Waveform Design for Target Information Maximization over a Complex Circle Manifold

Abstract

The cognitive radar framework presents a closed-loop adaptive processing paradigm that ensures the efficient acquisition of target information while exploring the environment and enhancing overall sensing performance. In this study, instead of mutual information, we employed the squared Pearson correlation coefficient (SPCC) to measure the target information in observations specifically considering only linear dependency. A waveform design method is proposed that simultaneously maximizes target information and minimizes the integrated sidelobe level (ISL) under the constant modulus constraint (CMC). To enhance computational efficiency, we reformulated the constrained problem as an unconstrained optimization problem by leveraging the inherent geometric property of CMC. Additionally, we present two conditional equivalences associated with waveform design in relation to target information. The simulation results validate the feasibility and effectiveness of the proposed method.

1. Introduction

Radar interacts with targets and the environment by means of transmitting signals and extracting information from echo signals. However, guaranteeing optimal capability for acquiring target information in complex dynamic electromagnetic environments remains challenging despite careful waveform design. Haykin [1] introduced the concept of cognitive radar (CR), which incorporates a feedback loop between the receiver and transmitter. This feedback loop allows the system to autonomously configure resources such as waveforms and antennas. As a consequence, CR could increase both in perception capability and resource utilization efficiency. More importantly, the establishment of a CR framework has prompted a shift in waveform design exploration, from focusing solely on the characteristics of waveform itself [2] to adaptation of different sensing requirements [3]. As is well known, various radar tasks have distinct demands for target and environmental information. For example, taking a Bayesian approach to detect a known signal in noisy measurement with binary hypotheses yields the maximum information of 1 bit, while more information would be required for estimation [4]. The architecture of the waveform optimization component in cognitive radar is depicted in Figure 1. As the figure shows, cognitive radar systems are capable of dynamically leveraging the acquired target signature to proactively adjust the transmitted waveforms, thereby enhancing their flexibility in information acquisition across various operational missions.

1.1. Motivations

Research on the topic of cognitive waveform design can generally be categorized as either task-driven or information-driven, with some exceptions for specialized issues. The objective function in the former type is always directly intertwined with the assessment of task performance, ensuring that the optimized outcomes can directly augment task performance. For instance, the possible optimization criteria for cognitive waveform design in the detection task are either the signal to interference-plus-noise power ratio (SINR) [5] or detection probability [6], while the waveform can be optimized based on the estimation error [7] in the tracking scenario. For classification purposes, the separability of different targets could be a criterion to guide the waveform design [8,9]. On the other hand, the information-driven paradigm adopts an information-theoretic perspective to enhance the information acquisition capability by improving the system capacity for target-of-interest information. The exemplary research on information-theory-based waveform design can be traced back to Bell’s work [10]. He proposed a water-filling method based on mutual information (MI), which entails constructing a measurement model between the received echo and the target impulse response (TIR) and leveraging their mutual information as a metric. Leshem [11] further boosted the above work by embedding multiple extended targets in Multiple-Input-Multiple-Output (MIMO) radar scenarios. Romero [12] proposed a frequency domain design methodology for information-theoretic waveforms for both deterministic and stochastic extended targets, employing the concept of matched illumination. Gu [13], in supplementary work, refined the target model and derived the analytical form for information-theoretic waveform design of a Gaussian mixture extended target. In the context of spectrum sharing with communication devices, the waveform optimization approaches can be accessed in [14,15,16]. Nonetheless, two problems remain in the information-driven radar waveform design.

• The aforesaid definition of design criteria associated with mutual information invariably necessitates an exact probability distribution at hand. Such knowledge is often ambiguous and only approximated through limited samples [17].
• The majority of the information-driven approaches primarily focus on optimizing the energy spectral density without taking into account the synthesis of the constant modulus waveform in the time domain, which is rendered arduous to implement in practical systems. Specifically, the constant modulus constraint (CMC) is the primary constraint on the probing waveform and serves as a fundamental guarantee to ensure optimal efficiency of the non-linear amplifier working in saturation mode. However, its derived feasible set has a non-convex nature, which is known to be NP-hard [18].

Fortunately, while mutual information (MI) can effectively capture the degree of information dependence between two variables, the Pearson correlation coefficient (PCC) is a more economical choice when considering only linear dependencies [19]. Moreover, Xu [20] has indeed demonstrated the practicality of using the PCC as a waveform optimization criterion to quantify target information and enhance classification performance. Although this work addressed the issue of high range resolution in waveforms, it did not account for the potential negative impact of amplitude-modulated time-domain waveforms on the transmitter. Building upon Xu’s research, we have considered a more generalized design criterion, which is the weighted sum of the squared Pearson correlation coefficient (SPCC) and the integral sidelobe level (ISL) to obtain target information waveforms with high range resolution. We have also fully accounted for the constant modulus constraint, ensuring that the designed waveforms are more suitable for real-world applications.

1.2. Major Contributions

The principal contributions of this paper are listed as follows:

• We address the problem of waveform design with a constant modulus constraint, aiming to maximize target information via the squared Pearson correlation coefficient (SPCC). Additionally, we simultaneously consider the integrated sidelobe level (ISL) to meet the requirements of autocorrelation and achieve high resolution. Simulations were performed to verify the feasibility and effectiveness of the proposed method.
• The constrained design problem is reformulated as an unconstrained problem based on the geometric characteristic of constant modulus constraint (CMC), and the complexity associated with resolving the optimization problem is diminished.
• A conditional equivalence is presented between two criteria when a wide-sense stationary-uncorrelated scattering (WSSUS) model is used to characterize the TIR. The first criterion involves minimizing the minimum mean-square error (MMSE) in estimating the TIR, while the second criterion involves minimizing the integrated sidelobe level of the waveform.
• Another conditional equivalence is illustrated for the case of zero-mean TIR, where maximizing the squared Pearson correlation coefficient (SPCC) is shown to be equivalent to maximizing the signal-to-noise ratio (SNR).

1.3. Notation and Organization

Boldface lowercase letters denote column vectors, while boldface uppercase letters denote matrices. Operators ${(·)}^{*}$, ${(·)}^{\mathrm{H}}$, and ${(·)}^{\mathrm{T}}$ denote the complex conjugate, conjugate transpose, and transpose, respectively. In addition, $\left|\mathbf{x}\right|$ represents the modulus of vector $\mathbf{x}$. ${\parallel ·\parallel }_p$ is the p-norm for matrices/vectors; specifically, ${\parallel \mathbf{A}\parallel }_F$ denotes the Frobenius norm for matrix $\mathbf{A}$. The n-dimensional real and complex column vector spaces are expressed as ${\mathbb{R}}^n$ and ${\mathbb{C}}^n$, respectively. For any positive integer n, ${\mathbb{Z}}_n$ stands for the set $\{1,\dots ,n\}$. The i-th entry of the vector $\mathbf{x}$ is denoted by $x_i$. $\Re \left(\mathbf{x}\right)$ denotes the real part of the vector $\mathbf{x}$, calculated element-wise. The inner product of two vectors $\langle \mathbf{x},\mathbf{z}\rangle$ represents ${\mathbf{x}}^{\mathrm{H}}\mathbf{z}$. $tr(·)$ and ${(·)}^{-1}$ are the trace and inverse operations of a matrix. $\mathcal{CN}(\mathit{\mu },\mathbf{\Sigma })$ denotes the complex Gaussian distribution with mean $\mathit{\mu }$ and covariance $\mathbf{\Sigma }$. The abbreviation “s.t.” stands for “subject to”. $\mathbb{E}$ and $\mathbb{D}$ represent the expectation and variance operators.

