Polyphase Sequences Research Articles

Design of polyphase sequence sets with good correlation properties under spectral distortion via majorization-minimization framework

Orthogonal polyphase sequence sets with desirable properties have numerous applications in sensing fields, e.g., a monostatic radar transmitting such waveforms can effectively counter the coherent repeater jamming (CRJ). However, due to imperfections in practical systems, the delicately designed sequences may suffer significant distortion and reduction in performance when transmitted as radar waveforms. In this paper, we focus on the problem of designing polyphase sequence sets with good correlation properties under spectral distortion. To address this issue, we first introduce a new metric called distortion correlation properties (DCorr), which quantifies the quality of such sequence sets. Subsequently, a majorization-minimization (MM)-based iterative algorithm is developed to solve the problem of minimizing DCorr. A number of numerical examples are provided to demonstrate the effectiveness of the proposed method in improving the correlation properties of polyphase sequence sets under spectral distortion. The results also validate that our proposed algorithms can effectively mitigate the performance reduction caused by the imperfections in practical radar systems simply.

Orthogonal Circulant Multiplexing for Bandwidth- Limited Direct-Detected Optical Communication Systems

We propose and investigate a novel orthogonal circulant multiplexing (OCM) technology for bandwidth-limited direct detection (DD) optical systems. We analytically derive that any OCM, i.e. using any orthogonal circulant matrix for multiplexing, can redistribute the noise uniformly across the subcarriers. However, the distribution of inter-symbol interference (ISI) and inter-carrier interference (ICI) strongly depends on the design of OCM. The previously proposed orthogonal chirp division multiplexing (OCDM), although showing better ISI/ICI tolerance than discrete-Fourier-transform spread (DFT-S) orthogonal frequency division multiplexing (OFDM), is a kind of OCM with poorer performance. We propose and analytically reveal that OCM based on chirp-like polyphase sequence (CLPS), general chirp sequence (GCS), sparse CLPS (SCLPS), and sparse GCS (SGCS) can achieve flatter ISI/ICI distribution across subcarriers and so better performance than OCDM in bandwidth-limited DD systems. The optimal parameter vector of the E-parameter CLPS is a Zadoff-Chu sequence while the parameter α in GCS should be larger. The optimal performance can be achieved by either CLPS (or GCS) using a larger E (or α) or SCLPS (or SGCS) with optimized sparsity. ∼110Gbit/s experiments using Mach-Zehnder modulator or electro-absorption modulator and receiver of 10-GHz photodiode or 8-GHz avalanche photodiode are conducted. The results agree with the theory and show that the proposed OCM outperforms conventional OFDM, DFT-S OFDM, and OCDM.

Designing Interference-Immune Doppler-Tolerant Waveforms for Radar Systems

Dynamic target detection using LFM waveform is challenging in the presence of interference for different radar applications. Degradation in SNR is irreparable and interference is difficult to mitigate in time and frequency domain. In this paper, a waveform design problem is addressed using the Majorization-Minimization (MM) framework by considering PSL/ISL cost functions, resulting in a code sequence with Doppler-tolerance characteristics of an LFM waveform and interference immune characteristics of a tailored polyphase sequence (unique phase code + minimal ISL/PSL). The optimal design sequences possess polynomial phase behavior of degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$Q$</tex-math></inline-formula> amongst its sub-sequences and obtain optimal ISL and PSL solutions with guaranteed convergence. By tuning the optimization parameters such as degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$Q$</tex-math></inline-formula> of the polynomial phase behavior, sub-sequence length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> and the total number of sub-sequences <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L$</tex-math></inline-formula> , the optimized sequences can be as Doppler tolerant as LFM waveform in one end, and they can possess small cross-correlation values similar to random-phase sequences in polyphase sequence on the other end. The numerical results indicate that the proposed method is capable to computationally design chirplike sequences which prior to this work, were obtained by mimicking phase variations of LFM waveform. An application of the proposed method for the automotive scenario is also illustrated in the numerical results.

