Continuous Wavelet Transform

Transcription

1 Chapter 2 Cntinuus Wavelet Transfrm 2. Intrductin t the -Dimensinal Cntinuus Wavelet Transfrm Wavelet transfrms are time-frequency transfrms which map the time-frequency plane in the manner f Figure. (c) using specific dilatins and translatins. Tilings f higher-frequency areas f the plane have larger bandwidth and thus, in accrdance with the uncertainty principle, shrter timespan. The bandwidth is prprtinal t the psitin f the tile alng the frequency axis. The relatinship between frequency psitin and bandwidth (and inversely, timespan) is in keeping with the Nyquist sampling thery which states that in rder t accurately capture the infrmatin abut a signal (s as t be able, fr instance, t recnstruct that signal), the signal must be sampled at a twice the value f the highest frequency in the signal. In ther wrds, the timespan cvered by the samples must be inversely prprtinal t the bandwidth f the signal. This makes wavelet tiling mre natural in a sense, and while it cannt guarantee the mst efficient representatin f any arbitrary signal, it will tend t be a gd representatin fr mst natural signals. It is imprtant t explain what I mean by a gd representatin. A vice-band recrding sampled at 8 khz can be called a gd representatin in the sense that it cntains all the infrmatin abut the signal necessary t recnstruct (e.g., play back) the recrding. On the ther hand, a sampled signal (withut any further transfrmatins) is rarely used fr speech recgnitin because the speech signal is encded int the time-dmain signal in cmplex ways that cannt be detected by a purely time-dmain analysis (the Dirac basis). Equivalently, a Furier transfrm (Furier basis) f the signal is f limited use. A windwed Furier transfrm des fairly well at capturing the essence f the speech recrding, but if the windw is f fixed length, then there will always be sme events which are t shrt in time t be accurately represented. Tw slutins t that prblem 4

2 may cme t mind: Either vary the size f the windw accrding t the demands f the mment, r perfrm the analysis in parallel with windws f several different lengths. The first slutin is adaptive, and while it is a very interesting slutin, it is difficult t cmpute and requires infrmatin abut the adaptatin t be kept with the decmpsed signal if the signal is t be analyzed r recnstructed. Advanced adaptive techniques are beynd the scpe f this thesis. The ther slutin, parallel analysis at different scales, wrks well, but the result is redundantly cmputed, which increases the bandwidth f the decmpsed signal n-fld fr n parallel analyses. If we remve the redundant parts f the decmpsitin, the part that is left is a wavelet decmpsitin. Parts f the decmpsitin capture features f the signal which ccur ver shrt time (transients) and ther parts capture features f the signal which have fine frequency structure (frmants). Natural signals, particularly acustic nes which are the primary tpic f the thesis, require the accurate detectin f bth transient-like and frmant-like features fr autmatic classificatin and interpretatin f the signal. Each tile f the time-frequency plane represents a single wavelet cefficient cmputed by applying a filtering functin centered n that area and having the crrect aspect rati between time span and bandwidth. The s-called mther functin describes a family f functins at different scales (a) and tempral ffsets (b) which determine the psitin and aspect rati f each tile cvering the time-frequency plane: a;b(t) = p a t b a : (2.) It is desirable fr the wavelet functin t have cmpact supprt; that is, the functin shuld be bunded r generally lcalized in time and frequency. In frmal terms, the wavelet shuld be able t meet the criteria [4] j (t)j c ( + jtj) (2.2) j (!)j c ( + j!j) (2.3) fr sme >. Existence f an inverse transfrm depends n the relatinship Z j (!)j 2 d! < +: (2.4) j!j Equatin (2.4) implies that (! = ) =, which in turn implies an scillatry functin in time. The Discrete Wavelet Transfrm (DWT) is a discrete-time functin which derives frm certain families f rthnrmal basis functins which satisfy the cnditins f (2.3) and (2.4). The 5

3 wavelet mther functin is bund is space by satisfying the strnger relatinship (t) = utside f a small regin f cmpact supprt. Bunding the frequency is a matter f filtering. Because the system is discrete-time, the filtering functins are FIR filters and the bandpass characteristic results frm sequential applicatin f a lwpass and a highpass filter [], a cmmn practice in signal prcessing. An elegant pyramid algrithm [] defines filtering ver different frequency scales in a recursive manner in a way that allws infrmatin frm higher frequencies t be pushed int lwer frequencies as the recursin prgresses. Using well-develped discrete-time filter thery, the FIR highpass and lwpass filters can be transfrmed int an efficient pipelined butterfly filter (als knwn as a lattice filter) [2]. The FIR filter cefficients r the equivalent lattice filter cefficients are uniquely determined by the mther wavelet basis functin. The length f the FIR r butterfly filter is the wavelet rder. Higher-rder wavelet functins typically yield better distributin f infrmatin amng the wavelet dilatins, leading t better data cmpressin when the wavelet system is used fr that purpse. 2.2 CWT vs. DWT The Cntinuus Wavelet Transfrm (CWT) is an analg filtering functin and is similar t what is knwn as the Gabr spectrgram [3]. Similarly t the Discrete Wavelet Transfrm, it requires peratins f lwpass and highpass filtering at different scales. Hwever, the filtering functins are perfrmed n the input in parallel, as a filterbank, with the lwpass and highpass functins cmbined int a single bandpass functin. The different scales are interpreted as adjacent bands in the frequency dmain, with the bandwidth increasing prprtinally with the center frequency f the band: thus the CWT is described by a cnstant-q filterbank. That descriptin is shwn graphically in Fig. 2.. The DWT prduces the equivalent result by starting with a blck f discrete data and perfrming successive high- and lwpass digital filtering. The filtering is repeated n the lwpass utput in rder t bandpass the signal in a series f stages called dilatins. Each dilatin divides the frequency space f the current interval in half while dubling the time span, thus keeping the time-frequency prduct cnstant. In cntrast, the CWT divides a signal int a set f lgarithmicallyscaled frequency bands by passing it thrugh a bank f cnstant-q bandpass filters. Bth the CWT and the DWT are filterbanks, althugh the CWT takes the mre intuitive frm f a physical filter with a transfer functin in the frequency and time dmains. DWT filters are carefully frmulated mathematical cnstructs whse transfer functins are recursive and which can nly be said t per- 6

4 Wavelet system utput channels Time Frequency Figure 2.: Output sampling. Pints marked represent center f the time-frequency area cvered by that sampled utput. Figure 2.2: Frequency-time representatin f the input as verlapping Gaussian filters. 7

5 frm lwpass and highpass functins in a vaguely-defined sense. Due t the cnstraints f the Discrete Wavelet Transfrm specificatin, all DWT filters have cmpact supprt. The fact that the DWT can be cnfined abslutely in time and frequency is prf that it cannt be implemented by physical filters in a cntinuus time dmain. By being physically realizable, the CWT cannt achieve perfect cmpact supprt. Instead, analgusly t the way elliptical filters such as the Chebyshev and Butterwrth are generated, CWT functins are cnstructed in rder t achieve maximal cmpactness with respect t sme criterin. A majr difference between the DWT and the CWT is that the CWT lses the cncept f cmpact supprt. The cntinuus-time nature f the transfrm implies the use f causal filters, whse frequency respnse cannt be perfectly limited t a given bandwidth. Instead, the filters must verlap. The shape and cmplexity f the filter determines the amunt f verlap and thus higher-rder filter functins can be used t describe higher-rder cntinuus wavelet transfrms, analgusly t the way that higher-rder functins perate in the discrete wavelet transfrm. 2.3 Gabr Lgns and Wavelets The beginnings f the thery f the cntinuus wavelet transfrm begin with a seminal wrk by Gabr [8], a paper with the rather bld title Thery f Cmmunicatin (946) which utlines the physical basis behind limitatins f time-dmain and Furier analysis and shws hw bth time and frequency can be incrprated int a functin prviding the ptimal tradeff in reslutin between the tw dmains, and hw the time-frequency plane can be effectively represented by tiling with these functins. Gabr calls the functins lgns, thse functins which have minimum!t. The simplest lgn frm (lwest rder CWT filter) is the Gaussian functin, (!) = e!2 =2 2 : (2.5) The laws f Furier transfrms dictate that a Gaussian in the time dmain is als a Gaussian in the frequency dmain, thus this functin is smth in all directins n the time-frequency plane. The prperties f Furier transfrms als dictate a symmetry between time and frequency shifts f a signal and multiplicatin by a cmplex sinusid in frequency and time, respectively: (!) = e j!ta e (!! b) 2 =2 2 b (2.6) (t) = e j!at e (t t b) 2 =2 2 b (2.7) 8