The rest of this paper is organized as follows. The signal model is developed in Section 2. The proposed design method via the Riemannian conjugate gradient based on complex circle manifold is derived in Section 3. Numerical results are presented in Section 4. The conclusions and discussion are given in Section 5.

2. Signal Model

We consider a phase-coded probing signal $s\left(t\right)$, consisting of N coding elements,

$$f\left(t\right)=\sum _{n=1}^{N}s\left(n\right)p_n\left(t\right)$$

(1)

where $p_n\left(t\right)$ is the shaping pulse that can be configured as an ideal rectangular shaping pulse or any other desired envelope. ${\left\{s\left(n\right)\right\}}_{n=1}^N$ denotes the coding elements, and we use a N-dimensional vector $\mathbf{s}=\left[s_1,\dots ,s_N\right]\in {\mathbb{C}}^N$ to represent these transmitted codes.

The target range extent greatly surpasses the radar resolution of the high-range resolution (HRR) radar, rendering the utilization of the point-like scatter target model inappropriate. The extended target in this case can be considered as a linear system, whose behavior is characterized by the TIR. The TIR depends primarily on the physical structure of the target, as well as the target aspect angle (TAA) with respect to the radar line-of-sight (LOS) and the radar-operating frequency [21]. However, precisely describing TIR is challenging due to the limited knowledge in non-cooperative sensing applications. A pragmatic approach to overcome this problem involves modeling a stochastic target with a specific probability distribution [10]. The WSSUS model [22] is a popular choice. The TIR vector is defined as $\mathbf{g}=\left[g_1,\dots ,g_M\right]\in {\mathbb{C}}^M$ and assumes $\mathbf{g}\sim \mathcal{CN}({\mathbf{m}}_g,{\mathbf{Q}}_g)$, where ${\mathbf{Q}}_g$ is a diagonal matrix and M is the target support interval, which means $g_m=0$ unless $m\in \{1,\dots ,M\}$. The discrete-time baseband representation of the measurements can be expressed as follows:

$$\mathbf{y}=\mathbf{g}★\mathbf{s}+\mathbf{\nu }$$

(2)

where $\mathbf{\nu }=\left[{\nu }_1,\dots ,{\nu }_L\right]$ is the Gaussian additive noise with zero-mean and circularly symmetric covariance matrix ${\mathbf{R}}_{\nu }={\sigma }_{\nu }^2{\mathbf{I}}_L$, $L=N+M-1$ and ★ denotes the convolution operator.

To rewrite (2) in a more compact form, the shift matrix ${\mathbf{J}}_m\in {\mathbb{C}}^M$ is defined [23],

$${\mathbf{J}}_q({\ell }_1,{\ell }_2)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}{\ell }_1-{\ell }_2=q\hfill \\ 0,\hfill & \mathrm{if}{\ell }_1-{\ell }_2\ne q\hfill \end{array}\right.$$

(3)

Then, $\mathbf{y}$ is given by

$$\mathbf{y}=\mathbf{G}\mathbf{s}+\mathbf{\nu }=\mathbf{S}\mathbf{g}+\mathbf{\nu }$$

(4)

where $\mathbf{G}={\sum }_{m=1}^M{\mathbf{g}}_m{\mathbf{J}}_{m-1}$ is the TIR convolution matrix and $\mathbf{S}={\sum }_{n=1}^N{\mathbf{s}}_n{\mathbf{J}}_{n-1}$ is the signal convolution matrix. A similar model is used in [24].

3. Constant Modulus Waveform Design for Maximizing Target Information with Low ISL

In this section, the target information and waveform autocorrelation will be used as design criteria. To simplify the problem, the multi-objective optimization problem is transformed into a single-objective optimization problem by using a weighted sum formulation [25]. Additionally, the geometric structure of the constant modulus constraint is included, and the Riemannian manifold optimization tool is employed to provide an efficient solution.

3.1. Design Criteria

3.1.1. Target Information

The PCC is a second-order statistics [26] that has been widely applied in many fields such as pattern recognition [27] and noise reduction [28]. However, serving as an approximation of MI for waveform design applications outlined in Section 1, the PCC can solely measure the strength and direction of a linear relationship between two random variables, and it can adequately encompass the pertinent information for a linear measurement model such as (4). As in [20], we first give the specific formulation of the PCC with respect to (4) as follows

$$\begin{array}{cc}\hfill \rho & =\frac{\mathbb{D}(\mathbf{y},\mathbf{z})}{\sqrt{\mathbb{D}(\mathbf{y},\mathbf{y})}·\sqrt{\mathbb{D}(\mathbf{z},\mathbf{z})}}\hfill \\ \hfill & =\frac{\sqrt{{\mathbf{s}}^{\mathrm{H}}\mathbb{E}\left[{\mathbf{G}}^{\mathrm{H}}\mathbf{G}\right]\mathbf{s}-{\left|{\mathbf{m}}_{\mathbf{y}}\right|}^2}}{\sqrt{{\sigma }_v^2+{\mathbf{s}}^{\mathrm{H}}\mathbb{E}\left[{\mathbf{G}}^{\mathrm{H}}\mathbf{G}\right]\mathbf{s}-{\left|{\mathbf{m}}_{\mathbf{y}}\right|}^2}}\hfill \end{array}$$

(5)

where $\mathbf{z}=\mathbf{G}\mathbf{s}$. The PCC is capable of discerning positive and negative correlations, which does not contribute to our method and incurs additional computational cost due to the root operator. In light of this, the SPCC is an alternative whose definition is directly the squared modulus of the PCC

$${\left|\rho \right|}^2=1-\frac{{\sigma }_{\nu }^2}{{\sigma }_{\nu }^2+{\mathbf{s}}^{\mathrm{H}}{\mathbf{R}}_g\mathbf{s}}$$

(6)

where ${\mathbf{R}}_g=\mathbb{E}\left[{\mathbf{G}}^{\mathrm{H}}\mathbf{G}\right]-\mathbb{E}{\left[\mathbf{G}\right]}^{\mathrm{H}}\mathbb{E}\left[\mathbf{G}\right]$. Obviously, the range of ${\left|\rho \right|}^2$ is $[0,1]$. The strength of the correlation between the two variables increases as ${\left|\rho \right|}^2$ approaches 1. Thus, ${\left|\rho \right|}^2$ can also be viewed as a linear-distortion index.

