This thesis deals with design algorithms for filter banks based on optimization. The design specifications consist of the perfect reconstruction and frequency response specifications for finite impulse response (FIR) analysis and synthesis filters. The perfect reconstruction conditions are formulated as a set of linear equations with respect to the analysis filters’ coefficients and the synthesis filters’ coefficients. Five design algorithms are presented. The first three are based on an unconstrained optimization of performance indices, which include the perfect reconstruction error and the error in the frequency specifications. The last two algorithms are formulated as constrained optimization problems with the perfect reconstruction error as the performance index and the frequency specifications as constraints. The performance of the five algorithms is evaluated and compared using six examples; these examples include uniform filter bank, compatible non-uniform filter bank and incompatible non-uniform filter bank designs. The evaluation criteria are based on distortion and aliasing errors, the magnitude response characteristics of analysis and synthesis filters, the computation time required for the optimization, and the convergence of the performance index with respect to the number of iterations. The results show that the five algorithms can achieve almost perfect reconstruction and can meet the frequency response specifications at an acceptable level. In the case of incompatible non-uniform filter banks, the algorithms have challenges to achieve almost perfect reconstruction. / Graduate

The use of the discrete wavelet transform (DWT) for the JPEG2000 image compression standard has sparked interest in the design of fast, efficient hardware implementations of the perfect reconstruction filter bank used for computing the DWT. The accuracy and efficiency with which the filter coefficients are quantized in a multiplierless implementation impacts the image compression and hardware performance of the filter bank. A high precision representation ensures good compression performance, but at the cost of increased hardware resources and processing time. Conversely, lower precision in the filter coefficients results in smaller, faster hardware, but at the cost of poor compression performance. In addition to filter coefficient quantization, the filter bank structure also determines critical hardware properties such as throughput and power consumption. This thesis first investigates filter coefficient quantization strategies and filter bank structures for the hardware implementation of the biorthogonal 9/7 wavelet filters in a traditional convolution-based filter bank. Two new filter bank properties—"no-distortion-mse" and "deviation-at-dc"—are identified as critical to compression performance, and two new "compensating" filter coefficient quantization methods are developed to minimize degradation of these properties. The results indicate that the best performance is obtained by using a cascade form for the filters with coefficients quantized using the "compensating zeros" technique. The hardware properties of this implementation are then improved by developing a cascade polyphase structure that increases throughput and decreases power consumption. Next, this thesis investigates implementations of the lifting structure—an orthogonal structure that is more robust to coefficient quantization than the traditional convolution-based filter bank in computing the DWT. Novel, optimal filter coefficient quantization techniques are developed for a rational and an irrational set of lifting coefficients. The results indicate that the best quantized lifting coefficient set is obtained by starting with the rational coefficient set and using a "lumped scaling" and "gain compensation" technique for coefficient quantization. Finally, the image compression properties and hardware properties of the convolution and lifting based DWT implementations are compared. Although the lifting structure requires fewer computations, the cascaded arrangement of the lifting filters requires significant hardware overhead. Consequently, the results depict that the convolution-based cascade polyphase structure (with "<i>z</i>₁-compensated" coefficients) gives the best performance in terms of image compression performance and hardware metrics like throughput, latency and power consumption. / Master of Science

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

In Chapters 1 and 2, we introduce the definition of interpolating refinable function vectors in dimension one and high dimensions, characterize such interpolating refinable function vectors in terms of their masks, and derive their sum rule structure explicitly. We study biorthogonal refinable function vectors from interpolating refinable function vectors. We also study the symmetry property of an interpolating refinable function vector and characterize a symmetric interpolating refinable function vector in any dimension with respect to certain symmetry group in terms of its mask. Examples of interpolating refinable function vectors with some desirable properties, such as orthogonality, symmetry, compact support, and so on, are constructed according to our characterization results. In Chapters 3 and 4, we turn to the study of general matrix extension problems with symmetry for the construction of orthogonal and biorthogonal multiwavelets. We give characterization theorems and develop step-by-step algorithms for matrix extension with symmetry. To illustrate our results, we apply our algorithms to several examples of interpolating refinable function vectors with orthogonality or biorthogonality obtained in Chapter 1. In Chapter 5, we discuss some possible future research topics on the subjects of matrix extension with symmetry in high dimensions and frequency-based non-stationary tight wavelet frames with directionality. We demonstrate that one can construct a frequency-based tight wavelet frame with symmetry and show that directional analysis can be easily achieved under the framework of tight wavelet frames. Potential applications and research directions of such tight wavelet frames with directionality are discussed. / Applied Mathematics

No description available.

