Real-time wavelet-transform spectrum analyzer for the investigation of 1/fα noise

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Real-time wavelet-transform spectrum analyzer for the investigation of 1/fα noise

Doriano Brogioli and Alberto VailatiDipartimento di Fisica and Istituto Nazionale per la Fisica della Materia,

Universita di Milano, via Celoria 16, 20133 Milano, Italy

A wavelet transform spectrum analyzer operating in real time within the frequency range 3 ×

10−5− 1.3 × 105Hz has been implemented on a low-cost Digital Signal Processing board operating

at 150MHz. The wavelet decomposition of the signal allows to efficiently process non-stationarysignals dominated by large amplitude events fairly well localized in time, thus providing the naturaltool to analyze processes characterized by 1/fα power spectrum. The parallel architecture of theDSP allows the real-time processing of the wavelet transform of the signal sampled at 0.3MHz. Thebandwidth is about 220dB, almost ten decades. The power spectrum of the scattered intensity isprocessed in real time from the mean square value of the wavelet coefficients within each frequencyband. The performances of the spectrum analyzer have been investigated by performing DynamicLight Scattering experiments on colloidal suspensions and by comparing the measured spectra withthe correlation functions data obtained with a traditional multi tau correlator. In order to asses thepotentialities of the spectrum analyzer in the investigation of processes involving a wide range oftimescales, we have performed measurements on a model system where fluctuations in the scatteredintensities are generated by the number fluctuations in a dilute colloidal suspension illuminated bya wide beam. This system is characterized by a power-law spectrum with exponent −3/2 in thescattered intensity fluctuations. The spectrum analyzer allows to recover the power spectrum witha dynamic range spanning about 8 decades. The advantages of wavelet analysis versus correlationanalysis in the investigation of processes characterized by a wide distribution of time scales andnon-stationary processes are briefly discussed.

PACS numbers: 07.50.Qx, 07.60.Rd, 02.70.Rr, 05.40.Ca, 05.45.Df

I. INTRODUCTION

The investigation of random processes characterizedby a wide range of timescales is becoming increasinglyimportant in many experimental fields ranging from theinvestigation of earthquakes in earth-science, to turbu-lence in fluids, or the price fluctuations in financial mar-kets [1, 2, 3, 4]. In particular, many random processesexhibit a scale invariant structure. The self-similarity ofthe signal is reflected in the power law behavior of thepower spectrum, which lacks characteristic time scales.

Traditional spectral decomposition techniques oftenfail when applied to a self similar signal. This is dueto the fact that such a signal is typically the superpo-sition of bursts occurring at many different timescales.Although rare, long bursts having a very large ampli-tude provide much of the energy content of the signal,while short bursts with small amplitude, although veryfrequent, give small contribution to the energy of the sig-nal. The energy is therefore fairly well localized in timeand the signal is non-stationary. This localization pre-vents Fourier Analisys to work as an effective tool forthis kind of signal.

However, a scale invariant signal can be efficiently an-alyzed by using the renormalization group approach de-vised in statistical physics to describe systems at a criticalpoint [5]. The self similar system goes through a consec-utive sequence of coarse grainings which generate a newsignal with statistical properties similar to those of theoriginal one.

In practice the renormalization group analisys of a sig-

nal is achieved by using the wavelet transform technique.Basically the signal is convolved with a chosen motherwavelet function. The mother wavelet consists in a trun-cated wave packet with central frequency ω and band-width Γ, localized at time t, with length l ∝ 1/Γ. Themean square amplitude of this component provides thepower spectral amplitude of the signal in the frequencyrange (ω − Γ, ω + Γ). An in-quadrature component ofthe signal is also simultaneously processed to recoverthe discarded lower frequency component, representingthe coarse grained signal. The mother wavelet is thenrescaled on a larger scale and the process iterated.

In this paper we describe a real-time wavelet transformspectrum analyzer (WTSA) working in the frequencyrange 3 × 10−5 − 1.3 × 105Hz. The spectrum analyzeris built around a Texas Instruments C6711 DSP StarterKit (DSK). This board incorporates a C6711 DSP run-ning at 150MHz together with a parallel port to commu-nicate with a Personal computer and various connectorsto interface the board to the real world. Solutions basedon wavelet analysis by means of the post processing on adedicated personal computer of the digitized signal haverecently been proposed [6, 7]. The highly parallel pro-cessing architecture of the DSP allows the real-time pro-cessing of the wavelet transform of the signal sampled at0.3MHz. The bandwidth spans almost ten decades.

We will present applications to Dynamic Light Scatter-ing (DLS) from colloidal suspensions and compare the re-sults obtained with the WTSA with the correlation func-tion obtained with a traditional multi tau correlator. Dy-namic Light Scattering has been used very extensively toinvestigate stochastic fluctuations in the intensity of the

2

light scattered by suspended particles undergoing brow-nian motion, aggregation processes, gelation processes,critical dynamics [8, 9, 10]. In recent times, processescharacterized by a wide distribution of relaxation times,often leading to stretched-exponential decay of the au-tocorrelation function, have attracted much interest (seeRefs. [11, 12] and references therein). The developmentof log-scale correlators has allowed the investigation ofprocesses extending from about 100ns up to hours. How-ever, large time lags require very long acquisitions timesto get good statistical accuracy in the correlation tails.

The wavelet analisys of the fluctuations in the intensityof the scattered light provides higher accuracy in shortertimes when compared to DLS when non-stationary selfsimilar signals are processed. This is due to the fact thatthe localized wavelets used to process the signal are veryefficient in isolating the well localized features of a self-similar signal, and this provides a shorter convergencetime of the processing to get results with a chosen statis-tical accuracy.

