Golay Sequences Research Articles

Almost two-valued periodic Golay sequence pairs derived from the Legendre symbol

Almost two-valued periodic Golay sequence pairs derived from the Legendre symbol

Constructions of mismatched binary periodic complementary pairs

Two mismatched binary sequences are called a mismatched binary periodic complementary pair (MB-PCP) if the sum of their periodic cross-correlation functions is a delta function. Such a pair is a generalization of the well-known Golay sequence pair, and has applications in the scenario where the mismatched filter is used. Up to now, there are only MB-PCPs of length $ 3q $ with peak value 4 or $ 2q-6 $, where $ q \equiv 3\; (\bmod \,4 ) $ is a prime. The objective of this paper is to construct four families of MB-PCPs of length $ pq $ via the generalized cyclotomy and the Chinese Remainder Theorem, where $ p $ and $ q $ are two different odd primes.

An Efficient NB-IoT Compatible GF-NOMA PHY Mechanism for mMTC

The third generation partnership project (3GPP) has proposed the narrowband Internet of things (NB-IoT) standard for serving IoT users. Improvements to the NB-IoT standard are being considered to efficiently support massive machine-type communication (mMTC) IoT applications. Subsequent releases of the NB-IoT standard envision to support mMTC using different access mechanisms. In this work, a grant-free (GF) non-orthogonal multiple access (NOMA) system compatible with the current NB-IoT standard is proposed, where the resources are divided into transmission opportunities (TOs). The TOs are further divided into multiple groups based on the repetitions adopted by the users in the system. Physical (PHY) layer procedures for preamble transmission/detection, channel estimation, data transmission/decoding, and acknowledgment (ACK) transmission/reception are described in detail for the proposed GF-NOMA system. Specifically, the preamble sequences are also used as spreading sequences for data transmission and three types of sequences -Zadoff-Chu, random Gaussian and Golay sequences are explored. Results are presented for per-user success probability, latency and energy consumption considering two NB-IoT compatible settings along with repetitions and retransmissions. It is demonstrated that the proposed GF-NOMA scheme meets the mMTC requirement recommended by the 3GPP with more than 99% per-user success probability and with the energy consumption reduced by at least 80% compared to NB-IoT, while being readily compatible with the current NB-IoT standard.

A Digital Frequency-Dependent I/Q Imbalance and Group Delay Estimation and Compensation Modules for mmWave Single Carrier Baseband Transceivers

This paper proposes the estimation and compensation modules of digital frequency-dependent (FD) I/Q imbalance and group delay for single carrier (SC) mode with 15 Gbps throughput rate in the millimeter-wave (mmWave) band. The proposed compensation module of a digital recursive filter can mitigate FD I/Q imbalance effects caused by the -3 dB corner imbalance of the reconstruction and anti-aliasing filters of I/Q branches. The proposed recursive filter with the appropriate initial coefficients based on the result of Golay sequence estimation can shorten the convergence time and precisely acquire the desired coefficients. The average image rejection ratio (IRR) performance after compensating by the proposed module can be improved from 21.97 dB to 47.54 dB. Besides, the group delay distortion in reconstruction and anti-aliasing filters will cause intersymbol interference to distort the in-band signal. We employ a digital equalization filter with the filter coefficients derived from the correlation results based on the Golay sequence to compensate the in-band group delay distortion. After the proposed group delay is pre-compensation at TX, the system-level performance of error vector magnitude (EVM) performance is improved by 6.21 dB to -27.78 dB to satisfy the IEEE 802.11ad/ay requirement in 64-QAM mode. The inner RX BER performance with the FD I/Q imbalance and the group delay compensation modules can achieve 3 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> at 23 dB and also satisfy the system requirement at SNR of required 25.52 dB. A four-times parallelism architecture is proposed to work at a clock rate of 625 MHz with 64-QAM mode. The gate count and power consumption of the proposed design are 154.4 K and 16.2 mW, respectively.

