Hadamard Transform Research Articles

Reducing peak to average power ratio (PAPR) for visible light communications (VLC) system using Walsh–Hadamard precoded discrete Hartley transforms

Abstract In visible light communication (VLC), intensity modulation with direct detection (IM/DD) based on DC-biased optical orthogonal frequency division multiplexing (DCO-OFDM) is used to transfer data at a high rate. Because OFDM signals are modulated using IM/DD, the peak of the transmitted signals is scaled up, resulting in a high peak-to-average power ratio (PAPR). In this article, the Walsh–Hadamard transform (WHT) is used as a precoder for discrete Hartley transform (DHT) to decrease the PAPR and the DC bias and achieve a good bit error rate (BER) performance in such a system. PAPR reduction results from the decrease of superposition of the transmitted signals owing to the use of WHT as a precoder. The proposed scheme and conventional DCO-OFDM indoor VLC system based on discrete Hartley transform (DHT) are compared to assure the new system’s benefits and efficiency.

A Robust image watermarking for copyright protection using Multiresolution Walsh-Hadamard transform and Schur using Differential evolution

\textit{Background} The proposed watermarking framework is based on the multi-resolution Walsh-Hadamard Transform (MR-WHT) along with Schur factorization. For embedding, the cover image is converted into a frequency domain using MR-WHT before Schur factorization is applied. \textit{Methods} The upper triangle's highest stable eigenvalues in the Schur factorization are employed as reliable embedding locations for the watermark. The singular values of these coefficients are updated with the singular values of the watermark, and the matrix array generated by the biggest eigenvalues is further factorised by singular value decomposition using multiple scaling factors and Differential evolution (DE). Later, an inverse SVD, inverse Schur, and inverse multiresolution Walsh-Hadamard transform is executed to obtain a watermarked image. \textit{Results} The simulation results show that the proposed method is imperceptible and robust against existing watermarking algorithms used in the literature.

A new bootstrap assisted test for checking second order stationarity

A new computationally driven method is proposed to check if a possibly nonlinear time series is second order stationary or not, which is important in time series modeling. The new test relies on blocks of blocks bootstrap covariance matrix estimates and Walsh transformations in order to capture the nonlinearity features of time series. The asymptotic normality of the Walsh coefficients and their asymptotic covariance matrix under the null hypothesis are derived for nonlinear processes. In addition, the asymptotic covariance matrix of an increasing dimension is shown to be consistently estimated by a blocks of blocks bootstrap procedure. In the framework of locally stationary nonlinear processes, it is shown that the proposed test is consistent under a sequence of local alternatives. A simulation study is conducted to examine the finite sample performance of the procedure. In many nonlinear time series settings, the proposed test works well while existing methods may have highly inflated type I error rates. The proposed test is applied to an analysis of a financial data set.

A hybrid encryption scheme of DNA and chaotic precoding for IMDD-FBMC systems

A new scheme that combines deoxyribonucleic acid (DNA) encryption algorithm and chaotic Walsh–Hadamard transform (WHT)/discrete Fourier transform (DFT) precoding to improve security performance is proposed for the first time in the filter bank multicarrier (FBMC) systems with intensity modulation and direct detection (IMDD). In our proposed scheme, we employed DNA and chaotic WHT/DFT precoding to achieve bit-level and symbol-level encryptions, respectively. With this hybrid encryption scheme, it is expected not only to enhance data transmission security but also to improve the bit error rate (BER) performance. We conducted experiments in the 16/32 quadrature amplitude modulation (QAM)-FBMC systems with IMDD. The relevant results show that our proposed scheme can improve receiver sensitivity by around 1 dB at the BER of 3.8e−3 compared with the scheme using only DNA encryption or without encryption. And a huge key space of 101806/101024 by DNA-chaotic WHT/DFT encryption is achieved. In addition, the net bit rate of transmission systems can be enhanced with a similar BER performance, compared to the optical orthogonal frequency division multiplexing (OFDM) systems.

