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

Verified compilation and optimization of floating-point kernels

When verifying safety-critical code on the level of source code, we trust the compiler to produce machine code that preserves the behavior of the source code. Trusting a verified compiler is easy. A rigorous machine-checked proof shows that the compiler correctly translates source code into machine code. Modern verified compilers (e.g. CompCert and CakeML) have rich input languages, but only rudimentary support for floating-point arithmetic. In fact, state-of-the-art verified compilers only implement and verify an inflexible one-to-one translation from floating-point source code to machine code. This translation completely ignores that floating-point arithmetic is actually a discrete representation of the continuous real numbers. This thesis presents two extensions improving floating-point arithmetic in CakeML. First, the thesis demonstrates verified compilation of elementary functions to floating-point code in: Dandelion, an automatic verifier for polynomial approximations of elementary functions; and libmGen, a proof-producing compiler relating floating-point machine code to the implemented real-numbered elementary function. Second, the thesis demonstrates verified optimization of floating-point code in: Icing, a floating-point language extending standard floating-point arithmetic with optimizations similar to those used by unverified compilers, like GCC and LLVM; and RealCake, an extension of CakeML with Icing into the first fully verified optimizing compiler for floating-point arithmetic.Bei der Verifizierung von sicherheitsrelevantem Quellcode vertrauen wir dem Compiler, dass er Maschinencode ausgibt, der sich wie der Quellcode verhält. Man kann ohne weiteres einem verifizierten Compiler vertrauen. Ein rigoroser maschinen-ü}berprüfter Beweis zeigt, dass der Compiler Quellcode in korrekten Maschinencode übersetzt. Moderne verifizierte Compiler (z.B. CompCert und CakeML) haben komplizierte Eingabesprachen, aber unterstützen Gleitkommaarithmetik nur rudimentär. De facto implementieren und verifizieren hochmoderne verifizierte Compiler für Gleitkommaarithmetik nur eine starre eins-zu-eins Übersetzung von Quell- zu Maschinencode. Diese Übersetzung ignoriert vollständig, dass Gleitkommaarithmetik eigentlich eine diskrete Repräsentation der kontinuierlichen reellen Zahlen ist. Diese Dissertation präsentiert zwei Erweiterungen die Gleitkommaarithmetik in CakeML verbessern. Zuerst demonstriert die Dissertation verifizierte Übersetzung von elementaren Funktionen in Gleitkommacode mit: Dandelion, einem automatischen Verifizierer für Polynomapproximierungen von elementaren Funktionen; und libmGen, einen Beweis-erzeugenden Compiler der Gleitkommacode in Relation mit der implementierten elementaren Funktion setzt. Dann demonstriert die Dissertation verifizierte Optimierung von Gleitkommacode mit: Icing, einer Gleitkommasprache die Gleitkommaarithmetik mit Optimierungen erweitert die ähnlich zu denen in unverifizierten Compilern, wie GCC und LLVM, sind; und RealCake, eine Erweiterung von CakeML mit Icing als der erste vollverifizierte Compiler für Gleitkommaarithmetik

公平性下における学習の理論的解析

筑波大学 (University of Tsukuba)201

An Efficient Polyphase Filter Based Resampling Method for Unifying the PRFs in SAR Data

Variable and higher pulse repetition frequencies (PRFs) are increasingly being used to meet the stricter requirements and complexities of current airborne and spaceborne synthetic aperture radar (SAR) systems associated with higher resolution and wider area products. POLYPHASE, the proposed resampling scheme, downsamples and unifies variable PRFs within a single look complex (SLC) SAR acquisition and across a repeat pass sequence of acquisitions down to an effective lower PRF. A sparsity condition of the received SAR data ensures that the uniformly resampled data approximates the spectral properties of a decimated densely sampled version of the received SAR data. While experiments conducted with both synthetically generated and real airborne SAR data show that POLYPHASE retains comparable performance to the state-of-the-art BLUI scheme in image quality, a polyphase filter-based implementation of POLYPHASE offers significant computational savings for arbitrary (not necessarily periodic) input PRF variations, thus allowing fully on-board, in-place, and real-time implementation

Design of quadrature mirror filter banks with canonical signed digit coefficients using genetic algorithms.

