1. Introduction
In the last decade, high-speed processors have enabled radar systems to take full advantages of received signals to obtain sensible performance improvements in their detection and anti-jamming capability [
1]. Meanwhile, modern multifunction phased array radar systems equipped with digital arbitrary waveform generators give the ability to generate high-accuracy, sophisticated, time-varying radar waveforms to achieve good anti-jamming performance and sensing objectives, such as improved detection performance with ideal ambiguity function (AF) shape [
2,
3]. As a consequence, in radar engineering, it has become crucial to optimize the transmitted waveform on a pulse-by-pulse basis to meet performance requirements, in terms of high detection possibility with good Doppler tolerance and low intercept probability with good orthogonal performance.
Linear frequency modulated (LFM) signal is well-known for its excellent characteristics, such as constant-envelope, easy generation, and good Doppler tolerance [
4,
5]. However, those anti-jamming methods based on LFM signal have poor performance, due to its simple and fixed form. Phase-coded (PC) signal (e.g., constant modulus sequence set) is another widely used waveform in several areas, for its high resolution and ranging accuracy. However, on account of its Doppler sensitivity, PC signal is usually restricted to detecting targets whose velocity ranges are a priori known. Taking into account the advantages of LFM and PC signals, we obtain the LFM-PC hybrid modulation signal. As a result, Doppler tolerance is extended, compared with a PC signal, and the complexity of the signal form is greatly increased to achieve better anti-jamming ability in radar systems.
It must be pointed out that the traditional LFM-PC signal is linear-frequency modulated within the pulse and coded by the sequence set between the pulses. In order to obtain better anti-jamming ability and wider Doppler tolerance, we investigate a new form of LFM-PC signal that is modulated by linear frequency and sequence set simultaneously within the pulse. Furthermore, considering the quantification of the digital to analog converter (DAC) of the radar waveform generator, we should optimize the constant modulus sequence set with a discrete phase constraint. In this paper, we pay attention to the unimodular sequence design of coherent waveform-agile radar on a predefined area of range-Doppler discrete ambiguity function (DAF) within the Doppler tolerance. In order to obtain better detection and orthogonal performance, the research was carried out in two scenarios; the first was a single pulse sequence design based on minimum INSL; the second was a multi pulse sequence set design based on minimizing the sum of NDAFSL and DCAF.
Over the last few decades, extensive research has investigated transmit waveform design in the radar research area. The optimal radar waveform is synthesized based on different performance objectives. Some studies have focused on shaping the AF of agile waveforms in sensing. In the works [
6,
7,
8,
9,
10,
11,
12,
13], the dynamic selection of waveforms for target tracking was considered, and the optimal waveform parameters were derived for tracking target motion using a linear/nonlinear observations model with detection in a clutter/clutter-free environment. In [
14], a kind of efficient gradient method was proposed for designing AFs for multistatic primary surveillance radar systems. Kerahroodi [
15,
16] applied the coordinates down (CD) to the optimization design of a PC signal. Agile waveform can also be designed from the viewpoint of orthogonality. Orthogonal waveforms are widely used in multiple input multiple output (MIMO) radar systems [
17] and anti-spoof jamming technology [
18]. In the literature [
19], the orthogonal waveform of MIMO radar is designed by constrained nonlinear programming based on the optimization criteria of minimizing the peak correlation sidelobe. Khan [
20] and Huang [
21] et al. considered the influence of Doppler frequency when designing orthogonal waveforms. Majumder [
22] proposed to use Walsh orthogonal code to realize intra pulse phase-coded modulation. Wu Yue [
23] used a LFM-PC signal to design an orthogonal waveform based on autocorrelation and cross-correlation AF. In [
24,
25], a CA-New (CAN) algorithm and a multi-CAN algorithm were proposed for peak sidelobe level (PSL) minimization and orthogonal waveform optimizing, respectively.
In the aforementioned works, the sequences designed by the methods mentioned above are arbitrary phase-coded signals. If we simply set the continuous phases to their closest discrete phase, the sequence performance will deteriorate drastically. To generate a sequence with a discrete phase set, many methods have been proposed and can be divided into two categories: the first is to use existing codes, such as Barker code or M code; the second is to use an intelligent search method to find the desired sequence. Genetic algorithm (GA) [
26,
27] is a search heuristic that is inspired by Charles Darwin’s theory of natural evolution. In [
28], a general and complicated problem of discrete-phase sequence design has already been addressed, and a GA approach is exploited for minimizing ISL in code sequence design. Discrete-phase sequence design with GA has also been successfully applied for orthogonal frequency division multiplexing (OFDM) radar [
29] and MIMO radars [
30]. In this paper, the performance of a GA approach is compared with the proposed algorithms.
The alternating direction method of multipliers (ADMM) [
31] is also a search algorithm that solves convex optimization problems by breaking them into smaller pieces, each of which are then easier to handle. It has recently found wide application in a number of areas. In [
32], Boyd showed that the ADMM is well suited to large-scale convex optimization problems. Liang successfully used the ADMM method to design a unimodular sequence with low autocorrelation sidelobes [
33] by solving a quadratic optimization problem with constraints. ADMM has also been successfully applied for radar in spectrally crowded environments [
34] and MIMO radars [
35,
36]. Considering the high dimension in LFM-PC sequence set design problem and the decomposability and superior convergence properties of ADMM, we designed SSD-ADMM and MSSD-ADMM methods to solve this problem.