Block Walsh–Hadamard Transform-based Binary Layers in Deep Neural Networks

Convolution has been the core operation of modern deep neural networks. It is well known that convolutions can be implemented in the Fourier Transform domain. In this article, we propose to use binary block Walsh–Hadamard transform (WHT) instead of the Fourier transform. We use WHT-based binary layers to replace some of the regular convolution layers in deep neural networks. We utilize both one-dimensional (1D) and 2D binary WHTs in this article. In both 1D and 2D layers, we compute the binary WHT of the input feature map and denoise the WHT domain coefficients using a nonlinearity that is obtained by combining soft-thresholding with the tanh function. After denoising, we compute the inverse WHT. We use 1D-WHT to replace the 1 × 1 convolutional layers, and 2D-WHT layers can replace the 3 × 3 convolution layers and Squeeze-and-Excite layers. 2D-WHT layers with trainable weights can be also inserted before the Global Average Pooling layers to assist the dense layers. In this way, we can reduce the number of trainable parameters significantly with a slight decrease in trainable parameters. In this article, we implement the WHT layers into MobileNet-V2, MobileNet-V3-Large, and ResNet to reduce the number of parameters significantly with negligible accuracy loss. Moreover, according to our speed test, the 2D-FWHT layer runs about 24 times as fast as the regular 3 × 3 convolution with 19.51% less RAM usage in an NVIDIA Jetson Nano experiment.

Use of Chaotic Oscillations for Precoding and Synchronization in OFDM

This paper proposes a novel linear precoding method for Orthogonal Frequency Division Multiplexing (OFDM) based on the employment of the chaotic waveforms generated by the fourth-order chaotic oscillator and orthonormalized by the Gram-Schmidt process. The proposed linear precoding method is aimed to increase resilience to the multipath propagation issues and reduce the Peak-to-Average Power Ratio (PAPR) of the transmitted signal. Moreover, the chaotic waveform enables novel timing synchronization methods to be implemented in the receiver. The modelling of baseband Linear Precoded OFDM (LP-OFDM) data transmission system with Rayleigh channel has been performed in Simulink environment to validate the proposed method and to compare the performance to the classic precoding methods, such as Walsh-Hadamard Transform (WHT). Experiments have shown that in a high Signal-to-Noise Ratio (SNR) scenario, the employment of the novel precoding scheme allows reducing Bit Error Ratio (BER) by several dB compared to non-precoded OFDM. The proposed precoding method leads to the reduction of PAPR; however, it is not as efficient as classical precoding schemes, such as WHT. Experimental evidence of synchronization of the chaotic oscillators within 50 samples long time interval is presented.

Transformation of Superposed Quantum States Using Measurement Operators

Quantum computation based on a gate model is described. This model initially creates a superposition | ψ 0 〉 consisting of N = 2 n states, and these states are labeled by an n qubit index value j . Two working qubits | 0 〉 w k 0 and | 0 〉 w k 1 are added for a measurement. Moreover, one marking qubit | 0 〉 m k is added to discriminate between states in a superposition. Thus, | ψ 1 〉 = 1 / N ∑ j = 0 N − 1 | j 〉 ⊗ | 0 〉 w k 0 ⊗ | 0 〉 w k 1 ⊗ | 0 〉 m k . The Hadamard transformation is applied to | 0 〉 w k 0 and | 0 〉 w k 1 . | ψ 2 〉 = 1 / N 1 / 2 2 ∑ j = 0 N − 1 | j 〉 ⊗ | 0 〉 w k 0 + | 1 〉 w k 0 ⊗ | 0 〉 w k 1 + | 1 〉 w k 1 ⊗ | 0 〉 m k . After a computation, a set of states is divided into two subsets; one is a subset bad (B) and the other is a subset good (G). | ψ 3 〉 = 1 / N 1 / 2 2 ∑ j ∈ B | j 〉 ⊗ | 0 〉 w k 0 + | 1 〉 w k 0 ⊗ | 0 〉 w k 1 + | 1 〉 w k 1 ⊗ | 0 〉 m k + ∑ j ∈ G | j 〉 ⊗ | 0 〉 w k 0 + | 1 〉 w k 0 ⊗ | 0 〉 w k 1 + | 1 〉 w k 1 ⊗ | 1 〉 m k . After a marking, a superposition is measured by POVM. The measurement is described by a collection of four measurement operators. The measurement transforms | ψ 3 〉 into | ψ 4 〉 = 1 / N ∑ j ∈ B | j 〉 ⊗ | 0 〉 w k 0 ⊗ | 0 〉 w k 1 ⊗ p 0 sin θ | 0 〉 m k + ∑ j ∈ G | j 〉 ⊗ | 0 〉 w k 0 ⊗ | 0 〉 w k 1 ⊗ p 1 cos θ | 1 〉 m k / D ; here, D = card B p 0 ∗ p 0 sin 2 θ + card G p 1 ∗ p 1 cos 2 θ , and p 0 ∗ p 0 sin 2 θ + p 1 ∗ p 1 cos 2 θ = 2 which is derived from the completeness equation. The state | 0 〉 m k and the state | 1 〉 m k before the measurement are transformed into p 0 sin θ | 0 〉 m k and p 1 cos θ | 1 〉 m k , respectively. This paper describes these measurement operators.