Since ${\sigma }_{\nu }^2$ in (6) has no impact on the optimization of $\mathbf{s}$, the problem of maximizing target information about $\mathbf{s}$ can be recast as

$${\mathcal{P}}_1:\left\{\begin{array}{cc}\hfill \underset{\mathbf{s}}{max}& {\mathbf{s}}^{\mathrm{H}}{\mathbf{R}}_g\mathbf{s}\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& {\mathbf{s}}^{\mathrm{H}}\mathbf{s}=E_s\hfill \end{array}\right.$$

(7)

where $E_s$ is the total transmitted energy of $\mathbf{s}$.

Notice that ${\mathcal{P}}_1$ is a Rayleigh quotient maximization problem, where the maximum value according to the largest eigenvalue and the optimal solution of $\mathbf{s}$ is represented by the eigenvector associated with this eigenvalue.

$${\mathbf{s}}_0=\sqrt{E_s}·\underset{max\left(\lambda \right)}{eig}\left({\mathbf{R}}_g\right)$$

(8)

where ${eig}_{\lambda }\left(·\right)$ denotes the operator that takes the eigenvector corresponding to the eigenvalue $\lambda$. In general, ${\mathbf{s}}_0$ exhibits a flexible form with modulated amplitude and phase, which would typically lead to a degradation in both bandwidth and autocorrelation performance, as shown in Figure 2. Additional constraints should be taken into account to design a more pragmatic waveform.

If optimization in terms of cost function paradigm is considered, ${\mathcal{P}}_1$ can be equivalently reformulated as a least-squares problem for the sake of consistency as below

$${\mathcal{P}}_2:\left\{\begin{array}{cc}\hfill \underset{\mathbf{s}}{min}& \gamma \left(\mathbf{s}\right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& {\mathbf{s}}^{\mathrm{H}}\mathbf{s}=E_s\hfill \end{array}\right.$$

(9)

where $\gamma \left(\mathbf{s}\right)={∥\mathbf{s}-{\mathbf{s}}_0∥}^2$.

Remark 1.

Xu [20] observes that there exists a distinction in the physical meaning between the objective function derived from the PCC and that from the SNR. However, it is noteworthy to mention that these two are equivalent when the target response obeys a zero-mean distribution [28],

$${\rho }^2=\frac{\mathrm{SNR}}{1+\mathrm{SNR}}$$

(10)

The reason is that the SNR metric quantifies the energy level, whereas the PCC measures second-order statistical information. By considering the definition of variance ($\mathbb{D}\left[X\right]=\mathbb{E}\left[X^{\mathrm{H}}X\right]$$-\mathbb{E}{\left[X\right]}^{\mathrm{H}}\mathbb{E}\left[X\right]$), it becomes apparent that signal energy equates to variance if and only if the mean is zero.

3.1.2. Correlation Properties

A common objective of probing waveform design is to minimize the integrated sidelobe level (ISL) or the peak sidelobe level (PSL) of the waveform autocorrelation. However, compared to the non-smooth infinity norm of the PSL, the ISL is preferred due to its differentiable Euclidean norm. Some studies have been also carried out with the joint consideration of CMC and orthogonality constraints [29,30,31] within the framework of waveforms possessing desired auto- and cross-correlation properties.

Given a sequence $\mathbf{s}$, its aperiodic autocorrelation function (AACF) is given by [29]

$$r_{\ell }=r_{-\ell }^{*}=\sum _{n=\ell +1}^N{s}_n{s}_{n-\ell }^{*},\ell =0,\dots ,N-1.$$

(11)

where $r_0$ is called the in-phase correlation and $r_0=E_s$ always holds. Then, the ISL of $\mathbf{s}$ can be represented by the $\mathrm{ISL}\left(\mathbf{s}\right)$, which has the form of

$$\mathrm{ISL}\left(\mathbf{s}\right)=2\sum _{\ell =1}^{N-1}{\left|r_{\ell }\right|}^2$$

(12)

It can be seen that the $\mathrm{ISL}\left(\mathbf{s}\right)$ is a quartic function of $\mathbf{s}$, which evidently is non-convex and typically too difficult to solve directly. Luckily, by leveraging Passaval equality, the completeness of Euclidean space, and the continuity of the Euclidean norm, as mentioned in [32], we can solve the following “almost equivalent” optimization problem instead:

$$\psi (\mathbf{s},\mathbf{v})=\sum _{m=1}^{2N}{\left|{\mathbf{f}}_m^{\mathrm{H}}\mathbf{s}-\sqrt{N}{e}^{j{\phi }_m}\right|}^2$$

(13)

where ${\mathbf{f}}_m^{\mathrm{H}}=\left[\begin{array}{c}1,e^{-j{\omega }_m},\cdots ,e^{-j(N-1){\omega }_m}\end{array}\right]\in {\mathbb{C}}^N$ is the Discrete Fourier Transformation (DFT) vector and ${\omega }_m$ takes the following form

$${\omega }_m=\frac{\pi }{N}(m-1),m=1,\dots ,2N$$

Then, (13) is reformulated into a compact form as

$$\psi (\mathbf{s},\mathbf{v})={∥{\mathbf{F}}^{\mathrm{H}}\mathbf{s}-\sqrt{N}\mathbf{v}∥}^2$$

(14)

where ${\mathbf{F}}^{\mathrm{H}}={\left[{\mathbf{f}}_1,{\mathbf{f}}_2,\dots ,{\mathbf{f}}_{2N}\right]}^{\mathrm{H}}$ and $\mathbf{v}={\left[e^{j{\phi }_1},e^{j{\phi }_2},\cdots ,e^{j{\phi }_{2N}}\right]}^{\mathrm{T}}\in {\mathbb{C}}^{2N}$ denote the first N columns of the DFT matrix and the auxiliary variable, respectively. More details can be found in [29]. Now, the waveform design problem using the ISL criterion is given by

$${\mathcal{P}}_3:\left\{\begin{array}{cc}\hfill \underset{\mathbf{s},\mathbf{v}}{min}& \psi (\mathbf{s},\mathbf{v})={∥{\mathbf{F}}^{\mathrm{H}}\mathbf{s}-\sqrt{N}\mathbf{v}∥}^2\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& |s_n|=1,\forall n\in {\mathbb{Z}}_N\hfill \\ \hfill & |v_n|=1,\forall n\in {\mathbb{Z}}_{2N}\hfill \end{array}\right.$$

(15)

The subsequent content will illustrate that the minimization of the ISL is tantamount to minimizing the MMSE of TIR in signal model (4) in a certain condition. We first apply the linear MMSE estimator of $\mathbf{g}$ to obtain $\widehat{\mathbf{g}}$, which is given by [24],

$$\widehat{\mathbf{g}}={\mathbf{m}}_g+{\mathbf{Q}}_g{\mathbf{S}}^{\mathrm{H}}{\left(\mathbf{S}{\mathbf{Q}}_g{\mathbf{S}}^{\mathrm{H}}+{\mathbf{R}}_n\right)}^{-1}\left(\mathbf{y}-\mathbf{S}{\mathbf{m}}_g\right)$$

