Integrated Radar and Communications Waveform Design Based on Multi-Symbol OFDM

Abstract

Integrated radar and communications (IRC) technology has become very important for civil and military applications in recent years, and IRC waveform design is a major challenge for IRC development. In this paper, we focus on the IRC waveform design based on the multi-symbol orthogonal frequency division multiplexing (OFDM) technique. In view of the defects resulting from high peak-to-mean envelope power ratios (PMEPRs) and high range sidelobes in IRC systems, an intelligent and effective IRC waveform design method jointly optimized with the PMEPR and peak-to-sidelobe ratio (PSLR) is proposed. Firstly, a flexible tone reservation (TR)-based IRC waveform structure is applied in both temporal and frequency domains, i.e., multi-symbol OFDM waveform. Secondly, the optimization problem considering PMEPR and PSLR and extending them to the $L_p$-norm form is reformulated. Then, the conjugate gradient of the objective function is analytically derived and the conjugate gradient algorithm (CGA) is presented to simultaneously improve the PMEPR and PSLR. Finally, the simulation results show that the proposed algorithm can efficiently generate IRC waveforms with an excellent PMEPR, PSLR, radar signal-to-noise ratio (SNR), and bit error rate (BER) performance.

1. Introduction

As two typical applications in modern radio technology, separately equipped radar and wireless communication systems suffer from problems such as heavy system weight, large size, high power consumption, and electromagnetic interference [1]. With the development of technology, these problems have become increasingly prominent and need to be solved urgently. In current technological development, radio-frequency front-end architectures in radar and communication technologies are becoming more and more similar, in particular, digital signal processing is gradually replacing the traditional hardware components. Therefore, an emerging and feasible trend is to integrate radar and wireless communication systems into one, i.e., to provide both radar and communication functions on a single hardware platform with a single waveform, and the resulting systems are known as “RadCom” [2], “Joint Radar Communication (JRC)” [3], “Dual-Function Radar-Communication (DFRC)” [4], “Integrate Radar and Communications (IRC)” [5], etc. In recent years, the research on IRC has attracted extensive attention, and it has very broad application prospects in military and civilian fields, such as intelligent transportation networks that need to establish communication links with other vehicles and perform active environmental sensing to ensure safety and radar sensor networks that perform information fusion to achieve efficient detection [2,3,4,5,6,7].

The main challenge in IRC development is to find suitable waveforms that can be applied to radar sensing and communication transmission simultaneously. At present, the IRC waveform design is mainly developed in two directions, one is based on radar waveforms and the other is based on communication waveforms. Radar waveforms are designed to create waveforms with the optimum autocorrelation properties to guarantee a high dynamic range of measurements when correlation processing is applied in the receiver. Linear frequency modulated (LFM) pulse [2] is the most popular example of this requirement, and is often used to design IRC waveforms [8,9]. However, this approach has a very significant disadvantage, i.e., the communication rate is very low, usually much lower than that achieved by traditional communication systems with the same bandwidth. Therefore, this approach is not optimal from the communications perspective. In order to achieve good communication performance, especially high communication rates, orthogonal frequency-division multiplexing (OFDM) in multicarrier modulation has attracted extensive attention in current communications systems [10,11]. It is remarkable that OFDM is already one of the best candidates for IRC waveforms, owing to the advantages of flexible subcarrier modulation, easy implementation with fast Fourier transform (FFT), high spectral efficiency, and the availability of processing gain at the receiver [2]. Additionally, its ability to provide high performance in radar and communications is very promising. Therefore, this paper mainly focuses on OFDM-based IRC waveform design. However, some extremely challenging problems such as high peak-to-mean envelope power ratio (PMEPR) and undesirable autocorrelation properties or peak sidelobe level (PSLL) also exist in OFDM-based IRC waveform design.

Since the OFDM signal is generated by the superposition of all subcarriers, it may have a high PMEPR. The high PMEPR will greatly reduce power amplifier (PA) efficiency or cause nonlinear distortion, thereby reducing signal energy and affecting radar detection capability or degrading communication bit error rate (BER) performance. Aiming at solving this problem, researchers have proposed various PMEPR reduction techniques, including the clipping techniques [12,13], probabilistic techniques [14], code techniques [15,16,17], and tone reservation (TR) techniques [18,19,20,21,22,23]. Among these state-of-the-art techniques, the TR technique is an attractive and widely used PMEPR reduction technique since it does not require the transmission of additional side information [10]. The basic idea of TR is that some of all subcarriers are used for data transmission and the remaining subcarriers are reserved for PMEPR reduction, it was first introduced in [19] and the TR ratio (TRR) is defined as the ratio of the number of reserved subcarriers to the number of all subcarriers. Since TRR determines the data transmission efficiency, its value generally takes 5–15% for OFDM systems in order to achieve good communication performance [21]. Many radar systems require waveforms with PMEPR below 3 dB [24]. However, this may be difficult to achieve for OFDM systems, even with some conventional PMEPR reduction techniques [10]. Therefore, a new PMEPR reduction method for IRC waveforms is urgently needed. In principle, to obtain excellent radar detection performance, such as good range resolution, the bandwidth allocated to radar systems may be much larger than that of traditional communication systems [6]. In view of the different bandwidth requirements between radar and communications, it is feasible to design the IRC waveform structure based on TR, a method with ultra-flexible bandwidth allocation. In other words, some subcarriers allocated in the IRC waveform structure are used for data transmission, and all contiguous subcarriers (including the allocated ones) constitute the entire detection bandwidth for radar detection. It is worth mentioning that the large TRR in this waveform structure is also beneficial to further reduce the PMEPR [22,23]. Therefore, in this paper, the TR-based PMEPR reduction for IRC waveforms is investigated.

Range sidelobes are a very important performance metric for radar [4]. High-range sidelobes are undesirable. They can severely interfere with targets in nearby range cells, and may even obscure targets with a small radar cross section or that are far away. Subcarrier complex weighting and phase coding methods are two popular waveform design methods and can be used not only to reduce PMEPR, but also to suppress sidelobes [16,25,26,27]. However, a lower PMEPR and better sidelobe suppression cannot be achieved simultaneously. The PSLL performance of the IRC waveform is improved by using direct spread spectrum sequence in [27] and a method is investigated to keep the channel capacity of the IRC waveform from degrading while suppressing the radar range sidelobe in [28], however, neither of them consider the PMEPR problem. In the existing literature, in addition to the waveform design method, sidelobe suppression can also be achieved by signal processing at the radar receiver, for instance, the authors of [2] proposed a signal processing method based on the “modulation symbol” domain (MSD), which eliminated the range sidelobe fluctuation caused by the randomness of the transmitted signal, ref. [29] proposed a sidelobe suppression method with low signal-to-noise ratio (SNR) loss based on the MSD method, and the authors of [30] proposed a joint weighted optimization method to design the filters for IRC waveforms, where the range sidelobe modulation, SNR loss, and sidelobe level of the filters are jointly minimized.

All the above-cited studies attempt to either reduce PMEPR or reduce sidelobes. However, little attention has been paid to jointly solving these two problems, especially given the randomness of the communication data. In [31], the optimal ambiguity function for OFDM radar signal is studied, and a method to improve PMEPR and PSLL is proposed. Although there is a certain effect, the effect is not very significant. In [6], the authors propose a new waveform design algorithm based on the majorization–minimization method for reducing the PMEPR of OFDM-based IRC waveforms, where, although the integrated sidelobe level improvement is involved, the actual PSLL is still poor, since oversampling is generally required in practical digital signal processing. In [32], the Gray code technique is used to reduce PMEPR while selecting the optimal cyclic sequence to reduce the sidelobes of the IRC waveform. However, the data rate is low. Therefore, we urgently need to jointly minimize the PMEPR and sidelobe level and reduce the influence of random communication data on radar detection capability, while considering communication performance such as data rate and BER.

For a TR-based IRC waveform structure with a fixed radar bandwidth (meeting radar resolution requirements), a tradeoff exists between the communication rate and PMEPR. For example, when a low PMEPR is required, the TRR can be increased (i.e., the total communication bandwidth is reduced) to reserve more subcarriers for PMEPR reduction. Additionally, when a high communication rate is required, TRR can be reduced (i.e., more bandwidth is allocated for communication). In this paper, in order to reduce PMEPR and achieve a high data rate at the same time, a multi-symbol OFDM-based IRC waveform is studied, in which one cyclic prefix (CP) is used for multiple OFDM symbols [33]. It is found that both the PMEPR and the maximum autocorrelation sidelobe have infinite-norm form, which can be approximated by the associated $L_p$-norm, where p is a large positive integer [23,34]. Therefore, in our joint optimization algorithm of PMEPR and peak-to-sidelobe ratio (PSLR), we first establish the joint optimization objective function based on the $L_p$-norm. Then, an efficient algorithm based on the classic conjugate gradient algorithm (CGA) [35] is proposed. The performance of the designed IRC waveform, including the PMEPR, PSLR, radar SNR, Doppler, and BER are analyzed by simulation experiments.

