at least 2.0 (i.e., bn, the number of bits assigned to each subcarrier, 51), and, furthermore, bn must be integer. Even though the requirement for single-carrier modulation is only that the SNR should be greater than unity, these extra constraints on multicarrier modulation are fairly inconsequential because:
* The capacity of the edges of the band beyond b(n) ˆ 1 and out to SNR ˆ 1 is very small, and a decision-feedback-equalized single-carrier modem can make use of those edges only with a large (perhaps impractically large) equalizer.
*We shall see how the log2(Á) can be rounded to the nearest integer with no increase in power or loss of capacity.
1So called by [Starr et al., 1999] because it is the gap between the practically achievable and the Shannon limit. For no margin and no coding gain and an error rate of 10À7 ˆ 9.0 (9.5 dB).
CHANNEL MEASUREMENT 81
The most important point about equations (5.3) and (5.4) is that the number of bits assigned to each subcarrier must be calculated from the SNR and sent back to the transmitter. This feedback from receiver to transmitter is analogous to the precoding ([Tomlinson, 1971] and [Harashima and Miyakawa, 1972]), that is used in single-carrier systems with severe channel distortion to avoid error propagation in the DFE; this is discussed in more detail in Sections 5.3 and 7.1.1.
By contrast, OFDM systems, which are used primarily for broadcastingÐ with no feedback possible from receiver(s) to transmitterÐuse a constant (or at least ®xed for a transmission session) bit loading. If this were used for transmission via the DSL, where the SNR varies widely across the band, then either the bit loading would have to be very conservative in order to protect the subcarriers with lowest SNRs, or the error rate on those subcarriers would be very high and would greatly degrade the performance.
5.2 CHANNEL MEASUREMENT
Calculation of the SNR requires two measurements for each subcarrier: of the channel response and of the variance about that response caused by noise, which is the sum of conventional noise, crosstalk, and residual (after equalization) channel distortion. The two measurements can be combined by the transmission of a pseudo-random sequenceÐusing all subcarriers2Ðthat subjects the transmitter/channel/receiver to all possible distorted sequences. The only question is how many blocks or symbols are needed. There are two requirements:
1. The error in the estimate of the response must be small enough that the “offset” (actual response minus assumed response) does not contribute signi®cantly to the total error during data transmission.
2. The error in the estimate of the SNR must be small enough that it does not signi®cantly affect the bit loading. If the aggregate noise is assumed to be Gaussian distributed, the standard deviation of the estimate of the SNR is 8.686/ . Using 4000 symbols, for example, would mean that there is a 1% chance that the actual SNR differs from the estimated SNR by more than 0.25 dB.
The ®rst requirement turns out to be much weaker than the second, so it can be ignored.
2 For channel estimation and bit loading there is, strictly speaking, no need to transmit subcarriers that can never be used for data transmission (e.g., those outside the band in an FDD system), but, as we shall see in Section 9.3, some of these subcarriers may be needed for accurate calculation of the channel impulse response.
5.3ADAPTIVE BIT LOADING: SEEKING THE 3“SHANNONGRI-LA” OF DATA TRANSMISSION
Several algorithms for calculating the b(n) have been described; the choice of the appropriate one depends mainly on whether the system is total power limited or PSD limited. The ®rst algorithm [Hughes-Hartogs, 1987] was developed for voiceband modems. These are total power limited because the important constraint is the power delivered to the multiplexing equipment at the COÐso many milliwattts regardless of the bandwidth used.
