Algorithm for Designing Waveforms Similar to Linear Frequency Modulation Using Polyphase-Coded Frequency Modulation

Abstract

Linear frequency modulation (LFM) waveforms have been widely adopted due to their excellent performance characteristics, such as good Doppler tolerance and ease of physical implementation. However, LFM waveforms suffer from high autocorrelation sidelobes (ACSLs) and limited design flexibility. Phase-coded frequency modulation (PCFM) waveforms can be used to design waveforms similar to LFM, offering greater design flexibility to optimize ACSLs. However, it has been found that the initial PCFM waveform experiences spectral expansion during the ACSL optimization process, which reduces its similarity to LFM. Therefore, this article jointly optimizes the ACSLs and spectrum of the initial PCFM waveform, establishes an optimized mathematical model, and then solves it using the heavy-ball gradient descent algorithm. Numerical experiments indicate that the proposed method effectively addresses the problem of waveform similarity degradation caused by spectral expansion while reducing waveform ACSLs. At the same time, a balance between reducing waveform ACSLs and preserving waveform similarity can be achieved by adjusting the parameters.

1. Introduction

The diversity of signal structures, radar waveforms and design methods continues to evolve (see, e.g., [1,2,3,4]). However, the initially proposed waveform using frequency modulation (FM), particularly linear frequency modulation (LFM) [5,6,7,8], remains a primary waveform used in many radar systems today. In fact, the simple but renowned LFM is a standard performance benchmark, widely employed because of its Doppler tolerance and ease of physical implementation. LFM also boasts an extremely wide bandwidth and employs stretch processing at the receiver [9,10]. However, a key limitation of LFM lies in its high sidelobes [11,12,13].

A common approach to reducing LFM sidelobes is through windowing, which broadens the mainlobe (degrading range resolution) because LFM amplitude weighting produces a “rounded off” spectral content rather than a relatively flat LFM spectrum [3]. Moreover, high-power transmitters often operate in saturation and need constant amplitude waveforms, which makes windowing unsuitable [14]. Numerous alternative waveforms with lower sidelobes have been proposed, such as nonlinear FM (NLFM) and phase-coded (PC) waveforms [4,8,15,16]. However, physical implementation and practical processing face many challenges. For example, polyphase-coded waveforms can cause distortion in the transmitter, specifically the abrupt transitions between adjacent chips in a code corresponding to out-of-band spectral content that cannot pass through the bandlimited transmitter [11]. Therefore, an ideal waveform should retain the desirable properties of LFM. In other words, the ideal waveform should exhibit high similarity to LFM and lower ACSLs. To meet these requirements, this paper designs a new waveform similar to LFM with lower sidelobes, based on the phase-coded frequency modulation (PCFM) waveform.

Adapting to the characteristics of radar systems, Blunt improved continuous phase modulation (CPM), commonly applied in the communication field, to obtain the PCFM waveform [17]. PCFM efficiently merges the benefits of PC and FM waveforms, offering greater design flexibility for reducing sidelobes [18]. As demonstrated in [1,2,19], segmented greedy search in the PCFM parameter space has proven to produced effective outcomes. However, its optimization speed is relatively slow because it searches across the entire solution space, hindering real-time modifications of the PCFM waveform with large bandwidths and time spans. To overcome these limitations, the continuous nature of PCFM waveform design will be employed to develop a gradient-based optimization method.

Gradient descent optimization algorithms have been employed in previous PCFM waveform designs to minimize sidelobe levels [20,21], but these approaches primarily focus on reducing sidelobe magnitude without considering the associated issues and solutions. In this article, we address the issue of PCFM sidelobe optimization potentially causing spectral expansion, which can lead to the designed waveform no longer resembling an LFM waveform. We incorporate the frequency template error [10,19,22] and the generalized integrated sidelobe ratio [21,23,24] as optimization objectives. Considering a matched filter receiver, we formulate a mathematical model for designing a new waveform. Due to the simplicity and effectiveness, the heavy-ball gradient descent algorithm has been widely applied in many waveform design scenarios [25,26,27,28,29,30]. In the later part of this article, we employ the heavy-ball gradient descent algorithm to accelerate the convergence speed of the waveform optimization process.

