Yes / A new wavelet transform (WT) is introduced based on the fractional properties of the traditional Fourier transform. The new wavelet follows from the fractional Fourier order which uniquely identifies the representation of an input function in a fractional domain. It exploits the combined advantages of WT and fractional Fourier transform (FrFT). The transform permits the identification of a transformed function based on the fractional rotation in time-frequency plane. The fractional rotation is then used to identify individual fractional daughter wavelets. This study is, for convenience, limited to one-dimension. Approach for discussing two or more dimensions is shown.

A definition of fractional Fourier transform as the generalization of ordinary Fourier transform is given at the beginning. Then due to optical reasons the fractional transform of a so-called chirp functions is considered in both theory and practical simulations. Because of a quadratic phase factor which is common in the definition of the transform and some optical concepts, a comparison between these concepts such as Fresnel diffraction, spherical wave, thin lens and free space propagation and the transform has been done. Finally an optical setup for performing the fractional transform is introduced.

No / In this work, we discuss two new multicarrier modulating kernels that can be adopted for multicarrier signaling. These multicarrier transforms are the fractional Forurier transform (FrFT) and discrete wavelet transforms (DWT). At first, we relate the transforms in terms of mathematical relationships, and then using numerical and simulation comparisons we show their performances in terms of bit error ratio (BER) for Multiple Input Multiple Output (MIMO) applications. Numerical results using BPSK and QPSK support that both can be applied for multicarrier signaling, however, it can be resource effective to drive the DWT as the baseband multicarrier kernel at the expense of the FrFT

No / Bifractional displacement operators, are introduced by performing two fractional Fourier transforms on displacement operators. They are shown to be special cases of elements of the group G , that contains both displacements and squeezing transformations. Acting with them on the vacuum we get various classes of coherent states, which we call bifractional coherent states. They are special classes of squeezed states which can be used for interpolation between various quantities in phase space methods. Using them we introduce bifractional Wigner functions A(α,β;θα,θβ)A(α,β;θα,θβ), which are a two-dimensional continuum of functions, and reduce to Wigner and Weyl functions in special cases. We also introduce bifractional Q-functions, and bifractional P-functions. The physical meaning of these quantities is discussed.

To a large extent mathematical transforms are applied on a signal to uncover information that is concealed, and the capability of such transforms is valuable for signal processing. One such transforms widely used in this area, is the conventional Fourier Transform (FT), which decomposes a stationary signal into different frequency components. However, a major drawback of the conventional transform is that it does not easily render itself to the analysis of non-stationary signals such as a frequency modulated (FM) or amplitude modulated (AM) signal. The different frequency components of complex signals cannot be easily distinguished and separated from one another using the conventional FT. So in this thesis an innovative mathematical transform, Fractional Fourier Transform (FRFT), has been considered, which is more suitable to process non-stationary signals such as FM signals and has the capability not only of distinguishing different frequency components of a multi-component signal but also separating them in a proper domain, different than the traditional time or frequency domain. The discrete-time FRFT (DFRFT) developed along with its derivatives, such as Multi-angle-DFRFT (MA-DFRFT), Slanted Spectrum and Spectrogram Based on Slanted Spectrum (SBSS) are tools belonging to the same FRFT family, and they could provide an effective approach to identify unknown signals and distinguish the different frequency components contained therein. Both artificial stationary and FM signals have been researched using the DFRFT and some derivative tools from the same family. Moreover, to accomplish a contrast with the traditional tools such as FFT and STFT, performance comparisons are shown to support the DFRFT as an effective tool in multi-component chirp signal analysis. The DFRFT taken at the optimum transform order on a single-component FM signal has provided higher degree of signal energy concentration compared to FFT results; and the Slanted Spectrum taken along the slant line obtained from the MA-DFRFT demonstration has shown much better discrimination between different frequency components of a multi-component FM signal. As a practical application of these tools, the motor current signal has been analyzed using the DFRFT and other tools from FRFT family to detect the presence of a motor bearing fault and obtain the fault signature frequency. The conclusion drawn about the applicability of DFRFT and other derivative tools on AM signals with very slowly varying FM phenomena was not encouraging. Tools from the FRFT family appear more effective on FM signals, whereas AM signals are more effectively analyzed using traditional methods like spectrogram or its derivatives. Such methods are able to identify the signature frequency of faults while using less computational time and memory.