This thesis is about the use of a genetic algorithm to design QMF bank with canonical signed digit coefficients. A filter bank has applications in areas like video and audio coding, data communication, etc. Filter bank design is a multiobjective optimization problem. The performance depends on the reconstruction error of the overall filter bank and the individual performance of the composing lowpass filter. In this thesis we have used reconstruction error of the overall filter bank as our main objective and passband error, stopband error, stopband and passband ripples and transition width of the individual lowpass filter as constraints. Therefore filter bank design can be formulated as single objective multiple constraint optimization problem. A unique genetic algorithm is developed to optimize filer bank coefficients such that the corresponding system\u27s response matches that of an ideal system with an additional constraint that all coefficients are in canonical signed digit (CSD) format. A special restoration technique is used to restore the CSD format of the coefficients after crossover and mutation operators in Genetic algorithm. The proposed restoration technique maintains the specified word length and the maximum number of nonzero digits in filter banks coefficients. Experimental results are presented at the end. It is demonstrated that the designed genetic algorithm is reliable, and efficient for designing QMF banks.Dept. of Electrical and Computer Engineering. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2004 .U67. Source: Masters Abstracts International, Volume: 43-05, page: 1785. Thesis (M.A.Sc.)--University of Windsor (Canada), 2004

A comparative study between the cubic spline and b-spline interpolation methods in free energy calculations

Numerical methods are essential in computational science, as analytic calculations for large datasets are impractical. Using numerical methods, one can approximate the problem to solve it with basic arithmetic operations. Interpolation is a commonly-used method, inter alia, constructing the value of new data points within an interval of known data points. Furthermore, polynomial interpolation with a sufficiently high degree can make the data set differentiable. One consequence of using high-degree polynomials is the oscillatory behaviour towards the endpoints, also known as Runge's Phenomenon. Spline interpolation overcomes this obstacle by connecting the data points in a piecewise fashion. However, its complex formulation requires nested iterations in higher dimensions, which is time-consuming. In addition, the calculations have to be repeated for computing each partial derivative at the data point, leading to further slowdown. The B-spline interpolation is an alternative representation of the cubic spline method, where a spline interpolation at a point could be expressed as the linear combination of piecewise basis functions. It was proposed that implementing this new formulation can accelerate many scientific computing operations involving interpolation. Nevertheless, there is a lack of detailed comparison to back up this hypothesis, especially when it comes to computing the partial derivatives. Among many scientific research fields, free energy calculations particularly stand out for their use of interpolation methods. Numerical interpolation was implemented in free energy methods for many purposes, from calculating intermediate energy states to deriving forces from free energy surfaces. The results of these calculations can provide insight into reaction mechanisms and their thermodynamic properties. The free energy methods include biased flat histogram methods, which are especially promising due to their ability to accurately construct free energy profiles at the rarely-visited regions of reaction spaces. Free Energies from Adaptive Reaction Coordinates (FEARCF) that was developed by Professor Kevin J. Naidoo has many advantages over the other flat histogram methods. iii Because of its treatment of the atoms in reactions, FEARCF makes it easier to apply interpolation methods. It implements cubic spline interpolation to derive biasing forces from the free energy surface, driving the reaction towards regions with higher energy. A major drawback of the method is the slowdown experienced in higher dimensions due to the complicated nature of the cubic spline routine. If the routine is replaced by a more straightforward B-spline interpolation, sampling and generating free energy surfaces can be accelerated. The dissertation aims to perform a comparative study between the cubic spline interpolation and B-spline interpolation methods. At first, data sets of analytic functions were used instead of numerical data to compare the accuracy and compute the percentage errors of both methods by taking the functions themselves as reference. These functions were used to evaluate the performances of the two methods at the endpoints, inflections points and regions with a steep gradient. Both interpolation methods generated identically approximated values with a percentage error below the threshold of 1%, although they both performed poorly at the endpoints and the points of inflection. Increasing the number of interpolation knots reduced the errors, however, it caused overfitting in the other regions. Although significant speed-up was not observed in the univariate interpolation, cubic spline suffered from a drastic slowdown in higher dimensions with up to 103 in 3D and 105 in 4D interpolations. The same results applied to the classical molecular dynamics simulations with FEARCF with a speed-up of up to 103 when B-spline interpolation was implemented. To conclude, the B-spline interpolation method can enhance the efficiency of the free energy calculations where cubic spline interpolation has been the currently-used method

