Perfect Periodic Autocorrelation Research Articles

Channel estimation with Zadoff–Chu sequences in the presence of phase errors

AbstractDue to their perfect periodic autocorrelation property, Zadoff–Chu sequences are often used as stimulus signals in the measurement of radio channel responses. In this letter, the cross‐correlation of a linear shift‐invariant system's response to a Zadoff–Chu sequence is derived for the case that a multiplicative perturbation such as phase noise is present at its output. The perturbation signal is shown to incur a spurious discrete Fourier transform, thus degrading the correlation's peak to sidelobe ratio. The implications for the performance limits of channel measurement systems are discussed.

Phase Noise Effects on Phase-Sensitive OTDR Sensors Using Optical Pulse Compression

We introduce a detailed theoretical, numerical, and experimental study of the effects of laser’s phase noise on the performance of phase-sensitive optical time-domain reflectometry ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\phi$</tex-math></inline-formula> -OTDR) sensors that use optical pulse compression (OPC). Pulse compression is a technique that can be used to improve the received signal amplitude by increasing the effective energy of the pulses that are launched into the fiber without degrading the spatial resolution of the measurements. Therefore, it is a valuable tool to extend the range of these sensors and mitigate fiber attenuation constraints. However, it has been observed that the limited coherence of the laser source has a degrading effect on the actual performance enhancement that this method can provide. Here, we derive a theoretical model that can be used to quantify this degradation for any type of OPC such as those based on either linear frequency modulation (LFM) pulses or perfect periodic autocorrelation (PPA) bipolar bit sequences. The model facilitates numerical estimation of the sensitivity of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\phi$</tex-math></inline-formula> -OTDR measurements. It also produces theoretical expressions for the mean and the variance of the phase-noise perturbed backscatter response. These results are validated via numerical simulations and experiments in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\phi$</tex-math></inline-formula> -OTDR setups using LFM as well as PPA OPC. Furthermore, we demonstrate the use of the model to investigate the basic trade-offs involved in the design of OPC <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\phi$</tex-math></inline-formula> -OTDR systems.

Ultrahigh scan-rate quasi-distributed acoustic sensing system using array match interrogation.

Inspired by compressed sensing techniques, a method for significantly enhancing the maximum allowable scan rate in quasi-distributed acoustic sensing (Q-DAS) is described and studied. Matching the scan parameters to the interrogated array facilitates orders of magnitude improvement in the scan rate and a corresponding increase in the maximum slew rate (SR) of differential phase variations which can be measured without ambiguity. The method is termed array matched interrogation (AMI). To improve the method's SNR, maximum number of sensing sections and maximum range, the interrogation pulse can be replaced by a perfect periodic autocorrelation (PPA) code. This version of the method is referred to as coded array matched interrogation (C-AMI). The implementation of C-AMI is not trivial and requires special design rules which are derived and tested experimentally. The design rules ensure that the 'folding' of the returning peaks of the Q-DAS array into a scan period, which is much shorter than the fiber's roundtrip time, will not lead to overlaps. The method demonstrated a scan rate of 20 times higher than the common limit and measurement of unprecedented slew-rate of 10.5 ×106 rad/s.

The sparsity of the response of a quasi-distributed fiber optic sensing system allows ‘overclocking’ its interrogation

The response of a quasi-distributed fiber optic sensing system to a short interrogation pulse can be considered as a sparse signal in the time domain. This attribute can be exploited to enhance its performance beyond the commonly accepted limits. The work described here is analogous to some compressed sensing systems, which can be adequately sampled at sub-Nyquist rates, based on knowledge of their sparsity in the spectral domain. Equivalently, it is shown here how the sparsity of the impulse-response of a Quasi-Distributed Acoustic Sensing (Q-DAS) system can be exploited to interrogate it at rates well above the limit dictated by the total duration of its impulse response. An additional improvement in performance is achieved by using a Perfect Periodic Auto-correlation (PPA) code, rather than a pulse, for interrogation of the system. Cross-correlating the raw response of the Q-DAS system with the original code retains the sparsity of the response and provides significant gain in SNR without losing spatial resolution. Experimentally, the method was utilized to implement a 76 km long acoustic sensing system and successfully detected 250 kHz ultrasound signal. This was achieved by interrogating the system at a rate of 971 kHz which is 737 times the inherent limit dictated by the duration of the impulse-response of the system.

