LGENERALIZED ORTHOGONALLY MULTIPLEXED COMMUNICATION VIA WAVELET PACKET BASES/
A Dissertation Presented to
The Faculty of the Russ College of Engineering and Technology
Ohio University
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
Alan R. Lindsey/ -* June 9, 1995 1995
Alan Ray Lindsey
All Rights Reserved Acknowledgements
I would like to sincerely thank Dr. Jeff Dill, my research advisor, for taking time
away from a very busy schedule to occasionally adjust my wayward direction. His practical insight and farmboy wisdom are enviable. I would also like to thank the members of my committee individually. Dr. Dennis Irwin has been a friend and colleague in addition to his role as mentor. His support, encouragement, and assistance over my tenure as a student at Ohio University have been a tremendous blessing. And speaking of blessings, Dr. John Tague has truly filled that role with numerous discussions about non-technical things like God, the Bible, and Christianity. However,
I am also grateful for his sharing of technical expertise in signal processing and communications. As the outside representative from mathematics, Dr. Jeff Connor contributed a tremendous amount of help in the theoretical developments of wavelets and wavelet packets. Without the Wavelet Research Group meetings, I would still be lost in a cesspool of notation, nested subspaces, p-norms, operators and Hilbert spaces! Many, many thanks to Dr. Connor for helping me organize and absorb that material. And finally, my sincere appreciation and respect are extended to Dr. Jerrell Mitchell, whose unique talent for symbiotically combining a practical approach with experience, and these v two with respectable analytical skills, makes him a great teacher and engineer. He also took time teaching me the art of concise and professional writing style.
Denise Ragan and the ECE office staff deserve much appreciation. I have not known a more organized and well-run administration. Thank you's go also to my colleagues in the control group: Enrique, Russ, Dave and Dan. Their friendship will be forever prized.
Special thanks go to the United States Air Force and the Communications branch at Rome Laboratory. Some specific people in this group who deserve mention are: Peter
Leong, my supervisor and friend, who led me gently through the barrage of paperwork and required form-filling, not to mention the consistently wise counsel on matters pertaining to both studies and life; John Patti, my PK mentor, whose technical prowess is to be esteemed; and Pauline Romano, the RL office manager who works so tirelessly.
My special intimate thanks go to my family, especially my wife Tammy. Her support and encouragement through the difficult days of writing, publishing, traveling, etc. have been a tremendous blessing. I continually give thanks to God for Tammy's companionship and love; and my children - Jared, Elisabeth, Benuel and Caleb will never know how I love being their daddy.
And most importantly, to The One who makes everything I do and say prosper or flounder according to His will and my heart, whose names include Jehovah Jireh, God my Provider; The God of Abraham, Isaac, and Jacob; The Creator of the universe who loves me infinitely more than I could ever love Him back; Jesus the Savior of the world;
My Father in Heaven. Table of Contents
Acknowledgements ...... iv
Table of Contents ...... vi
List of Figures ...... viii
List of Abbreviations and Symbols ...... ix
Chapter 1: Introduction ...... 1 1.1 Contributions Leading to this Work ...... 2 1.2 Overview ...... 3
Chapter 2: Review of Previous Work ...... 6 2.1 Quadrature Amplitude Modulation (QAM) ...... 7 2.1.1 QAM Signal ...... 8 2.1.2 Multi-dimensional Signaling ...... 10 2.2 Multi-Scale Modulation ...... 11 2.2.1 Essential MRA Results ...... 11 2.2.2 Tiling Diagram ...... 13 2.2.3 Figures of Merit ...... 15 2.3 M-Band Wavelet Modulation ...... 18 2.3.1 Essential M-Band Wavelet System Results ...... 18 2.3.2 Tiling Diagram ...... 20 2.3.3 Figures of Merit ...... 21
Chapter 3: Wavelet Packet Modulation ...... 23 3.1 Construction of Wavelet Packet Bases ...... 25 3.2 Example of Theorem 3.2 ...... 31 3.3 Waveform Development ...... 32 3.3.1 Dimensionality and Special Partitions ...... 35 3.4 An Example ...... 36 3.5 Waveform Figures of Merit ...... 38 3.5.1 Power Spectral Density ...... 38 3.5.2 Bandwidth Efficiency ...... 42
Chapter 4: Implementation ...... 44 4.1 The Discrete Wavelet Packet Transform and its Inverse ...... 45 4.1.1 The WPM Transceiver ...... 47 4.2 Translation Between Tiling Diagram and Filter Bank ...... 51 4.2.1 Gray Coding of Frequency Bins ...... 51 4.2.2 Translation Computations ...... 53 4.2.3 Why Gray Coding? ...... 55
Chapter 5: Supersymbol Tuning ...... 57 5.1 T-F Diagram Principles ...... 58 5.2 Supersymbol Tuning Rules ...... 59 5.3 Wavelet Packet Library Combinatorics ...... 61 5.4 The Simplest T-F Jammer . 111 ...... 62 5.5 N, / N, Channels .A General Supersymbol Tuning Algorithm . . 65 5.6 Enhancements to the General Tuning Algorithm ...... 75 5.6.1 111 Noise Scenarios Provide Predictable Best-Level ..... 76 5.6.2 Efficiency Improvement via Another Reverse-Tuning Stage ...... 78
Chapter 6: Conclusions and Future Work ...... 80 6.1 Conclusions ...... 80 6.2 Future Work ...... 83 6.2.1 Timing and Synchronization ...... 83 6.2.2 Parameterization of Scaling Filter Coefficients ...... 84 6.2.3 Coding Via Variable Constellation Geometries With Fixed Symbol Counts ...... 85 6.2.4 Generalized P-adic Filter Banks and Associated Packets . . 85 6.2.5 Improvement of Frequency Localization ...... 86 6.2.6 Continued Development of Supersymbol Tuning ...... 86 6.2.6.1 Other Noise Classes ...... 87 6.2.6.2 Better Cost Functions ...... 87 6.2.7 Spread-Spectrum Application of WPM ...... 88 6.3 Epilogue ...... 88
References ...... 89
Appendix A ...... 103 List of Figures
Figure 2.1. Low-Pass Equivalent Modulation Diagram ...... 9 Figure 2.2. Tiling Diagram for MultiScale Modulation ...... 15 Figure 2.3. Tiling Diagram for M-Band Wavelet Modulation ...... 21 Figure 3.1: Uniform Analysis Filter Bank corresponding to basic wavelet packet bases with m=3 . Nonuniform bank corresponding to standard wavelet bases is shown in grey.shades ...... 24 Figure 3.2. Analysis filters for signal decomposition ...... 26 Figure 3.3. Synthesis filters used in signal expansion ...... 26 Figure 3.4. Wavelet packet decomposition by two-channel filter bank ...... 28 Figure 3.5. Generalized subspace decomposition ...... 29 Figure 3.6: Example application of theorem 3.2, moving from partition to filter bank . Translation to the tiling diagram is shown for completeness ..... 32 Figure 3.7: Comparison of modulation methods for a given interference environment consisting of time impulse and tone jammers . Grey-shaded areas indicate corrupted symbols ...... 37 Figure 4.1: Single stage of the (a) Wavelet Packet Transform (b) Inverse Wavelet Packet Transform ...... 47 Figure 4.2. WPM Transmitter 1 Receiver Model ...... 48 Figure 4.3: Connection between natural ordering of filter bank nodes and sequency or Gray coded ordering of frequency bins ...... 54 Figure 5.1. Tuning procedure illustrated step by step ...... 63 Figure 5.2. Flow Diagram for General Tuning Algorithm ...... 67 Figure 5.3. Best Level Supersymbol for N=6 ...... 71 Figure 5.4. Forward-Tuned Supersymbol ...... 72 Figure 5.5. Optimally Tuned Supersymbol: Cost =30 ...... 73 Figure 5.6. Optimally Tuned Partition for N=8: Cost =58 ...... 74 Figure 5.7. Optimally Tuned Partition, N =9: Cost =245 ...... 75 List of Abbreviations and Symbols
1 2(2) The space of square-summable sequences L 2(R) The space of all square-integrable functions on 1W z The set of integers 0' Partition defining a supersymbol RMS bandwidth Af At RMS duration @ Mutually exclusive vector space sum, i.e., "Direct Sum" AWGN Additive white Gaussian noise bps Bits per second BPSK Binary Phase Shift Keying DSPN Direct Sequence Pseudo Noise DWPT Discrete Wavelet Packet Transform IDWPT Inverse Discrete Wavelet Packet Transform IS1 Inter-symbol interference LPIID Low Probability of InterferenceIDetection MAXLEVELS Maximum level of the wavelet packet filter bank MBWS M-Band Wavelet System MCM Multi-Carrier Modulation MRA Multiresolution Analysis MSM Multi-Scale Modulation MWM M-Band Wavelet Modulation PSD Power spectral density QAM Quadrature Amplitude Modulation QMF Quadrature mirror filter SNR Signal-to-noise ratio WPM Wavelet Packet Modulation Zntrodu ction
The study of communication, i.e., conveying a message from sender to receiver, has been an ongoing matter for centuries, even millennia, and the more subtle issues of message degradation, enhancement, detection, interception, etc. have been around just as long. Modern communication systems where electronic messages traverse copper wire or free-space are only one way of tackling this gargantuan problem. Though even interplanetary communication links are accomplished successfully, this seemingly unbounded capability does have its limits. There are constraints on bandwidth for different applications, infinite time is not accessible, physical laws mandate certain impenetrable boundaries, and nature always adds her noisy messages. These limitations continue to be conquered, but with each victory comes a new challenge - in actuality, just a new, slightly extended boundary.
For some time, multidimensional signals have been successfully applied to this problem of signaling for channel exploitation, increasing message reliability, covertness, 2 etc., and many (even introductory) texts devote space to the theory behind the concept.
In this work the state of the art in communication systems is advanced with respect to this problem. In particular, building on the relatively recent work of a few researchers, the fertile field of wavelet theory is studied and new connections are presented which significantly further the concepts of orthogonal signalling.
1.1 Contributions Leading to this Work
The material presented here really represents the union of two fields of science; communication theory from engineering, and wavelet theory from mathematics. In the history of communications, orthogonal signalling methods herald a good deal of work from which to draw. On the mathematical side, from Haar7spair of orthogonal square pulses to the already familiar sinusoidal functions, periodically orthogonal at 90" phase shifts and at integral multiples of frequency, there is a rich heritage of ideas applicable to the orthogonal signalling problem. But Morlet's work in 1982 and his collaboration with Grossman in 1984 [76], from which came the beautiful construction of wavelet bases, were pioneering efforts which spawned the fascinating field of wavelet theory.
From this humble beginning, along with the works of Haar [77], Gabor [71], Allen &
Rabiner [43], and Portnoff [124], arose work by Vetterli [I461 and Smith & Barnwell
[134,135] on multirate filter banks, Daubechies [68,11] on orthonormal wavelet bases, and Mallat [I10,1111 on multiresolution analysis. Cohen [60], Chui [52], Strang [138], and Nguyen & Vaidyanathan [120,142] and others contributed significantly to the early development of wavelets as well. 3
From 1989 to 1994, several papers and at least two books appeared by Coifman
[61,62,63,64,65,66], Wickerhauser [40,154,155], and Meyer [28], which introduced the extension of wavelets to wavelet packets and built solidly on the concept. Though Chui
[4,5], Hess-Nielsen [83], and others discuss the topic and make significant contributions, the labors of the former should be highlighted. Much of the mathematical work presented here has been derived from the labors of these men.
Regarding the communication issues, material has been extended or derived from a number of sources too numerous to list here, although the references contain them.
There are a few of special mention: namely Shannon [132], Proakis [32], Couch [9],
Oppenheim [30], Simon [33], Sklar [34], Bingham [46], Hirosaki [86], and Jones [92].
1.2 Overview
The content of this dissertation is divided into 6 chapters, a reference list, and an appendix. Chapter 1 is what you are reading. Chapter 2 contains a review of work done previous to this contribution and from which this work was derived. It includes an overview of multiresolution analysis and wavelet bases leading to an orthogonally multiplexed modulation scheme known as multi-scale modulation or MSM. The chapter continues with the review of M-band wavelet systems and the M-band wavelet modulation scheme known as MWM. The concept of a supersymbol or tiling diagram is introduced as a convenient way to visualize the time-frequency dimensionality of the
MSM or MWM signals.
Chapter 3 begins with a review of wavelet packet bases, their construction from quadrature mirror filters and recursive two-scale relations, the subspaces they generate, 4 the efficient and convenient filter bank structure, and notation. The rest of the chapter is devoted to the analytical development of the title topic - wavelet packet modulation or
WPM. First the signal itself is derived by synthesizing the definition of wavelet packets with an applicable theorem using dyadically related disjoint partitions of an interval. An example is then given to illustrate this concept. Finally, the important figures of merit, power spectral density and bandwidth efficiency, are derived.
Chapter 4 deals with the issues of implementation. The WPM transceiver is developed, utilizing an inverse wavelet packet transform in conjunction with a single pulse shaping filter to accomplish the signal generation, and a matched filter, sampler, and wavelet packet transform for reception and symbol reconstruction. -4lso of importance in this chapter is the translation between partition elements (wavelet packet numbers) and frequency bins, or rather, between the filter bank and tiling diagram. This topic leads to the discussion of Gray-coding which turns out to be the system by which one is obtained from the other. An example illustrates the use of the Gray-code in the translation procedure.
Chapter 5 deals with the development of a supersymbol tuning algorithm that optimally localizes tones and impulses to a minimal number of supersymbol atoms. First some basic principles of time-frequency diagrams are reviewed, and then two supersymbol tuning rules are presented which govern the action of the search algorithm.
A brief look at the explosive combinatorics involved in the wavelet packet libraries gives the reader an appreciation for the importance of an efficient search. A simple jammer scenario is worked out first, and a simple algorithm which works for this case is 5 developed using pseudo-code as the drawing board. From this initial concept-rendering algorithm the general case of arbitrary numbers and positions of tones and impulses is addressed. The general algorithm incorporates bi-directional searching, a best-level computation, and forward and reverse locking for improved efficiency.
