abstract: A signal with time-varying frequency content can often be expressed more clearly using a time-frequency representation (TFR), which maps the signal into a two-dimensional function of time and frequency, similar to musical notation. The thesis reviews one of the most commonly used TFRs, the Wigner distribution (WD), and discusses its application in Fourier optics: it is shown that the WD is analogous to the spectral dispersion that results from a diffraction grating, and time and frequency are similarly analogous to a one dimensional spatial coordinate and wavenumber. The grating is compared with a simple polychromator, which is a bank of optical filters. Another well-known TFR is the short time Fourier transform (STFT). Its discrete version can be shown to be equivalent to a filter bank, an array of bandpass filters that enable localized processing of the analysis signals in different sub-bands. This work proposes a signal-adaptive method of generating TFRs. In order to minimize distortion in analyzing a signal, the method modifies the filter bank to consist of non-overlapping rectangular bandpass filters generated using the Butterworth filter design process. The information contained in the resulting TFR can be used to reconstruct the signal, and perfect reconstruction techniques involving quadrature mirror filter banks are compared with a simple Fourier synthesis sum. The optimal filter parameters of the rectangular filters are selected adaptively by minimizing the mean-squared error (MSE) from a pseudo-reconstructed version of the analysis signal. The reconstruction MSE is proposed as an error metric for characterizing TFRs; a practical measure of the error requires normalization and cross correlation with the analysis signal. Simulations were performed to demonstrate the the effectiveness of the new adaptive TFR and its relation to swept-tuned spectrum analyzers. / Dissertation/Thesis / M.S. Electrical Engineering 2012

Methods for digital image compression have been the subject of much study over the past decade. Advances in wavelet transforms and quantization methods have produced algorithms capable of surpassing the existing image compression standards like the Joint Photographic Experts Group (JPEG) algorithm. For best performance in image compression, wavelet transforms require filters that combine a number of desirable properties, such as orthogonality and symmetry. However, the design possibilities for wavelets are limited because they cannot simultaneously possess all of these desirable properties. The relatively new field of multiwavelets shows promise in removing some of the limitations of wavelets. Multiwavelets offer more design options and hence can combine all desirable transform features. The few previously published results of multiwavelet-based image compression have mostly fallen short of the performance enjoyed by the current wavelet algorithms. This thesis presents new multiwavelet transform methods and measurements that verify the potential benefits of multiwavelets. Using a zerotree quantization scheme modified to better match the unique decomposition properties of multiwavelets, it is shown that the latest multiwavelet filters can give performance equal to, or in many cases superior to, the current wavelet filters. The performance of multiwavelet packets is also explored for the first time and is shown to be competitive to that of wavelet packets in some cases. The wavelet and multiwavelet filter banks are tested on a much wider range of images than in the usual literature, providing a better analysis of the benefits and drawbacks of each. NOTE: (03/2007) An updated copy of this ETD was added after there were patron reports of problems with the file. / Master of Science

<p>The initial goal of this report was to implement and compare cosine modulated filter banks. Because of time limitations, focus shifted towards the implementation. Filter banks and multirate systems are important in a vast range of signal processing systems. When implementing a design, there are several considerations to be taken into account. Some examples are word length, number systems and type of components. The filter banks were implemented using a custom made software, especially designed to generate configurable gate level code. The generated code was then synthesized and the results were compared. Some of the results were a bit curious. For example, considerable effort was put into implementing graph multipliers, as these were expected to be smaller and faster than their CSDC (Canonic Signed Digit Code) counterparts. However, with one exception, they turned out to generate larger designs. Another conclusion drawn is that the choice of FPGA is important. There are several things left to investigate, though. For example, a more thorough comparison between CSDC and graph multipliers should be carried out, and other DCT (Discrete Cosine Transform) implementations should be investigated.</p>

The initial goal of this report was to implement and compare cosine modulated filter banks. Because of time limitations, focus shifted towards the implementation. Filter banks and multirate systems are important in a vast range of signal processing systems. When implementing a design, there are several considerations to be taken into account. Some examples are word length, number systems and type of components. The filter banks were implemented using a custom made software, especially designed to generate configurable gate level code. The generated code was then synthesized and the results were compared. Some of the results were a bit curious. For example, considerable effort was put into implementing graph multipliers, as these were expected to be smaller and faster than their CSDC (Canonic Signed Digit Code) counterparts. However, with one exception, they turned out to generate larger designs. Another conclusion drawn is that the choice of FPGA is important. There are several things left to investigate, though. For example, a more thorough comparison between CSDC and graph multipliers should be carried out, and other DCT (Discrete Cosine Transform) implementations should be investigated.