The subsequent sections of this article are as follows. Section 2 introduces the definition and generation process of PCFM, elaborating on the issues of low sidelobes and similarity in PCFM waveform design. Subsequently, a mathematical model of the problem, namely the problem formulation, is established. Section 3 focuses on optimizing and solving the problem proposed in Section 2. First, the problem formulation is discretized, and then the derivative of the discretized expression is obtained to outline the process of heavy-ball gradient descent process. Section 4 verifies the effectiveness of the proposed method through numerical simulation experiments, analyzes the influence of different parameters on low sidelobe performance and similarity, and finally proves the superiority of the proposed algorithm through comparative experiments with other algorithms. Finally, the conclusions drawn in this article are summarized in Section 5.

2. Problem Formulation

An LFM signal is one of the earliest studied and widely used pulse compression signal types. An LFM signal specifically refers to those obtained through nonlinear phase modulation or linear frequency modulation, offering a large time-bandwidth product. An LFM signal can be expressed as

$$u\left(t\right)=A\mathrm{rect}\left({\displaystyle \frac{t}{T}}\right)\exp \left(j\pi k{t}^2\right)\exp \left(j2\pi f_{c}t\right)$$

(1)

where A represents the signal amplitude; rect(t) is the unit rectangular window function; T is the pulse width; B is the signal bandwidth; k is the modulation slope, defined as $k=B/T$; and $f_c$ refers to the carrier frequency.

The phase function is a key determinant of waveform performance, and its form critically influences the specific waveform type. For the linear frequency-modulated waveform discussed, the phase function is a quadratic function of t, which limits the degrees of freedom in the design of waveform optimization. In contrast, phase-coded waveforms alter their phase at the onset of each sub-pulse while maintaining a constant phase within, thereby introducing more design flexibility in waveform optimization. However, this approach leads to lower spectral efficiency and increased energy loss during transmission.

The PCFM signal structure is derived from the continuous phase modulation (CPM) framework commonly used in the communications field [31]. PCFM achieves continuous phase modulation by incorporating smooth phase jumps, combining the benefits of both frequency-modulated and phase-coded signals. It exhibits lower spectral side lobes and higher sidelobe attenuation compared to conventional approaches.

The generation of PCFM waveforms is shown in Figure 1, where an uninterrupted sequence of N pulses, each with a duration of $T_p$, is formed, resulting in a total duration of the pulse sequence of $T=N{T}_c$. The weighted coefficient for the nth pulse is denoted as ${\alpha }_n$, which represents the phase difference between two consecutive phase-coded chips. This difference is defined as

$${\alpha }_n=\psi \left({\tilde{\alpha }}_n\right)=\left\{\begin{array}{cc}{\tilde{\alpha }}_n,\hfill & |{\tilde{\alpha }}_n|⩽\pi \hfill \\ {\tilde{\alpha }}_n-2\pi sgn\left({\tilde{\alpha }}_n\right),\hfill & |{\tilde{\alpha }}_n|>\pi \hfill \end{array}\right.$$

(2)

where

$${\tilde{\alpha }}_n={\theta }_n-{\theta }_{n-1},n=1,\cdots N$$

(3)

where $\mathrm{sgn}(·)$ refers refers to the sign function and ${\theta }_n$ is the phase value of the nth chip in the $N+1$ length polyphase code. The expression of the PCFM waveform can be written as

$$\begin{array}{cc}\hfill s(t;\mathbf{x})& =exp\left\{j\left({\int }_0^{t}g\left(\tau \right)\ast \left[\sum _{n=1}^N{\alpha }_n\delta (t-(n-1)T_p)\right]d\tau \right)\right\}\hfill \\ \hfill & =exp\left(j\varphi \right(t;\mathbf{x}\left)\right)\hfill \end{array}$$

(4)

where $g\left(\tau \right)$ is a time-domain pulse-shaping filter function and $\delta \left(t\right)$ is the unit pulse function. The generation of the PCFM waveform can be viewed as the weighted sum of N phase-difference-encoded pulses, where each pulse has a duration of $T_p$. This weighted pulse sequence is then convolved with the function $g\left(\tau \right)$, which has support in the interval $[0,T_p]$, and is integrated to obtain the continuous phase function $\varphi (t;\mathbf{x})$ over time.