A cost function related to signal-to-noise ratio (SNR), which counts the number of atoms intersected by tones or impulses, is used for the search criterion, along with the added constraint that the optimal supersymbol have minimal frequency bins. Several examples are given for N=6, 8, and 9, with different jammer scenarios for each. The chapter ends with some enhancements to the general algorithm that provide additional improvements in efficiency.
Chapter 6 summarizes the results presented in chapters 3, 4, and 5, and then discusses numerous future extensions to this work. The extensive reference section follows, listing all works known by the author incorporating wavelets in communication systems, in addition to a substantial amount of related material not directly cited in the text. Finally the appendix lists the documented MATLAB"m-files that implement the general supersymbol tuning algorithm. Review of Previous Work
The purpose of this chapter is to provide a foundation for understanding wavelets and their connection to multidimensional signaling by reviewing the pertinent concepts of communication systems and current wavelet-based modulation schemes. Most notation is defined here, and the groundwork is laid for developing an enhanced modulation scheme based on wavelet packets in chapter 3. As this work is primarily an extension and generalization of [92], most of the results presented in this section are covered in much greater detail there. In the true spirit of review, details and proofs are omitted, except where they foster clarity or understanding. The acute reader should refer to the appropriate references.
To begin, the operation and construction of orthogonally multiplexed communication signals are reviewed. The Quadrature Amplitude Modulated (QAM) signal is visited briefly, since it is the connecting point for the wavelet-based modulation schemes, with the wavelet basis functions taking the place of the pulse shaping function 7 in the definition. The connections between QAM and Jones' Multi-Scale Modulation
(MSM) and M-Band Wavelet Modulation (MWM) are then reviewed, developing the analytical forms of the respective signals. Enough Multiresolution Analysis (MRA) theory is presented so as to establish the notation for MSM, and a similar quantity of
M-Band Wavelet System (MBWS) theory is given to establish notation for MWM. Both
MSM and MWM are then examined in light of the basic communication figures of merit: power spectral density, bandwidth efficiency, and performance in additive white Gaussian noise (AWGN). The tiling diagram, which is a new concern arising out of the time- frequency dimensionality of these schemes, follows. This latter concept will be very important in the understanding of Wavelet Packet Modulation (WPM), which is the generalization of MSM and MWM, and the topic of Chapter 3. In this novel format the partition defining the tiling diagram is a variable to tune for optimizing channel exploitation.
2.1 Quadrature Amplitude Modulation (QAM)
Digital modulation is the process of mapping binary information into analog waveforms that are designed to match channel characteristics. Typically a k-bit block from a sequence {d,) of binary digits is mapped to a complex symbol a taken from a constellation of M = 2k symbols. The pattern of symbol placement in the constellation defines the type of modulation. These symbols are each mapped to one of M deterministic, finite energy signals, sa(t); hence the name M-ary signaling. Binary signaling is the simplest example, where the initial bit stream is used directly, as in 8
Binary Phase Shift Keying (BPSK)'. If the rate of the digital data is R bits per second
(bps) then the symbol rate is R symbols per second. Equivalently, each symbol occupies a T=f second time interval.
2.1.1 QAM Signal
The analog waveforms representing the data are generally bandpass and are described by [32]
sa(t) = Re{a u(t)e ihf:} (2.1) where fc is the carrier frequency. The quantity a u(t) is the equivalent information- bearing lowpass waveform, where u(t) defines the shape of the lowpass pulse anda holds the information in the form of a complex symbol drawn from the constellation.
When this complex symbol varies in both phase and amplitude, the carrier signal is said to be amplitude-phase modulated, or more commonly, because the complex exponential is really the complex sum of the orthogonal cosine and sine functions, Quadrature
Amplitude Modulated. The bandwidth requirement for the modulation format is determined by the pulse shape, so it is particularly important in signal design.
In this dissertation the issues of carrier or symbol synchronization are not a concern. Thus consideration is only given to the standard lowpass form [32], [34], [36] of the QAM signal,
%his is not to be confused with multi-dimensional signaling which we will define and discuss later in this review. composed of an infinite sum of individual signals delayed by integral multiples of the symbol period T. In order to simplify later analytical developments, an equivalent representation in which the symbol period is normalized to unity is
Though the time axes for these two representations are different, the concept of summing integral delayed versions of a weighted lowpass pulse is the same. Thus, all results quantified in this unconventional framework have exact analogs to the former.
A most useful modification to the previous representation is to clearly indicate the signal power as a scale factor, utilizing symbols and pulse shapes with unit average energy. This allows for accurate comparison to other modulation formats. LettingEs be the average symbol energy, the operative definition of a general QAM signal is
where $ takes the place of u to emphasize the normalization of the pulse-shape energy.
Map to Complex Symbol I 1 I Pulse-Shape Gain GarySequence
I Impulse i- t , L--' Generator '-' T I
Figure 2.1: Low-Pass Equivalent Modulation Diagram 10
This lowpass equivalent modulation process is perhaps more appropriately described graphically. In the block diagram of figure 2.1 a digital bit stream is sectioned into k-bit blocks, each of which is mapped to a complex symbol from the QAM constellation (that is assumed to have unit average energy over all symbols). Each element in this sequence of complex numbers becomes a scale factor for a discrete impulse whose delay is exactly that of the symbol index. This chain of scaled impulses convolves with the pulse shaping filter (also unit-energy), and the result is a summed sequence of delayed pulse shaping functions weighted by the corresponding symbols.
The gain block simply applies the desired power, giving the modulated output signal, which can then be frequency shifted by a carrier for electromagnetic propagation if desired.
2.1.2 Multi-dimensional Signaling
In equation (2.1), there are two orthogonal signals on which the digital data are placed, namely sin(2zfct) and cos(2zfCt). In effect, the carrier amplitude and phase corresponding to a symbol in the two dimensional signal constellation are modulated.
The dimensionality can easily be increased via multiple strategically spaced carrier frequencies or by partitioning the symbol interval and placing orthogonal pulses in these shorter spaces, or a combination of both [32]. The dimensionality refers to the number of elements in the corresponding vector describing the aggregate signal.
In the case of QAM, two orthogonal basis signals utilize a single frequency, so the QAM signal, occupying a finite non-zero bandwidth B, can be repeated N times in a frequency interval NB , at appropriate carrier frequencies. This scheme is referred to as Multi-Carrier Modulation (MCM) 1461. Each QAM symbol corresponds to 2 dimensions and these symbols are independently placed on each of N carriers yielding a 2N-dimensional signal. The 2N carrier signals form an orthogonal basis of the signal space.
2.2 Multi-Scale Modulation
In this format, orthonormal dyadic wavelet basis functions from a multiresolution analysis (MRA) are used as pulse shapes upon which independent QAM symbols are placed [92]. These basis functions provide a nonuniform partition of the data bandwidth and have dimensionality in both time and frequency. Thus a whole new class of pulse shapes for (2.4) are available to the designer, making it a very useful signal indeed. The essential concepts for this format are now reviewed, essentially repeating what is written in [92], following the notation of [I101 and [I 111.
2.2.1 Essential MRA Results
Consider the space L 2(R) of all square-integrable functions having domain R , where R is the set of real numbers. An MRA is a sequence of closed subspaces of
L 2(R) , denoted V, , satisfying the following:
3. U V2, is dense in L '(R) j€Z 4. n v,, = {o} j€Z 5. 3 gE V, 3 g(x-n), VnEZ, is a Riesz basis for V, where Z is the set of integers. Theorem 2.1 Let (V ) be an MRA. Then 3 ! @ E Vl;l @$(2jx -n), Vn 'n Z, is an 2'jez orthonormal basis for V,.
The function @ is called the scaling function. A very useful consequence of Theorem
2.1 is that the family of functions @(x-n), VnEZ,is an orthonormal basis for V,. This is precisely the connection between multidimensional QAM signals and wavelets. If the pulse shaping function @(t)in (2.4) is a valid scaling function, then due to orthogonality, the adjacent symbols do not suffer from inter-symbol interference (ISI).
Definition: A vector space V is said to be the DIRECT SUM of W, and W2,denoted by V = W1 @ W,, if W, and W2 are subspaces of V such that W, fl W2 = {0} and W, + W2 = V.
Defining W,, to be the subspace of V,,,- that satisfies
the following theorem results:
Theorem 2.2 Let (V } be an MRA. Then 3 ! I) E V,3 @7+9(2jx -n), VnE 2, is an 2' jEz orthonormal basis for W,, . Furthermore, @7+9(2jx -n) , Vnj E Z ', is an orthonormal basis for L 2(R).
We call I) the wavelet function, and it is actually determined by the scaling function for the MRA, which is consequential when discussing the implementation but is not instructive for this review
Now because of the way W,, is defined, and the definition of MRA, it is evident that for any J r 2, This decomposition into orthogonal subspaces has a striking consequence: Becauses(t) in (2.4) is a linear combination of basis functions for V,, it resides in this space, and by
(2.7), it can be written as a linear combination of the basis functions for the decomposed subspaces. This new waveform is,
where the a:, k E 10,.. . J-1) are complex valued QAM symbols, each from a possibly different constellation. This new waveform has been christened Multi-Scale Modulation because the information bearing data symbols have been placed at different scales. It has the very attractive feature of zero IS1 since the basis functions are orthogonal, and, because the wavelet subspaces are mutually orthogonal by definition, the individual signals at each scale are also orthogonal, thereby alleviating interference problems.
Indeed this 2J-1-dimensional signal has very useful properties, especially when one considers that the dimensionality is not constrained to either the time or frequency domains, but actually utilizes both.
2.2.2 Tiling Diagram
The mechanism for picturing the time-frequency decomposition of the MSM signal is the tiling diagram. This useful tool indicates the mean-square energy concentration [13, 41 in the time-frequency plane of the basis functions forming the 14 aggregate signal. For MSM, the signal is composed of a family of dyadically scaled pulses, with the high scales having long duration (low frequency) and lower symbol rates and the low scales shorter duration (high frequency) and higher symbol rates. Hence this time-frequency partitioning is nonuniform in both time and frequency.
Figure 2.2 illustrates this very clearly. The grey-shaded area represents one
"supersymbol," which is defined as the set of 2J-1 coefficients arising from a complete
MSM decomposition as per (2.8). Each rectangle represents one basis function and is referred to as a time-frequency "atom." The areas in bold lines correspond to the coefficients for a particular m. However, caution must be used; this subscript, which from equation (2.8) indicates position in time, must be interpreted in accordance with the scale. The coefficients for the scaling function and the mother wavelet function, a: and a,,,1 respectively, occur at the supersymbol rate L,where T is the source symbol period, Y1T but the coefficients for the next wavelet space occur at twice this rate. Each successively higher wavelet space carries QAM symbols at twice the rate of the previous. Thus the subscripts on the a's must be divided by the symbol rate for that channel to arrive at an absolute time location. The significant suggestion of figure 2.2 is that e~ery2~-'T seconds, another supersymbol is generated with dimensionality in both time and frequency. This is exactly what MSM does, and its usefulness against certain channel impairments has been demonstrated as an effective compromise between total dimensionality in time or total dimensionality in frequency. Time
Figure 2.2: Tiling Diagram for MultiScale Modulation.
2.2.3 Figures of Merit
Recalling the QAM signal for a moment, it is well known that when the am are independent with zero mean, the power spectral density (PSD) of this waveform is [32]
= El@0l2 (2.9) where $(t) and @(f)is a Fourier Transform pair. It has been shown [92] that the MSM waveform has a power spectral density
sM,yMu> = EI@O12 16 i.e., MSM and QAM perform equally with respect to bandwidth requirements.
Interestingly, even though the MSM waveform is made up of mostly wavelet functions, the PSD is only dependent on the scaling function. Hence the designer need only be concerned with a single pulse shaping filter.
In addition to spectral density, another valuable figure of merit for a modulation scheme is bandwidth efficiency, defined as the ratio of bit rate to data bandwidth in which the units are bits per second per Hertz (bps/Hz). For MSM, independent QAM symbols from separate constellations are placed on the various basis functions, and the bandwidth efficiency for this format has been shown to be
where K, represents the number of bits per symbol of the 1 'th QAM constellation containing 2Ki symbols, and P is the percentage of excess bandwidth required over nyquist rate signaling. When considering equivalent QAM constellations at all scales, so that K,=K, 1E (0,. .. J-1) , then (2.11) simplifies to
This is precisely the bandwidth efficiency obtained using the conventional QAM signal in (2.4) [32] and, as Jones has correctly pointed out, is fortunately what should happen, since theoretical bandwidth efficiency does not increase for orthogonal pulse shapes [92].
Yet another figure of merit for a modulation scheme is performance in the presence of Gaussian noise. This is highly dependent on the receiver, so a decision as to the optimum receiver for the MSM signal must be made. Since MSM is actually a
composition of orthonormal weighted basis functions, the optimum receiver is a bank of
filters matched to these functions and a logic unit for decisions. Thus any noise
projected onto these basis functions via the matched filters will be uncorrelated with zero
mean and variance -.No Thus Jones [92] was able to show that for MSM, the average 2 probability of symbol error, q,is
where pSkis the probability of symbol error for the k 'th constellation. If consideration
is only given to those applications where the constellation is the same for all channels,
then P:=P~, so (2.13) becomes q=Ps,implying that average symbol error probability
for the MSM signal is determined by the average symbol error probability for a single
constellation. This is the intuitive expectation in this case. For the case where different
constellations are desired for each scale, simulations must be used, but Jones has
provided some bounds for comparison.
Perhaps the most attractive property of this modulation method is the
implementation. Making use of the discrete wavelet transform and properties of the filter
bank structure, all the analog processing required for this scheme can be replaced by digital signal processing. In chapter 3 this result will be developed in great detail when
Wavelet Packet Modulation is presented. Since MSM is a special case of this more
general format, there is no reason to belabor it here. 2.3 M-Band Wavelet Modulation
In partnership with Multi-scale Modulation, which provides a nonuniform
partitioning of the data bandwidth, Jones 1921 also developed M-Band Wavelet
Modulation (MWM) which provides a uniform partitioning, useful for certain types of
interferences which are not effectively mitigated with MSM. This scheme utilizes the
basis functions associated with M-Band Wavelet Systems, and, as in MSM, these
functions work as pulse shapes for complex QAM symbols. The essential concepts for
this format are now reviewed.
