Efficient Spectral Analysis of Time Series

Abstract

This work focuses on efficient, joint time-frequency analysis of time series data. Joint time-frequency analysis is based on the sliding window. There are two major contributions of this thesis. Firstly, we haveThis work focuses on efficient, joint time-frequency analysis of time series data. Joint time-frequency analysis is based on the sliding window. There are two major contributions of this thesis. Firstly, we have introduced a notion of “aggregate spectrogram (AS)” which is a unimodal distribution at each time instant. The AS is extremely useful and computationally efficient when we are interested in a few spectral features and not the entire spectrum. Properties/characteristics of the AS have been listed. A para- metric method, based on a second order autoregressive model of the signal, for the construction of the AS, has been described. Of all the existing spectral estimation tools, the AS has the least computational complexity. Based on the AS, instan- taneous frequency estimation for multicomponent signals with equal amplitudes has been achieved. The AS does not require Goertzel filters in dual tone multi frequency detection applications. The AS finds many potential application. A few examples are voice activity detection, edge detection, motion vector estimation etc. Secondly, the problem of estimating the instantaneous frequency and band- width for multicomponent signals with time varying amplitudes has been solved by employing a new peak detection algorithm. The algorithm has been shown to outperform existing algorithms when the frequencies and amplitudes of the multi- component noisy signals are time-varying. Other contributions of the thesis include: low computational cost algorithms for the sliding discrete Fourier transform, and algorithms for its extension to spectral interpolation through zero padding and window padding. A low cost, optimized iii split-radix FFT architecture for zero-padded signals is also proposed. The Wiener-Khintchine theorem (WKT) yields better spectral estimates of Gaussian signals as compared to the discrete Fourier transform (DFT). Higher order spectra find utility in case of additive colored noise or the signals are non- Gaussian. Due to high computational complexities, the WKT and higher order spectra are avoided in the sliding window based spectral analysis. We have devel- oped recursive forms of the WKT, bispectrum and trispectrum whose computa- tional complexities have reduced to linear, quadratic and cubic orders, respectively introduced a notion of “aggregate spectrogram (AS)” which is a unimodal distribution at each time instant. The AS is extremely useful and computationally efficient when we are interested in a few spectral features and not the entire spectrum. Properties/characteristics of the AS have been listed. A para- metric method, based on a second order autoregressive model of the signal, for the construction of the AS, has been described. Of all the existing spectral estimation tools, the AS has the least computational complexity. Based on the AS, instan- taneous frequency estimation for multicomponent signals with equal amplitudes has been achieved. The AS does not require Goertzel filters in dual tone multi frequency detection applications. The AS finds many potential application. A few examples are voice activity detection, edge detection, motion vector estimation etc. Secondly, the problem of estimating the instantaneous frequency and band- width for multicomponent signals with time varying amplitudes has been solved by employing a new peak detection algorithm. The algorithm has been shown to outperform existing algorithms when the frequencies and amplitudes of the multi- component noisy signals are time-varying. Other contributions of the thesis include: low computational cost algorithms for the sliding discrete Fourier transform, and algorithms for its extension to spectral interpolation through zero padding and window padding. A low cost, optimized split-radix FFT architecture for zero-padded signals is also proposed. The Wiener-Khintchine theorem (WKT) yields better spectral estimates of Gaussian signals as compared to the discrete Fourier transform (DFT). Higher order spectra find utility in case of additive colored noise or the signals are non- Gaussian. Due to high computational complexities, the WKT and higher order spectra are avoided in the sliding window based spectral analysis. We have devel- oped recursive forms of the WKT, bispectrum and trispectrum whose computa- tional complexities have reduced to linear, quadratic and cubic orders, respectively