Recent Advances in Variable Digital Filters

Variable digital filters are widely used in a number of applications of signal processing because of their capability of self-tuning frequency characteristics such as the cutoff frequency and the bandwidth. This chapter introduces recent advances on variable digital filters, focusing on the problems of design and realization, and application to adaptive filtering. In the topic on design and realization, we address two major approaches: one is the frequency transformation and the other is the multi-dimensional polynomial approximation of filter coefficients. In the topic on adaptive filtering, we introduce the details of adaptive band-pass/band-stop filtering that include the well-known adaptive notch filtering

On the design and implementation of FIR and IIR digital filters with variable frequency characteristics

This paper studies the design and implementation of finite-impulse response (FIR) and infinite-impulse response (IIR) variable digital filters (VDFs), whose frequency characteristics can be controlled continuously by some control or tuning parameters. A least squares (LS) approach is proposed for the design of FIR VDFs by expressing the impulse response of the filter as a linear combination of basis functions. It is shown that the optimal LS solution can be obtained by solving a system of linear equations. By choosing the basis functions as piecewise polynomials, VDFs with larger tuning range than that of ordinary polynomial based approach results. The proposed VDF can be efficiently implemented using the familiar Farrow structure. Making use of the FIR VDF so obtained, an Eigensystem Realization Algorithm (ERA)-based model reduction technique is proposed to approximate the FIR VDF by a stable IIR VDF with lower system order. The advantages of the model reduction approach are: 1) it is computational simple which only requires the computation of the singular value decomposition of a Hankel matrix; 2) the IIR VDF obtained is guaranteed to be stable; and 3) the frequency response such as the phase response of the FIR prototype is well preserved. Apart from the above advantages, the proposed IIR VDF does not suffer from undesirable transient response during parameter tuning found in other approaches based on direct tuning of filter parameters. For frequency selective VDFs, about 40% of the multiplications can be saved using the IIR VDFs. The implementation of the proposed FIR VDF using sum-of-powers-of-two (SOPOT) coefficient and the multiplier block (MB) technique are also studied. Results show that about two-third of the additions in implementing the multiplication of the SOPOT coefficients can be saved using the multiplier block, which leads to significant savings in hardware complexity.link_to_subscribed_fulltex

Time-Interleaved Analog-to-Digital Converter (TIADC) Compensation Using Multichannel Filters

Published methods that employ a filter bank for compensating the timing and bandwidth mismatches of an M-channel time-interleaved analog-to-digital converter (TIADC) were developed based on the fact that each sub-ADC channel is a downsampled version of the analog input. The output of each sub-ADC is filtered in such a way that, when all the filter outputs are summed, the aliasing components are minimized. If each channel of the filter bank has N coefficients, the optimization of the coefficients requires computing the inverse of an MN times MN matrix if the weighted least squares (WLS) technique is used as the optimization tool. In this paper, we present a multichannel filtering approach for TIADC mismatch compensation. We apply the generalized sampling theorem to directly estimate the ideal output of each sub-ADC using the outputs of all the sub-ADCs. If the WLS technique is used as the optimization tool, the dimension of the matrix to be inversed is N times N. For the same number of coefficients (and also the same spurious component performance given sufficient arithmetic precision), our technique is computationally less complex and more robust than the filter-bank approach. If mixed integer linear programming is used as the optimization tool to produce filters with coefficient values that are integer powers of two, our technique produces a saving in computing resources by a factor of approximately (100.2N(M- 1)/(M-1) in the TIADC filter design.published_or_final_versio

Adaptive beamforming using frequency invariant uniform concentric circular arrays