Combining PRF staggering with phase‐coded inter‐pulse overlay to combat delay–Doppler ambiguities

Pulse repetition frequency (PRF) staggering is a well-established technique in the pulse-Doppler radar, used in order to increase the unambiguous range and Doppler and to minimise blind spots. The study offers a new staggering dimension that can increase the number of different staggering choices. The new approach is to add an inter-pulse coherent phase-coded overlay. The code is shorter than the number of pulses in each PRF batch and is repeated an integer number of times within the batch. A required property of the phase code is perfect periodic autocorrelation, namely zero autocorrelation sidelobes. In order to minimise the added complexity, the overlay codes use a two-phase or three-phase alphabet. As in a conventional PRF staggering, there are P coherent batches, each using a different PRF, with its matched phase-coded overlay. The performance analysis uses a conventional binary integration rule that declares detection if there were H , or more, individual detections in the P batches.

Sequences with Perfect Periodic Auto and Cross Correlation Properties

Sequences with Perfect Periodic Auto and Cross Correlation Properties

Distributed Dynamic Strain Sensing via Perfect Periodic Coherent Codes and a Polarization Diversity Receiver

Rayleigh scattering-based dynamic strain sensing with high spatial resolution, fast update rate, and high sensitivity is highly desired for applications such as structural health monitoring and shape sensing. A key issue in dynamic strain sensing is the tradeoff between spatial resolution and the Signal-to-Noise Ratio (SNR). This tradeoff can be greatly relaxed with the use of coding. A sequence of optical pulses is injected into the fiber and the detected backscattered signal is cross correlated with the original signal. With the use of coding, SNR is indeed improved, but if the sequence is not well chosen, the resulting Peak to Sidelobe Ratio (PSR) can be rather low. An excellent choice of codes are biphase Legendre sequences which offer near Perfect Periodic Autocorrelation (PPA). Other common issues in Rayleigh scattering-based sensing techniques are signal fading and dynamic range. The former issue can occur due to destructive interference between lightwaves that are scattered from the same spatial resolution cell and, in coherent detection schemes, when the polarization states of the backscattered light and the reference light are mismatched. The latter issue is a concern in phase sensitive schemes which require signal jumps not to exceed 2π. In this paper, a biphase Legendre sequence with 6211 pulses is used in conjunction with polarization diversity scheme and a PM fiber. The setup provides two independent measurements of the sensing fiber complex profile and achieves highly sensitive, distributed dynamic strain sensing with very low probability of fading. In addition, the system can handle both very large perturbation signals and very small perturbation signals. The system operated at a scan rate of ~107 kHz and achieved spatial resolution of ~10 cm and sensitivity of ~1.1mrad/ √Hz. The ratio between the powers of the maximum and minimum excitations that can be measured by the system is 136 dB.

Large Families of “Grey” Arrays with Perfect Auto-correlation and Optimal Cross-Correlation

Digital watermarking applications have a voracious demand for large sets of distinct 2D arrays of variable size that possess both strong auto-correlation and weak cross-correlation. We use the discrete Finite Radon Transform to construct “perfect” \(p \times p\) arrays, for p any prime. Here the array elements are comprised of the integers \(\{0,\pm 1,+2\}\). Each array exhibits perfect periodic auto-correlation, having peak correlation value \(p^2\), with all off-peak values being exactly zero. Each array, by design, contains just \(3(p-1)/2\) zero elements, the minimum number possible when using this “grey” alphabet. The grey alphabet and the low number of zero elements maximises the efficiency with which these perfect arrays can be embedded into discrete data. The most useful aspect of this work is that large families of such arrays can be constructed. Here the family size, M, is given by \(M = p^2-1\). Each of the \(M(M-1)/2\) intra-family periodic cross-correlations is guaranteed to have one of the three lowest possible merit factors for arrays with this alphabet. The merit factors here are given by \(v^2/(p^2-v^2)\), for \(v = 2\), 3 and 4. Whilst the strength of the auto-correlation rises with array size p as \(p^2\), the strength of the many (order \(p^4\)) cross-correlations between all M family members falls as \(1/p^2\).

On the structure of quadratic Gauss sums in the Talbot effect.