6 The cmplex expnential in either transfrm can be split int real and imaginary parts, in which case the functin appears as a sinusid mdulated by a Gaussian envelpe, as shwn in Figure 2.3. A family f curves can be generated n this basis, using the Gaussian as an envelpe arund a sinusidal signal f differing frequencies. The meaning f these curves shuld be intuitively clear: The mre cycles f a sinusid fit int the Gaussian envelpe, the better the frequency is defined, but the prer the time is defined. A pure sinusid f infinite duratin represents ne extreme, fr which frequency is knwn exactly but time is nt knwn at all. Likewise, the Gaussian itself represents the ther extreme: the csine functin fr which time is knwn exactly but fr which frequency is entirely unknwn, having n reference signal t which it can be cmpared. The wavelet tiling f the time-frequency plane (Figure. (c)) dictates the rati f! a, the mdulating frequency, t b, the width f the Gaussian envelpe. As ne gets bigger, the ther gets smaller..8 Sine (dd) Csine (even) Figure 2.3: Gabr sine and csine lgns, r wavelets. The Furier transfrm prperty f time and frequency shifts yields an interesting insight abut the cnnectin t dilatins and shifts f the discrete wavelet transfrm, since they are manifestatins f the Furier and inverse Furier transfrms. Unlike the discrete wavelet transfrm families, the Furier transfrm pair f the Gabr lgn family is beautifully symmetric. 9

7 Other useful cntinuus wavelet filter functins can be derived starting frm different criteria fr cmpactness. One such case is the chirplet [6], which allws rtatin f the filter with respect t the time and frequency axes. The frequency shift f the Gabr lgn, which is a multiplicatin by a cmplex sinusid in the time dmain, amunts t a signal demdulatin, where in rder t maintain the symmetry between the time and frequency sides f the transfrm, the cmplex nature f the frequency shift must be maintained; that is, the mdulatin must be perfrmed with bth a sine and a csine t represent the real and imaginary parts f the multiplicatin with the cmplex expnential. 2.4 Cmplex Demdulatin Cntinuus-time bandpass filter functins with arbitrarily high Q values are ntriusly difficult t design with precisin (a fact which will be discussed in detail in Chapter 3, Sectin 3.5). It is nt especially difficult t build a secnd-rder filter sectin which can be tuned externally t the desired specificatins, but managing t get an entire filterbank f 6 r 32 independent sectins t all match within a given tlerance can be difficult r impssible, depending n the strictness f the specificatins. Adding a requirement f lw pwer cnsumptin cmpunds the prblem. Generally, the nly practical circuit slutin is t leave the cntinuus-time dmain and instead enter the discrete-time dmain, with the use f switched capacitr (S-C) circuits. The result is accurate (fr analg) cmputatin with mdest pwer cnsumptin at the expense f die area, which tends t be high due t the use f large numbers f capacitrs. We did, in fact, resrt t switched capacitr architectures fr all f ur wavelet filter functins. Hwever, it was nt necessary t stp ptimizing at the architectural (circuit) level. On the algrithmic level, we were further able t make better bandpass filters using a methd called cmplex demdulatin [9]. Demdulatin is a well-established methd used widely in mdems and radis, based n the principle that the frequency cntent f a signal can be shifted up r dwn by multiplying it by a pure sinusidal carrier signal and then filtering apprpriately. In thse applicatins, the signal t be bradcast is first mdulated n a carrier signal t push it int a highfrequency bradcast band, and then demdulated by the receiver t retrieve the riginal signal. The purpse f mdulating is twfld: First, the high-frequency signal can be bradcast much further withut significant attenuatin, and secnd, a large number f signals can be transmitted at the same time by assigning each ne a nn-verlapping prtin f the electrmagnetic spectrum. 2

8 Demdulatin f a signal by a carrier An input time-varying signal x(t) is multiplied by a sinusidal reference signal s(t) = cs(! t). As viewed in the frequency dmain, the result splits the input arund the reference t generate the sum (reference + input) and difference (reference input) cmpnents. This result can be easily seen by cnsidering a input cnsisting f a single sinusidal functin f arbitrary phase relatinship t the reference, fr instance x(t) = cs(!t + ). The mdulatin is then Nw, adding the trignmetric identities tgether gives x(t)s(t) = cs(!t + )cs(! t): (2.8) cs(u + v) = cs(u)cs(v) sin(u)sin(v) (2.9) cs(u v) = cs(u)cs(v) + sin(u)sin(v) (2.) cs(u + v) + cs(u v) = 2cs(u)cs(v) (2.) such that, when used with Equatin (2.8), we get x(t)s(t) = 2 (cs((! +!)t + ) + cs((!!)t )) (2.2) which cntains ne sinusid describing the sum! +! f the carrier and signal frequencies, and ne describing the difference!!. Extending the result t arbitrary inputs invlves viewing the arbitrary input as a Furier series f sine cmpnents; since the mdulatin multiplicatin is a linear functin, superpsitin applies, and every Furier cmpnent f the arbitrary input is split int sum and difference cmpnents acrss the carrier frequency. Filtering the resultant signal with a highpass filter which accepts the sum cmpnent but rejects the difference cmpnent is the prcess knwn as mdulatin. Filtering the same signal instead with a lwpass filter which accepts the difference cmpnent but rejects the sum cmpnent is the prcess knwn as demdulatin. A useful applicatin f the principle f signal mdulatin invlves perfrming the demdulatin first. In such case, the signal t be encded is multiplied by an in-band carrier frequency in rder t shift the desired frequency band dwn t zer. The resulting signal is lwpass filtered t remve the cmpnent representing the sum f carrier and mdulatr. If desired, the signal can then be mdulated back int its riginal band, at which pint the result is a bandpassed signal. Nt nly is it bandlimited, but the band t which it is limited can be made arbitrarily small, since there is n limit (ther than the practical limits f nise, jitter f the carrier frequency, etc.) t the cutff frequency f the lwpass functin. The tight band limit translates t an arbitrarily large effective value f Q. The psitin f the band is placed at the carrier frequency, s it can be made very accurate (particularly since in ur architecture the carrier frequency is driven by a quartz scillatr). 2

9 The methd just described is als a well-established methd, used in subband cders and t make lck-in amplifiers able t analyze miniscule frequency bands fr signal and nise analysis. There is ne small mathematical catch t this methd. When a signal is demdulated t zer frequency, parts f the signal extend int the negative frequency dmain, which in terms f the actual measured signal means that these frequencies are flded back (aliased) int the psitive frequency dmain. There they wuld be inseparable except fr the cmplex part f cmplex demdulatin. The methd requires tw carrier signals, ne f which is ffset frm the ther by 9 degrees phase. Thus if ne carrier describes sin(!t), then the ther describes cs(!t). Tw demdulatins are perfrmed in parallel, each with ne f the tw carriers. The tw resulting signals bth have frequencies aliased int apparent mush, but between the tw signals is all the infrmatin necessary t separate ut the aliased parts after subsequent mdulatin. In fact, it can be shwn that this requires nthing mre than separately mdulating the tw results, again with the mdulatin carrier signals ffset in phase, and adding the tw mdulatin results tgether. N filtering is necessary fr the mdulatin step, which is anther cnsequence f the math. Cmplex Demdulatin and Recnstructin f a signal by a carrier Returning t ur previus example: An input time-varying signal x(t) is multiplied by tw sinusidal reference signals s (t) = cs(! t) and s 2 (t) = sin(! t). Chsing fr x(t) the simple frm f a sinusid f arbitrary phase : x(t)s (t) = cs(!t + )cs(! t) (2.3) x(t)s 2 (t) = cs(!t + )sin(! t) (2.4) Applying the fllwing trignmetric identities yields the fllwing expressins: cs(u + v) + cs(u v) = 2cs(u)cs(v) (2.5) sin(u + v) + sin(u v) = 2sin(u)cs(v) (2.6) x(t)s (t) = 2 (cs((! +!)t + ) + cs((!!)t )) (2.7) x(t)s 2 (t) = 2 (sin((! +!)t + ) + sin((!!)t )) : (2.8) Nw perfrm a lwpass filter by assuming an ideal filtering functin, h(t), having a Furier transfrm H(!), which perfectly rejects the sum f frequencies while perfectly passing the difference f frequencies (see Figure 2.4): h(t) x(t)s (t) = 2 cs((!!)t ) (2.9) h(t) x(t)s 2 (t) = 2 sin((!!)t ): (2.2) 22

10 S(ω) = δ(ω ω ) H(ω) X(ω) ω ω ω 2ω 3ω 5ω 4ω ω 2ω 3ω 4ω 5ω Figure 2.4: Demdulatin f an input X in the frequency dmain with a perfect sine wave S and ideal mdulatin filter H. Recnstructin invlves first multiplying each functin by the same sine and csine carriers: h(t) (x(t)s (t))s (t) = cs(! t) 2 cs((!!)t ) (2.2) h(t) (x(t)s 2 (t))s 2 (t) = sin(! t) 2 sin((!!)t ): (2.22) Finally, withut filtering, these tw parts are added tgether t prduce the recnstructin (in this case, exact recnstructin, due t the use f perfect filters): x (t) = h(t) (x(t)s (t))s (t) + h(t) (x(t)s 2 (t))s 2 (t) (2.23) = cs(! t) 2 cs((!!)t ) +sin(! t) 2 sin((!!)t ): (2.24) The trignmetric identity (2.) applies directly, giving the final result: x (t) = 2 cs((!! +!)t + ) (2.25) = 2 cs(! t + ) (2.26) = x(t) (2.27) 2 shwing that the recnstructin is exact except fr the required applicatin f a gain f tw. Again, this example can be extended t arbitrary inputs by the applicatin f Furier series and superpsitin. 2.5 Cmplex Demdulatin in the Cntinuus Wavelet Prcessr The architecture used fr cmplex demdulatin in the wavelet prcessr is depicted in Figures 2.5 and 2.6 and described belw. 23

