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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

7

OTHER TYPES OF MCM

As we saw in Section 6.3, the big problem with DMT and OFDMÐand any other MCM system that uses rectangular pulsing of sinusoidal carriers (“tones”)Ðis the sidelobes. In summary, the high-level sidelobes of DMT:

1. Increase the sensitivity to channel distortion.
2. Increase noise enhancement in all linear equalizers and therefore make a guard period essential.
3. Make the problem of digital RFI cancellation (see Section 10.3) much harder.
4. Increase ICI and ISI at the band edges of an FDD system (this is really another manifestation of effect 1: for the wideband DMT signals, the sharp cutoff of the ®lters is an extreme form of channel distortion).
5. Increase the sensitivity to frequency offset [Armstrong, 1998]; this is not important in DSL systems, but it is very important in broadcast wireless systems, which use unmatched up and down conversion stages.

A guard periodÐand its most common form, the cyclic pre®xÐdoes not reduce the sidelobes in any way, but it does ameliorate their effects; it helps with effects 1, 2, and 4, and a shaped cyclic pre®x helps a little with RFI cancellation. The 8% loss of capacity incurred in ADSL, however, represents approximately 1.5 dB loss in margin. Purists might consider such a big loss unacceptable, but a combination of sidelobe reduction with a very short guard period (<3% perhaps?) that allowed the amplitude spectrum of the SIR to approximately match that of the lineÐand thereby reduced the noise enhancement of the equalizerÐcould be very useful.

Most of the (non-DMT) systems discussed in this chapter use one or more of the following methods to reduce the sidelobes:

^* Frequency-domain spreading (i.e., modulating each symbol onto a weighted sum of tones, which then constitute a “subcarrier”); this is often described as maintaining orthogonality on a distorted channel.

111
Figure 7.1 Frequency-domain spreading by matrix transformation of the IFFT input.

^* Filtering of the separate parallel constellation-encoded symbols by means of a weighted sum over several intervals.
^* Time-domain shaping of each symbol.

7.1 FREQUENCY-DOMAIN SPREADING

Figure 7.1 shows the essentials of this method. The memoryless matrix M, which is simply the transparent identity matrix if no spreading is used, generates a set of basis functions (“sub-carriers”) as weighted sums of sinusoids. Very general forms of the method have been described as vector coding [Kasturia and Ciof®, 1988] and structured channel coding [Lechleider, 1989], but we will con®ne our attention to simpler forms in which each row of M has at most two off-diagonal terms.

7.1.1 Frequency-Domain Partial Response

[Schmid et al., 1969] showed that partial-response coding and shaping, which is well known in the time domain (e.g., [Lender, 1964] and [Kretzmer, 1965]), can be applied with almost perfect duality in the frequency domain. The partialresponse transfer functions that were of most interest in the time domain¹were (1 ‡ D) and (1ÀD²), where D is the symbol delay operator, but the frequencydomain function of most interest to us now²is (1ÀÁ), where Á is the frequency shift operator. This rather imprecise de®nition will become clearer with an example: A 7 Â 7 matrix³M becomes

¹ Kretzmer called these partial response classes I and IV, and the names have been used since.
²In Section 11.1 we consider (1ÀÁ/2ÀÁ²/2).
³The exemplary size is not a power of 2, in order to emphasize that the operation is performed only on the used tones.

FREQUENCY-DOMAIN SPREADING 113
²1 À1 0 0 000^3
601 À1 0 000⁷
6 7
⁶00 1 À1 000⁷
6 7

M
ˆ
⁶0 001 À10 0⁷

_{6 7} …7:1† ⁶0 000 1 À10⁷
6 7
40 000 01 À15
0 000 001
NOTES:

1. The only departure from duality is that time-domain partial-response encoding is continuous in time, whereas this encoding begins anew with each symbol.

2. Time-domain encoding must go forward in time, but frequency-domain encoding can go either way. Equation (7.1), for reasons that will be apparent later, shows the highest used subcarrier as a pure tone and the encoding proceeding downward thereafter. This will, somewhat arbitrarily, be designated as (1ÀÁ) coding. Coding proceeding upward will be designated as (1ÀÁ^À1).

The subcarrier that is used for data symbol a_mis, in terms of the normalized variable x ˆ t/T,
C_m ˆ exp…j2mx†À exp‰ j2…m ‡ 1†xŠ…7:2† ˆÀexp‰j2…m ‡¹†xŠ2j sin…x†…7:3†₂

which is offset by Áf/2 from the conventional tones, and shaped by a halfsinusoid. The spectrum of this subcarrier is shown in Figure 7.2 for m ˆ 24, together with the spectrum of a rectangularly pulsed tone 24. It can be seen that the sidelobes of the partial-response subcarrier are much suppressed.

Sensitivity to Distortion. Figure 7.3 shows the c₁ coef®cients, as de®ned in (6.22b), compared with those for a rectangular envelope; the sensitivity to distortionÐparticularly to the large terms (i small)Ðis greatly reduced.

Receiver Processing. The simplest method of detection would be a frequency-domain equivalent of a DFE: Make a decision on one tone, and subtract its effect from the next. Because the feedback tap for partial response is unity, however, this could lead to severe error propagation. This can be avoided by precoding⁴ in the transmitter, and detecting a 2^k-level signal by slicing a (2^k‡1 À 1)-level eye (see, e.g., [Bingham, 1988]). The big disadvantage of frequency-domain partial response (FDPR) is the 3-dB loss in margin incurred

⁴Originally described in [Lender, 1964] and later generalized in [Tomlinson, 1971] and [Harashima and Miyakawa, 1972].
Figure 7.2 Spectra of tone 24 withand without FDPR.

by the partial-response encoding and the DFE. This loss can be retrieved by Viterbi detection⁵or by the error detection and correction algorithm (EDCA) described in [Bingham, 1988], but it not clear whether either is compatible with trellis coding. FDPR is considered again in Sections 11.1 and 11.6.

7.1.2 Polynomial Cancellation Coding

Polynomial cancellation coding (PCC) [Armstrong, 1998], in its simplest form, groups the tones in pairs and applies a (1ÀÁ) operation to the pair. It achieves the same low sidelobes as FDPR but reportedly, has lower sensitivity to frequency shift and channel distortion caused by multipath. It uses only half the possible number of subcarriers, however, so its data rate is only one-half that of all the other methods. The simplest form is of interest only as an introduction to PCC with time overlap (see Section 7.3.3).

⁵ See [Nasiri-Kenari et al., 1995] and references therein for more ideas on the use of Viterbi detection of partial-response coding.
FILTERING 115
Figure 7.3 Coef®cients of h² for FDPR correlation._i

7.2 FILTERING thereby increases the sensitivity to channel distortion. I have seen no comparisons of the relative performances of ®ltered and un®ltered systems when the latency is ®xed.

Early systems used ®lters that completely separated the subbands in order to maintain orthogonality, with a consequent loss of bandwidth ef®ciency. Several authors (see [Filt1]±[Filt5]) then described multiple staggered QAM (MSQAM), in which the ®lters had only 3-dB attenuation at the crossover points, thereby maintaining full bandwidth ef®ciency. One disadvantage is the complexity of the ®lters, but nevertheless, interest and progress in this method has continued (particularly for wireless broadcasting); the use of ®lter banks derived from those originally developed for transmultiplexers has reduced the complexity almost⁶down to that of the un®ltered approach (see [Filt 6,7]).

Another disadvantage is the extra latency caused by the ®lters. If a system has a maximum allowable latency,⁷ the maximum symbol duration is the latency divided by the number of symbol periods spanned by the ®lter. Reducing the symbol duration by this factor effectively widens the ®lter bandwidth and

⁶ I apologize for using such an imprecise word, but (1) I have seen con¯icting estimates of the complexity, and (2) measures of complexity differ enormously depending on whether the implementation is by DSP or ASIC!
⁷For ADSL it is 1.5 ms, but for broadcast there is no limitation, which probably explains the greater interest in this method for DAB and DTV.

7.3 TIME-DOMAIN SHAPING
Time-domain shaping is really just a way of FIR ®ltering; variations include:

^* The shaping may be applied just to the cyclic pre®xÐa minor modi®cation to DMT discussed in Section 6.5Ðor to the whole envelope pulse of duration T (Sections 7.3.1 and 7.3.2).

^* The “whole pulse” shaping methods are either synchronized (Section 7.3.1), in which all the 2N_car (real plus imaginary) inputs to the modulator(s) change simultaneously, or staggered (Section 7.3.2), in which half of the inputs are delayed by T/2.

7.3.1 Whole Pulse Shaping with Synchronized Inputs

The advantage of the synchronized method would be that a cyclic pre®x could still be used. The disadvantage would be that the only pulse shape that can achieve orthogonality and zero loss of margin is a rectangle: that is, no pulse shaping.⁸ [Slimane, 1998] described a synchronized system that maintained orthogonality but appeared to lose 3 to 6 dB of margin. We do not discuss such methods further.

7.3.2 Whole Pulse Shaping with Staggered Inputs: SMCM

The systems described by [Mallory, 1992], Chaffee⁹ and [Vahlin and Holte, 1994], were natural extensions to pulses of ®nite duration of the ®ltered systems of [Chang, 1966], [Saltzberg, 1967], and [Hirosaki, 1981]. They were also natural attempts to correct the problems of FDPR: namely, the 3-dB loss and the fact that the transmit power envelope has nulls. The Mallory and Chaffee (M & C) pulses were limited in duration to one symbol, and were raised-cosine shaped; Vahlin and Holte (V & H) showed pulses of one, one and a half, and two symbols duration,¹⁰ and the shapes were derived from prolate spheroids that jointly minimized distortion and out-of-band energy. The sidelobes generated by M & C’s raised-cosine pulse fall off very fast (Á1/n³for n large), and those for V & H’s pulse of duration T are very similar; we discuss only the simpler M & C pulses here.

⁸ I have not seen this proved, but I am fairly certain that it is true.
⁹Unpublished paper and private conversations.
¹⁰They called them two, three, and four symbols, but their T was half of the symbol period used here.

TIME-DOMAIN SHAPING 117

For greatest immunity to noise the raised-cosine shaping should be split equally between transmitter and receiver. It is therefore more informative to de®ne the pulse as beginning at t ˆ 0 and refer to it as half-sine-squared shaping, which is split into half-sine  half-sine.

NOTE: If the shaping is split thus the transmit spectra for M & C pulses are the same as for FDPR.

Implementation. The ®gures in the earlier papers (Saltzberg, etc.) showed what might be called alternating staggering; this suggests a method of implementation for SMCM. The real inputs to the even-numbered channels and the imaginary inputs to the odd channels (destined to the even cosines and odd sines) form one set; they are modulated to generate a time-domain sequence x₁, which is then shaped and passed undelayed to the output. The other inputs (destined to the odd cosines and even sines) form a second set; they are modulated to generate x₂, which is shaped and delayed by T/2. This would, however, require two IFFTs with 8N log N real multiplies each. Clearly, a ®rst priority is to condense to one IFFT.

The implementation of a transmitter was described by Chaffee. All the data are modulated in one IFFT to generate a temporary sequence x^H, and then the two sets of time-domain samples x₁and x₂are constructed by using the facts that x₁ is symmetrical about t ˆ T/4 and 3T/4, and x₂is antimetrical. That is,

x₁…t†ˆ x^H…t†‡ x^H…T=2 À t† for 04 t4 T=2
ˆ x^H…t†‡ x^H…T À t† for T=24 t4 T …7:4† x₂…t†ˆ x^H…t†À x^H…T=2 À t† for 04 t4 T=2
ˆ x^H…t†À x^H…T À t† for T=24 t4T …7:5†

The steps can be described succinctly as IDFT±separate±shape±delay±add, and in the receiver they are reversed: separate±delay±shape±merge±DFT.

Orthogonality. How orthogonality is achieved can be understood from Figure 7.4, which shows the simplest cosines and sines: tones 1, 2, and 3. Within the ®rst set (even cosines and odd sines) the sin²(t=T) shaping is symmetrical about T/2, and orthogonality is achieved in the conventional way with integration of the products from 0 to T. Similarly, in the second set (odd cosines and even sines) the shaping is symmetrical about T, and the integration is from T/2 to 3T/2. Between the sets the sin(2t=T) shaping is symmetrical about both T/4 and 3T/4, and the integration is zero both from 0 to T/2 and from T/2 to T.

A More Ef®cient Implementation of the Raised-Cosine Special Case? As we saw in Section 7.2, FDPR using the operator (1ÀÁ) shapes the time-domain waveform with a half sinusoid. Could it not, therefore, be used instead of the two s(t)’s to save a further 2N multiplies? [Mallory, 1992] describes a

Figure 7.4 Orthogonality between tones 1, 2, and 3.
method to do this, but unfortunately I do not understand how the process in the receiver achieves orthogonality, so I can neither verify nor contradict it.

NOTE: With the ®rst (time-domain shaping) method the modulated tones are centered halfway between the tone frequencies n Áf, but with the second method they would be centered around the tones. Time-domain shaping and frequency-domain spreading (partial response, correlation, etc.) are not exactly equivalent.