To asses the potentialities of the WTSA in investigat-ing time-scale invariant systems we have devised a simplemodel system giving rise to a power-law spectrum withexponent −3/2. This system is made up by stronglydiluted brownian particles diffusing within a large sam-pling volume. We show that the WTSA is able to mea-sure the power spectrum within a dynamic range as largeas 8 decades. We also show that the WTSA requires ameasurement time roughly 100 times smaller than thatneeded by a correlator to get the same statistical accu-racy at large lag times.

II. WAVELET ANALYSIS

The wavelet analisys of signals is becoming increas-ingly popular within the scientific community. With re-spect to traditional Fourier decomposition techniques,wavelet analisys allows to characterize signals having alarge bandwidth by using a small number of coefficients.

Basically, instead of decomposing the signals intoarmonic oscillations as in Fourier analisys, the signalgets decomposed into the superposition of wave packets(WP). The WP are localized both in time and in fre-quency. These wave packets are obtained by rescaling achosen Mother Wavelet (MW) Ψ (t) with a proper scales so as to obtain a wavelet function ϕt0,s (t) = Ψ

(

t−t0s

)

.The wavelets exhibit the same functional form of theMW, but are centered around time t0 and their band-width is a factor s larger than that of the MW. Therefore,if the signal has a bump at time t with a certain dura-tion ∆t the wavelet decomposition will likely give rise toa wave packet localized at t with a scale s proportional tothe duration of the bump. This allows a very efficient de-composition of spiky signals and non-stationary signalsin general. In practice the wavelet analysis is usuallyperformed by using the popular Discrete Wavelet Trans-form (DWT) algorithm [13]. To apply this algorithm,

one starts with a sampled discrete signal y0n made up by

2N samplings, where N is an integer. The signal under-goes successive coarse grainings and band-pass filteringsby convolution with a chosen wavelet set. The scale s ofthe wavelets used during the coarse grainings changes bya factor 2 at each step, so that the scaling of the signalat the k-th step is given by sk = s02

k. On the contrary,the number of samplings in the coarse grained signal de-creases as 2N/2k. Therefore, as the scale increases due tothe successive scale doublings, the resolution in the tem-poral localization of the wavelet decreases accordingly.In the following we will give a brief and more rigorousoverview on how the power spectrum is processed fromthe sampled signal.

Consider a discrete signal y0n sampled at a given sample

frequency fs. According to Nyquist theorem the higherangular frequency it contains is ω0 = πfs, that corre-sponds to the function y0

n = (−1)n. The original signal

y0n is passed through a high-pass filter, giving the wavelet

coefficients c0n, representing the high frequency details of

the signal in the range (ω0/2, ω0), and through a low-passfilter, giving a coarse grained signal y1

n. Both the filtersare implemented by convolving the signal with suitablefilter coefficients gn (the MW) and hn; they are localizedin n so that the only non vanishing elements are the oneswith 0 ≤ n < M :

{

c0j =

∑

n y0ngn−2j

y1j =

∑

n y0nhn−2j

. (1)

Both the filtered signals c0n and y1

n are sampled at half theoriginal frequency fs. The two filter coefficients hn andgn are such that the coarse grained signal y1

n, along withthe details c0

n, allow to recover the whole original sig-nal y0

n: they are referred to as quadrature mirror filters.In the measurements we performed we used Daubechieswavelets [13].

The procedure is then iterated, using y1n as the new sig-

nal; a high frequency component c1n is obtained together

with a coarse grained signal y2n, and so on. In general,

the following recursion formulas apply:

{

ckj =

∑

n ykngn−2j

yk+1j =

∑

n yknhn−2j

. (2)

At step k, the signal ykn is coarse grained to obtain

yk+1n ; the high frequency details, spanning the angular

frequency range(

ω0/2k+1, ω0/2k)

are the wavelet coeffi-

cients ckn.

The power spectrum of the signal can then be derived

from the mean square average of the details⟨

∣

∣ckn

∣

∣

2⟩

n.

With the traditional Fourier transform, the analyzedwavelengths are discretized, due to the finiteness of thesample; the spacing between two consecutive wavenum-bers is constant. Unlikely, with wavelet analysis, thespacing is exponential: the mean square amplitude ofthe wavelet coefficients represents the power containedin an octave.

3

The exponential spacing is particularly suited to mea-sure phenomena spanning many decades in wavenumber.This feature is analogous to the multi tau analysis doneby the best commercial correlators. Though the powerspectrum and the correlation function are connected bythe well known Wiener-Kintchine theorem, in many ac-tual cases it is not possible to recover the power spectrumby the measurement of the correlation function. In gen-eral, the exponential spaced correlation function givenby a multi-tau correlator cannot be safely Fourier trans-formed into the power spectrum, over many decades.Moreover, in some cases Wiener-Kintchine theorem doesnot apply. This is the case of some scale invariant andnon-stationary signals; in Sect. VB we present an exam-ple in which the power spectrum obtained by the waveletanalysis correctly shows a power law behaviour, whilea multi tau correlator seemingly says that the signal isdelta correlated.

The analogy between the wavelet analysis and themulti tau correlation also includes the possibility of per-forming the required processing in real time, as the sam-ple is acquired. With Fourier analysis, as the samplebecomes larger, we must analyze more and more wave-lengths, and this requires that all the sample is recordedand re-analyzed. On the contrary, as time goes on, themulti tau correlator averages the older samples; as a new,longer delay time is available, the correlation function isevaluated from only a few registers. This allows to ex-tend the dynamic range in time delay over many decades,without the need to record the samples with the same,huge dynamic range. The same thing happens with theWTSA. The iterative procedure is implemented by usingN , almost identical levels k. The memory needed scalesas N while the time needed by the processing does notchange at all, as we will show in Sect. IV; on the otherhand, the frequency range scales as 2N .