Software radios (SWR) have been proposed for wireless communication systems to enable them to operate according to incompatible wireless communication standards by implementing most analog functions in the digital section on software-reprogrammable hardware. However, this significantly increases the required computations for SWR functionality, mainly because of the digital front-end computationally intensive filtering functions, such as sample rate conversion (SRC), channelization, and equalization. For increasing the computational efficiency of SWR systems, two new SRC methods with better performance than conventional SRC methods are presented. In the first SRC method, we modify the conventional CIC filters to enable them to perform SRC on slightly oversampled signals efficiently. We also describe a SRC method with high efficiency for SRC by factors greater than unity at which SRC in SWR systems may be computationally demanding. This SRC method efficiently increases the sample rate of wideband signals, especially in SWR base station transmitters, by applying Lagrange interpolation for evaluating output samples hierarchically using a low-rate signal that is computed with low cost from the input signal. A new channelizer/synthesizer is also developed for extracting/combining frequency multiplexed channels in SWR transceivers. The efficiency of this channelizer/synthesizer, which uses modulated perfect reconstruction (PR) filter banks, is higher than polyphase filter banks (when applicable) for processing few channels, and significantly higher than discrete filter banks for processing any number of variable-bandwidth channels where polyphase filter banks are inapplicable. Because the available methods for designing modulated PR filter banks are inapplicable due to the required number of subchannels and stopband attenuation of the prototype filters, a new design method for these filter banks is introduced. This method is reliable and significantly faster than the existing methods. Modulated PR filter banks are also considered for implementing a frequency-domain block blind equalizer capable of equalizing SWR signals transmitted though channels with long impulse responses and severe intersymbol interference (ISI). This blind equalizer adapts by using separate sets of weights to correct for the magnitude and phase distortion of the channel. The adaptation of this blind equalizer is significantly more reliable and its computational requirements increase at a lower rate compared to conventional time-domain equalizers making it efficient for equalizing long channels that exhibit severe ISI.

In system identification scenario, the linear approximation of the system modelled by its impulse response, is estimated in real time by gradient type Least Mean Square (LMS) or Recursive Least Squares (RLS) algorithms. In recent applications like acoustic echo cancellation, the order of the impulse response to be estimated is very high, and these traditional approaches are inefficient and real time implementation becomes difficult. Alternatively, the system is modelled by a set of shorter adaptive filters operating in parallel on subsampled signals. This approach, referred to as subband adaptive filtering, is expected to reduce not only the computational complexity but also to improve the convergence rate of the adaptive algorithm. But in practice, different subband adaptive algorithms have to be used to enhance the performance with respect to complexity, convergence rate and processing delay. A single subband adaptive filtering algorithm which outperforms the full band scheme in all applications is yet to be realized. This thesis is intended to study the subband adaptive filtering techniques and explore the possibilities of better algorithms for performance improvement. Three different subband adaptive algorithms have been proposed and their performance have been verified through simulations. These algorithms have been applied to acoustic echo cancellation and EEG artefact minimization problems. Details of the work To start with, the fast FIR filtering scheme introduced by Mou and Duhamel has been generalized. The Perfect Reconstruction Filter Bank (PRFB) is used to model the linear FIR system. The structure offers efficient implementation with reduced arithmetic complexity. By using a PRFB with non adjacent filters non overlapping, many channel filters can be eliminated from the structure. This helps in reducing the complexity of the structure further, but introduces approximation in the model. The modelling error depends on the stop band attenuation of the filters of the PRFB. The error introduced due to approximation is tolerable for applications like acoustic echo cancellation. The filtered output of the modified generalized fast filtering structure is given by (formula) where, Pk(z) is the main channel output, Pk,, k+1 (z) is the output of auxiliary channel filters at the reduced rate, Gk (z) is the kth synthesis filter and M the number of channels in the PRFB. An adaptation scheme is developed for adapting the main channel filters. Auxiliary channel filters are derived from main channel filters. Secondly, the aliasing problem of the classical structure is reduced without using the cross filters. Aliasing components in the estimated signal results in very poor steady state performance in the classical structure. Attempts to eliminate the aliasing have reduced the computation gain margin and the convergence rate. Any attempt to estimate the subband reference signals from the aliased subband input signals results in aliasing. The analysis filter Hk(z) having the following antialiasing property (formula) can avoid aliasing in the input subband signal. The asymmetry of the frequency response prevents the use of real analysis filters. In the investigation presented in this thesis, complex analysis filters and real'synthesis filters are used in the classical structure, to reduce the aliasing errors and to achieve superior convergence rate. PRFB is traditionally used in implementing Interpolated FIR (IFIR) structure. These filters may not be ideal for processing an input signal for an adaptive algorithm. As third contribution, the IFIR structure is modified using discrete finite frames. The model of an FIR filter s is given by Fc, with c = Hs. The columns of the matrix F forms a frame with rows of H as its dual frame. The matrix elements can be arbitrary except that the transformation should be implementable as a filter bank. This freedom is used to optimize the filter bank, with the knowledge of the input statistics, for initial convergence rate enhancement . Next, the proposed subband adaptive algorithms are applied to acoustic echo cancellation problem with realistic parameters. Speech input and sufficiently long Room Impulse Response (RIR) are used in the simulations. The Echo Return Loss Enhancement (ERLE)and the steady state error spectrum are used as performance measures to compare these algorithms with the full band scheme and other representative subband implementations. Finally, Subband adaptive algorithm is used in minimization of EOG (Electrooculogram) artefacts from measured EEG (Electroencephalogram) signal. An IIR filterbank providing sufficient isolation between the frequency bands is used in the modified IFIR structure and this structure has been employed in the artefact minimization scheme. The estimation error in the high frequency range has been reduced and the output signal to noise ratio has been increased by a couple of dB over that of the fullband scheme. Conclusions Efforts to find elegant Subband adaptive filtering algorithms will continue in the future. However, in this thesis, the generalized filtering algorithm could offer gain in filtering complexity of the order of M/2 and reduced misadjustment . The complex classical scheme offered improved convergence rate, reduced misadjustment and computational gains of the order of M/4 . The modifications of the IFIR structure using discrete finite frames made it possible to eliminate the processing delay and enhance the convergence rate. Typical performance of the complex classical case for speech input in a realistic scenario (8 channel case), offers ERLE of more than 45dB. The subband approach to EOG artefact minimization in EEG signal was found to be superior to their fullband counterpart. (Refer PDF file for Formulas)

