We have presented a first application of the wavelet transform in searching for monopoles in previous years reports. As we have seen, the wavelets arise in time-frequency localization problems by offering new bases for analyzing functions. In analogy to the Fourier transform (whose coefficients represent the function over its entire support), the wavelet transform takes the inner product of a function f(t) with a doubly indexed family of functions psi(a,b)=|a|^{-1/2}psi[(t-b)/a] (the ``wavelets functions'') which are obtained by dilations (by a) and translations (by b) of the ``mother wavelet'' psi. What makes the wavelet transform a powerful tool for analyzing functions is its ability to adjust the resolution of the analyzing functions in order to match their scale (frequency): small values of a yield high frequency spectral information and in order to give better accuracy the resolution (time interval) is finer; on the other hand, for high values of a which yield low-frequency information, the resolution becomes coarser. Among the many choice of wavelet bases, we have used one of the simplest and actually the oldest example of a discrete wavelet basis to process MACRO waveforms: the so-called Haar [10] basis. The Haar ``mother wavelet'' is defined by:
while the Haar ``wavelet family'' is given by:
Besides this one, wavelet analyses based on Daubechies' D4,D8 and D16 [8, 9] wavelet bases were also performed. The advantages of these bases over the Haar one is currently under study. As with our wavelet-based monopole analysis of the six-month run [6, 7] data, we have used similar features of a waveform's wavelet transform in order to define our software monopole trigger. A key issue is obtaining the wavelet transform of a candidate waveform is defining its size, i.e., the number of samples that will be fed into the algorithm at a time. In this preliminary work we have defined our single wavelet ``frame'' to be composed of 4096 (2^{12}) samples. Given that each sample corresponds to 5nsec, every frame reflects of PMT history.
Given a waveform frame composed of 4096 samples we proceed as follows. At each resolution scale m -with the exception of the last one- we detect and record the two absolute highest value h_m1, h_m2, and their locations (m,n1), (m,n2) , of the Haar coefficients |W(m,n1)f|,|W(m,n2)f|. The maximum h_i1 among all the resolution scales (h_m1) we define as the Haar global maximum while the scale i1 at which this occurs we define as the Haar global focus. In analogy to the Fourier transform, we introduce as a measure of the ``energy content'' of a scale m the quantity (h_m1)^2. The energy contrast is then constructed by taking the difference of the sum of the energy content of the five lower scales minus the sum of the energy content of the five higher scales. We will say that our signal is ``low frequency biased'' if the above quantity is negative or otherwise we will say that it is ``high frequency biased''.
The same quantities for the second maxima are also constructed: the maximum h_i2 among all h_m2 we define as the Haar second maximum while the scale i2 at which this occurs we define as the Haar second focus. We also define the ``secondary energy content'' of a scale m the quantity (h_m2)^2. The secondary energy contrast is then constructed by taking the difference of the sum of the secondary energy content of the five lower scales minus the sum of the secondary energy content of the five higher scales.
Our preliminary monopole candidate filter has been established with the following requirements:
The Haar approach we have just outlined has been studied on Monte Carlo simulated monopole data covering all expected range of velocities and ionization yields. As we can see in figure 4, the algorithm shows efficiency above 90% for the slowest monopole we expect to record. Let us mention that theefficiency is a steeply varying function at that monopole velocity and lightyield regimes. Moreover, we still have a pleathora of wavelet characteristics that we have not employed in the software trigger definition. This is work in progress that we expect to lead to higher efficiencies for even less ionizing monopole signatures via a combination of the full Haar features with other wavelet bases.
Figure 4:
Main features of the Haar transform of 1000 simulated monopole
waveforms generated according to the Monte Carlo described in the
previous section.
The pulse duration was 6600nsec within
which an average of 100 photoelectrons was generated.
The top two plots show the resolution scale at which the global
Haar maximum occurs together with the corresponding (i.e. the global
max) energy contrast.
The same quantities for the ``secondaries'' is shown in the middle
two plots.
The bottom plot shows the photoelectron distribution for
the events whose waveforms fulfilled the wavelet monopole criteria
(scale>5, energy contrast<100).
The monopole filter as used herein displays an efficiency
at the level of 92%.