abstract: For synthetic aperture radar (SAR) image formation processing, the chirp scaling algorithm (CSA) has gained considerable attention mainly because of its excellent target focusing ability, optimized processing steps, and ease of implementation. In particular, unlike the range Doppler and range migration algorithms, the CSA is easy to implement since it does not require interpolation, and it can be used on both stripmap and spotlight SAR systems. Another transform that can be used to enhance the processing of SAR image formation is the fractional Fourier transform (FRFT). This transform has been recently introduced to the signal processing community, and it has shown many promising applications in the realm of SAR signal processing, specifically because of its close association to the Wigner distribution and ambiguity function. The objective of this work is to improve the application of the FRFT in order to enhance the implementation of the CSA for SAR processing. This will be achieved by processing real phase-history data from the RADARSAT-1 satellite, a multi-mode SAR platform operating in the C-band, providing imagery with resolution between 8 and 100 meters at incidence angles of 10 through 59 degrees. The phase-history data will be processed into imagery using the conventional chirp scaling algorithm. The results will then be compared using a new implementation of the CSA based on the use of the FRFT, combined with traditional SAR focusing techniques, to enhance the algorithm's focusing ability, thereby increasing the peak-to-sidelobe ratio of the focused targets. The FRFT can also be used to provide focusing enhancements at extended ranges. / Dissertation/Thesis / M.S. Electrical Engineering 2011

Dans cette thèse, l'holographie numérique dans l'axe (DIH) est la principale méthode optique utilisée pour analyser des inclusions dans une gouttelette. L'holographie numérique dans l'axe est utilisée pour caractériser des inclusions du point de vue de leur taille, leur position 3D et leur trajectoire à l'intérieur de la gouttelette. Comme les particules sont situées à l'intérieur d'une gouttelette, le front d'onde incident sur l'inclusion est modifié avant qu'il l'illumine. Le défi de ce travail est double : premièrement de prendre en compte la forme de la gouttelette dans le modèle d'holographie et deuxièmement d'étendre l'analyse aux inclusions transparentes (type objet de phase). Pour décrire l'hologramme enregistré par le capteur CCD, l'intégrale d'Huygens-Fresnel et le formalisme des matrices ABCD ont été utilisés. Dans ce modèle, nous introduisons les polynômes de Zernike pour décrire la fonction de transmission d'une particule. Pour l'analyse des hologrammes, l'outil mathématique de la transformation de Fourier fractionnaire 2D (2D-FRFT) est utilisé pour restituer l'image des inclusions et dans ce cas une mesure la taille de l'inclusion et de sa position 3D sont réalisées. Les trajectoires des inclusions dans la goutte est possible avec un long temps de pose de l'obturateur du capteur CCD. Nous avons également proposé un nouveau modèle pour décrire des objets de phases quelconque et des particules opaques. Pour ce nouveau modèle, les mêmes procédés ont été utilisés. Dans le cas d'inclusions filiformes à l'intérieur d'une géométrie cylindrique comme un canal, une méthode de simulation d'imagerie interférométrique multi-coeurs est proposée. Dans ce cas, une somme de distributions de Dirac, localisées le long d'une droite, introduite dans l'intégrale de Fresnel généralisée (c'est-à-dire le formalisme des matrices ABCD et l'intégrale de Fresnel) permet d'obtenir un bon degré de similitude entre l'expérience et la simulation. / In this thesis, the digital in-line holography (DIH) is the main optical method used to analyze inclusions in a droplet. The digital in-line holography is used to characterize the inclusions in terms of of their size, their 3D position, and their trajectories inside the droplet. Since the particles are located within a droplet, the incident wavefront is changed before it illuminates the inclusions. The challenge of this work has two points : first to take into account the shape of the droplet in the holographic model and secondly to extend the analysis to the transparent inclusions (phase object). To describe the hologram recorded by the CCD sensor, the Huygens-Fresnel integral and the ABCD matrix formalism were used. In this model, we introduce the Zernike polynomials to describe the transmission function of a particle. For the analysis of holograms, the2D fractional Fourier transformation (2D-FRFT) is used to reconstruct the image of inclusions and in this case the size and their 3D position of the inclusions are performed.The trajectories of the inclusions in the drop are possible tracked with a long exposure shutter speed of the CCD. We also proposed a new simulation to describe objects of any phases and opaque particles. For this simulation, the same methods of reconstruction were used. In the case of micro-channel inclusions inside a cylindrical geometry such as a pipe, the interferometric imaging of multi-core pipe is proposed. In this case, summation of Dirac delta distribution, located along a line, introduced into the generalized Fresnel integral allows us to get a good agreement between the experiment and the simulation.

