Degree Name

Doctor of Philosophy


Department of Electrical and Computer Engineering


This thesis deals with the accurate estimation of phase, amplitude and frequency of sinusoids buried in noise. Several algorithms are proposed to determine these parameters.

A Constrained Notch Fourier Transform (CNFT) is proposed for estimating the Fourier coefficients of noise corrupted harmonic signals given a priori knowledge of the signal frequencies. The proposed method provides bandwidth controlled bandpass filters in contrast to the conventional Notch Fourier Transform (NFT) [Tadokoro and Abe (1987)] and its equivalent real valued Frequency Sampling (FS) structures which utilise fixed bandwidth bandpass filters. New sliding algorithms have been derived for both the NFT and CNFT for the purpose of estimating the Fourier coefficients of the sinusoidal components. A similar algorithm to the CNFT is also proposed for estimating the coefficients of sinusoids at arbitrary known frequencies. The main feature of the modified CNFT is that it uses a second order IIR bandpass filter whose centre frequency parameter is decoupled from the bandwidth parameter. In these techniques, the bandwidth control aspect provides the user with an efficient means of achieving the required resolution as well as reducing spectral leakage. In general, the proposed approach leads to considerable reduction in terms of acquisition time and memory storage.

The sliding CNFT algorithm is extended to the Generalised Frequency Sampling (GFS) filter bank whose parametisation is derived based on the Least Means Square (LMS) spectrum analyser. The GFS filter bank possesses the desired characteristics that its resonant frequencies and nulls are arbitrarily set. This feature is used to effectively reduce the leakage problem. The use of GFS filter bank together with the CNFT algorithm provides faster acquisition time when compared with the conventional FS filter bank. Further, it is computationally more efficient than the direct use of the LMS spectrum analyser.

The merits and demerits of the conventional Goertzel algorithm are evaluated when it is applied for the task of estimating sinusoidal parameters. A sliding Goertzel algorithm is then developed based on parallel second order digital resonators that are tuned at the input spectral lines. This approach exhibits good performance in low Signal to Noise Ratio (SNR) as verified by extensive simulation tests. Further, unlike the modified and conventional Goertzel algorithms which require a complete signal period to accurately compute the phase and amplitude of the input sinusoids, it computes Fourier coefficients in less than one signal period.

The conventional FS structure was modified to obtain a new modular IIR FS filter bank with reduced spectral overlap as well as minimal spectral hole between adjacent bandpass filters. The IIR FS structure together with the self-orthogonalising LMS algorithm is employed for Adaptive Line Enhancer (ALE) applications. The proposed method provides faster convergence than the conventional adaptive FS methods. The Performance characteristics such as the minimum Mean-Squared-Error (MSE), steady-state excess MSE and convergence conditions of the adaptive FS filter bank is analysed. A performance comparison of three adaptive techniques (the ITR FS structure, conventional FS structure and the Tapped Delay Line (TDL)) is carried out to establish the merits of each algorithm.

Finally, the conventional IIR Parallel Adaptive Line Enhancer (PALE) which is comprised of a parallel second order IIR bandpass filter is modified such that the convergence to local minima or saddle points is avoided. Error surface analysis is carried out to establish the convergence behaviour of the proposed technique. It is shown that the convergence speed of the proposed structure is the same as the serial configuration. However, it provides superior performance in terms of reduced distortion in amplitude and phase associated with each of the enhanced sinusoids.



Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.