2.3.1 Essential M-Band Wavelet System Results
Similar to section 2.2.1, the results of Jones [92] are summarized and a notation which will lead nicely into the generalization to Wavelet Packet Modulation is established. For M12, an M-Band Wavelet System is a sequence of closed scaling and wavelet subspaces of L '(R) , denoted VM, and w;, , 1 E (1,..., M-1) , respectively, which satisfy the following:
1. v,,., = v,, e w;, e ... a3 WE-' 2. VM, 1 w;, 1 ... 1 wZ-l 3. 3 @EVl 3 @(x-n), VnEZ, is an orthonormal basis for V, (2.14) 4. For each lE(1,..., M-I), 3v1€w:3pl(x-n), Vn'nEZ, is an orthonormal basis for w:. where the direct sum symbol is defined on page 12. The following theorem establishes the basis for the scaling space: Theorem 2.3 Let (V } be a sequence of scaling subspaces in an M-Band Wavelet MJ jeZ System. Then m@(Mjx-n), Vn€ 2, is an orthonormal basis for V,, .
The symbol @ is called the M-band scaling function. A similar theorem for the wavelet subspaces is:
Theorem 2.4 Let {W;,,) be the sequence of 1 'th wavelet subspaces in an M-Band I €2 Wavelet System, 1 = { 1, . . . , M - 1 Then ~II,~(M~-n) , VnE 2, is an orthonormal basis for w:,.
The symbol q!J[ is called the 1 'th M-band wavelet function, but deviating from the MRA case where M=2, is not, in general, uniquely specified by the scaling function.
It is noteworthy at this point to remark that theorems 2.3 and 2.4 are exactly those of theorems 2.1 and 2.2 with M=2, since an M-band wavelet system is a multiresolution analysis in this case. Thus, the results of section 2.2.1 could be tailored to fit into this framework. However, this is not done because of a very important assumption made about the M-band decomposition: The V, space is decomposed once and only once into M subspaces. In MSM, each successive scaling space gets decomposed into a finer scaling space and the corresponding orthogonally complementary wavelet space. Were this same process applied to MWM, only one space out of M would be refined. When M=2 (MSM) this is useful, but for larger M, it is not, recalling that the purpose of MWM is to provide a uniform decomposition of the data bandwidth; this would not be the case with further division. 20
Now, the first property in (2.14) for some M 1 2 can be written
V, = v,., e w;., e ... e W;-;' (2.15) where the particular case j = -1 has been used. Thus if @ E V, in the general QAM signal
(2.4) is an M-band scaling function, and hence lies in V,, it can be expanded into the basis functions on the right of (2.15). The new multidimensional signal is
where the a,,k kE (0, ...,M-1) are complex valued QAM symbols, each from a possibly different constellation; this signal is designated as M-Band Wavelet Modulation. Just as in MSM, this signal is free of IS1 and cross-channel interference, but it is different in that the symbol rates for each channel are equal at 'th the source symbol rate, and the
2M-dimensionality is gained totally in frequency.
2.3.2 Tiling Diagram
The time-frequency tiling diagram for MWM is shown in figure 2.3. The signal is composed of a family of single-scale pulses operating at equal rates so this partitioning is uniform in frequency. One supersymbol in this case, shown with duration MT, is the set of M coefficients arising from a complete MWM decomposition as per (2.16).
Contrary to MSM, where the subscripts on the a:'s had to be divided by the symbol rate for that channel to arrive at an absolute time location, the subscripts for the MWM coefficients suggest thc actual time location. The suggestion of figure 2.2 is that every T seconds, another supersymbol is generated with dimensionality in frequency. This is exactly what MWM does, and its usefulness against stationary narrowband interference
has been demonstrated in [92].
Time
Figure 2.3: Tiling Diagram for M-Band Wavelet Modulation.
2.3.3 Figures of Merit
Jones showed that the power spectral density of MWM is exactly the same as
QAM and MSM, i.e.,
SMWM(~)= EP(fT)I2 (2.17) where the PSD is only dependent on the M-band scaling function and not on any of the wavelet functions, just as in MSM.
The bandwidth efficiency for MWM was shown to be M-I
where again K, is the number of bits per symbol of the 1 'th QAM constellation containing 2K1 symbols, and /3 is the percentage of excess bandwidth required over nyquist rate signaling. Assuming equivalent QAM constellations at all scales, so that
Kr=K, 1 E (0,. . . J-1) , (2.18) simplifies to
This result is the same as for conventional QAM and MSM.
The performance of MWM in Gaussian noise is given by
which for equiprobable symbol errors for each constellation, becomes P=Ps,just as in
MSM. Again, if it is desired that (2.20) be evaluated specifically, simulations must be used. At the very least established bounds must be resorted to.
The implementation for MWM has been shown to be transferrable to a completely digital form, utilizing uniform synthesis/analysis filter banks based on an extension of the discrete wavelet transform. In order to avoid redundancy, the detailed discussion is deferred to chapter 3 when Wavelet Packet Modulation is introduced, since MWM is a special case of this more general format. Wavelet Packet Modulation
At the beginning of the decade, Coifman, Meyer, Wickerhauser and others published a series of articles introducing the notion of wavelet packets [5, 61, 62, 63,
64, 65, 661, and they are discussed as well in [4] and [48]. These functions arise from the natural concept of filling out the nonuniform binary tree used in wavelet decompositions.
The wavelet transform is implemented as a nonuniform filter bank, and is depicted by the grey shaded blocks of figure 3.1. Coifman and Meyer expanded on this concept by establishing the theory for libraries of orthonormal bases which were obtained by filling out the binary tree to some uniform depth m, as in figure 3.1. The functions so obtaincd arc given the name "basic" wavelet packets [64], implying a constant level of decomposition.
One of the obvious consequences of this construction was rapid computation of
M-band wavelets which are discussed in [I, 38, 1371. These M-Band Systems were Figure 3.1: Uniform Analysis Filter Bank corresponding to basic wavelet packet bases with m=3. Nonuniform bank corresponding to standard wavelet bases is shown in grey-shades. shown in chapter 2 to have application in Jones' MWM communication signals [92], and they are also utilized in tree-structured sub-band coding [I, 134, 135, 142, 1461.
Wickerhauser [I541 (published in preprint form in 1989) actually applied this "first- generation" wavelet packet theory in acoustic signal compression. The next logical step in this historical sequence was to generalize the algorithm by establishing that any
"arbitraryu2pruning of the full binary tree would indeed give rise to a basis for 12(Z).
Coifman, et. al. [61,64] accomplished this, and Wickerhauser [40] treats the subject from a practical perspective. Mcyer 1281 devotes a chapter to the subject, and the more general case of p-adic decompositions is given in [63].
2The word arbitrary is used rather loosely herc, referring to a subset of branches that if decomposed to some maximum level for all, a uniform binary tree would result. I.e., this "arbitrary" pruning must be a valid one. 25
3.1 Construction of Wavelet Packet Bases
Following [61], a pair of quadrature mirror filters (QMF) of length L7 hk and
gk= h~-l-k 7 Provide the starting point. Two operators, H and G , which depend on
the QMF7sand act on the space of square-summable sequences 1 2(Z)3 are introduced as
These operators correspond to the convolution-subsampling operations shown in figure
3.2. H and G have adjoints (a "dual" relation) given by
corresponding to the upsampling-anticonvolution operations shown in figure 3.3. These four operators have been shown to satisfy the perfect reconstruction conditions
where o indicates operator composition and I is the identity operator in Z2(Z). Thus these operations are useful for analysis/compression applications as well as the transceiver (synthesis/analysis) problem from communication theory.
The sequence of functions, p,(x) , defined recursively as
3~.e.,those {x,},,, such that (x, l2 < m . This vector space is infinite-dimensional. When k EZ considering signals with finite length, say N=2", one need only consider the space z2(N). Figure 3.2: Analysis filters for signal decomposition.
Figure 3.3: Synthesis filters used in signal expansion. is the set of wavelet packets arising from the given QMF pair. The function p,(x) is the unique fixed point of the first two-scale equation above and is exactly the scaling function q4 from a MRA, i.e., the function which forms the basis for the subspace Vl . Similarly,p,(x) is the corresponding wavelet function II, by the second equation and thus, the relationsp, = q4 and pl = II, are assigned at the outset. Indeed, MRA and the wavelet transform (and
M-band wavelets at the first decomposition) are special cases of this general construction.
These functions have some very useful properties, as given by Chui [4]. The first one is
(pJ P-= aj,, , j,kE (3.5) which says that each individual wavelet packet is orthogonal to itself at all nonzero translates. Also, meaning that pairs of packets coming from the same parent4 packet are orthogonal at all
translates.
The details of how a wavelet packet function is affected as it traverses this binary
filter structure are now examined. The result will admit a general expression for the
functions at a particular node of the dyadic tree, given the scaling function p,(x) at the
root node. Rewriting the first equation in (3.4), modifying the argument and amplitude-
scaling, then changing variables yields the following new form in the operator H:
j+l where 2Tpn(2j+1x-rn) is actually a sequence indexed on m, with x and n fixed. This
expression can be interpreted as a one parameter family of discrete sequencess, where
each sequence depends on the value of x [163]. The fact that t or x come from an
uncountable set is not an issue. It is this interpretation which admits the "discrete
filtering of a continuous function." Applying the same steps to the second equation in
(3.4) gives
'he same function pn is used to generate both pBz and Thus we refer to the former as the "parent" of the latter "children".
h his is similar to the concept of a time-varying matrix exponential eAifrom system theory, which is an uncountable family of matrices in which each member matrix depends on the real number t. 28
These equations are easily interpreted in terms of a filter bank as figure 3.4 shows. The
input "signal" gets decomposed into two orthogonal "signals" at a smaller scale (smaller
scales increase width - decreasing time resolution.) Again, these signals are actually
discrete sequences for a given x, which when considered together form an orthogonal
family of functions of x.
Figure 3.4: Wavelet packet decomposition by two-channel filter bank.
At this point it is necessary to modify the notation for subspaces of L2(R)due to
the generality of the wavelet packet construction; the V,W notation used in MRA7sfor
scaling and wavelet spaces respectively, is altered to more easily include other branches of the tree. Define W: h V,, and w,' nW,, so that w:+ = W: @ w,' where the direct sum symbol @ is defined in chapter 2, page 12. This subspace relationship is provided by the MRA theory as per equation (2.6). Since the two summands in this relation are the spaces for the first two branches of some fixed level in a binary tree, it is natural to continue the pattern so that W71,, 1 2 0, 0 5 n 5 2'-1 describes the spaces representing all 2' branches in the 1 'h level. This generalized notation is shown figure 3.5.
Recall that in an MRA, each V,- is the closed linear span of the functions
2j1"(2~x-k), kEZ, and each W,- is the closed linear span of the functions Figure 3.5: Generalized subspace decomposition.
2jJ29!(2jx-k), kEZ. The more general definition, which accounts for other subspaces generated at arbitrary nodes in the filter bank is
where (* ) indicates closed linear span. From (3.4), both pk and p,,, are expansions in the scaled function pn(2x), implying that the spaces generated by these two functions are both subspaces of that generated by the parent. That is,
Furthermore, the orthogonality properties of (3.5) and (3.6) are sufficient to admit the decomposition relation, 30 which is a generalization of the orthogonal decomposition w:., = W: @ W: mentioned above. The proof of (3.11) is provided in [4]. Thus every parent space is decomposed into orthogonal subspaces. From the filter bank perspective, every node can have two branches where, if a function from the parent space is input to the two-channel filter bank, the functions produced at the output channels are orthogonal to each other, and their spans sum to the parent space directly (completely and with no redundancy).
It has been shown 1611, that the set of wavelet packets, p,(x -k), n,kEZ form an orthonormal basis for L2(R). Moreover, the following theorem establishes the set of basis functions for an arbitrary pruning of the infinite binary tree.
Theorem 3.1 If the collection 6={(l,n)) is such that the dyadic intervals I,= [2'n, 2'(n +l))form a disjoint covering of [0, m), then the set of wavelet packets 2"2p,(2'x -k) , k E Z, (1 ,n) E 6, form a complete orthonormal basis for L "R).
For example, the collection ((1, I)), 1E % , gives rise to the wavelet basis for L '(R) . The symbol P is referred to as a "frequency partition" in that it defines how to divide up the interval into dyadic subintervals determining the frequency localization of the basis.
However, a subset of L2(IW) functions are of more practical interest; namely those that can be represented as expansions in the basis set oC thc scaling space, V, (which in the present notation is w;). Functions outside this space do not meet one or more of the criterion for practical communication signals, in particular the bandwidth, which is intimately related to the sampling rate and defines the initial scaling space from the start. 31
Thus the "finite" version of this theorem which is credited to Coifman, et. al. in
[64] is applied.
Theorem 3.2 Let wiNCL '(R) be equipped with orthonormal basis 2N12$(2N~ -k), k€ 2. If the collection PNCP={(l,n))is such that the dyadic intervals I,= [2'n , 2'(n + 1))form a disjoint covering of [0,2,)) then the set of wavelet packets 2'1"n(2[x -k), k E2, (1 ,n) E PN, form a complete orthonormal basis for w:.