ADSL modems, on the other hand, are PSD limited because it is necessary to limit the crosstalk induced in other pairs. VDSL modems may be either total power or PSD limited. For both constraints the b(n) may be chosen to achieve any one of the following:
1. Maximum data rate at a de®ned error rate, margin, and coding gain: that is, at a de®ned
2. Minimum error rate at a de®ned data rate, margin, and coding gain: that is, maximum at a de®ned data rate
3. Maximum data rate that is an integer multiple of some N Â the symbol rate4under the same conditions as item 1.
5.3.1 Adaptive Loading with a PSD Limitation
Maximum Data Rate at a De®ned c.The algorithm for this requirement would appear to be very simple: the b(n) can be calculated from (5.6), and then R from (5.3). The single value of calculated from (5.5) is exact only for square constellations [i.e., for b(n) even]; for b(n) odd and 55 the error is less than 0.2 dB and can be ignored; for b(n) ˆ 1 and 3 should be increased by factors of 1.5 and 1.29 respectively. The continuously variable b(n) must, however, be rounded to integers, while maintaining the equality of error rate on all subcarriers in order to minimize the overall error rate. This must be done by scaling the transmit levels so as to result in new SNR values given by
log2…1‡…SNRH= †† ˆ ‰b…n†Š …5:7†log2…1‡…SNR= b…n† 3 I know, I used this one in [Bingham, 1990], but I cannot resist repeating it! ADAPTIVE BIT LOADING
where [b(n)] is the rounded value of b(n). The scaling parameters, called g(n)in T1.413, are then given by
g…n†ˆ2‰b…n†Š À 1 % 2‰b…n†ŠÀb…n† …5:8†2b…n† À 1
4T1.413 speci®es that the minimum increments of data rate should be 32 kbit/s, which is 8 Â the symbol rate.
Figure 5.2 Transmit gain adjustments, g(n), for a monotonically decreasing SNR.
The g(n) for the simple case of monotonically decreasing SNR and b(n) (from 12 on tone 30 to 4 on tone 255) are shown in Figure 5.2. The saw-toothed shape between 0.84 and 1.19 (Æ 1.5 dB) occurs because as the frequency increases, the rounding changes from rounding down to rounding up, and the g increases to compensate. This means that the PSD does exceed the limit in some narrow bands, but is within the limit when averaged over the range of any one value of b.
Maximum c at a De®ned Data Rate.For the second requirement it might seem that (5.6) and (5.3) could be solved to express as a function of B, but the nonlinear operation of rounding interferes. The following iteration is needed:
1. Calculate B1, a ®rst value of B from (5.3) and (5.6) using 1derived from the maximum acceptable error rate.
2. If B1< the desired Bdes, then reporting to, and renegotiation with, the higher layers may be needed.
3. If B1>Bdes, calculate k‡1from
…BkÀBdes†=Ncark for k ˆ 1;FFFk‡1 ˆk  2
where Ncarkis the number of subcarriers used on the kth iteration. 4. Repeat steps 1 to 3 as needed.
Because of the rounding operation this algorithm may oscillate about the desired B. Every programmer will have his or her favorite and proprietary way of avoiding this!
Maximum Data Rate That Is an Integer Multiple of Some N Â the Symbol Rate.The smallest increment of data rate for the mod/demod part of an MCM system is the symbol rate (adding one bit on one subcarrier), but the R-S FEC code words, which usually are locked to the symbol rate, are made up of bytes. Therefore, B is usually constrained to be an integer multiple of 8 Â fs. Then the preceding algorithm should be modi®ed:
1a. Calculate B0from (5.3) and (5.6) using 0derived from the maximum acceptable error rate.
1b. Truncate B0to B1ˆ the nearest multiple of 8 Â fs, and continue as previously.
5.3.2 Adaptive Loading with a Total Power Constraint
The basic principle of this algorithm, which is similar to, but slightly simpler than, that in [Hughes-Hartogs, 1987], is that the loading is increased one bit at a time, and each time the new bit is added to the subcarrier that requires the least additional power. This ensures that any accumulated data rate is transmitted by the minimum power.