The rest of this paper is organized as follows. The multi-symbol OFDM-based IRC signal model and the IRC system are illustrated in Section 2. In Section 3, the joint optimization objective function of PMEPR and PSLR is established. In Section 4, an efficient algorithm based on CGA is presented. The PMEPR and PSLR simulation results obtained by the proposed algorithm are analyzed in Section 5. Some discussions on the evaluation of radar detection and communication performance are given in Section 6. Finally, Section 7 summarizes the conclusions.

2. Multi-Symbol OFDM-Based IRC Systems and Signal Model

In this section, the multi-symbol OFDM-based IRC signal model is first introduced. Then, the IRC system framework is illustrated.

2.1. Multi-Symbol OFDM-Based IRC Signal Model

In this paper, it is assumed that there is an arbitrary number of (potentially non-contiguous) subcarriers for communication purposes lying within one OFDM symbol, which corresponds to the TR technique in conventional OFDM systems. Assume that the set of OFDM subcarriers is $\{-N/2,\dots ,N/2-1\}$, where N denotes the number of subcarriers. Let $\mathcal{R}=\{i_0,i_1,\dots ,i_{N_r-1}\}$ be the index set of reserved subcarriers over which the waveform is optimized, where $N_r$ denotes its cardinality. Additionally, the complementary set ${\mathcal{R}}^c$ of $\mathcal{R}$ is the index set of communication subcarrier with $N-N_r$ as its cardinality. TRR is defined as $R_N=N_r/N$$(1\le N_r<N)$. Therefore, each OFDM symbol computed by the inverse discrete Fourier transform (IDFT) of the N-point complex modulation sequence is

$$s\left(q\right)=\sum _{n=-N/2}^{N/2-1}\left(X_n+C_n\right)e^{j2\pi n\frac{q}{N}}$$

(1)

where $q=0,1,\dots ,N-1$. $X_n$ and $C_n$ represent the n-th modulated complex element in the communication symbol vector $\mathbf{X}\in {\mathbb{C}}^N$ and the reserved symbol vector $\mathbf{C}\in {\mathbb{C}}^N$, respectively. It should be noted that at any subcarrier position, there must be a zero value between $\mathbf{X}$ and $\mathbf{C}$, i.e., $X_n=0$ for $n\in \mathcal{R}$ and $C_n=0$ for $n\in {\mathcal{R}}^c$.

The IRC waveform with M consecutive OFDM symbols can be generated according to (1) as

$$s\left(q\right)=\sum _{m=0}^{M-1}\sum _{n=-N/2}^{N/2-1}\left(X_{n,m}+C_{n,m}\right){\varphi }_m(q,n)$$

(2)

where $q=0,1,\dots ,MN-1$, and ${\varphi }_m(q,n)=\left\{\begin{array}{cc}{e}^{j2\pi n\frac{q-mN}{N}}\hfill & mN\le q\le (m+1)N-1\hfill \\ 0\hfill & elsewhere\hfill \end{array}\right..$

From (2), we can see that the IRC waveform is the splicing of the M OFDM symbols in the time domain. Thus, denoting $\mathbf{s}={\left[{\mathbf{s}}_0^T,{\mathbf{s}}_1^T,\dots ,{\mathbf{s}}_{M-1}^T\right]}^T\in {\mathbb{C}}^{MN}$, where “${\left(\right)}^T$” denotes the matrix transpose, and ${\mathbf{s}}_m^T,m=0,1,\dots ,M-1$ is the m-th OFDM symbol. Let $\mathbf{X}={\left[{\mathbf{X}}_0^T,{\mathbf{X}}_1^T,\dots ,{\mathbf{X}}_{M-1}^T\right]}^T\in {\mathbb{C}}^{MN}$ and $\mathbf{C}={\left[{\mathbf{C}}_0^T,{\mathbf{C}}_1^T,\dots ,{\mathbf{C}}_{M-1}^T\right]}^T\in {\mathbb{C}}^{MN}$, where ${\mathbf{X}}_m^T$ and ${\mathbf{C}}_m^T$ are the communication symbol vector and the reserved symbol vector of the m-th OFDM symbol, respectively. Then, (2) is expressed in matrix form as

$$\mathbf{s}=\Phi \left(\mathbf{X}+\mathbf{C}\right)$$

(3)

where the IDFT matrix $\Phi ={\left[{\varphi }_0,{\varphi }_1,\dots ,{\varphi }_{M-1}\right]}_{MN\times MN}$.

At the transmitter, the IRC waveform with M consecutive OFDM symbols is generated before CP insertion, and the waveform structure in discrete time-frequency domain is shown in Figure 1. Here, one common CP is used for multiple OFDM symbols [33].

2.2. Multi-Symbol OFDM-Based IRC System

Based on the multi-symbol OFDM IRC waveform, the signal processing flow in IRC systems is shown in Figure 2. At the transmitter, the binary data are modulated by a quadratic-amplitude-modulated (QAM) modulator to generate $\mathbf{X}$. The communication waveform ${\mathbf{s}}_X=\Phi \mathbf{X}$ is then passed through the waveform optimization module proposed in Section 4 to generate the IRC waveform $\mathbf{s}$. The CP is then inserted into $\mathbf{s}$ for transmission via the PA. It is noted that the CP duration is related to the multipath delay, i.e., it needs to be longer than the maximum multipath delay, otherwise, inter-symbol interference (ISI) will be caused [33]. If the ISI is eliminated by CP, then at the receiver, the received vector in the discrete-time domain is obtained after CP removal.

For communication processing, the received vector in the discrete-time domain is processed symbol by symbol after serial–parallel conversion. Firstly, it is first converted to the discrete frequency domain by DFT calculation. Then, the data in the communication subcarriers is extracted, and the binary data is obtained after QAM demodulation, so far the information reception is completed.

For radar processing, assuming that the IRC signal is reflected by L objects, ignoring noise, the received echo is written as [32]

$$s_r\left(t\right)=\sum _{l=0}^{L-1}{A}_{l}s(t-{\tau }_l)$$

(4)

where $A_l$ is the attenuation factor, ${\tau }_l=2R_l/c$, $R_l$ is the range from the l-th object to radar, and c is speed of light.

The range profile is obtained by echo pulse compression [36]. The impulse response of the matched filter is expressed as

$$H\left(t\right)=s^{*}(-t)$$

(5)

where “${\left(\right)}^{*}$” is the complex conjugate operation. Thus, the matched filter output of the echo in (4) is given by

$$y\left(t\right)=s_r\left(t\right)\otimes H\left(t\right)={\int }_{-\infty }^{\infty }{s}_r\left(t\right)s^{*}(\tau -t)$$

(6)

where ⨂ is convolution computation.

Assuming the Fourier transform of $s\left(t\right)$ and $s_r\left(t\right)$ are $S\left(f\right)$ and $S_r\left(f\right)$, respectively, then the Fourier transform of $y\left(t\right)$ can be achieved as

$$Y\left(f\right)=S_r\left(f\right)S^{*}\left(f\right)=\sum _{l=0}^{L-1}{A}_l{\left|S\left(f\right)\right|}^2{e}^{-j2\pi f{\tau }_l}$$

(7)

Furthermore, the range profile obtained by an inverse Fourier transform of $Y\left(f\right)$ is

$$y\left(t\right)=\overline{R}\left(t\right)\otimes \sum _{l=0}^{L-1}{A}_l\delta (t-{\tau }_l)=\sum _{l=0}^{L-1}{A}_l\overline{R}(t-{\tau }_l)$$

(8)

where $\overline{R}\left(t\right)$ is the inverse Fourier transform of ${\left|S\left(f\right)\right|}^2$, and is also the autocorrelation function of $s\left(t\right)$. Therefore, the radar signal processing algorithm is implemented based on the Hadamard product between the discrete received vector ${\mathbf{S}}_r$ and the discrete reference vector ${\mathbf{S}}^{*}$. The range can be obtained from the IDFT results of this Hadamard product.

3. IRC Waveform Design Objective

In this section, we first review the PMEPR and PSLL problems of IRC waveforms. Then, the joint optimization objective function of PMEPR and PSLR in IRC waveform design is formulated.

3.1. PMEPR

The PMEPR is often used to measure the envelope fluctuations of waveforms. For a discrete-time IRC waveform $s\left(q\right)$, the PMEPR is defined as the ratio of the maximum power to the average power [10], i.e.,

$${\mathrm{PMEPR}}_{s\left(q\right)}=\frac{\underset{q\in \mathcal{T}}{max}{\left|s\left(q\right)\right|}^2}{E[{\left|s\left(q\right)\right|}^2]}$$

(9)

where $\mathcal{T}$ is the time interval over which the PMEPR is calculated, and $E[{\left|s\left(q\right)\right|}^2]=P_X+\frac{1}{M}{\displaystyle \sum _{m=0}^{M-1}}{\displaystyle \sum _{n\in \mathcal{R}}}{\left|C_{n,m}\right|}^2$ is the average power of the waveform $s\left(q\right)$, and $P_X=\frac{1}{M}{\displaystyle \sum _{m=0}^{M-1}}{\displaystyle \sum _{n\in {\mathcal{R}}^c}}{\left|X_{n,m}\right|}^2$.