Probabilistically-Shaped DMT for IM-DD Systems with Low-Complexity Fast WHT-Based PDSP

Transmission capacity and receiver sensitivity of an intensity-modulation direct detection (IM-DD) optical discrete multi-tone (DMT) system can be improved by using the probabilistically shaping (PS) technique. However, different probabilistic distributions will be required owing to the unbalanced signal-to-noise ratio (SNR) among data-carrying subcarriers (SCs) induced by the imperfect frequency response of optical/electrical devices, which can increase the implementation complexity of the PS-DMT transceiver. In this work, different signal pre-processing schemes including pre-equalization, Walsh–Hadamard transform (WHT)-based full data-carrying SCs precoding (FDSP) and fast WHT-based partial data-carrying SCs precoding (PDSP) are investigated for SNR equalization in a short-reach PS-DMT transmission system. After transmission over 50 km single-mode fiber, the experimental results indicated that three pre-processed signals have almost the same generalized mutual information (GMI) performance and receiver sensitivity improvements. The proposed fast WHT-based PDSP scheme may be a good option for the implementation of the PS-DMT transmission systems with a large SC SNR fluctuation regarding computational complexity.

On the Performance of Precoded OFDM Systems in the Presence of Symbol Timing Offset

This correspondence considers the performance evaluation of precoded orthogonal frequency division multiplexing (P-OFDM) based on Walsh-Hadamard Transform (WHT) in the presence of symbol timing offset (STO). The signal-to-interference plus noise ratio (SINR) is evaluated for flat and frequency-selective fading channels (FSC), where closed-form analytical expressions are derived for the flat fading case. In FSC, the SINR is obtained using Monte Carlo simulation due to the analysis intractability. The bit error rate (BER) performance of OFDM and P-OFDM based on WHT, Haar, and Zadoff-Chu matrix transforms are considered to demonstrate the STO impact on the BER using various precoding transforms. The obtained results show that the impact of STO on P-OFDM is substantially different from conventional OFDM. In particular, it is found that STO affects the SINR of each subcarrier differently, where the SINR of certain subcarriers is severely degraded while others are immune. Furthermore, the diversity gain advantage of the precoding process may vanish for large STOs.

Estimation and Compensation of IQ Imbalance for OTSM Systems

Recently, orthogonal time sequency multiplexing (OTSM) waveform was proposed using Walsh Hadamard transform (WHT) for doubly-dispersive channels at a lower modulation/demodulation complexity than orthogonal time frequency space (OTFS) waveform. At high frequencies, hardware imperfections such as in-phase and quadrature (IQ) imbalance is prominent at the transceiver. In this letter, the input-output (I/O) relation in the delay-sequency domain is derived under these impairments at the receiver. The impairment parameters are estimated in the delay-time domain and are compensated in the time domain using proposed dual pilot approach. The normalized mean square error (NMSE) of the estimated imbalance parameters is evaluated and it is observed that NMSE reduces with increase in SNR and with reduction in mobility. The results are evaluated with OTFS system and it is observed that the NMSE performance of OTSM is similar to OTFS under high pilot power and high SNR at no excess pilot power. The bit error rate performance of the compensated system is evaluated and its performance is close to the ideal system.

Analysis of Stray Light and Enhancement of SNR in DMD-Based Spectrometers.

Due to advantages such as the high efficiency of light utilization, small volume, and vibration resistance, digital micro-mirror device (DMD)-based spectrometers are widely used in ocean investigations, mountain surveys, and other field science research. In order to eliminate the stray light caused by DMDs, the stray light in DMD-based spectrometers was first measured and analyzed. Then, the stray light was classified into wavelength-related components and wavelength-unrelated components. Moreover, the noise caused by the stray light was analyzed from the perspective of encoding equation, and the de-noising decoding equation was deduced. The results showed that the accuracy range of absorbance was enhanced from [0, 1.9] to [0, 3.1] in single-stripe mode and the accuracy range of absorbance was enhanced from [0, 3.8] to [0, 6.3] in Hadamard transform (HT) multiple-stripe mode. A conclusion can be drawn that the de-noising strategy is feasible and effective for enhancing the SNR in DMD-based spectrometers.