An interesting small difference between the PSD-limited and total powerlimited cases is that for the latter one-bit constellations need not be considered. The power needed for one 4QAM subcarrier is the same as for two two-point subcarriers, so it is better to use the narrower band signal.5
The algorithm is therefore initialized by calculating the power needed for two bits on each subcarrier. The SNR needed for two bits is 3 . Therefore, if the noise power measured on tone k is (k)2, the received signal power needed for two bits is 3 (k)2, and the power that must be transmitted to deliver this is
P
…
1
;
k
†ˆ
3 …k†2
2jH…k†j…5:9† ˆ
3 Psc
SNR…k†
where Pscis the power transmitted per subcarrier during channel measurement (ˆ 4312.5 Â 10À4mW for ADSL). Then the incremental transmit powers needed for subsequent bits on that subcarrier can be de®ned by successive
5The bene®t from using just one one-bit subcarrier at the edge of the band is insigni®cant. SCM/MCM DUALITY 85 m P 6 10 20 42 82 170 330 682 ÁP 4 4 10 22 40 88 160 352 (m) 4.0 1.0 2.5 2.2 1.818 2.2 1.818 2.2
TABLE 5.1 Powers and Incremental Powers for Multipoint Constellations
ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ 3 4 5 6 78 910
ÁP…m; k† P…m; k†À P…m À 1; k†…5:10† …m†ÁP…m À 1; k†
The constellations de®ned in T1.413 are alternating squares and crosses with a suboptimal one used for three bits.6The normalized powers of these and the (m) for m ˆ 3 to 10 are shown in Table 5.1; thereafter the s alternate between 2.2 and 1.818. The (m) can be stored in ROM, and the incremental power updated after each bit addition by just the one multiplication shown in (5.10).
NOTE: The principle of always adding onto the “cheapest” subcarrier automatically generates the saw-toothed PSD caused by the g(n) of the previous algorithm.
This algorithm can be used for all three maximum data rate/minimum error rate combinations considered above. It can also be used with a PSD constraint; the process stops on each subcarrier if the next allocation would push that subcarrier over the allowed PSD.
This algorithm is better than the one in Section 5.3.1 in that it is more versatile and is guaranteed to converge to the optimum; it is worse in that it is slowerÐthe search over all the usable subcarriers for the smallest ÁP may have to be performed as many as 1500 timesÐand may be covered by the Hughes-Hartogs patent.
5.4 SCM/MCM DUALITY
Time-frequency duality was discussed in [Bello, 1964], and recently much has been made of the supposed duality of SCM and MCM systems: what SCM does in the time domain MCM does in the frequency domain, and vice versa. Examples that have been cited (and argued about) include:
1. A single tone of interference (“in the frequency domain”) was originally
6A star constellation is more ef®cient for three bits, but the sub-optimal one has the advantage that all the points lie on the square grid, which simpli®es coding and decoding.
said to be the dual of an impulse of noise (“in the time domain”), but [Werner and Nguyen,1996] pointed out that it really is the dual of a repeated sequence of impulses. Without any measures to correct for it, a single tone of suf®cient amplitude would wipe out a few subcarriers and cause a very high error rate; similarly, a sequence of impulses would wipe out a single-carrier system.
2. Conversely, a sequence of small impulses of noise would be spread evenly over all subcarriers, in the same way that a single interfering tone would be spread harmlessly over all time in a single-carrier system; neither would cause errors.
3. On the other hand, a sequence of large impulses would cause errors on all subcarriers in the same way that a large interfering tone would cause errors in all single-carrier symbols.
4. Ideal SCM pulses are limited in bandwidth and in®nite in duration; they maintain orthogonality because they are zero at regular sampling instants. Ideal MCM pulses that are generated by an IDFT (see Chapter 6) are limited in time and in®nite in bandwidth; they maintain orthogonality because they are zero at regular frequency intervals.
5. In Chapter 7 we consider a system that uses partial response in the frequency domain; this is a dual of the well-known SCM time-domain partial response systems described in [Lender, 1964] and [Kretzmer, 1965].
It must be recognized, however, that the concept of duality is useful only as a tool for early learning and perhaps later inspiration, and furthermore, only to those for whom intuition is an important part of understanding. For any particular problem (e.g., analysis of the effects of impairments) MCM must be analyzed with the same degree of rigor that has been applied to SCM problemsÐwithout invoking the fact (or, more probably, the opinion) that the problem is or is not the dual of one in SCM.