At the baseband, OFDM symbols sampled at the Nyquist rate have no equivalent PMEPR to consecutive symbols. Therefore, oversampling at least four times is usually required [10]. In this paper, the J-times oversampling operation is implemented by inserting zeros, i.e., inserting $(J-1)N$ zeros to the middle of $X_{n,m}+C_{n,m}$, and the IDFT operation is applied to the extended vector $X_{n,m}^{\prime }+C_{n,m}^{\prime }=[X_{-N/2,m}+C_{-N/2,m},\dots ,X_{-1,m}+C_{-1,m},\underset{(J-1)Nzeros}{\underbrace{0,\dots ,0}},X_{0,m}+C_{0,m},\dots ,X_{N/2-1,m}+C_{N/2-1,m}]$. For this case, the interval $\mathcal{T}$ in (9) is $[0,MQ-1]$, where $Q=JN$.

A commonly used measure of the PMEPR is to consider the complementary cumulative distribution function (CCDF) defined as

$$\mathrm{CCDF}=\mathrm{Pr}\left(\mathrm{PMEPR}>\zeta \right)$$

(10)

where $\zeta$ is the PMEPR threshold.

The real and imaginary parts of the input communication symbol $X_n$ are assumed to be independent and identically distributed with zero mean and variance ${\sigma }^2$. Based on the central limit theorem, the real and imaginary parts of the initial waveform $s_X\left(q\right)$ for large N ($N\ge 64$ [37]) are independent and identically distributed Gaussian random processes with zero mean and variance ${\sigma }^2$. Then its amplitude $|s_X\left(q\right)|$ obeys the Rayleigh distribution, which means that the PMEPR of the initial waveform is large [11], as shown in Figure 3, where the CCDFs of the PMEPR for OFDM waveforms with different subcarrier numbers are shown. However, a PMEPR exceeding 11 dB is not practical and will negatively affect the performance of the IRC systems. Therefore, the PMEPR optimization of such a waveform is necessary and a topic of the paper.

3.2. PSLR

According to (8), the range sidelobes depend on the autocorrelation sidelobes property. The aperiodic autocorrelations of a waveform $s\left(q\right),q=0,1,\dots ,MQ-1$ are defined as

$$R\left(k\right)=\sum _{q=0}^{MQ-1-k}s\left(q\right)s^{*}(q+k)$$

(11)

It can be seen from (11) that the peak of the mainlobe $R\left(0\right)$ satisfies $R\left(0\right)={\displaystyle \sum _{q=0}^{MQ-1}}{\left|s\left(q\right)\right|}^2={MQE\left[\right|s\left(q\right)|}^2]$ which is the total energy of $s\left(q\right)$.

When low autocorrelation sidelobes are required, waveform design issues for good autocorrelation properties are usually considered. A commonly used metric to measure the autocorrelation properties of a waveform is PSLL, i.e.,

$$\mathrm{PSLL}=\underset{k\ne 0}{max}\left\{\right|R\left(k\right)\left|\right\}$$

(12)

The randomness of the communication data may cause the IRC waveforms to have different total energies. In this paper, PSLR is used to measure the PSLL of autocorrelation function [30]. The PSLR is defined as

$$\mathrm{PSLR}=\frac{\left|R\right(0\left)\right|}{\underset{k\in \Omega }{max}\left\{\right|R\left(k\right)\left|\right\}}$$

(13)

where $\Omega$ is the index set of peaks of all sidelobes.

From (13), we know that increasing PSLR can get a good autocorrelation property and improve radar detection performance, so in this paper, the method of improving PSLR is studied.

3.3. Joint Optimization of PMEPR and PSLR

High PMEPR and poor PSLR are two inherent problems for OFDM-based IRC waveforms, and both can be improved by waveform design. It can be seen from (9) and (13) that both PMEPR and PSLR are expressions of the waveform $s\left(q\right)$, and according to (3), we also know that if the communication symbol vector $\mathbf{X}$ is given, $s\left(q\right)$ is uniquely determined by the reserved symbol vector $\mathbf{C}$. Therefore, PMEPR and PSLR are the images of the reserved symbol vector $\mathbf{C}$ in two functions respectively, i.e.,

$$f_1:{\mathbb{C}}^{MN}\to \mathbb{R},\mathbf{C}\mapsto \mathrm{PMEPR}{f}_2:{\mathbb{C}}^{MN}\to \mathbb{R},\mathbf{C}\mapsto \mathrm{PSLR}$$

(14)

It is obvious from (14) that there is a new mapping O as

$$O:{\mathbb{C}}^{MN}\to \mathbb{R},\mathbf{C}\mapsto \alpha ·10{log}_{10}\mathrm{PMEPR}-10{log}_{10}{\mathrm{PSLR}}^2$$

(15)

where $\alpha (\alpha >0)$ is a constant weight to balance the relative magnitude between PMEPR and PSLR.

Based on (15), a new joint optimization objective function is proposed, namely,

$$\begin{array}{cc}\hfill O\left(\mathbf{C}\right)& =\alpha ·10{log}_{10}\mathrm{PMEPR}-10{log}_{10}{\mathrm{PSLR}}^2\hfill \\ \hfill & =\alpha ·10{log}_{10}\frac{\underset{0\le q\le MQ-1}{max}{\left|s\left(q\right)\right|}^2}{E[{\left|s\left(q\right)\right|}^2]}+10{log}_{10}\frac{\underset{k\in \Omega }{max}{\left\{\right|R\left(k\right)|}^2\}}{{\left(MQ\right)}^2{(E[{\left|s\left(q\right)\right|}^2])}^2}\hfill \end{array}$$

(16)

where PMEPR and PSLR are both calculated in dB.

Therefore, the problem of interest in this paper is the following $O\left(\mathbf{C}\right)$ minimization problem:

$$\underset{\mathbf{C}\in {\mathbb{C}}^{MN}}{min}O\left(\mathbf{C}\right)$$

(17)

which includes both the PMEPR minimization problem and the PSLR maximization problem, and the performance of PMEPR and PSLR can be traded off by $\alpha$.

It can be found from (16) that the objective function is a complex function of $s\left(q\right)$, and finding the exact solution of (17) is a tedious task. Thus, it is necessary to convert the objective function to an easily solvable form.

Since the instantaneous power ${\left|s\left(q\right)\right|}^2$ is a real-valued vector, the $L_p$-norm for it is defined as

$${\parallel \left|s\left(q\right)\right|}^2{\parallel }_p={\left(\sum _{q=0}^{MQ-1}{\left({\left|s\left(q\right)\right|}^2\right)}^p\right)}^{1/p},\forall p\ge 1,p\in \mathbb{R}$$

(18)

Note that the above expression is monotonically decreasing versus p, and has the commonly adopted peak power measure as its special case:

$${\parallel \left|s\left(q\right)\right|}^2{\parallel }_{\infty }=\underset{p\to \infty }{lim}{\parallel \left|s\left(q\right)\right|}^2{\parallel }_p=\underset{0\le q\le MQ-1}{max}{\left|s\left(q\right)\right|}^2$$

(19)

Therefore, the objective function $O\left(\mathbf{C}\right)$ can be transformed by an approximation method based on the $L_p$-norm approximating the infinite-norm, and then the new optimization problem is given by

$$\begin{array}{cc}\hfill \underset{\mathbf{C}\in {\mathbb{C}}^{MN}}{min}\tilde{O}\left(\mathbf{C}\right)=& \alpha ·10{log}_{10}\frac{{\left({\displaystyle \sum _{q=0}^{MQ-1}}{\left|s\left(q\right)\right|}^{2p}\right)}^{1/p}}{E[{\left|s\left(q\right)\right|}^2]}+10{log}_{10}\frac{{\left({\displaystyle \sum _{k\in \Omega }}{\left|R\left(k\right)\right|}^{2p}\right)}^{1/p}}{{\left(MQ\right)}^2{(E[{\left|s\left(q\right)\right|}^2])}^2}\hfill \\ \hfill =& \frac{10}{p}\left(\alpha {log}_{10}\left(\sum _{q=0}^{MQ-1}{\left|s\left(q\right)\right|}^{2p}\right)+{log}_{10}\left(\sum _{k\in \Omega }{\left|R\left(k\right)\right|}^{2p}\right)\right)-\hfill \\ \hfill & 10(\alpha +2){log}_{10}\left(E\right[\left|s\left(q\right){|}^2\right])-20{log}_{10}\left(MQ\right)\hfill \end{array}$$

(20)

where the peak power of $R\left(k\right)$ in (16) is also approximated by $L_p$-norm.

