Data-recovery algorithm and circuit for cyclic convolution based on FNT

Abstract

Abstract:The existing cyclic convolution based on Fermat number transform (FNT) can't produce the same results as the conventional one based on fast Fourier transform (FFT) due to the impact of modular operation in many cases. To overcome the problem, this paper proposes a novel data-recovery algorithm (DRA) for the cyclic convolution based on FNT with transform kernel 2 or its integer power in the ordinary binary number system. Then an efficient data-recovery circuit (DRC) is designed to implement the algorithm in the diminished-1 number systems, involving parallel-prefix carry computation units. The DRA and the DRC can eliminate the impact of modular operation and make the cyclic convolution based on FNT produce the correct resulting sequences. Synthesis results show that delay and area of the n-bit DRC are both less than the ones of an n-bit parallel-prefix adder (PPA).

Synthesis of fast-operating devices for digital signal processing based on the number­theoretic transforms

“…An analysis of the methods for finding convolutions and ACF with the help of NTT shows that the main arithmetic operations that need to be performed are addition and multiplication based on the selected simple module, that is, calculating the remainder of the result of arithmetic operations. For an arbitrary module, these operations are rather laborious; therefore, some moduli of a special form are of interest for which the modulo addition operation is much simpler [4].…”