5.5 DISTORTION, EFFICIENCY, AND LATENCY
Transients occur only at the beginning and end of a multicarrier symbol, so for a channel with an impulse response of a given duration, the effects of distortion can be diluted by increasing the symbol length. This effect is quanti®ed for the “®lterless” implementation of MCM in Section 6.1. The processing time through a multicarrier transmitter and receiver is typically about ®ve symbols.7 The maximum length of a symbol is therefore limited to about 0.2 times the permissible latency. Other methods of reducing the effects of distortion are:
7It can be shortened by clever use of buffering and memory, but not by much. THE PEAK/AVERAGE RATIO PROBLEM 87
* Use of a cyclic pre®x (Section 6.2)
*Time-domain equalization (Sections 7.3.4 and 9.3) *Sidelobe suppression (Sections 9.2, 9.3, and 9.4)
5.6 THE PEAK/AVERAGE RATIO PROBLEM
If the N subcarriers of a multicarrier signal each have unit average power and are each modulated with just 4QASK (two bits), the root-mean-square andp p
maximum output samples arepN and N respectively; the peak/average ratio (PAR) that results is 2 . If the carriers were all modulated with
p
multipoint QASK constellations, which themselves have a PAR that approachesp
for large constellations, the output PAR would be .
For the downstream ADSL signal and for both down and up VDSL signals, N ˆ 512, which would result in a theoretical PAR 9 55; the summation of so many individual sine waves should, however, ensure that the central limit theorem applies, and the amplitude distribution of the signal for all probabilites
p
of interest can be considered to be Gaussian. For the upstream ADSL signal, N
ˆ 64 and the absolute PAR ˆ 384ˆ 19.6. This would again seem to be large enough to ensure a Gaussian distribution over the interesting range of amplitudes,8but I have heard reports that the real distribution is broader than Gaussian (i.e., the tails are higher). Checking this would, however, require either the simulation or the measurement of many millions of samples of the output signal (both very tedious), and I have seen neither con®rmation nor refutation of this; in the interests of simplicity we will assume a Gaussian distribution.
Such a distribution would, of course, have an in®nite PAR, but the peaks would occur only once in an aeon! All MCM systems must therefore decide on some PAR and be prepared to deal with the clips that occur if the signal exceeds that. PARs values less than 3.0 are almost certainly not practically attainable; PARs greater than 7.0 are, as we shall see, expensive and unnecessary. Choosing a number in that 3 to 7 range is an important preliminary task in the design of multicarrier systems.
NOTES:
by
p
1. SCM PARs are typically calculated in the baseband and must be increased (3 dB) to account for the modulation into a passband. It is sometimes said (e.g., in [Saltzberg, 1998]) that MCM PARs must be similarly increased, but this is incorrect. It is the output samples that are Gaussian distributed; whether they are considered baseband or passband signals is irrelevant.
8Typically for probabilities >10À9, that is, out to about the 8 point.
2. Whether the PAR of MCM signals is higher, signi®cantly higher, orÐas claimed in some of the more partisan writingsÐdisastrously higher than that of SCM signals is a very controversial subject. [Saltzberg, 1998] points out that the sharp bandlimiting ®lters (typically 15% excess bandwidth) used in SCM systems may increase the PAR calculated from simple modulation of a baseband constellation by as much as 6 dB: bringing the two systems to about the same PAR values. Ciof®9has suggested that this can be thought of as a Gaussian distribution induced by the multiple taps of the bandlimiting ®lters.
The main disadvantage of the high PAR of any signals, MCM or SCM, is that all components, analog and digital, must have a wide dynamic range. In order of increasing importance, this affects:
* The DSP. Providing one more bit of internal processing precision may, depending on the processor, be either inconsequential or extremely irksome.