This theorem can be interpreted in terms of the direct sum of the subspaces defined by the elements in the relevant partition CP,. That is,
where 1 3.2 Example of Theorem 3.2 Consider the N=O case of Theorem 3.2 with a hypothetical partition of [O,l) given by It is easily seen that these ordered pairs do actually provide a disjoint covering of the interval as per the theorem. The unique filter bank associated with this partition is Figure 3.6: Example application of theorem 3.2, moving from partition to filter bank. Translation to the tiling diagram is shown for completeness. shown in figure 3.6, where the ordered pairs on each node represent the coordinates (412)that would be present in the partition if that node were an output channel. In this light the filter bank outputs are exactly the coordinates of partition (3.13). The corresponding tiling diagram is also illustrated, and the association of filter bank outputs to frequency bins is given by the dotted lines. The reasons for this unconventional ordering of the frequency bins and the rules for translation between tiling diagram and filter bank are discussed in chapter 4. 3.3 Waveform Development The task at hand is to design a communication signal having "desirable" properties for a given channel. A desirable signal is one which maximizes the information transferred from sender to observer with minimal distortion in minimal time. 33 The distortion and time criterion translate into figures of merit for the signal characteristics. To accomplish the task described, the signal must stay within the frequency bandwidth allotted, filling it with as much information as possible while maintaining a relatively small number of erroneous info-atoms. A very good way to do this is to look at the time-frequency plane introduced in chapter 2 and see where interferences in the channel show up. Then a variety of techniques which utilize time-frequency methods can be utilized to optimize communication. One idea would be to "communicate around" the interferences - that is design the signal in such a way that it has no time-frequency components competing with interference components. This method relies on a knowledge of the channel before transmission, in which case sophisticated modulation schemes are unnecessary. Another technique is to decompose the received signal into some basis set, excising those components of the received signal which are unusually high, supposedly indicating an interference source [91, 115, 116, 1171. This method relies on designing a signal that does not have components in the given basis with unusually high amplitudes. The focus of this work is to utilize functions with time-frequency flexibility to construct a signal which will minimize the effect of noise in the channel. Where excision methods require analysis first, then synthesis for reconstruction (as do all signal compression and/or processing procedures,) the modulation scheme developed here reverses this process, synthesizing a signal for transmission first, with decomposition at the receiver. 34 Jones' work was a substantial step in this direction, but the two methods he introduced are just special cases of a much more general scheme which allows for a variety of time-frequency representations, giving the designer substantial freedom to "tune" the supersymbol for the channel. For MSM, a high frequency interferer will do more damage than a low frequency interferer due to the longer bandwidths of the basis functions in those subchannels. For MWM, any time-impulsive noise is bad because all the basis functions spread out over an entire super-symbol. Clearly, a signal with a T-F representation that can be specified arbitrarily (see footnote 2, page 24) would minimize the effects of a broader class of interference sources. Wavelet Packet Modulation is one answer to this problem. This development begins by recalling the QAM signal of equation (2.4), where Es is the average signal energy over a period of T seconds of a QAM symbolam pulse-shaped by q5 . Now suppose q5 E w:. The w:~space can be decomposed, as per theorem 3.2, into a finite set of orthogonal subspaces defined by the collection ' = {(117nl)?(12>n2)? "' > (1J7nJ)) as where li < 0, and the direct sum symbol @ is defined in chapter 2, page 12. The QAM signal can then be rewritten in terms of the basis functions of these decomposed spaces, obtaining the multidimensional signal, where a; are the complex QAM signals on the ithchannel of the associated filter bank 1 which has scale 2( (i.e. the channel symbol rate is -) and has its spectrum concentrated 2" at the band determined by ni. In chapter 4 the Gray-coding scheme that determines the bin number of the frequency band is discussed. Since data is placed on orthogonal functions arising from a wavelet packet decomposition of the W: space, the signal defined in (3.16) is called Wavelet Packet Modulation or WPM. The orthogonality across scales and within fixed scales of the constituent packets insures a zero IS1 waveform, which is also free of cross channel interference. 3.3.1 Dimensionality and Special Partitions It is very important to understand that the dimensionality, i.e. the number of T-F atoms in a WPM supersymbol, is fixed in advance as 2N for some N, and has nothing to do with the number of symbols in the QAM constellation in general. The number of channels J upon which to multiplex the data can vary from 1 to 2N. Of course the original QAM signal, with no decomposition performed means each symbol occupies the entire bandwidth, but with smaller durations. This corresponds to the partition P,, = (0,O). The partition is the decomposition of (2.7) defining the Multiscale Modulation scheme of (2.8), ~here2~-' is the number of QAM symbols representing J=N+ 1 dyadic frequency bands in the T-F diagram. For the same dimensionality, the partition is the decomposition of (2.15) and so defines the M-Band Wavelet Modulation scheme of (2.16), where M=J= 2N, in which all symbols occur at the super-symbol rate, but occupy equivalent bandwidths much smaller than the data bandwidth. Hence these formats are special cases of the more general WPM scheme. 3.4 An Example Obviously, because of the inherent flexibility of WPM, it will perform better in many types of interference environments, while keeping the utility of the two special cases MSM and MWM. I.e., there may be an environment which is optimally handled with MSM, and in that case, WPM will become MSM. The advantage of the WPM is demonstrated in environments where other T-F decompositions would be preferred over MSM or MWM. For instance, consider the case where a narrowband jammer is operating in the high frequency portion of the data bandwidth in conjunction with an impulsive time domain interferer somewhere in the supersymbol duration. Let N =4, pr0viding2~=16 T-F atoms in the decomposition. There will be N+ 1=5 dyadic frequency bands in MSM and 16 equal-width bands in MWM, illustrated by the top two blocks in figure 3.7. 37 The normal QAM construction is given in the bottom left and a Wavelet Packet construction is shown in the bottom right. Notice that for MSM, the tone jammer corrupts 8 of the coefficients and the F MSM f C WPM Figure 3.7: Comparison of modulation methods for a given interference environment consisting of time impulse and tone jammers. Grey-shaded areas indicate corrupted symbols. impulse corrupts 5. One of these is affected by both, thus 12 of the 16 coefficients are noisy. The MWM case has all 16 coefficients affected by the impulse and QAM has all 16 affected by the tone. Thus none of these methods is ideal for these noise sources. However, the WPM diagram in the bottom right, which is optimized for this environment, isolates the interferers to 5 of the 16 coefficients - a significant improvement over the previous three. It is precisely this flexibility to isolate certain 38 channel impairments which makes wavelet packet modulation an attractive time-frequency method for multidimensional signaling. 3.5 Waveform Figures of Merit In order to utilize WPM as a practical signaling strategy, several key properties of the waveform, namely bandwidth requirements (PSD), bit error rate, and performance in a Gaussian noise environment, must be quantified. The cases of MSM and MWM are presented in the review of chapter 2, and in more detail in [92], where it was shown that both cases have exactly the same spectral density as conventional QAM. This is really no surprise as all the schemes are variations of the same time-frequency area. However, it is interesting that this property is a function solely of the scaling function and is not dependent on any wavelet function. It turns out that this is true in the general case of wavelet packets as well. For bandwidth efficiency, the special cases were again shown to be equivalent to QAM, and indeed, since orthogonal pulse shapes are utilized, the results are expected to be consistent. Since the analysis for signals in the presence of noise is prohibitive even for relatively simple waveforms, bounds and simulation should be used for quantifying this important signal property. This is not necessarily a problem however, since these are the measures that are used in the actual implementation. 3.5.1 Power Spectral Density Recall from chapter 2 that the conventional QAM signal has power spectral density where @(t)and B(n are a Fourier Transform pair. In order to derive the power spectral density of the WPM waveform, the following lemma is required: Lemma: Let pn(x) be a wavelet packet function and Pn(f) its corresponding Fourier transform. Then Vn €2, Proof of Lemma: Recall the first equation in (3.4) Taking the Fourier transform gives, where H(i)is the half scale Discrete Fourier Transform of the hk sequence. Squaring the magnitudes of both sides gives By a similar process, the second equation in (3.4) provides so that adding (3.21) and (3.22) gives But h and g are quadrature mirror filters, so they possess the well-known "power complementary property" [I] [38], which when substituted into (3.23) yields (3.18). The PSD of the wavelet packet modulated signal can now be derived. Consider the partition P ={(ll,nl), . . . , (l,,n,)} , and introduce the notation T,(t)A 1 pnO)1 ' for convenience. WPM affords the capability of assigning to each channel (basis function) a completely different symbol constellation with possibly varying geometry and/or symbol energies - though they must have the same number of symbols. In this way it is possible for every channel to draw its symbols from different sources (a really attractive feature for coding.) However, the derivation of spectral density in this case is quite involved, and without a closed-form solution. Thus it is assumed that the source is generating independent data and the resulting symbols are then modulated onto J channels (wavelet packets) where the symbols on each channel are identically distributed from the same constellation. The PSD of the WPM waveform is The proof of this step is exactly the same as that used to show the PSD of ordinary QAM as shown in numerous digital modulation texts, most notably [32]. Now if P is sorted to facilitate manipulation such that the ordered pairs follow the rules 1. l1s 125 ... 5 l,, 2. If li= li+l, then ni , then because of this ordering and the dyadicity of the underlying structure, it must be true that I,-,= I,, and nJ-,= n,-1 . Thus (3.25) can be rewritten as where the last line is allowed by the lemma. This relation is actually the PSD of another / / WPM waveform with partition ' = {(n),. , (lJ1nJl)} where (I:-, , nJ-,)I = (I,-1 , -),n~-l and this new partition can again reordered as per (3.26). This 2 process can be continued J-1 times, yielding a final partition 8' = { (0,O) ) . Thus the equivalent expression for the PSD of WPM becomes Also, recalling that the zero'th wavelet packet is just the scaling function from an MRA at the same scale, yields which is precisely the power spectral density of the QAM signal. The significance of this result is that not only do the special cases of WPM - MSM and MWM - have bandwidth requirements equivalent to QAM, but every WPM waveform shares this property. Thus any WPM signal can be utilized on channels currently supporting traditional QAM. 42 3.5.2 Bandwidth Efficiency For this figure of merit it is instructive to first consider the general case of separate constellations for each channel. Let the i-th channel carry symbols a: from a 24-QAM constellation, where Bi is the number of bits per symbol and Riis the symbol rate. Since the PSD of (3.29) has some bandwidth where T is the period of the pulse-shaping function @(t),and p is the percent excess bandwidth required beyond the Nyquist signaling, the genera1 expression for the bandwidth efficiency of channel i is and therefore the bandwidth efficiency of WPM is Now, from the WPM waveform (3.16), the period for each symbol on channel i is 2l' 2-! T, so the i '"channel rate is - , i .e., T and hence (3.32) becomes This is as much simplification as is possible when considering the general case of different QAM constellations for each channel. However, for the case where Ri = I?, 43 i.e., all channels have the same symbol length (and therefore the same number of symbols in each constellation, although no constraint has been placed on the geometry) Then because the li 's come from a dyadic partition of the interval [O, I), it must be that xSl2( = 1, leaving the bandwidth efficiency for WPM to be, which is precisely that for QAM using the same pulse shaping filter. Hence, the theoretically expected result: increasing dimensionality by orthogonal decomposition of the pulse shaping function does not improve bandwidth efficiency. Implementation In the previous chapter, the novel modulation scheme, WPM, based on wavelet packets was developed in detail, but it remains to be seen how this format can be practically implemented. It is certainly expected that a transceiver similar to that developed by Jones in [92] would accomplish the task, and it turns out that this is indeed the case. However, in order to justify this digital implementation of WPM which will be developed in the next section, it is necessary that the Discrete Wavelet Packet Transform (DWPT) relative to some partition P and the corresponding inverse operation (IDWPT) be illuminated in sufficient detail. The reader is perhaps familiar with the standard wavelet transform and its associated non-uniform filter bank structure, or the M-Band wavelet (and fixed level wavelet packet) transform and the associated uniform filter bank structure. What is presently not so widely known is the generalized wavelet packet transform and the associated generalized filter bank. From the development of chapter 3 it is easy to deduce the heuristic operation of this structure, but sufficient mathematical exactitude is provided so as lo satisfy the astute reader, and in the process, the additional notation needed for further developments is established. 