The Walsh–Fourier Transform on the Real Line

The Walsh–Fourier Transform on the Real Line

Functional Acoustic Metamaterial Using Shortcut to Adiabatic Passage in Acoustic Waveguide Couplers

Shortcut to adiabatic passage (STAP), initially proposed to accelerate adiabatic quantum-state transfer, has been widely explored and applied in quantum optics and integrated optics. Here we bring STAP into the field of acoustics to design compact couplers and functional metamaterial. The space-varying coupling strengths of acoustic waveguides (WGs) are tailored by means of dressed states in a three-level system, accounting for the desirable acoustic energy transfer among three WGs with short length. We show that the acoustic coupler has a one-way feature when loss is introduced into the intermediate WG. More uniquely, when the propagation of acoustic wave is designed to mimic a Hadamard transformation, an acoustic metamaterial can be constructed by arraying several couplers, possessing the beam-splitting function and unidirectional, broadband performances. Our work bridges STAP and the acoustic coupler as well as metamaterial, which may have profound impacts on exploring quantum technologies for promoting advanced acoustic devices.

Parallel Hardware Implementation of Walsh Hadamard Transform

The Walsh Hadamard Transform is a powerful notion in digital signal processing. This paper explains the construction of parallel hardware architecture using the mathematical concept of Kronecker product based approach to Walsh Hadamard Transform and its simulation using Verilog. This architecture is simulated here using Field Programmable Gate Array (FPGA) technology in Verilog Spartan 3e platform. Furthermore, this paper illustrates the fast algorithm and parallel computational result of both one-dimensional and two-dimensional transforms using the Kronecker product.This algorithm can be used to implement a systolic array based dedicated hardware for computation of the transform. Our proposed hardware design for the Walsh Hadamard Transform will be used in various digital signal processing applications. The systematic derivation of parallel architecture design using the concept of Kronecker product and stride permutation would depict the real time processing rather than conventional way and reducing time complexity using minimal resources is a challenging task.

Design Switchable Precoded Space-Time Parallel Two-Path OFDM Systems

A 2x2 space-time (ST) orthogonal frequency division multiplexing (OFDM) system is proposed with parallel cancellation (PC) two-path transmission to form a 2x2 STPC system. The 2x2 STPC-OFDM systems mitigate the inter-carrier interference (ICI) in the mobile channels with a better BER than that of a 2x2 ST-OFDM system. The design is switchable between two operation modes: either high data rate with ST-OFDM or high reliability with STPC in mobile fading channels having lower bit error rate (BER). Furthermore, a generalized STPC architecture is designed with multiple receiver antennas. It offers higher diversity order and better BER. Two orthogonal transforms, such as Zadoff-Chu transform (ZCT) and Walsh–Hadamard transform (WHT), are employed as the precoder to enhance the BER further. Although the proposed STPC-OFDM systems require two-path transmission with the cost of reducing bandwidth efficiency, simulations show that the proposed 2x2 STPC-OFDM systems are robust to various channel environments and have a better BER performance for fair data rate comparison with 2x2 ST-OFDM.

A Blind Video Copyright Protection Technique in Maximum and Minimum Energy Frames Based on The Fast Walsh Hadamard Transform (FWHT) and Discrete Wavelet Transform (DWT) and Arnold Map

Video copyright protection is the most generally acknowledged method of preventing data piracy. This paper proposes a blind video copyright protection technique based on the Fast Walsh Hadamard Transform (FWHT), Discrete Wavelet Transform (DWT), and Arnold Map. The proposed method chooses only frames with maximum and minimum energy features to host the watermark. It also exploits the advantages of both the fast Walsh Hadamard transform (FWHT) and discrete wavelet transforms (DWT) for watermark embedding. The Arnold map encrypts watermarks before the embedding process and decrypts watermarks after extraction. The results show that the proposed method can achieve a fast embedding time, good transparency, and robustness against various attacks.

Performance analysis of novel precoding matrix techniques for optical OFDM-based visible light communication systems

Visible Light Communication (VLC) exploits the renowned modulation scheme like orthogonal frequency division multiplexing (OFDM) to accomplish high data rate transmission. However, the optical OFDM system like DC biased optical OFDM (DCO-OFDM) is vulnerable to high peak to average power ratio (PAPR) which results in detrimental clipping distortion which in turn degrades the overall system performance. The limited dynamic range of the LEDs further agitates this issue. On these grounds, PAPR reduction techniques like precoding matrix schemes have been proposed for reducing the high amount of PAPR in DCO-OFDM system. In this paper, discrete Hartley Matrix transform (DHMT) precoded discrete Hartley transform (DHT)-based DCO-OFDM system is proposed for reducing PAPR. In addition, we comparatively investigate the PAPR performance of a new precoding technique like T- transform which is a hybrid combination of Walsh Hadamard transform (WHT) and Zadoff chu transform (ZCT) with other precoding matrix techniques like discrete cosine matrix transform (DCMT), discrete Fourier matrix transform (DFMT), Walsh Hadamard Matrix transform (WHMT) and DHMT. Furthermore, this paper derives the mathematical analysis corresponding to the time-domain signal formats for the precoded DCO-OFDM system which is exploiting complex Fourier signal processing and real trigonometric transform like DHT. From the simulated results, it can be surmised that when compared with other precoding techniques, DHMT precoded DHT-based DCO-OFDM system not only achieves a significant reduction in PAPR but also achieves a better bit error rate (BER) and spectral efficiency performance. Furthermore, less computational complexity is required to implement it.