New Family of Polyphase Sequences with Low Correlation from Galois Rings

New Family of Polyphase Sequences with Low Correlation from Galois Rings

Sequences With Good Correlations Based on Circular Florentine Arrays

Sequences and their correlation properties have been extensively studied due to their broad applications. In this paper, we develop a connection between sequences and well-studied combinatorial objects, circular Florentine arrays. This connection allows us to derive two types of sequences with good correlation properties. The first type consists of sequences having optimal correlation with respect to the Sarwate bound. Our constructions are based on perfect polyphase sequences. The number of perfect sequences with optimal correlation depends on the existence of circular Florentine arrays, which improves the previous known results. The second type is about multiple ZCZ sequence sets with low inter-set cross-correlation. Each generated ZCZ sequence set is optimal with respect to the Tang-Fan-Matsufuji bound, and each sequence in each set is perfect. In addition, any two sequences from distinct ZCZ sequence sets possess optimal inter-set cross-correlation with respect to the Sarwate bound. Compared with the previous results, the number of ZCZ sequence sets with optimal inter-set cross-correlation property is improved, because of the existence of circular Florentine arrays.

Chirp-Like Polyphase Sequence Based Discrete Circulant Transform Spread OFDM for Bandwidth-Limited IM/DD Systems

We propose and investigate chirp-like polyphase sequence (CLPS) based discrete circulant transform spread (DCrT-S) OFDM in bandwidth-limited intensity-modulation and direct-detection systems. This scheme can control the distribution of residual inter-symbol interference (ISI) and inter-carrier interference (ICI) across subcarriers by adjusting the CLPS's parameters. We also propose a sparse CLPS (SCLPS) scheme, by inserting zeros between the elements of the CLPS, to further balance the effects of the ISI/ICI and the noise and improve the performance by optimizing both the CLPS's parameters and the sparsity. We reveal the optimal design rule of CLPS/SCLPS analytically and investigate the performance of the proposed schemes using two-parameter CLPS in ∼110-Gbit/s simulations and experiments. It is shown that the optimal parameter vector of the CLPS is a Zadoff-Chu sequence that spreads the ISI/ICI more evenly over subcarriers. The performance is further improved by optimizing the sparsity. The optimal CLPS and sparsity are insensitive to the received optical power, the system bandwidth, the length of cyclic prefix, the chirp parameter and the signal data rate, and so can be pre-set without channel information. Both CLPS- and SCLPS-based DCrT-S-OFDM outperform Nyquist PAM, conventional OFDM, discrete-Fourier-transform spread (DFT-S) OFDM, and orthogonal-circulant-transform precoded (OCT-P) OFDM. By using the proposed scheme, the data rate is enhanced by 6.7% and 3.2% compared to DFT-S-OFDM and OCT-P-OFDM at a bit error rate (BER) of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−3</sup> , while the complexity is reduced to 57.9% of OCT-P-OFDM.

Design of Polyphase Sequences With Low Integrated Sidelobe Level for Radars With Spectral Distortion via Majorization-Minimization Framework

Polyphase sequences with low aperiodic autocorrelation levels, assessed in terms of the integrated sidelobe level (ISL), can be applied in many engineering fields, such as radars, sonars, and code-division multiple access communication systems. In practice, a polyphase sequence that is optimal in some sense (for example, minimum ISL) could be significantly distorted, thus reducing the detection performance of the radar system. Hence in this article, we focus on designing ISL minimization-based polyphase sequences with spectral distortion taken into consideration, where we first derive an expression for the ISL under spectral distortion, i.e., distorted ISL (DISL), and then we propose an iterative algorithm based on majorization-minimization framework to solve the DISL minimization problem. The proposed algorithm is proved to be nonincreasing and is guaranteed to converge to a stationary point. A variety of design results utilizing the proposed algorithm in different scenarios are presented and compared with those by other methods or some known sequences. The proposed algorithm is proved to be effective to design polyphase sequences with low DISL.

A new family of polyphase sequences with low correlation

A new family of polyphase sequences with low correlation

A new class of polyphase pseudo-random sequences with a nonlinear generation algorithm for secure radio communication link between unmanned vehicles and dispatch centers of the "smart city"

