Representation and design of wavelets using unitary circuits

The representation of discrete, compact wavelet transformations (WTs) as circuits of local unitary gates is discussed. We employ a similar formalism as used in the multiscale representation of quantum many-body wave functions using unitary circuits, further cementing the relation established in the literature between classical and quantum multiscale methods. An algorithm for constructing the circuit representation of known orthogonal, dyadic, discrete WTs is presented, and the explicit representation for Daubechies wavelets, coiflets, and symlets is provided. Furthermore, we demonstrate the usefulness of the circuit formalism in designing WTs, including various classes of symmetric wavelets and multiwavelets, boundary wavelets, and biorthogonal wavelets

Comparative study and performance evaluation of MC-CDMA and OFDM over AWGN and fading channels environment

Η απαίτηση για εφαρμογές υψηλής ταχύτητας μετάδοσης δεδομένων έχει αυξηθεί σημαντικά τα τελευταία χρόνια. Η πίεση των χρηστών σήμερα για ταχύτερες επικοινωνίες, ανεξαρτήτως κινητής ή σταθερής, χωρίς επιπλέον κόστος είναι μια πραγματικότητα. Για να πραγματοποιηθούν αυτές οι απαιτήσεις, προτάθηκε ένα νέο σχήμα που συνδυάζει ψηφιακή διαμόρφωση και πολλαπλές προσβάσεις, για την ακρίβεια η Πολλαπλή Πρόσβαση με διαίρεση Κώδικα Πολλαπλού Φέροντος (Multi-Carrier Code Division Multiple Access MC-CDMA). Η εφαρμογή του Γρήγορου Μετασχηματισμού Φουριέ (Fast Fourier Transform,FFT) που βασίζεται στο (Orthogonal Frequency Division Multiplexing, OFDM) χρησιμοποιεί τις περίπλοκες λειτουργίες βάσεως και αντικαθίσταται από κυματομορφές για να μειώσει το επίπεδο της παρεμβολής. Έχει βρεθεί ότι οι μετασχηματισμένες κυματομορφές (Wavelet Transform,W.T.) που βασίζονται στον Haar είναι ικανές να μειώσουν το ISI και το ICI, που προκαλούνται από απώλειες στην ορθογωνιότητα μεταξύ των φερόντων, κάτι που τις καθιστά απλούστερες για την εφαρμογή από του FFT. Επιπλέον κέρδος στην απόδοση μπορεί να επιτευχθεί αναζητώντας μια εναλλακτική λειτουργία ορθογωνικής βάσης και βρίσκοντας ένα καλύτερο μετασχηματισμό από του Φουριέ (Fourier) και τον μετασχηματισμό κυματομορφής (Wavelet Transform). Στην παρούσα εργασία, υπάρχουν τρία προτεινόμενα μοντέλα. Το 1ο, ( A proposed Model ‘1’ of OFDM based In-Place Wavelet Transform), το 2ο, A proposed Model ‘2’ based In-Place Wavelet Transform Algorithm and Phase Matrix (P.M) και το 3ο, A proposed Model ‘3’ of MC-CDMA Based on Multiwavelet Transform. Οι αποδόσεις τους συγκρίθηκαν με τα παραδοσιακά μοντέλα μονού χρήστη κάτω από διαφορετικά κανάλια (Κανάλι AWGN, επίπεδη διάλειψη και επιλεκτική διάλειψη).The demand for high data rate wireless multi-media applications has increased significantly in the past few years. The wireless user’s pressure towards faster communications, no matter whether mobile, nomadic, or fixed positioned, without extra cost is nowadays a reality. To fulfill these demands, a new scheme which combines wireless digital modulation and multiple accesses was proposed in the recent years, namely, Multicarrier-Code Division Multiple Access (MC-CDMA). The Fourier based OFDM uses the complex exponential bases functions and it is replaced by wavelets in order to reduce the level of interference. It is found that the Haar-based wavelets are capable of reducing the ISI and ICI, which are caused by the loss in orthogonality between the carriers. Further performance gains can be made by looking at alternative orthogonal basis functions and finding a better transform rather than Fourier and wavelet transform. In this thesis, there are three proposed models [Model ‘1’ (OFDM based on In-Place Wavelet Transform, Model ‘2’ (MC-CDMA based on IP-WT and Phase Matrix) and Model ‘3’ (MC-CDMA based on Multiwavelet Transform)] were created and then comparison their performances with the traditional models for single user system were compared under different channel characteristics (AWGN channel, flat fading and selective fading). The conclusion of my study as follows, the models (1) was achieved much lower bit error rates than traditional models based FFT. Therefore these models can be considered as an alternative to the conventional MC-CDMA based FFT. The main advantage of using In-Place wavelet transform in the proposed models that it does not require an additional array at each sweep such as in ordered Fast Haar wavelet transform, which makes it simpler for implementation than FFT. The model (2) gave a new algorithm based on In-Place wavelet transform with first level processing multiple by PM was proposed. The model (3) gave much lower bit error than other two models in additional to traditional models