BER Performance of Pre-Coded Space-Time Conjugate Two-Path OFDM Systems

It’s critical to design a robust wireless communication system that achieves an improved bit error rate (BER) in mobile channels. In this paper, a 2x2 space-time (ST) orthogonal frequency division multiplexing (OFDM) system is proposed with conjugate cancellation (CC) two-path transmission scheme. The 2x2 STCC-OFDM system features in the excellent inter-carrier interference (ICI) mitigation in the mobile channels with a better BER than that of a conventional 2x2 ST-OFDM system. Five orthogonal transforms are employed as the precoder to enhance the BER further, such as Zadoff-Chu transform (ZCT), slant transform (ST), discrete Hartley transform (DHT), Haar transform (HT), and Walsh–Hadamard transform (WHT). ZCT row wise precoder provide the lowest peak-to-average power ratio and good BER performance. Simulations demonstrate that the proposed STCC-OFDM systems are robust to various channel environments Although the STCC-OFDM systems improve BER with the cost of reducing bandwidth efficiency when compared to a ST-OFDM, they can be compensated by a higher QAM constellation size.

The Sufficient Condition for Ensuring the Reliability of Perception of the Steganographic Message in the Walsh-Hadamard Transform Domain

In view of the high compatibility of the Walsh-Hadamard transform to the architecture of modern computing facilities, it is sensible to use it in methods of steganography and steganalysis. Now, there are some efficient steganographic methods that use the Walsh-Hadamard transform domain for information embedding presented in the literature. It is known that to ensure the reliability of the steganographic message perception, information should be embedded in its high-frequency components; nevertheless, the issues of the correspondence of the Walsh-Hadamard transformants to the frequency components of the original matrix of the cover image are poorly researched. The purpose of this paper is to develop a formal sufficient condition for ensuring the reliability of perception of steganographic messages in the Walsh-Hadamard transform domain. This purpose was achieved by establishing the relationship between the Walsh-Hadamard transformants and the discrete cosine transform transformants, which was theoretically substantiated in two ways, and also experimentally confirmed. Based on the established relationship between the Walsh-Hadamard transform and the discrete cosine transform, as well as the components of the singular value decomposition of the corresponding matrices, a sufficient condition has been developed to ensure the reliability of the perception of steganographic messages in the Walsh-Hadamard transform domain. The sufficient condition consists in the fact that after embedding of additional information using any steganographic method, those WalshHadamard transformants that correspond to the singular triples of the matrix corresponding to small (smallest) in value singular numbers, should change.

An over view of geometric function theory (GFT) and analytic properties of meromorphic functions

This article presents an over view of Geometric Function Theory (GFT) and utilized the analytic properties of meromorphic functions. Geometric Function Theory is a branch of mathematics that focuses on the geometric interpretations and implications of analytic functions defined in the complex plane. Our exploration begins with an itemized discussion of key concepts within GFT, emphasizing their relevance and theoretical underpinnings. Central to our study is the investigation of meromorphic functions, which are functions that are analytic except for isolated singularities where they may have poles. We examined various classes of meromorphic functions and elucidate their properties, including their behavior near singularities and their broader geometric implications. A significant portion of our inquiry involves the Hadamard product of functions. This operation allows us to explore the combined effect of two analytic functions, considering their series expansions and how their product transforms under this operation. By studying the Hadamard transformation, we uncover analogues and interesting results that shed light on the interplay between analytic functions and their geometric representations. We also provide detailed diagrammatic descriptions of fundamental geometric shapes such as circles, open unit disks, and closed unit disks. These diagrams serve to visually illustrate key concepts and relationships within GFT, aiding in the understanding of how analytic functions behave in different spatial configurations. Our article offers a comprehensive exploration of Geometric Function Theory, emphasizing its foundational concepts and their applications in analyzing analytic and meromorphic functions.