4.1 The Discrete Wavelet Packet Transform and its Inverse Recall from (3.4) the recursively defined wavelet packet functions where the h, g sequences, in the context of the current discussion, are interpreted as the series coefficients for the functions p,(x), and p,+,(x) (which again, for the n=O case, are the scaling and wavelet functions from an MRA.) That is and similarly for g(k) with p,+, . Now because of the orthonormality of the packet basis functions, any function f E W: has a corresponding wavelet packet series associated with the partition 6,given by where the coefficients are computed as inner products with the basis functions, i.e., These coefficients constitute the Discrete Wavelet Packet Transform off relative to 6, and the following 'single node decomposition7 result, where H and G are the filter and down-sample operations introduced in chapter 3, provides for the efficient calculation of the coefficients without the potentially costly inner product. The signal processing operations for the relations of (4.5) are shown in figure 4. la, where the signal lines are labeled with the relative coefficient sequences. The relations are proven in numerous references, e.g. [I1, 48, 1111. It should be noted that in this general setting the complete wavelet packet filter bank defined by the partition has no pre-programmable structure, i.e., the decomposition is not necessarily to some uniform level nor is it necessarily a wavelet transform. Therefore the actual algorithm which computes the coefficients must be slightly more involved from a bookkeeping perspective, keeping track of when a particular terminating node has been reached, as well as the partition coordinates. If the current node is not one of the partition coordinates, then another stage of decomposition is performed at that node. The generalized composition (synthesis) result is and provides an equally efficient method of calculating the IDWPT for the given partition. This relation represents the signal processing of figure 4. lb. In the communication signal design problem, this latter building block is used to generate the WPM signal, and the former is used for reception and detection. This is Figure 4.1: Single stage of the (a) Wavelet Packet Transform (b) Inverse Wavelet Packet Transform. a twist on the vast majority of the literature which concentrates on signal analysis by processing the wavelet coefficients, or signal compression by thresholding the wavelet coefficients. Once the objectives are accomplished the signal is reconstructed - perfectly in the case of signal analysis and near-perfectly in the case of signal compression. In this application, the objective is to "build" a custom signal that exploits the channel properties and avoids nowGaussian noise if possible. Hence, the operations are reversed. 4.1.1 The WPM Transceiver Now that the requisite concepts are understood, the actual generation of a WPM signal can be considered. It is entirely plausible that WPM be implemented directly with analog pulse shaping filters for each partition coordinate, requiring a complex scheme for tracking the symbols, and their different rates. Additionally the receiver would require matched filters for every pulse shape, which is exponentially significant for large dimensionality. The preferred method, however, is to take advantage of the structure built in the inverse wavelet packet transform to construct a signal determined by the QAM symbols. Figure 4.2 shows how this is accomplished. The sequence of complex QAM data symbols, a(k) , is used as the source for wavelet packet coefficients and, after demultiplexing this symbol stream into J channels at appropriate rates, the substreamsai are applied to the synthesis process of the IDWPT. This forms a sequence of "coded" complex symbols, b(k), which are intimately related to the QAM data. These new symbols do not map to a QAM constellation but do sustain the information via the inverse transform. The symbols become weights for impulses which are then pulse- shaped by the scaling function for transmission. At the receiver, a single filter matched to the scaling function followed by a sampler provides the constructed symbol estimates,j(k) and the analysis process of the DWPT (for the same partition as that used in the synthesis) expands these noisy symbols, resulting in estimates, ri(k), of the original QAM data. il Gain Matched Filter Figure 4.2: WPM Transmitter / Receiver Model The process can be shown mathematically starting with the b(k) sequence out of the IDWPT. The signal y(t) is a weighted train of impulses given by which after convolution with the pulse-shaping filter and gain factor becomes Changing variables as u =1 = dz=T du , gives T yielding, by the sifting property of delta functions, which looks a great deal like the original QAM signal of (2.4). In fact, the only difference in these two signals is the fact that the complex symbols b(k) can take on immensely more values than the 2k values of the original QAM symbol set. As a matter of record, output sequence range has cardinality Nh = 2kLJ where k is the number of bits per symbol in the QAM constellation, L is the length of the QMF filters, and J is the number of input channels in the filter bank. For example, with a BPSK constellation, 2-tap Daubechies QMF's, and a two-channel filter bank, the 50 output sequence can take on any of 2(3(2)=16 values. For posterity, a 16-QAM constellation, 37-tap square-root raised cosine QMF7s, and an 8-channel bank provide 16(37)(16)= 16592> possible output values - a virtual continuum. At the receiver, the transmitted signal is corrupted additively by noise, n(t), in the channel so that the output of the matched filter is where nF(t)is the filtered noise. Substituting (4.10) into the above yields which after a change of variables as above is Sampling at t = kT gives and by the orthonormality of the scaling function at integer shifts, which is a sequence of estimates of the original symbols out of the inverse wavelet packet transform. Decoding these estimates is accomplished with a forward wavelet packet transform and demultiplexer, the result being a sequence resembling the original QAM 5 1 data. Just how well these output symbols represent the original ones is a function of the noise and the partition chosen for the WPM expansion. For instance, in the example scenario of section 3.4, the partition chosen guarantees that 11 of the 16 symbols in the supersymbol will be reliably represented by the output estimates since they are unaffected by the noise (assuming ideal filters). 4.2 Translation Between Tiling Diagram and Filter Bank Every tiling diagram is associated with a partition defining the actual basis functions utilized in the signal which generates it. Actually, the partition and the tiling diagram are equivalent ways of describing a signal composed of wavelet packet bases - the difference manifested mainly in the way the description is interpreted. A tiling diagram describes the basis functions in terms of their frequency localization and symbol rate, and the partition describes the basis functions directly as output nodes of a dyadic filter bank whose nodes are labelled as ordered pairs. The goal of this section is to establish a method of translating from one to the other, and in the course of this development, the concept of gray-coding is utilized. At this point it is necessary to review the notation and generation of gray-codes. 4.2.1 Gray Coding of Frequency Bins Gray encoding is a binary coding scheme which was originally developed to circumvent the effects of large transient errors in electrical counters. Communication systems use the coding method for building in immunity to large bit error in certain signaling systems. The idea behind gray codes is that any symbol is only one bit 52 different from its nearest neighbors in the code. Therefore, since most errors are due to a one-symbol discrepancy, the code provides lower bit error rates. However, reducing errors is not the only application of gray codes. It turns out that many situations which involving the use of binary notation and/or dyadic structure are ripe for a gray code application. In the case of dyadic perfect-reconstruction filter banks, the structure of the banks admits a natural application. Consider the sequence of sets, G,, in which first three elements in the sequence are These three sets are the first three terms in the more general recursion 53 The set G, describes a gray coded ordering of the numbers { 0, -., 2k-1}, and is actually a bijective mapping in which the domain is { 0, ..-, 2,-1) and the range is the set of all k-bit binary sequences. In this light, G, has an inverse given by (4.19) Gk-l(dbaS.2) LA n 3 Gk(n)=dbase , which follows from the definition. Now, the Paley or natural ordering for the labels of the filter bank nodes has already been chosen. At level -k, the nodes are labeled in order from top to bottom, starting at 0 and proceeding through 2,-1. If the data bandwidth is divided into2, subintervals and the n7thbin is labelled (starting at 0) by the decimal equivalent of G,(n) , a direct mapping from the filter bank node n at level -k to the n7th frequency bin results. Figure 4.3 illustrates this important connection. 4.2.2 Translation Computations Given the above information it is a simple task to translate a coordinate pair in some partition to its frequency bin in the bandwidth. To be very specific, if N is the maximum level of the wavelet packet filter bank (MAXLEVELS) and B, T are the data bandwidth and QAM symbol duration respectively, then other important information such as bin width and symbol rate for the channel can be found. Clearly the supersymbol duration is Tss = 2NT. Now consider the partition coordinate (-1,n) , where 1 2 0 for convenience. For this coordinate the bandwidth is divided into 2' subintervals and the filter bank nodes are labelled according to the Gray coding procedure above. The packet described by this coordinate has the following frequency parameters LEVEL -4 - 0 ...... 0 natural Order FILTER BANK +1 ------1.----.------1 \ Gray WedOrder Figure 4.3: Connection between natural ordering of filter bank nodes and sequency or Gray coded ordering of frequency bins. frequency bin # = ~,-'(n,~,) A b B bin width = - 2' 2b+l center frequency = -B . 31+1 The time parameters of the T-F atoms generated at this node of the filter bank are T 1 time width = 22 = 2'T T-F atom rate = - . (4.21) 2N-I 2'~ To complete the idea, consider a specific partition coordinate, say (-3,7), of a filter bank with N=4. This coordinate is the last node at level -3 in the filter bank, and describes a set of T-F atoms with the following parameters bin width = -B - -B 23 8 center frequency = 2(5) + 1B = -B11 23+1 16 24-3 2 rate = - - - Ts, Tss The translation from frequency bin to wavelet packet number is simple: The decimal representation of the gray code for the bin is the wavelet packet number. The scale or level number -1 is directly associated with the atom-rate for the channel and the value of N, as seen from (4.21), so that 1 = N-log,(rate T_). Thus a systematic method of moving from the tiling diagram to the filter bank (i.e. basis functions), and vice-versa, has been established. It is interesting to note that if the operators H and G are substituted for 0 and 1 respectively, in G,(b), the resulting sequence, call it F,(b), is exactly the filter sequence necessary to obtain frequency localization in bin b at level -1. 4.2.3 Why Gray Coding? The answer to this excellent question is revealed by recognizing an important property of the decimation blocks in the filter bank - Jones [92] calls it "band-shuffling. " If h,, gk are discrete lowpass and highpass quadrature mirror filters that for a given sampling rate Fs have frequency localizations such that the h-filter has bandwidth [O, $), and the g-filter [L,F i),F then the decimation blocks cause a certain exploitable 4 - form of aliasing in which the lowpass and highpass bands of the output of a g-filter and decimator actually switch positions relative to the original bandwidth. Thus two consecutive G operations effectively filter a signal to the band [L, ) and a GH 4 8 5 6 operation filters the same signal to the highpass band [T,:). The proof of this interesting phenomenon is provided in [lo]. Since this only occurs at the output of the G-filters, and since the complete filter bank is made up of iterated two-channel filter banks, the gray coded frequency localization comes naturally. Supersymbol Tuning In chapter 3 it was established that some wavelet packet signals are better than others in a given noise environment, thereby introducing an optimization process which will now be mechanized. Depending on the type of channel interference, the complexity of this process can be relatively simple or extremely difficult, the more complex optimization algorithms holding little hope of any practical implementation in real time. For the purposes of this work, narrowband frequency and short-duration time interferences are considered, which when encountered jointly constitute a formidable challenge for the receiver. These are two noiseljammer types commonly encountered in practice, and they also happen to be the interference environments that allow wavelet packet modulation to show its superiority over traditional methods to date. Within this class of channel interferences there are infinitely many possible combinations of tones and impulses - the simplest being a singular tone and singular impulse. Henceforth this combination is denoted as the 111 channel - implying 1 58 frequency tone and 1 time impulse and it will be the first topic of discussion. A simple algorithm will be developed for this 111 channel that efficiently mitigates the noise and illustrates the supersymbol tuning concepts. From this point, the generalN, / N, combination follows. The goal of this section is to develop a general algorithm which will systematically arrive at a partition which optimizes the supersymbol performance for the given noise environment. Since there are several measures of performance, a choice must be made as to which is most suitable for this application. In this work, the signal- to-noise ratio (SNR) at the receiver is chosen as the performance measure for two reasons: It is the most obvious choice for comparing information and noise for practical systems, and the directly related manifestation in the supersymbol as the number of T-F atoms which intersect the tones and impulses (henceforth referred to as the partition cost) makes it intuitive and easily quantified. 5.1 T-F Diagram Principles In order to gain insight into the unique problem of supersymbol tuning, a paraphrase of the Uncertainty Principle is helpful: The time-bandwidth product of a given signal decomposition is always positive and nonzero. That is, one cannot know, to arbitrary accuracy, both the time and frequency localization of a signal. When the quantitative measures of RMS duration At and RMS bandwidth Af are applied to the localizations, the well known [4, 13, 711 lower bound on the "area" of a time- frequency cell, 59 results, where equality (though still nonzero) is obtained with a Gaussian window function. In essence this guarantees some minimum area for T-F atoms of any signal decomposition. In addition, orthogonal transforms like Fourier, Walsh, Mellin, and the Wavelet Transform, all have the constraint that the T-F atom area is equal for all basis functions. With the flexibility of wavelet packets, this constraint is loosened somewhat so that the shape can change, and a basic time-frequency property for these functions is that a doubling of the frequency localization for a particular packet induces a halving of the time localization for the same. But because of the structure of the packet generation, i.e. a bank of decimators and filters, changing the frequency localization also changes the filter bank structure, possibly resulting in different channel rates (the reference here is to filter bank channels or outputs and not communication channels). Thus the halving of the time localization dictates a doubling of the symbol rate for the channel and the formation of one channel out of two (commonly termed composition). In like manner, doubling At compels the formation of two channels (decomposition), each with half the bandwidth, in which the symbol rate for each is half that of the original. 