III. THE WAVELET TRANSFORM SPECTRUM

ANALYZER

The spectrum analyzer is built around a Texas Instru-ments C6711 DSP Starter Kit (DSK), graciously pro-vided by Texas Instruments DSP University Program.

This board incorporates a C6711 DSP running at150MHz, together with a parallel port to communicatewith a Personal computer and various connectors to in-terface the board to the real world. The software canbe developed on a Personal Computer by using TI CodeComposer Studio and then downloaded on the board.The board and the bundled software are very cheap com-pared to a real time correlator board, the purchase pricebeing of the order of 300$ for educational institutions.

The core of the board is made up by a Digital Sig-nal Processor, a single-chip programmable device includ-ing a CPU specifically suited for digital signal processing[14]. The DSP is a TMS320C6711, built by Texas Instru-ments, one of the fastest processors currently available.

It is based on VelociTI, a high-performance, advancedvery-long-instruction-word architecture, and its CPU canperform operations on double precision (64 bit) floatingpoint numbers.

The CPU is divided into two almost identical sides;each of them includes 16 registers, one word (32 bit)long. Couples of registers can be used to represent onefloating point, double precision register. Each section in-cludes four ALUs, operating simultaneously; each ALUcan perform only a given set of operations. Data pathsfrom registers to memory and from a register in one sideto an ALU in the other are also provided.

The CPU works at 150MHz. At each clock cycle, up toeight instructions are fetched, dispatched and decoded,that is, up to one instruction for each ALU. All thefixed point instructions are executed in one clock cycle;this means that the CPU can perform 1.2 × 109 instruc-tions per second, a figure comparable to the number ofMFLOPS available on a 2GHz Pentium IV processor.Some floating point instructions require more than oneclock cycle to return the results (execution latency); gen-erally, the same unit can start a new instruction aftera few clock cycles (functional unit latency), before theresults of the previous operations are availabe. For ex-ample, the functional unit latency of a double precisionfloating point multiplication is 4 clock cycles, and theexecution latency is 9 clock cycles. Consider, for ex-ample, an ALU starting a multiplication at t = 0: att = 4 clock cycles it can start a new multiplication andat t = 9 clock cycles the results of the first multiplicationare available on the registers. Since there are two ALUsable to perform floating point multiplications, the CPUcan do 60×106 multiplications per second, and, simulta-neously, much more floating point additions, using othertwo ALUs. This structure requires a sophisticated, par-allel programming technique, resulting in extremely fast,real time programs.

The DSP includes a two level memory [15]. The firstlevel is composed by two 4Kbytes cache memories, L1Pfor the program instructions and L1D for the data. Thesecond level, L2, is a 64Kbytes RAM, that can be config-ured partially as a SRAM or as a chache for an externalmemory. Since our program and data fit in the SRAM,we didn’t use the slower external RAM and we config-ured all the L2 as SRAM. Reads and writes in the L1Pand L1D memories are performed in four clock cycles;obviously, L2 is slower and a miss in the L1 cache resultsin a CPU stall.

The DSP includes many other peripherals; amongthem, two counter-timers, TIMER1 and TIMER2 [15].They are broadly configurable through a configurationregister. The input can be connected to the system clockor to an external pin, driven by a CMOS signal, thusselecting if the device is acting as a counter of externalevents or as a timer. The count register can be read andused by the program. Based on the counts and the valueof a period register, an output is driven, in order to gen-erate square waves of different duty cycle. Moreover, the

4

output can drive an interrupt. We used TIMER1 as acounter of external events. In this way, the input of thespectrum analyzer can be driven by a discrete pulse train,as in the application to DLS discussed below, where thetimer is connected to the output of a photon countingphotomultiplier. Alternatively, by using a voltage to fre-quency converter, the input of the timer could be drivenby an analog signal, such as the one from a photodiode.The other timer, TIMER2, generates a periodic inter-rupt, in order to define the integration period: the in-terrupt service routine reads the counts of TIMER1 andresets it, passing the number of counts to the processingprogram.

Texas Instruments provides a DSK (DSP Starter Kit),including a board with the DSP and some peripherals;among them, a device that allows to connect the boardto a PC through a parallel port. The PC can read andwrite the memory of the DSP, without halting the CPU,and can run a program. The PC uploads the programthat evaluates the power spectrum of the signal in theSRAM, then runs it, and periodically reads the results.

The software consists of two parts, one for the DSPand one for the host computer. Texas Instruments pro-vides Code Composer Studio (CCS), that includes a Ccompiler and an assembler for the DSP. The program forthe DSP has been written in assembler code, since in thisway a high processing speed is obtained in conjunctionwith an accurate timing. Moreover, the provided soft-ware includes some libraries for building host programsthat interact with the DSP, that is, to upload a program,run it, and read some memory locations in the DSP [16].All the software we developed is freely available. [17]

IV. DISCRETE WAVELET TRANSFORM

PROGRAM

With respect to the hardware correlators traditionallyused within the Dynamic Light Scattering community,the use of a DSP allows the flexible implementation ofmany processing algorithms simply by dowloading a dif-ferent processing code onto the board. These algorithmsinclude FIR and IIR filters, cross correlation, convolutionfilters etc.

In this section, we discuss the implementation of thediscrete, real time wavelet algorithm on the DSK board.

The overall data structure is shown in Fig. 1, alongwith a simplified description of the data flow. The higestsampling frequency is about 0.3MHz and the lowest fre-quency corresponds to a period of some hours. Every fre-quency octave represents a level in the wavelet processingalgorithm. Figure 1 shows only levels 0 and 1 and partof level 2. The entire structure is made up by N = 32levels, each corresponding to a measured frequency. Ev-ery level contains a circular queue yk

j of double precisionfloating points to store the last M values of the coarsegrained signal, where M is the length of the MW. TheN queues are managed by using the circular addressing

mode of the DSP; this allows only values of M that arepowers of two. Therefore, the Daubechies wavelets ourprogram can use are those with length M = 4, 8, 16, 32and 64. We will discuss the implementation of the Daub4M = 4 transform.