The article discusses an algorithm for generating a new class of polyphase pseudo-random sequences with a nonlinear generation algorithm for use in secure communication systems with spread spectrum technology. In particular, the proposed sequences are relevant to use for generating signals in a secure command radio link between unmanned vehicles located in different environments, and dispatch centers of the "smart city". An analysis of the existing ensembles and an assessment of their main characteristics are carried out. Namely, the size of the ensemble, which affects the security of the communication system, and the level of maximum emissions of the cross-correlation function. The existing ensembles of polyphase pseudo-random sequences do not simultaneously have satisfactory values of the considered indicators. Therefore, an urgent task is to develop an ensemble that finds a compromise between these two characteristics and has a nonlinear generation algorithm. The essence of the algorithm for the formation of the proposed new class of polyphase pseudo-random sequences is to expand the classical polyphase sequences, for example, Frank, Chu, or Milewski, which have good correlation properties, de Bruijn sequences with a large ensemble and a nonlinear formation algorithm. Comparative analysis of the proposed method for forming a class of polyphase sequences shows the advantage of its use. In particular, an improvement in cross-correlation characteristics and a significant increase in the size of the ensemble of new sequences compared to the original sequences are shown. These properties can be used to increase the structural secrecy of command radio links.

Rudin-Shapiro-Like Sequences With Maximum Asymptotic Merit Factor

Borwein and Mossinghoff investigated the Rudin-Shapiro-like sequences, which are infinite families of binary sequences, usually represented as polynomials. Each family of Rudin-Shapiro-like sequences is obtained from a starting sequence (which we call the seed) by a recursive construction that doubles the length of the sequence at each step, and many sequences produced in this manner have exceptionally low aperiodic autocorrelation. Borwein and Mossinghoff showed that the asymptotic autocorrelation merit factor for any such family is at most 3, and found the seeds of length 40 or less that produce the maximum asymptotic merit factor of 3. The definition of Rudin-Shapiro-like sequences was generalized by Katz, Lee, and Trunov to include sequences with arbitrary complex coefficients, among which are families of low autocorrelation polyphase sequences. Katz, Lee, and Trunov proved that the maximum asymptotic merit factor is also 3 for this larger class. Here we show that a family of such Rudin-Shapiro-like sequences achieves asymptotic merit factor 3 if and only if the seed is either of length 1 or is the interleaving of a pair of Golay complementary sequences. For small seed lengths where this is not possible, the optimal seeds are interleavings of pairs that are as close as possible to being complementary pairs, and the idea of an almost-complementary pair makes sense of remarkable patterns in previously unexplained data on optimal seeds for binary Rudin-Shapiro-like sequences.

Generalization and Improvement of the Levenshtein Lower Bound for Aperiodic Correlation

This paper deals with lower bounds on aperiodic correlation of sequences. It intends to solve two open questions. The first one is on the validity of the Levenshtein bound for a set of sequences other than binary sequences or those over the roots of unity. Although this result could be a priori extended to polyphase sequences, a formal demonstration is presented here, proving that it does actually hold for these sequences. The second open question is on the possibility to find a bound tighter than Welch’s, in the case of a set consisting of two sequences $M = 2$ . By including the specific structure of correlation sequences, a tighter lower bound is introduced for this case. Besides, this method also provides in the cases $M = 3$ and $M = 4$ a tighter bound than the up-to-now tightest bound provided by Liu et al.

AMPLITUDE AND PHASE MODULATED PULSE TRAINS FOR CLUTTER REJECTION IN RADAR

In radar systems the very low side-lobes that can be achieved with amplitude and phase modulated pulse trains make their use particularly attractive in systems requiring a large dynamic range. Moreover, the excellent self-clutter rejection performance is obtained without sacrificing the ability for the separation of close targets (no main lobe widening). The additional expense of encoding and decoding in amplitude and phase may be justified for radars that must cope with land clutter or operate in a dense-target environment. This paper presents a method for generating finite length sequences that have high energy ratio and low sidelobe energy in their autocorrelation function. The method is computer based and generates the desired sequence of any given length in an iterative manner starting with a sequence that has relatively low sidelobes in its autocorrelation function. It is shown that energy ratio of polyphase sequences that have low sidelobe energy , is improved through the process of clipping.