11 frequency cntrl ω c Signal in f in (t) Gaussian Filter Banks Sine h(t) Sine Out Csine h(t) Csine Out 2 Sine h(t) Sine Out 2 Csine h(t) Csine Out Figure 2.5: Cmplex demdulatin (2 channels shwn). 24

12 frequency cntrl ω c Sine Sine In Csine In Csine 2 Recnstructed Output f^in(t) Sine Sine In 2 Csine In 2 Csine Figure 2.6: Cmplex mdulatin recnstructin (2 channels shwn). 25

13 The input functin is designated by f in (t). In rder t demdulate it with respect t sme given frequency! c (ne f the center frequencies f the N channels f the wavelet decmpsitin), we use the fllwing multiplicatin: f ut (t) = 2h(t) (f in (t) (cs! c t + j sin! c t)) (2.28) where the lwpass filter time-dmain transfer functin h(t) is shwn withut reference t a specific equatin and is assumed fr the sake f argument t be a filter which passes all frequencies belw its cutff unattenuated, and blcks all frequencies abve its cutff. Since the lwpass filter is assumed t be real, and the input f in (t) is real, then the utput f ut (t) f Equatin (2.28) must necessarily be cmplex-valued, and can be represented by separating it int real and imaginary parts: f ut (t) f real (t) + jf imag (t): (2.29) Therefre, and f real (t) = 2h(t) (f in (t) cs! c t) (2.3) f imag (t) = 2h(t) (f in (t) sin! c t) : (2.3) Nte that Equatins (2.3) and (2.3) are themselves bth real-valued. Bth results are btained easily by multiplying the input by tw sinusids which are 9 ut f phase with each ther. Each part (sine and csine) f the demdulatin prcess prduces a new signal which cntains the sum and the difference f the riginal and carrier frequencies. Since we are demdulating the carrier frequency dwn t zer, we are interested nly in the difference, s we use the lwpass filter h(t) t get rid f the part cntaining the sum f the tw signal frequencies. Frm the remaining difference, ne cannt separate signals n ne side f! c frm thse n the ther since! c is nw at zer and negative frequencies have n physical meaning. In the real-valued signal, negative frequencies are flipped ver the frequency axis and alias int the psitive frequency spectrum. Hwever, the infrmatin necessary t separate the psitive frm the negative frequency cmpnents is preserved in the phase relatinship between f real (t) and f imag (t), as shwn by the exact recnstructin belw. In rder t remdulate the signal back t its riginal frequency, we perfrm the fllwing multiplicatin: ^f in (t) = f ut (t)(cs! c t j sin! c t): (2.32) 26

14 This is the same functin as the demdulatin (2.28) except fr the change in sign and the lack f a lwpass filter functin. We multiply ut the real and imaginary parts f this equatin t get a purely real result: ^f in (t) = f real (t) cs! c t + f imag (t) sin! c t: (2.33) This step separates the negative frm the psitive frequency cmpnents as it pushes the center frequency up frm zer t its riginal value. The result is the exact recnstructin f the riginal input (within the limits f a physical lwpass filter t apprximate the ideal ne used here). The signs have wrked ut such that the remdulating sinusids have exactly the same phase relatin as the demdulating sinusids. This fact suggests that an efficient architecture shuld make use f the same hardware t perfrm bth the demdulatin and the remdulatin. In ther wrds, a single chip can be cnfigured either as the functin decmpser r as the functin recnstructr. In the instance f the cntinuus wavelet transfrm, the lwpass filter fr the demdulatin can be cmbined with the Gaussian filter f the transfrmatin such that n additinal filter is required. 2.6 Pst-prcessing Subsequent t cmplex demdulatin and Gaussian filtering the system utputs are in a frm useful fr signal prcessing: fr instance, analg methds f cmpressin wherein signal bands with energy less than a critical threshld can be eliminated t save bandwidth prir t transmissin and recnstructin. As described thus far, the Gaussian CWT is a band equalizer, but with all the utputs ccupying frequency space arund zer frequency. Thus the utputs are nt in a frm suitable fr efficient transmissin, as they all verlap in the frequency dmain. There are tw ways t arrange the utputs fr transmissin t the recnstructin system:. Mdulate the signals int nnverlapping frequency bands 2. Sample the system and time-multiplex the samples int an efficient representatin. The first methd is the methd f recnstructin, althugh the separated channels can be mdulated t any desired transmissin frequency and rdered in any manner r cmpressed by the scheme mentined abve t reduce the ttal transmissin bandwidth. The secnd methd allws mre flexibility in the representatin by interleaving samples. It is als mre faithful t the idea f tiling the time-frequency plane: the bandpass filterbank quantizes the frequency dmain; a sampler 27

15 quantizes the time dmain. A true wavelet transfrm needs t quantize bth. A bandpass filterbank whse utputs are nt sampled is nt, strictly speaking, a wavelet transfrmer. Since the bandwidth f each channel is prprtinal t the center frequency as required in a cnstant-q system, and the cntents f each band have been shifted dwn in frequency until the band is centered arund zer frequency, the mst efficient descriptin in terms f the time-frequency uncertainty relatin in a sampled-data representatin requires that each frequency band be sampled at a rate prprtinal t its bandwidth. The result f the sampling is that each rectangle in the timefrequency plane shwn in Figure 2. has an equal area representing the effective time and frequency bandwidths f the Gaussian filter (with sme verlap). The verlap f filter functins is depicted in Figure 2.2. If the filter channels are sampled in the binary-tree fashin shwn when the channels are centered n a lg 2 scale, the samples can be easily time-divisin multiplexed int a single utput stream [7]. 2.7 An Analg CWT Prcessr The first attempt t build a Cntinuus Wavelet Transfrming prcessr cnsisted primarily f a cntinuus-time, subthreshld analg design fabricated in a standard CMOS prcess. The analg circuits were based n the analg VLSI techniques described by Carver Mead in Analg VLSI and Neural Systems []. The underlying idea was t generate a set f expnentially-spaced clck (square) wavefrms which wuld then be shaped (via filtering) int sine and csine pairs, and multiplied directly with the cntinuus-time, cntinuus-valued input using a translinear (analg) multiplier. This chip was designed in subthreshld analg as an alternative t using a cmputer r DSP system t perfrm the same transfrm. The advantages f this apprach are reduced size and pwer cnsumptin. The resulting implementatin is inflexible in terms f ability t reprgram the type r rder f the wavelet functin, and requires dealing with the prblems f temperature sensitivity, nnlinearity f analg cmputatin, variable prcess parameters, and nise injectin thrughut the circuit (particularly that caused by the digital circuits generating the square wave). As will be seen presently, nt all f these prblems culd be vercme sufficiently, resulting in a mve tward a mre mixed-mde architecture (Chapter 2 Sectin 2.5). 28

16 2.8 Generating carrier sinusids This chip, as mentined abve, used square waves (clck signals) as its basis fr generating sinusids at specific frequency and phase. Specifically, the architecture called fr each channel center frequency t be half the value f the neighbring channel. Additinally, the architecture was made such that each wavelet prcessr cre wuld generate center frequencies fr six channels. The master clck is applied t the system mst cnveniently as a tw-phase, nn-verlapping signal which becmes the first input t a cascade f tggle flip-flps (T-FFs) made in the standard way frm tw transparent latches in series. The tw-phase flip-flps cnveniently give utputs frm the first and secnd latches which are 9 ut f phase with each ther (this is necessarily true fr the input f the secnd stage and beynd because each f the tw-phase utputs f the previus stage is frced t be 25% duty cycle, and the tw phases are exactly 8 apart. The same can be ensured fr the first stage by dubling the master clck frequency and preceding the first stage with anther tggle flip-flp). The flip-flp cnfiguratin is shwn in Figure 2.7. Signals cs(in) and sin(in) are the pulse trains crrespnding t the frequency and phase f the sine and csine cmpnents f the preceding channel. Signals cs(ut) and sin(ut) are the pulse trains which are shaped by filtering and becme the mdulating signals fr the current channel. The reset lgin ensures that the sine and csine parts have the crrect phase relative t each ther upn initializatin f the circuit. φ φ cs(in) sin(in) φ2 φ2 Vdd φ2 cs(ut) φ Vdd φ φ2 Reset φ2 Reset φ φ sin(ut) φ2 Figure 2.7: Circuit diagram f the frequency-divisin tggle flip-flps. 29

