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Thesis (MEng) -- Stellenbosch University, 2014. / ENGLISH ABSTRACT: Beamforming is a technique used to combine signals from an array of antennas to effectively synthesize a single aperture and beam. In the Radio Astronomy community the technique is used to obtain a desirable beam pattern as well as to electronically point the beam of an array. Next generation radio telescopes such as the Square Kilometre Array (SKA) surpass current technology and will extensively make use of beamforming techniques. Various factors determine the output of a beamformer; however, given an array with a fixed configuration only the weights applied to the incoming signals affect the synthesized aperture and beam. Furthermore, the incoming data must be processed in real-time, at a rate equal to the input-output rate of the processor. Both the weighting function and the real-time implementation of beamforming, are the primary subjects of this thesis. In this thesis various deterministic weighting functions are investigated. The algorithms are implemented in a matlab program, serving as a simulation tool for investigating the techniques. The program is verified by comparing the expected theoretical outcomes to the simulated output. For the program the following functionalities are included: a steering technique, spectral weighting, Dolph-Chebychev, and the Least Square Error algorithm. Applications of these techniques is investigated and their prominence in the Radio Astronomy community is established. For the real-time beamformer implementation, the UniBoard platform configured with beamformer firmware, is investigated. This is important as the UniBoard is an excellent example of a beamformer implementation within the Radio Astronomy community. The architecture is used to emulate a linear array by implementing a python control script, where the output corresponded accurately with the expected theoretical values. The thesis also constitutes the design and implementation of a digital frequency domain beamformer on the ROACH board. This processing board is employed by the Karoo Array Telescope (KAT-7) in South Africa. This work is therefore important as it demonstrates a beamformer implementation on an architecture in use by the Radio Astronomy community. An antenna array is designed and built for the verification of the beamformer design. Results with a good degree of accuracy were obtained and where errors exist they are discussed. / AFRIKKANSE OPSOMMING: Bundelvorming is ’n tegniek waarmee die seine van ’n antenna samestelling gekombineer word om ’n enkele effektiewe stralingsvlak en stralingspatroon te sintiseer. In die Radio Astronomie gemeenskap word die tegniek gebruik om ’n gewenste stralingspatroon te sintiseer sowel as om die rigting van die patroon elektronies te beheer. Die Square Kilometre Array (SKA) is ’n toekomstige radioteleskoop en sal grootliks gebruik maak van bundelvorming tegnieke. Die uitset van bundelvormers word geaffekteer deur verskeie faktore, maar vir ’n gegewe samestelling is dit net die gewigsfunksies wat toegepas word op die inkomende seine wat die gesintiseerde patroon kan beheer. Verder moet die inkomende data verwerk word teen ’n tempo gelykstaande aan die inset-en-uitsetkoers van die verwerker. Die gewigsfunksie so wel as die implementasie van die bundelvormer is albei primêre onderwerpe van die tesis. ’n Verskeindenheid van deterministiese bundelvormingstegnieke sal ondersoek word in hierdie tesis. Die algoritmes is in ’n matlab program geïmplementeer vir simulasie doeleindes. Die program is geverifieër deur die uitset te vergelyk met die verwagte teoretiese waardes. Die program sluit die volgende funksies in: ’n rigting beheer algoritme, spektraalgewigte, Dolph-Chebychev, en die minste vierkantsfout algoritme. Hierdie tegnieke is van belang weens hul toepassing in die Radio Astronomie gemeenskap. Vir die implementasie van ’n bundelvormer is die UniBoard hardeware, geprogrameer in ’n bundelvormings modus, van gebruik gemaak. Hierdie aspek is belangrik omdat die Uni- Board ’n goeie voorbeeld van ’n geïmplementeerde bundelvormer in die Radio Astronomie gemeenskap is. Die UniBoard word gebruik om ’n lineêre samestelling te emuleer deur in python ’n beheer skrip te skryf, waar die uitset van die emuleerder akkuraat ooreenstem met die verwagte waardes. Die tesis behels ook die ontwerp en implementasie van ’n digitale frekwensiegebied bundelvormer op die ROACH platform. Hierdie verwerker word tans gebruik in die Karoo Array Telescope (KAT-7) in Suid-Afrika. Hierdie werk is dus belangrik omdat dit die implementasie van ’n bundelvormer op tegnologie wat huidiglik in die Radio Astronomie gemeenskap gebruik word demonstreer. Daarbenewens is ’n antenna samestelling ontwerp en gebou om die bundelvormer te verifieër. Die resultate is akkuraat tot ’n redelike mate. Waar daar ’n fout onstaan het word dit in die tesis bespreek.

