Engineering Applications of Prolate Spheroidal Wave Functions and Sequences and Legendre Polynomials: Filtering and Beamforming in 1D and 2D

Abstract

This dissertation uses the Prolate Spheroidal Wave Functions (PSWFs) and the Legendre Polynomials for several applications in electrical engineering ranging from signal filtering, image denoising, to antenna beamforming. The PSWFs form a basis in which each function is both time limited and energy concentrated within a frequency band and with decreasing energy value. This property of dual concentration in time and frequency domain is very relevant in the field of signal processing because physical signals are band limited. The energy concentration can be set through parameter c, which restricts the time duration or length and bandwidth. Different approaches are used to generate the basis set in its continuous and discrete versions, respectively. For the continuous set, a generation method is based on the combination of Legendre Polynomials.

In the filtering application, we present results for practical design of a digital filter based on the window method using the first order PSWF. This method is suboptimal but it is computationally simple. The design equations allow the user to obtain a filter with specified cutoff frequency and sidelobe attenuation in the frequency domain for a specified filter length. The characteristics of the filter are compared with similar filters generated using the Chebyshev and Kaiser windows, respectively, calculating the merit of each case. The filter characteristics obtained with the PSWF window are closer to the ones obtained with optimal methods (such as the Remez method), than when the other two windows are used.

This filter approach is extended to 2D to be used for image filtering. We present the mathematical generation of the filter and results of application over an image to enhance the signal-to-noise level.

We also present a different type of filter for the denoising of images using the property that the PSWFs constitute a basis for band limited functions to build a basis for a 2D space. We focus on images obtained with Magnetic Resonance Imaging technology, which are affected by non-additive Gaussian noise. This technology is used by medical doctors to assess various conditions in the organs of their patients and also in food science to measure the water content and other physical parameters. The contrast or other distortions in the image may be biased, depending on Signal-to-Noise Ratio (SNR) that may affect the intensity of the pixels, which may limit the accuracy of the diagnosis or the measurement of the physical parameter. Post-processing methods are important to image quality without increasing the time of image acquisition, which poses an additional benefit for the patient. In our approach, we use the PSWF basis set to represent the image and we filter the coefficients in order to reduce the distortion measured through the Peak Signal-to-Noise ratio (PSNR), Structural Similarity Index Measure (SSIM), and Quality Index Local Variance (QILV) merit parameters using a soft threshold cutoff strategy applied to each single coefficient of the representation. Our threshold strategy is based on an empirical study of the variance of the PSWF coefficients representation yielding denoising results that are similar to results obtained with the Discrete Cosine Transform for standard images and does better than DCT for an image acquired by Magnetic Resonance Imaging technology in the case of a fixed c < 2 parameter. However, results are better than DCT if the c parameter is chosen adaptive to the noisy image block characteristics. Different approaches based on sparsity measures have been attempted in order to find a good guess.

The application of PSWFs to antenna array beamforming is based on the apodization of the excitation by a window that is based, as in the case of the previous application of filtering, on the value of the first order PSWF window. This application is mathematically similar to digital filtering. The parameters of merits considered are the beamwidth and the directivity of the array beam pattern. Using the PSWF window as input excitation, the far-field pattern are in between the ones obtained by Kaiser and Chebyshev input excitations, but the far-field pattern has the maximum ratio between the power radiated in the main lobe and the power radiated in the observed pattern. The importance of this feature is that there is a better use of the radiated energy in the desired direction.

A very different beamforming strategy is presented using the Legendre Polynomials to obtain the coordinates of a linear curved array with broadband properties. The algorithm calculates the coordinates of each individual antenna in the array subject to no exceed a maximum total length, so that the broadside far-field pattern value is as constant (or broadband) as possible within a given frequency range of operation selected by the user as well.