Algorithmic Structures for Realizing Short-Length Circular Convolutions with Reduced Complexity

Abstract

A set of efficient algorithmic solutions suitable to the fully parallel hardware implementation of the short-length circular convolution cores is proposed. The advantage of the presented algorithms is that they require significantly fewer multiplications as compared to the naive method of implementing this operation. During the synthesis of the presented algorithms, the matrix notation of the cyclic convolution operation was used, which made it possible to represent this operation using the matrix–vector product. The fact that the matrix multiplicand is a circulant matrix allows its successful factorization, which leads to a decrease in the number of multiplications when calculating such a product. The proposed algorithms are oriented towards a completely parallel hardware implementation, but in comparison with a naive approach to a completely parallel hardware implementation, they require a significantly smaller number of hardwired multipliers. Since the wired multiplier occupies a much larger area on the VLSI and consumes more power than the wired adder, the proposed solutions are resource efficient and energy efficient in terms of their hardware implementation. We considered circular convolutions for sequences of lengths N= 2, 3, 4, 5, 6, 7, 8, and 9.