(16)

and the corresponding MMSE is

$$\begin{array}{cc}\hfill MMSE& =\mathbb{E}\left({∥\mathbf{g}-\widehat{\mathbf{g}}∥}_2^2\right)\hfill \\ \hfill & =tr\left[{\mathbf{Q}}_{\mathrm{g}}-{\mathbf{Q}}_{\mathrm{g}}{\mathbf{S}}^{\mathrm{H}}{\left(\mathbf{S}{\mathbf{Q}}_{\mathrm{g}}{\mathbf{S}}^{\mathrm{H}}+{\mathbf{R}}_{\mathrm{n}}\right)}^{-1}\mathbf{S}{\mathbf{Q}}_{\mathrm{g}}\right]\hfill \\ \hfill & =tr\left[{\left({\mathbf{Q}}_{\mathrm{g}}^{-1}+{\mathbf{S}}^{\mathrm{H}}{\mathbf{R}}_{\mathrm{n}}^{-1}\mathbf{S}\right)}^{-1}\right]\hfill \end{array}$$

(17)

where the matrix inversion lemma (${(\mathbf{A}+\mathbf{BCD})}^{-1}={\mathbf{A}}^{-1}-{\mathbf{A}}^{-1}\mathbf{B}{\left(\mathbf{D}{\mathbf{A}}^{-1}\mathbf{B}+{\mathbf{C}}^{-1}\right)}^{-1}\mathbf{D}{\mathbf{A}}^{-1}$) is employed in obtaining the third equality. To address the trace of an inverse matrix, it is imperative to introduce a pertinent lemma beforehand.

Lemma 1.

For a $N\times N$ positive-definite matrix $\mathbf{A}$ with $(m,n)$th entry $a_{mn}$, it holds that

$$tr\left({\mathbf{A}}^{-1}\right)\ge \sum _{n=1}^N\frac{1}{a_{nn}}$$

(18)

where the equality is attained if and only if $\mathbf{A}$ is diagonal.

See Appendix A in [33] for the proof of Lemma 1. Lemma 1 indicates that to minimize the MMSE, the matrix ${\mathbf{Q}}_{\mathrm{g}}^{-1}+{\mathbf{S}}^{\mathrm{H}}{\mathbf{R}}_v^{-1}\mathbf{S}$ in (17) should be diagonal, then we have the following proposition.

Proposition 1.

The minimization of the MMSE of $\widehat{\mathbf{g}}$ for a stochastic target, whose TIR is assumed to be a Gaussian distribution and the corresponding covariance matrix (i.e., second-order statistics) takes on a diagonal form (due to the wide sense stationary-uncorrelated scattering model [22]), is equivalent to $\mathbf{\Omega }={\mathbf{S}}^{\mathrm{H}}{\mathbf{R}}_v^{-1}\mathbf{S}$, being a diagonal matrix with positive elements ${\mathbf{\Omega }}_{ii}>0,\forall i\in {\mathbb{Z}}_M$. When ${\mathbf{R}}_v={\sigma }_v^2{\mathbf{I}}_L$, the following equality holds,

$${\mathbf{S}}^{\mathrm{H}}\mathbf{S}=E_s{\mathbf{I}}_N$$

(19)

According to Proposition 1, minimizing the value of MMSE in estimating the TIR means radar should transmit the ideal waveform with a zero autocorrelation sidelobe. We further reformulate (19) into a least-squares problem:

$${\mathcal{P}}_{\mathrm{MMSE}}:\left\{\begin{array}{cc}\hfill \underset{\mathbf{s}}{min}& \Pi \left(\mathbf{s}\right)={∥{\mathbf{S}}^{\mathrm{H}}\mathbf{S}-E_s{\mathbf{I}}_N∥}_F^2\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& {\mathbf{s}}^{\mathrm{H}}\mathbf{s}=E_s.\hfill \end{array}\right.$$

(20)

Based on the definition of the Frobenius norm (${\parallel \mathbf{A}\parallel }_F^2={\sum }_{i=1}^m{\sum }_{j=1}^n{\left|a_{ij}\right|}^2$), we have

$$\Pi \left(\mathbf{s}\right)\propto \mathrm{ISL}\left(\mathbf{s}\right)$$

(21)

To date, it can be concluded that the minimization of MMSE in TIR estimation for a specific target necessitates minimizing the ISL of the transmitted waveform.

Remark 2.

Solving ${\mathcal{P}}_{\mathrm{MMSE}}$ involves performing a singular value decomposition (SVD) operation in each iteration step [34] with a computational complexity of $\mathcal{O}\left(N^3\right)$. In contrast, the cyclic algorithm-new (CAN) [29] algorithm only requires fast Fourier transform (FFT) during each iteration, resulting in a computational complexity of $\mathcal{O}\left(NlogN\right)$. Consequently, our method adopts the “almost equivalent” criterion in (13) as the ISL description of the waveform.

The reasons behind selecting the ISL as the criterion for waveform design can be summarized in three key aspects, as follows:

• The ISL is a widely used criterion for representing the autocorrelation quality of the waveform, enabling the comprehensive measurement of the sidelobe energy level. Compared to the PSL, the “almost equivalent” form of the ISL is convex and smooth, facilitating ease of solutions.
• The MMSE minimization in TIR estimation for a specific target requires minimizing the ISL of the transmitted waveform.
• A phase-coded sequence designed based on the ISL would behave similarly to white noise, and consequently, the modulus of its spectrum should be nearly constant. In other words, the devised sequence would best utilize the bandwidth.

3.2. Problem Formulation

It can be observed that the directly designed waveform from ${\mathcal{P}}_1$ often demonstrates poor autocorrelation performance. To provide sufficient resolution and potentially enhance TIR estimation performance, it becomes necessary to incorporate autocorrelation requirements into the objective function. We formulated the design problem as a bi-objective optimization under constant modulus constraints,

$$\left\{\begin{array}{cc}\hfill \underset{\mathbf{s},\mathbf{v}}{min}& \psi (\mathbf{s},\mathbf{v}),\gamma \left(\mathbf{s}\right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& |s_n|=1,\forall n\in {\mathbb{Z}}_N\hfill \\ \hfill & |v_n|=1,\forall n\in {\mathbb{Z}}_{2N}\hfill \end{array}\right.$$

(22)

It is typically intractable to find a feasible solution that simultaneously minimizes all objective functions within the context of a multi-objective optimization framework [35]. Accordingly, the aim of solving (22) has been modified to find the Pareto-optimal solutions, which also generally poses a daunting task. One feasible strategy for accessing the Pareto front is through the implementation of the scalarization technique [18]. This approach involves utilizing a specific weighted sum of $\psi (\mathbf{s},\mathbf{v})$ and $\gamma \left(\mathbf{s}\right)$ as the objective function. Specifically, let the Pareto weight $\eta \in [0,1]$, and the (22) is reformulated as

$${\mathcal{P}}_4:\left\{\begin{array}{cc}\hfill \underset{\mathbf{s},\mathbf{v}}{min}& \xi (\mathbf{s},\mathbf{v})\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& |s_n|=1,\forall n\in {\mathbb{Z}}_N\hfill \\ \hfill & |v_n|=1,\forall n\in {\mathbb{Z}}_{2N}\hfill \end{array}\right.$$

