IIR Digital Filter Design Using Convex Optimization

Digital filters play an important role in digital signal processing and communication. From the 1960s, a considerable number of design algorithms have been proposed for finite-duration impulse response (FIR) digital filters and infinite-duration impulse response (IIR) digital filters. Compared with FIR digital filters, IIR digital filters have better approximation capabilities under the same specifications. Nevertheless, due to the presence of the denominator in its rational transfer function, an IIR filter design problem cannot be easily formulated as an equivalent convex optimization problem. Furthermore, for stability, all the poles of an IIR digital filter must be constrained within a stability domain, which, however, is generally nonconvex. Therefore, in practical designs, optimal solutions cannot be definitely attained. In this dissertation, we focus on IIR filter design problems under the weighted least-squares (WLS) and minimax criteria. Convex optimization will be utilized as the major mathematical tool to formulate and analyze such IIR filter design problems. Since the original IIR filter design problem is essentially nonconvex, some approximation and convex relaxation techniques have to be deployed to achieve convex formulations of such design problems. We first consider the stability issue. A sufficient and necessary stability condition is derived from the argument principle. Although the original stability condition is in a nonconvex form, it can be appropriately approximated by a quadratic constraint and readily combined with sequential WLS design procedures. Based on the sufficient and necessary stability condition, this approximate stability constraint can achieve an improved description of the nonconvex stability domain. We also address the nonconvexity issue of minimax design of IIR digital filters. Convex relaxation techniques are applied to obtain relaxed design problems, which are formulated, respectively, as second-order cone programming (SOCP) and semidefinite programming (SDP) problems. By solving these relaxed design problems, we can estimate lower bounds of minimum approximation errors, which are useful in subsequent design procedures to achieve real minimax solutions. Since the relaxed design problems are independent of local information, compared with many prevalent design methods which employ local search, the proposed design methods using the convex relaxation techniques have an increased chance to obtain an optimal design

Design and implementation of computationally efficient digital filters

Ph.DDOCTOR OF PHILOSOPH

NATURAL ALGORITHMS IN DIGITAL FILTER DESIGN

Digital filters are an important part of Digital Signal Processing (DSP), which plays vital roles within the modern world, but their design is a complex task requiring a great deal of specialised knowledge. An analysis of this design process is presented, which identifies opportunities for the application of optimisation. The Genetic Algorithm (GA) and Simulated Annealing are problem-independent and increasingly popular optimisation techniques. They do not require detailed prior knowledge of the nature of a problem, and are unaffected by a discontinuous search space, unlike traditional methods such as calculus and hill-climbing. Potential applications of these techniques to the filter design process are discussed, and presented with practical results. Investigations into the design of Frequency Sampling (FS) Finite Impulse Response (FIR) filters using a hybrid GA/hill-climber proved especially successful, improving on published results. An analysis of the search space for FS filters provided useful information on the performance of the optimisation technique. The ability of the GA to trade off a filter's performance with respect to several design criteria simultaneously, without intervention by the designer, is also investigated. Methods of simplifying the design process by using this technique are presented, together with an analysis of the difficulty of the non-linear FIR filter design problem from a GA perspective. This gave an insight into the fundamental nature of the optimisation problem, and also suggested future improvements. The results gained from these investigations allowed the framework for a potential 'intelligent' filter design system to be proposed, in which embedded expert knowledge, Artificial Intelligence techniques and traditional design methods work together. This could deliver a single tool capable of designing a wide range of filters with minimal human intervention, and of proposing solutions to incomplete problems. It could also provide the basis for the development of tools for other areas of DSP system design

Efficient algorithms for arbitrary sample rate conversion with application to wave field synthesis