Optimal Families of Perfect Polyphase Sequences from Cubic Polynomials

Optimal Families of Perfect Polyphase Sequences from Cubic Polynomials

Optimal Mutually Orthogonal ZCZ Polyphase Sequence Sets

Optimal Mutually Orthogonal ZCZ Polyphase Sequence Sets

Merit factor analysis of polyphaser sequences using cyclic algorithm new with good correlation properties

Polyphase sequences such as Pn (n=1, 2, 3, 4, x), Golomb, Frank, and the Chu are with good correlation properties, lower sidelobe levels and large merit factor values are helpful in applications like radar, sonar and channel estimation and communications. The goodness of a sequence obtained from merit factor. The transmitted and received signal may not be the same due to noise. The correlation function of given sequence is expressed by ISL (Integrated Sidelobe Level) by minimizing the ISL metrics the performance parameter merit factor is improved. To make this possible the ISL metric is expressed in the frequency domain and minimized to its most recent values and fixing at their most recent value until the predefined threshold satisfied. Because of FFT operations, the Cyclic Algorithm New applied to very long length sequences say N~106. In this paper, the Merit factor and correlation levels compared with standard, and cyclic algorithm new initialized with Polyphase sequences for lengths 102~104. Moreover, the observations made for four consecutive even and odd integer lengths say 162, 172, 182, and 192. CAN (P3, Golomb) exhibits merit factor improvement of 3.77%. These sequences of sidelobe levels reduced.

A Family of Polyphase Sequences With Asymptotically Optimal Correlation

Sequences with low correlation have important applications in communications, radar, and cryptography. In this paper, a simple construction of polyphase sequences using additive and multiplicative characters over the finite field ${\mathrm {F}}_{q}$ is proposed. The construction works for any finite field ${\mathrm {F}}_{q}$ with $q>2$ and generates a family of $q-1$ sequences with period $q-1$ and maximum correlation $\sqrt {q}$ . This family is asymptotically optimal with respect to the well-known Welch bound. Most notably, the maximum autocorrelation magnitude of each sequence in this family is equal to 1, and every two distinct sequences are orthogonal to each other. The distribution of the correlation magnitudes of this family is also established.

The Optimal Assignment of Orthogonal Polyphase Sequences in CDMA Systems

In this letter, ternary semi-bent functions and a complex Hadamard matrix are used to construct spreading sequences for code-division multiple-access systems. An efficient assignment of these sequences to a lattice of regular hexagonal cells, which are mutually orthogonal within each cell and the adjacent cells, is achieved and most notably the number of sequences per cell is maximum possible. The large sequence reuse distances, being $\sqrt {27}$ , and low cross-correlation between non-orthogonal sequences (in non-adjacent cells) make these sequences appealing for practical applications.

A polyphase sequences with low autocorrelations

This paper gives numerical method for unimodular sequences design with low peak sidelobe (PSL) level of aperiodic autocorrelation function (AACF). The result of numerical optimization for design of unimodular sequences of length N = 1024 with smallest today PSL ≈ 2.618 is shown. Such sequences are useful for resolution and parameter estimation in radar applications due to their large signal-to-noise ratio.

Application of P4 Polyphase codes pulse compression method to air-coupled ultrasonic testing systems

Air-coupled ultrasonic testing systems are usually restricted by low signal-to-noise ratios (SNR). The use of pulse compression techniques based on P4 Polyphase codes can improve the ultrasound SNR. This type of codes can generate higher Peak Side Lobe (PSL) ratio and lower noise of compressed signal. This paper proposes the use of P4 Polyphase sequences to code ultrasound with a NDT system based on air-coupled piezoelectric transducer. Furthermore, the principle of selecting parameters of P4 Polyphase sequence for obtaining optimal pulse compression effect is also studied. Successful results are presented in molded composite material. A hybrid signal processing method for improvement in SNR up to 12.11dB and in time domain resolution about 35% are achieved when compared with conventional pulse compression technique.

Orthogonal superimposed training design for doubly selective channel estimation using basis expansion models

AbstractIn this paper, we consider channel estimation for single‐carrier block transmission over doubly selective fading channel, where basis expansion models are used to describe the time‐varying channel. The affine precoding model is used as the transmission strategy, wherein the training sequence is added on the top of the precoded data prior to transmission. By enforcing a special form of orthogonality between training sequence and precoded matrix, we propose orthogonal superimposed training design for doubly selective channel that makes channel estimation decouple from symbol detection, and then the affine precoding model based on orthogonal polyphase sequence set is designed under the orthogonal conditions. To further improve the estimation performance, we exploited a joint iterative approach is used, where the detected symbols, to enhance the estimation performance. Simulations results show that the proposed scheme can yield better detection performance than the data‐dependent superimposed training and can be competitive with time‐multiplexed training scheme with higher bandwidth efficiency.