The constant terms 10 and $20{log}_{10}\left(MQ\right)$ in (20) can be ignored since they do not contribute to the optimization problem. Then, the optimization problem in (20) can be rewritten as

$$\begin{array}{cc}\hfill & \underset{\mathbf{C}\in {\mathbb{C}}^{MN}}{min}\tilde{O}\left(\mathbf{C}\right)\hfill \\ \hfill =& \frac{1}{p}\left[\alpha {log}_{10}\left(\sum _{q=0}^{MQ-1}{\left|s\left(q\right)\right|}^{2p}\right)+{log}_{10}\left(\sum _{k\in \Omega }{\left|R\left(k\right)\right|}^{2p}\right)\right]-(\alpha +2){log}_{10}\left(E\right[\left|s\left(q\right){|}^2\right])\hfill \\ \hfill =& \frac{1}{p}\left[\alpha {log}_{10}\left(\sum _{q=0}^{MQ-1}{\left|s\left(q\right)\right|}^{2p}\right)+{log}_{10}\left(\sum _{k\in \Omega }{\left|R\left(k\right)\right|}^{2p}\right)\right]-\hfill \\ \hfill & (\alpha +2){log}_{10}\left(P_X+\frac{1}{M}\sum _{m=0}^{M-1}\sum _{n\in \mathcal{R}}{\left|C_{n,m}\right|}^2\right)\hfill \end{array}$$

(21)

It is found from (21) that the objective function $\tilde{O}\left(\mathbf{C}\right)$ is an analytical expression about $\mathbf{C}$. This motivates us to consider a gradient analysis-based approach to this unconstrained optimization problem. Since the CGA [35,38] is a classical algorithm for solving such unconstrained optimization problems, in the next section, we will develop a conjugate gradient-based iterative algorithm to minimize $\tilde{O}\left(\mathbf{C}\right)$.

4. Proposed Algorithm and Its Complexity

In this section, we minimize Equation (21) via CGA. In the following, we first derive the conjugate gradient of the objective function in (21) with respect to $\mathbf{C}$, the step size, and the descent direction which are the keys of CGA. Then the algorithm is summarized and a complexity analysis is performed.

4.1. Conjugate Gradient Analysis

The conjugate gradient of $\mathbf{C}$ will be derived below, and for the convenience of derivation, let $f\left(\mathbf{C}\right)={\displaystyle \sum _{q=0}^{MQ-1}}{\left|s\left(q\right)\right|}^{2p}$, $g\left(\mathbf{C}\right)={\displaystyle \sum _{k\in \Omega }}{\left|R\left(k\right)\right|}^{2p}$, and $L\left(\mathbf{C}\right)=P_X+\frac{1}{M}{\displaystyle \sum _{m=0}^{M-1}}{\displaystyle \sum _{n\in \mathcal{R}}}{\left|C_{n,m}\right|}^2$. Then, the objective function $\tilde{O}\left(\mathbf{C}\right)$ is rewritten as

$$\underset{\mathbf{C}\in {\mathbb{C}}^{MN}}{min}\tilde{O}\left(\mathbf{C}\right)=\frac{1}{p}\left[\alpha {log}_{10}f\left(\mathbf{C}\right)+{log}_{10}g\left(\mathbf{C}\right)\right]-(\alpha +2){log}_{10}L\left(\mathbf{C}\right)$$

(22)

For ease of presentation, the conjugate gradient ${\nabla }_{{\mathbf{C}}^{*}}\tilde{O}\left(\mathbf{C}\right)$ is first computed mono-symbolically and then converted to a vector, i.e., we first derive the gradient of $\tilde{O}\left(\mathbf{C}\right)$ with respect to $C_{u,r}^{*}$$(u\in \mathcal{R},r=0,1,\dots ,M-1)$, which is the conjugation of $C_{u,r}$. Using matrix calculus and the chain rule, the gradient $\partial \tilde{O}\left(\mathbf{C}\right)/\partial C_{u,r}^{*}$ is given by

$$\frac{\partial \tilde{O}\left(\mathbf{C}\right)}{\partial C_{u,r}^{*}}=\frac{1}{ln10}\left[\frac{1}{p}\left(\frac{\alpha }{f\left(\mathbf{C}\right)}\frac{\partial f\left(\mathbf{C}\right)}{\partial C_{u,r}^{*}}+\frac{1}{g\left(\mathbf{C}\right)}\frac{\partial g\left(\mathbf{C}\right)}{\partial C_{u,r}^{*}}\right)-\frac{\alpha +2}{L\left(\mathbf{C}\right)}\frac{\partial L\left(\mathbf{C}\right)}{\partial C_{u,r}^{*}}\right]$$

(23)

In the following, $\partial f\left(\mathbf{C}\right)/\partial C_{u,r}^{*}$, $\partial g\left(\mathbf{C}\right)/\partial C_{u,r}^{*}$ and $\partial L\left(\mathbf{C}\right)/\partial C_{u,r}^{*}$ will be further specifically derived. Based on the expression of $f\left(\mathbf{C}\right)$, we have

$$\frac{\partial f\left(\mathbf{C}\right)}{\partial C_{u,r}^{*}}=\sum _{q=0}^{MQ-1}\frac{\partial {\left|s\left(q\right)\right|}^{2p}}{\partial C_{u,r}^{*}}=p\sum _{q=0}^{MQ-1}{\left|s\left(q\right)\right|}^{2(p-1)}\frac{\partial {\left|s\left(q\right)\right|}^2}{\partial C_{u,r}^{*}}$$

(24)

According to (2), we know that

$$\frac{\partial {\left|s\left(q\right)\right|}^2}{\partial C_{u,r}^{*}}=\frac{\partial s\left(q\right)s^{*}\left(q\right)}{\partial C_{u,r}^{*}}=s\left(q\right)e^{-j2\pi u\frac{q}{Q}},q\in [(r-1)Q,rQ-1]$$

(25)

By substituting (25) into (24), the gradient $\partial f\left(\mathbf{C}\right)/\partial C_{u,r}^{*}$ is given by

$$\frac{\partial f\left(\mathbf{C}\right)}{\partial C_{u,r}^{*}}=p\sum _{q=(r-1)Q}^{rQ-1}\tilde{s}\left(q\right)e^{-j2\pi u\frac{q}{Q}}=p{\left({\varphi }_{r-1}\right)}^H\tilde{\mathbf{s}}=p{\tilde{C}}_{u,r}$$

(26)

where “${\left(\right)}^H$” is the conjugate transpose, and $\tilde{\mathbf{s}}=\check{\mathbf{s}}\circ \mathbf{s}$, $\check{\mathbf{s}}={\left|\mathbf{s}\right|}^{2(p-1)}$, “∘” is the Hadamard product.

According to the expression of $g\left(\mathbf{C}\right)$, we have

$$\begin{array}{cc}\hfill \frac{\partial g\left(\mathbf{C}\right)}{\partial C_{u,r}^{*}}& =\sum _{k\in \Omega }\frac{\partial {\left|R\left(k\right)\right|}^{2p}}{\partial C_{u,r}^{*}}=p\sum _{k\in \Omega }{\left|R\left(k\right)\right|}^{2(p-1)}\frac{\partial {\left|R\left(k\right)\right|}^2}{\partial C_{u,r}^{*}}\hfill \\ \hfill & =p\sum _{k\in \Omega }{\left|R\left(k\right)\right|}^{2(p-1)}\left(\frac{\partial R\left(k\right)}{\partial C_{u,r}^{*}}{\left(R\left(k\right)\right)}^{*}+R\left(k\right)\frac{\partial {\left(R\left(k\right)\right)}^{*}}{\partial C_{u,r}^{*}}\right)\hfill \end{array}$$

(27)

According to (11), we know that

$$\frac{\partial R\left(k\right)}{\partial C_{u,r}^{*}}=\sum _{q=0}^{MQ-1-k}s\left(q\right)\frac{\partial {\left(s(q+k)\right)}^{*}}{\partial C_{u,r}^{*}}=\left\{\begin{array}{cc}{\displaystyle \sum _{q=0}^{rQ-1-k}}s\left(q\right)e^{-j2\pi u\frac{q+k}{Q}}\hfill & (r-1)Q\le k\le rQ-1\hfill \\ {\displaystyle \sum _{q=(r-1)Q-k}^{rQ-1-k}}s\left(q\right)e^{-j2\pi u\frac{q+k}{Q}}\hfill & 1\le k\le (r-1)Q-1\hfill \\ 0\hfill & otherwise\hfill \end{array}\right.$$

