Date of Award


Publication Type

Doctoral Thesis

Degree Name



Electrical and Computer Engineering


Engineering, Electronics and Electrical.



Creative Commons License

Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.


This work presents the development of complex digital signal processing algorithms using number theoretic techniques. Residue number principles and techniques are applied to process complex signal information in Finite Impulse Response (FIR) and Infinite Impulse Response (IIR) digital filters. Residue coding of complex samples and arithmetic for processing complex data have been presented using principles of quadratic residues in the Residue Number System (RNS). In this work, we have presented modifications to the Quadratic Residue Number System (QRNS), which we have termed the Modified Quadratic Residue Number System (MQRNS), to process complex integers. New results and theorems have been obtained for the selection of operators to code complex integers into the new MQRNS representation. A novel scheme for residue to binary conversion has been presented for implementation using both the QRNS and MQRNS. Hardware implementations of multiplication intensive complex nonrecursive and recursive digital filters have been presented where the QRNS and MQRNS structures are realized using a bit-slice architectural approach. The computation of Complex Number Theoretic Transforms (CNTTs) and the hardware implementation of a radix-2 NTT butterfly structure, using high density ROM arrays, are presented in both the QRNS and MQRNS systems. As an illustration, the computation of the CNTT developed in this work, is used to compute Cyclic Convolution for complex sequences. These results are verified by computer programs. The recursive FIR filter structure for uniformly spaced frequency samples on the unit circle developed by adapting the Complex Number Theoretic z-transform, has been implemented using the QRNS and MQRNS. In this work, the filter structure is extended for non-uniformly spaced frequency samples and has been termed the generalized number theoretic filter structure. It is shown that for the implementation of this generalized structure, the MQRNS is more efficient than the conventional RNS; the QRNS does not support appropriate fields for the generalized structure.Dept. of Electrical and Computer Engineering. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1985 .K757. Source: Dissertation Abstracts International, Volume: 46-08, Section: B, page: 2757. Thesis (Ph.D.)--University of Windsor (Canada), 1985.