(23)

where

$$\begin{array}{cc}\hfill \xi (\mathbf{s},\mathbf{v})& =(1-\eta )·\psi (\mathbf{s},\mathbf{v})+\eta ·\gamma \left(\mathbf{s}\right)\hfill \\ \hfill & =(1-\eta ){∥{\mathbf{F}}^{\mathrm{H}}\mathbf{s}-\sqrt{N}\mathbf{v}∥}^2+\eta {∥\mathbf{s}-{\mathbf{s}}_0∥}^2\hfill \end{array}$$

Notice that both the variables $\mathbf{s}$ and $\mathbf{v}$ are restrained by CMC, indicating the possibility of merging them into a unified variable. Let $\mathbf{x}={\left[{\mathbf{s}}^{\mathrm{T}},{\mathbf{v}}^{\mathrm{T}}\right]}^{\mathrm{T}}\in {\mathbb{C}}^{3N}$, then $\xi (\mathbf{s},\mathbf{v})$ could be rewritten as

$$\xi \left(\mathbf{x}\right)=(1-\eta ){∥\mathbf{A}\mathbf{x}∥}^2+\eta {∥\mathbf{B}\mathbf{x}-{\mathbf{x}}_{\mathrm{opt}}∥}^2$$

(24)

where

$$\begin{array}{cc}\hfill \mathbf{A}& =\left[{\mathbf{F}}^{\mathrm{H}},-\sqrt{N}{\mathbf{I}}_{2N}\right]\in {\mathbb{C}}^{2N\times 3N}\hfill \\ \hfill \mathbf{B}& =\left[{\tilde{\mathbf{I}}}_{2N\times N},{\mathbf{0}}_{2N\times 2N}\right]\in {\mathbb{C}}^{2N\times 3N}\hfill \\ \hfill {\mathbf{x}}_{\mathrm{opt}}& =\left[{\mathbf{s}}_0;{\mathbf{0}}_{N\times 1}\right]\in {\mathbb{C}}^{2N}\hfill \end{array}$$

We can further expand Equation (24) and obtain that

$$\xi \left(\mathbf{x}\right)={\mathbf{x}}^{\mathrm{H}}\mathrm{\Sigma }\mathbf{x}+2\Re \left({\mathbf{q}}^{\mathrm{H}}\mathbf{x}\right)+\mathbf{C}$$

(25)

where

$$\begin{array}{cc}\hfill \mathrm{\Sigma }& =(1-\eta ){\mathbf{A}}^{\mathrm{H}}\mathbf{A}+\eta {\mathbf{B}}^{\mathrm{H}}\mathbf{B}\hfill \\ \hfill \mathbf{q}& =-\eta {\mathbf{B}}^{\mathrm{H}}{\mathbf{x}}_{\mathrm{opt}}\hfill \\ \hfill \mathbf{C}& =\eta {\mathbf{x}}_{\mathrm{opt}}^{\mathrm{H}}{\mathbf{x}}_{\mathrm{opt}}\hfill \end{array}$$

Note that the independence of $\mathbf{C}$ from $\mathbf{x}$ allows us to exclude it from the objective function for convenience, and then the equivalent form of ${\mathcal{P}}_4$ can be given by ${\mathcal{P}}_5$ as

$${\mathcal{P}}_5:\left\{\begin{array}{cc}\hfill \underset{\mathbf{x}}{min}& \xi \left(\mathbf{x}\right)={\mathbf{x}}^{\mathrm{H}}\mathrm{\Sigma }\mathbf{x}+2\Re \left({\mathbf{q}}^{\mathrm{H}}\mathbf{x}\right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& |x_n|=1,\forall n\in {\mathbb{Z}}_{3N}\hfill \end{array}\right.$$

(26)

The objective function of ${\mathcal{P}}_5$ has a definite inhomogeneous quadratic form with $\mathrm{\Sigma }$ being a positive semi-definite matrix. The non-convexity of ${\mathcal{P}}_5$ mainly arises from the CMC, which can be addressed through several available methods, such as semi-definite relaxation (SDR) [36], the projected gradient method [37], or the alternating direction method of multipliers (ADMM) [38,39]. In our study, the optimization method on the complex circle manifold is included.

3.3. Optimization over Complex Circle Manifold

Generally speaking, the superiority of manifold optimization lies in the capability to convert constrained problems into unconstrained ones over a manifold $\mathcal{M}$ that is constructed with the feasible set of the constraints. This processing paradigm can effectively exploit the geometric attributes of constraints to bolster the exploration of more efficient optimization techniques.

3.3.1. Complex Circle Manifold

To handle the CMC, there exist at least two kinds of methods. One involves approximating or relaxing the solution and subsequently normalizing the resulting sequence [40], whereas the other entails directly solving the CMC [36,41], which can yield good performance albeit at a significantly increased computational cost. In this study, the geometric properties of constant modulus constraints (CMC) are utilized to directly solve for waveform sequences, as detailed in the following.

Each entry $x_n$ of a constant modulus sequence $\mathbf{x}$ can be explained as a point located on the unit circle within the complex plane, while the entire sequence $\mathbf{x}$ can be regarded as a point residing on the unit hypersphere in a $3N$-dimensional space. This particular hypersphere is referred to as a complex circular manifold (CCM). Numerous works have explored optimization methods for CCM and primarily involve extending manners from Euclidean space to manifolds, like Riemannian steepest descent, Riemannian conjugate gradient, quasi-Newton [42] and trust region methods [43]. These techniques have been extensively employed across various disciplines, with a particular emphasis on radar waveform design [44,45,46].

The constant modulus constraint in ${\mathcal{P}}_5$ forms a $3N$-dimensional CCM $\mathcal{M}$, and the constrained problem ${\mathcal{P}}_5$ can be rewritten as

$${\mathcal{P}}_6:\left\{\begin{array}{cc}\hfill \underset{\mathbf{x}}{min}& \xi \left(\mathbf{x}\right)={\mathbf{x}}^{\mathrm{H}}\mathrm{\Sigma }\mathbf{x}+2\Re \left({\mathbf{q}}^{\mathrm{H}}\mathbf{x}\right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \mathbf{x}\in \mathcal{M}\hfill \end{array}\right.$$

(27)

where

$$\mathcal{M}=\left\{\mathbf{x}\in {\mathbb{C}}^{3N}:|x_1|=\cdots =|x_{3N}|=1\right\}$$

(28)

To establish Riemannian first-order methods, a preliminary concept of Riemannian gradients on smooth manifolds is essential. In this context, we exploit the fact that smooth manifolds can be locally linearized around the point $\mathbf{x}$ to define the tangent space. The tangent space, denoted as $T_{\mathbf{x}}\mathcal{M}$, comprises all tangent vectors associated with curves at point $\mathbf{x}$ [47], as illustrated in Figure 3. Then, the tangent space for CCM is given by

$$T_{\mathbf{x}}\mathcal{M}=\left\{{\varrho }_{\mathbf{x}}\in {\mathbb{C}}^{3N}:\Re \left\{{\varrho }_{\mathbf{x}}\odot {\mathbf{x}}^{*}\right\}=\mathbf{0}\right\}$$

(29)

where ⊙ denotes the Hadamard product.

