Real-Time Wavelet-transform spectrum analyzer for the investigation of   1/f^\alpha noise
  A wavelet transform spectrum analyzer operating in real time within the frequency range 3X10^(-5) - 1.3X10^5 Hz 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-stationary signals dominated by large amplitude events fairly well localized in time, thus providing the natural tool to analyze processes characterized by 1/f^alpha power spectrum. The parallel architecture of the DSP allows the real-time processing of the wavelet transform of the signal sampled at 0.3MHz. The bandwidth is about 220dB, almost ten decades. The power spectrum of the scattered intensity is processed in real time from the mean square value of the wavelet coefficients within each frequency band. The performances of the spectrum analyzer have been investigated by performing Dynamic Light Scattering experiments on colloidal suspensions and by comparing the measured spectra with the correlation functions data obtained with a traditional multi tau correlator. In order to asses the potentialities of the spectrum analyzer in the investigation of processes involving a wide range of timescales, we have performed measurements on a model system where fluctuations in the scattered intensities are generated by the number fluctuations in a dilute colloidal suspension illuminated by a wide beam. This system is characterized by a power-law spectrum with exponent -3/2 in the scattered intensity fluctuations. The spectrum analyzer allows to recover the power spectrum with a dynamic range spanning about 8 decades. The advantages of wavelet analysis versus correlation analysis in the investigation of processes characterized by a wide distribution of time scales and non-stationary processes are briefly discussed. 
Doriano Brogioli and Alberto Vailati