Level 0 is driven by TIMER1, configured as a counter,which counts the digitized pulses at the input of the spec-trum analyzer within a selected bin. The integration timeis defined by TIMER2, configured as a timer, so that itsoutput signal generates an interrupt at fs = 375KHz,corresponding to the sampling frequency. At each sam-pling time the number of pulses counted by TIMER1 isfed to the first element of the shift register y0

n in Level 0and the other cells are shifted down. Every two sampletimes the content of the shift register is convolved withthe two wavelet coefficient banks gn and hn, the MWand the smoothing filter coefficients respectively. Theresult of the convolution with the bank hn feeds the shiftregister y1

n in Level 1 and gives rise to a coarse grainedreplica of the signal at half the sampling frequency. Thein-quadrature component obtained from the convolutionwith bank gn contains the wavelet coefficients c0

n whichspecify the local amplitude of the band-pass filtered sig-nal in the frequency range (ω0/2, ω0), where ω0 = πfs.The process is repeated in cascade for all the levels.

The processing of N levels requires to store additionalinformations: the sum and the counter needed to eval-uate the mean of

∣

∣ckn

∣

∣

2, the current address inside the

queue and the status of the queue.

The coefficients ckn feed the register

∑

∣

∣ckn

∣

∣

2, which

stores the mean square amplitude of the high-pass fil-tered signal representing the power spectrum S (ω) inthe angular frequency range

(

ω0/2k+1, ω0/2k)

.The status of the queue can be 0 (the queue has just

been processed), 1 (the queue had one input) or 2 (thequeue is full and requires processing); it is incrementedby one each time an element is put in the queue and isreset to zero when it is processed. At the initialization,the status is set to −M + 2, so that the queue is filledat least with M elements before it is processed the firsttime.

As outlined above, Level 1 is updated only when twonew samplings are available in Level 0. Therefore theupdate of Level 1 occurs with a frequency fs/2, and itsprocessing with a frequency fs/4, which corresponds tothe update frequency of Level 2, and in general Level k isupdated at a frequency fs/2k and processed at fs/2k+1.The sum of all the processing frequencies for every queue,∑

n fs/2k+1, converges to fs as N → +∞. This meansthat the frequency at which all the queues are processedis smaller than the sampling frequency fs. At each sam-pling time, our algorithm processes the lowest-level fullqueue, if any. This ensures that, by processing exactlyone queue at each sampling time, the output values yk

n

never go to an already full queue. For the Daubechieswavelets with M = 4 all the operations can be performedin about 2µs.

Beyond the processing sofware on the DSP board two

5

y0

n−3

y0

n−2

y0

n−1

y0

n×

×

gj

hj

Status=2?+

Counter

∑

j

∣

∣c0

j

∣

∣

2

·2

Level 0

y1

m

c0

m

y1

m−3

y1

m−2

y1

m−1

y1

m×

×

gj

hj

Status=2?+

Counter

∑

j

∣

∣c1

j

∣

∣

2

·2

Level 1

y2

l

c1

l

Level 2

TIMER2

TIMER1

ISR

Input

FIG. 1: Data structure and data flow of the wavelet transform spectrum analyzer.

host programs run on the host computer: the first up-loads the DSP programs and runs it; the second readsthe memory of the DSP, without halting the CPU. Thisprogram is used to get a real time display of the processedspectrum. The read program accesses the DSP memorythrough the parallel port. Inside the DSK board, data areread from memory by using a low priority DMA channel,and thus without interfering with the CPU operations.

Another processing method, not yet implemented, in-volves the processing of the cross spectrum of two signals.This is very useful to get rid of noise sources such as af-terpulsing from a photomultiplier detector. This methodinvolves the parallel use of two processing structures likethe one shown in Fig. 1. Each detector signal feeds one ofthe TIMER input. The output of each structure is a setof wavelet coefficients ck

an and ckbn. The averaged product

of the coefficients∑

ckanck

bn gives the desired cross spec-trum. Due to the simultaneous operation of two pro-cessing structures, the minimum sampling frequency isreduced by a factor of two.

V. EXPERIMENTAL RESULTS

To assess the performances of the Spectrum Analyzerwe have performed test measurements on a colloidal sus-pension and on a model system giving rise to a power-lawpower spectrum.

The experimental setup is based on the traditionalDLS one. A diagram of the setup is shown in Fig. 2.The beam coming from a JDS Uniphase 35mW CW HeNeLASER model 1145P is spatially filtered and focused bythe lens F at the center of the sample contained in thecuvette C. Light scattered from the sample is collectedby the lens L and by the diaphragms D1 and D2. Thescattered radiation eventually impinges onto the photo-cathode of a EMI 9863 B04/350 photon counting Pho-

LASER

C F

Spatialfilter

PMTD2

D1

L

ALVPM-PD

HM8035 DSP PC

HM8021-3

ALV5000 PC

FIG. 2: Diagram of the experimental setup.

tomultiplier Tube (PMT). The scattering angle can bechanged by rotating the collecting optics and the PMTaround the cuvette by means of a goniometer. The cur-rent pulses from the PMT are amplified and filtered by anALV PM-PD preamplifier-discriminator, which also dig-itizes the pulse train at TTL levels. In the traditionalDLS apparatus the signal is then fed to an ALV5000multi tau hardware correlator hosted in a personal com-puter. Alternatively, the digitized signal at the outputof the amplifier-discriminator can be routed to a timerinput of the DSP board, which is connected to a per-sonal computer for the configuration and visualizationof measurements. The DSP board is also connected toa HAMEG 1.6 GHz frequency counter model HM8021-3and a HAMEG 20 MHz pulse generator model HM8035for testing purposes.