After establishing the tangent space, it becomes feasible to formulate the operational rules on the manifold. Initially, we selected the canonical complex inner product as the Riemannian metric of $T_{\mathbf{x}}\mathcal{M}$ for convenience, which is $\vartheta (\varrho ,\mathbf{d})=〈\varrho ,\mathbf{d}〉=\Re \left({\varrho }^{\mathrm{H}}\mathbf{d}\right)$.

Riemannian Gradient

The Riemannian gradient $grad\xi \left(\mathbf{x}\right)$ of $\xi \left(\mathbf{x}\right)$ is defined as the unique element of $T_{\mathbf{x}}\mathcal{M}$ satisfying

$$\mathcal{D}\xi \left(\mathbf{x}\right)\left[{\varrho }_{\mathbf{x}}\right]=〈grad\xi \left(\mathbf{x}\right),{\varrho }_{\mathbf{x}}〉$$

(30)

where $\mathcal{D}\xi \left(\mathbf{x}\right)\left[{\varrho }_{\mathbf{x}}\right]$ denotes the directional derivative of the objective function $\xi \left(\mathbf{x}\right)$ at point $\mathbf{x}$ along direction ${\varrho }_{\mathbf{x}}\in T_{\mathbf{x}}\mathcal{M}$. Particularly, the Riemannian gradient $grad\xi \left(\mathbf{x}\right)$ can be calculated by projecting the Euclidean gradient $Grad\xi \left(\mathbf{x}\right)$ of $\xi \left(\mathbf{x}\right)$ onto the tangent space $T_{\mathbf{x}}\mathcal{M}$, which is

$$\begin{array}{cc}\hfill grad\xi \left(\mathbf{x}\right)& ={\mathcal{P}}_{\mathbf{x}}\left(Grad\xi \left(\mathbf{x}\right)\right)\hfill \\ \hfill & =Grad\xi \left(\mathbf{x}\right)-\Re \left\{Grad\xi \left(\mathbf{x}\right)\odot {\mathbf{y}}^{*}\right\}\odot \mathbf{y}\hfill \end{array}$$

(31)

where ${\mathcal{P}}_{\mathbf{x}}\left({\varsigma }_{\mathbf{x}}\right)={\varsigma }_{\mathbf{x}}-〈\mathbf{x},{\varsigma }_{\mathbf{x}}〉\mathbf{x}$ denotes the orthogonal projection operator and $Grad\xi \left(\mathbf{x}\right)$ is calculated with

$$\begin{array}{cc}\hfill Grad\xi \left(\mathbf{x}\right)& =\frac{\partial }{\partial \mathbf{x}}\left({\mathbf{x}}^{\mathrm{H}}\mathrm{\Sigma }\mathbf{x}+2\Re \left({\mathbf{q}}^{\mathrm{H}}\mathbf{x}\right)+\mathbf{C}\right)\hfill \\ \hfill & =2\mathrm{\Sigma }\mathbf{x}+2\Re \left(\mathbf{q}\right)\hfill \end{array}$$

(32)

Retraction and Vector Transport

It is crucial to emphasize that the effectiveness of manifold optimization in practical applications mainly stems from the introduction of the retraction concept, which enables computational feasibility [43]. The retraction operator ${\mathcal{R}}_{\mathbf{x}}:T_{\mathbf{x}}\mathcal{M}\to \mathcal{M}$ is a local mapping to preserve constraints with low computational complexity, defined as

$${\mathcal{R}}_{\mathbf{x}}\left({\varrho }_{\mathbf{x}}\right)=\frac{\mathbf{x}+{\varrho }_{\mathbf{x}}}{\parallel \mathbf{x}+{\varrho }_{\mathbf{x}}\parallel }$$

(33)

Direct addition of two vectors belonging to different tangent spaces over the CCM is not feasible. To address this issue, the process of vector transport is introduced to perform the adding operation. In detail, the vector transport operator is a smooth mapping that transports the tangent vector ${\varrho }_{\mathbf{x}}$ from a point $\mathbf{x}$ to another point ${\mathbf{x}}^{\prime }$ on the manifold, which is expressed as

$${\mathcal{T}}_{{\mathbf{x}}^{\prime }\leftarrow \mathbf{x}}\left({\varrho }_{\mathbf{x}}\right)={\varrho }_{\mathbf{x}}-\Re \left\{{\varrho }_{\mathbf{x}}^{*}\odot {\mathbf{x}}^{\prime }\right\}\odot {\mathbf{x}}^{\prime }$$

(34)

3.3.2. Riemannian Conjugate Gradient Method

Without consideration of the constraint, one of the simplest ways to search for the minimum of ${\xi }_{\mathbf{x}}$ is to employ a steepest-descent (SD) method in Euclidean space. However, we prefer the conjugate gradient (CG) algorithm due to the fact that if the matrix $\mathrm{\Sigma }$ is ill-conditioned, the SD approach would be slow, while the CG provides a faster search by utilizing the gradient information from the previous iteration. A Riemannian conjugate gradient (RCG) algorithm has been proposed in [48] for the Riemannian manifold to solve the nonlinear least-squares (NLS) problem and has shown a satisfactory performance. Although we have not introduced any improvements on this work, a brief introduction to the RCG procedure will be given to ensure the completeness of this paper.

A linear RCG method mainly consists of two steps: First, the search point and search direction are updated on the basis of the following update formula,

$${\mathbf{x}}^{k+1}={\mathcal{R}}_{{\mathbf{x}}^k}\left({\alpha }_k·{\mathbf{d}}^k\right)$$

(35)

$${\mathbf{d}}^{k+1}=-\mathrm{grad}f\left({\mathbf{x}}^{k+1}\right)+{\beta }_{k+1}{\mathcal{T}}_{{\mathbf{x}}^{k+1}\leftarrow {\mathbf{x}}^k}\left({\mathbf{d}}^k\right)$$

(36)

where $\left\{{\mathbf{d}}^k\right\}$ is the gradient-related sequence, denoting the search direction. ${\alpha }_k$ is the step length and its selection is crucial for ensuring convergence. A common scheme is the Armijo backtracking procedure [49]. ${\beta }_k$ can be chosen as the Fletcher–Reeves or Polak–Ribière parameter, where the Polak–Ribière parameter is defined as

$${\beta }_k=\frac{{\left(grad\xi \left({\mathbf{x}}^k\right)\right)}^{\mathrm{T}}\left[grad\xi \left({\mathbf{x}}^k\right)-grad\xi \left({\mathbf{x}}^{k-1}\right)\right]}{{\left(grad\xi \left({\mathbf{x}}^{k-1}\right)\right)}^{\mathrm{T}}grad\xi \left({\mathbf{x}}^{k-1}\right)}$$

(37)

Readers can find more details in [43].

The pseudocode of the waveform design for target information maximization based on CCMRCG is summarized in Algorithm 1, and the relevant procedures are visualized in Figure 4.

