Mathematical Problems in Engineering

Volume 2018, Article ID 2860809, 13 pages

https://doi.org/10.1155/2018/2860809

## Optimal Unimodular Sequences Design Method for Active Sensing Systems

Science and Technology on Electromechanical Dynamic Control Laboratory, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to Xiaopeng Yan; nc.ude.tib@gnepoaixnay

Received 7 December 2017; Accepted 22 March 2018; Published 10 May 2018

Academic Editor: Kishin Sadarangani

Copyright © 2018 Ze Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In active sensing systems, unimodular sequences with low autocorrelation sidelobes are widely adopted as modulation sequences to improve the distance resolution and antijamming performance. In this paper, in order to meet the requirements of specific practical engineering applications such as suppressing certain correlation coefficients and finite phase, we propose a new algorithm to design both continuous phase and finite phase unimodular sequences with a low periodic weighted integrated sidelobe level (WISL). With the help of the transformation matrix, such an algorithm decomposes the -dimensional optimization problem into one-dimensional optimization problems and then uses the iterative method to search the optimal solutions of the one-dimensional optimization problems directly. Numerical experiments demonstrate the effectiveness and the convergence property of the proposed algorithm.

#### 1. Introduction

Since random sequences with low autocorrelation sidelobes can greatly improve the distance resolution of the detection signal, they are widely adopted as modulation sequences in active sensing systems, like radar, communication, proximity fuze, and so on [1–4]. Moreover, due to the limitations of hardware conditions (such as digital-to-analog converters, power amplifiers), it is usually more desirable to transmit unimodular sequences (i.e., constant modulus) to make full utilization of the transmitter’s power [5, 6]. So far, the most common modulation sequences applied in practical engineering are the pseudorandom code binary sequences, like sequence, Gold sequence, chaotic sequence, and so on. Based on a binary phase shift keying modulation scheme, the pseudorandom code sequence usually corresponds to simple system hardware, but it is easy to crack. For instance, the sequence can be easily cracked by estimating its primitive polynomial with the help of third-order correlation function [7]. Therefore, in order to improve both the distance resolution and antijamming performance of active sensing systems, it is necessary to carry on waveform design to obtain unimodular sequences with good autocorrelation property.

According to the definition, the autocorrelation can be distinguished from periodic autocorrelation and aperiodic autocorrelation. Let denote the unimodular sequence to be designed. The periodic () and aperiodic () autocorrelations of are defined asThen the integrated sidelobe level (ISL) which can be used to measure the quality of the autocorrelation property of a sequence is defined asThe smaller the ISL of a sequence, the better the autocorrelation property of this sequence. The above formula shows the definition of ISL metric for periodic autocorrelations, and the ISL metric for the aperiodic autocorrelations can be defined similarly. It is worth noting that the ISL metric is equivalent to another very important measure, the merit factor (MF), and the relationship between the two is as follows:Correspondingly, the higher the MF of a sequence, the better the autocorrelation property of this sequence.

The simple analysis described above indicates that to improve the autocorrelation property of a sequence it is necessary to optimize each code of the sequence, which means that the optimization problem is essentially an -dimensional optimization problem, and its optimal solution cannot be obtained directly. In order to solve this problem, the majority of scholars have proposed a number of algorithms to obtain the optimal solution [8–22]. Since the -dimensional optimization problem may have multiple local optimal solutions, the simulated annealing [8] and stochastic optimization methods [9, 10] were suggested to obtain the global optimal solution at the early stage. However, as the length of the sequence () grows, the computational burdens of these methods increase exponentially. Therefore, in order to design a sequence with a long length, both the effectiveness and convergence property of the algorithm need to be taken into account. Recently, two groups of typical algorithms [11–18] with superiorities in the above two aspects have attracted extensive attention from scholars. The work in [11] proposed an iterative algorithm called CAN (Cyclic Algorithm-New) to design a unimodular sequence with low aperiodic ISL. The algorithm deduces the relationship between the aperiodic ISL and the Fourier spectrum of the sequence, which makes it possible to significantly improve the computational efficiency of such an algorithm by implementing fast Fourier transform (FFT) operations in the process of seeking the optimal solution. In addition, differently modified or extended versions of such algorithms have been applied to generate unimodular sequences with low periodic ISL (PeCAN) [12], unimodular sequences with low aperiodic weighted ISL (WeCAN) [11], and complementary sets of sequences (CSS) with both good autocorrelation and cross-correlation properties for MIMO radar systems (CANARY) [13–15]. The work in [16] proposed another iterative algorithm called MISL (monotonic minimizer for ISL). Based on the general majorization-minimization scheme, the MISL algorithm constructs the upper limit function of objective function twice so as to transform the complex ISL optimization problem into a simple optimization problem. Meanwhile, it is pointed out in [16] that by replacing the corresponding FFT matrix the MISL algorithm can also be applied to deal with the periodic ISL optimization problem. Subsequently, the authors improved the MISL algorithm (MWISL) in [17] to deal with the aperiodic weighted ISL optimization problem and provided several extended versions of the MISL algorithm in [18] for the design of CSS with both good aperiodic autocorrelation and cross-correlation properties. In addition to the two groups of typical algorithms mentioned above, some novel algorithms were proposed in the same period. For instance, [19] introduces a computational framework based on an iterative twisted approximation (ITROX) which can deal with various sequence design problems. However, ITROX needs to do eigenvalue decomposition at each iteration, which limits the application of such algorithms in the design of long sequences. The work in [20] proposed an iteration direct search optimization algorithm to design a constant modulus sequence set with low aperiodic WISL. Such an algorithm solves the optimization problem by optimizing each code of the sequence one by one directly, which greatly reduces the number of iterations and ensures the convergence property. Besides the correlation properties, some of the latest algorithms have also begun to take into account the other properties of the sequence. The work in [21] proposed a gradient-based algorithm named gradient-weighted correlation-SFW (Gra-WeCorr-SFW) to design unimodular sequences sets with both good aperiodic correlation and stopband properties. Furthermore, [22] proposed the design of unimodular sequences whose aperiodic autocorrelation and aperiodic complementary autocorrelation vanish for a given set of lags.