(28)

$$\frac{\partial {\left(R\left(k\right)\right)}^{*}}{\partial C_{u,r}^{*}}={\displaystyle \sum _{q=0}^{MQ-1-k}}s(q+k)\frac{\partial {\left(s\left(q\right)\right)}^{*}}{\partial C_{u,r}^{*}}=\left\{\begin{array}{cc}{\displaystyle \sum _{q=(r-1)Q}^{MQ-1-k}}s(q+k)e^{-j2\pi u\frac{q}{Q}}\hfill & k\le MQ-1-(r-1)Q\hfill \\ 0\hfill & otherwise\hfill \end{array}\right.$$

(29)

By substituting (28) and (29) into (27), the gradient $\partial g\left(\mathbf{C}\right)/\partial C_{u,r}^{*}$ is given by

$$\frac{\partial g\left(\mathbf{C}\right)}{\partial C_{u,r}^{*}}=p{\widehat{C}}_{u,r}$$

(30)

where

$$\begin{array}{cc}\hfill {\widehat{C}}_{u,r}& =\sum _{k\in \cap \left\{\Omega ,[1,MQ-1-(r-1\left)Q\right]\right\}}\left[\tilde{R}\left(k\right)\sum _{q=(r-1)Q}^{MQ-1-k}s(q+k)e^{-j2\pi u\frac{q}{Q}}\right]\hfill \\ \hfill & +\sum _{k\in \cap \{\Omega ,[(r-1)Q,rQ-1\left]\right\}}\left[{(\tilde{R}\left(k\right))}^{*}\sum _{q=0}^{rQ-1-k}s\left(q\right)e^{-j2\pi u\frac{q+k}{Q}}\right]\hfill \\ \hfill & +\sum _{k\in \cap \{\Omega ,[1,(r-1)Q-1\left]\right\}}\left[{(\tilde{R}\left(k\right))}^{*}\sum _{q=(r-1)Q-k}^{rQ-1-k}s\left(q\right)e^{-j2\pi u\frac{q+k}{Q}}\right]\hfill \end{array}$$

(31)

and $\tilde{\mathbf{R}}=\check{\mathbf{R}}\circ \mathbf{R}$, $\check{\mathbf{R}}={\left|\mathbf{R}\right|}^{2(p-1)}$.

According to the expression of $L\left(\mathbf{C}\right)$, we have

$$\frac{\partial L\left(\mathbf{C}\right)}{\partial C_{u,r}^{*}}=\frac{\partial }{\partial C_{u,r}^{*}}\left[\frac{1}{M}\sum _{m=0}^{M-1}\sum _{n\in \mathcal{R}}{\left|C_{n,m}\right|}^2\right]=\frac{1}{M}{C}_{u,r}$$

(32)

By substituting (26), (30), and (32) into (23), the gradient $\partial \tilde{O}\left(\mathbf{C}\right)/\partial C_{u,r}^{*}$ is given by

$$\frac{\partial \tilde{O}\left(\mathbf{C}\right)}{\partial C_{u,r}^{*}}=\frac{1}{ln10}\left(\alpha \frac{{\tilde{C}}_{u,r}}{f\left(\mathbf{C}\right)}+\frac{{\widehat{C}}_{u,r}}{g\left(\mathbf{C}\right)}-(\alpha +2)\frac{C_{u,r}}{M·L\left(\mathbf{C}\right)}\right)$$

(33)

Further, the gradient is expressed in vector form. Let $\tilde{\mathbf{C}}={\left[{\tilde{\mathbf{C}}}_0^T,{\tilde{\mathbf{C}}}_1^T,\dots ,{\tilde{\mathbf{C}}}_{M-1}^T\right]}^T$ and $\widehat{\mathbf{C}}={\left[{\widehat{\mathbf{C}}}_0^T,{\widehat{\mathbf{C}}}_1^T,\dots ,{\widehat{\mathbf{C}}}_{M-1}^T\right]}^T$. Then, according to (33), the gradient vector denoted by $\mathbf{G}$ can be written as

$$\mathbf{G}={\nabla }_{{\mathbf{C}}^{*}}\tilde{O}\left(\mathbf{C}\right)=\frac{1}{ln10}\left(\alpha \frac{\tilde{\mathbf{C}}}{f\left(\mathbf{C}\right)}+\frac{\widehat{\mathbf{C}}}{g\left(\mathbf{C}\right)}-(\alpha +2)\frac{\mathbf{C}}{M·L\left(\mathbf{C}\right)}\right)$$

(34)

4.2. Update Rules and Step Size

It can be seen from (34) that it is an analytical expression for $\mathbf{C}$, which provides strong support for CGA to be used to solve the joint optimization problem we established.

In the iterative process, the l-th iteration point is assumed to be ${\mathbf{C}}^l$, then it is updated by the new iteration as

$${\mathbf{C}}^{l+1}={\mathbf{C}}^l+{\mu }^l{\mathbf{d}}^l$$

(35)

where ${\mu }^l$ represents the step size and ${\mathbf{d}}^l$ represents the descent direction.

The step size is recalculated at each iteration and plays a crucial role in the convergence rate of our algorithm. The classical line search method needs to compute the objective function many times to obtain the step size, which makes it have a high time cost [38]. In this paper, we will directly solve the step size by establishing a minimization problem about the step size, i.e., minimize the objective function $\tilde{O}\left(\mathbf{C}\right)$ at the $(l+1)$-th iteration point to obtain the optimal step size, as follows:

$$\begin{array}{cc}\hfill \underset{{\mu }^l}{min}W\left({\mu }^l\right)& =\tilde{O}\left({\mathbf{C}}^{l+1}\right)\hfill \\ \hfill & =\frac{1}{p}[\alpha {log}_{10}f\left({\mathbf{C}}^{l+1}\right)+{log}_{10}g\left({\mathbf{C}}^{l+1}\right)]-(\alpha +2){log}_{10}L\left({\mathbf{C}}^{l+1}\right)\hfill \end{array}$$

(36)

Similar to (23), the first derivative $W^{\prime }\left({\mu }^l\right)$ is given by

$$W^{\prime }\left({\mu }^l\right)=\frac{1}{ln10}\left[\frac{1}{p}\left(\alpha \frac{f^{\prime }\left({\mathbf{C}}^{l+1}\right)}{f\left({\mathbf{C}}^{l+1}\right)}+\frac{g^{\prime }\left({\mathbf{C}}^{l+1}\right)}{g\left({\mathbf{C}}^{l+1}\right)}\right)-(\alpha +2)\frac{L^{\prime }\left({\mathbf{C}}^{l+1}\right)}{L\left({\mathbf{C}}^{l+1}\right)}\right]$$

(37)

According to the expressions of $f\left(\mathbf{C}\right)$, $g\left(\mathbf{C}\right)$, and $L\left(\mathbf{C}\right)$, the derivatives of $f\left({\mathbf{C}}^{l+1}\right)$, $g\left({\mathbf{C}}^{l+1}\right)$ and $L\left({\mathbf{C}}^{l+1}\right)$ with respect to ${\mu }^l$ are obtained as follows, respectively,

$$f^{\prime }\left({\mathbf{C}}^{l+1}\right)=p\sum _{q=0}^{MQ-1}{\left|s^{l+1}\left(q\right)\right|}^{2(p-1)}\frac{d|s^{l+1}{\left(q\right)|}^2}{d{\mu }^l}=2p\sum _{q=0}^{MQ-1}{\check{s}}^{l+1}\left(q\right)s_{re}\left(q\right)$$

(38)

where $s_{re}\left(q\right)=\mathrm{Re}\left\{s^{l+1}\left(q\right){\left(D^l\left(q\right)\right)}^{*}\right\}$ and $D^l\left(q\right)={\Phi }_q{\mathbf{d}}^l$, ${\Phi }_q$ represents the q-th row of $\Phi$.

$$g^{\prime }\left({\mathbf{C}}^{l+1}\right)=p\sum _{k\in \Omega }{\left|R^{l+1}\left(k\right)\right|}^{2(p-1)}\frac{d|R^{l+1}\left(k\right){|}^2}{d{\mu }^l}=2p\sum _{k\in \Omega }{\check{R}}^{l+1}\left(k\right)R_{re}\left(k\right)$$

(39)

where $R_{re}\left(k\right)=\mathrm{Re}\left\{{\left(R^{l+1}\left(k\right)\right)}^{*}\left(R_{s,D^l}\left(k\right)+R_{D^l,s}\left(k\right)\right)\right\}$, $R_{s,D^l}\left(k\right)={\displaystyle \sum _{q=0}^{MQ-1-k}}{s}^{l+1}\left(q\right){\left(D^l(q+k)\right)}^{*}$ is the cross-correlation function of $s^{l+1}\left(q\right)$ and $D^l\left(q\right)$, and $R_{D^l,s}\left(k\right)={\displaystyle \sum _{q=0}^{MQ-1-k}}{\left(s^{l+1}(q+k)\right)}^{*}{D}^l\left(q\right)$ is the cross-correlation function of $D^l\left(q\right)$ and $s^{l+1}\left(q\right)$.    

$$\begin{array}{cc}\hfill L^{\prime }\left({\mathbf{C}}^{l+1}\right)& =\frac{1}{M}\sum _{m=0}^{M-1}\sum _{n\in \mathcal{R}}{\left(|C_{n,m}^{l+1}{|}^2\right)}^{\prime }=\frac{1}{M}\sum _{m=0}^{M-1}\sum _{n\in \mathcal{R}}\left({C_{n,m}^{l+1}}^{\prime }{C_{n,m}^{l+1}}^{*}+C_{n,m}^{l+1}{\left({C_{n,m}^{l+1}}^{*}\right)}^{\prime }\right)\hfill \\ \hfill & =\frac{2}{M}{A}_d\hfill \end{array}$$

(40)

where $A_d={\displaystyle \sum _{m=0}^{M-1}}{\displaystyle \sum _{n\in \mathcal{R}}}\mathrm{Re}\left\{C_{n,m}^{l+1}{d_{n,m}^l}^{*}\right\}$.