Algorithm 1 Riemannian Conjugate Gradient over Complex Circle Manifold (CCMRCG) for Target Information Maximization

Input: The cost function $\xi \left(\mathbf{x}\right)$, initial sequence ${\mathbf{x}}_0$ and a pre-defined threshold value $ϵ$.

Output: An optimal designed sequence ${\mathbf{x}}^{†}$ over the complex circle manifold ${\mathcal{M}}_{\mathbf{x}}$.  1. Set $k=0$.  2. $k=k+1$.  3. Calculate step size ${\alpha }_k$.  4. Update ${\mathbf{x}}^{k+1}$ according to (35).  5. Calculate $grad\xi \left({\mathbf{x}}^{k+1}\right)$ according to (31).  6. Transport ${\mathbf{d}}_k$ to $T_{{\mathbf{x}}^{k+1}}\mathcal{M}$ using (36).  7. Choose ${\beta }_k$ according to (37).  if  $\left|\xi \left({\mathbf{x}}_{k+1}\right)-\xi \left({\mathbf{x}}_k\right)\right|<ϵ$  then   Output ${\mathbf{x}}^{†}={\mathbf{x}}^k$.  else   Return to step 2.  end if

3.3.3. Convergence and Computation Complexity Analysis

Convergence

To obtain conclusions regarding the convergence of the proposed algorithm, it is imperative to introduce a lemma pertaining to Riemannian manifolds.

Lemma 2.

Let $\left\{{\mathbf{x}}_k\right\}$ be an infinite sequence of iterates generated using line-search methods on the manifold. Assume that the level set $\mathcal{L}=\{\mathbf{x}\in \mathcal{M}:\xi \left(\mathbf{x}\right)\le \xi \left({\mathbf{x}}_0\right)\}$ is compact. Then, ${lim}_{k\to \infty }\parallel grad\xi \left(\mathbf{x}\right)\parallel =0$ [43].

According to Lemma 2, considering the compactness of complex circle manifold ${\mathcal{M}}_{\mathbf{x}}$ itself, the sequence $\left\{{\mathbf{x}}_k\right\}$ obtained through line-search methods on ${\mathcal{M}}_{\mathbf{x}}$ is also compact and always converges to a stationary point. However, it cannot be guaranteed that the optimized result in one trial is globally optimal due to RCG’s reliance solely on first-order gradient information and its inherent difficulty in circumventing the influence of saddle points (if existing).

Computation Complexity

The primary computational complexity of the proposed method is directly proportional to both the number of iterations, denoted as I, and the number of elements in $\mathbf{s}$. The computational complexity of each iteration will be briefly demonstrated first.

• Termination Conditions: The dominant computational cost for (25) derives from ${\mathbf{x}}^{\mathrm{H}}\mathrm{\Sigma }\mathbf{x}$, which requires a numerical cost of $\mathcal{O}\left(9N^2\right)$. The term $\Re \left({\mathbf{q}}^{\mathrm{H}}\mathbf{x}\right)$ involves the cost of $\mathcal{O}\left(3N\right)$. Therefore, the total cost of ${\xi }_{\mathbf{x}}$ is $\mathcal{O}\left(9N^2\right)$.
• Riemannian Gradient: The computation of the Riemannian gradient involves two aspects. First, the computational cost derives from computing (32) with $\mathcal{O}\left(9N^2\right)$. Second, The computational cost of the project operator is $\mathcal{O}\left(3N\right)$.
• Retraction Operator: The computational cost of the retraction operator is $\mathcal{O}\left(3N\right)$.
• Vector Transport Operator: The computational cost of the vector transport operator is $\mathcal{O}\left(3N\right)$.

In summary, the total computational cost of the proposed method is $\mathcal{O}\left(I{N}^2\right)$. This is a general complexity result for using first-order algorithms over CCM with objective function having inhomogeneous quadratic form. As it circumvents the need for matrix inversion, its computational complexity is significantly diminished. In order to facilitate comparative analysis, we have compiled a concise overview of the computational complexities associated with SDR [36], ADPM [39], NCADMM [38] and CAN [29] in

Table 1. Overall computational complexities, where I is the required total iteration number, $N_r$ denoting the number of randomization trials in SDR.

Method	Computational Complexity
MTIC	$\mathcal{O}\left(N^3\right)$
SDR	$\mathcal{O}\left(N^{3.5}\right)+\mathcal{O}\left(N_r{N}^2\right)$
ADPM	$\mathcal{O}\left(N^3\right)+\mathcal{O}\left(I{N}^2\right)$
NCADMM	$\mathcal{O}\left(N^3\right)+\mathcal{O}\left(I{N}^2\right)$
CAN	$\mathcal{O}(INlogN)$
CCMRCG	$\mathcal{O}\left(I{N}^2\right)$

.

4. Numerical Results

Numerical simulations were performed to validate the feasibility and effectiveness of the proposed method. Performance analysis was carried out from two perspectives. Firstly, by fixing $\eta$, we evaluated the trade-off performance between the autocorrelation and target information acquisition through the ISL, PSL, and spectra similarity to the optimal waveform. MTIC [20], SDR [36], and CAN [29] were regarded as the subjects of comparison. In the second part, we analyzed the impact of different optimization algorithms and $\eta$ on these methods. To this end, ADPM [39] and NCADMM [38] are included here for comparison, and 500 Monte Carlo trials were performed to obtain an average performance assessment.

4.1. Configurations

For the sake of notational simplicity, the normalized frequency within the range of 0 to 1 Hz was considered. Unless explicitly specified, the code length of the sequence was set to $N=128$, and a random phase sequence was employed for initialization. The target impulse response was a zero-mean complex Gaussian process with the covariance matrix ${\mathbf{I}}_M$, and 1000 random samples were generated to compute ${\mathbf{R}}_g$ for verifying the target information acquisition. For clarity, in the subsequent content, the proposed method optimized on CCM is abbreviated as CCMRCG, the waveform designed using the method in [20] is abbreviated as MTIC, and ${\mathbf{s}}_0$ calculated from (8) is called the OPT waveform.

Although the SDR algorithm is a classical approach to solving the CMC problem, a further detailed explanation may be needed for ${\mathcal{P}}_5$ due to its inhomogeneous quadratic form. A certain reformulation of the objective function was implemented as below [50]:

$${\mathcal{P}}_{\mathrm{SDR}}:\left\{\begin{array}{cc}\hfill \underset{x}{min}& tr\left(\mathbf{Q}\mathbf{Y}\right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \mathbf{Y}=\mathbf{y}{\mathbf{y}}^{\mathrm{H}},diag\left(\mathbf{Y}\right)=\mathbf{1}\hfill \\ \hfill & \mathbf{y}={\left[{\mathbf{x}}^{\mathrm{T}},t\right]}^{\mathrm{T}},\mathbf{Y}⪰0\hfill \end{array}\right.$$

(38)

where

$$\mathbf{Q}=\left[\begin{array}{cc}\mathrm{\Sigma },& \mathbf{q}\\ {\mathbf{q}}^{\mathrm{H}},& 0\end{array}\right]$$

Remark 3.