6

10-3

10-2

10-1

100

101

10-7 10-5 10-4 10-3 10-2 10-1 100

C(∆

t) (

Arb

itrar

y un

its)

∆t (s)

Afterpulsing

FIG. 3: Correlation function of the intensity fluctuationsof light scattered at 30◦ by a colloidal suspension, inthe Dynamic Light Scattering regime, measured with ALVequipement. Circles represent the experimental data. Thecontinuous line represents a fit of the experimental data upto the second cumulant.

A. Colloidal suspensions

The test on the colloidal suspension were performedon calibrated polystyrene latex of 0.115µm diameter, di-luted at a volume fraction of about φ = 1.5×10−6. For adilute suspension of monodisperse brownian particles theHomodyne time autocorrelation function of the intensityfluctuations in the scattered light has the usual exponen-tial form C (t) = 〈I〉2 [1 + A exp (−Γt)], where Γ = 2Dq2

and D = kBT/(6πηa) is the Stokes-Einstein Diffusion co-efficient, η is the shear viscosity for the solvent and a theparticle radius. The correlation function measured at ascattering angle of 30◦ is plotted in Fig. 3 as a functionof the lag time. The experimental data show the decay ofthe correlation function within a dynamic range of about2.5 decades. The run duration was 2700s.

At the longer delay times the correlation function isdominated by noise due to the poor statistical sample

accumulated. The additional contribution at small lagtimes is due to the afterpulsing of the PMT. The after-pulses of the PMT used to performed the measurementhave a characteristic range of correlation times between0.1µs and 3.2µs, which give rise to a stretched exponen-tial decay at small lags. The decay at lags larger thanabout 3.2µs is determined by the intensity fluctuationsdue to the brownian motion of the colloid. Data canbe fit with an exponential function exp (−Γt). A morerefined analisys of the experimental data involves the in-clusion of polidispersity effects in the fitting procedure.This is traditionally performed by a cumulant analisysof the correlation function [8]. In the case of a smallpolidispersity the correlation function up to the secondcumulant is given by

C (t) ∝ (3)

exp

(

−Γ0 |t| +1

2σ2t2

)

∝∫

exp (−Γ |t|) exp

[

− (Γ − Γ0)2

2σ2

]

dΓ,

where the second equality shows that polidispersity ef-fects give rise to the superposition of exponential decayscentered around a central linewidth Γ0, these decays be-ing weighted by a gaussian function with variance σ rep-resenting the polidispersity. The best fit of the experi-mental data with Eq. (3) is indicated by the solid line inFig. 3 and yields Γ0 = 140Hz and σ = 42Hz.

We performed the same experiment, by using the DSPWTSA and the same homodyne detection scheme out-lined above. According to the Wiener-Kintchine theo-rem, the power spectrum of the fluctuations in the inten-sity of light scattered by the colloidal suspension is theFourier transform of the correlation function C (t). Fordiluted monodisperse brownian particle the power spec-trum has a Lorenzian shape: S (ω) ∝ Γ/

(

Γ2 + ω2)

.

Figure 4 shows data for the measured power spectrumas a function of angular frequency ω. Raw data are rep-resented with crosses. At the smaller frequency the effectof noise due to the reduced statistical sample is evident.This occurs at angular frequencies roughly smaller than1rad/s, corresponding to a time delay of the order of6s. However, this same noise affects the correlation func-tion shown in Fig 3 starting from about t = 10−1s. Asthe duration of the experiment is in both cases 2700s,the wavelet processing allows a better convergence atlarger delay times: the timescale range across which thepower spectrum can be reliably determined is roughlytwo decades more extended than that of the correlator.At larger frequency the power spectrum in Fig. 4 fallsdown up to a constant base line, represented by a hor-izontal dashed line. This baseline is determined by theshot noise. In fact, it is well known that the detectionof photons is a random process characterized by a Pois-son distribution and gives rise to a delta-correlated whitenoise, whose power spectrum corresponds to the average

7

10-2

10-1

100

101

102

103

10-2 10-1 100 101 102 103 104 105 106

S(ω

) (A

rbitr

ary

units

)

ω (s-1)

Shot noise

FIG. 4: Power spectrum of the intensity fluctuations of lightscattered at 30◦ by a colloidal suspension, in the DynamicLight Scattering regime, measured with WTSA. Crosses rep-resent the raw data. The dashed line is the baseline due to theshot noise. Circles represent the power spectrum obtained bysubtracting the baseline from the raw data. The continuousline is the best fit of the experimental data up to the secondcumulant.

number of photons detected within the sampling time [8].

C (t) ∝ 〈I (t′) I (t′ + t)〉〈I2 (0)〉 − 1 + 〈n〉 δ (t) . (4)

In the frequency space this gives rise to a constant term〈n〉, represented by the dashed line in Fig. 4, which addsup to the power spectrum. The average number of pho-tons 〈n〉 can be easily processed in real time and sub-tracted.

The circles in Fig. 4 show experimental data for thepower spectrum after the subtraction of the shot noisecontribution. The data span a dynamic range of morethan 4 decades. Data can be represented by a Lorenziancurve Γ/

(

ω2 + Γ2)

. In this case also, a more refinedanalisys involves the inclusion of polidispersity effects.

By taking the Fourier transform of Eq. (3) we get:

S (ω) ∝∫

Γ

ω2 + Γ2exp

[

− (Γ − Γ0)2

2σ2

]

dΓ. (5)

Therefore the power spectrum is the superposition oflorenzians weighted by the same gaussian function in-troduced to describe the polidispersity in the correlationanalysis. We evaluated numerically the integral of Eq.(5), in order to fit the results in Fig. 4. However, each cir-cle in Fig. 4 represents the integral of the power spectrumacross an octave; this smooths even more the bell-shapedmeasured curve; this effect has been considered in the fitprocedure. We obtained the best fit with Γ0 = 158Hzand σ = 37Hz, which favourably compare with the re-sults obtained with the correlator. The best fitting curveis represented by the solid line.