5.2 Supersymbol Tuning Rules What does this all mean when applied to supersymbol tuning? The answer lies in recognizing two important rules for optimizing performance in the noise environments considered here. In the following, composition of two packets into one is called the reverse tuning direction and decomposition of one packet into two the forward tuning direction. There are two basic rules that govern the search for an optimal partition: Supersymbol Tuning Rule #I: If reverse forward) tuning reduces the partition cost, then forward (reverse) tuning of the same elements will raise that cost. This first rule simply states the obvious - "If a step is taken which improves the cost, then going back to the previous state is not smart." Since T-F cells get narrow in time with composition, improvement via reverse tuning indicates the presence of a time- domain impulse, and therefore spreading out the symbols in time by forward tuning will only increase the likelihood of symbol corruption from this source. Similarly, improvement via decomposition, or forward tuning, indicates the presence of a frequency-domain tone, so that spreading out the symbols in frequency by reverse tuning will increase the likelihood of symbol corruption from this source. Supersymbol Tuning Rule #2: If forward (reverse) tuning raises the partition cost, then no further forward (reverse) tuning will reduce the cost. This rule asserts that when a T-F atom is spread out in frequency or time and the performance is degraded, further spreading only increases the likelihood of jammer intersection. If the spreading of T-F atoms is in frequency, then degradation of cost implies the presence of a tone. Further spreading in frequency cannot decrease the cost. At best, the performance will remain constant with no improvement. A similar argument holds for spreading in time and the presence of impulses. Together these rules suggest the idea of locking out partition elements from the tuning process in one direction or the other. Reverse-locking an element keeps it from being composed during a reverse-tuning operation, and forward-locking similarly 6 1 prevents decomposition of a node during forward-tuning. The next step is to develop an algorithm which takes the above principles into account, starting with the simplest example to develop insight and then generalizing. First the combinatorial issue is explored in order to get a feel for the efficiency of the search algorithm. 5.3 Wavelet Packet Library Combinatorics The goal here is to see just what expanse the search for an optimal partition covers. This will depend entirely on the value chosen for MAXLEVELS, but just how big is this "library1' of bases? Denote the cardinality of the WPM library for N=MAXLEVELS by IP,I . The library for N=O has exactly one partition, namely QAM, so (Po( = 1 . When N= 1, the partition from N=O plus the single available partition at level 1 yield I P, I = 2. When N = 2, there are three new partitions available for the library so IP,I = 5. Each increase of N by one, for the purposes of combinatorial analysis, effectively takes two copies of the previous bank and connects them at the root node. The previous bank had IPA possible partitions, and there are two copies, so this makes I PN1' possible partitions which when the possibility of the root node alone is included, yields the following recursion for the cardinality of dyadic wavelet packet libraries: IPN+ll = IPN12 + . (5.2) For reference, the first seven values in this sequence are 1, 2, 5, 26, 677, 458330, ~2.1x 1011 . Thus, for a filter bank constrained to six or fewer levels, there are over 210 billion possible wavelet packet partitions. It is plain to see that even for relatively small values 62 for MAXLEVELS, the optimization will be difficult at best, unless a very efficient search algorithm is developed. This is the motivation for the work of this chapter. 5.4 The Simplest T-F Jammer - 111 The 111 noise environment is comprised of a single frequency tone somewhere in the data bandwidth in conjunction with a single time impulse somewhere within the duration of the supersymbol. This scenario is exactly that of the WPM example in section 3.4, where the WPM partition used in figure 3.7 turns out to be the optimal one (when MAXLEVELS = 4). It turns out that in the case of 111 noise, the supersymbol tuning procedure is simplified a great deal. Starting with the partition CP = ((0,O)) , tune forward, decomposing each node of the current partition into its children one node at a time, and the evaluating the partition cost at each step. If the cost is not less than the current minimum cost the element is forward-locked, excluding it from the search hence. If the cost is better than the minimum cost, that element becomes part of the partition. This process is continued until all elements in the partition are forward-locked at which point the current partition is the optimal one for the given noise conditions. It is profitable to illustrate this process in two ways: First with a step by step graphical illustration, then with pseudo-code that mechanizes the graphical procedure. The noise scenario of the aforementioned example will be used for the illustration. Figure 5.1 shows the steps involved in this MAXLEVELS=4 example, where the supersymbol for each step is shown on the left, followed by its corresponding filter bank skeleton, noting the coordinates in both (remember the gray-coding in the notation for the supersymbol coordinates). On the right are the highlights of pivotal points in the Q1 = { (-1 10) (-111) 1 C-9~16-Cmin-9 -1" ,O :::: 8' * 8,, - continue with J=2 Qf = { (-2,O)(-2,l) (-1,l) } c = 10 < 9 * 6'l = {(-1,O)(-Ill)} -2,l -2,o QFL { 61 to) 1 -I,O Qlm { (-1lo) (-22) (-23) } C=6<9- Cmin=6 -2,3 8' * 8,, - continue with J-3 8' = { (-1,O) (-22) (-3,6) (-3,7) 1 C=5<6-Cmin=5 8' + PFL-- continue with J=4 t Q'= 8,,+ END Figure 5.1: Tuning procedure illustrated step by step. procedure. In this particular example eight steps are required to reach an optimum partition - significantly less than the 677 possible partitions with MAXLEVELS=4. In fact, the forward only algorithm will never require more than 2N steps to conclude. Thus this algorithm does a very good job of taming an explosive combinatorial problem. The following C-like structured pseudo-code implements the above procedure where it is assumed that the constant MAXLEVELS has been assigned in advance. The reader may follow the code line by line with the above example and see that figure 5.1 depicts the convergence of the initial supersymbol to the optimum one exactly Supersymbol Optimization Pseudocode for 111 Channels Initialize Cmin= Cost( 6' ) Compute initial partition cost while ( 6' # P, ) { Until all partition elements are forward-locked Notation for the elements in current partition for i = 1 to J { Run through all words in current partition If not forward locked and If not decomposable FonvardLock( 6,, li, ni ) Add this word to forward locked partition 1 else { Must be decomposable so Decompose( CP', li, n, ) Replace the chid coords by parent C = Cost( 6' ) Compute cost of new partition ifC CC.min { If cost improved Cmin= C Set improved cost to minimum > else { Else cost not improved, so FonvardLock( 6,, li, ni ) Add this coord to forward locked partition Compose( @'I, li, ni ) And put the parent back 65 The resulting 6' is the optimal partition for the hypothesized noise. Since this algorithm reduces a search from 677 steps to 8, it is tempting to use it for more general noise scenarios, but as is shown in the next section, this forward-only algorithm affords ill solutions in more general cases of multiple tones and impulses. 5.5 N, 1N, Channels - A General Supersymbol Tuning Algorithm If two frequency tones are present in the data bandwidth and are sufficiently separated so that each is in its own half, then the 111 channel algorithm of the previous section fails, since the first decomposition will not yield any cost improvement and the procedure will mechanically assume that improvement is therefore impossible. But improvement is plainly available at the next level. Thus the algorithm must be modified to handle this more general case. The first step in this modification is finding the best uniform level. This is the level at which the partition representing a uniform filter bank at that level has minimum cost. This effectively jump-starts the tuning algorithm in the middle of the T-F compromise. At this point however, another problem arises; the actual optimum partition could have elements at levels in front and behind the best level. It is therefore necessary to search both ways using reverse-locking as well as forward-locking to keep the search efficient. The next question is obvious: After computing the best level, which direction should be searched first (or rather, last, as will be shown to be a more appropriate question)? First instincts would suggest that it should not matter. However, it does make a difference when the constraint placed on the previous forward-only algorithm is incorporated, namely that the algorithm produce the partition that optimizes the cost, but with the minimum number of elements. If the order is chosen such that the last stage is a forward search, the result is a larger partition due to the nature of the forward direction adding elements (decomposition of one parent into two children.) However, searching in reverse last, yields a smaller partition - the desired goal Once the forward direction is complete, the rest of the general algorithm is simply a completion of the search by reverse locking the elements that were improved in the forward direction and then reverse tuning for the optimal partition. The whole search is complete when the current partition is not changed by composition or reverse locking. The steps are described more clearly in the flow diagram of figure 5.2 and the following narrative summary. General Supersymbol Tuning Algorithm 1. FIND BEST LEVEL Starting at 2MAXCEwLSdimensions, iterate through all levels to find the level BL and its associated uniform partition, PBLwhich has the lowest cost (preferring thc partition with lesser elements in the case of equivalent costs for different levels). Set the optimum partition IP', to PBL. If step 1 produces BL=O (QAM) or BL= MAXLEVELS (MWM) then stop. 2. FORWARD-TUNE Find improved frequency dimensionality by splitting (decomposing) all nodes one at a time, updating the optimum partition and minimum cost laterally, and forward-locking all nodes that degrade the cost (by rule #2) or have reached MAXLEVELS. Repeat step 2 until all are forward-locked, i .c., the current partition matches the forward-locked set. Choose MAXLEVELS r,,,,Compute EL, 8 BL, C L,Update 8',C* I Compute P,, 1 REVERSE-TUNE I / Update P1. Cmin 1 Figure 5.2: Flow Diagram for General Tuning Algorithm. 3. REVERSE-LOCK FORWARD-IMPROVED NODES All nodes that have been improved by decomposition, by rule #I, should not be considered for reverse improvement through composition. 4. REVERSE-TUNE Find improved time dimensionality by composing all sibling pairs not in the reverse-locked set to the previous level, one pair at a time, updating the optimum partition laterally with those coordinates that do not degrade the minimum cost and reverse-locking those that do (by rule #2.) Repeat step 4 until all nodes in current partition are either at level -1 or are not eligible for composition either because they are reverse-locked or are not part of a pair. The termination of this step yields the partition which minimizes the cost over all partitions, relative to the given noise scenario. The following pseudocode implements the general supersymbol tuning algorithm just outlined. BEST LEVEL (MAXLEVELS is a pre-determined quantity) Initialize minimum cost and Best Level for i = 0 to MAXLEVELS { Iterate through all levels P = { (-i,O) , . . . , (-i7p-1)} Partition at uniform level, 1 C = Cost (P) Compute cost of this partition if C < Cmin{ If this level is an improvement Cmin= C Update minimum cost and best level BL = -i } 1 6 = {(BL,O), ... , (BL,~"-1)) Establish the partition at the best level, BL FORWARD-TUNING (Requires 6,, Cmin) while ( 6' # 6, ) { Until all partition element3 are forward-locked 3 8' = U(li,ni) Notation for the elements in current partition i=l for i = 1 to J { Run through all coords in current partition if (li,~i)@ PFL{ I£ not forward locked if( li = -(h&YZEI"ELS-l) ) { and If not decomposable FonvardLock( 6,, li, ni ) ~ddthis word to forward locked partition } else { MUS~be decomposable SO Decompose( 6' , li, ni ) Replace the parent by children C = Cost( 6' ) Compute wst of new partition if C < Cmin{ ~f cost improved Cmin= C Set improved cost to minimum P unP = Php U (li,ni) U (li,ni+l) Add children to "improved" set } elseif C > Cmin{ ELse cost degraded, SO FonvardLock( 6,, li, ni ) Add this coord to forward locked partition Compose( 6', [li-1, 2ni], [li-1, 2ni+l] ) And put the parent back REVERSE-LOCKING of FORWARD-IMPROVED ELEMENTS (Requires 0" , Pkp) Set of reverse locked coordinates initially empty Notation for current partition for i = 1 to J { Run through each word in current partition if (li7ni) € Php, An improved element @', @', = 6, U (li,ni) Add to reverse-locked set } REVERSE-TUNING (Requires P' , P,, Cmin) Initialization while (CHANGED = = TRUE) { Go until improvements cease CHANGED = FALSE; Notation for the elements in current partition for i = 1 to J { Run through all words in current partition Make sure the current coordinate is the first child of two and the successive child is also in the current partition (not decomposed) and the current level is greater than 1 so that composition will not yield QAM and the cbildren aren't reverse-locked if (n, is even) and (li,ni+l)E G' and (li,ni) @P, and li < -1 { Compose( 6',[li,ni], [/,,ni+l] ) Replace children by parent in partition C = Cost( u" ) Compute wst of new partition ifC>C.mu1 { Degraded wst ReverseLock( 6,, [li,ni], [li,ni+l] ) child words degrade WS~in reverse Decompose( G",li+17 ) Put children back 1 else { Nondegraded wst cmi,= C; Update wst CHANGED = TRUE; Something changed 1 } > } SUB-ROUTINES The arguments in routines Compose and ReverseLock describe children in partition 6 to be composed, while the arguments in Decompose describe the parent node to be decomposed, and the arguments in ForwardLock describe the node itself. Remove children, add parent Decompose (6,l,n) { Remove parent, add children 6 = 6 \ (1,n) U {(I-1,2n), (1-1,2n+l)) } ReverseLock (6, l,, n,, I,, n,) { Add children to RL partition OJ = @ U (117n1>, (127n,> > 1 ForwardLock (6,l,n) { Add c,n) to FL partition P = 6 U (1,n) } This pseudocode has been implemented in the form of MATLAB" m-files which are included in Appendix A. Using this working model, the performance of the algorithm can be examined when subjected to complicated noise environments. Referring to figure 5.3, with MAXLEVELS =6, the data bandwidth and supersymbol duration have been divided into 64 (26)bins. The 412 noise scenario for this figure is comprised of tones located in frequency bins 4, 16, 31, 57 and impulses located in time bins 17 and 38. This figure represents the supersymbol for the best level computation which yielded BL=-3. The cost for this partition is 40 atoms. Even at this crude stage the noise has been isolated to less than 213 of the total symbols. Figure 5.3: Best Level Supersymbol for N =6 The next step is to tune forward for frequency selectivity, where decompositions that improve the cost are tracked, so that they may be reverse locked. This stage produces the supersymbol of figure 5.4, where it is obvious that the forward tuning is localizing the tones while maintaining the same or lower cost - lower in this case at 32 symbols. This cost is an improvement of 8 symbols over the best level partition. It should be observed here that the forward-tuned partition is not minimal, since the long thin atoms around the tones, which are all corrupted by noise, have the same performance as shorter thicker atoms - describable by one partition element instead of Figure 5.4: Forward-Tuned Supersymbol two at each tone. This is another purpose of the reverse-tuning stage which produces the optimal partition of figure 5.5. Indeed, it is clear that the reverse-tuning "backs off" the frequency localization in many bands in order to either improve cost or decrease partition size. As mentioned above, the bands around the tones are filled with thicker atoms since the cost is the same, and the large band in the middle is widened to localize the impulses since there is no tonal energy there. The effect is a decrease of 2 more symbols in the cost, and a final optimal partition cost of 30 symbols. This is less than half of the total 64 symbols - remarkably good for the quantity of noise in the supersymbol. Figure 5.5: Optimally Tuned Supersymbol: Cost=30 As another example with larger dimensionality, consider the case where MAXLEVELS=8 (256 atoms) and another noise source is placed in the supersymbol. Figure 5.6 shows the optimally tuned partition for this scenario, in which tones have been arbitrarily placed in bins 50, 68, and 193 and impulses placed in bins 14, 79, 85, and 230. Again it is clearly shown how tuning the supersymbol localizes the tones while maintaining a cost compromise with the impulses and widens the atoms where no tonal energy exists to localize the impulses - again maintaining compromise. In this case the cost of the optimally tuned partition is 58 atoms - less than one-fourth of the total. Figure 5.6: Optimally Tuned Partition for N =8: Cost =58 A final example illustrates the power of the time-frequency flexibility in wavelet packet modulation. In figure 5.7 a MAXLEVELS = 9 supersymbol has been subjected to a total of 19 noise sources - 9 tones placed arbitrarily in the bandwidth, along with 10 tones placed arbitrarily in the supersymbol period. The optimally tuned partition shown effectively localizes these 19 tones and impulses to 245 atoms, less than one-half of all the 512 atoms in the supersymbol! It is evident that increasing the dimensionality (within the constraints of practicality) will continue to improve the ratio of cost to total atoms. The cost of this improvement is speed and computational complexity. Figure 5.7: Optimally Tuned Partition, N = 9: Cost =245 5.6 Enhancements to the General Tuning Algorithm It is possible to make enhancements to the general algorithm to increase the efficiency. For instance, if a 111 noise situation is known in advance, it is possible to capitalize on some analytical results arising in this case to make the general algorithm as efficient as the forward only algorithm. Also, redundancy in the current scheme can be decreased by incorporating another reverse-tuning before the forward-tuning stage, where any elements that improve in this direction first are forward-locked. There are other efficiencies to be gained as well, but only these two are examined in this work. 76 5.6.1 111 Noise Scenarios Provide Predictable Best-Level To improve the efficiency in 111 noise, the best level is computed analytically, since this particular noise type admits a predictable cost in uniform partitions. To show this result, consider the uniform level 1 in an N = MAXLEVELS filter bank. There are 2N-' symbols corrupted by the tone, 2' corrupted by the impulse and, one symbol common to both. Letting 6, be the I -level uniform partition, the cost of this partition is C(6,) = 2N-1 + 2' - 1 (5.3) which is easily minimized over 1 , yielding an expression for the best level in terms of N. Taking the derivative of (5.3) with respect to 1 yields Setting (5.4) equal to zero and letting BL denote the solution value, the result, after some elementary algebra is, At this point, nothing in the analysis has constrained the fields for 1 and N, i.e., they have been assumed to be real numbers. However, for this application, 1 and N are integers, so the modification is necessary, where the additional constraint of choosing the partition with the smallest cardinality forces the L-1 "floor" operation. Thus the BEST LEVEL stage of the general algorithm can be bypassed when the 111 noise environment is known in advance. 77 This gives the general algorithm the same efficiency as the forward-only procedure developed specifically for this case. It is profitable to note several things at this point to as to make a statement about the optimal partition for this case. First, the optimal partition in 111 noise will always localize the tone to a channel at MAXLEVELS. Second, half of the data bandwidth will always be free of tonal noise in this case and each successive halving of the corrupted portion will result in one atom in that channel being affected by the impulse. The optimal partition cost will therefore always be Cmi, = MAXLEVELS+l. Since the total number of T-F atoms in the supersymbol is 2MAXLEwLS,it is clear that increasing MAXLEVELS will decrease the costltotal atoms ratio ad infinitum, proving that larger N improves performance always. It is possible to extend this same process to the more general case where there are N, tones and N, impulses present, except that a bound replaces the equality in the cost computation. Using the same reasoning as above the partition cost is bounded by C(6,) I: N, 2N-1+ N, 2' - NP, . (5.7) The relation becomes an equality for certain noise geometries in which the interferences are spread appropriately throughout the time and frequency domains. The minimization in this case is similar to the previous. The derivative gets set to zero and BL represents the optimal solution. The result, skipping the details, is where again, the floor function satisfies the constraint that BL be an integer. It is noted here that this result is consistent with the previous special case where N, = N, = 1 so that BL = 1~/2].The practicality of this result is questionable from the standpoint that it is really only useful when the noise is known fully, in which case other means can be used to mitigate the noise entirely. In the 111 result, the only thing assumed known was the presence of exactly one tone and one impulse; their positions were not part of the BEST LEVEL result. The one thing this result does provide is insight into the idiosyncrasies of this noise class, and in chapter 6 these are discussed briefly for future work to develop these ideas into a more cohesive theory. 5.6.2 Efficiency Improvement via Another Reverse-Tuning Stage Presently, the supersymbol tuning algorithm first finds a best level, then tunes forward until elements reach MAXLEVELS or are forward locked due to cost degradation. This forward tuning, if continued with very little forward locking, can grow exponentially in its computation requirements, since every additional level doubles the number of partition elements, and hence the number of cost computations. After this forward tuning process is complete, the reverse tuning then, somewhat redundantly, reduces the partition size by composing elements that do not degrade cost in their composition. For many scenarios this amounts to doing unnecessary work not only once but twice, because if it is the case that reverse tuning from the best level would result in a smaller partition without decreasing the cost, this would be an improvement. 79 Forward tuning these elements first, only means that they must be reverse tuned again, back to the best level (two unnecessary steps), before they are improved in reverse. Hence the efficiency of the algorithm can be improved by introducing an intermediate step of reverse tuning just after the best level calculation to find any elements that improve in this direction. These elements would then be forward locked, so as to exclude them from the forward tuning step, thereby eliminating unnecessary work. The disadvantage of this seemingly simple enhancement is that the bookkeeping is quite complex, since improved elements must be tracked, potentially over several levels, to their ends where they are recorded as the final reverse tuned partition element. The MATLAB" m-files in the appendix incorporate this extra step for the interested reader. Conclusions and Future Work As with all technical results, there are many things about the generalized orthogonally multiplexed communication scheme, Wavelet Packet Modulation, developed in this work that are useful. There are many other things that are not useful, and there are some ideas, properties, and issues that still need to be addressed. It is the author's view that any technical achievement which does not raise new questions in the course of answering old ones is ultimately futile, for it would imply an end. Philosophically speaking, this is an absurdity - there must be more to everything than that which is understood, even though new advances are made. In this light, the following summary the work presented here highlights those items which are significant contributions, and then a discussion of ideas for future developments of these ideas is given. 6.1 Conclusions In this work of fundamental communication research, contributions to a practically implementable system employing new orthogonally multiplexed signalling technology 81 have been made. Working from the foundations (multiresolution analysis, m-band wavelets, wavelet packet bases, multi-scale modulation, and M-band wavelet modulation) developed by several esteemed professionals and colleagues and reviewed in chapter 2 and the first part of chapter 3, the wavelet packet modulated signal, which uses orthogonal wavelet packets as the pulse shaping filters of a standard QAM format was constructed. The basic communication signal figures-of-merit, power spectral density and bandwidth efficiency, were derived and shown to be exactly the same as a regular QAM signal when the QAM pulse shape is a wavelet packet scaling function. It was shown that this signal is a generalized form of the MSM and MWM signals previously derived, allowing improved performance in tone and impulsive noise situations, where a smaller fraction of the supersymbol atoms localized the noise sources. The implementation of WPM was developed in chapter 4, where the inverse wavelet packet transform and the forward wavelet packet transform were utilized in a transmultiplexer arrangement, with a wavelet packet scaling function as the pulse shaping filter and a filter matched to this function providing transmission and reception capabilities. The fact that the filter banks used to generate the wavelet packets do not yield a Paley (natural) ordering of the frequency bins in the bandwidth was also discussed and developed. The gray-encoding algorithm provided the connection between the wavelet packet level and number and the associated bin containing its primary frequency localization. An example of translating from partition to frequency bins and back 82 afforded some insight into the process, and an explanation of why this permutation of the ordering occurs was given. In chapter 5 the time-frequency flexibility of the WPM signal was utilized in the "supersymbol tuning algorithm" for tone and impulse noise scenarios which searched the space of partitions to arrive at an optimal partition for the given noise. The principles behind the T-F diagram established two supersymbol tuning rules that naturally led to the ideas of traversing the filter bank according to these rules in both the forward and reverse directions while tracking and locking elements that exhibited desirable behavior. A detour through wavelet packet library combinatorics provided a view of the potentially massive size of the search space, giving a clear understanding of the importance of the algorithm efficiency. By first examining the case of a single tone and impulse in the supersymbol, some insight into the process of supersymbol tuning was gained. From the forward-only algorithm developed for this case, a move to the more general algorithm came naturally, incorporating a best-level calculation followed by forward-tuning to localized tones, and then reverse-tuning to remove unnecessary partition elements and simultaneously localize impulses. Pseudo-code was presented to methodically organize the concept without confusing bookkeeping. Several examples gave credence to the claim of taming the combinatorial issue, while arriving at an optimal partition. Finally, two enhancements to the general algorithm were presented; a simple calculation which makes the general algorithm as efficient as the forward-only algorithm for 111 noise and the incorporation 83 of a reverse-tuning stage before the forward-tuning stage to decrease the amount of redundancy in the search. 6.2 Future Work There are a number of issues which must be addressed in the continuing development of WPM. From symbol timing and synchroni.zation to optimizing the QMF's for the application, several questions are still unanswered. This section mentions several of these issues so as to provide a starting point for future research. 6.2.1 Timing and Synchronization The obvious next step in developing wavelet packet modulation as a practical tool for communication is to solve the problem of synchronization and timing. Ideally, this algorithm will be continuously updating the supersymbol so that channel changes are reflected in the modulation of the data, within some reasonable time constant not exceeding the rate of change for the channel. This would require that the transmitter and receiver both have access to critical information determining their operation. I.e., the transmitter must be given the partition determined by the adaptation circuit, and it must be able to change the parameters of the modulation to accomodate the new supersymbol. The receiver, on the other hand, must know what to expect for each supersymbol, and equally important, it must have timing information for accurate sampling and decoding. This problem is handled very differently depending on whether a feedback path exists. Without feedback, a preamble is necessary before any change in the supersymbol can be correctly interpreted by the receiver. Also the transmitter has no information on 84 how to correct the tiling diagram, so it basically adapts in ignorance unless some channel estimation processing is integrated in the system. This may increase the system complexity to prohibitive levels and thus the supersymbol tuning algorithm may be unacceptable without feedback, depending on the system requirements. When feedback is possible, the receiver can make the decisions about how to change the partition and communicate them to the transmitter easing the synchronization problem, but the timing of the supersymbols and the sampling must still be worked out. This may be feasibly addressed with a PLL-type circuit which tracks timing attributes common to all packets, or a correlator implementation of the matched filter with a phase locking device that lines up the signals for maximum correlation performance. 6.2.2 Parameterization of Scaling Filter Coefficients In the development of wavelet packet modulation, the standard pulse shaping filter, $, is assumed to be a valid scaling function. There are numerous scaling functions available to the designer which have varying properties, some of which are useful and some of which are not. For example, the square-root raised cosine pulse, which is a Meyer scaling function [Ill, performs much better for orthogonally multiplexed communication applications where frequency selectivity is important, than does any of the Daubechies family of scaling functions, though both are valid pulse shapes. It is desired to optimize the scaling function with respect to the criterion for good communication signals, and it is over the space of independent parameters specifying the h(n) sequences that optimal wavelets for a particular problem or class of signals are found [48]. 85 6.2.3 Coding Via Variable Constellation Geometries With Fixed Symbol Counts In the analysis presented in chapters 2 and 3, it was mentioned that each symbol stream utilized in the MSM, MWM, and ultimately the WPM signals, could feasibly be drawn from different complex constellations. Since this complicated the concept initially, it was decided that only the case where the constellation was the same for all symbol streams would be examined. However, it is entirely possible and indeed desirable to utilize different constellation geometries (with the same number of symbols throughout) as a coding platform. The Low Probability of Interference/Detection (LPIID) properties of a technique such as this are very attractive, but in order to validate the properties, the power spectral density and bandwidth efficiency calculations of chapter 3 must be revisited. 6.2.4 Generalized P-adic Filter Banks and Associated Packets This work was strictly concerned with dyadic filters, where p=2. Though very useful, it somewhat limits the ability to "arbitrarily" choose the supersymbol pattern. The theory is already established to generalize this to p=3 and beyond, where even mixing the split counts at each node is possible, though a nightmare of notation. With these results it is possible to have an unbounded variety of frequency and time localization schemes that can effectively contract one frequency band, which is not dependent on a predefined structure, dyadic or otherwise, while expanding the neighboring band. The effect is to loosen the constraints on the supersymbol structure significantly. 