Omnidirectional precoding for massive MIMO with uniform rectangular array in presence of mutual coupling

The massive Multiple-Input and Multiple-Output (MIMO) system is one of the technologies in 6-Generation (6G) communication networks. Omnidirectional or broad coverage precoding is used in massive MIMO to broadcast the common signals from public channels to all active and non-active users so that the radiation power of these signals is constant in all spatial directions. On the other hand, in large-scale array antennas, the presence of mutual coupling leads to deteriorating the performance of MIMO systems. In this paper, we propose omnidirectional precoding of the massive MIMO equipped with a Uniform Rectangular Array (URA) in presence of the mutual coupling. In the proposed method, omnidirectional transmission and maximum achievable ergodic rate conditions are developed for the case that mutual coupling exists, then an omnidirectional precoding matrix is designed to satisfy three constraints: 1) Omnidirectional transmission, 2) Equal average power transmission per antenna, and 3) Maximum achievable ergodic rate. For this purpose, we show that an omnidirectional matrix can be reconstructed using a pair of vectors with elements' absolute value equal to one that forms a Complementary Set (CS), and the Complete Complementary Codes (CCC) that satisfied some conditions. The intended CS is selected from the Golay sequence and the intended CCC is obtained by solving a gradient algorithm with projection on the Grassmann and Stiefel manifolds. Simulation results confirm the expected results.

Constructions of almost complementary pairs based on non-standard Golay sequences

Constructions of almost complementary pairs based on non-standard Golay sequences

Golay-Encoded Ultrasound Monitoring of Simultaneous High-Intensity Focused Ultrasound Treatment: A Phantom Study.

Ultrasound (US) imaging has high potential in monitoring high-intensity focused US (HIFU) treatment due to its superior temporal resolution. However, US monitoring is often hindered by strong HIFU interference, which overwhelms the echoes received by the imaging array. In this study, a method of Golay-encoded US monitoring is proposed to visualize the imaged object for simultaneous HIFU treatment. It effectively removes HIFU interference patterns in real-time B-mode imaging and improves the metrics of image quality, such as peak signal-to-noise ratio (PSNR), structural similarity (SSIM), and contrast ratio (CR). Compared to the pulse-inversion sequence, the N -bit Golay sequence can boost the echo magnitude of US monitoring by another N times and, thus, exhibits higher robustness. Simulations show that a sinusoidal HIFU waveform can be fully eliminated using Golay decoding when the bit duration of the N -bit Golay sequence ( N is the power of 4) coincides with either odd (Case I) or even (Case II) integer multiples of the HIFU quarter period. Experimental results also show that the Golay decoding with Case II can increase the PSNR of US monitoring images by more than 30 dB for both pulse- and continuous-wave HIFU transmissions. The SSIM index also effectively improves to about unity, indicating that the B-mode image with HIFU transmission is visually indistinguishable from that acquired without HIFU transmission. Though Case I is inferior to Case II in the elimination of even-order HIFU harmonic, they together enable a more flexible selection of imaging frequencies to meet the required image resolution and penetration for Golay-encoded US monitoring.

Wideband Channel Sounding Simulation Tool and Sounding Sequences Evaluation

This work presents the Base-Band Sounding Tool (BSOT), a wideband channel sounding simulator with different scenarios and configurations, providing the channels characterization in terms of delay and Doppler spreading. BSOT follows COST 207 statistical channel model and the pulse compression technique with direct cross-correlation at the receiver. Besides its validation results, the paper presentes a performance evaluation of 6 types of sounding sequences (Random, Kasami, Gold, m-Sequence, Zadoff-Chu and Golay), including a robustness evaluation with different lengths, sampling rates, and SNR values. The simulated outputs are coherent with the theoretical modeling, and the investigation of sounding sequences reveals a better sounding performance obtained with the Zadoff-Chu and Golay sequences.

Signal-to-noise ratio of diverging waves in multiscattering media: Effects of signal duration and divergence angle.

In this paper, SNR maximization in coded diverging waves is studied, and experimental verification of the results is presented. Complementary Golay sequences and binary phase shift keying modulation are used to code the transmitted signal. The SNR in speckle and pin targets is maximized with respect to chip signal length. The maximum SNR is obtained in diverging wave transmission when the chip signal is as short a duration as the array permits. We determined the optimum diverging wave profile to confine the transmitted ultrasound energy in the imaging sector. The optimized profile also contributes to the SNR maximization. The SNR performances of the optimized coded diverging wave and conventional single-focused phased array imaging are compared on a single frame basis. The SNR of the optimized coded diverging wave is higher than that of the conventional single-focused phased array imaging at all depths and regions.

