Synthesis methods for linear-phase FIR filters with a piecewise-polynomial impulse response

his thesis concentrates on synthesis methods for linear-phase finite-impulse response filters with a piecewise-polynomial impulse response. One of the objectives has been to find integer-valued coefficients to efficiently implement filters of the piecewise-polynomial impulse response approach introduced by Saram¨aki and Mitra. In this method, the impulse response is divided into blocks of equal length and each block is created by a polynomial of a given degree. The arithmetic complexity of these filters depends on the polynomial degree and the number of blocks. By using integer-valued coefficients it is possible to make the implementation of the subfilters, which generates the polynomials, multiplication-free. The main focus has been on finding computationally-efficient synthesis methods by using a piecewise-polynomial and a piecewise-polynomial-sinusoidal impulse responses to make it possible to implement high-speed, low-power, highly integrated digital signal processing systems. The earlier method by Chu and Burrus has been studied. The overall impulse response of the approach proposed in this thesis consists of the sum of several polynomial-form responses. The arithmetic complexity depends on the polynomial degree and the number of polynomial-form responses. The piecewise-polynomial-sinusoidal approach is a modification of the piecewise-polynomial approach. The subresponses are multiplied by a sinusoidal function and an arbitrary number of separate center coefficients is added. Thereby, the arithmetic complexity depends also on the number of complex multipliers and separately generated center coefficients. The filters proposed in this thesis are optimized by using linear programming methods

Optimal FIR filter design

The design of Finite Impulse Response (FIR) digital filters that considers both phase and magnitude specifications is investigated. This dissertation is divided into two parts. In Part I we present our implementation of an algorithm for the design of minimum phase filters. In Part II we investigate the design of FIR filters in the complex domain and develop a new powerful design method for digital FIR filters with arbitrary specification of magnitude and phase;Part I considers the design of minimum-phase filters. The method presented uses direct factorization of the transfer function of a companion Parks-McClellan linear-phase filter of twice the length of the desired minimum-phase filter. The minimum-phase filter is derived with excision of half the zeros of the companion linear-phase filter. The zeros of the prototype filter are found using Laguerre\u27s method. We will present our implementation of the design method, and describe some practical aspects and problems associated with the design of minimum-phase filters;Part II investigates the design of optimal Chebychev FIR filters in the complex domain. The design of FIR filters with arbitrary specification of magnitude and phase is formulated into a problem of complex approximation. The method developed is capable of designing filters with real or complex coefficients. Complex impulse response designs are an extension of the real coefficient case based on a proper selection of the approximating basis functions;The minimax criterion is used and the complex Chebychev approximation is posed as a minimization problem in linear optimization. The primal problem is converted to its dual and is solved using an efficient, quadratically convergent algorithm developed by Tang (14). The relaxation of the linear-phase constraint results in a reduction of the number of coefficients compared to linear-phase designs. Linear-phase filters are a special case of our filter design approach. We examine the design of frequency selective filters with or without the conjugate symmetry, the design of one-sided, two-sided, narrowband and fullband Hilbert Transformers and differentiators

Generalizations of the sampling theorem: Seven decades after Nyquist

The sampling theorem is one of the most basic and fascinating topics in engineering sciences. The most well-known form is Shannon's uniform-sampling theorem for bandlimited signals. Extensions of this to bandpass signals and multiband signals, and to nonuniform sampling are also well-known. The connection between such extensions and the theory of filter banks in DSP has been well established. This paper presents some of the less known aspects of sampling, with special emphasis on non bandlimited signals, pointwise stability of reconstruction, and reconstruction from nonuniform samples. Applications in multiresolution computation and in digital spline interpolation are also reviewed

Design Of Polynomial-based Filters For Continuously Variable Sample Rate Conversion With Applications In Synthetic Instrumentati