We report on the detailed derivation of the Gauss sums leading to the weighting phase factors in the fractional Talbot effect. In contrast to previous approaches, the derivation is directly based on the two coprime integers p and q that define the fractional Talbot effect so that, using standard techniques from the number theory, the computation is reduced, up to a global phase, to the trivial completion of the exponential of the square of a sum. In addition, it is shown that the Gauss sums can be reduced to only two cases, depending on the parity of integer q. Explicit and simpler expressions for the two forms of the Talbot weighting phases are also provided. The Gauss sums are presented as a discrete Fourier transform pair between quadratic phase sequences showing perfect periodic autocorrelation and a connection with the theory of biunimodular sequences is presented. In addition, the Talbot weighting factors of orders 1/q and 2/q are reduced to a closed form, and the equivalence to existing characterizations of Talbot weighting phases is also discussed. The relationship with one-dimensional multilevel phase structures is exemplified by the study of Talbot array illuminators. These results simplify and extend the description of the role played by Gauss sums in the fractional Talbot effect, providing a compact synthesis of previous results.

A Pre-Compression Algorithm for Optimal Acquisition of Modernized GNSS Signal

The acquisition of GNSS signal is the most challenging part of any GNSS receiver. With the modernized GNSS system having longer code length and wider bandwidth, the size of the data block becomes large. To reduce the computational load of the processor, the reduction in the size of the data block without compromise in the acquisition loop performance provides overall improvement to the system. Here, a model is developed for acquisition of GPS signal L1 and L2C by compressing the input raw GPS signal with sequences of perfect periodic autocorrelation properties such as the Ipatov sequence. A compression matrix of the required dimension is constructed for compressing the input GPS signal. The improvement in the acquisition performance of the model is discussed.

Almost multiphase sequences based on Chu sequences

A new family of almost multiphase sequences (AMPSs) with perfect periodic autocorrelation function (PACF) is offered. The obtained sequences are based on Chu sequences and differ from them by the structure, the peak-factor which is nearly equal to one, and by fewer phases for the same periods.

Optimal design of perfect DFT sequences

In this paper, we propose a novel scheme to construct a large set with N(N+1) perfect sequences of length N, derived from an Inverse Discrete Fourier Transform (IDFT) of Chu and maximum-length sequences. This optimum set of perfect periodic autocorrelation sequences has a maximum absolute value of periodic cross-correlation strictly lower than N and close to the well known lower bound SQRT(N). Moreover, we present a method to transform these perfect sequences into orthogonal sequences. A similar method is also proposed to obtain optimum bipolar codes derived from an alternative set of N2 perfect sequences.In the design of perfect sequences, the difficulty is to achieve both low cross-correlation and low peak-to-average power ratio (PAPR). Many of the proposed perfect DFT sequences should have low PAPR and thus can be applied in an OFDM–CDMA (Orthogonal Frequency-Division Multiplexing–Code Division Multiple Access) communication system or in a simple CDMA communication system. Alternatively, the proposed perfect DFT sequences may be useful for pre-coded mapping in OFDM communication systems or for the design of radar waveform diversity sets.

Orthogonal perfect discrete Fourier transform sequences

In this study, the authors propose a method to obtain many different sets of N orthogonal perfect periodic autocorrelation sequences, with length N, derived from Walsh-Hadamard codes and orthogonal Gold codes. Moreover, these orthogonal perfect discrete Fourier transform (DFT) sequences can be transformed into real orthogonal perfect sequences or into bipolar codes with better correlation properties than other well known bipolar codes used in code division multiple access (CDMA) systems.

Phase-Coded Interrupted CW Signals and Processors

High resolution radars require signals with large time-bandwidth product such as CW signal and coherent pulse train (CPT). We discuss a phase-coded interrupted CW (ICW) signal which is the combination of CW signal and CPT. Phase codes used here are with perfect periodic autocorrelation. The periodic ambiguity function of ICW signals is studied including single-carrier signal and multi-carrier signal. It is interesting that the gate function has different effects on two signals and contributes to a multi-carrier ICW signal which yields nearly perfect autocorrelation. Meanwhile we also suggest an efficient receiver approach to ICW signals, which can reduce the computational burden of the processor and utilize the good properties of P3 and P4 codes.