Efficient Cell-Specific Beamforming for Large Antenna Arrays

We propose an efficient method for designing broad beams with spatially flat array factor and efficient power utilization for cell-specific coverage in communication systems equipped with large antenna arrays. To ensure full power efficiency, the method is restricted to phase-only weight manipulations. Our framework is based on the discovered connection between dual-polarized beamforming and polyphase Golay sequences. Exploiting this connection, we propose several methods for array expansion from smaller to larger sizes, while preserving the radiation pattern. In addition, to fill the gaps in the feasible array sizes, we introduce the concept of ϵ-complementarity that relaxes the requirement on zero side lobes of the sum aperiodic autocorrelation function of a sequence pair. Furthermore, we develop a modified Great Deluge algorithm (MGDA) that finds ϵ-complementary pairs of arbitrary length, and hence enables broad beamforming for arbitrarily-sized uniform linear arrays. We also discuss the extension of this approach to two-dimensional uniform rectangular arrays. Our numerical results demonstrate the superiority of the proposed approach with respect to existing beam-broadening methods.

Self-Interference-Resistant IEEE 802.11ad-Based Joint Communication and Automotive Radar Design

The IEEE 802.11ad based joint communication and radar sensing (JCRS) has attracted great attention for vehicles in recent years. Existing studies mostly assume full-duplex radio for radar receiver based on perfect self-interference cancellation. However, the self-interference may not be fully cancelled in practice. In this paper, imperfect self-interference cancellation is considered. To combat the strong self-interference, a novel JCRS is developed based on IEEE 802.11ad orthogonal frequency division multiplex physical-layer (OFDM PHY) for multi-target sensing. More specifically, the received radar signal is designed to be synchronized with self-interference, and two radar sensing functions are developed for short-range sensing and long-range sensing separately. The synchronization with self-interference resolves the difficulty in self-interference calibration in short-range sensing and prevents the inter-subcarrier-interference (ICI) from self-interference in long-range sensing. The Golay sequences in the preamble are leveraged to combat self-interference for short-range sensing, thanks to the relatively strong reflection signal. For long-range sensing, the pilot signal and a few self-interference-free OFDM symbols at the end of the data frame are leveraged to combat self-interference. Particularly, a novel pilot signal design is first developed to achieve subcarrier-wise pulse radar for accurate velocity and coarse range estimation, and then the self-interference-free OFDM symbols are utilized to achieve accurate range estimation. A strategy of frame transmissions is finally developed to integrate these two functions while balancing the performance between communications and sensing. Simulation results show that cm/s level velocity estimation and cm level range estimation can be achieved for up to 200-meter radar sensing.

Three-phase Golay sequence and array triads

3-phase Golay sequence and array triads are a natural generalisation of 2-phase Golay sequence pairs, yet their study has until now been largely neglected. We present exhaustive search results for 3-phase Golay sequence triads for all lengths up to 24, showing that the existence pattern is much richer than that for 2-phase Golay sequence pairs. We give an elementary proof that there is no 3-phase Golay sequence triad whose length is congruent to 4 modulo 6. We give two construction methods for 3-phase Golay array triads, and show how to project 3-phase Golay array triads to lower-dimensional Golay triads. In this way we explain much of the existence pattern found by exhaustive search.

Binary Golay Spreading Sequences and Reed-Muller Codes for Uplink Grant-Free NOMA

