DIGITAL PROCESSING OF LOW FREQUENCY RANDOM SIGNALS USING ORTHOGONAL FUNCTIONS

Abstract

Digital signal processing (DSP), emerged as a growing and dynamic discipline in mid 1960s. With the phenomenal progress in computer technology during the past two decades, DSP has become a major tool for signal/data processing in diverse and wide variety of fields. The main advantages of DSP are its higher speed, flexibilty, and a desired level of accuracy and resolution. Introduction of computationally efficient algorithms like fast Fourier transform (FFT), fast Walsh and fast Hartley transforms has revolutionised the whole gamut of signal processing. The main objective of this thesis is to develop need based and dedicated algorithms for analyses and processing of low frequency random signals. The signals selected for analyses are seismic (i.e earthquake) and electroencephalogram (EEG). Both of these are non-stationary signals and have been treated as wide-sense stationary for analysis and processing. Orthogonal functions have been adopted for use in the analysis of signals because orthogonal sets of functions can be made to synthesize completely any time-function or time-series to a required degree of accuracy. Apart from the Fourier transforms, highly economical (computer time and memory wise) functions and transforms like Walsh, Haar and Hartley have been employed for the analyses. Walsh and Haar functions form an ordered set of rectangular wave-forms which take only two amplitude values and are thus dyadic in nature. Hence, these are eminently suited to computer working. Moreover, Walsh transforms involve only additions and subtractions and no multiplications, making them really fast. Hartley transform depends on the definition of a cas function (cas 9 = cos 6 + sin 8), doing away with the bothersome imaginary quantity j. So Hartley transforms are highly suitable for analysis of real data. Before taking up the processing of raw data it is desirable to pre-filter the data for the purpose of band-limiting. This also provides anti-aliasing character to the filtered signal. To this end finite impulse response (FIR) filters have been adopted due to their inherent stability and linear phase properties. These FIR filters have been designed for both, the seismic signal and the electroencephalogram, by different methods and the best ones are selected for actual filtering of the signals. Seismic or earthquake signals are shock waves (i.e., P and S waves ) and are detected by seismometers at the recording stations and a record of seismic waveform is obtained. The seismic signals vary in period-content from milliseconds to hours with corresponding displacement amplitudes varying from nanometres to metres. From frequency point of view seismic signals can be classified into teleseismic, regional, local and micro earthquake signals. Only regional earthquake signals, covering a frequency range of 1 to 4 Hz have been considered in this thesis. Here the aim is to detect the P-wave arrival, hence, a fast algorithm to detect this event from a single trace has been developed. This uses the conventional short-term, long-term averages to eliminate noise but statistics of the time-series of the seismic signal have been exploited for event detection. The algorithm has been designed only for regional earthquakes i.e., those having a frequency of 1 to 4 Hz. The concept of threshold filtering has been used to isolate the desired frequency band of 1 to 4 Hz. The algorithm has been run with both simulated data and actual seismic digital data. The relative efficacy of different transforms, namely, Walsh, Haar and Hartley for event detection using the developed fast algorithm has been worked out. The success rate of detection of an event as defined above, by this algorithm using Hartley transform is more than 95%. EEG is the time-varying voltage observed between two electrodes placed at appropriate points on the human scalp. The magnitudes of these signals vary from 20 to 100 micro volts. The predominant and significant frequency range is from 0 to 30 Hz sub-divided into four main categories i.e., delta (0 to 4 Hz), theta (4 to 8 Hz), alpha ( 8 to 13 Hz), and beta ( 13 to 30 Hz). The work presented in this thesis deals with both, the simulated EEG signal and the actual EEG. Epoch-wise classification of the EEG signal has been carried out. Each epoch considered is of 1.28s duration. Classification has been made from the point of view of frequency band under which a particular epoch falls i.e., whether it is alpha, beta, theta or delta. A particular EEG trace, generally, has a train of one type of epochs, signifying the predominant frequency content of the EEG signal. Again Walsh and Hartley transforms, along with threshold filtering have been employed for this classification. The simplified method developed gives above 90% accurate results of epoch classification. Also auto-correlation function and power spectrum density for the EEG signals have been computed using developed software. These confirmed the random nature of the signal as well as the low frequency characteristics. All the above have, finally, been incorporated into an EEG classification package. All the algorithms are written in Fortran IV/77 and have been run on DEC-2050 System and PC/AT. These algorithms are amenable to implementation for on-line/real-time processing.