At the higher frequencies the spectrum is dominatedby the contribution of afterpulses to the intensity fluctu-ations which give rise to the tail in the range 104−106Hzin Fig. 4.

The wavelet analysis of the data presents some ad-vantages when compared to correlation techniques. Anotable advantage is represented by the way backgroundsubtraction is achieved. In correlation techniques a con-stant baseline has to be subtracted from the measuredcorrelation function to recover data like those shown inFig. 3. The baseline represents the correlation functionat large la times, where correlations disappear. The in-trinsic contrast of the correlation function, defined bythe ratio between the amplitude of the completely cor-related part at t = 0 and the uncorrelated part at largelag times, [C (0) − C (+∞)] /C (∞), cannot exceed thevalue of two. Therefore, the signal and the backgroundare of the same order of magnitude and the dynamicrange is strongly limited by the accuracy in the base-line. The accuracy in the baseline can be determinedfrom the arguments presented in Ref. [18] and summa-rized below. Let us call τL the largest decay time in thecorrelation function and T the duration of the run. Sam-ples acquired at time lags smaller than τL are correlated,therefore the number of independent samples acquiredduring T is Nc = T/τL. This independent samples av-erage out according to a normal distribution to give riseto the baseline. Therefore, the accuracy in the base-

line is of the order of 1/N1/2c . In the case of the results

presented in Fig. 3, τL = 40ms, T = 2700s, thus giv-

ing 1/N1/2c = 4 × 10−3, which corresponds fairly well to

the dynamic range of about 2.5 decades of the measure-ment. The same argument can be applied to the accuracyof the baseline in the power spectrum determined withthe WTSA and presented in Fig. 4. In this case, how-ever, the baseline is generated by the shot noise, whichis correlated on a time of the order of the inverse of thesampling frequency ω0. The number N of independentsamples corresponds to the total number of samples ac-cumulated at ω0. For the results presented in Fig. 4N = 109 and the accuracy of the baseline is of the order

8

of 1/N1/2 = 3 × 10−5. By eliminating the contributionof afterpulsing to the spectrum as described in Sect. IV,the dynamic range could potentially be extended aboutfive decades above the intrinsic contrast of the measure-ment, which in this case amounts to about 103. It has tobe remarked that in this case there’s no upper limit forthe contrast, therefore a dynamic range as high as 1010

could potentially be obtained.

B. Model system

The ultimate task of the WTSA is the characteriza-tion of signals characterized by a wide distribution oftimescales.

To assess its performances we have chosen a sim-ple model system made up by strongly diluted diffus-ing brownian particles illuminated by a wide collimatedlaser beam. This system gives rise to a nice power-lawpower spectrum with the exponent −3/2. A detailedderivation of the predicted power spectrum is presentedin Appendix. The sample is a colloidal suspension ofpolystyrene latex spheres of 10µm diameter, suspendedin water at a concentration of roughly 104particles/cm

3.

The solvent is a mixture of equal volumes of water anddeuterated water, so that its density is matched to thatof the colloid within 0.1%. The sample is illuminated bya gaussian laser beam with a diameter at 1/e2 of about1mm. The scattering volume is delimited by two slitsperpendicular to the beam. Therefore the scattering vol-ume is determined in one direction by the hard edge ofthe slits and in the other direction by the gaussian profileof the beam. The scattered light is collected at 90◦ withrespect to the main beam.

Due to the strongly diluted sample and to the widebeam illumination, fluctuations in the intensity of thescattered light are mostly determined by changes in thenumber of particles in the scattering volume. By usinga wide laser beam, the coherence areas onto the photo-catode of the PMT are very small. Therefore, the pho-tocatode collects the averaged contribution coming frommany coherence areas. In this way the component ofthe correlation function due to the diffusion of particleshas a vanishingly small contrast, when compared to theaveraged intensity measured by the photodetector. Thesignal onto the PMT thereby represents the superposi-tion of the intensities scattered by single particles. As itis shown in the Appendix, intensity fluctuations spanninga wide frequency range are excited when particles crossthe hard edge at the boundary of the scattering volume.

Figure 5 shows the correlation function of the fluctu-ations in the scattered intensity. The correlation func-tion shows an almost featureless structure and a narrowdynamic range. The correlation function is almost flatwithin about six decades in lag time, due to the longtails associated with the slowest modes. The peak atsmall lags still represents the contribution of afterpulses.The correlation function eventually decays at times of

10-3

10-2

10-1

100

10-7 10-5 10-4 10-3 10-2 10-1 100 101 102 103

C(∆

t) (

Arb

itrar

y un

its)

∆t (s)

Afterpulsing

FIG. 5: Correlation function of the intensity fluctuations inthe model system, measured with ALV equipement.

the order of 60s. However, this decay is an artifact dueto the finite measurement time. In fact, correlation anal-ysis assumes that the correlation function decays to zeroat large delays, once its baseline has been subtracted.The baseline is usually evaluated from the value of thecorrelation function at very large delays. However, fortime-scale invariant processes, the baseline still containsthe contribution of the slowest modes. Therefore, an in-crease of the acquisition time involves a decay at a largertime. As a rule of thumb, experimental data for the cor-relation function are reliable up to delays roughly 100times smaller than the acquisition time. This is a featureof correlation techniques which strongly limits their abil-ity in characterizing processes where an upper time scaleis absent.