This paper proposes new adaptive beamforming algorithms for a class of uniform concentric circular arrays (UCCAs) having near-frequency invariant characteristics. The basic principle of the UCCA frequency invariant beamformer (FIB) is to transform the received signals to the phase mode representation and remove the frequency dependence of individual phase modes through the use of a digital beamforming or compensation network. As a result, the far field pattern of the array is electronic steerable and is approximately invariant over a wider range of frequencies than the uniform circular arrays (UCAs). The beampattern is governed by a small set of variable beamformer weights. Based on the minimum variance distortionless response (MVDR) and generalized sidelobe canceller (GSC) methods, new recursive adaptive beamforming algorithms for UCCA-FIB are proposed. In addition, robust versions of these adaptive beamforming algorithms for mitigating direction-of-arrival (DOA) and sensor position errors are developed. Simulation results show that the proposed adaptive UCCA-FIBs converge much faster and reach a considerable lower steady-state error than conventional broadband UCCA beamformers without using the compensation network. Since fewer variable multipliers are required in the proposed algorithms, it also leads to lower arithmetic complexity and faster tracking performance than conventional methods. © 2007 IEEE.published_or_final_versio

Hybrid DDS-PLL based reconfigurable oscillators with high spectral purity for cognitive radio

Analytical, design and simulation studies on the performance optimization of reconfigurable architecture of a Hybrid DDS – PLL are presented in this thesis. The original contributions of this thesis are aimed towards the DDS, the dithering (spur suppression) scheme and the PLL. A new design of Taylor series-based DDS that reduces the dynamic power and number of multipliers is a significant contribution of this thesis. This thesis compares dynamic power and SFDR achieved in the design of varieties of DDS such as Quartic, Cubic, Linear and LHSC. This thesis proposes two novel schemes namely “Hartley Image Suppression” and “Adaptive Sinusoidal Interference Cancellation” overcoming the low noise floor of traditional dithering schemes. The simulation studies on a Taylor series-based DDS reveal an improvement in SFDR from 74 dB to 114 dB by using Least Mean Squares -Sinusoidal Interference Canceller (LM-SIC) with the noise floor maintained at -200 dB. Analytical formulations have been developed for a second order PLL to relate the phase noise to settling time and Phase Margin (PM) as well as to relate jitter variance and PM. New expressions relating phase noise to PM and lock time to PM are derived. This thesis derives the analytical relationship between the roots of the characteristic equation of a third order PLL and its performance metrics like PM, Gardner’s stability factor, jitter variance, spur gain and ratio of noise power to carrier power. This thesis presents an analysis to relate spur gain and capacitance ratio of a third order PLL. This thesis presents an analytical relationship between the lock time and the roots of its characteristic equation of a third order PLL. Through Vieta’s circle and Vieta’s angle, the performance metrics of a third order PLL are related to the real roots of its characteristic equation

Time-delay interferometric ranging for LISA: Statistical analysis of bias-free ranging using laser noise minimization