In this article, we propose an ADMM algorithm to design a discrete phase-coded sequence set on range-Doppler plane within Doppler tolerance. The rest of this work is organized as follows.
Section 2 discusses the mathematical model and derives the DAF of LFM-PC signal.
Section 3 proposes two optimization methods for optimizing the shape of DAF based on ADMM, i.e., SSD-ADMM and MSSD-ADMM.
Section 4 demonstrates the proposed algorithms by extensive simulation experiments. Finally, concluding remarks and directions for future research are presented in
Section 5.
2. LFM-PC Signal Model
In the following paragraphs the boldface upper case letters denote matrices; boldface lower case letters denote column vectors, and italics denote scalars. C denotes the complex field, and Z denotes the integer field. The superscripts ${(.)}^{T}$, ${(.)}^{\ast}$, and ${(.)}^{H}$ denote transpose, complex conjugate, and conjugate transpose, respectively. $\otimes $ denotes the Kronecker product of matrices. $\mathrm{arg}(.)$ denotes the argument of a complex number. $\mathrm{argmax}(.)$ and $\mathrm{argmin}(.)$ stand for the argument of the maximum and minimum, respectively. $diag(.)$ denotes the diagonal matrix formed by the entries, and $\Vert .\Vert $ indicates the Frobenius norm, $<.,.>$ denotes inner product.
We assume a waveform agile radar transmits a coherent burst of
M slow-time pulses. The baseband transmit signal can be formulated as the following general form:
where
${u}_{i}\left(t\right)$ is the complex envelope of the
ith transmit pulse, and
${T}_{R}$ is the pulse repetition interval (PRI). We assume that
$T$ is the pulse duration and that
${T}_{R}$ is much larger than
$T$.
The complex envelope of the
ith LFM-PC transmit pulse
${u}_{i}\left(t\right)$ is expressed as [
4]:
where
$\mu $ is the slope of frequency modulation;
$N$ is the sub-pulse length;
${c}_{i}\left(n\right)$ is the modulating code sequence of
ith pulse, and
${p}_{n}(t)$ is an ideal rectangular shaping pulse of time length
${T}_{P}$, in which
${T}_{P}=\frac{T}{N}$ stands for the code element width.
Let,
${\mathit{c}}_{i}={[{c}_{i}\left(0\right),{c}_{i}\left(1\right),\cdots ,{c}_{i}\left(N-1\right)]}^{T}$ denote the modulating code sequence of the
ith transmit subpulse, and code element
${c}_{i}\left(n\right)$ is modulated by discrete phase and expressed as
where
K denotes the number of discrete phases.
According to the literature [
37], the AFs of the
ith subpulse is expressed as:
where
$\tau $ is the time delay, and
$f$ denotes the Doppler frequency shift. Substituting (2) into the definition of ambiguity function in (4), we can obtain:
In order to derivate the discrete ambiguity function (DAF) of the
ith transmit pulse, we suppose that there are
${N}_{S}$ sampling points within each code element and that the sampling period is
${T}_{S}=\frac{{T}_{P}}{{N}_{S}}$; the total number of sampling points in a pulse is
${N}^{\prime}=N\cdot {N}_{S}^{}$. The signal in (2) can be rewritten as:
where
${c}_{i}{}^{\prime}={[{c}_{i}{}^{\prime}\left(0\right),{c}_{i}{}^{\prime}\left(1\right),\cdots ,{c}_{i}{}^{\prime}\left({N}^{\prime}-1\right)]}^{T}={\mathit{c}}_{i}\otimes {1}_{Ns}$,
${1}_{Ns}={\left[1,1,\mathrm{1...1}\right]}^{T}$ represents the
ith code sequence after sampling;
${{p}^{\prime}}_{n}\left(t\right)$ indicates ideal rectangular pulse with the width
${T}_{S}$. Thus, AF in (5) can be rewritten as:
We set
$\begin{array}{cc}\tau =k{T}_{S},& -({N}^{\prime}-1)\le k\le {N}^{\prime}-1\end{array}$ to discretize the delay time axis, and we also set
$\begin{array}{cc}f=\mu p{T}_{S},& -({N}^{\prime}-1)\le p\le {N}^{\prime}-1\end{array}$ to discretize the Doppler shift axis and obtain:
We write
${\chi}_{i}\left(k{T}_{S},\mu p{T}_{S}\right)$ as
${\chi}_{i}\left(k,p\right)$ for short, and define:
Then the DAF of LFM-PC signal can be abbreviated as:
The above range-Doppler DAF defined on discrete range and Doppler plane is widely used in the designing of radar waveforms. Since the shape of DAF is related to the modulation sequence codes, ${\mathit{c}}_{i}{}^{\prime}$, then we can shape the DAF by optimizing ${\mathit{c}}_{i}{}^{\prime}$ within the Doppler tolerance on discrete range and Doppler plane to satisfy the requirement in sensing objectives.