Arbitrary sample rate conversion (ASRC) is used in many fields of digital signal processing to alter the sampling rate of discrete-time signals by arbitrary, potentially time-varying ratios. This thesis investigates efficient algorithms for ASRC and proposes several improvements. First, closed-form descriptions for the modified Farrow structure and Lagrange interpolators are derived that are directly applicable to algorithm design and analysis. Second, efficient implementation structures for ASRC algorithms are investigated. Third, this thesis considers coefficient design methods that are optimal for a selectable error norm and optional design constraints. Finally, the performance of different algorithms is compared for several performance metrics. This enables the selection of ASRC algorithms that meet the requirements of an application with minimal complexity. Wave field synthesis (WFS), a high-quality spatial sound reproduction technique, is the main application considered in this work. For WFS, sophisticated ASRC algorithms improve the quality of moving sound sources. However, the improvements proposed in this thesis are not limited to WFS, but applicable to general-purpose ASRC problems.Verfahren zur unbeschränkten Abtastratenwandlung (arbitrary sample rate conversion,ASRC) ermöglichen die Änderung der Abtastrate zeitdiskreter Signale um beliebige, zeitvarianteVerhältnisse. ASRC wird in vielen Anwendungen digitaler Signalverarbeitung eingesetzt.In dieser Arbeit wird die Verwendung von ASRC-Verfahren in der Wellenfeldsynthese(WFS), einem Verfahren zur hochqualitativen, räumlich korrekten Audio-Wiedergabe, untersucht.Durch ASRC-Algorithmen kann die Wiedergabequalität bewegter Schallquellenin WFS deutlich verbessert werden. Durch die hohe Zahl der in einem WFS-Wiedergabesystembenötigten simultanen ASRC-Operationen ist eine direkte Anwendung hochwertigerAlgorithmen jedoch meist nicht möglich.Zur Lösung dieses Problems werden verschiedene Beiträge vorgestellt. Die Komplexitätder WFS-Signalverarbeitung wird durch eine geeignete Partitionierung der ASRC-Algorithmensignifikant reduziert, welche eine effiziente Wiederverwendung von Zwischenergebnissenermöglicht. Dies erlaubt den Einsatz hochqualitativer Algorithmen zur Abtastratenwandlungmit einer Komplexität, die mit der Anwendung einfacher konventioneller ASRCAlgorithmenvergleichbar ist. Dieses Partitionierungsschema stellt jedoch auch zusätzlicheAnforderungen an ASRC-Algorithmen und erfordert Abwägungen zwischen Performance-Maßen wie der algorithmischen Komplexität, Speicherbedarf oder -bandbreite.Zur Verbesserung von Algorithmen und Implementierungsstrukturen für ASRC werdenverschiedene Maßnahmen vorgeschlagen. Zum Einen werden geschlossene, analytischeBeschreibungen für den kontinuierlichen Frequenzgang verschiedener Klassen von ASRCStruktureneingeführt. Insbesondere für Lagrange-Interpolatoren, die modifizierte Farrow-Struktur sowie Kombinationen aus Überabtastung und zeitkontinuierlichen Resampling-Funktionen werden kompakte Darstellungen hergeleitet, die sowohl Aufschluss über dasVerhalten dieser Filter geben als auch eine direkte Verwendung in Design-Methoden ermöglichen.Einen zweiten Schwerpunkt bildet das Koeffizientendesign für diese Strukturen, insbesonderezum optimalen Entwurf bezüglich einer gewählten Fehlernorm und optionaler Entwurfsbedingungenund -restriktionen. Im Gegensatz zu bisherigen Ansätzen werden solcheoptimalen Entwurfsmethoden auch für mehrstufige ASRC-Strukturen, welche ganzzahligeÜberabtastung mit zeitkontinuierlichen Resampling-Funktionen verbinden, vorgestellt.Für diese Klasse von Strukturen wird eine Reihe angepasster Resampling-Funktionen vorgeschlagen,welche in Verbindung mit den entwickelten optimalen Entwurfsmethoden signifikanteQualitätssteigerungen ermöglichen.Die Vielzahl von ASRC-Strukturen sowie deren Design-Parameter bildet eine Hauptschwierigkeitbei der Auswahl eines für eine gegebene Anwendung geeigneten Verfahrens.Evaluation und Performance-Vergleiche bilden daher einen dritten Schwerpunkt. Dazu wirdzum Einen der Einfluss verschiedener Entwurfsparameter auf die erzielbare Qualität vonASRC-Algorithmen untersucht. Zum Anderen wird der benötigte Aufwand bezüglich verschiedenerPerformance-Metriken in Abhängigkeit von Design-Qualität dargestellt.Auf diese Weise sind die Ergebnisse dieser Arbeit nicht auf WFS beschränkt, sondernsind in einer Vielzahl von Anwendungen unbeschränkter Abtastratenwandlung nutzbar