Unknown wireless devices that use receiver architectures with a mixer may be detected and located using stimulated emissions. Transmitting a known stimulation signal and correlating leaked mixer products allows measurement of the TOF and thus range. The FRFT improves the detection of the stimulated emissions by compressing the energy of the stimulated emissions to a single axis value. The stimulation signal has many parameters that may be optimized for maximum detection distance or minimum range error or somewhere in between. The primary limiting factor for the parameters is the processing time, as the algorithm to compute the discrete FRFT is computationally intensive at the time of this report. The tests performed in this investigation achieved 45+meters detection distance with < 3 meters of range error, with potential for farther detection distance.

Context: We identify a material or thing that can be seen and touched in the world as having structures at both coarser and finer levels of scale. Scaling problem presents in a branch of science concerned with the description, prediction understanding of natural phenomena and visual arts. A moon, for instance, may appear as having a roughly round shape is much larger than stars when seen from the earth. In the closer look, the moon is much smaller than the stars. The fact that objects in the world appear in different ways depending upon the scale of observation has important implications when analyzing measured data, such as images, with automatic methods [1]. The type of information we are seeking from a one-dimensional signal or two-dimensional image is only possible when we have the right amount of scale for the structure of an image or signal data. In many modern applications, the right scale need not be obvious at all, and we all need a complete mathematical analysis on this scaling problem. This thesis is shown how a mathematical theory is formulated when data or signal is describing at different scales. Objectives: The subtle patterns deforming in data that can foretell of a scaling problem? The main objectives of this thesis are to address the dynamic scaling pattern problem in computers and study the different methods, described in the latest issue of Science, are designed to identify the patterns in data. Method: The research methodology used in this thesis is the Fractional Fourier Transform. To recognize the pattern for a different level of scale to one or many components, we take the position and size of the object and perform the transform operation in any transform angle and deform the component by changing to another angle which influences the frequency, phase, and magnitude.  Results: We show that manipulation of Fractional Fourier transform can be used as a pattern recognition system. The introduced model has the flexibility to encode patterns to both time and frequency domain. We present a detailed structure of a dynamic pattern scaling problem. Furthermore, we show successful recognition results even though one or many components deformed to different levels using one-dimensional and two-dimensional patterns. Conclusions: The proposed algorithm FrFT has shown some advantages over traditional FFT due to its competitive performance in studying the pattern changes. This research work investigated that simulating the dynamic pattern scaling problem using FrFT. The Fractional Fourier transform does not do the scaling. Manipulating the Fractional Fourier transform can be helpful in perceiving the pattern changes. We cannot control the deformation but changing the parameters allow us to see what is happening in time and frequency domain.

This work concentrates on the application of diversity techniques and space time block coding for future high speed mobile wireless communications on multicarrier systems. At first, alternative multicarrier kernels robust for high speed doubly-selective fading channel are sought. They include the comparisons of discrete Fourier transform (DFT), fractional Fourier transform (FrFT) and wavelet transform (WT) multicarrier kernels. Different wavelet types, including the raised-cosine spectrum wavelets are implemented, evaluated and compared. From different wavelet families, orthogonal wavelets are isolated from detailed evaluations and comparisons as suitable for multicarrier applications. The three transforms are compared over a doubly-selective channel with the WT significantly outperforming all for high speed conditions up to 300 km/hr. Then, a new wavelet is constructed from an ideal filter approximation using established wavelet design algorithms to match any signal of interest; in this case under bandlimited criteria. The new wavelet showed better performance than other traditional orthogonal wavelets. To achieve MIMO communication, orthogonal space-time block coding, OSTBC, is evaluated next. First, the OSTBC is extended to assess the performance of the scheme over extended receiver diversity order. Again, with the extended diversity conditions, the OSTBC is implemented for a multicarrier system over a doubly-selective fading channel. The MIMO-OFDM systems (implemented using DFT and WT kernels) are evaluated for different operating frequencies, typical of LTE standard, with Doppler effects. It was found that, during high mobile speed, it is better to transmit OFDM signals using lower operating frequencies. The information theory for the 2-transmit antenna OSTBC does not support higher order implementation of multi-antenna systems, which is required for the future generation wireless communications systems. Instead of the OSTBC, the QO-STBC is usually deployed to support the design of higher order multi-antenna systems other than the 2-transmit antenna scheme. The performances of traditional QO-STBC methods are diminished by some off-diagonal (interference) terms such that the resulting system does not attain full diversity. Some methods for eliminating the interference terms have earlier been discussed. This work follows the construction of cyclic matrices with Hadamard matrix to derive QO-STBC codes construction which are N-times better than interference free QO-STBC, where N is the number of transmit antenna branches.