Figure 6 shows the power spectrum of the fluctuationsmeasured by means of the WTSA. The crosses representthe raw data, the horizontal line the background leveldue to shot noise and the circles the subtracted data rep-resenting the power spectrum. The most prominent fea-ture of the results in Fig. 6 is the impressive dynamic

9

10-3

10-2

10-1

100

101

102

103

104

105

10-2 10-1 100 101 102 103 104 105 106

S(ω

) (A

rbitr

ary

units

)

ω (s-1)

Shot noise

ω-3/2

FIG. 6: Power spectrum of the intensity fluctuations in themodel system, measured with WTSA. Crosses represent theraw data. The dashed line is the baseline due to the shot noise.Circles represent the power spectrum obtained by subtractingthe baseline from the raw data. The continuous line representsa power-law with exponent −3/2 for comparison.

range spanning about 8 decades. The measured powerspectrum decays approximately as a power law with anexponent of the order of −1.2. The straight line repre-sents a power law with exponent −3/2 for comparison.At the larger frequency, the contribution of afterpulses isapparent.

Deviations from the power law behavior at intermedi-ate frequencies can be attributed to a residual contribu-tion of the diffusion of particles to the correlation func-tion. Such a contribution cannot be completely elimi-nated, even by collecting a large number of coherence ar-eas onto the multiplier. As outlined above, the diffusionof particles gives rise to fluctuations in the scattered in-tensity at 90◦, characterized by lorentzian spectrum witha linewidth of the order of Γ = 20Hz. Deviations formthe power-law behavior occur at frequencies roughly cen-tered around Γ. However, a more refined analysis of thedata, including the evaluation of contributions due to dif-

fusion and due to number fluctuations to the spectrum,is beyond the aim of this paper.

The wavelet transform analysis is particularly effectivefor processes where the power law exponent α is largerthan 1, like the model system just described. In fact,correlation techniques completely fail when applied tosuch systems. For exponents smaller than one the spec-tral behavior can be characterized either by measuringthe power spectrum S (ω) ∝ 1/ωα or by measuring thecorrelation function, which also exhibits a power-law be-havior C (t) ∝ t1−α. In the limit α → 1 the correlationfunction becomes flat. For α > 1 the correlation func-tion is still flat and time independent. Naively speaking,this is due to the strong divergence of the power spec-trum at small frequency which behaves as a delta shapedspectrum. The diverging energy of the small frequencymodes gets uniformly distributed along the lag time axisof the correlation function, which becomes flat. Morerigorously, the correlation function can be obtained byFourier transforming the power-law spectrum, regular-ized to avoid its divergence at ω = 0:

C (t) ∝∫

limǫ→0

1

ǫ + ωαexp (−iωt)dω. (6)

For α > 1, the spectrum has an integrable power law tailfor ω → +∞. It approximates a delta function δ (ω) asǫ → 0. Therefore, for α ≥ 1, C (t) is a constant: a cor-relation function like that shown in Fig. 3 is completelyfeatureless and useless.

An important application of the WTSA is the time de-pendent analisys of transient behavior during processessuch as gelation [6, 19] and colloidal aggregation [20]. Ingeneral, during such transients the highest frequenciesare excited at first, and lower frequencies get graduallyexcited as time goes by. This introduces a characteristiclag time τc marking the boundary between excited andnon-excited modes. The proper characterization of thisprocesses involves the measurement of the power spec-trum (or correlation function) from data sampled alonga duration time T larger then τc, so to be able to resolveτc. In principle a longer T allows to measure the powerspectrum or correlation function with a higher accuracy,due to the increase of the number of independent sampleprocessed. However the lag time τc changes during thetransient. Therefore, a compromise has to be reachedbetween the need to keep T not much larger than τc andthe desire to measure the spectrum with a high accuracy.

As far as the WTSA is concerned, the same argumentused to estimate the accuracy of the baseline can be usedto get an order of magnitude estimate of the number ofsamplings needed to get a chosen accuracy. Suppose thatwe want to obtain the power spectrum at a frequency ωk

with an accuracy better than a. As it was shown in sec-tion II, the power spectrum is calculated by averaging

the squared wavelet coefficients: S (ωk) ∝ ∑

n

∣

∣ckn

∣

∣

2. We

recall here that the coefficients ckn represent the wavelet

coefficients evaluated at subsequent times n. Althoughthis argument is not rigorous, for a generic random sig-

10

nal, it is reasonable to assume that the coefficients eval-uated at different times are independent. Therefore the

percentual error on S (ωk) scales as 1/N1/2k , where Nk

is the number of samplings of the signal accumulated atthe frequency ωk. In this way, a 1% in S (ωk) is achievedby sampling the signal up to frequencies of the order of10000ωk. However, an order of magnitude estimate of thepower spectrum can be obtained from a few samplings atωk.

Therefore, a very important feature of the waveletanalysis, when compared to correlation techniques, is itsability to characterize the power spectrum at small fre-quencies, without introducing artifacts related to the fi-nite acquisition time. This can be appreciated from Fig.6, where the smallest experimental angular frequency isof the order of 10−2Hz, roughly corresponding to the ac-quisition time of 2700s. The wavelet analysis only yieldsan order of magnitude estimate of the amplitude of thismode, due to the absence of a significative statisticalsample. However, the wide dynamic range involved inthe power spectrum makes it almost insensitive to back-ground subtraction at small frequencies, so that no char-acteristic times associate with the duration of the experi-ment are introduced in the subtraction. Therefore, whencompared with correlation techniques, wavelet analysisallows a dramatic increase of the investigated frequencyrange, this increase amounting to almost two decades.This feature makes the WTSA the ideal tool to investi-gate slow processes, non-stationary processes, time-scaleinvariant processes and processes characterized by a widedistribution of time-scales in general.

Acknowledgments

We thank Texas Instruments DSP University programfor the gift of the DSK board and for technical support.We also thank M. Giglio for discussion and support.