Correspondence between Multiwavelet Shrinkage/Multiple Wavelet Frame Shrinkage and Nonlinear Diffusion

There are numerous methodologies for signal and image denoising. Wavelet, wavelet frame shrinkage, and nonlinear diffusion are effective ways for signal and image denoising. Also, multiwavelet transforms and multiple wavelet frame transforms have been used for signal and image denoising. Multiwavelets have important property that they can possess the orthogonality, short support, good performance at the boundaries, and symmetry simultaneously. The advantage of multiwavelet transform for signal and image denoising was illustrated by Bui et al. in 1998. They showed that the evaluation of thresholding on a multiwavelet basis has produced good results. Further, Strela et al. have showed that the decimated multiwavelet denoising provides superior results than decimated conventional (scalar) wavelet denoising. Mrazek, Weickert, and Steidl in 2003 examined the association between one-dimensional nonlinear diffusion and undecimated Haar wavelet shrinkage. They proved that nonlinear diffusion could be presented by using wavelet shrinkage. High-order nonlinear diffusion in terms of one-dimensional frame shrinkage and two-dimensional frame shrinkage were presented in 2012 by Jiang, and in 2013 by Dong, Jiang, and Shen, respectively. They obtained that the correspondence between both approaches leads to a different form of diffusion equation that mixes benefits from both approaches. The objective of this dissertation is to study the correspondence between one-dimensional multiwavelet shrinkage and high-order nonlinear diffusion, and to study high-order nonlinear diffusion in terms of one-dimensional multiple frame shrinkage also well. Further, this dissertation formulates nonlinear diffusion in terms of 2D multiwavelet shrinkage and 2D multiple wavelet frame shrinkage. From the experiment results, it can be inferred that nonlinear diffusion in terms of multiwavelet shrinkage/multiple frame shrinkage gives better results than a scalar case. On the whole, this dissertation expands nonlinear diffusion in terms of wavelet shrinkage and nonlinear diffusion in terms of frame shrinkage from the scalar wavelets and frames to the multiwavelets and multiple frames

Highly Symmetric Multiple Bi-Frames for Curve and Surface Multiresolution Processing

Wavelets and wavelet frames are important and useful mathematical tools in numerous applications, such as signal and image processing, and numerical analysis. Recently, the theory of wavelet frames plays an essential role in signal processing, image processing, sampling theory, and harmonic analysis. However, multiwavelets and multiple frames are more flexible and have more freedom in their construction which can provide more desired properties than the scalar case, such as short compact support, orthogonality, high approximation order, and symmetry. These properties are useful in several applications, such as curve and surface noise-removing as studied in this dissertation. Thus, the study of multiwavelets and multiple frames construction has more advantages for many applications. Recently, the construction of highly symmetric bi-frames for curve and surface multiresolution processing has been investigated. The 6-fold symmetric bi-frames, which lead to highly symmetric analysis and synthesis bi-frame algorithms, have been introduced. Moreover, these multiple bi-frame algorithms play an important role on curve and surface multiresolution processing. This dissertation is an extension of the study of construction of univariate biorthogonal wavelet frames (bi-frames for short) or dual wavelet frames with each framelet being symmetric in the scalar case. We will expand the study of biorthogonal wavelets and bi-frames construction from the scalar case to the vector case to construct biorthogonal multiwavelets and multiple bi-frames in one-dimension. In addition, we will extend the study of highly symmetric bi-frames for triangle surface multiresolution processing from the scalar case to the vector case. More precisely, the objective of this research is to construct highly symmetric biorthogonal multiwavelets and multiple bi-frames in one and two dimensions for curve and surface multiresolution processing. It runs in parallel with the scalar case. We mainly present the methods of constructing biorthogonal multiwavelets and multiple bi-frames in both dimensions by using the idea of lifting scheme. On the whole, we discuss several topics include a brief introduction and discussion of multiwavelets theory, multiresolution analysis, scalar wavelet frames, multiple frames, and the lifting scheme. Then, we present and discuss some results of one-dimensional biorthogonal multiwavelets and multiple bi-frames for curve multiresolution processing with uniform symmetry: type I and type II along with biorthogonality, sum rule orders, vanishing moments, and uniform symmetry for both types. In addition, we present and discuss some results of two-dimensional biorthogonal multiwavelets and multiple bi-frames and the multiresolution algorithms for surface multiresolution processing. Finally, we show experimental results on curve and surface noise-removing by applying our multiple bi-frame algorithms

Sampling—50 Years After Shannon

This paper presents an account of the current state of sampling, 50 years after Shannon's formulation of the sampling theorem. The emphasis is on regular sampling where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we re-interpret Shannon's sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of "shift-invariant" functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) pre-filters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., non-bandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multi-wavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned