Design of Ultraspherical Window Functions with Prescribed Spectral Characteristics
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Design of Ultraspherical Window Functions with Prescribed Spectral Characteristics Stuart W. A. Bergen Department of Electrical and Computer Engineering, University of Victoria, P.O. Box 3055 STN CSC, Victoria, BC, Canada V8W 3P6 Email: [email protected]
Andreas Antoniou Department of Electrical and Computer Engineering, University of Victoria, P.O. Box 3055 STN CSC, Victoria, BC, Canada V8W 3P6 Email: [email protected] Received 7 April 2003; Revised 17 January 2004; Recommended for Publication by Hideaki Sakai A method for the design of ultraspherical window functions that achieves prescribed spectral characteristics is proposed. The method comprises a collection of techniques that can be used to determine the three independent parameters of the ultraspherical window such that a specified ripple ratio and main-lobe width or null-to-null width along with a user-defined side-lobe pattern can be achieved. Other known two-parameter windows can achieve a specified ripple ratio and main-lobe width; however, their side-lobe pattern cannot be controlled as in the proposed method. A comparison with other windows has shown that a difference in performance exists between the ultraspherical and Kaiser windows, which depends critically on the required specifications. The paper also highlights some applications of the proposed method in the areas of digital beamforming and image processing. Keywords and phrases: window functions, ultraspherical window, beamforming, image processing, digital filters.
1.
INTRODUCTION
Windows are time-domain weighting functions that are used to reduce Gibbs’ oscillations resulting from the truncation of a Fourier series. Their roots date back over one-hundred years to Fejer’s averaging technique for a truncated Fourier series and they are employed in a variety of traditional signal processing applications including power spectral estimation, beamforming, and digital filter design. Despite their maturity, windows functions (or windows for short) continue to find new roles in the applications of today. Very recently, windows have been used to facilitate the detection of irregular and abnormal heartbeat patterns in patients in electrocardiograms [1, 2]. Medical imaging systems, such as the ultrasound, have also shown enhanced performance when windows are used to improve the contrast resolution of the system [3]. Windows have also been employed to aid in the classification of cosmic data [4, 5] and to improve the reliability of weather prediction models [6]. With such a large number of applications available for windows that span a variety of disciplines, general methods that can be used to design windows with arbitrary characteristics are especially useful. Windows can be categorized as fixed or adjustable [7]. Fixed windows have only one independent parameter,
namely, the window length which controls the main-lobe width. Adjustable windows have two or more independent parameters, namely, the window length, as in fixed windows, and one or more additional parameters that can control other window
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