VLSI architecture for adaptive digital filtering utilising number theoretic transform

This thesis investigates the design of a frequency adaptive digital filter in Very Large Scale Integration using the Number Theoretic Transform. The properties of residue numbering systems are investigated, and particularly the possible advantages occured from parallelism, operations without the need for carry, and the absence of round-off errors. The conclusion is reached that this numbering system is in some circumstances more suitable for high speed processing in Very Large Scale Integration. Transform techniques in a finite field are then examined to determine how they could perform filtering operation more efficiently in terms of the number of arithmetic operations required compared with other techniques such as Fast Fourier Transform. Adaptive filtering in the frequency domain using the Number Theoretic Transform, and in particular Fermat Number Transform, with appropriate formulae for filter weight adaptation using the Least Mean Square algorithm are presented. Several result show that the frequency mean square error as a performance index results in convergence to an optimal solution. A complexity' ratio is used to ascertain that frequency adaptive digital filters need less computational power (number of arithmetic operations) than time domain adaptive filters. A design of special purpose processor using Very Large Scale Integration technology is described, several structures using pipelining and systolic arrays are presented which support the main Very large Scale Integration design features. A table look-up approach using Programmable Logic Arrays (PLAs) for processing elements and a measurements of system performance regarding time/area complexity' are described. Finally, it is concluded that, with suitable further development, a Very' Large Scale Integration architecture for frequency adaptive digital filter using number theoretic transform which has high sampling rate, regular internal structure and capability to parallel devices could likely be achieved.