Application-Specific Number Representation

Reconfigurable devices, such as Field Programmable Gate Arrays (FPGAs), enable application- specific number representations. Well-known number formats include fixed-point, floating- point, logarithmic number system (LNS), and residue number system (RNS). Such different number representations lead to different arithmetic designs and error behaviours, thus produc- ing implementations with different performance, accuracy, and cost. To investigate the design options in number representations, the first part of this thesis presents a platform that enables automated exploration of the number representation design space. The second part of the thesis shows case studies that optimise the designs for area, latency or throughput from the perspective of number representations. Automated design space exploration in the first part addresses the following two major issues: ² Automation requires arithmetic unit generation. This thesis provides optimised arithmetic library generators for logarithmic and residue arithmetic units, which support a wide range of bit widths and achieve significant improvement over previous designs. ² Generation of arithmetic units requires specifying the bit widths for each variable. This thesis describes an automatic bit-width optimisation tool called R-Tool, which combines dynamic and static analysis methods, and supports different number systems (fixed-point, floating-point, and LNS numbers). Putting it all together, the second part explores the effects of application-specific number representation on practical benchmarks, such as radiative Monte Carlo simulation, and seismic imaging computations. Experimental results show that customising the number representations brings benefits to hardware implementations: by selecting a more appropriate number format, we can reduce the area cost by up to 73.5% and improve the throughput by 14.2% to 34.1%; by performing the bit-width optimisation, we can further reduce the area cost by 9.7% to 17.3%. On the performance side, hardware implementations with customised number formats achieve 5 to potentially over 40 times speedup over software implementations

Signal processing using short word-length

Recently short word-length (normally 1 bit or bits) processing has become a promising technique. However, there are unresolved issues in sigma-delta modulation, which is the basis for 1b/2b systems. These issues hindered the full adoption of single-bit techniues in industry. Among these problems is the stability of high-order modulators and the limit cycle behaviour. More importantly, there is no adaptive LMS structure of any kind in 1b/2b domain. The challenge in this problem is the harsh quantization that prevents straightforward LMS application. In this thesis, the focus has been made on three axes: designing new single-bit DSP applications, proposing novel approaches for stability analysis, and tacking the unresolved problems of 1b/2b adaptive filtering. Two structures for 1b digital comb filtering are proposed. A ternary DC blocker structure is also presented and performanc e is tested. We also proposed a single-bit multiplierless DC-blocking structure. The stability of a single-bit high-order signma-delta modulator is studied under dc inputs. A new approach for stability analysis is proposed based on analogy with PLL analysis. Finally we succeeded in designing 1b/2b Wiener-like filtering and introduced (for the first time) three 1b/2b adaptive schemes

Theory of polar domains in moir\'e heterostructures

The discovery of unconventional ferroelectric behavior in twisted bilayers has prompted the consideration of moir\'e heterostructures as polar materials. However, misconceptions about the nature and origin of the observed ferroelectricity indicate that a better theoretical understanding of the polar properties of moir\'e heterostructures is needed. In this paper, it is proposed, and verified with first-principles calculations, that all moir\'e heterostructures exhibit an out-of-plane moir\'e polar domain (MPD) structure. In transition metal dichalcogenide bilayers, an interlayer charge transfer occurs due to the change in stacking arrangements throughout the moir\'e superlattice, leading to a local out-of-plane dipole moment, with the magnitude and shape of the MPDs being dominated by the chalcogen atoms. While the MPDs in all heterostructures are sensitive to the moir\'e period, it is only in the homo-bilayers that they can be tuned with an out-of-plane electric field. The misconceptions about ferroelectricity in moir\'e heterostructures are addressed, and it is proposed that the only scenario in which the MPDs can be considered ferroelectric domains is via a global van der Waals sliding by one third of a unit cell in a homo-bilayer. Finally, a general theoretical discussion of the polar properties of moir\'e heterostructures is provided.Comment: Accepted versio