By substituting (38)–(40) into (37), we have

$$W^{\prime }\left({\mu }^l\right)=\frac{2}{ln10}\left(\alpha \frac{{\displaystyle \sum _{q=0}^{MQ-1}}{\check{s}}^{l+1}\left(q\right)s_{re}\left(q\right)}{f\left({\mathbf{C}}^{l+1}\right)}+\frac{{\displaystyle \sum _{k\in \Omega }}{\check{R}}^{l+1}\left(k\right)R_{re}\left(k\right)}{g\left({\mathbf{C}}^{l+1}\right)}-\frac{\alpha +2}{M}\frac{A_d}{L\left({\mathbf{C}}^{l+1}\right)}\right)$$

(41)

In (41), $W^{\prime }\left({\mu }^l\right)$ is a cumbersome high-order function with respect to ${\mu }^l$, which makes it a difficult task to directly compute the step size. Thus, the Newton’s downhill method [23] is used to solve the approximate solution of the step size.

On the basis of (37), the second derivative $W^{\left(2\right)}\left({\mu }^l\right)$ is expressed as

$$\begin{array}{cc}\hfill W^{\left(2\right)}\left({\mu }^l\right)=& \frac{1}{ln10}\left[\frac{1}{p}\left(\alpha \left(\frac{f^{\left(2\right)}\left({\mathbf{C}}^{l+1}\right)}{f\left({\mathbf{C}}^{l+1}\right)}-{\left|\frac{f^{\prime }\left({\mathbf{C}}^{l+1}\right)}{f\left({\mathbf{C}}^{l+1}\right)}\right|}^2\right)+\frac{g^{\left(2\right)}\left({\mathbf{C}}^{l+1}\right)}{g\left({\mathbf{C}}^{l+1}\right)}-{\left|\frac{g^{\prime }\left({\mathbf{C}}^{l+1}\right)}{g\left({\mathbf{C}}^{l+1}\right)}\right|}^2\right)-\right.\hfill \\ \hfill & \left.(\alpha +2)\left(\frac{L^{\left(2\right)}\left({\mathbf{C}}^{l+1}\right)}{L\left({\mathbf{C}}^{l+1}\right)}-{\left|\frac{L^{\prime }\left({\mathbf{C}}^{l+1}\right)}{L\left({\mathbf{C}}^{l+1}\right)}\right|}^2\right)\right]\hfill \end{array}$$

(42)

The second derivative of $f\left({\mathbf{C}}^{l+1}\right)$, $g\left({\mathbf{C}}^{l+1}\right)$ and $L\left({\mathbf{C}}^{l+1}\right))$ with respect to ${\mu }^l$ can be obtained from (38)–(40), respectively, which is

$$f^{\left(2\right)}\left({\mathbf{C}}^{l+1}\right)=2p\sum _{q=0}^{MQ-1}\left(2(p-1)|s^{l+1}{\left(q\right)|}^{2(p-2)}|s_{re}{\left(q\right)|}^2+{\check{s}}^{l+1}\left(q\right){\left|D^l\left(q\right)\right|}^2\right)$$

(43)

$$\begin{array}{cc}\hfill g^{\left(2\right)}\left({\mathbf{C}}^{l+1}\right)=& 2p\sum _{k\in \Omega }\left(2(p-1)|R^{l+1}{\left(k\right)|}^{2(p-2)}{\left|R_{re}\left(k\right)\right|}^2+\right.\hfill \\ \hfill & \left.{\check{R}}^{l+1}\left(k\right)\left(2\mathrm{Re}\left\{{\left(R^{l+1}\left(k\right)\right)}^{*}{R}_{D^l}\left(k\right)\right\}+{|R_{s,D^l}\left(k\right)+R_{D^l,s}\left(k\right)|}^2\right)\right)\hfill \end{array}$$

(44)

where $R_{D^l}\left(k\right)={\displaystyle \sum _{q=0}^{MQ-1-k}}\left(D^l\left(q\right){\left(D^l(q+k)\right)}^{*}\right)$ is the autocorrelation function of $D^l\left(q\right)$.

$$L^{\left(2\right)}\left({\mathbf{C}}^{l+1}\right)=\frac{2}{M}{A}_d^{\prime }=\frac{2}{M}{d}_{cons}$$

(45)

where $d_{cons}={\displaystyle \sum _{m=0}^{M-1}}{\displaystyle \sum _{n\in \mathcal{R}}}{\left|d_{n,m}^l\right|}^2$.

By substituting (38)–(40) and (43)–(45) into (42), we have

$$\begin{array}{cc}\hfill W^{\left(2\right)}\left({\mu }^l\right)=& \frac{1}{ln10}\left[\frac{1}{p}\left(\alpha \left(\widehat{f}\left({\mathbf{C}}^{l+1}\right)-{\left|\tilde{f}\left({\mathbf{C}}^{l+1}\right)\right|}^2\right)+\widehat{g}\left({\mathbf{C}}^{l+1}\right)-{\left|\tilde{g}\left({\mathbf{C}}^{l+1}\right)\right|}^2\right)-\right.\hfill \\ \hfill & \left.(\alpha +2)\left(\widehat{L}\left({\mathbf{C}}^{l+1}\right)-{\left|\tilde{L}\left({\mathbf{C}}^{l+1}\right)\right|}^2\right)\right]\hfill \end{array}$$

(46)

where $\tilde{f}\left({\mathbf{C}}^{l+1}\right)=\frac{f^{\prime }\left({\mathbf{C}}^{l+1}\right)}{f\left({\mathbf{C}}^{l+1}\right)}$, $\tilde{g}\left({\mathbf{C}}^{l+1}\right)=\frac{g^{\prime }\left({\mathbf{C}}^{l+1}\right)}{g\left({\mathbf{C}}^{l+1}\right)}$, $\tilde{L}\left({\mathbf{C}}^{l+1}\right)=\frac{L^{\prime }\left({\mathbf{C}}^{l+1}\right)}{L\left({\mathbf{C}}^{l+1}\right)}$, $\widehat{f}\left({\mathbf{C}}^{l+1}\right)=\frac{f^{\left(2\right)}\left({\mathbf{C}}^{l+1}\right)}{f\left({\mathbf{C}}^{l+1}\right)}$, $\widehat{g}\left({\mathbf{C}}^{l+1}\right)=\frac{g^{\left(2\right)}\left({\mathbf{C}}^{l+1}\right)}{g\left({\mathbf{C}}^{l+1}\right)}$, and $\widehat{L}\left({\mathbf{C}}^{l+1}\right)=\frac{L^{\left(2\right)}\left({\mathbf{C}}^{l+1}\right)}{L\left({\mathbf{C}}^{l+1}\right)}$.

Based on the expressions of $W^{\prime }\left({\mu }^l\right)$ and $W^{\left(2\right)}\left({\mu }^l\right)$, the search direction of ${\mu }^l$ can be easily determined, i.e., $-W^{\prime }\left({\mu }^l\right)⁄W^{\left(2\right)}\left({\mu }^l\right)$. Then ${\mu }^l$ after $i+1$ updates is represented as

$${\mu }_{i+1}^l={\mu }_i^l-\lambda \frac{W^{\prime }\left({\mu }_i^l\right)}{W^{\left(2\right)}\left({\mu }_i^l\right)}$$

(47)

where $\lambda \in (0,1]$ represents the downhill factor. Note that $\lambda$ may cause ${\mu }^l$ to jump out of the local optimal range due to its large value in the iterative process, so $\lambda$ needs to be adjusted in time to ensure that $|W^{\prime }\left({\mu }^l\right)|$ decreases monotonically.