A new two-dimensional transform is derived and implemented that is able to discriminate with respect to angular and radial frequency. This octave-band directional filter bank (OBDFB) is maximally decimated, has a separable polyphase implmentation, provides perfect reconstruction, and can be implemented in a tree structure allowing for a somewhat arbitrary number of angular and radial divisions. This decomposition is based on the directional filter bank (DFB) and is compared to other transforms with similar properties. Additionally, the OBDFB is used in three applications. Texture segmentation results are provided with comparisons to both decimated and undecimated transforms. With hyperspectral data, the OBDFB is used to increase classification accuracy using texture augmentation and likelihood score combination. Finally, ultrasound despeckling is addressed with respect to real-time implementations, and subjective test results are presented. A non-uniform two-dimensional transform is also designed that is a modified version of the OBDFB. It is rationally sampled and maximally decimated, but it provides both angular and radial frequency passbands from the initial stage instead of making separate divisions like the OBDFB. It also does not create subband boundaries on the principal frequency axes and allows for further decomposition as well.

The work is focused on multitone modulation implemented by filter banks, which are alternative modulations to DMT and OFDM modulations. The paper describes the two proposed modulation schemes, non-overlapped FMT and half-overlapped FMT. This modulations utilize the benefits of implementing FFT algorithm in conjunction with a bank of polyphase filters that are derived from the prototype filter. To design the prototype filter a Matlab tool is developed. For both modulations the simulation models were created for xDSL systems and for wireless transmission channel. Non-overlapped FMT modulation proved to be a suitable modulation scheme for channels with narrowband interference and sharp edges in the transmission function of the channel. It was allowed by perfect separation of each subchannel and with preserving the orthogonality between subchannels using the FFT algorithm and utilization of appropriate filters. Half-overlapped FMT modulation allows as well as the DMT modulation optimal utilization of provided transmittion band, because the power spectral density of the signal is flat. But it is not suitable for transmission channels with roughly shaped by changes in the transmission function and narrowband interference. Optimization implementations were designed using parallel processing on the DSP Texas Instruments TMS320C67x series. It thus achieved almost ten-fold reduction in computational complexity. In FMT modulation both options ensure increased signal to noise ratio and thus allow the use of constellation diagrams with a large number of signal elements with the same transmit power and also higher transmission speed.

State of the art filter bank structures achieve practically perfect reconstruction with very high computational efficiency. However, the increase in computational requirements due to the need to process increasingly wider band signals is paramount. New filter bank structures that provide extra information about a signal while achieving the same level of required efficiency, and perfect reconstruction properties, need to be developed. In this work a new filter bank structure, the interpolated FIR (IFIR) filter bank is developed. Such a structure combines the concepts of filter banks, and interpolated FIR filters. The filter design procedures for the IFIR filter bank are developed and explained. The resulting structure was compared with the non-maximally-decimated filter bank (NMDFB), achieving the same performance in terms of the number of multiplications required per sample and the overall distortion introduced by the system, when operating with Nyquist prototype filters. In addition, the IFIR filter is tested in both simulated and real communication environments. Performance, in terms of bit-error-rate, was found to not be degraded significantly when using the IFIR filter bank system for transmission and reception of QPSK symbols. / Master of Science

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In this thesis we develop algorithms for the processing of textures and images using a ladder-based biorthogonal directional filter bank (DFB). This work is based on the DFB originally proposed by Bamberger and Smith. First we present a novel implementation of this filter bank using ladder structures. This new DFB provides non-trivial FIR perfect reconstruction systems which are computationally very efficient. Furthermore we address the lack of shift-invariance in the DFB by presenting a novel undecimated DFB that preserves the computational simplicity of its maximally decimated counterpart. Finally, we study the use of the DFB in combination with pyramidal structures to form polar-separable image decompositions. Using the proposed filter banks we develop and evaluate algorithms for texture classification, segmentation and synthesis. We perform a comparative study with other image representations and find that the DFB provides some of the best results reported on the data sets used. Using the proposed directional pyramids we adapt wavelet thresholding algorithms. We find that our decompositions provide better edge and contour preservation than the best results reported using the undecimated discrete wavelet transform. Finally, we apply the developed algorithms to the analysis and processing of synthetic aperture radar (SAR) imagery. SAR image analysis is impaired by the presence of speckle noise. Our first objective will be to study the removal of speckle to enhance the visual quality of the image. Additionally, we implement land cover segmentation and classification algorithms taking advantage of the textural characteristics of SAR images. Finally, we propose a model-based SAR image compression algorithm in which the speckle component is separated from the structural features of a scene. The speckle component is captured with a texture model and the scene component is coded with a wavelet coder at very low bit rates. The resulting decompressed images have a better perceptual quality than SAR images compressed without removing speckle.