* The converters. The digital-to-analog (DAC) and analog-to-digital (ADC) converters may need to accommodate one more bit than does a singlecarrier un®ltered signal. There are, however, ways of ameliorating this problem; these are described in Section 7.2.5.
* The quality of POTS service with some telephone handsets if no splitter is used. This problem is discussed in more detail in Chapter 9.
* The analog front-end circuitry. Increasing the peak output voltage does not signi®cantly increase either the output power or the active power consumed in the line drivers. The quiescent power consumed in the bias circuits of conventional drivers is, however, proportional to the square of this peak voltage. There are ways of reducing this bias power (many of them proprietary), but there is, nevertheless, a very strong motivation to reduce the output PAR.
Methods of reducing the PAR are discussed in Section 8.2.11; for the moment we concentrate on de®ning the problem.
5.6.1 Clipping
If the PAR is set at k, the probability of a clip is
…I 2=2 …5:11†clip ˆp eÀxPr 2=
k 9Private conversation.
CLIPPING 89
and the average energy in a clip (assuming unit signal energy per sample10)is
p
2eÀxEclip ˆ 2= x
…I 2=2 …5:12†
k
Early ADSL systems used a PAR of about 6.0 (15.6 dB); 4.0 (12 dB) is probably about the maximum that will be acceptable for a second-generation system, 3.5 (10.9 dB) is a reasonable number to strive for, and 3.0 (9.5 dB) is an aggressive goal for a G.lite system. We will therefore consider k values over a range 3.0 to 6.0.
For these values of k,Eclipis very small and the signal/clip energy ratio (SCR) is high. It might seem, therefore, that the contribution of clip energy to the total noise would be insigni®cant. This is, however, very misleading; the clip energy must not be averaged over all symbols, but only over those symbols in which a clip occurs. The probability of a clip occurring in a symbol is
Prclipsymb ˆ 1À…1 À Prclip†N …5:13†
and since the signal energy per sample is normalized to unity, the signal/ conditional clip ratio is
SCCR ˆPrclipsymb …5:14†Eclip
Prclipsymband SCCR are shown in Table 5.2 for values of k from 3.0 to 6.0 for the case N ˆ 512.
In every symbol in which any number of clips occur, noise will be spread evenly across the entire band and will eventually cause errors on all subcarriers for which the loading is such that [3b(n)À3) > SCCR. As an example, if the maximum loading is 12 bits, errors will occur in all symbols for which SCCR < 33 dB. It must be noted, however, that the SCCRs in Table 5.2 are average values; the total clip energy and the resultant SCCR will vary widely from one
TABLE 5.2 Clip Probabilities and SCCR Values for Various PARs PAR ( k) Pr(clipsymb) 0.75 0.21 3.2 Â 10À2 3.5 Â 10À3 2.9 Â 10À4 1.9 Â 10À5 10À6 Average 32.7 35.8 37.1 38.0 38.8 39.5 40.2 SCCR (dB)
ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ 3.0 3.5 4.0 4.5 5.0 5.5 6.0
10Normalizing with respect to the signal energy rather than power is simpler because it makes the explanation independent of the sample rate.
clipped symbol to the next. The statistical distribution of the SCCRs is dif®cult to calculate, but preliminary calculations show that the 1 percentile, for example, may be 12 dB or more below the average. This suggests that it would be almost impossible to prevent errors when clips occur and that the only feasible strategy is to set the PAR high enough to make the clip rate acceptably low. Most ®rst-generation ADSL transmitters played it safe and used PARs of 6.0 (15.6 dB) or more, but the methods of PAR reduction and clip shaping discussed in Section 8.2.11 should improve matters greatly for second-generation ADSL and VDSL.
Entropy of a Clip. Clips are very tantalizing because as discussed in more detail in Chapter 7 of [Starr et al., 1999] and many other papers referenced therein, the loss of capacity that would be incurred by detecting a clip in the transmitter and conveying all information about it to the receiver is very small. Even with a PAR of 10 dB, the theoretical loss is only about 3%.
Không có nhận xét nào:
Đăng nhận xét