Random numbers (in one sense or another) have applications in computer simulation, Monte Carlo integration, cryptography, randomized computation, radar ranging, and other areas. It is impractical to generate random numbers in real life, instead sequences of numbers (or of bits) that appear to be ``random" yet repeatable are used in real life applications. These sequences are called pseudorandom sequences. To determine the suitability of pseudorandom sequences for applications, we need to study their properties, in particular, their statistical properties. The simplest property is the minimal period of the sequence. That is, the shortest number of steps until the sequence repeats. One important type of pseudorandom sequences is the sequences generated by feedback with carry shift registers (FCSRs). In this dissertation, we study statistical properties of N-ary FCSR sequences with odd prime connection integer q and least period (q-1)/2. These are called half-ℓ-sequences. More precisely, our work includes: The number of occurrences of one symbol within one period of a half-ℓ-sequence; The number of pairs of symbols with a fixed distance between them within one period of a half-ℓ-sequence; The number of triples of consecutive symbols within one period of a half-ℓ-sequence. In particular we give a bound on the number of occurrences of one symbol within one period of a binary half-ℓ-sequence and also the autocorrelation value in binary case. The results show that the distributions of half-ℓ-sequences are fairly flat. However, these sequences in the binary case also have some undesirable features as high autocorrelation values. We give bounds on the number of occurrences of two symbols with a fixed distance between them in an ℓ-sequence, whose period reaches the maximum and obtain conditions on the connection integer that guarantee the distribution is highly uniform. In another study of a cryptographically important statistical property, we study a generalization of correlation immunity (CI). CI is a measure of resistance to Siegenthaler's divide and conquer attack on nonlinear combiners. In this dissertation, we present results on correlation immune functions with regard to the q-transform, a generalization of the Walsh-Hadamard transform, to measure the proximity of two functions. We give two definitions of q-correlation immune functions and the relationship between them. Certain properties and constructions for q-correlation immune functions are discussed. We examine the connection between correlation immune functions and q-correlation immune functions.

This thesis studies the techniques of small-signal impedance measurement in three-phase power systems. Stability issue has become critically important since power electronics are highly applied in power distribution and conversion systems. Controlled output systems cause the risk of instability. In order to obtain the impedance model, an impedance extraction in D-Q reference frame algorithm is developed. This paper also applied Interpolated Fast Fourier Transform to increase accuracy of impedance model. Based on the voltage injection, Phase-Locked Loop, Park Transform, D-Q reference frame, and IPFFT. Three-phase system has been realigned on D-Q coordinate and impedance model is extracted in this form. Firstly, impedance extraction algorithm is designed by MATLAB/Simulink, the algorithm includes PLL, D-Q transform, and IPFFT is used to obtain magnitude and phase angle in frequency domain. Impedance matrices in D-Q frame may be solved through the relation between currents and voltages. Impedance model is made through various tests. Secondly, using the algorithm to test RL circuit to verify with real bode plot of the circuit. Then apply the algorithm on sophisticated circuit model. Finally, implement the algorithm on LabView/Multisim for future hardware tests. This paper clearly describes the objective of the research, the research problem and approaches, and experiment setup. This paper presents work conducted at the Smart Grid Power Systems Laboratory at University of South Florida.