4.3. Descent Direction

A modified descent direction, i.e., the classic Polak–Ribière–Polyak (PRP) descent direction [39], is adopted in this paper to ensure the efficient execution of our algorithm, which is expressed as

$${\mathbf{d}}^l=\left\{\begin{array}{cc}-{\mathbf{G}}^l\hfill & l=0\hfill \\ -{\mathbf{G}}^l+\frac{{\left({\mathbf{G}}^l-{\mathbf{G}}^{l-1}\right)}^T{\mathbf{G}}^l}{\parallel {\mathbf{G}}^{l-1}{\parallel }^2}{\mathbf{d}}^{l-1}\hfill & l\ge 1\hfill \end{array}\right.$$

(48)

4.4. Algorithm Summary and Complexity Analysis

By integrating the above derivation, a conjugate gradient-based iterative algorithm for the joint optimization of the PMEPR and PSLR (CGIA-JOPP) of multi-symbol IRC waveforms is summarized in Algorithm 1. In Algorithm 1, the first is the initialization process. We set the initial reserved symbol vector to $\mathbf{0}$, calculate the initial gradient vector ${\mathbf{G}}^0$ according to (34), and obtain the initial descent direction ${\mathbf{d}}^0$ according to (48). The second is the iterative process, and the last is the convergence criterion. In the iterative process of Algorithm 1, appropriate step size is what CGIA-JOPP pursues. In our algorithm, the optimal step size is obtained by solving the minimization problem (36), and its calculation process is summarized in Algorithm 2. Newton’s downhill method is applied here. Specifically, the downhill factor is determined, the search direction is calculated according to (41) and (46), and then based on the update rule (47), the optimal step size required for each iteration in Algorithm 1 is obtained. It should be noted that in the iterative process of Algorithm 2, the search for the downhill factor $\lambda$ will stop when $\left|W^{\prime }\left({\mu }_{i+1}^l\right)\right|<\left|W^{\prime }\left({\mu }_i^l\right)\right|$ and $W\left({\mu }_{i+1}^l\right)<W\left({\mu }_i^l\right)$ are satisfied to prevent the step size from falling into other local optimal ranges due to the excessively large value of $\lambda$. After obtaining the optimal step size, the new reserved symbol vector is calculated according to the update rule (35), and the new IRC waveform is obtained, so the new gradient vector and descent direction are calculated again. In this way, the iterations are repeated until the convergence condition is satisfied.

Complexity Analysis: The number of real multiplications is used to measure the computational complexity of our algorithm. One complex multiplication counts as four real multiplications. A Q-point IFFT/FFT pair requires $\mathcal{O}\left(Q{log}_{2}Q\right)$ complexity. Since the initialization step in Algorithm 1 occurs only once, its computational complexity is ignored. In Algorithm 1, the rest of the steps except Step 6 are only composed of the Hadamard product of the vector, the $MQ$-point IFFT/FFT pair, and the Q-point IFFT/FFT pair, then the complexity is $\mathcal{O}\left(MQ{log}_2\left(MQ\right)\right)$. In Step 6, the complexity is mainly brought by the calculation of $\widehat{\mathbf{C}}$ in (34). It can be seen from (31) that each part of the three-part summation term can be regarded as a combination of FFT and matrix Hadamard product. Assuming that the maximum length of the three-part summation index sets for k is $\overline{K}$, then the complexity of Step 6 is $\mathcal{O}\left(\overline{K}Q{log}_{2}Q\right)$. Note that the number of reserved subcarriers is also related to the complexity of Algorithm 1, but is ignored because it is too small relative to the $MQ$-point IFFT/FFT pair.

Therefore, for each iteration, the computational complexity of the proposed algorithm is mainly determined by the $MQ$-point IFFT/FFT pair and the complexity from (34), which shows that the proposed algorithm can be effectively implemented by IFFT/FFT pair and Hadamard product.   

Algorithm 1: CGIA-JOPP

1: Consider p, N, Q, $R_N$, M. Store transmit data ${\mathbf{X}}_m$: $X_{n,m}$, $n\in {\mathcal{R}}^c,m=0,1,\dots ,M-1$. Initialization    conditions: $l=0$, ${\mathbf{C}}_m^0$: $C_{n,m}^0=0(n\in \mathcal{R})$, ${\mathbf{G}}^0={\nabla }_{{\mathbf{C}}^{*}}\tilde{O}\left({\mathbf{C}}^0\right)$ and ${\mathbf{d}}^0=-{\mathbf{G}}^0$. 2: $\mathbf{Repeat}$ 3: Compute step size ${\mu }^l$ according to Algorithm 2. 4: Update the reserved symbol vector: ${\mathbf{C}}^{l+1}={\mathbf{C}}^l+{\mu }^l{\mathbf{d}}^l$. 5: ${\mathbf{s}}^{l+1}=\Phi (\mathbf{X}+{\mathbf{C}}^{l+1})$. 6: Compute the conjugate gradient ${\mathbf{G}}^{l+1}$ according to (34). 7: Compute the descent direction ${\mathbf{d}}^{l+1}$ according to (48). 8: $l=l+1$. 9: $\mathbf{Until}$ a stopping condition is met, e.g., the sum of changes in the objective function value over 10 consecutive iterations does not exceed ${10}^{-4}$, then, transmit $\mathbf{s}$.

Algorithm 2: Compute step size

1: Initialization conditions: $i=0$, ${\mathbf{C}}^l$, ${\mathbf{d}}^l$, ${\mu }_0^l=0$, $W^{\prime }\left({\mu }_0^l\right)$,$W^{\left(2\right)}\left({\mu }_0^l\right)$. 2: $\mathbf{Repeat}$ 3: $\lambda =1$. 4:    Update the step size: ${\mu }_{i+1}^l={\mu }_i^l-\lambda W^{\prime }\left({\mu }_i^l\right)/W^{\left(2\right)}\left({\mu }_i^l\right)$. 5:    ${\mathbf{C}}^{l+1}={\mathbf{C}}^l+{\mu }_{i+1}^l{\mathbf{d}}^l$ 6:    ${\mathbf{s}}^{l+1}=\Phi (\mathbf{X}+{\mathbf{C}}^{l+1})$. 7:    Compute $W\left({\mu }_{i+1}^l\right)=\tilde{O}\left({\mathbf{C}}^{l+1}\right)$ and $W^{\prime }\left({\mu }_{i+1}^l\right)$ according to (22) and (41), respectively. 8:    if $\left|W^{\prime }\left({\mu }_{i+1}^l\right)\right|<\left|W^{\prime }\left({\mu }_i^l\right)\right|$ and $W\left({\mu }_{i+1}^l\right)<W\left({\mu }_i^l\right)$ is not met, let $\lambda =1/2\lambda$, and repeat Steps 4–7. 9: Compute $W^{\left(2\right)}\left({\mu }_{i+1}^l\right)$ according to (46). 10: $i=i+1$. 11: $\mathbf{Until}$ a stopping condition is met, e.g., $\left|W^{\prime }\left({\mu }_i^l\right)\right|<\left|W^{\prime }\left({\mu }_0^l\right)\right|/{10}^3$.

5. Simulation Results

In this section, we simulate and verify the effectiveness of the proposed algorithm. In our simulations, $\mathcal{R}$ is randomly selected and $R_N$ = 50% is set, i.e., half of the entire bandwidth used for radar detection is used to transmit communication data. Unless otherwise specified, the randomly generated input data is modulated by 4 QAM. The parameter settings are shown in

Table 1. Simulation parameters.

Parameter	Value
Baseband bandwidth	128 MHz
Sampling frequency	512 MHz
Number of subcarriers	512
Modulation	4 QAM/16 QAM
Single OFDM symbol duration	4 μs
Cyclic prefix duration	10 μs
Number of OFDM symbols	3
Pulse width	22 μs
Tone reservation ratio	$50\%$
p in $L_p$-norm	10
$\alpha$	$[0.2,100]$

.

To examine the effectiveness of the proposed algorithm from both the obtained PMEPR and PSLR performances, ${10}^5$ initial waveforms were randomly generated. In the following, we first study the effectiveness of the proposed algorithm with a fixed weight $\alpha =5$, and then analyze the impact of weight $\alpha$ on the performance of PMEPR and PSLR, since $\alpha$ can change the proportion of PMEPR and PSLR in the objective function.

5.1. PMEPR and PSLR Performance of IRC Waveform

For a fixed weight $\alpha =5$, an IRC waveform is obtained by joint optimization of the PMEPR and PSLR, and then the PMEPR, objective function of (21) and PSLR are calculated with respect to the number of iterations and are shown in Figure 4. It can be observed from Figure 4a that the objective function of (21) shown on the right axis is monotonically decreasing, and the PMEPR on the left axis and the PSLR in Figure 4b generally show decreasing and increasing trends, respectively. Another observation is that the obtained IRC waveform has the desired PMEPR and PSLR performances, reaching 2.3 dB and 29.6 dB, respectively.