Digital Filters

The new technology advances provide that a great number of system signals can be easily measured with a low cost. The main problem is that usually only a fraction of the signal is useful for different purposes, for example maintenance, DVD-recorders, computers, electric/electronic circuits, econometric, optimization, etc. Digital filters are the most versatile, practical and effective methods for extracting the information necessary from the signal. They can be dynamic, so they can be automatically or manually adjusted to the external and internal conditions. Presented in this book are the most advanced digital filters including different case studies and the most relevant literature

DOCSIS 3.1 cable modem and upstream channel simulation in MATLAB

The cable television (CATV) industry has grown significantly since its inception in the late 1940’s. Originally, a CATV network was comprised of several homes that were connected to community antennae via a network of coaxial cables. The only signal processing done was by an analogue amplifier, and transmission only occurred in one direction (i.e. from the antennae/head-end to the subscribers). However, as CATV grew in popularity, demand for services such as pay-per-view television increased, which lead to supporting transmission in the upstream direction (i.e. from subscriber to the head-end). This greatly increased the signal processing to include frequency diplexers. CATV service providers began to expand the bandwidth of their networks in the late 90’s by switching from analogue to digital technology. In an effort to regulate the manufacturing of new digital equipment and ensure interoperability of products from different manufacturers, several cable service providers formed a non-for-profit consortium to develop a data-over-cable service interface specification (DOCSIS). The consortium, which is named CableLabs, released the first DOCSIS standard in 1997. The DOCSIS standard has been upgraded over the years to keep up with increased consumer demand for large bandwidths and faster transmission speeds, particularly in the upstream direction. The latest version of the DOCSIS standard, DOCSIS 3.1, utilizes orthogonal frequency-division multiple access (OFDMA) technology to provide upstream transmission speeds of up to 1 Gbps. As cable service providers begin the process of upgrading their upstream receivers to comply with the new DOCSIS 3.1 standard, they require a means of testing the various functions that an upstream receiver may employ. It is convenient for service providers to employ cable modem (CM) plus channel emulator to perform these tests in-house during the product development stage. Constructing the emulator in digital technology is an attractive option for testing. This thesis approaches digital emulation by developing a digital model of the CMs and upstream channel in a DOCSIS 3.1 network. The first step in building the emulator is to simulate its operations in MATLAB, specifically upstream transmission over the network. The MATLAB model is capable of simulating transmission from multiple CMs, each of which transmits using a specific “transmission mode.” The three transmission modes described in the DOCSIS 3.1 standard are included in the model. These modes are “traffic mode,” which is used during regular data transmission; “fine ranging mode,” which is used to perform fine timing and power offset corrections; and “probing” mode, which is presumably used for estimating the frequency response of the channel, but also is used to further correct the timing and power offsets. The MATLAB model is also capable of simulating the channel impairments a signal may encounter when traversing the upstream channel. Impairments that are specific to individual CMs include integer and fractional timing offsets, micro-reflections, carrier phase offset (CPO), fractional carrier frequency offset (CFO), and network gain/attenuation. Impairments common to all CMs include carrier hum modulation, AM/FM ingress noise, and additive white Gaussian noise (AWGN). It is the hope that the MATLAB scripts that make up the simulation be translated to Verilog HDL to implement the emulator on a field-programmable gate array (FPGA) in the near future. In the event that an FPGA implementation is pursued, research was conducted into designing efficient fractional delay filters (FDFs), which are essential in the simulation of micro-reflections. After performing an FPGA implementation cost analysis between various FDF designs, it was determined that a Kaiser-windowed sinc function FDF with roll-off parameter β = 3.88 was the most cost-efficient choice, requiring at total of 24 multipliers when implemented using an optimized structure