Non-orthogonal multiple access (NOMA) is an emerging technology for massive connectivity in machine-type communications (MTC). In code-domain NOMA, non-orthogonal spreading sequences are uniquely assigned to all devices, where active ones attempt a grant-free access to a system. In this paper, we study a set of user-specific, non-orthogonal, binary spreading sequences for uplink grant-free NOMA. Based on binary Golay complementary sequences, each spreading sequence provides the peak-to-average power ratio (PAPR) of at most 3 dB for multicarrier transmission. Exploiting the theoretical connection to Reed-Muller codes, we conduct a probabilistic analysis to search for a permutation set for Golay sequences, which presents theoretically bounded low coherence for the spreading matrix. Simulation results confirm that the maximum PAPR of transmitted multicarrier signals via the spreading sequences is significantly lower than those for random bipolar, Gaussian, and pseudo-random sequences. Thanks to the low coherence, the performance of compressed sensing (CS) based joint channel estimation and multiuser detection using the spreading sequences turns out to be superior or similar to those for random bipolar, Gaussian, pseudo-random, and Zadoff-Chu sequences. The binary Golay spreading sequences have only two phases regardless of the sequence length, which can make them suitable for low cost MTC devices.

Time-variance of 60 GHz vehicular infrastructure-to-infrastructure (I2I) channel

In this paper, we study the time-variance of a roadside infrastructure to infrastructure (I2I) channel operating at 60 GHz millimetre wave (mmWave) band, where the time-variance is caused by moving vehicles acting as scatterers. At first, measurement data is obtained by placing the transmitter (TX) and the receiver (RX) at different heights to emulate a link between two nonidentical roadside units (RSUs), and time-domain channel sounding is performed by sending complementary Golay sequences from the TX to the RX. A linear piece-wise interpolation of the corresponding temporal auto-correlation function (ACF) is used to find the Doppler spread of the I2I channel, where our interpolation method compensates for a slower sampling rate. Next, a framework is presented for time-variant channel impulse response (CIR) simulation which focuses on moving scatterers only and validates the linear piece-wise ACF model. The framework is useful for time-variant vehicular I2I channel simulation and in speed estimation related vehicular applications. Finally, a double-slope curve-fitted analytical model for ACF is proposed as a generalisation to the linear piece-wise model. The proposed model and its Doppler spectrum are found to be in agreement with the analytical results for fixed-to-fixed (F2F) channels with moving scatterers and matches perfectly with the measured data.

Accurate Angular Inference for 802.11ad Devices Using Beam-Specific Measurements

Due to their sparsity, 60GHz channels are characterized by a few dominant paths. Knowing the angular information of their dominant paths, we can develop various applications, such as the prediction of link performance and the tracking of 802.11ad devices. Although they are equipped with phased arrays, the angular inference for 802.11ad devices is still challenging due to their limited number of RF chains and limited phase control capabilities. Considering the beam sweeping operation and the high communication bandwidth of 802.11ad devices, we propose variation-based angle estimation (VAE), called VAE-CIR, by utilizing beam-specific channel impulse responses (CIRs) measured under different beams and the directional gains of the corresponding beams to infer the angular information of dominant paths. Unlike state-of-the-arts, VAE-CIR exploits the variations between different beam-specific CIRs for angular inference and provides a performance guarantee in the high signal-to-noise-ratio region. To evaluate VAE-CIR, we generate the beam-specific CIRs by simulating the beam sweeping of 802.11ad devices with the beam patterns measured on off-the-shelf 802.11ad devices. The 60GHz channel is generated via a ray-tracing-based simulator and the CIRs are extracted via channel estimation based on Golay sequences. Through extensive experiments, VAE-CIR is shown to achieve more accurate angle estimation than existing schemes.

English

Optical microscopic analysis of honey is time consuming due to the period needed to prepare samples. Time reduction could be achieved with ultrasound microscopy. This paper investigates the behavior of 125 MHz signal modulated by 16-bits Golay code spread out through a honey sample containing pollen. A bipolar phase shift keying (BPSK) modulation of 125 MHz frequency by 16-bits Golay code was implemented in Simulink/Matlab environment. The frequency implemented was set up considering the acoustic properties of honey containing pollen, the thickness of the sample and the size of pollen. At this frequency, the evaluated attenuation coefficient of honey containing pollen was 0.135 dB/&micro;m/MHZ(&gamma; =1); it depends on the power factor &gamma; related to the scattering medium, and the delay induces by the size of pollen. The impact of these parameters, added to 5dB White Gaussian Noise on 200 V magnitudes BPSK Golay sequences, decreased the autocorrelation function magnitude from 8&times;104 V to 1.5&times;104 V. The width (Wd) decreases from 4 ns at 0.135 dB/&micro;m/MHz(&gamma; =1) to 3.5 ns at 16.875 dB/&micro;m/MHz(&gamma; =2), when the Pulse Side lobe Level (PSL) increases from -22.79 dB at 1.509 dB/&micro;m/MHz(&gamma; =1) to -9.54 dB at 16.875 dB/&micro;m/MHz(&gamma; =2). Key words: Ultrasound, honey, attenuation coefficient, Golay code, BPSK modulation, autocorrelation function. &nbsp;

