Generalized discrete Fourier transform with non-linear phase : theory and design

Constant modulus transforms like discrete Fourier transform (DFT), Walsh transform, and Gold codes have been successfully used over several decades in various engineering applications, including discrete multi-tone (DMT), orthogonal frequency division multiplexing (OFDM) and code division multiple access (CDMA) communications systems. Among these popular transforms, DFT is a linear phase transform and widely used in multicarrier communications due to its performance and fast algorithms. In this thesis, a theoretical framework for Generalized DFT (GDFT) with nonlinear phase exploiting the phase space is developed. It is shown that GDFT offers sizable correlation improvements over DFT, Walsh, and Gold codes. Brute force search algorithm is employed to obtain orthogonal GDFT code sets with improved correlations. Design examples and simulation results on several channel types presented in the thesis show that the proposed GDFT codes, with better auto and cross-correlation properties than DFT, lead to better bit-error-rate performance in all multi-carrier and multi-user communications scenarios investigated. It is also highlighted how known constant modulus code families such as Walsh, Walsh-like and other codes are special solutions of the GDFT framework. In addition to theoretical framework, practical design methods with computationally efficient implementations of GDFT as enhancements to DFT are presented in the thesis. The main advantage of the proposed method is its ability to design a wide selection of constant modulus orthogonal code sets based on the desired performance metrics mimicking the engineering .specs of interest. Orthogonal Frequency Division Multiplexing (OFDM) is a leading candidate to be adopted for high speed 4G wireless communications standards due to its high spectral efficiency, strong resistance to multipath fading and ease of implementation with Fast Fourier Transform (FFT) algorithms. However, the main disadvantage of an OFDM based communications technique is of its high PAPR at the RF stage of a transmitter. PAPR dominates the power (battery) efficiency of the radio transceiver. Among the PAPR reduction methods proposed in the literature, Selected Mapping (SLM) method has been successfully used in OFDM communications. In this thesis, an SLM method employing GDFT with closed form phase functions rather than fixed DFT for PAPR reduction is introduced. The performance improvements of GDFT based SLM PAPR reduction for various OFDM communications scenarios including the WiMAX standard based system are evaluated by simulations. Moreover, an efficient implementation of GDFT based SLM method reducing computational cost of multiple transform operations is forwarded. Performance simulation results show that power efficiency of non-linear RF amplifier in an OFDM system employing proposed method significantly improved

Design of Block Transceivers with Decision Feedback Detection

This paper presents a method for jointly designing the transmitter-receiver pair in a block-by-block communication system that employs (intra-block) decision feedback detection. We provide closed-form expressions for transmitter-receiver pairs that simultaneously minimize the arithmetic mean squared error (MSE) at the decision point (assuming perfect feedback), the geometric MSE, and the bit error rate of a uniformly bit-loaded system at moderate-to-high signal-to-noise ratios. Separate expressions apply for the ``zero-forcing'' and ``minimum MSE'' (MMSE) decision feedback structures. In the MMSE case, the proposed design also maximizes the Gaussian mutual information and suggests that one can approach the capacity of the block transmission system using (independent instances of) the same (Gaussian) code for each element of the block. Our simulation studies indicate that the proposed transceivers perform significantly better than standard transceivers, and that they retain their performance advantages in the presence of error propagation.Comment: 14 pages, 8 figures, to appear in the IEEE Transactions on Signal Processin

Group Frames with Few Distinct Inner Products and Low Coherence

Frame theory has been a popular subject in the design of structured signals and codes in recent years, with applications ranging from the design of measurement matrices in compressive sensing, to spherical codes for data compression and data transmission, to spacetime codes for MIMO communications, and to measurement operators in quantum sensing. High-performance codes usually arise from designing frames whose elements have mutually low coherence. Building off the original "group frame" design of Slepian which has since been elaborated in the works of Vale and Waldron, we present several new frame constructions based on cyclic and generalized dihedral groups. Slepian's original construction was based on the premise that group structure allows one to reduce the number of distinct inner pairwise inner products in a frame with $n$ elements from $\frac{n(n-1)}{2}$ to $n-1$. All of our constructions further utilize the group structure to produce tight frames with even fewer distinct inner product values between the frame elements. When $n$ is prime, for example, we use cyclic groups to construct $m$-dimensional frame vectors with at most $\frac{n-1}{m}$ distinct inner products. We use this behavior to bound the coherence of our frames via arguments based on the frame potential, and derive even tighter bounds from combinatorial and algebraic arguments using the group structure alone. In certain cases, we recover well-known Welch bound achieving frames. In cases where the Welch bound has not been achieved, and is not known to be achievable, we obtain frames with close to Welch bound performance

Quantum algorithms for highly non-linear Boolean functions

Attempts to separate the power of classical and quantum models of computation have a long history. The ultimate goal is to find exponential separations for computational problems. However, such separations do not come a dime a dozen: while there were some early successes in the form of hidden subgroup problems for abelian groups--which generalize Shor's factoring algorithm perhaps most faithfully--only for a handful of non-abelian groups efficient quantum algorithms were found. Recently, problems have gotten increased attention that seek to identify hidden sub-structures of other combinatorial and algebraic objects besides groups. In this paper we provide new examples for exponential separations by considering hidden shift problems that are defined for several classes of highly non-linear Boolean functions. These so-called bent functions arise in cryptography, where their property of having perfectly flat Fourier spectra on the Boolean hypercube gives them resilience against certain types of attack. We present new quantum algorithms that solve the hidden shift problems for several well-known classes of bent functions in polynomial time and with a constant number of queries, while the classical query complexity is shown to be exponential. Our approach uses a technique that exploits the duality between bent functions and their Fourier transforms.Comment: 15 pages, 1 figure, to appear in Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'10). This updated version of the paper contains a new exponential separation between classical and quantum query complexit

Biangular vectors

viii, 133 leaves ; 29 cmThis thesis introduces unit weighing matrices, a generalization of Hadamard matrices. When dealing with unit weighing matrices, a lot of the structure that is held by Hadamard matrices is lost, but this loss of rigidity allows these matrices to be used in the construction of certain combinatorial objects. We are able to fully classify these matrices for many small values by defining equivalence classes analogous to those found with Hadamard matrices. We then proceed to introduce an extension to mutually unbiased bases, called mutually unbiased weighing matrices, by allowing for different subsets of vectors to be orthogonal. The bounds on the size of these sets of matrices, both lower and upper, are examined. In many situations, we are able to show that these bounds are sharp. Finally, we show how these sets of matrices can be used to generate combinatorial objects such as strongly regular graphs and association schemes