6.2.5 Improvement of Frequency Localization An important disadvantage of the wavelet packet constructions developed in this work is the declining precision of the frequency localization at each level of the filter bank. This is important since the supersymbol tuning concept is heavily dependent on the localization of tones since spectral leakage caused by poor localization affects adjacent frequency bins. At least two methods of controlling this spreading have been presented in the literature 1401, 1831, the one most appealing to this application being the implementation of longer filters with better frequency resolution. A good compromise between speed and resolution is to modify the filter lengths at each level so as to improve frequency localization as it is needed. It is even possible to construct a filter family with differing lengths at each level so that the wavelet packets are uniformly bounded and therefore have equal precision in their frequency localization 1831. It is important in future work to consider this issue in more detail, modifying the filter banks used in the implementation to improve supersymbol tuning performance. 6.2.6 Continued Development of Supersymbol Tuning The ideas presented in chapter 5 were actually just the beginning of this fascinating concept. Several issues should necessarily be investigated for possible improvements to the supersymbol tuning algorithm including the examination of other noise classes, and new and even more realistic cost functions. 6.2.6.1 Other Noise Classes The supersymbol tuning algorithm is really a tool to illustrate the abstract concept of noise mitigation through modification of time-frequency localization. Though tones and impulses exist in practice, they are far from the only noise scenarios encountered; hence some amount of future work should be devoted to extending the concepts of supersymbol tuning to other noise classes. Some examples which are not beyond the scope of achievability are: narrowband jammers (this is potentially a direct application of the proposed algorithm) and short duration pulses, and wideband frequency domain and long duration time-domain interferences with low energy. These last two noise types could possibly require an appropriate change in the cost function used to describe the signal to noise ratio. This issue is discussed in the next section. 6.2.6.2 Better Cost Functions The cost function used to develop the SS tuning algorithm is very crude but serves the purpose of simplifying the abstract concepts. New and better cost functions that more appropriately handle varying noise scenarios should be developed. One that immediately follows is a cost dependent on noise energy that assumes utilization of all symbols, but weighs the symbols with less noise energy more heavily in the symbol decisions. This is opposed to an excision strategy which just annihilates corrupted symbols. 6.2.7 Spread-Spectrum Application of WPM A most important application of wavelet packet modulation is in direct sequence spread spectrum communications, where the symbols determined by the pseudo noise sequence are positioned in the supersymbol in some pseudo random fashion. In traditional DSPN (Direct Sequence Pseudo Noise), the wide bandwidth of the symbols causes problems in the presence of frequency-domain noise where all symbols are corrupted, but the time-frequency dimensionality has immediate advantages in mitigating the effects of narrowband jammers and impulses, where only a fraction of the symbols are corrupted. 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Jones, and others, Wavelets: Advanced Topics in Real Analysis, Seminar-Math 869, Ohio University, Athens, OH, January 1994-present. [162] W.W. Jones, "Multiresolution Analysis and Filter Banks, " notes from private lectures, 1994. [163] D. Lawrence, Discussion on parameterized families of discrete sequences, November, 1994. Appendix A The following is the actual m-file code used to implement the general supersymbol tuning algorithm in MATLAB" 4.2. function CPopt,Coptl = sstune< WLNELS, Tones, Iqulses % [Popt,Coptl = SSTUNE( MAXLEVELS, Tones, Impulses ) % % Script for automating the supersymbol tuning algorithm. The % maximum filter bank Level is specified in MAXLEVELS and the noise % scenario is specified in the vectors Tones and Impulses. Each tone % and impulse position is an integer entry in the range 1:2^MAXLEVELS. % % SSTUNE provides some intermediate data visualization by first outputting % the best level partition, Pbl, and cost, Cbl, with a plot. If there % are tones and impulses present, BL will be between 0 and -MAXLEVELS so % it then outputs the reverse-tuned partition, Prt, and cost, Crt, % with a plot. The forward-tuned partition, Pft, and cost, Cft, come next % with plot, and the Last stage of reverse tuning produces Popt,Copt % and a plot. These last two are also return variables. % Initialize N=MAXLEVELS; % Best Level [Pbl, BL, Cbll = bestlevl( N, Tones, Impulses ) ssplot( Pbl, N, Tones, Impulses ); pause(1) if BL == -MAXLEVELS I EL == 0, Popt = Pbl; Copt = Cbl; else, % partition needs tuning % Reverse Tune from best level (no reverse locked elements) to pick % up improved elements - forward locking them. [Prt, Crt, Pfll = revtune( Pbl, [I, N, Tones, Impulses) ssplot( Prt, N, Tones, Impulses ); pause(1) % Forward Tune the resulting partition without messing with the forward % Locked elements, while tracking forward improved elements and reverse % locking them for the Last stage. [Pft, Cft, Prll = fortune( Prt, Pf1, N, Tones, Impulses ) ssplot( Pft, N, Tones, Impulses ); pause(1) % Reverse Tune Final Stage subject to the forward improved elements CPopt, Copt, Pnewf 11 = revtune( Pft, Prl, N, Tones, Impulses) ssplot( Popt, N, Tones, Impulses 1; end function [Pbl,BL,Cminl = bestLevL< IIAXLEVELS. Tones, Impulses % [Pbl,BL, Cminl = BESTLEVL( MAXLEVELS, Tones, Impulses % % Finds the best uniform filter bank Level, BL, and its associated % minimum cardinality uniform partition, Pbl, where "best" is measured % in terms of minimum supersymbol cost, Cmin - which is also returned. % % MAXLEVELS - constant describing the maximum filter bank Level % Tones - vector of integers in the range l:ZAN describing frequency % bins containing sinusoidal interference. % Impulses - vector of integers in the range ?:2^N describing time % bins containing impulsive interference. N = MAXLEVELS; Cmin = ZAN; BL = 0; for i=O:N, P= [-i*ones(ZAi ,I)(O:Zni-I)'] ; %Build uniform partition at Level i C=sscost(P,N,Tones, Impulses); %Compute cost if C < Cmin, %Ifimproved Cmin = C; % adjust Cmin BL = -i; % adjust best Level Pbl = P; % reset best level partition end end function CPrt, Crt, Pf 11 = revtune( P, PrL, MAXLEVELS, Tones, IquLses % [Prt, Crt, Pfll = REVTUNE( P, C, MAXLEVELS, Tones, Impulses ) % % Performs a reverse tuning operation on the partition, PI subject to % the reverse locked partition, Prl - which is empty if revtune is called % after BESTLEVL, but is not empty if called after FORTUNE. % An improved partition, Prt, and its associated cost, Crt, are returned % along with improved elements recorded in Pfl, the forward locked partition. % The global constant MAXLEVELS must be passed as an argument. The % cost, Cmin, of P relative to Tones and Impulses, must be computed as a % starting point for cost improvement. A % Tones - vector of integers in the range l:ZAN describing frequency % bins containing sinusoidal interference. % Impulses - vector of integers in the range l:ZAN describing time % bins containing impulsive interference. % % % Initializations Rootlevel=O; N=MAXLEVELS; Pnew = P; Primp = [I; Cmin = sscost(P,N,Tones,Impulses); % compute initial cost CHANGED=? ; % Reverse tune until no change occurs in partition. whi le CHANGED, CHANGED = 0; Pcur = Pnew; CJ,xl = size( Pcur 1; for i = l:J, 1 = Pcur(i,l); n = Pcur(i,Z); if (iseven(n1 & isinp(Pcur,l,n+l) & (L < Rootlevel-1) & -isinp(Prl,L,n)), Pnew=compose(Pnew,l,n,L,n+l); % compose to parent C=sscost(Pnew,N,Tones,ImpuLses); if C > Cmin, Prl = revlock(Prl,L,n,L,n+l); % add (L,n),(L,n+l) to Prl Pnew = decompos(Pnew,l+l,n/Z); % put children back elseif C < Cmin, Cmin = C; % update cost Primp = [Primp; 1+1 n/21; % add (L+l,n/2) to rev improved prtn CHANGED = 1; % something changed else CHANGED = 1; end end end end % Forward lock the improved elements. Since Primp potentially has a tree % of descendants for a band that improved several times, Pnew is utilized % to discern which of the elements in Primp are the 'Ieldest." Pfl = [I; [J,xl=size(Pnew); for i = l:J, 1 = Pnew(i,l); n = Pnew(i,2); if isinp(Primp,l,n), Pf 1 = forlock( Pfl,l,n 1; end end Prt = Pnew; Crt = Cmin; function CPft,Cft,Prll=FORNNE(P,Pf L,RAXLNELS,Tones,Iqulses) % ~Pft,Cft,Prll=FORTUNE(P,Pfl,MAXLEVELS,Tones,ImpuLses~ % % Finds the partition, Pft, and it associated cost, Cft, relative to the % noise modelled in the Tones and Impulses vectors, that minimizes the % supersymbol cost in the forward direction - starting with P and subject % to the forward locked partition, Pfl. % MAXLEVELS is the maximum Level for the filter bank. % Initialization Pnew = P; N=MAXLEVELS; Cmin = sscost(P,N,Tones,Impulses); while -isequal(Pnew,Pfl), Pcur = Pnew; [J,xl = size( Pcur ); for i = l:J, 1 = Pcur(i,l); n = Pcur(i,2); if -isinp(Pf 1, L,n), if 1 == -(MAXLEVELS-I), % can't go beyond this Pf1 = orderp( forlock( Pf 1, 1, n) ); else Pnew = decompos( Pnew, I, n ); C = sscost( Pnew, N, Tones, Impulses); if C < Cmin, Cmin = C; % update minimum cost Pf imp = [Pfimp; 1-1 2*nl; % add chi Ldren of (1,n) to "improved" Pfimp = [Pf imp; 1-1 2*n+ll ; % elements. elseif C > Cmin Pfl = forlock( Pfl, L, n ); Pnew = compose( Pnew, 1-1,2*n, 1-1,2*n+l ); end end end end Pnew = orderp(Pnew); % order elements for comparison end % Reverse lock the improved elements. Since Pfimp potentially has a tree % of parents that improved several times, Pnew is utilized % to discern which of the elements in Pfimp are the "youngest" for i = l:J, 1 = Pnew(i,l); n = Pnew(i,2); if isinp(Pfirnp, l,n), Prl = CPrL; 1 nl; end end Pft = Pnew; Cft = Cmin; function y=sscost(P, mxlevels,Tones, Iqulses) % y = SSCOST(P, maxlevels,Tones, Impulses) % % P - partition matrix -> (1,n)'s % maxlevels - Maximum filter bank level - 2^maxlevels blocks % Tones - vector of integers in the range 1:2^N describing frequency % bins containing sinusoidal interference. % Impulses - vector of integers in the range l:ZAN describing time % bins containing impulsive interference. % % Shorthand N=max levels; RootLevel=O; % Find number of partition elements, J, which is the # of rows of P [J,xl = size(P); % Order the elements as would appear from top down in filter bank. P = orderp(P); % Convert partition elements to sorted Linear partition of an interval LinearP = [2 .^ P(:,I).*P(:,2); 2^RootLevell.*2^N; % Form noise template and model tones and impulses as one-vectors % in rows (Tones) and columns (Impulses) SS = zeros(2^N); SS(Tones,:)=ones(Length(Tones),2~N); SS(:,Imp~lses)=ones(2~N,Length(Impulses)); % Compute cost for this partition Cost = 0; for i = l:J, rate(i)=ZA(N+P(i ,I)); for j = l:rate(i); atom = ~~(Linear~(i)+l:linearP(i+l),l+((j-l)*2~N/rate~i~~:~*2~N/rate~i~~; if sum(sum(atom)) -= 0, Cost=Cost+l; end end end function y=sspLot(P, mxlevels,Tones, Iqulses) % y=SSPLOT(P, maxlevels,Tones, Impulses) / /a % P - partition matrix -> (1,n)'s % maxlevels - Maximum filter bank level - 2^maxlevels blocks % Tones - vector of integers in the range 1:2^N describing frequency % bins containing sinusoidal interference. % Impulses - vector of integers in the range 1:2-N describing time % bins containing impulsive interference. % % Shorthand and initializations N=maxlevels; RootLevel=O; % Find number of partition elements % Convert partition elements to sorted linear partition of an interval % and get the index vector from the sort for rearranging P. [LinearP,II = sort(C2 .^ P(:,I).*P(:,2); 2^RootLevell.*2^N); % Rearrange partition to match order of sorted linear partition, % keeping in mind that the linear partition vector has a 2"N appended % for bookkeeping purposes. P = P(I(I:J),:); % Allow for a noise border. and Fix axes so that lines come out on % integers. c b3 noiseborder = 2^N/8; axis( [O 2^N+noi seborder 0 ZAN+noiseborderl ) ; % Plot horizontal channel Lines and right and Left borders %set(line([zeros(J+I,l) ones(J+1,1).*2^NI ', [LinearP LinearPl '),'Colorl,'w'); plot([zeros(J+I,l) ones(J+1,1).*2^NI',[linearP linearP1'); %set(Line([O 2^N;O 2^N1, [O 0;2*N 2"NI),'Color','wO; plot( [O 2^N;O 2^N1, [O 0;2^N 2"NI); % Plot vertical block divisions in each channel for i=l:J rateci )=2^(N+P(i ,I)); for j=l:rate(i)-I, fromx=j*2^N/rate(i); % set(line([fromx fromxl, [linearP(i) linearP(i+l)l),'Color','wl); plot( [fromx fromxl, CLinearP(i) LinearP(i+l)l); end end % Center tones and impulses in their bands for plotting Tones=Tones-.5*ones(size(Tones)); Impulses=Impulses-.5*ones(size(Impulses)); % Plot indicator lines for tones Nt = length(Tones); Ni = Length(Impu1ses); xdata = [2^N*ones(Nt,l) (Z^N+noiseborder)*ones(Nt,l)l'; ydata = [Tones( : ) Tones(: )I' ; %set(1ine(xdata,ydata),'Color','w1~ plot(xdata,ydata); % and impulses xdata = [Impulses(:) Impulses(: )I '; ydata = [2^N*ones(Ni,l) (2^N+noiseborder)*ones(Ni,l)l'; %set(line(xdata,ydata), 'Color', 'w') plot(xdata,ydata); function Pnew=colpose(P, L,n, ls,ns) % % Replaces the (1,n) "childN and (ls,ns) "sibling" in P with the element % (l+l,n/2) "parent" effecting a composition. % if n > ns, tmp=ns; ns=n; n=tmp; % switch ns and n so that n < ns end if -iseven(n) I (I-=Ls) I ns-n-=I, error; end [J,xl = size(P); % Compute number of partition elements y=findelem(P, 1,n); % Find the row number of (1,n) in P Pnew=P([I:y-I y+l:J],:); % delete (L,n) from P y=findelem(Pnew,ls,ns); % Find the row number of (Ls,ns) in Pnew Pnew=Pnew([I:y-I y+l:J-11,:); % delete (ls,ns) from Pnew Pnew= [Pnew; L+1 n/21; % add (ltl,n/2) to Pnew function Pn&ews(P, L,n) % % Replaces the element (1,n) in P with the elements (1-1,2n) and (1-1,2n+l) % The element (1,n) describes the parent node of partition P % J = length(P); % Compute number of partition elements y=findelem(P,l,n); % Find the row number of (1,n) in P Pnew=P(CI:y-I y+l:JI,:); % delete (1,n) pnew= [pnew; 1-1 2*n; 1-1 2*n+l] ; % add ( 1-1,2n) and ( 1-1,2n+l) function y=revLock(PrL, L,n, Ls,ns) % y=REVLOCK(Prl, L,n) % % Adds the (1,n) "child" and (Ls,ns) "sibling" elements to the partition of % reverse Locked elements. They are both added at the same time % because reverse locking means forbidding a composition of two % children (namely those in question) into one parent. % if n > ns, % in case the order is backward tmp=ns; ns=n; n=tmp; % switch ns and n so that n < ns end if -iseven(n) I (1-=ls) I ns-n-=I, error; end y = CPrl; 1 n; Is nsl; function neuPf L=forLock(Pf L, L,n) % newPf l=FORLOCK(Pf 1, 1,n) % % Adds the (1,n) "parent" element to the partition, Pfl, of % forward-locked elements so that no future decomposition will be % performed on that element. % newPfl = [Pfl; 1 nl; function y = orderp(P); % y = ORDERP(P); % Rearranges the partition matrix P such that the elements successively % describe a linear partition of an interval. % Convert partition elements to sorted Linear partition of an interval % and get the index vector from the sort for rearranging P. [linearP,Il = sort( 2 .^ P(:,l) .* P(:,2) 1; % Rearrange partition to match order of sorted linear partition, y = PCI,:); function y=findeLem(P,L,n) % y=FINDELEM( P,l,n ) % Finds the row of P in which the partition element (1,n) resides. % % Form a matrix with two columns indicating presence of matching % values in the partition column - first column contains 1's in % all positions where P has an 1 in the first column, and second % column contains 1's everywhere that P has an n in the second column. nodeindexmatrix = [P(:,l)==l P(: ,2)==nl; % "find" the index where the sum of the rows is 2. This is the % row where both L and n are together. y = find(sum(nodeindexmatrix'==l)==2); function y=iseqwl(A,B) % y = ISEQUAL( A, B ) % % Determines if A and B are exactly the same matrices. % y=O; % assume not equal CmA,nAl = size(A); CmB,nBI = size(B); if mA == mB & nA == nB, % Check for conformable dimension first if A==B, % compare y=l ; end end function y=iseven( n ) % y=ISEVEN( n % Determines if n is an even number. /o y = 0; % Assume odd if n/2-floor(n/2) == 0, % Division minus integer part of division. y = I; end function y=isinp( P, 1, n % y = ISINPC P, 1, n ) % % Determines if the (L,n) element is part of the partition, P. A % Returns 1 for true and 0 for false. % y=O; % Assume false if -isempty(P), % Possibly true if -isempty(f indelem(P, L,n)), % Definitely true y=l ; end end