Designing unique words for near‐optimal training in block transmission systems

AbstractIn this paper, we consider the problem of designing unique words (UWs) in single‐carrier block transmission systems for the purpose of performing bandwidth efficient channel estimation. By first deriving the Cramér‐Rao lower bound (CRLB) for the UW‐based channel estimator, we show that designing optimal UWs reduces to a problem of constructing sequences of length Q ≤ N/2 that, when padded with zeros to make a length‐N sequence, have good periodic autocorrelation properties. We then show that the only class of short (length‐Q) sequences that, when padded with N − Q zeros, have perfect periodic autocorrelation properties is the class of Kronecker delta sequences. Due to the fact that these sequences typically suffer from a high peak‐to‐average power ratio (PAPR), we then formulate two optimisation procedures to find near‐optimal UWs under certain constraints. Finally, we show that the sequences that are found with the proposed technique perform significantly better than Kronecker delta sequences that have been scaled to meet a given PAPR constraint. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Almost perfect sequences based on cyclic difference sets

Abstract The perfect sequences are so ideal that all out-of-phase autocorrelation coefficients are zero, and if the sequences are used as the coding modulating signal for the phase-modulated radar, there will be no interference of side lobes theoretically. However, it has been proved that there are no binary perfect sequences of period 4 < n ≤ 12100. Hence, the almost perfect sequences with all out-of-phase autocorrelation coefficients as zero except one are of great practice in engineering. Currently, the cyclic difference set is one of most effective tools to analyze the binary sequences with perfect periodic autocorrelation function. In this article, two characteristic formulas corresponding to the autocorrelation and symmetric structure of almost perfect sequences are calculated respectively. All almost perfect sequences with period n, which is a multiple of 4, can be figured out by the two formulas. Several almost perfect sequences with different periods have been hunted by the program based on the two formulas and then applied to the simulation program and practical application for ionospheric sounding.

Two-valued frequency-coded waveforms with favorable periodic autocorrelation

There are known phase-coded (two-valued or polyphase) CW radar signals that exhibit perfect periodic autocorrelation function (PACF). A PACF is perfect when all its out-of-phase autocorrelation values are identically equal to zero. This paper investigates periodic, two-valued, frequency-coded signals. While none could be found with perfect PACF, we present examples with nearly perfect PACF. Their relationship to binary phase-coded signals is also considered. These signals should be attractive for CW radars because of their simple implementation, clean spectrum, and the favorable range response of their matched receiver.

Mismatched filtering of periodic quadriphase and 8-phase sequences

Quadriphase and 8-phase sequences with good autocorrelation properties are required in higher order digital modulation schemes, e.g., for channel estimation or synchronization. Since such sequences with perfect periodic autocorrelation function exist only for a few lengths, mismatched filtering is discussed. Mismatched filtering entirely suppresses all sidelobes of the correlation function at the expense of a slightly reduced signal-to-noise ratio (SNR). We present new construction methods for quadriphase and 8-phase sequences which exist for many lengths and exhibit significantly lower SNR losses than any other known sequences.

Digital multi-carrier spread spectrum versus direct sequence spread spectrum for resistance to jamming and multipath

We compare single user digital multi-carrier spread spectrum (MC-SS) modulation with direct sequence (DS) SS (with a modified implementation) in the presence of narrowband interference (NBI) and multipath fading. We derive closed-form expressions for the symbol error probability for both the linear MMSE receiver as well as the conventional matched-filter receiver under different scenarios: additive white Gaussian noise (AWGN) channel with NBI, multipath channel with or without NBI. We show that DS-SS can achieve the same performance as MC-SS if the spreading code is carefully designed to have perfect periodic autocorrelation function (PACF). On the other hand, MC-SS is more robust to narrowband interference and multipath fading than is DS-SS with the widely used spreading codes that do not possess perfect PACE. Our analysis reveals that the performance improvement of MC-SS is precisely due to the implicit construction of an equivalent spreading code having nonconstant amplitude but possessing perfect periodic autocorrelation.

Two-dimensional perfect quaternary arrays

We study two-dimensional (2-D) arrays of fourth roots of unity which have all out-of-phase periodic autocorrelations equal to zero. Generalizing the concept of a perfect binary array, we call these arrays perfect quaternary arrays. We establish connections with combinatorial design theory and exhibit large families of such arrays. Increasing the alphabet from size two to size four greatly increases the flexibility one has in choosing the dimensions for 2-D arrays with perfect periodic autocorrelation. For example, we show that the number of entries in a 2-D perfect quaternary array may be divisible by any Mersenne prime and indeed by many other primes, whereas 2-D perfect binary arrays are only known to exist with size equal to a power of two times a power of three.