Bingo game is a popular and interesting game. This paper considers some interesting properties of the Bingo game played in Taiwan. We discuss how to use the computer to calculate some interesting probability value for various sizes of bingo games. For example, the expectation of the calls to hit a Bingo and the expectation of the Bingo number after the $k$th number is called. Some interesting results are also discussed.

abstract: Redundant Binary (RBR) number representations have been extensively used in the past for high-throughput Digital Signal Processing (DSP) systems. Data-path components based on this number system have smaller critical path delay but larger area compared to conventional two's complement systems. This work explores the use of RBR number representation for implementing high-throughput DSP systems that are also energy-efficient. Data-path components such as adders and multipliers are evaluated with respect to critical path delay, energy and Energy-Delay Product (EDP). A new design for a RBR adder with very good EDP performance has been proposed. The corresponding RBR parallel adder has a much lower critical path delay and EDP compared to two's complement carry select and carry look-ahead adder implementations. Next, several RBR multiplier architectures are investigated and their performance compared to two's complement systems. These include two new multiplier architectures: a purely RBR multiplier where both the operands are in RBR form, and a hybrid multiplier where the multiplicand is in RBR form and the other operand is represented in conventional two's complement form. Both the RBR and hybrid designs are demonstrated to have better EDP performance compared to conventional two's complement multipliers. The hybrid multiplier is also shown to have a superior EDP performance compared to the RBR multiplier, with much lower implementation area. Analysis on the effect of bit-precision is also performed, and it is shown that the performance gain of RBR systems improves for higher bit precision. Next, in order to demonstrate the efficacy of the RBR representation at the system-level, the performance of RBR and hybrid implementations of some common DSP kernels such as Discrete Cosine Transform, edge detection using Sobel operator, complex multiplication, Lifting-based Discrete Wavelet Transform (9, 7) filter, and FIR filter, is compared with two's complement systems. It is shown that for relatively large computation modules, the RBR to two's complement conversion overhead gets amortized. In case of systems with high complexity, for iso-throughput, both the hybrid and RBR implementations are demonstrated to be superior with lower average energy consumption. For low complexity systems, the conversion overhead is significant, and overpowers the EDP performance gain obtained from the RBR computation operation. / Dissertation/Thesis / M.S. Electrical Engineering 2011

This research focuses on a novel integrated approach for computing and representing complex numbers as a single entity without the use of any dedicated multiplier for calculating the fast Fourier transform algorithm (FFT), using the Distributed Arithmetic (DA) technique and Complex Binary Number Systems (CBNS). The FFT algorithm is one of the most used and implemented technique employed in many Digital Signal Processing (DSP) applications in the field of science, engineering, and mathematics. The DA approach is a technique that is used to compute the inner dot product between two vectors without the use of any dedicated multipliers. These dedicated multipliers are fast but they consume a large amount of hardware and are quite costly. The DA multiplier process is accomplished by shifting and adding only without the need of any dedicated multiplier. In today's technology, complex numbers are computed using the divide and conquer approach in which complex numbers are divided into two parts: the real and imaginary. The CBNS technique however, allows for each complex addition and multiplication to be computed in one single step instead of two. With the combined DA-CBNS approach for computing the FFT algorithm, those dedicated multipliers are being replaced with a DA system that utilize a Rom-based memory for storing the twiddle factor 'wn' value and the complex arithmetic operations being represented as a single entity, not two, with the CBNS approach. This architectural design was implemented by coding in a very high speed integrated circuit (VHSIC) hardware description language (VHDL) using Xilinx ISE design suite software program version 14.2. This computer aided tool allows for the design to be synthesized to a logic gate level in order to be further implemented onto a Field Programmable Gate Array (FPGA) device. The VHDL code used to build this architecture was downloaded on a Nexys 4 DDR Artix-7 FPGA board for further testing and analysis. This novel technique resulted in the use of no dedicated multipliers and required half the amount of complex arithmetic computations needed for calculating an FFT structure compared with its current traditional approach. Finally, the results showed that for the proposed architecture design, for a 32 bit, 8-point DA-CBNS FFT structure, the results showed a 32% area reduction, 41% power reduction, 59% reduction in run-time, 42% reduction in logic gate cost, and 66% increase in speed. For a 28 bit, 16-point DA-CBNS FFT structure, its area size, power consumption, run-time, and logic gate, were also found to be reduced at approximately 30%, 37%, 60%, and 39%, respectively, with an increase of speed of approximately 67% when compared to the traditional approach that employs dedicated multipliers and computes its complex arithmetic as two separate entities: the real and imaginary.

Sequences often occur in many branches of applied mathematics. Recurrence relation is a powerful tool to characterize and study sequences. Some commonly used methods for solving recurrence relations will be investigated. Many examples with applications in algorithm, combination, algebra, analysis, probability, etc, will be discussed. Finally, some well-known contest problems related to recurrence relations will be addressed.