Design of the Ultraspherical Window Function and Its Applications

Window functions are used to reduce Gibbs’ oscillations resulting from the truncation of a Fourier series and they are employed in a variety of signal processing applications including power spectral estimation, beamforming, and digital filter design. In this dissertation, the application of window functions based on the ultraspherical window is explored. First, two methods for evaluating the coefficients of the ultraspherical window are presented. An efficient formulation for one of the methods is proposed which requires significantly less computation than that required for the Kaiser window. Next, a method for selecting the three independent parameters of the ultraspherical window so as to achieve prescribed spectral characteristics is proposed. The method can be used to achieve a specified ripple ratio and either a main-lobe width or null-to-null width along with a user-defined side-lobe pattern. The side-lobe pattern in other known twoparameter windows cannot be controlled as in the proposed method. Applications of the proposed method in digital beamforming and image processing are explored. A closed-form method for the design of nonrecursive digital filters using the ultraspherical window is developed. The method can be used to design lowpass, highpass, bandpass, and bandstop filters as well as digital differentiators and Hilbert transformers that would satisfy prescribed specifications. The method yields lower-order filters relative to designs obtained with other windows such as the Kaiser, Saramäki, and Dolph-Chebyshev windows. Alternatively, for a fixed filter length, the ultraspherical window can provide reduced passband ripple and increased stopband attenuation. In addition, it entails reduced computational complexity which renders it suitable for applications where the design must be carried out in real or quasi-real time. An efficient closed-form method for the design of -channel cosine-modulated filter banks using the ultraspherical window that would yield prescribed stopband attenuation in iii the subbands and channel overlap is proposed. On the average, the method yields prototype filters with the shortest length and least design computational complexity while the Kaiser window yields filter banks with the smallest reconstruction error. When compared with other methods, the proposed method yields filter banks that have prototype filters of the same length, increased average maximum amplitude error, and the same average aliasing error and average total aliasing error. The dissertation also considers the application of the ultraspherical window along with the short-time discrete Fourier transform method for gene identification based on the well known period-three property. The ultraspherical window is employed to suppress spectral noise originating from noncoding regions in the DNA sequence. A method for tailoring the independent parameters of the ultraspherical window for the identification of a particular gene is proposed. Comparisons show that the ultraspherical, Kaiser, and Saramäki windows yield values for a gene-identification measure that are approximately the same, and that they are 13.72% better than that achieved when using the rectangular window. iv Examining Committee: Dr. A. Antoniou, Supervisor (Dept. of Elec. and Comp. Eng.) Dr. P. Agathoklis, Department Member (Dept. of Elec. and Comp. Eng.) Dr. D. Shpak, Department Member (Dept. of Elec. and Comp. Eng.) Dr. D. Olesky, Outside Member (Dept. of Comp. Sci.) Dr. M. O. Ahmad, External Examiner (Dept. of Elec. and Comp. Eng., Concordia Univ.)