A new broadband beamformer composed of nested arrays (NAs), multi-dimensional (MD) filters, and multirate techniques is proposed for both linear and planar arrays. It is shown that this combination results in frequency-invariant response. For a given number of sensors, the advantage of using NAs is that the effective aperture for low temporal frequencies is larger than in the case of using uniform arrays. This leads to high spatial selectivity for low frequencies. For a given aperture size, the proposed beamformer can be implemented with significantly fewer sensors and less computation than uniform arrays with a slight deterioration in performance. Taking advantage of the Noble identity and polyphase structures, the proposed method can be efficiently implemented. Simulation results demonstrate the good performance of the proposed beamformer in terms of frequency-invariant response and computational requirements. The broadband beamformer requires a filter bank with a non-compatible set of sampling rates which is challenging to be designed. To address this issue, a filter bank design approach is presented. The approach is based on formulating the design problem as an optimization problem with a performance index which consists of a term depending on perfect reconstruction (PR) and a term depending on the magnitude specifications of the analysis filters. The design objectives are to achieve almost perfect reconstruction (PR) and have the analysis filters satisfying some prescribed frequency specifications. Several design examples are considered to show the satisfactory performance of the proposed method. A new blind multi-stage space-time equalizer (STE) is proposed which can separate narrowband sources from a mixed signal. Neither the direction of arrival (DOA) nor a training sequence is assumed to be available for the receiver. The beamformer and equalizer are jointly updated to combat both co-channel interference (CCI) and inter-symbol interference (ISI) effectively. Using subarray beamformers, the DOA, possibly time-varying, of the captured signal is estimated and tracked. The estimated DOA is used by the beamformer to provide strong CCI cancellation. In order to alleviate inter-stage error propagation significantly, a mean-square-error sorting algorithm is used which assigns detected sources to different stages according to the reconstruction error at different stages. Further, to speed up the convergence, a simple-yet-efficient DOA estimation algorithm is proposed which can provide good initial DOAs for the multi-stage STE. Simulation results illustrate the good performance of the proposed STE and show that it can effectively deal with changing DOAs and time variant channels. / Graduate / 0544 / imanmoaz@uvic.ca