Analysis of exponentially decaying pulse signals and weak unsteady signals using statistical approach

Abstract

Using statistical test in engineering is becoming common in these few years. In this thesis, several statistical techniques are investigated to solve acoustical engineering problems. A stochastic volatility model incorporated the exponential power distributions and Student - t distributions are adopted in this thesis to analyze exponentially decay pulses in the presence of background noises of various magnitudes. It is found that the present stochastic volatility model can retrieve the instant of the pulse initiation and the decay constant within engineering tolerance even when the noise is slightly stronger that the pulse amplitude. The results suggest that both these distributions can give accurate recovery of the instants when the abrupt changes take place if the background noise level is lower than that of the changes by 3dB. They also indicate that the exponential-power distribution is more useful when the signal-to-noise ratio falls below OdB. The results are compared with those obtained by the conventional short-time Fourier transform and its performance is considerably better than that of the latter when the frequency of the decay pulse fluctuates. To recover the initialization of an exponentially growing wave embedded inside a stationary background noise is very important especially in building services engineering where the early detection of very small alien signal is crucial to the smooth operation of machines. A parameter derived from two statistical tests, namely the Jarque-Bera and the D'Agostino's tests, which are used for checking data normality, is introduced. It is found that the newly derived parameter is very sensitive to the change incurred by the wave to the background noise statistics and is very helpful in locating the instant of the wave initialization even when the signal-to-noise ratio drops to -30dB. The corresponding accuracy of the recovery can be as low as 3 time steps. A simple numerical function together with the Fourier Transform analysis in the detection of very weak sinusoidal signals embedded in a non-stationary random broadband background noise is proposed in the present study. Its performance is studied through the use of two numerical examples. It is found that the present method enables good recovery of the sinusoidal signals and the instants of their initiations even when the signal-to-noise ratio is down to -17dB.