Performance. The primary aim of all the “other” MCM methods discussed in this chapter is to reduce the sidelobes, but one of the reasons for doing this is, of course, to reduce the sensitivity to channel distortion. SMCM certainly reduces the sidelobes, but unfortunately, it only slightly reduces the sensitivity to distortion. The “intraset” distortion (loss of orthogonality) is much reduced by the gradual turn-on of the sin²(t=T) shaping, but the “interset” distortion is only slightly reduced by the sin(2t=T) shaping.

A SMCM system is therefore slightly less sensitive to channel distortion than a DMT system without a cyclic pre®x, but much more sensitive than one with. This raises the interesting question: Is there a future for SMCM? Here are some ideas that might be explored:

*
modulation (DWMT), although the name overlapped discrete multitone modulation is also used, in tribute to a key property that is discussed later.

Let us return to Figure 5.1, the simpli®ed block diagram of a MCM system, shown there with the IFFT implementation, (i.e., as DMT). The fundamental nature of the DWMT system is exactly the same as that of the MCM (or DMT) system, shown in this ®gure. The only difference between the two systems is that the modulation is implemented with an inverse fast wavelet transform rather than the IFFT, where, of course, the word “fast” designates the use of a fast algorithm for calculating the inverse discrete wavelet transform. Naturally the block diagram of a DWMT receiver would employ a fast direct (or forward) wavelet transform (FWT) in the appropriate place as well. A simpli®ed block diagram of the entire DWMT system is shown in Figure 7.5, where this difference between the two systems is seen. At the transmitter, the outputs of the constellation encoders are used to amplitude modulate the basis elements (orthonormal collection of signals) of some wavelet transform, as indicated in this ®gure.

The symbolism is an important factor in understanding these systems, so a review of previously de®ned symbols is in order at this juncture, as well as changes in symbols and precise de®nitions of new ones. This is done in conjunction with the block diagram of the transmitter, shown in Figure 7.5. Let us recall that the input data to the S/P converter consists of a serial TDM data stream that is divided into frames of B bits each. If the frame duration is T seconds, the input data rate is R_b ˆ B=T. If, for example, T ˆ 125 ms and B ˆ 256 bits/frame, the rate is 2.048 Mbit/s.

Recall next that with the input to the S/P converter being frames of B bits each, the output of each parallel port is a symbol (group of bits) with b(n) bits each and

X
B ˆ b…n†…7:6†
n4M
Figure 7.5 Block diagram of a DWMT system: (a) transmitter; (b) receiver.

where M (this symbol is replacing N_carat this juncture) is the number of groups or parallel ports, and also the number of eventual subcarriers and subchannels. The groups of b(n) bits each are then constellation-encoded and modulated into the subcarriers. The number of bits in a given group is equal to the maximum number of bits that its intended subchannel can support with an acceptable symbol error rate. This is based on channel measurements made during an initialization or training period (see also Section 5.2). Some of the groups may have zero bitsÐchannels with narrowband noise can be avoided that way.

Now let s^m represent the symbol in group m that came from frame i (s21, for_i
example, would denote the symbol that originated in the second group of the ®rst frame of the serial data sequence and is headed to the second subchannel). Now we let ^m (t) denote the bandpass analog signal in channel m whose origin was_i
frame i. If this signal is sampled, it would be denoted as ^m …l†, where l is the_i
sampling instance. All the u(t) signals are added in time to produce the composite signal, which is sent to the channel. Let us denote by N the number of samples of this composite signal. In the DWMT case M ˆ N, whereas in the DMT case N ˆ M ‡ , where is the length of the cyclic pre®x. If the rate of sampling of this composite signal is f^s, the distance between samples is 1=f^s, and the duration of its samples for a given frame is N=f^s.

At the DWMT receiver, the FDMed signals are demodulated using the forward (or direct) fast wavelet transform (FWT). Each of the several sequences of the detector outputs usually undergoes equalization before decisions are made for the channel symbols. The decoded data sequences are then converted back to a single TDM stream.

Before we proceed to a detailed analysis of the DWMT system, we shall need some background on wavelets. Knowledge of wavelet transforms at the level of [Burrus et al., 1998] would be quite helpful for complete understanding of what follows. We proceed with the assumption of this knowledge. Even so, what we sketch below is also a minimal review of what is needed for the explanation of DWMT.