Root-Hadamard transforms and complementary sequences

In this paper we define a new transform on (generalized) Boolean functions, which generalizes the Walsh-Hadamard, nega-Hadamard, 2k-Hadamard, consta-Hadamard and all HN-transforms. We describe the behavior of what we call the root-Hadamard transform for a generalized Boolean function f in terms of the binary components of f. Further, we define a notion of complementarity (in the spirit of the Golay sequences) with respect to this transform and furthermore, we describe the complementarity of a generalized Boolean set with respect to the binary components of the elements of that set.

Multi-frame motion deblurring using coded exposure imaging with complementary fluttering sequences

Motion blurring is one of the most common but unpleasant phenomenons along with the camera imaging process, resulting in image resolution degradation. Coded exposure technique is an effective computational imaging approach to handle with this problem by introducing extra shutter control within the exposure time and help the deblurring problem become well-posed. In this paper, a multi-frame motion deblurring framework is presented using coded exposure imaging with complementary fluttering sequences. A set of complementary sequences are well designed to compensate for the spatial losses due to motion blurring in each frame based on the complementary theory of Golay matrix pair. Hadamard matrix is served as the Golay sequence pair to generate complementary fluttering sequences to capture multi-frame images for the following deblurring processing. The joint modulation transfer function (MTF) of the complementary code sequences has a more flat spectrum curve with smaller variance value and higher MTF value. For multi-frame deblurring implementation, a unified multi-image deblurring framework is performed by introducing Total Variation (TV) regularization prior with each selected code sequence in the complementary set integrated in the energy function. A real imaging system is built and various groups of synthetic and real-world imaging experiments are carried out to verify the effectiveness of the method. Experimental results demonstrate that the deblurrring images obtained by the proposed approach are of high quality with improved objective image evaluation values and better negative artifacts suppression.

Generalizing pairs of complementary sequences and a construction of combinatorial structures

Pairs of complementary sequences such as Golay pairs have zero sum autocorrelation at all non-trivial phases. Several generalizations are known where conditions on either the autocorrelation function, or the entries of the sequences are altered. We aim to unify most of these ideas by introducing autocorrelation functions that apply to any sequences with entries in a set equipped with a ring-like structure which is closed under multiplication and contains multiplicative inverses. Depending on the elements of the chosen set, the resulting complementary pairs may be used to construct a variety of combinatorial structures such as Hadamard matrices, complex generalized weighing matrices, and signed group weighing matrices. We may also construct quasi-cyclic and quasi-constacyclic linear codes which over finite fields of order less than 5 are also Hermitian self-orthogonal. As the literature on binary and ternary Golay sequences is already quite deep, one intention of this paper is to survey and assimilate work on more general pairs of complementary sequences and related constructions of combinatorial objects, and to combine the ideas into a single theoretical framework.

Optimization of a new Golay sequences shaping for low sidelobe radar ambiguity function

This work addresses the radar waveform design problem using phase-coded waveforms especially Golay complementary pair (GCP). Thus, a new shaping of GCP is proposed and the good performance of this shaping in terms of ambiguity function sidelobe (SL) rejection is shown. With the new shaping, a gain of at least 10 dB is easily obtained in comparison with the standard GCP. In order to improve the performance of the proposed waveform, two optimization algorithms are proposed to deal with low sidelobe ambiguity function problem, a very hard combinatorial optimization problem. The first one is the simulated annealing algorithm, a local search algorithm. The second represents the genetic algorithm-based global search strategy. Simulation results are provided and demonstrate that both types of algorithms decrease the SL rejection of the ambiguity function for all tested instances.