Radix-2² Algorithm for the Odd New Mersenne Number Transform (ONMNT)

\ua9 2023 by the authors. This paper introduces a new derivation of the radix- (Formula presented.) fast algorithm for the forward odd new Mersenne number transform (ONMNT) and the inverse odd new Mersenne number transform (IONMNT). This involves introducing new equations and functions in finite fields, bringing particular challenges unlike those in other fields. The radix- (Formula presented.) algorithm combines the benefits of the reduced number of operations of the radix-4 algorithm and the simple butterfly structure of the radix-2 algorithm, making it suitable for various applications such as lightweight ciphers, authenticated encryption, hash functions, signal processing, and convolution calculations. The multidimensional linear index mapping technique is the conventional method used to derive the radix- (Formula presented.) algorithm. However, this method does not provide clear insights into the underlying structure and flexibility of the radix- (Formula presented.) approach. This paper addresses this limitation and proposes a derivation based on bit-unscrambling techniques, which reverse the ordering of the output sequence, resulting in efficient calculations with fewer operations. Butterfly and signal flow diagrams are also presented to illustrate the structure of the fast algorithm for both ONMNT and IONMNT. The proposed method should pave the way for efficient and flexible implementation of ONMNT and IONMNT in applications such as lightweight ciphers and signal processing. The algorithm has been implemented in C and is validated with an example

Analysis and correction of the helium speech effect by autoregressive signal processing

SIGLELD:D48902/84 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

New Decimation-In-Time Fast Hartley Transform Algorithm

This paper presents a new algorithm for fast calculation of the discrete Hartley transform (DHT) based on decimation-in-time (DIT) approach. The proposed radix-2^2 fast Hartley transform (FHT) DIT algorithm has a regular butterfly structure that provides flexibility of different powers-of-two transform lengths, substantially reducing the arithmetic complexity with simple bit reversing for ordering the output sequence. The algorithm is developed through the three-dimensional linear index map and by integrating two stages of the signal flow graph together into a single butterfly. The algorithm is implemented and its computational complexity has been analysed and compared with the existing FHT algorithms, showing that it is significantly reduce the structural complexity with a better indexing scheme that is suitable for efficient implementation

Theory and realization of novel algorithms for random sampling in digital signal processing

Random sampling is a technique which overcomes the alias problem in regular sampling. The randomization, however, destroys the symmetry property of the transform kernel of the discrete Fourier transform. Hence, when transforming a randomly sampled sequence to its frequency spectrum, the Fast Fourier transform cannot be applied and the computational complexity is N(^2). The objectives of this research project are (1) To devise sampling methods for random sampling such that computation may be reduced while the anti-alias property of random sampling is maintained : Two methods of inserting limited regularities into the randomized sampling grids are proposed. They are parallel additive random sampling and hybrid additive random sampling, both of which can save at least 75% of the multiplications required. The algorithms also lend themselves to the implementation by a multiprocessor system, which will further enhance the speed of the evaluation. (2) To study the auto-correlation sequence of a randomly sampled sequence as an alternative means to confirm its anti-alias property : The anti-alias property of the two proposed methods can be confirmed by using convolution in the frequency domain. However, the same conclusion is also reached by analysing in the spatial domain the auto-correlation of such sample sequences. A technique to evaluate the auto-correlation sequence of a randomly sampled sequence with a regular step size is proposed. The technique may also serve as an algorithm to convert a randomly sampled sequence to a regularly spaced sequence having a desired Nyquist frequency. (3) To provide a rapid spectral estimation using a coarse kernel : The approximate method proposed by Mason in 1980, which trades the accuracy for the speed of the computation, is introduced for making random sampling more attractive. (4) To suggest possible applications for random and pseudo-random sampling : To fully exploit its advantages, random sampling has been adopted in measurement Random sampling is a technique which overcomes the alias problem in regular sampling. The randomization, however, destroys the symmetry property of the transform kernel of the discrete Fourier transform. Hence, when transforming a randomly sampled sequence to its frequency spectrum, the Fast Fourier transform cannot be applied and the computational complexity is N"^. The objectives of this research project are (1) To devise sampling methods for random sampling such that computation may be reduced while the anti-alias property of random sampling is maintained : Two methods of inserting limited regularities into the randomized sampling grids are proposed. They are parallel additive random sampling and hybrid additive random sampling, both of which can save at least 75% , of the multiplications required. The algorithms also lend themselves to the implementation by a multiprocessor system, which will further enhance the speed of the evaluation. (2) To study the auto-correlation sequence of a randomly sampled sequence as an alternative means to confirm its anti-alias property : The anti-alias property of the two proposed methods can be confirmed by using convolution in the frequency domain. However, the same conclusion is also reached by analysing in the spatial domain the auto-correlation of such sample sequences. A technique to evaluate the auto-correlation sequence of a randomly sampled sequence with a regular step size is proposed. The technique may also serve as an algorithm to convert a randomly sampled sequence to a regularly spaced sequence having a desired Nyquist frequency. (3) To provide a rapid spectral estimation using a coarse kernel : The approximate method proposed by Mason in 1980, which trades the accuracy for the speed of the computation, is introduced for making random sampling more attractive. (4) To suggest possible applications for random and pseudo-random sampling : To fully exploit its advantages, random sampling has been adopted in measurement instruments where computing a spectrum is either minimal or not required. Such applications in instrumentation are easily found in the literature. In this thesis, two applications in digital signal processing are introduced. (5) To suggest an inverse transformation for random sampling so as to complete a two-way process and to broaden its scope of application. Apart from the above, a case study of realizing in a transputer network the prime factor algorithm with regular sampling is given in Chapter 2 and a rough estimation of the signal-to-noise ratio for a spectrum obtained from random sampling is found in Chapter 3. Although random sampling is alias-free, problems in computational complexity and noise prevent it from being adopted widely in engineering applications. In the conclusions, the criteria for adopting random sampling are put forward and the directions for its development are discussed