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The Hartley transform generalizes to the fractional Hartley transform (FRHT) which gives various uses in different fields of image encryption. Unfortunately, the available literature of fractional Hartley transform is unable to provide its inversion theorem. So accordingly original function cannot retrieve directly, which restrict its applications. The intension of this paper is to propose inversion theorem of fractional Hartley transform to overcome this drawback. Moreover, some properties of fractional Hartley transform are discussed in this paper.

This paper investigates optical orthogonal frequency division multiplexing (O-OFDM) systems with index modulation (IM) based on discrete Hartley transform (DHT) for visible light communications (VLC). An interesting trade-off between spectral efficiency (SE), energy efficiency, peak-to-average power ratio (PAPR) and bit error rate (BER) performance can be made by using the IM technique. And the DHT-based O-OFDM-IM systems can achieve higher SE compared with the DFT (discrete Fourier transform)-based counterparts. Employing DHT instead of DFT enables the removal of the Hermitian symmetry requirement, which contributes to transmitting more index bits and results in SE improvement. To acquire the same SE, the proposed systems can significantly lower the constellation order leading to better BER performance by contrast with the DFT-based counterparts. We employ DC (direct current) biasing and AC (Asymmetrically clipped) techniques to investigate the O-OFDM-IM systems in this paper. Simulation results indicate that SNR gains can be obtained by the DHT-based DCO/ACO-IM in comparison to the DFT-based counterparts. We also propose a novel reduced-complexity ML (maximum likelihood) detector specifically applicable to VLC. The new detector exhibits the same BER performance as the ML detector and performs better than the LLR (log-likelihood ratio) detector with reduced computational complexity.

This research paper proposes a secure opto-audio cryptosystem using XORing mask and Hartley transform (HT). The digital one dimensional (1D) plainaudio data is converted into two dimensional plainaudio map (2D PAM) and after that, the 2D PAM is divided into 2D plainaudio blocks (2D PABs). The basic idea of the proposed opto-audio encryption system depends on XORing each block of the 2D PABs with a single image selected from a personal image database that can be viewed as a secret key in the proposed opto-audio cryptosystem. Each block of the mixed 2D PABs is then transposed using the chaotic logistic adjusted sine map (LASM) and optically encrypted with HT. The XOR and LASM are implemented digitally while HT implemented optically. An additional XORing mask step helps to remove residual intelligibility from the 2D PABs, fill in speechless gaps in spoken conversations, and destroy both the pitch and format details. The utilization of chaotic LASM allows efficient noise immunity. A comparative study is held between the proposed opto-audio encryption system and other related audio encryption systems in terms the standard well known encryption metrics. The results have confirmed the efficiency of the introduced opto-audio encryption system. The proposed opto-audio encryption system security is explored from an accurate encryption point of view, and tests confirmed the superiority of the proposed opto-audio encryption system from the encryption point of view.

In this paper, we introduce a new type of singular first-order differential-difference operator of Dunkl type on the real line. This operator is obtained as a limiting case from both the first-order Dunkl-type operators corresponding to Bannai-Ito and Big 1-Jacobi orthogonal polynomials. We provide an explicit expression for the eigenfunction of this operator in terms of Bessel functions. The obtained kernel is called the Big Hartley function, which is a generalization of the usual Hartley kernel and the little Hartley function studied in Bouzeffour [The generalized Hartley transform. Integral Transforms Spec Funct. 2014;25(3):230–239]. Additionally, we present a new product formula for the little Hartley function, which is related to the Kingman-Bessel hypergroup and the Rosler-Dunkl signed hypergroup. Finally, we investigate inversion formulae for the transforms of both the little Hartley function and the big Hartley function. [ABSTRACT FROM AUTHOR]

Abstract In recent years, there has been an increasing interest in the use of the discrete Hartley transform (DHT) for orthogonal frequency-division multiplexing (OFDM). Previous studies have not reported an improvement in bit error rate (BER) when using a DHT-based OFDM without substantial complexity increase due to the fact that the DHT cannot diagonalize a circulant channel matrix but rather produces a unique structured matrix that has only a main diagonal and anti-diagonal. In this paper, the coupling between symmetric carriers is exploited by introducing a frequency-domain subcarrier diversity receiver for DHT-based OFDM systems. The proposed simple receiver structure takes advantage of the coupling to enhance the BER performance of the DHT-OFDM system by increasing the diversity gain between subcarriers. A detailed statistical and performance analysis of the system is presented, in which closed-form expressions of the average BER were derived. The proposed system performance was compared to the conventional discrete Fourier transform OFDM (DFT-OFDM), the generalized DHT-OFDM, and the DHT-precoded OFDM systems in terms of average BER and peak-to-average power ratio (PAPR). The proposed DHT-OFDM system outperforms the generalized DHT-OFDM system by 12 dB and outperforms the conventional DFT-OFDM and the DHT-precoded OFDM systems by 17 dB at an average BER of 10−5. In terms of PAPR, the proposed DHT-OFDM system suffers from an approximately 5.5, 2, and 1 dB increase in PAPR when compared to the DHT-precoded OFDM, the conventional DFT-OFDM, and the generalized DHT-OFDM systems, respectively.