The Nested Periodic Subspaces: Extensions of Ramanujan Sums for Period Estimation

In the year 1918, the Indian mathematician Srinivasa Ramanujan proposed a set of sequences called Ramanujan Sums as bases to expand arithmetic functions in number theory. Today, exactly a 100 years later, we will show that these sequences re-emerge as exciting tools in a completely different context: For the extraction of periodic patterns in data. Combined with the state-of-the-art techniques of DSP, Ramanujan Sums can be used as the starting point for developing powerful algorithms for periodicity applications. The primary inspiration for this thesis comes from a recent extension of Ramanujan sums to subspaces known as the Ramanujan subspaces. These subspaces were designed to span any sequence with integer periodicity, and have many interesting properties. Starting with Ramanujan subspaces, this thesis first develops an entire family of such subspace representations for periodic sequences. This family, called Nested Periodic Subspaces due to their unique structure, turns out to be the least redundant sets of subspaces that can span periodic sequences. Three classes of new algorithms are proposed using the Nested Periodic Subspaces: dictionaries, filter banks, and eigen-space methods based on the auto-correlation matrix of the signal. It will be shown that these methods are especially advantageous to use when the data-length is short, or when the signal is a mixture of multiple hidden periods. The dictionary techniques were inspired by recent advances in sparsity based compressed sensing. Apart from the l1 norm based convex programs currently used in other applications, our dictionaries can admit l2 norm formulations that have linear and closed form solutions, even when the systems is under-determined. A new filter bank is also proposed using the Ramanujan sums. This, named the Ramanujan Filter Bank, can accurately track the instantaneous period for signals that exhibit time varying periodic nature. The filters in the Ramanujan Filter Bank have simple integer valued coefficients, and directly tile the period vs time plane, unlike classical STFT (Short Time Fourier Transform) and wavelets, which tile the time-frequency plane. The third family of techniques developed here are a generalization of the classic MUSIC (MUltiple SIgnal Classification) algorithm for periodic signals. MUSIC is one of the most popular techniques today for line spectral estimation. However, periodic signals are not just any unstructured line spectral signals. There is a nice harmonic spacing between the lines which is not exploited by plain MUSIC. We will show that one can design much more accurate adaptations of MUSIC using Nested Periodic Subspaces. Compared to prior variants of MUSIC for the periodicity problem, our approach is much faster and yields much more accurate results for signals with integer periods. This work is also the first extension of MUSIC that uses simple integer valued basis vectors instead of using traditional complex-exponentials to span the signal subspace. The advantages of the new methods are demonstrated both on simulations, as well as real world applications such as DNA micro-satellites, protein repeats and absence seizures. Apart from practical contributions, the theory of Nested Periodic Subspaces offers answers to a number of fundamental questions that were previously unanswered. For example, what is the minimum contiguous data-length needed to be able to identify the period of a signal unambiguously? Notice that the answer we seek is a fundamental identifiability bound independent of any particular period estimation technique. Surprisingly, this basic question has never been answered before. In this thesis, we will derive precise expressions for the minimum necessary and sufficient datalengths for this question. We also extend these bounds to the context of mixtures of periodic signals. Once again, even though mixtures of periodic signals often occur in many applications, aspects such as the unique identifiability of the component periods were never rigorously analyzed before. We will present such an analysis as well. While the above question deals with the minimum contiguous datalength required for period estimation, one may ask a slightly different question: If we are allowed to pick the samples of a signal in a non-contiguous fashion, how should we pick them so that we can estimate the period using the least number of samples? This question will be shown to be quite difficult to answer in general. In this thesis, we analyze a smaller case in this regard, namely, that of resolving between two periods. It will be shown that the analysis is quite involved even in this case, and the optimal sampling pattern takes an interesting form of sparsely located bunches. This result can also be extended to the case of multi-dimensional periodic signals. We very briefly address multi-dimensional periodicity in this thesis. Most prior DSP literature on multi-dimensional discrete time periodic signals assumes the period to be parallelepipeds. But as shown by the artist M. C. Escher, one can tile the space using a much more diverse variety of shapes. Is it always possible to account for such other periodic shapes using the traditional notion of parallelepiped periods? An interesting analysis in this regard is presented towards the end of the thesis.</p

Advances in the Field of Electrical Machines and Drives

Electrical machines and drives dominate our everyday lives. This is due to their numerous applications in industry, power production, home appliances, and transportation systems such as electric and hybrid electric vehicles, ships, and aircrafts. Their development follows rapid advances in science, engineering, and technology. Researchers around the world are extensively investigating electrical machines and drives because of their reliability, efficiency, performance, and fault-tolerant structure. In particular, there is a focus on the importance of utilizing these new trends in technology for energy saving and reducing greenhouse gas emissions. This Special Issue will provide the platform for researchers to present their recent work on advances in the field of electrical machines and drives, including special machines and their applications; new materials, including the insulation of electrical machines; new trends in diagnostics and condition monitoring; power electronics, control schemes, and algorithms for electrical drives; new topologies; and innovative applications

Treatise on Hearing: The Temporal Auditory Imaging Theory Inspired by Optics and Communication

A new theory of mammalian hearing is presented, which accounts for the auditory image in the midbrain (inferior colliculus) of objects in the acoustical environment of the listener. It is shown that the ear is a temporal imaging system that comprises three transformations of the envelope functions: cochlear group-delay dispersion, cochlear time lensing, and neural group-delay dispersion. These elements are analogous to the optical transformations in vision of diffraction between the object and the eye, spatial lensing by the lens, and second diffraction between the lens and the retina. Unlike the eye, it is established that the human auditory system is naturally defocused, so that coherent stimuli do not react to the defocus, whereas completely incoherent stimuli are impacted by it and may be blurred by design. It is argued that the auditory system can use this differential focusing to enhance or degrade the images of real-world acoustical objects that are partially coherent. The theory is founded on coherence and temporal imaging theories that were adopted from optics. In addition to the imaging transformations, the corresponding inverse-domain modulation transfer functions are derived and interpreted with consideration to the nonuniform neural sampling operation of the auditory nerve. These ideas are used to rigorously initiate the concepts of sharpness and blur in auditory imaging, auditory aberrations, and auditory depth of field. In parallel, ideas from communication theory are used to show that the organ of Corti functions as a multichannel phase-locked loop (PLL) that constitutes the point of entry for auditory phase locking and hence conserves the signal coherence. It provides an anchor for a dual coherent and noncoherent auditory detection in the auditory brain that culminates in auditory accommodation. Implications on hearing impairments are discussed as well.Comment: 603 pages, 131 figures, 13 tables, 1570 reference