Optimisation of multiplier-less FIR filter design techniques

This thesis is concerned with the design of multiplier-less (ML) finite impulse response (FIR) digital filters. The use of multiplier-less digital filters results in simplified filtering structures, better throughput rates and higher speed. These characteristics are very desirable in many DSP systems. This thesis concentrates on the design of digital filters with power-of-two coefficients that result in simplified filtering structures. Two distinct classesof ML FIR filter design algorithms are developed and compared with traditional techniques. The first class is based on the sensitivity of filter coefficients to rounding to power-of-two. Novel elements include extending of the algorithm for multiple-bands filters and introducing mean square error as the sensitivity criterion. This improves the performance of the algorithm and reduces the complexity of resulting filtering structures. The second class of filter design algorithms is based on evolutionary techniques, primarily genetic algorithms. Three different algorithms based on genetic algorithm kernel are developed. They include simple genetic algorithm, knowledge-based genetic algorithm and hybrid of genetic algorithm and simulated annealing. Inclusion of the additional knowledge has been found very useful when re-designing filters or refining previous designs. Hybrid techniques are useful when exploring large, N-dimensional searching spaces. Here, the genetic algorithm is used to explore searching space rapidly, followed by fine search using simulated annealing. This approach has been found beneficial for design of high-order filters. Finally, a formula for estimation of the filter length from its specification and complementing both classes of design algorithms, has been evolved using techniques of symbolic regression and genetic programming. Although the evolved formula is very complex and not easily understandable, statistical analysis has shown that it produces more accurate results than traditional Kaiser's formula. In summary, several novel algorithms for the design of multiplier-less digital filters have been developed. They outperform traditional techniques that are used for the design of ML FIR filters and hence contributed to the knowledge in the field of ML FIR filter design

Waveform Design for 5G and beyond Systems

5G traffic has very diverse requirements with respect to data rate, delay, and reliability. The concept of using multiple OFDM numerologies adopted in the 5G NR standard will likely meet these multiple requirements to some extent. However, the traffic is radically accruing different characteristics and requirements when compared with the initial stage of 5G, which focused mainly on high-speed multimedia data applications. For instance, applications such as vehicular communications and robotics control require a highly reliable and ultra-low delay. In addition, various emerging M2M applications have sparse traffic with a small amount of data to be delivered. The state-of-the-art OFDM technique has some limitations when addressing the aforementioned requirements at the same time. Meanwhile, numerous waveform alternatives, such as FBMC, GFDM, and UFMC, have been explored. They also have their own pros and cons due to their intrinsic waveform properties. Hence, it is the opportune moment to come up with modification/variations/combinations to the aforementioned techniques or a new waveform design for 5G systems and beyond. The aim of this Special Issue is to provide the latest research and advances in the field of waveform design for 5G systems and beyond

Analysis and Design of Low-Cost Waveguide Filters for Wireless Communications

The area of research of this thesis is built around advanced waveguide filter structures. Waveguide filters and the waveguide technology in general are renowned for high power capacity, low losses and excellent electromagnetic shielding. Waveguide filters are important components in fixed wireless communications as well as in satellite and radar systems. Furthermore, their advantages and utilization become even greater with increase in frequency, which is a trend in modern communication systems because upper frequency bands offer larger channel capacities. However, waveguide filters are relatively bulky and expensive. To comply with more and more demanding miniaturization and cost-cutting requirements, compactness and economical design represent some of the main contemporary focuses of interest. Approaches that are used to achieve this include use of planar inserts to build waveguide discontinuities, additive manufacturing and substrate integration. At the same time, waveguide filters still need to satisfy opposed stringent requirements like small insertion loss, high selectivity and multiband operation. Another difficulty that metal waveguide components face is integration with other circuitry, especially important when solid-state active devices are included. Thus, improvements of interconnections between waveguide and other transmission interfaces are addressed too. The thesis elaborates the following aspects of work: Further analysis and improved explanations regarding advanced waveguide filters with E-plane inserts developed by the Wireless Communications Research Group, using both cross coupled resonators and extracted pole sections (Experiments with higher filter orders, use of tuning screws, degrees of freedom in design, etc. Thorough performance comparison with competing filter technologies) - Proposing novel E-plane filter sections with I-shaped insets - Extension of the E-plane filtering structures with metal fins to new compact dual band filters with high frequency selectivity and miniaturized diplexers. - Introduction of easy-to-build waveguide filters with polymer insert frames and high-performance low-profile cavity filters, taking advantage of enhanced fabrication capabilities when using additive manufacturing - Developing new substrate integrated filters, as well as circuits used to transfer signals between different interfaces Namely, these are substrate integrated waveguide to metal waveguide planar transitions that do not require any modifications of the metal waveguides. Such novel transitions have been designed both for single and orthogonal signal polarizations

Analogue filter networks: developments in theory, design and analyses

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