Date of Award
Doctor of Philosophy
Electrical and Computer Engineering
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.
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