Note that ${\mathcal{P}}_{\mathrm{SDR}}$ and ${\mathcal{P}}_5$ share the same optimal value. If ${\mathbf{y}}^{†}={\left[{\left({\mathbf{x}}^{\mathrm{T}}\right)}^{†},t^{†}\right]}^{\mathrm{T}}$ is optimal for ${\mathcal{P}}_{\mathrm{SDR}}$, then ${\mathbf{x}}^{†}/t^{†}$ is optimal for ${\mathcal{P}}_5$.

When excluding the rank one constraint, ${\mathcal{P}}_{\mathrm{SDR}}$ transforms into a convex problem that can be solved with the CVX toolbox in MATLAB, and then the rank one decomposition [51] of the output is subsequently executed again to obtain the optimized sequence. In this paper, Gaussian randomization steps [36] were repeated 2000 times, and the randomized feasible solution that resulted in the minimum objective function value was selected as the approximate solution.

The multiplier vector ${\mathit{\lambda }}^0=\mathbf{0}$, ${ϵ}_{\mathrm{abs}}={10}^{-6}$, ${ϵ}_{\mathrm{rel}}={10}^{-7}$, ${\delta }_{1,c}=0.9995$, ${\delta }_{2,c}=1.005$, and $\nu ={10}^4$ were used for ADPM (see [39] for more detail). A fixed $\varrho =0.8$ was used for NCADMM. The maximum number of iterations of both ADPM and NCADMM was set to 2000.

To quantify the resemblance of spectra, we introduce cosine similarity as an indicator to assess the degree of similarity between the spectrum of the designed waveform and the optimal spectrum, which is defined as

$${\rm Y}(\mathbf{x},\mathbf{z})=\frac{〈\mathbf{x},\mathbf{z}〉}{\parallel \mathbf{x}\parallel ·\parallel \mathbf{z}\parallel }$$

(39)

Finally, all the simulations were performed, and the running time was recorded using MATLAB on a standard laptop with an Intel Core i9-12900H and 32 GB of RAM.

4.2. Performance Analysis

The autocorrelation function and spectra of the devised waveforms are present in Figure 5 and Figure 6, respectively, when $\eta =0.55$. Compared with the OPT waveform, the waveform based on CCMRCG shows improved performance in terms of autocorrelation sidelobes in spite of the sacrifice of spectrum match performance. Additionally, the CCMRCG-based waveform can produce a flatter spectrum outside of the OPT spectral peak, achieving a wider bandwidth and higher resolution. In contrast, while SDR can attain a waveform displaying superior spectral matching, this advantage is accompanied by a trade-off in terms of compromised sidelobe performance. Simultaneously, the waveform devised using MTIC exhibits a comparable manifestation to that of CCMRCG, which distinguishes itself through its non-constant modulus envelope.

The cosine similarity in (39) was employed to quantitatively assess the correlation properties and spectra resemblance among distinct designed waveforms in

Table 2. Evaluation indicator with $\eta =0.55$.

Method	ISL (dB)	PSL (dB)	$\mathit{{\rm Y}}(\mathit{x},{\mathit{x}}_{\mathbf{opt}})$
OPT	61.45	21.05	/
MTIC	40.99	12.12	0.0910
CAN	30.23	7.11	0.0914
SDR	53.64	17.21	0.6581
CCMRCG	39.91	11.35	0.2500

. It is shown that CCMRCG has a commendable capacity to achieve equilibrium.

Figure 7 shows the normalized ambiguity function results of various methods. Evidently, with the exception of the OPT waveform, the waveforms demonstrate a spike shape with different degrees, indicating a good ability for estimation but limited Doppler tolerance. Thereby, CCMRCG has the potential to enhance the Doppler tolerance to a certain extent through the utilization of diverse initial sequences, akin to CAN. On the whole, however, the proposed algorithm is suitable for designing waveforms intended for estimation tasks instead of searching tasks, which typically necessitate higher Doppler tolerance.

For the convergence analysis, Figure 8 provides the cost function value versus the number of iterations, which exhibits a monotonically decreasing trend as anticipated and intuitive linear convergence.

Next, we conducted an analysis of the impact of different Pareto weights on algorithm performance. In order to eliminate any potential influence from random target impulse response, we have performed 500 Monte Carlo trials and calculated the corresponding average as an evaluation metric.

Figure 9 presents curves depicting different evaluation metrics in relation to $\eta$. It is evident that when $\eta$ is small, the algorithm mainly focuses on optimizing the ISL of the waveform. As $\eta$ increases, there is a gradual shift towards attaching importance to target information. When $\eta >0.75$, the cost function value of SDR is lower than that of CMMRCG. The reason is that SDR relaxes the rank-one constraint and uses the Gaussian randomization technique for exploring solutions within the feasible solution space. The higher the $\eta$ is, the closer to ${\mathbf{s}}_0$ the designed sequence is, resulting in the reduction in the reliance on the ISL. Additionally, it seems that ADPM and NCADMM have no pronounced inclination towards target information with different $\eta$ compared to CCMRCG and SDR. This can be attributed to the modification of the update order of ADPM and NCADMM, which may lead to a bias towards maintaining the CMC rather than optimizing the objective function before achieving full convergence. This implies that the obtained sequence may not inherently contain updates pertaining to the target information from the objective function. This discrepancy also suggests that achieving convergence to a stationary point for these aforementioned methods may not necessarily be guaranteed under the condition of 2000 maximum iterations and the selected penalty coefficient. The presence of outliers on the curve of ADPM and NCADMM at $\eta =0.5$ and $\eta =0.75$ further substantiates our statements. Specially, the parameter values associated with the penalty coefficient need to be carefully chosen. However, as the primary focus of this paper does not revolve around ADPM and NCADMM, an extensive analysis was not conducted in this work; instead, they will be further discussed in ongoing research endeavors.

Finally, we performed a statistical analysis on the mean computation CPU time of 500 Monte Carlo trials, as shown in Figure 10. CCMRCG shows a similar complexity to CAN, implying its potential applicability for virtually any practically relevant values of N. However, the computational burden of SDR, NCADMM and ADPM is exceedingly intractable in most practical scenarios.

5. Conclusions

This work investigated the trade-off between waveform autocorrelation performance and target information acquisition performance when designing phase-coded sequences under constant modulus constraints. We addressed this challenge by converting multiple objective functions into a single objective optimization problem using weighted summation. Additionally, we exploited the geometric structure of the complex circle manifold formed by the constant modulus constraint to iteratively solve the problem using the Riemannian conjugate gradient method. Importantly, this approach eliminates the need for matrix inversion operations during iteration, resulting in reduced computational complexity for solving optimization problems. Numerical simulation results demonstrate that the proposed algorithm effectively achieves a balance between target information acquisition performance and waveform autocorrelation performance.

It is important to acknowledge that we have consistently made assumptions about the underlying conditions of the target information. However, practical applications often present challenges in accurately acquiring prior information. Therefore, future research should focus on addressing concerns regarding robustness by improving the capability to extract target information under imprecise prior conditions. Additionally, simultaneously achieving real-time optimization of the waveform remains a significant challenge.