A principal component approach to space-based gravitational wave astronomy

The current approach to data analysis for the Laser Interferometry Space Antenna (LISA) depends on the time delay interferometry observables (TDI) which have to be generated before any weak signal detection can be performed. These are linear combinations of the raw data with appropriate time shifts that lead to the cancellation of the laser frequency noises. This is possible because of the multiple occurrences of the same noises in the different raw data. Originally, these observables were manually generated starting with LISA as a simple stationary array and then adjusted to incorporate the antenna's motions. However, none of the observables survived the flexing of the arms in that they did not lead to cancellation with the same structure. The principal component approach is another way of handling these noises that was presented by Romano and Woan which simplified the data analysis by removing the need to create them before the analysis. This method also depends on the multiple occurrences of the same noises but, instead of using them for cancellation, it takes advantage of the correlations that they produce between the different readings. These correlations can be expressed in a noise (data) covariance matrix which occurs in the Bayesian likelihood function when the noises are assumed be Gaussian. Romano and Woan showed that performing an eigendecomposition of this matrix produced two distinct sets of eigenvalues that can be distinguished by the absence of laser frequency noise from one set. The transformation of the raw data using the corresponding eigenvectors also produced data that was free from the laser frequency noises. This result led to the idea that the principal components may actually be time delay interferometry observables since they produced the same outcome, that is, data that are free from laser frequency noise. The aims here were (i) to investigate the connection between the principal components and these observables, (ii) to prove that the data analysis using them is equivalent to that using the traditional observables and (ii) to determine how this method adapts to real LISA especially the flexing of the antenna. For testing the connection between the principal components and the TDI observables a 10x 10 covariance matrix containing integer values was used in order to obtain an algebraic solution for the eigendecomposition. The matrix was generated using fixed unequal arm lengths and stationary noises with equal variances for each noise type. Results confirm that all four Sagnac observables can be generated from the eigenvectors of the principal components. The observables obtained from this method however, are tied to the length of the data and are not general expressions like the traditional observables, for example, the Sagnac observables for two different time stamps were generated from different sets of eigenvectors. It was also possible to generate the frequency domain optimal AET observables from the principal components obtained from the power spectral density matrix. These results indicate that this method is another way of producing the observables therefore analysis using principal components should give the same results as that using the traditional observables. This was proven by fact that the same relative likelihoods (within 0.3%) were obtained from the Bayesian estimates of the signal amplitude of a simple sinusoidal gravitational wave using the principal components and the optimal AET observables. This method fails if the eigenvalues that are free from laser frequency noises are not generated. These are obtained from the covariance matrix and the properties of LISA that are required for its computation are the phase-locking, arm lengths and noise variances. Preliminary results of the effects of these properties on the principal components indicate that only the absence of phase-locking prevented their production. The flexing of the antenna results in time varying arm lengths which will appear in the covariance matrix and, from our toy model investigations, this did not prevent the occurrence of the principal components. The difficulty with flexing, and also non-stationary noises, is that the Toeplitz structure of the matrix will be destroyed which will affect any computation methods that take advantage of this structure. In terms of separating the two sets of data for the analysis, this was not necessary because the laser frequency noises are very large compared to the photodetector noises which resulted in a significant reduction in the data containing them after the matrix inversion. In the frequency domain the power spectral density matrices were block diagonals which simplified the computation of the eigenvalues by allowing them to be done separately for each block. The results in general showed a lack of principal components in the absence of phase-locking except for the zero bin. The major difference with the power spectral density matrix is that the time varying arm lengths and non-stationarity do not show up because of the summation in the Fourier transform.

Bull's eye leaky wave antenna for terrestrial and space applications

This work presents the study of Bull's eye antenna designs, a type of leaky wave antenna (LWA), operating in the 60 GHz band. This band emerged as a new standard for specific terrestrial and space applications because the radio spectrumbecomes more congested up to the millimetre-wave band, starting at 30 GHz. Built on existing Bull's eye antenna designs, novel structures were simulated, fabricated and measured, so as to provide more exibility in the implementation of wireless solutions at this frequency. Firstly, the study of a 60 GHz Bull's eye antenna for straightforward integration onto a CubeSat is presented. An investigation of the design is carried out, from the description of the radiation mechanism supported by simulation results, to the radiation pattern measurement of a prototype which provides a gain of 19.1 dBi at boresight. Another design, based on a modified feed structure, uses a microstrip to waveguide transition to provide easier and inexpensive integration of a Bull's eye antenna onto a planar circuit. Secondly, the design of Bull's eye antennas capable of creating beam deflection and multi-beam is presented. In particular, a detail study of the deflection mechanism is proposed, followed by the demonstration of a Bull's eye antenna generating two separate beams at ±16° away from the boresight. In addition, a novel mechanically steerable Bull's eye antenna, based on the division of the corrugated area in paired sectors is presented. A prototype was fabricated and measured. It generated double beams at ±8° and ±15° from the boresight, and a single boresight beam. Thirdly, a Bull's eye antenna capable of generating two simultaneous orbital angular momentum (OAM) modes l = 3 is proposed. The design is based on a circular travelling wave resonator and would allow channel capacity increase through OAM multiplexing. An improved design based on two stacked OAM Bull's eye antennas capable of producing four orthogonal OAM modes l = (±3,±13) simultaneously is presented. A novel receiving scheme based on discretely sampled partial aperture receivers (DSPAR) is then introduced. This solution could provide a lower windage and a lower cost of implementation than current whole or partial continuous aperture.