17 2.9 Analg multiplicatin Purely analg multiplicatin is very prblematic due t the limited range f linearity in such circuits, especially when the circuits are realized in subthreshld CMOS technlgy. A standard circuit fr analg multiplicatin f vltages is the Gilbert multiplier (Figure 2.8). The Gilbert multiplier is based n the translinear principle and in terms f linearity and nise perfrmance is best implemented using biplar transistrs rather than MOSFETs fr the critical cmpnents f the translinear lp (fr a full discussin f translinear circuits and the tradeffs between MOSFET and biplar devices, see Chapter 3). Vdd Vdd Vdd Vdd Vdd Vdd V ref V in I ut V ref2 V in2 V in2 V ref2 V bias Figure 2.8: Gilbert multiplier with cascdes. The Gilbert multiplier circuit executes a nnlinear functin which, fr input signals clse t the reference, can be described by the linear apprximatin I ut / (V in V ref )(V in2 V ref2 ). As with mst subthreshld MOS translinear circuits, the input vltage swing is limited t a few kt =q, r abut 5 mv. and is difficult t extend by mre than a factr f tw r s thrugh the use f increasingly cmplicated linearizatin techniques [5]. A simple derivatin f this limiting vltage can be fund in Appendix A. 2. Wavelet Gaussian Functin In additin t allwing simple sampling f each utput channel, the use f cmplex demdulatin simplifies the prcess f designing bandpass filters that maintain a Gaussian shape while 3

18 allwing variable center frequency and width. Because the signals are first demdulated until the center f the frequency band f interest is placed at zer frequency, it is nly necessary t create a Gaussian lwpass filter, which is ne half f a Gaussian functin placed at zer frequency. Tw identical filters are required fr the sine and csine parts f the cmplex demdulatin. The signals are remdulated t their respective center frequencies during recnstructin using the same methd (and preferably the same hardware, if feasible). The resulting utput will be reflected symmetrically acrss the mdulatin frequency, behaving as if it had been passed directly thrugh a bandpass filter with Gaussian characteristics. Fr demdulatin used in the cntext f the cntinuus wavelet transfrm, the lwpass filter fr the demdulatin can be cmbined with the Gaussian filter required by the wavelet transfrmatin. Nte that, accrding t Equatin 2.33, recnstructing the signal des nt require any filtering subsequent t remdulatin. We based the design f the circuit which apprximates the half-gaussian lwpass filter functin (as described by Grssman [3]) n a prbability argument. First, I present Equatin (2.34) which describes the filter transfer functin f the circuit: H(s) = V ut V in = s + n : (2.34) This filter functin describes a cascade f n first-rder lwpass filter sectins in series. Althugh Equatin (2.46) cnverges t a delta functin in the limit as n! fr cnstant, it can be shwn that when is replaced by an expressin which maintains cnstant bandwidth, the transfer functin appraches the Gaussian functin (2.5) as n!. A prf f this equatin can be fund in [3] which shws that the Gaussian shape is an example f the central limit therem f prbability: In ther wrds, it is a result f the use f cascaded stages and is relatively independent f the shape f the filter. Fr n sectins in cascade, the relatinship between and the Gaussian bandwidth frm the wavelet mther functin, Equatin (2.5), is = p n ln(2) : (2.35) The circuit is shwn in part in Fig. 2.. Althugh nt an architectural necessity, we chse t implement each first-rder filter as a transcnductance-c filter using transcnductance amplifiers perating in the subthreshld regin. The transcnductance amplifiers are cnnected as vltage fllwers, which in cnjunctin with the capacitr at the utput is a cnfiguratin called a fllwer-integratr. Due t cnsideratins f signal-t-nise rati and the ability t generate a 3

19 Nrmalized Amplitude (db) Frequency (units fs) Figure 2.9: Frequency-dmain transfer functin f the final utput f n cascaded lwpass filters as a functin f the number f stages n. Impulse Respnse (nrmalized) ss2 s3... s Time (arb. units) Figure 2.: Impulse respnse f the final utput f n cascaded lwpass filters as a functin f the number f stages n. 32

20 given cnstant frm a vltage applied t the fllwer-integratrs, we chse a cascade rder f five sectins. V gaussh V in (i) V gauss (i) R stage stage 2 stage 3 stage n C C C C + V ut (i) V gaussl Figure 2.: n cascaded stages f a filter apprximating a half-gaussian functin, using cntinuus-time transcnductance-c filters. The center frequencies f the filters in the filter bank are spaced n a lg 2 scale. Assuming speech-quality bandwidth fr the input signal, we decided that six utputs wuld be sufficient, giving typical center frequencies f 9kHz, 4.5kHz, 2.25kHz,.25kHz, 562.5Hz, and 28.25Hz (unless mre than ne system is interleaved in the manner described in Sectin 2.22). The center frequency f the highest-frequency filter is determined by an scillatr which can be generated either n-chip fr a vltage-cntrlled frequency, r ff-chip fr a stable frequency. The center frequencies f the rest f the filters are determined by dividing dwn the scillatr apprpriately. The bandwidth f each Gaussian filter is set autmatically with respect t the thers with the exceptin f the first and last filters, which have widths adjustable using tw cntrl vltages V gaussh and V gaussl. In terms f the transfer functin (2.46) fr the filter, the parameters f the highest- and lwest-frequency filters are fixed by these cntrl inputs. shuld be calculated t assure that the width f each Gaussian is prprtinal t the value f the center frequency, as it is nt determined autmatically frm the ther system parameters. Mre recently, this idea has been expanded upn by Harris et al. [22] fr the creatin f s-called gamma-tne filters and filter structures which use carefully calculated weighted feedback and feedfrward cnnectins t cause the filter transfer functin t reach a given accuracy f apprximatin f the Gaussian shape in significantly fewer stages. 33

21 2. Wavelet chip slice The parallel nature f the analg wavelet cmputatin allws the VLSI layut t be generated easily by abutting slices f circuitry. The filtering functins are identical fr the sine and csine parts f the transfrm, s slices are lgically gruped in pairs, with each slice cntaining ne transparent latch such that the pair f slices frms the flip-flp fr dividing dwn the clck input t that sectin. Thus the path f the frequency stepping runs vertically acrss all channels while the signal path runs frm left t right. In additin t generating the divide-by-tw scillatr, each slice is respnsible fr shaping (filtering) the scillatr signal t prduce a smth sine (r csine) wave, multiplying this mdulating signal with the input, and filtering the result thrugh a half-gaussianshaped lwpass functin. Finally, depending n whether the sectin is cnfigured fr decmpsitin r recnstructin, the circuit samples the filter utputs and multiplexes them int a stream (decmpsitin), r else aggregates all the utputs tgether t prduce the final result (recnstructin). A blck diagram f decmpsitin and recnstructin fr a wavelet transfrm prcessr sine/csine pair (single channel, r slice ) is shwn in Figures 2.2 and 2.3, respectively. The entire analg Wavelet Transfrm chip is shwn in Fig Chip Specificatins Pwer Supply: +5 V DC 5% Input mean value: 2.5 V.5 V Input p-p amplitude:.4 V Input frequency range: 8 Hz t khz Number f utput channels: 2 (6 pairs) Silicn area:.96 6 µm 2 (in a 2. micrn CMOS prcess) Chip package: 68-pin PLCC The technlgies used t fabricate the several versins f the analg Cntinuus Wavelet Transfrm prcessr are as fllws: 34

22 Oscillatr signal in Channel i Input signal value 2 flip-flp Attenuate and bias LPF Gaussian Filter Real-part Output (csine) phase-shifted 2 flip-flp Attenuate and bias LPF Gaussian Filter Imaginary-part Output (sine) Figure 2.2: Wavelet decmpsitin blck diagram fr a single sine/csine pair. Oscillatr signal in Channel i 2 flip-flp phase-shifted 2 flip-flp Attenuate and bias Attenuate and bias LPF LPF + + Real-part Input (csine) Imaginary-part Input (sine) Recnstructin utput Figure 2.3: Wavelet recnstructin blck diagram fr a single sine/csine pair. Table 2.: Technlgies used fr the Wavelet prcessrs. Chip name Fundry Min. feature size Well type Ply WaveChip2 Zilg 2. micrn twin-well single Z89c55ba Zilg.2 micrn twin-well single WaveChip5a Orbit 2. micrn N-well duble WaveChip5b Orbit 2. micrn P-well duble 35

23 Oscillatr signal in Input signal value... 2 flip-flp phase-shifted 2 flip-flp 2 flip-flp phase-shifted 2 flip-flp Attenuate and bias Attenuate and bias Attenuate and bias Attenuate and bias LPF LPF LPF LPF Decmpsitin Gaussian Filter Gaussian Filter Gaussian Filter Gaussian Filter Real-part Output (csine) Recnstructed utput Imaginarypart Output (sine) Real-part Output (csine) Imaginarypart Output (sine) Recnstructin Real-part Input (csine) Imaginarypart Input (sine) Real-part Input (csine) Imaginarypart Input (sine) + 2 flip-flp phase-shifted 2 flip-flp Attenuate and bias Attenuate and bias LPF LPF Gaussian Filter Gaussian Filter Real-part Output (csine) Imaginarypart Output (sine) Real-part Input (csine) Imaginarypart Input (sine) Figure 2.4: Wavelet Transfrm Chip blck diagram. 2.3 Limitatins f the Architecture The majr drawback f this analg architecture invlves the use f filtered square wave signals as the mdulating sinusids. The use f a square wave as a mdulatin carrier is welldcumented (see, fr example, Hrwitz & Hill, 2nd ed., p. 437) [28]. Hwever, its use is restricted t mdulatin in the usual sense wherein a lw-frequency signal is pushed up int a high-frequency band. The Furier series describing a square wave cntains an infinite series f dd harmnics beginning with the fundamental, which attenuates nly as =n, where n is the harmnic number (, 3, 5,... ). The mdulatin perfrmed by multiplying the square wave by the input signal can be brken dwn by superpsitin int the multiplicatin f each f the square wave harmnics with the input. If the destinatin band (carrier frequency) is large enugh, then all f the intermdulatin prducts (multiplicatin f the input by all harmnics greater than the fundamental) are very far ut f band and can be easily attenuated with a simple filter. Unfrtunately, if the functin perfrmed is demdulatin instead f mdulatin, then by definitin the carrier frequency is in the band f the signal and s any f its harmnics may be als. Cnsequently, intermdulatin prducts will be in-band and cannt be filtered ut. Shaping a square wave int a sinusid by filtering ut the higher harmnics is a difficult prspect at best invlving cmplicated high-rder elliptic filters; the filter cutff must be prhibitively sharp t pass the fundamental frequency but attenuate the third harmnic t reasnable levels fr clean signal prcessing (at least 4 t 5 db fr mst audi 36