Figure 5 and Figure 6 show the CCDF curves of the PMEPR and PSLR for the IRC waveforms with $\alpha =5$ processed by our proposed algorithm, and give the results obtained by [23] (i.e., only the PMEPR is optimized in (20)) and [31] for comparison. The PMEPR and PSLR performance are evaluated at $\mathrm{CCDF}={10}^{-4}$. It is found from Figure 5 and Figure 6 that the proposed CGIA-JOPP algorithm achieves 10 dB PMEPR reduction, which is 1.1 dB worse than only optimizing PMEPR, but improves the PSLR by 10.4 dB, which is a significant improvement effect. Compared with [31], the PMEPR reduction and PSLR improvement obtained by our proposed algorithm reach 2.6 dB and 14.8 dB, respectively. This shows that the proposed algorithm has obvious advantages in both the PMEPR and PSLR, which fully demonstrates its effectiveness. In addition, the input data modulated by 16 QAM are also used to generate the IRC waveform, and the obtained PMEPR and PSLR performances are also analyzed in Figure 5 and Figure 6; the results show that the proposed algorithm is also applicable to higher order modulations.

5.2. Tradeoff between PMEPR and PSLR

In practical applications, low radar SNR loss and low range sidelobe levels are desirable for IRC systems. By selecting a specific $\alpha$ under different backgrounds, one can obtain the corresponding waveform by using the proposed algorithm. Therefore, the impact of weight $\alpha$ on the characteristic of the waveform needs to be analyzed.

Since the PMEPR and PSLR in the objective function (21) are of different magnitudes, the weights need to be searched in a large range. We take the weight range as $[0.2,100]$ because it is sufficient to analyze the problem. To analyze the impact of weights on PMEPR and PSLR performance, a standard Monte Carlo technique with the same 3000 independent trails is employed for each weight, and the results are presented specifically in Figure 7.

As expected, both the PMEPR and PSLR decrease with the weight $\alpha$ increasing, i.e., the optimal PMEPR is obtained with a large weight, while the optimal PSLR is obtained with a small weight. Moreover, the PMEPR of the IRC waveform for a large weight $(\alpha >100)$ approximates the PMEPR obtained by only optimizing PMEPR [23], i.e., the PSLR is basically not optimized. Additionally, the PSLR of the IRC waveforms for a small weight $(\alpha \le 0.2)$ is very good, but the PMEPR is particularly poor, even worse than that of the initial waveform. At this time, we consider the obtained IRC waveform to be unqualified, because the meaning of PMEPR optimization is lost. Furthermore, note that individual initial waveforms in Monte Carlo with small weights may get trapped in local optima during the iterative process, resulting in the obtained IRC waveforms with suboptimal PMEPR and PSLR performance. Then, an important observation from Figure 7 is that there is a tradeoff between the two performance evaluation indicators of PMEPR and PSLR. Therefore, the optimal weights need to be selected to meet the requirements for the PMEPR and PSLR in practical applications.

6. Discussion

To further characterize the IRC waveform obtained by the proposed algorithm, we will specifically discuss the radar and communication performance of the IRC waveform here, where in addition to the PSLR performance mentioned above, the radar SNR, sidelobes in the low Doppler domain, and communication BER are mainly discussed, which are also very typical performance metrics in IRC systems. The IRC waveforms to be analyzed are generated at $\alpha =5$. The solid-state power amplifier (SSPA) with Rapp’s model [40] is considered.

The IRC waveform is passed through an SSPA. The input/output relationship of the SSPA is modeled by [40]

$$\tilde{G}\left[s\left(q\right)\right]=\frac{s\left(q\right)}{{\left(1+\left(\right|s\left(q\right)|/A_{sat}{)}^{2b}\right)}^{\frac{1}{2b}}}$$

(49)

where $A_{sat}$ is defined as a saturation level, and b determines the output sharpness parameter. In this paper, to compare the BER performance of the IRC systems with that obtained from traditional communication systems, we define the input communication power back-off (ICPBO) as

$$\mathrm{ICPBO}=\frac{A_{sat}^2}{P_{in}}$$

(50)

where $P_{in}$ is the input communication average power of SSPA. In our simulations, the SSPA parameters are $b=3$ and $A_{sat}=11.3$ dB compared to the average signal power before optimization.

6.1. Radar Performance of IRC Waveform

The radar SNR performances of the IRC waveform at different ICPBOs are shown in Figure 8. It can be observed that the radar SNR loss is smaller at lower ICPBO, and the radar SNR performance of the IRC waveform is better than that of the initial waveform. More importantly, compared with [31], at ICPBO = 5 dB, the proposed algorithm can obtain radar SNR gains of 2.3 dB at ICPBO = 6 dB and 0.8 dB at ICPBO = 8 dB.

One of the focuses of this paper is the sidelobe suppression along the zero-Doppler axis. In the following, we analyze the PSLR performance of Doppler in the range of [−31.25 KHz, 31.25 KHz]. Figure 9 presents the PSLR performance of the waveforms before and after optimization along the zero-Doppler axis and non-zero Doppler domain, respectively. It can be observed that the PSLR of the IRC waveform along the non-zero Doppler domain is very high, much higher than that of the initial waveform, and the difference from zero-Doppler is only within 3.5 dB, which indicates that the proposed algorithm can be well applied to the detection of low-speed moving targets. Sidelobe suppression along the high Doppler domain is complex, which is not discussed in this paper, but will be our future research direction.

6.2. Communication Performance of IRC Waveform

In this section, the communication performance is analyzed for noises (such as additive white Gaussian noise (AWGN) and Middleton Class B model [41]) whose time samples are independent and identically distributed.

The communication BER performance of the IRC waveform at different ICPBOs under an AWGN channel is first discussed. Figure 10 shows some BER performance comparisons of the initial waveform and the IRC waveform generated by our proposed algorithm and the one in [31], respectively, over the same SSPA. Note that the “Ideal PA” results obtained over a linear PA are also plotted for comparison. To achieve BER = ${10}^{-3}$, the value of $E_b/N_0$ needs to be 7.62 dB (ICPBO = 6 dB) and 7.05 dB (ICPBO = 8 dB) for CGIA-JOPP, 7.11 dB (ICPBO = 6 dB) and 6.9 dB (ICPBO = 8 dB) for the initial waveforms, and 7 dB (ICPBO = 5 dB) for the algorithm in [31], respectively. It is found that the BER performance of our proposed algorithm is slightly worse than that of the initial waveforms. However, as can be seen from Figure 8, the proposed IRC waveform shows a great advantage in radar SNR performance. As shown in Figure 8 and Figure 10, the BER performance of the proposed IRC waveform at ICPBO = 8 dB is almost identical to that of the initial waveform at ICPBO = 6 dB and that of the IRC waveform generated in [31], while the radar SNR loss is improved by 2.8 dB and 0.8 dB, respectively. Therefore, combined with the PSLR performance in Section 5, the proposed IRC waveform has very good radar detection and communication performance.

In practice, channel noise may suffer from both impulsive noise and Gaussian noise, i.e., the noise is heavy-tailed. Then, the non-Gaussian noise such as the Middleton Class B model [41] that simultaneously describes the impulsive noise and Gaussian noise is discussed. The characteristic function of the Middleton Class B model is expressed as

$$F\left(\theta \right)=\exp \left\{-{\nu }^{\alpha }{\left|\theta \right|}^{\alpha }-\frac{1}{2}{\sigma }^2{\theta }^2\right\}$$

(51)

where $\nu$ is the dispersion, $\alpha \in (0,2]$ is the characteristic exponent, which controls the heaviness of the tails of the distribution, and ${\sigma }^2$ is the variance of Gaussian noise.

Here, the simulation parameters are set: the ratio of signal power to Gaussian noise power ${\sigma }^2$ is 6.5 dB, ${\sigma }^2/\nu =2.4$, and $\alpha =1.3$. Figure 11 shows the signal constellation diagram with the noise modeled by Middleton Class B added, and the calculated BER is $1.3\times {10}^{-4}$. This shows that the IRC waveform generated by the proposed algorithm can also work well under non-Gaussian noise channels such as the Middleton Class B model. In addition, there is also a certain research value for noises (such as [42]) whose samples are correlated in time, which will be considered in our future work.

7. Conclusions

This paper dealt with two major characteristics of optimization in multi-symbol OFDM-based IRC waveform design, PMEPR and PSLR, and proposes an efficient IRC waveform design algorithm, i.e., CGIA-JOPP, to jointly optimize the PMEPR and PSLR. Our proposed CGIA-JOPP algorithm is inspired by an interesting observation that both the PMEPR and PSLR of the waveform can be transformed with the $L_p$-norm, and its joint optimization implies that IRC waveforms with both low PMEPR and high PSLR can be obtained by optimizing the reserved subcarriers. In our proposed algorithm, after the gradient analysis model of the objective function is derived, PRP-CGA is applied to deal with the established joint optimization problem. Simulation results show that both PMEPR and PSLR are effectively improved by the proposed algorithm, and the designed waveform achieves excellent radar detection performance and communication BER performance simultaneously.