Wavelet theory can be developed using various approaches, the most common of which is to consider the wavelet transform (actually a series) as an expansion. We start our wavelet discussion here with this approachÐit is the foundational approach of the theory. However, we quite rapidly move on to a second approach, the ®lter bank approach, which views the expansion as the sum of the outputs of a bank of ®lters, whose impulse responses are related to the basis elements of the expansion. The ®lter bank “realization” of a wavelet expansion is more useful in the implementation of the expansion. In presenting this ®lter bank approach, we concentrate our discussion on a special subcase: that of ®lter banks with the property of perfect reconstruction. The requirement that the ®lters of the bank possess this property is equivalent to the requirement that the wavelet basis is complete, which leads to the property that an expanded function is equal to its expansion. Within this subcase of ®lter banks with the perfect reconstruction property we will rapidly zero in on a special case, the cosinemodulated ®lter banks (CMFBs) [Koilpillai and Vaidyanathan, 1992]. The CMFBs appear to be the easiest vehicles for fast implementation of the wavelet expansion, although not trivial by any stretch of the imagination.

With the two paragraphs above serving as prolegomena, we start our discussion with the expansion approach. Given a signal y(t) that meets certain conditions (we omit the details of “expandability” here), it can be represented as

XX
y…t†ˆ a_{j;k j;k}…t†…7:7†
k j
where j and k are both integer indices, and the ‘s are the wavelet expansion functions that usually form an orthogonal basis. The ‘s are given by
_j;k…t†ˆ 2^j=2 …2^jt À k†…7:8†

where …t†, the function that is parameterized in two dimensions (shifts and scalings) to create the basis, is called the generating wavelet or mother wavelet. Quite obviously, it is the mother wavelet that completely speci®es the wavelet system. There are many wavelet systems (orthogonal bases) generated by picking some mother wavelet that satis®es some desired condition or restriction. If the wavelet system is not only an orthogonal but also an orthonormal basis, the coef®cients of the expansion (7.7) would be the inner products of y(t) with the basis elements, as is always the case with orthonormal bases.

With the foregoing brief discussion of wavelets in mind, it becomes easier to see that the outputs of the constellation encoders would amplitude modulate the wavelet basis, and this would be accomplished by taking a fast discrete inverse wavelet transformation of the wavelet basis chosen for the system, with the eventual forward discrete wavelet transformation performed at the receiver, as shown in Figure 7.5. The expression for the signal in each subchannel would involve the information signals multiplying the corresponding basis element in the wavelet expansion, and this would presumably place the spectrum of the down-sampled signals at the appropriate channel subbands.

Well, what wavelet basis is used in the DWMT system? This can be speci®ed by giving the mother wavelet of the expansion, since it alone completely describes the entire system. So what is the mother wavelet used in the DWMT system? How does this single mother wavelet, and the resulting orthonormal basis that it creates, accomplish spectral shifts in the system? And what do the spectra look like in the subchannels?

We are not about to give the reader the wavelet basis at this point, since the approach we will take in outlining implementation of the DWMT needs the ®lter bank point of view, as mentioned earlier. This point of view visualizes the wavelet basis elements as the impulse responses of ®lters, so instead of searching for ways to ®nd the basis elements, one seeks to ®nd ®lters with impulse responses that are the basis elements. Both approaches are considered important in understanding wavelets. The reason that we must turn to the ®lter bank point of view is that most of the designed DWMT systems are based on Mallat’s algorithm [Mallat, l989a, l989b] for the structure of ®lters and up-samplers/ down-samplers used to calculate the discrete wavelet transform, and this algorithm is based on ®lter bank theory. But besides providing ef®cient computational techniques for wavelet implementation, this theory also gives many valuable insights into the construction of wavelet bases, as well as some of the deeper aspects of wavelet theory.

At this point, then, we brie¯y review the theory of M-band multirate ®lter banks (a.k.a. discrete basis functions with arbitrary overlap) as given in [Malvar, 1992] and Chapter 8 of [Burrus et al., 1998]. These structures allow a signal to be decomposed (analyzed) into subsignals, usually at lower sampling rates, and later recomposed (synthesized) at the original rates. We limit our brief discussion to the special case of perfect reconstruction ®lter banks, for which the reconstituted signal is a delayed replica of the original.