It is to be hoped that the methods described in the first part of this thesis will be extensively applied to imaging devices and to photographic emulsions, since this is the only way in which an unambiguous figure for the performance of the detector may be derived. Lack of attention to such concepts has produced many pitfalls, notably the enormous (and imaginary) gains reported earlier for various types of image tube, which, when they were not realised in practice, produced some considerable despondency. The work of Shaw(60) on aerial films is an out- standing example of thorough application of Information Theory to the photographic process. His results are formally the same as those of Chapter I of this thesis. The results derived enable a prediction of signal-to-noise-ratio (accuracy) and information gain for the image tube under any conditions of exposure. The results also point to the necessity of knowing, for instance, the IMF of the spectrograph used with the detector, in order to fully exploit the storage capacity of the detector. Part II of this work has described the manner in which the Spectracon, as a particular example of an astronomical image tube, may be brought into routine use as part of the observational astronomer's instrument capability. Much is still to be achieved in this field. In particular, the emulsions are in no way comparable in standard (understandably, they are still a laboratory - rather than commercial - commodity) with the best astronomical emulsions. Similarly, such is the experimental complexity in building a tube, that to produce a flawless cathode and window ( and both are focal surfaces) is asking a great deal. But it is not asking for the impossible, and it is foreseeable that in time such difficulties will be eliminated. There remains the problem of the small area of the cathode. In effect, while the information storage capacity of the image tube is considerably greater than that of astronomical emulsions, its limited area rather nullifies this advantage in that, with suitable coding, all the information in the small cathode area could be spread onto one large astronomical plate. However, to do this the exposing intensity must be reduced, and this, because of the nonlinearity of the photographic process, reduces the efficiency. Thus the crucial advantage in using an image tube must be in exploiting its higher detecting efficiency. If programs consist in looking at small regions of spectrum (or of sky), then the advantages of the image tube are obvious. Some of this work has already been reported in a paper read to the 3rd Symposium on Photoelectronic Imaging Devices.

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<p>Both the Laser Interferometer Gravitational Wave Observatory (LIGO) and the Laser Interferometer Space Antenna (LISA) will over the next decade detect gravitational waves emitted by the motion of compact objects (e.g. black hole and neutron star binaries). This thesis presents methods to improve (i) LIGO detector quality, (ii) our knowledge of waveforms for certain LIGO and LISA sources, and (iii) models for the rate of detectability of a particular LISA source.</p> <p>1) Plunge of compact object into a supermassive black hole: LISA should detect many inspirals of compact objects into supermassive black holes (~ 10⁵-10⁷ M_⊙). Since the inspiral of each compact object terminates shortly after the inspiralling object reaches its last stable orbit, the late-stage inspiral waveform provides insight into the location of the last stable orbit and strong-field relativity. I discovered that while LISA will easily see the overall inspiral (consisting of many cycles before plunge), the present LISA design will just miss detecting the waves emitted from the transition from inspiral to plunge.</p> <p>2) Scheme to reduce thermoelastic noise in advanced LIGO: After its first upgrade, LIGO will have its sensitivity limited by thermoelastic noise. [Thermoelastic noise occurs because milimeter-scale thermal fluctuations in the mirror bulk expand and contract, causing the mirror surface to shimmer.] The interferometer's sensitivity could be enhanced substantially by reducing thermoelastic noise. In collaboration with Kip Thorne, Erika d'Ambrosio, Sergey Vyatchanin, and Sergey Strigin, I developed a proposal to reduce thermoelastic noise in advanced-LIGO by switching the LIGO cavity optics from simple spherical mirrors to a new, Mexican-hat shape.</p> <p>3) Geometric-optics-based analysis of stability of symmetric-hyperbolic formulations of Einstein's equations: Einstein's equations must be evolved numerically to predict accurate waveforms for the late stages of binary black hole inspiral and merger. But no matter which representation of Einstein's equations is used, numerical simulations rarely run long. For examle, for first-order symmetric-hyperbolic (FOSH) formulations of Einstein's evolution equations, sometimes exact but unphysical solutions grow so large that the evolution fails. For FOSH formulations, I found easily-understood solutions (wave packets) and used them to predict which formulations will be particularly ill-behaved.</p>