24 applicatins; 6 t 7 db fr high fidelity). Cmplicated filter architectures were nt addressed in this series f wavelet prcessrs, and s results did nt reach acceptable levels f perfrmance. 2.4 Variatins n an Architecture One interesting variatin n this design was created and develped by Mreira-Tamay et al. [23, 24] at Texas A&M University. Using primarily the same cmplex demdulatin technique based n ur architecture, the authrs translated the entire prblem int the time dmain. In place f the wavelet Gaussian functin, they used a rectified csine (als knwn as a Hanning windw, a functin which is reasnably easy t generate in the time dmain and which is reasnably clse t the Gaussian in its time and frequency supprt. Its time-frequency prduct is apprximately.53, r 2.6% larger than the minimum area (/2) f the Gaussian. The rectified csine itself becmes the envelpe functin; the wavelet functin itself is generated by multiplying the rectified csine by a sinusid f higher frequency. Effectively, this is a duble demdulatin, and simplifies the implementatin smewhat by making use f the same hardware architecture fr generating bth the Gaussian (r in this case, Hanning) envelpe and the wavelet mdulating functin. Their wavelet family can be described by the functin: (t) = Ae j!ct ( + m cs (! p t)) ; <! p t < (2.36) where this functin is repeated (chained) in time, and the system repeated ver dilatins f the frequency. The wavelet cmputatin n an input functin f(t) fr frequency scale a and time shift b is written CWT (f; a; b) = p a Z `f `i f(t) g t b a t b v dt (2.37) a where g(t) = exp ( j! c t) is implemented as a cmplex demdulatin by separating the functin int sin (! c t) and cs (! c t), and v(t) = :5( + m cs (! p t)). The architecture f Mreira-Tamay et al. als fllws ur architecture in its use f a master clck divided dwn by flip-flps fr generatin f multiple frequency square wave signals subsequently filtered t generate the sinusid mdulatr. Hwever, the system is built at the cmpnent level rather than as a VLSI prcessr. The analg multiplicatins are implemented with MC494 analg multiplier integrated circuits. The integratin peratin in Equatin (2.37) is implemented by a transcnductance-c circuit. A slightly simplified blck diagram f ne channel f the system is shwn in Figure 2.5. Only the wavelet decmpsitin was reprted in [23]. Cmpare Figure 2.5 t the blck diagram f the decmpsitin half f ur wavelet decmpsitin architecture, Figure

25 Oscillatr signal in Channel i Input signal value 2 flip-flp Attenuate and bias DC (m) Sample & Hld Real-part Output (csine) k flip-flp Attenuate and bias Σ delay trigger phase-shifted 2 flip-flp Attenuate and bias Sample & Hld Imaginary-part Output (sine) Figure 2.5: Architecture f Mreira-Tamay et al. The Mreira-Tamay architecture quite nicely demnstrates the duality f the wavelet transfrm: Because it incrprates bth the time dmain and the frequency dmain, efficient architectures can be realized in either ne. The respnse f bth systems t the same input is the same. 2.5 A Mixed-Mde Wavelet Prcessr Several hardware implementatins f the Cntinuus Wavelet Transfrm and related bandpass filter bank architectures have been reprted in recent years [7, 9, 23], including the analg architecture discussed at the beginning f this chapter. The remaining sectins f the chapter highlight a nvel architecture we develped fr the cntinuus wavelet transfrm prcessr which is unique in its encding f the decmpsitin utput and use f versampling techniques. The gal f the new architecture was t vercme the bvius drawbacks f the analg subthreshld MOS circuits, namely the quality f the carrier sinusid signal and the linearity f the mdulatin multiplicatin. The new architecture relies mre n digital prcessing and as such is mre apprpriately cnsidered a mixed-mde design. Other than the nvel circuit methds used, the new design retains the essential characteristics f the riginal wavelet architecture. T recap: The prcessr perfrms a demdulatin f the audi-frequency input signal in parallel acrss N channels, where the channels are adjacent with minimal verlap, cver the audi frequency band f interest, and are centered n a lgarithmic scale. The prcess f demdulatin shifts the signal frequencies frm each channel center frequency 38

26 dwn t zer. Each demdulatin result is then filtered by a lwpass versin f the wavelet functin, which als serves as the pstmultiplicatin filter fr the demdulatr. Signal infrmatin n bth sides f the channel center frequency is preserved by using cmplex demdulatin, in which each channel is split int tw parts, with the mdulating sinusid f ne being 9 ut f phase with that f the ther. The CWT is invertible: The recnstructin prcess cnsists f mdulating the channel utputs back t their respective center frequencies, and summing them all tgether. The decmpsitin methd is shwn in Figure 2.5 and the recnstructin methd in Figure 2.6. The wavelet nature f the utput allws the channel utputs t be efficiently encded. The utputs f the decmpsitin are time sampled at the Nyquist rate f each channel, and all the sampled channel utputs are time-multiplexed int a single stream. The recnstructin prcessr decdes the data stream prir t recnstructin. As we develped the versampling architecture described in the previus sectin, it became clear that, as the carrier signal was bth binary and discrete-time based n a well-defined synchrnus clck, the mst efficient architecture wuld maintain the discrete-time nature f the carrier, and retain a digital mde f prcessing thrughut as much f the architecture as pssible. Fllwing this line f reasning leads t an elegant and extremely efficient design fr a cmplex demdulatin multiplier and Gaussian filter. In the intrductin t this chapter, I mentined that by using cmplex demdulatin, very accurate center frequencies fr the bandpass functin derive directly frm the carrier signal frequency, which itself is derived frm a quartz scillatr. In the previus sectin, the wavelet transfrm architecture was develped, but the circuits using that architecture culd nt achieve acceptable perfrmance due t intermdulatin prducts caused by prly attenuated harmnics f the carrier signal. Analg multipliers built frm analg circuits by nature have a limited range f peratin due t nnlinearities in the circuit functin. It remains t be explained hw t get frm the frequencyaccurate, digital-dmain square wave clck signal t an accurate and repeatable sinusidal signal. That is the purpse f this sectin. We develped a methd fr sinusidal mdulatin f analg signals which des nt require an explicit multiplicatin, and hinges n generatin f accurate analg sine wave using the mixed-mde technique f versampling. As demnstrated by repeated failure f perfrmance f previusly fabricated versins f the wavelet prcessr, btaining a sinusidal signal by lwpass filtering a square wave intrduces t much distrtin t make the system usable. The mdulatin functin is sensitive t distrtin harmnics, which get multiplied by the input signal and act like many separate mdulatins abut 39

27 many different frequency axes. Each prduces its wn sum and difference cmpnents, and sme are bund t end up in the signal band at the system utput. In rder t get predictable system perfrmance, it is necessary t have a sinusid carrier signal with a fixed and cntrllable amunt f harmnic distrtin. T this end we cnsidered the pssibility f generating an versampled versin f a sinusid using a delta-sigma mdulatr. The delta-sigma circuit f Lu et al. [26], an excellent example f the methd, begins with a standard resnatr system cnsisting f a simple lp f tw integratrs. Implemented in the digital (z) dmain, each integratr requires a delay and accumulate peratr, and ne multiplicatin by a cnstant cefficient. The delta-sigma methd eliminates the multi-bit multiplicatins by inserting a delta-sigma mdulatr int the lp which renders part f the resnatr circuit a single-bit value, where the single bit represents an versampled sinusidal scillatin. Figure 2.6 shws the architecture, where n actual multiplicatins are required: One multiplicatin is reduced t a multiplexing peratin, and the ther in the digital dmain is a bit shift peratin. The remainder f the system still requires precisin digital delay and accumulate peratins, as des the implementatin f the secnd-rder delta-sigma mdulatr. + Σ LPF Output a 2 =2 L z z + +a 2 a 2 Figure 2.6: The analg scillatr architecture f Lu et al. [26] cntaining a 2nd-rder delta-sigma mdulatr. Eventually it became bvius t us that ur system is much mre specific than the general delta-sigma methd described abve. Mre t the pint, the delta-sigma methd is rather a bit f verkill fr the purpse f generating a sine wave, particularly when nly a finite set f frequencies is required rather than a full range f arbitrary values. We need t generate a number f sinusids f knwn fixed frequency and amplitude. Presumably there exists a fixed sequence f bits which describes the ptimal n-times versampled sinusid in the cntext f ur system fr a given value f n. If the ptimal sequence can be determined, then it has a fixed amunt f harmnic distrtin which shuld decrease with increasing n as the versampled sequence cntains increasing amunt 4