In this work, the design and application of Polynomial-Based Filters (PBF) for continuously variable Sample Rate Conversion (SRC) is studied. The major contributions of this work are summarized as follows. First, an explicit formula for the Fourier Transform of both a symmetrical and nonsymmetrical PBF impulse response with variable basis function coefficients is derived. In the literature only one explicit formula is given, and that for a symmetrical even length filter with fixed basis function coefficients. The frequency domain optimization of PBFs via linear programming has been proposed in the literature, however, the algorithm was not detailed nor were explicit formulas derived. In this contribution, a minimax optimization procedure is derived for the frequency domain optimization of a PBF with time-domain constraints. Explicit formulas are given for direct input to a linear programming routine. Additionally, accompanying Matlab code implementing this optimization in terms of the derived formulas is given in the appendix. In the literature, it has been pointed out that the frequency response of the Continuous-Time (CT) filter decays as frequency goes to infinity. It has also been observed that when implemented in SRC, the CT filter is sampled resulting in CT frequency response aliasing. Thus, for example, the stopband sidelobes of the Discrete-Time (DT) implementation rise above the CT designed level. Building on these observations, it is shown how the rolloff rate of the frequency response of a PBF can be adjusted by adding continuous derivatives to the impulse response. This is of great advantage, especially when the PBF is used for decimation as the aliasing band attenuation can be made to increase with frequency. It is shown how this technique can be used to dramatically reduce the effect of alias build up in the passband. In addition, it is shown that as the number of continuous derivatives of the PBF increases the resulting DT implementation more closely matches the Continuous-Time (CT) design. When implemented for SRC, samples from a PBF impulse response are computed by evaluating the polynomials using a so-called fractional interval, µ. In the literature, the effect of quantizing µ on the frequency response of the PBF has been studied. Formulas have been derived to determine the number of bits required to keep frequency response distortion below prescribed bounds. Elsewhere, a formula has been given to compute the number of bits required to represent µ to obtain a given SRC accuracy for rational factor SRC. In this contribution, it is shown how these two apparently competing requirements are quite independent. In fact, it is shown that the wordlength required for SRC accuracy need only be kept in the µ generator which is a single accumulator. The output of the µ generator may then be truncated prior to polynomial evaluation. This results in significant computational savings, as polynomial evaluation can require several multiplications and additions. Under the heading of applications, a new Wideband Digital Downconverter (WDDC) for Synthetic Instruments (SI) is introduced. DDCs first tune to a signal\u27s center frequency using a numerically controlled oscillator and mixer, and then zoom-in to the bandwidth of interest using SRC. The SRC is required to produce continuously variable output sample rates from a fixed input sample rate over a large range. Current implementations accomplish this using a pre-filter, an arbitrary factor resampler, and integer decimation filters. In this contribution, the SRC of the WDDC is simplified reducing the computational requirements to a factor of three or more. In addition to this, it is shown how this system can be used to develop a novel computationally efficient FFT-based spectrum analyzer with continuously variable frequency spans. Finally, after giving the theoretical foundation, a real Field Programmable Gate Array (FPGA) implementation of a novel Arbitrary Waveform Generator (AWG) is presented. The new approach uses a fixed Digital-to-Analog Converter (DAC) sample clock in combination with an arbitrary factor interpolator. Waveforms created at any sample rate are interpolated to the fixed DAC sample rate in real-time. As a result, the additional lower performance analog hardware required in current approaches, namely, multiple reconstruction filters and/or additional sample clocks, is avoided. Measured results are given confirming the performance of the system predicted by the theoretical design and simulation

Design of sparse FIR filters with low group delay

The aim of the work is to present the method for designing sparse FIR filters with very low group delay and approximately linear-phase in the passband. Significant reduction of the group delay, e.g. several times in relation to the linear phase filter, may cause the occurrence of undesirable overshoot in the magnitude frequency response. The method proposed in this work consists of two stages. In the first stage, FIR filter with low group delay is designed using minimax constrained optimization that provides overshoot elimination. In the second stage, the same process is applied iteratively to reach sparse solution. Design examples demonstrate the effectiveness of the proposed method

Design of sparse FIR filters with low group delay

The aim of the work is to present the method for designing sparse FIR filters with very low group delay and approximately linear-phase in the passband. Significant reduction of the group delay, e.g. several times in relation to the linear phase filter, may cause the occurrence of undesirable overshoot in the magnitude frequency response. The method proposed in this work consists of two stages. In the first stage, FIR filter with low group delay is designed using minimax constrained optimization that provides overshoot elimination. In the second stage, the same process is applied iteratively to reach sparse solution. Design examples demonstrate the effectiveness of the proposed method