Die Laser Interferometer Space Antenna (LISA) ist eine Mission der europäischen Weltraumagentur (ESA) zur Detektion von Gravitationswellen im Frequenzbereich zwischen 10^-4 Hz und 1 Hz. Gravitationswellen induzieren relative Abstandsänderungen, die LISA mithilfe von Laserinterferometrie mit Picometerpräzision misst. Ein großes Problem hierbei ist das Frequenzrauschen der Laser. Um dieses zu unterdrücken, ist es notwendig, mithilfe eines Algorithmus namens TDI (engl. time-delay interferometry), virtuelle Interferometer mit gleichlangen Armen zu konstruieren, wie z.B. das klassische Michelson-Interferometer. In dieser Arbeit untersuchen wir die Performanz von TDI unter realistischen Bedingungen und identifizieren verschiedene Kopplungsmechanismen des Laserfrequenzrauschens. Als erstes betrachten wir die Datenverarbeitung an Bord der Satelliten, die benötigt wird, um die Abtastrate der interferometrischen Messungen zu reduzieren. Hierfür sind Anti-Alias-Filter vorgesehen, die der Faltung von Laserrauschleistung in das Beobachtungsband vorbeugen. Außerdem wirkt sich die Ebenheit der Filter auf die Effektivität von TDI aus (engl. flexing-filtering-effect). Dieser Effekt ist bereits in der Literatur beschrieben und wir demonstrieren in dieser Arbeit die Möglichkeit, ihn mithilfe von Kompensationsfiltern effektiv zu reduzieren. Als zweites betrachten wir Kopplungsmechanismen von Laserfrequenzrauschen im TDI-Algorithmus selbst. Fehler in der Interpolation der interferometrischen Messungen und Ungenauigkeiten in den absoluten Abstandsmessungen zwischen den Satelliten führen ebenfalls zu einer unzureichenden Reduzierung des Laserfrequenzrauschens. Wir beschreiben die oben genannten Kopplungsmechanismen analytisch und validieren die zugrundeliegenden Modelle mithilfe von numerischen Simulationen. Das tiefere Verständnis dieser Residuen ermöglicht es uns, geeignete instrumentelle Parameter zu wählen, die von hoher Relevanz für das Missionsdesign von LISA sind. Des Weiteren beschäftigen wir uns in dieser Arbeit mit der möglichst genauen Bestimmung der absoluten Abständen zwischen den Satelliten, die für den TDI Algorithmus erforderlich sind. Hierfür werden die Abstandsinformationen aus den Seitenbändern und der PRN-Modulation (engl. pseudo-random noise) kombiniert. Wir zeigen, dass die PRN-Messung von systematischen Verzerrungen betroffen ist, die zu Laserrauschresiduen in den TDI-Variablen führen. Um diesen Fehler zu korrigieren, schlagen wir als zusätzliche Abstandsmessung TDI-Ranging (TDI-R) vor. TDI-R ist zwar ungenauer, aber frei von systematischen Verzerrungen und kann daher zur Kalibrierung der PRN-Messungen herangezogen werden. Wir präsentieren in dieser Arbeit eine ausführliche statistische Studie, um die Performanz von TDI-R zu charakterisieren. Dafür formulieren wir die Likelihood-Funktion der interferometrischen Messungen und berechnen die Fisher-Informationsmatrix, um die theoretisch mögliche untere Grenze der Schätzvarianz zu finden. Diese verhält sich invers proportional zur Integrationszeit und dem Verhältnis von Sekundärrauschleistung, die die interferometrische Messung fundamental limitiert, und Laserrauschleistung. Zusätzlich validieren wir die analytische untere Grenze der Schätzvarianz mithilfe von numerischen Simulationen und zeigen damit, dass unsere Implementierung von TDI-R optimal ist. Der entwickelte TDI-R-Algorithmus wird Teil der Datenverarbeitungspipeline sein und Konsistenzprüfungen und Kalibrierung der primären Abstandsmessmethoden ermöglichen.The Laser Interferometer Space Antenna (LISA) is a future ESA-led space-based observatory to explore the gravitational universe in the frequency band between 10^-4 Hz and 1 Hz. LISA implements picometer-precise inter-satellite ranging to measure tiny ripples in spacetime induced by gravitational waves (GWs). However, the single-link measurements are dominated by laser frequency noise, which is about nine orders of magnitude larger than the GW signals. Therefore, in post-processing, the time-delay interferometry (TDI) algorithm is used to synthesize virtual equal-arm interferometers to suppress laser frequency noise. In this work we identify several laser frequency noise coupling channels that limit the performance of TDI. First, the on-board processing, which is used to decimate the sampling rate from tens of megahertz down to the telemetry rate of a few hertz, requires careful design. Appropriate anti-aliasing filters must be implemented to mitigate folding of laser noise power into the observation band. Furthermore, the flatness of these filters is important to limit the impact of the flexing-filtering effect. We demonstrate that this effect can be effectively reduced by using compensation filters on ground. Second, the post-processing delays applied in TDI are subject to interpolation and ranging errors. We study these laser and timing noise residuals analytically and perform simulations to validate the models numerically. Our findings have direct implications for the design of the LISA instrument as we identify the instrumental parameters that are essential for successful laser noise suppression and provide methods for designing appropriate filters for the on-board processing. In addition, we discuss a dedicated ranging processing pipeline that produces high-precision range estimates that are the input for TDI by combining the sideband and pseudo-random noise (PRN) ranges. We show in this thesis that biases in the PRN measurements limit the laser noise suppression performance. Therefore, we propose time-delay interferometric ranging (TDI-R) as a third ranging sensor to estimate bias-free ranges that can be used to calibrate the biases in the PRN measurements. We present a thorough statistical study of TDI-R to evaluate its performance. Therefore, we formulate the likelihood function of the interferometric data and use the Fisher information formalism to find a lower bound on the estimation variance of the inter-satellite ranges. We find that the ranging uncertainty is proportional to the inverse of the integration time and the ratio of secondary noise power, that limits the interferometric readout, to the laser noise power. To validate our findings we implement prototype TDI-R pipelines and perform numerical simulations. We show that we are able to formulate optimal estimators of the unbiased range that reach the Cramér-Rao lower bound previously expressed analytically. The developed TDI-R pipeline will be integrated into the ranging processing pipeline to perform consistency checks and ensure well-calibrated inter-satellite ranges