28 f infrmatin abut the encded signal. The abve phrase in the cntext f ur system is critical in that it is nt t be assumed that the same sequence is ptimal fr every pssible implementatin. The frequency-dmain representatin f the riginal (unfiltered) single-bit versampled sequence can be determined by cnverting the bit sequence directly t a sequence f vltages in the time dmain accrding t the fllwing rules: bits are represented by sme fixed psitive vltage bits are represented by sme fixed negative vltage each vltage is applied fr a fixed length f time t such that the entire sequence f n bits has length T = nt. We then cmpute the FFT f this sequence. A sequence which accurately encdes a sinusid shuld shw a large FFT cmpnent at the frequency f the sinusid (which withut lss f generality we cnsider t be the inverse f the sequence length, f = =T ), shw a highly attenuated respnse at all frequencies clse t the fundamental, with the harmnic distrtin increasing at large frequencies (an inherent prperty f versampled signals). The quality f the sinusid in the cntext f the system cannt be determined frm this FFT result. The versampled sequence must be lwpass filtered t retrieve the encded signal, and s the quality f the resulting signal depends bth n the binary sequence used and the prperties f the lwpass functin used t retrieve the analg signal frm its versampled representatin. Furthermre, the input signal must be prefiltered with a lwpass t attenuate cmpnents which wuld mix with the versampled sequence and add t the resulting harmnic distrtin. The pint is that the harmnic distrtin is determined by the prcess f the mdulatin itself specifically, the way that it causes frequencies f the input t mix with harmnic distrtin cmpnents f the carrier t becme errr cmpnents at the utput and s the binary sequence required fr ptimal perfrmance is intimately tied t the system itself. One fallut f this cnsideratin is that the bit sequence used t encde the sinusid cannt be determined until the system is knwn in detail. On the ther hand, it is nt pssible t knw the exact requirements fr the filtering system (fr instance, size requirement f the VLSI layut) until we knw what the sequence is. Sme iteratins can be expected befre a slutin is agreed upn. This fact makes it necessary t have a methd which can fairly rapidly determine the ptimal sequence. 4

29 It shuld be clear at nce that a full search f the n-bit space f the sequence is an nphard prblem, and furthermre that each search invlves a large amunt f cmputatin fr simulating the antialiasing filter and utput filter, cmputing an FFT, and measuring harmnic distrtin. Fr the purpse f finding the ptimal sequence, we define harmnic distrtin as the rati f the amplitude f the largest distrtin prduct t the amplitude f the fundamental frequency. On the ther hand, it shuld als be clear that there are a number f factrs in ur favr:. Inverted sequences have the same respnse as their nn-inverted frms, and d nt need t be investigated. This is als true f reversed sequences. 2. Sinusids have quarter-wave symmetry, and therefre the sequences shuld als (symmetric signals can be clsely apprximated by nn-symmetric sequences, as is generally the case in a 2nd-rder delta-sigma system, but there is n reasn t explre this space). This cuts the search space by a factr f fur. 3. Only a small subset f 2 n sequences lk anything like a sinusid, allwing the pssibility f heuristic appraches. The first cnsideratin invlves generating a quarter sequence, and then generating the ther three quarters by (in turn) reversing the sequence, inverting and reversing, and reversing nce again. A reasnable heuristic apprach is t start with a knwn sequence which lks smething like a sinusid, namely a square wave (fr which the first quarter sequence is all bits). Frm that pint we need an iterative methd which will explre the lcal space f nearby bit sequences ( nearby meaning in a Hamming distance sense) which will (hpefully) cnverge quickly t a slutin withut encuntering lcal minima alng the way. As is ften true fr these srts f methds, we d nt present (nr have we even attempted) a prf f cnvergence. Our methd successfully fund bit sequences which exceeded the system cnstraints and which met with ur apprval (i.e., the sequence was shrt enugh that it was cnceivable t design a simple sequence generatr t fit the wavelet system n a 2mm 2mm layut). 2.6 Details f the Bit-Sequence-Finding Algrithm It is pssible t ascertain sme prperties f the slutin befre running the algrithm. Fr instance, the average f nes and zers in the bit sequence must match the integratin under the 42

30 sinusid, and the difference between the tw gives a rugh indicatin f the minimum distrtin pssible using a length-n sequence. The way we have defined the system in Sectin 2.5, a DC value f zer is represented exactly by a bit sequence f alternating nes and zers ( vltage). A DC value f ne is represented exactly by a bit sequence f all nes. Given this cnsideratin, the expected rati f nes t zers in a bit sequence in the first case is /2, and in the secnd case. Fllwing this line f reasning, the expected rati f nes t zers in the first quarter f the sine sequence is Z 2 ( + :5 sin x)dx: (2.38) The ttal integratin divided by the interval cmes t =2 + =, r.883. This multiplied by 64 bits, fr example, yields an expected value f between 52 and 53 ne bits, leaving 2 r zer bits. This is what we can expect the makeup f near-ptimal sequences t be. Cmparing the integratin under the curve t the instantaneus average f the bit sequence at each step and then determining the next bit f the sequence accrdingly is exactly the methd f sigma-delta mdulatin. The bit sequence prduced is always changing frm cycle t cycle due t the residual errr between the filtered bit sequence and the integral under the sine curve, which is never zer. A sequence f n bits which is repeated exactly n every cycle can never achieve the accuracy f an n-bit delta-sigma mdulatr. On the ther hand, as n increases, s des the accuracy, s that if the system requirement is a maximum fixed value f ttal harmnic distrtin, there exists a fixed bit sequence f sme minimum length n which will meet that requirement. The prblem in finding the sequence is this: A sigma-delta system is deterministic: The next bit in the utput sequence is an exact functin f the current state f the system. But a sigmadelta-like methd cannt be used t find an ptimal fixed sequence, because changing any bit in the fixed sequence changes the past, present, and future state f the system. Instead, it is necessary t search the space f 2 n bit sequences t find ne which achieves the desired accuracy. Knwing the expected number f ne and zer bits in the sequence still desn t help much in finding the ptimal sequence: The set f all cmbinatins f m zer bits in a sequence f length n remains cmputatinally intractable fr reasnable values f n (such as 64). described by The mdulatin system fr the CWT, when using the versampled sequence methd, is y(t) = (x(t) g(t)) s (t) h(t); (2.39) 43

31 where g(t) the impulse respnse f the lwpass prefilter and h(t) is the pstmultiplicatin filter as befre (Sectin 2.4). The binary sequence s (t) is assumed peridic with quarter-wave symmetry, and therefre can be described in the frequency dmain by a Furier series cntaining nly harmnics f dd rders in! : S (!) = X k= S k(!) ; (2.4) S k(!) = jc k [ (! (2k + )! ) (! + (2k + )! )] : (2.4) The fundamental cmpnent S (!) crrespnds t the desired sinusidal signal s(t). Figure 2.7 shws hw the tw systems perate under the assumptin that the lwpass filter functins H(!) and G(!) are ideal, i.e., flat in the passband and with infinite rllff at the cutff frequency. Under this assumptin, it can be seen frm the figures that an arbitrary input spectrum X(!) crrespnding t the time-dmain input x(t) prduces the same utput y(t) fr bth systems if and nly if the prefilter bandwidth BW(G(!)) is cnstrained by! + BW(H(!)) < BW(G(!)) < 3! BW(H(!)) : (2.42) if the last inequality is nt satisfied, cnvlutin prducts f X(!)S k(!) fr k > will be aliased int the utput. In reality, the filters H(!) and G(!) have finite rllffs, and the equivalence between the systems in Figure 2.7 is nly apprximate. The quality f the apprximatin depends n the harmnic cefficients c k crrespnding t the binary sequence, which can be ptimized fr minimum distrtin. There is a trade-ff between the cmplexity f the sequence and that f the filters G and H, as illustrated in Sectin Sequence generatin The utput spectrum generated by the mdulatin scheme cntains intermdulatin prducts between the prefiltered input G(!) X(!) and the harmnics f S k(!), with terms f the frm jc k G (! (2k + )! ) X (! (2k + )! ) H(!): (2.43) Only the fundamental term k = is desired, and distrtin arises frm the higher-rder intermdulatin prducts, k >. T reduce distrtin, the cefficients c k need t be small fr k >, 44

