#### Title

Improving the efficiency and accuracy of spectral analysis with applications to harmonic analysis

#### Year

1995

#### Degree Name

Doctor of Philosophy

#### Department

Department of Electrical and Computer Engineering

#### Recommended Citation

Xi, Jiangtao, Improving the efficiency and accuracy of spectral analysis with applications to harmonic analysis, Doctor of Philosophy thesis, Department of Electrical and Computer Engineering, University of Wollongong, 1995. https://ro.uow.edu.au/theses/1344

#### Abstract

This thesis deals with spectral analysis. A number of new techniques are proposed which improve the computational efficiency of discrete orthogonal transform algorithms, and the accuracy of spectral analysis with applications to harmonic signal analysis.

The techniques for computing discrete orthogonal transforms using adaptive filtering are systematically investigated. Firsdy, a general relationship between orthogonal transforms and adaptive filtering is established, which sets the foundation for these techniques. Secondly, the issue of computing block-based discrete orthogonal transforms using the adaptive Least Mean Square (LMS) algorithm is examined. Sufficient conditions for implementing block LMS-based discrete orthogonal transforms are proposed, and the performance of the techniques for computing block LMS-based discrete Walsh transforms and discrete cosine transforms is analysed. Thirdly, the thesis proposes LMS-based techniques for computing running orthogonal transforms, including running discrete Hartley transform, running discrete Cosine and Sine transforms, as well as running discrete W transforms. Finally, the thesis examines the possibility of orthogonal analysis using other adaptive processing algorithms. It is shown that the sample matrix inversion (SMI) algorithm can be used to compute all the discrete orthogonal transforms, while the adaptive Howells-Applebaum loop can be used to implement a spectral analyser for continuous signals.

The running computation of discrete orthogonal transforms based on their shift properties is studied in detail. A number of discrete orthogonal transforms, including the discrete Hartley transform, the discrete cosine and sine transforms, and the discrete W transforms are considered. The shift properties of these transforms are developed, which are in effect recursive equations that connect the previous and updated transform coefficients. Both the first order shift properties and the second order shift properties are proposed for these transforms. As expected the first order shift properties are in the form of first order difference equations. These first order difference equations involve two transform coefficients of different transforms (for example, a discrete cosine transform and its corresponding discrete sine transform). This is a source of extra computational burden. For some transforms such as the discrete Hartley transform and the discrete W transform, this extra computational burden can be eliminated by using the reverse symmetrical properties of the transform coefficients. However, for the discrete cosine and sine transforms, the computation associated with the first order shift properties is not very efficient. The second order shift properties (that is, second order difference equations) are proposed, which can independently update transform coefficients thus reducing the computational burden. It is shown that for the discrete cosine and sine transforms, the computational burden associated with second order shift properties can be considerably reduced in comparison to the first order shift properties.

A time domain interpolation pre-processing algorithm is proposed in an effort to reduce leakage effects associated with the DFT analysis of periodic signals. The leakage effect refers to the spreading of energy from one frequency bin into adjacent ones. To avoid leakage, the sampling frequency should be an integer multiple of the signal frequency. The basic idea of the proposed time domain interpolation pre-processing algorithm is to modify the actual samples towards an ideal sample sequence whose sampling frequency is an integer multiple of the signal frequency. The algorithm is based on a first order approximation of the Taylor's series. It is shown that the proposed algorithm can reduce quite significantly both the DFT leakage and the truncation error associated with digital wattmeter power measurement.

Finally, frequency estimation techniques based on adaptive IIR notch filtering are considered. In order to improve the steady state error performance of existing techniques, a block-gradient based adaptive algorithm is proposed for adaptive IIR filtering. The input signal sequence is arranged into data blocks and the filter coefficients are kept constant within each block. The gradients are evaluated for each complete block of data and the filter coefficients are updated on a block by block basis. Application of the proposed block gradient algorithm to the problem of sinusoidal frequency estimation is studied. It reveals that the proposed algorithm is characterised by a lower steady state error as well as reduced computational complexity.