Most of the above literature is focused on the aperiodic correlation property of sequences with continuous phase, and the sequence with low aperiodic correlation is mainly used in the active sensing system with intrapulse coded modulation such as the pulse compression radar [23]. However, some active sensing systems such as proximity fuzes [24] and multiple modulation detectors [25] adopt interpulse coded modulation and therefore consider the periodic correlation property of modulation sequence rather than the aperiodic correlation property. In addition, for some practical engineering applications, it is not yet possible to achieve continuous phase modulation, mainly in regard to the hardware condition of detectors, especially miniaturization detectors, and the Doppler tolerance deterioration caused by continuous phase modulation. Consequently, for cases of specific applications, finite phase unimodular sequence with good periodic correlation property is demanded.

However, to our knowledge, less attention is paid to the research on the periodic correlation property of sequences with finite phase. In this paper, considering cases of specific applications, we propose a new algorithm called the Periodic correlation Weighted Cyclic Iteration Algorithm (PWCIA), which can be used to generate both continuous phase and finite phase unimodular sequences with low periodic weighted ISL. With the help of the transformation matrix, the PWCIA decomposes the -dimensional sequence optimization problem into one-dimensional sequence optimization problems and uses the iterative method to search the optimal solutions of the one-dimensional optimization problems directly. The convergence of the algorithm is analyzed theoretically, and the FFT operations are implemented to ensure the actual convergence speed of the algorithm. Numerical experiments indicate that compared with other algorithms the proposed PWCIA has the advantages of both periodic autocorrelation property and convergence property.

The rest of the paper is organized as follows. In Section 2, we formulate the optimization problem and review the existing algorithm. In Section 3, we deduce and present the PWCIA. In Section 4, we evaluate the performance of the PWCIA via a number of numerical experiments. Finally, in Section 5, we provide the conclusions.

*Notation*. Boldface uppercase letters denote matrices, and boldface lowercase letters denote column vectors. and denote the real and imaginary part. , and denote complex conjugate, transpose, and conjugate transpose, respectively. denotes the phase of a complex number. denotes the th element of the vector . and denote the Frobenius norm and the modulus.

#### 2. Problem Formulation and Existing Methods

In order to obtain good distance resolution and antijamming performance, it needs to improve the processing gain of active sensing systems. For active sensing systems which adopt interpulse coded modulation, such a goal can be achieved by designing unimodular sequences with low periodic autocorrelation sidelobes as modulation sequences. Hence, the issue of concern in this paper is actually to design the unimodular sequence with good periodic autocorrelation property.

##### 2.1. Problem Formulation

In the introduction, we let denote the unimodular sequence to be designed (i.e., ); then any code in the sequence can be expressed aswhere denotes the phase of corresponding code . We define as the phase vector of the unimodular sequence. For the unimodular sequence with arbitrary continuous phase, the value of can be any real number in the interval, that is, , while for the unimodular sequence with finite phase the value of is discrete and its range is , where denotes the number of finite phases. Thus, according to formula (1), the periodic autocorrelation of sequence can be expressed as

Combining (3) and (6), the weighted integrated sidelobe level (WISL) can be defined aswhere denotes the weight of autocorrelation coefficient corresponding to different lag time . By setting the value of , we can arbitrarily adjust the side lobe level at any lags to meet different actual requirements (such as suppressing the interference signal within a certain distance). In particular, ISL metric can be regarded as a special case of WISL metric, by simply taking .

The problem of interest in this paper is the following WISL minimization problem:

So far, the research work on unimodular sequences with a good aperiodic autocorrelation property has been carried out a lot, but the typical algorithm for designing unimodular sequences with a good periodic autocorrelation property is just PeCAN (Periodic correlation Cyclic Algorithm-New), which we will briefly review before we introduce the proposed algorithm in this paper.

##### 2.2. Existing Methods

The CAN (Cyclic Algorithm-New) and WeCAN (Weighted Cyclic Algorithm-New) were proposed in [11] for separately designing unimodular sequences with low ISL and low WISL. The above two algorithms concern the aperiodic correlation property of the sequence, but in this paper, according to the actual engineering requirements, we consider the periodic correlation property of the sequence. It can be seen from (1) and (2) that the periodic autocorrelation coefficient and the aperiodic autocorrelation coefficient greatly differ on the definition, and thus CAN and WeCAN are not suitable for the periodic correlation case.

In the follow-up research work, the authors proposed a PeCAN similar to CAN specifically to design a unimodular sequence with low ISL of periodic correlation [12]. Instead of minimizing the ISL metric directly, the authors proposed solving the following simpler problem:wheredenotes the finite Fourier transform of the sequence , and are auxiliary variables. The authors had proved that problem (9) is “almost equivalent” to the original problem (8) with .

Let denote the FFT matrix. The objective function of problem (9) can be rewritten in the following more compact form:where , . The PeCAN then minimizes (11) by alternating between and . For given , compute the FFT of , that is, , and the minimization of (11) with respect to yields

Similarly, for given , compute the IFFT of , let us say , and the minimization of (11) with respect to is given by

The PeCAN for minimizing of the periodic correlation ISL metric in (9) is summarized as shown in Algorithm 1.