32 S(ω) = δ(ω ω ) H(ω) X(ω) ω ω ω 2ω 3ω 5ω 4ω S(ω) k= = δ(ω ω ) G(ω) H(ω) X(ω) ω 2ω 3ω 4ω 5ω ω 2ω 3ω 4ω 5ω ω ω 2ω 3ω 4ω 5ω ω Figure 2.7: Tp: Demdulatin f an input X in the frequency dmain with a perfect sine wave S and ideal mdulatin filter H. Bttm: Demdulatin f an input X in the frequency dmain with an versampled sine wave S using an ideal smthing filter G and mdulatin filter H. except fr large values f k fr which the terms f (2.43) are reasnably small due t the attenuatin by the prefilter G. In ther wrds, the lw-frequency cmpnents f the binary sequence s (t) need t apprximate the sine wave s(t) as clsely as pssible. Qualitatively, this crrespnds t an versampled nise-shaped sine wave, as prduced by the delta-sigma mdulatr methd [26]. Techniques fr deriving peridic sequences with several zer r small harmnic cmpnents f c k are presented in [33]. We frmulate the prblem f finding an ptimal binary sequence directly frm a minimum distrtin criterin n the intermdulatin cmpnents (2.43). In general, the amunt f distrtin is input dependent, and assumptins need t be made n X(!) t frmulate an ptimizatin criterin. Our criterin is t maximize the rati f energy in the fundamental harmnic mdulatin cmpnent (k = ) t the cmbined energy f the distrtin cmpnents (k > ). Assuming a narrw bandwidth f H(!) and an input spectrum X(!) which in the wrst case is flat in amplitude, the criterin becmes: Maximize : X k= c 2 jg(! )j 2 c 2 k jg ((2k + )! )j 2 : (2.44) which is equivalent t minimizing the harmnic distrtin f the sequence s (t), filtered with the same prefilter G(!). The criterin can be applied, in principle, t select the ptimal bit sequence 45

33 s (nt ), fr n = : : : N. With quarter-wave symmetry, nly N=4 bits (ne quadrant) need t be determined. Still, this prblem has a cmbinatrial cmplexity, and becmes intractable fr large N. We btain apprximate slutins using a technique f iterative blck ptimizatin, where a full search is repeatedly cnducted ver randmly selected blcks f cnsecutive bits in the sequence. Appendix B lists a shrt Matlab prgram which perfrms the blck-iterative sequence search. The prcedure fr determining the harmnic distrtin is as fllws: A quarter-wave bit sequence is expanded by inverting and reversal int a full wave and described in terms f values + and. An FFT is applied t this sequence and its magnitude cmputed. The resulting spectrum is multiplied by the frequency-dmain transfer functin f the lwpass filter (described as an attenuatin cefficient per FFT bin). Then we can directly cmpute ttal harmnic distrtin as the magnitude f the secnd FFT bin (the signal) divided by the sum f the magnitudes f all the ther FFT bins (the distrtin). In lieu f cmputing every ne f the 2 n pssible sequences, we cmbine exhaustive search with a randm perturbative methd. The exhaustive search is perfrmed ver a tractable subspace f m bits, where m < n (generally, m < 6 t cmpute in reasnable time), where the remaining n m bits are held fixed. On each iteratin, the starting pint f the subsequence t exhaustively search is chsen at randm, the search space f 2 m sequences cmputed fr a minimum, then the minimizing sequence is chsen as the new sequence and the prcess is repeated. There is n guarantee f success, and in practice the errr surface is rife with lcal minima. Usually, hwever, the algrithm prduces acceptable results, meaning ttal harmnic distrtin f 6 t 7 db, which suffices fr mst applicatins. The frmulatin f the algrithm leaves pen the pssibility f variants based n genetic algrithms: at every iteratin f the algrithm, the best z slutins are kept rather than than the single best slutin, allwing ppulatin statistics t determine the curse f the ptimizatin. By searching a brader slutin space at each step, the system is less likely t becme trapped in lcal minima, and the cnvergence time is significantly reduced. 2.8 Results and implementatin We demnstrate the principle with the fllwing example and describe a simple and elegant implementatin. The filter G(!) is third rder, implemented as a cascade f three single-ple filter stages, each ple lcated at z = 5=6. The ttal sequence length is N = 256. Figure 2.9 shws the 64 bits f the first quarter f the sequence btained using the iterative blck ptimizatin methd with blck size 8. Figure 2.2 shws the FFT results fr the filtered and unfiltered binary 46

34 sequences. All harmnics f the filtered sequence are mre than 6 db belw the fundamental. The magnitude f the prime harmnic f the sequence is apprximately.2, which is within 2% f unity. This indicates that if the binary sequence is made f vltage levels V seq, then the resulting sine wave will have a zer-t-peak amplitude within 2% f V seq. Sine sequence and filtered sine wave utput.5 Nrmalized magnitude.5 Binary sine sequence Simulated filtered utput Measured utput Time (us) Figure 2.8: The versampled sine sequence. s (t) t Figure 2.9: Optimized 64-bit versampled sine sequence, first quadrant. The advantages f using an versampled mdulatin sequence rather than a simpler binary sequence can be appreciated by fllwing cmparisn. T btain the same 6 db linearity perfrmance with, say, a square wave mdulatr, a premultiplicatin filter G with much sharper rllff such as a furth-rder Chebyshev r a 6th-rder Butterwrth wuld be required t cmpen- 47

35 x Nrmalized Amplitude (db) x x x x x x x x x x x x x x x Frequency Index (2k+) Figure 2.2: Frequency dmain prperties f the raw () and filtered (x) bit sequences. sate fr the sizable harmnics in the square wave. Such a filter wuld be mre cmplicated t build than the simple cascade f single-ple filters in the abve example, and wuld be mre sensitive t mismatches in the implementatin. On the ther hand, the versampled sequence is fairly easy t generate, as utlined belw. The first-quadrant sequence f Figure 2.9 cnsists f zers and 53 nes, as expected frm the integral calculatin in Equatin The asymmetry allws a simple implementatin using a sparse address decder. A binary cunter cunts frm t r, where r is the length f ne quarter f the full binary sequence. The address decder, a wired- r implementatin f nmos transistrs and a pmos pullup device with a small layut ftprint, generates the zer bits at the prper pints f the cunt. The inversin and reversing peratins needed t btain the rest f the full sequence can be elegantly realized by using a gray-cde cunt rather than a binary cunt. In an n-bit gray cde, as illustrated in Figure 2.2, the lwer n 2 bits describe a sequence which cunts ut frwards and then backwards. The inversin f the sequence is determined thrugh an exclusive-r peratin f the sequence bit with the n-th bit f the gray-cde cunter. It is als quite straightfrward t generate the addresses f a sequence 9 ut f phase with the riginal, fr 48

36 cmplex mdulatin with bth sine and csine cmpnents, by perfrming inversin as abve but using bit n f the cunter. One technique is t use a sigma-delta mdulatr t transfrm an arbitrary signal int binary frm [25, 26]. Hwever, in the case f mdulatin the (carrier) signal is knwn a priri, s a sigma-delta mdulatr is unnecessary and can be replaced by a simple digital circuit which prvides the multiplexer with a predetermined fixed sequence. quadrant quadrant 2 quadrant 3 quadrant 4 sign bit fr sine sign bit fr csine Figure 2.2: Use f Gray cde t generate sine and csine sequences. 2.9 Mdulatin Multiplier The previus sectin described a methd by which an accurate sine wave can be generated by filtering an versampled binary sequence. Utilizing this technique immediately alleviates ne f the tw majr prblems f the analg wavelet prcessr. This sectin addresses the ther prblem, that f prducing a linear multiplicatin f the input with the sinusidal carrier signal. Because implementatin f the demdulatin multiplicatin requires multiplicatin f an 49

37 arbitrary cntinuus-valued input with a knwn peridic functin (the sine wave), we are able t use the versampled sine sequence directly t achieve a highly linear analg multiplier. Simple but practical mdulatin schemes make use f multiplicatin f a cntinuusvalued signal and a binary-valued signal. An exact multiplicatin f an arbitrary input by a binaryvalued () functin can be realized as a multiplexer which is cntrlled by the functin and alternates between the input and its inverse. This is shwn in Figure In the figure, the multiplexer is Cntrl Input Mux Multiply LPF Output Figure 2.22: Multiplexing vs. Multiplying. cntrlled by a square wave [28]. The square wave is used as an apprximatin t a sine wave with all harmnics ther than the fundamental cnsidered t be errr terms. As mentined in the previus sectin, this wrks as lng as the intermdulatin prducts f the input signal and these errr harmnics fall well utside f the passband f the final result after the usual lwpass filtering. In the same manner as the square wave example, any arbitrary binary sequence can be used as a cntrl, insfar as any unwanted intermdulatin prducts fall utside the passband f the pstmultiplicatin filter. Since the purpse f versampled representatins f wavefrms is t push unwanted harmnics as far away as pssible frm the harmnics f the desired wavefrm, binaryvalued versampled representatins wrk extremely well in place f the square wave in Figure It is assumed that the versampling nise is negligible in the frequency band f interest, and can therefre be filtered frm the multiplicatin utput. 2.2 Switch-Cap Wavelet Gaussian Functin The circuit which apprximates a Gaussian filter is based n the same architecture as described in Sectin 2. (shwn in Figure 2.), which is in turn based n the Central Limit Therem f statistics: A half-gaussian prfile is prduced by an infinite cascade f lwpass filters. 5