Algorithms and VLSI architectures for parametric additive synthesis

A parametric additive synthesis approach to sound synthesis is advantageous as it can model sounds in a large scale manner, unlike the classical sinusoidal additive based synthesis paradigms. It is known that a large body of naturally occurring sounds are resonant in character and thus fit the concept well. This thesis is concerned with the computational optimisation of a super class of form ant synthesis which extends the sinusoidal parameters with a spread parameter known as band width. Here a modified formant algorithm is introduced which can be traced back to work done at IRCAM, Paris. When impulse driven, a filter based approach to modelling a formant limits the computational work-load. It is assumed that the filter's coefficients are fixed at initialisation, thus avoiding interpolation which can cause the filter to become chaotic. A filter which is more complex than a second order section is required. Temporal resolution of an impulse generator is achieved by using a two stage polyphase decimator which drives many filterbanks. Each filterbank describes one formant and is composed of sub-elements which allow variation of the formant’s parameters. A resource manager is discussed to overcome the possibility of all sub- banks operating in unison. All filterbanks for one voice are connected in series to the impulse generator and their outputs are summed and scaled accordingly. An explorative study of number systems for DSP algorithms and their architectures is investigated. I invented a new theoretical mechanism for multi-level logic based DSP. Its aims are to reduce the number of transistors and to increase their functionality. A review of synthesis algorithms and VLSI architectures are discussed in a case study between a filter based bit-serial and a CORDIC based sinusoidal generator. They are both of similar size, but the latter is always guaranteed to be stable

Design techniques for sigma-delta modulators in communications applications

Specialised design techniques for sigma-delta modulators are described in this thesis with all of the examples coming from modern communications systems. The noise shaping and the signal transfer functions can be optimised using a weighted least squares approach. Numerical problems arising in the optimisation as a result of high oversampling rates are overcome through the use a simple transformation. The application to digitising audio is discussed, with the conclusion that Butterworth response noise shaping is preferable to inverse Chebyshev noise shaping for audio applications. An example of optimising the signal transfer function to provide immunity to instability brought about by large out-of-band signals is also presented. The use of redundant arithmetic in the implementation of very high speed sigma-delta modulators is introduced, together with a DAC / filter combination suitable for reconstructing an analogue signal from the redundant arithmetic SDM. An improved topology for a speech compander is described which offers a number of significant advantages over existing published methods. This uses no external components for ac coupling or setting the response time-constant, yet is robust and insensitive to parasitic components and process variations. This has been integrated on a CMOS IC process and the results are compared with the high level simulations. A simulation method which allows the verification of switched-capacitor schematics with several orders of magnitude speed improvements over commercially available simulation tools is discussed. The method assumes ideal components, with internally controllable switches and reduces the schematic netlist to the few key equations that an experienced designer would derive manually. This process is fully automated and consequently is useful for providing confidence in implementations of complex SC systems

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