APPENDIX A: POWER SPECTRUM OF THE

NUMBER FLUCTUATIONS OF BROWNIAN

PARTICLES IN A GIVEN VOLUME.

In this appendix we will derive the power spectrumdetermined by the number fluctuations of brownian par-ticles diffusing within a scattering volume delimited bysharp edges on two sides and by the gaussian profile ofa beam on the two perpendicular sides. We will assumethat the sample is illuminated by a wide collimated beam,so that spatial coherence effects can be neglected. Someof the arguments we use in the following are also used inthe derivation of fluorescence correlation spectroscopy.[8]

Given N particles, at position ~xn, the measured inten-sity It at time t is the sum of the intensities scattered byeach particle, since light is collected over several speckles,and interference plays no role. Moreover, the intensity

scattered by each particle is proportional to the intensityI (~x) of the main beam at the position of the particle:

It ∝N−1∑

n=0

I [~xn (t)]. (A1)

Roughly speaking, It is proportional to the number ofparticles inside the main beam, weighted by the intensityin that point. The temporal correlation function is thus:

CI (τ) = 〈ItIt+τ 〉 ∝N−1∑

n,m=0

〈I [~xn (t)] I [~xm (t + τ)]〉 ,

(A2)where the brackets represent the average over t. By as-suming that the particles are non-interacting, their posi-tions are not correlated:

CI (τ) ∝ N (N − 1) I2 + N 〈I [~x (t)] I [~x (t + τ)]〉 , (A3)

where the bar represents the average over the volumeV of the whole cell. From this equation, we derive therelative mean square value of the fluctuations of It:

⟨

I2t

⟩

− 〈It〉2

〈It〉2=

CI (0) − CI (+∞)

CI (+∞)=

1

N

I2 − I2

I2. (A4)

Roughly speaking,(

I2 − I2)

/I2 is the ratio V/Vb, whereVb is the volume inside the main beam. This statementcan be easily proved for an hypotetic beam with constantintensity inside Vb, vanishing outside. By calling Nb thenumber of particles inside the beam:

⟨

I2t

⟩

− 〈It〉2

〈It〉2=

1

Nb. (A5)

This equation is analogous to the relation between num-ber fluctuation and mean number for Poisson distribu-tion. In order for the fluctuations to be strong, only afew particles must be inside the beam simultaneously.

The diffusion of each particle can be described as arandom walk; Pτ (~y − ~x), the probability that a particlein ~x goes in ~y in a time τ , is thus:

Pτ (∆~x) =1

[√2πσ (τ)

]3 e− ∆~x2

2σ2 (τ) , (A6)

where σ (τ) =√

D |τ | and D is the diffusion coefficient.By using this probability distribution, we can explicitlyrewrite Eq. (A3):

CI (τ) ∝ c +

∫

I (~x) I (~y)Pτ (~y − ~x) d~xd~y, (A7)

where c is a constant. The convolution operator is diag-onalized by Fourier transform:

CI (τ) = c +

∫

|I (~q)|2 Pτ (~q) d~q, (A8)

11

where Pτ (~q) and I (~q) are the Fourier transforms in ~x ofPτ (~x) and I (~x):

Pτ (~q) = e−1

2q2σ2 (τ)

. (A9)

By Fourier transforming Eq. (A8) in τ :

SI (ω) =

∫

|I (~q)|2 Pω (~q) d~q, (A10)

where Pω (~q) is the Fourier transform in τ of Pτ (~q) asdefined by Eq. (A9):

Pω (~q) =1

π

Dq2

ω2 + D2q4. (A11)

Now, we explicitly give the intensity distribution I (~x)within the scattering volume:

I (x, y, z) = I0χ[−L/2,L/2] (z) e−x2 + y2

2R2 , (A12)

where L is the length of the portion of the beam weobserve, R is the radius of the main beam at e−1/2 andI0 is the intensity at the center of the main beam. TheFourier transform is:

I (qx, qy, qz) =4πI0

QLQ2R

sin (qz/QL)

qz/QLe−

q2x + q2

y

2Q2R , (A13)

where QL = 2/L e QR = 1/R. By inserting the explicitexpressions of Eq. (A9) and Eq. (A13) in Eq. (A10):

SI (ω) =16πI2

0

Q2LQ4

R

∫

sin2 (qz/QL)

(qz/QL)2 e

−q2x + q2

y

Q2R × (A14)

D(

q2x + q2

y + q2z

)

ω2 + D2(

q2x + q2

y + q2z

)2 dqxdqydqz .

Now, we impose that ω ≫ DQ2R and ω ≫ DQ2

L. Thismeans that ω corresponds to times much shorter thanthe time needed by a particle to travel across the mainbeam. Under this condition, we can neglect qx and qy atthe denominator, since the gaussian part in the integrandimposes |qx| . QR and |qy| . QR; then, we integrate overqx and qy:

SI (ω) =16πI2

0

Q2LQ2

R

∫

sin2 (qz/QL)

(qz/QL)2D

(

q2z + 2Q2

R

)

ω2 + D2q4z

dqz.

(A15)The sin2 (qz/QL) has fast oscillations, and averages to1/2 for qz ≫ QL:

SI (ω) =8πI2

0

Q2R

∫

Q2L

q2z + Q2

L

D(

q2z + 2Q2

R

)

ω2 + D2q4z

dqz . (A16)

The leading term in the integral is the q−2z coming from

the sinc function. This means that the behaviour of themeasured intensity is mainly due to particles crossing thesharp edges of the slits, moving along the main beam,while the movements inside and outside the beam can beneglected.

Since QR and QL are of the same order of magnitude:

SI (ω) = 8πI20D

∫

1

ω2 + D2q4z

dqz . (A17)

The integral converges; by substituting qz = x√

ω:

SI (ω) ∝ ω−3/2. (A18)

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