38 A finite cascade f simple filters will apprximate the Gaussian transfer functin t the degree f accuracy required, prvided that there is enugh space in the VLSI layut t accmmdate the number f filters required. Once it was determined that the utput f the mdulatin multiplicatin wuld be a synchrnus discrete-time signal, the decisin was made t change the cntinuus time filters int switched capacitr filters. This is dne fr three reasns: The discrete-time system scales crrectly t any clck speed, s the system can be run accurately either with real-time inputs r with nn-real-time inputs, such as test inputs generated by a cmputer. Nn-real-time inputs d nt even need t be strictly synchrnus. The switched capacitr architecture has much larger linear input and utput ranges than the transcnductance-c filters used in the analg architecture. The bandwidth is cntrlled digitally and s can be cmputed directly frm the system clck driving the center-frequency scillatrs, as ppsed t being an independently-cntrlled variable. As with the cntinuus-time filter architecture, it is nly necessary t create ne half f a Gaussian centered arund zer. The same argument relating t the Central Limit Therem applies, s the switched capacitr filter retains the architecture f a set f cascaded first-rder lwpass sectins. Figure (2.23) shws a sectin f the filter we designed. This circuit is an RC lwpass filter using a simple switched capacitr simulated resistr [27]. The filter design is a discrete-time circuit and directly implements the (z-transfrmed) lwpass functin thrugh distributin f charge: = ; (2.45) + V ut = V in z 2 + ( ) V ut z (2.46) where in the figure and 2 are nnverlapping clck signals f perid T which maps t the z-dmain unit delay. This circuit maps t an equivalent (s-dmain) RC lwpass filter by RC = T. The Gaussian lwpass filter cnsists f a cascade f n sectins, which have a cmbined transfer functin which cnverges t a Gaussian in the limit n!. The chice f the number f sectins is a tradeff between the accuracy f the filter with respect t a true Gaussian and the delay (and als nise) incurred by a signal passing thrugh the cascade. We chse a cascade f eight 5

39 φ φ 2 V ut V in + C αc Figure 2.23: Single discrete-time lwpass filter sectin. sectins due t a primary cnstraint f accuracy. Fr eight stages, the resulting transfer functin matches a true Gaussian t abut 6 db. The center frequencies f the effective bandpass functins are determined by the frequency f the mdulatr sine waves (described belw), which are spaced n a lg 2 scale. The bandwidth f each wavelet filter is determined by the cutff frequency f the switched capacitr lwpass filter, and therefre is prprtinal t the perid f its driving tw-phase clck. The frequency f this clck signal, like the clck which generates the versampled carrier signal, is spaced n a lg 2 scale. The fixed relatinship between the carrier and the lwpass cutff gives the wavelet filterbank a cnstant-q characteristic. The clck frequency can be derived frm the clck driving the mdulatr sequence generatr (in ur implementatin, this is accmplished by dividing dwn the master clck ff-chip). The bandwidths typically are set s that adjacent channels verlap at the half-magnitude pint f the Gaussian functin. 2.2 Output Time Multiplexing Outputs f the Gaussian filter sectins shuld be sampled in a manner which generates the time-frequency distributin shwn in Figure 2.. The binary-tree time multiplexing shwn is quite easy t generate frm a Gray cde. As shwn in Figure 2.24, cntrl signals fr taking the utput frm each wavelet channel can be achieved by a cunter which uses tw-phase clcking and generates Gray cde as utput. This cntrl signal is the trigger signal, 2, fr each Gray cde digit Q i. It can be seen that if 2 cntrls sampling f Channel, 22 cntrls sampling f Channel 2, etc., then Channel is sampled twice as ften as Channel 2, which is sampled twice as ften as Channel 3, and s frth, and nne f the signals verlap. One time slt per sequence f utputs is 52

40 unused, but culd be used t encde the DC level f the input. Nte that a Gray cde cunter is as simple t design as a binary cunter. In the usual implementatin f a binary cunter as a cascade f tggle flip-flps where each tggle flip-flp is a pair f transparent latches in series, the utput f the first latch in each pair encdes a Gray cde while the utput f the secnd latch in each pair encdes the binary sequence. φ φ 2 Q φ 2 φ 22 Q 2 φ 3 φ 23 Q 3 φ 4 φ 24 Q 4 Output Sync Figure 2.24: A scheme fr cntrlling time-multiplexing f the utputs. The same methd is used thrughut the chip t generate clcks which ensure that the mdulatr frequencies and filter bandwidths differ by a factr f 2 fr each channel. An additinal bnus f using Gray cde is that digital nise is kept lw because nly ne channel is clcked at a time, and pwer cnsumptin is kept t a minimum due t the event-driven nature f the prcess Wavelet chip slice The parallel nature f the analg wavelet cmputatin makes the chip easy t create using abutting slices f circuitry. We built a bank f eight slices (channels), each cntaining the lgic t divide dwn the incming clck signals, generate the sine and csine sequences, multiply these mdulating signals with the chip inputs, filter the result with a Gaussian-shaped functin, and timemultiplex the utputs n tw buses (ne fr sine, ne fr csine parts). I devised a simple way t expand the system frm eight t sixteen channels cvering the same ttal frequency span: In a 53

41 sixteen channel system, each channel has a center frequency value which is p 2 f the neighbring channel, with the bandwidth f each channel narrwed by a factr f tw t accunt fr the fact that there are twice as many channels squeezed int the same ttal frequency span. The value f p 2 = :442 : : : is apprximated reasnably well by the integer fractin 7=5 = :4, which is nly abut.% lw. The sixteen-channel system is enabled by starting with a master clck which is first divided dwn by five and by seven, each result becming the master clck fr ne eight-channel sectin. The utputs f each eight-channel sectin are interleaved, and the sampling scheme f Figure 2. must be mdified s that the timesteps are halved s as t allw bth sectins t be sampled. The sixteen-channel architecture is shwn in Figure Wavelet decmpsitins f any multiple f eight channels can be realized in a similar manner. Master 2-phase clck φ 8kHz φ khz 5.74 khz khz.429 khz 74Hz 357Hz 79Hz 89Hz khz 8kHz 4kHz 2kHz khz 5 Hz 25 Hz 25 Hz Figure 2.25: Sixteen-channel architecture using the 7-t-5 frequency rati. Decmpsitin and recnstructin functins are similar and therefre are able t share the same multiplier and filter circuitry, and the chip can be cnfigured fr either functin. Recnstructin requires sample and hld circuitry n the frnt end t demultiplex the sine and csine inputs, and a capacitive adder at the utput. During recnstructin, the Gaussian filters d nt shape the 54

42 signal but are used nly t reject the high-frequency nise utput f the multiplier. A blck diagram shwing tw channels f the Wavelet Transfrm chip is shwn in Figure Sine In Csine In Signal CLK In In Select CLK2 CLK3 CLK4 CLK5 Sine LPF S&H LPF LPF Csine LPF S&H LPF LPF Sine 2 2 LPF S&H LPF LPF Csine 2 LPF S&H LPF LPF Recnstructed Output Time-Multiplexed: Sine Out Csine Out Figure 2.26: The wavelet chip, blck diagram. The circuit layut f the CWT prcessr fits tw sectins f eight channels each n a single 4 mm6 mm die in a 2 µm CMOS p-well prcess, packaged in an 84-pin PGA. Figure 2.27 is a phtmicrgraph f the integrated circuit. Test results reprted here are frm this chip and a separate test chip cntaining a single channel Experimental Results In a test f the mdulatin multiplicatin (including the pstmultiplicatin filter), the sinusid mdulatr is multiplied by a cnstant input. A binary sequence representing the versampled sinusid is prduced at the multiplexer utput, and is smthed int a sine wave by the lwpass filter. Figure 2.2 shws the FFT f the test chip mdulatin utput befre ( ) and after ( x ) 55

43 Figure 2.27: Phtmicrgraph f the mixed-mde cntinuus wavelet transfrm prcessr, a 4 mm6 mm die size fabricated in a 2 µm CMOS p-well prcess. 56

44 lwpass filtering. Distrtin cmpnents f the binary sequence are attenuated by the filter t belw 6 db. Figure 2.28 is an scillscpe phtgraph shwing the demdulating hardware in actin. The tp trace is the input signal, a sine wave generated by a functin generatr fr the purpses f evaluating the system. The middle trace is the result f multiplying the input by the binary sequence. The functin flips rapidly frm psitive t negative vltages, which is t fast t be captured in the scillscpe phtgraph where it appears t be tw verlapping sinusids. The bttm trace is the utput after filtering thrugh the Gaussian filters, and is a sinusid f a frequency which is the difference between the input and the carrier. The bttm trace is jagged due t the discrete-time nature f the switched capacitr hardware. Because the utput is sampled at the same rate prir t time-divisin multiplexing int the utput stream, there is n need at this pint t apply a smthing filter t the utput. Figure 2.28: Signal demdulatin using the wavelet chip. The tp trace is the input signal. The middle trace is the input multiplied by the binary sequence. The bttm trace is the utput after filtering. Figures 2.29 and 2.3 shw the measured transfer functin (magnitude and phase) f the Gaussian filter, a cascade f eight single-ple switched capacitr filters, as cmpared t the predicted result fr an ideal eight-stage lwpass filter cascade. Bandwidth is nrmalized t the clck frequency f the filter switches, shwing that the shape f the filter is independent f the crner frequency. 57