Generally, an M-band multirate ®lter bank is speci®ed by two collections of ®lters: the M analysis ®lters h(k) and the M synthesis ®lters f (k), (k ˆ0,FFF,MÀ1). All these ®lters are FIR ®lters with length gM, where g is the genus or overlap factor.¹¹ The M ®lters h(k), which are shown in block diagram form in Figure 7.6, are chosen to approximate the ideal frequency response shown in Figure 7.7; they partition the frequency domain into M subbands (channels) of interest.

A signal x(n) is passed through the bank of analysis ®lters h_i(n), and the M output signals are decimated (down-sampled) by a factor of M; that is, only every Mth sample is retained. If the input signal x(n) is transmitted at a rate R, the ®lter outputs are transmitted at the rate R/M, hence the name multirate ®lter bank. When the decimation ratio is the same as the number of ®lters M, the ®lter bank is called a critically decimated system.

Figure 7.6 Critically decimated M-channel ®lter bank (analysis/synthesis). ¹¹This terminology is established, but it is misleading; g ˆ 1 means zero overlap. Figure 7.7 Ideal frequency response of an M-band ®lter bank.

To resynthesize the M signals back into one signal, each of them is ®rst upsampled by inserting MÀ1 zero between its samples and then passed through the synthesis ®lters f_i(n). The M signals of this up-sampling-and-®ltering operation are then added to yield the reconstructed signal y(n). Under perfect reconstruction this signal is a delayed version of the original x(n), that is,

y…n†ˆ x…n À n⁰†…7:9†
A suf®cient condition for perfect reconstruction [Vaidyanathan, 1992] is that the analysis and synthesis ®lters satisfy
h_k‰nŠf_k‰n ‡ lMŠˆ M‰k À k^HŠ‰lŠ…7:10†

It is of interest for our discussion to consider a subclass of these ®lter banks for which the synthesis ®lters are time-reversed versions of the analysis ®lters,¹²that is,

f_k…n†ˆ h_k…L À n À 1†…7:11†
where L ˆ gM is the length of the ®lters. Under this restriction on the synthesis ®lters, (7.10) becomes
^Xh_k…n†h_k…L À n À 1 ‡ lM†ˆ M…k À k^H†…l†…7:12†
n

That is, the ®lters are orthogonal under shifts by M. Such a ®lter bank is called para-unitary [Vaidyanathan, 1992] by analogy with unitary matrices. This important subclass includes the wavelet transform [Steffen et al., 1993]; that is, the impulse responses of the ®lters make up a wavelet basis. That is why such a ®lter bank is often referred to as an M-band wavelet transform, and the analysis and synthesis ®lter banks as the direct and inverse transforms, respectively.

The DWMT system uses such a ®lter bank, and the de®nition of the analysis ®lters, which in turn specify the synthesis ®lters, is the key to implementation of
¹²Malvar considered the general case of the f complex, so that h is its complex-conjugate time reverse, but our f values are real.
Figure 7.8 M-band implementation of the wavelet transform (synthesis/analysis).

the DWMT system. Of course, when the ®lter bank is actually used in the system, the up-sampling and the synthesis ®lters (which implement the inverse wavelet transform) are placed in the transmitter, with the corresponding analysis ®lters and the down-sampling in the receiver. This arrangement is shown conceptually in Figure 7.8. Quite clearly, the arrangement creates the needed inverse wavelet transform in the transmitter and the direct one in the receiver.

Returning now to our discussion of the ®lter bank, how are the analysis ®lters constructed using the foregoing conditions? The method for doing so is quite complicated. Filter bank design typically involves optimization of the ®lter coef®cients to maximize some goodness measure, subject always to the perfectreconstruction constraint of (7.12). In such a constrained nonlinear programming problem, numerical optimization leads to local minima, and the problem gets very messy when there are very many subbands, as can be the case in ADSL systems, for example.

To ease these dif®culties, we usually try to impose additional structural constraints on the ®lters, constraints that may lead to simpler designs. The most popular such constraint is that in (7.10), all analysis (and synthesis) ®lters are obtained by sinusoidal modulation of a single low-pass prototype analysis (synthesis) ®lter. This is the key idea behind the well-known cosine-modulated ®lter banks, which we take up later. It is interesting to note, though, that whereas under the “expansion” approach to wavelets, one must come up with one signalÐthe mother wavelet. Under the ®lter bank approach, the problem has been reduced to ®nding a single ®lterÐthe prototype ®lter. This low-pass prototype ®lter is sometimes called a window, although it differs from the usual time windows by having the overlap property.

For such cosine-modulated ®lter banks (CMFBs), the synthesis ®lters are given by [Malvar, 1992, p. 107]