A discrete set of basic parameters in PCFM offers a convenient framework for waveform optimization, as initially explored in [1,2,19]. By performing convolution and integration, the phase element in Equation (4) is correspondingly denoted as

$$\varphi (t;\mathbf{x})=\sum _{n=1}^N{\alpha }_n{b}_n\left(t\right)$$

(5)

where the basic function $b_n\left(t\right)$ is denoted as

$$b_n\left(t\right)={\int }_0^{t}g(\tau -(n-1)T_p)d\tau$$

(6)

where $b_n\left(t\right)$ is the result of integrating the function $g\left(\tau \right)$ over integer multiples of the delay. The nth basic function corresponds to a sloping function with respect to the delay

$$b_n\left(t\right)=\left\{\begin{array}{c}\hfill 0,0\le t\le \left(n-1\right)T_p\hfill \\ \hfill \left(t-\left(n-1\right)T_p\right)/T_p,\left(n-1\right)T_p\le t\le n{T}_p\hfill \\ \hfill 1,n{T}_p\le t\le N{T}_p\hfill \end{array}\right.$$

(7)

In contrast to PC waveforms, the phase change vector $\mathbf{x}$ does not have a physical meaning in PCFM waveforms. Consequently, optimizing PCFM waveforms necessitates working with their discrete-time waveform sequences. The length M of the discrete PCFM sequence, the “oversampling factor” K, and the oversampling factor L, together with their corresponding ranges, are introduced in the original PCFM formula in order to produce a specific sequence. The oversampling factor L refers to dividing an original coding into L sections over the pulse duration $T_p$ of the pulse-shaping filter, resulting in a total of N coding elements within a pulse, where $N=L\left(BT\right)$. K denotes the number of sampling points within each sub-pulse $T_p$, which correlates closely with the 3dB bandwidth B. The choice of K depends on the desired spectral roll-off rate of the waveform; typically, selecting 2 or 3 is adequate to effectively mitigate the inevitable aliasing. Consequently, the length of the PCFM sequence can be calculated as $M=KN/L=K\left(BT\right)$. Therefore, the discretized PCFM waveform can be represented as

$$\mathbf{s}=exp\left(j\mathbf{Bx}\right)$$

(8)

where $\mathbf{B}$ is an $M\times N$ matrix composed of the M-sampled length versions of N basic functions and can be denoted as

$$\mathbf{B}=\left[\begin{array}{cccc}{b}_1\left(1\right)& b_2\left(1\right)& \cdots & b_N\left(1\right)\\ b_1\left(2\right)& b_2\left(2\right)& \cdots & b_N\left(2\right)\\ ⋮& ⋮& \ddots & ⋮\\ b_1\left(M\right)& b_2\left(M\right)& \cdots & b_N\left(M\right)\end{array}\right]$$

(9)

The autocorrelation function of the PCFM waveform $s\left(t\right)$ is

$$r_a\left(\tau \right)={\int }_{-\infty }^{+\infty }s\left(t\right)s^{*}\left(t+\tau \right)dt$$

(10)

where ${(·)}^{*}$ denotes the complex conjugate.

We define the maximum cross-correlation coefficient between the optimized PCFM waveform $s\left(t\right)$ and the initial LFM waveform $s_0\left(t\right)$ as a measure of their similarity, which is expressed as

$$I_{sim}=max\left\{\left|r_c\left(\tau \right)\right|\right\}=max\left\{\left|{\int }_{-\infty }^{+\infty }s\left(t\right)s_0^{*}\left(t+\tau \right)dt\right|\right\}$$

(11)

where max$\{·\}$ is the maximum operation; $|·|$ is the modulo operation; and $I_{sim}\in [0,1]$. The closer $I_{sim}$ is to 1, the higher the similarity between the waveforms; conversely, if $I_{sim}$ is not close to 1, it indicates lower similarity.

The two most well-known metrics for evaluating side-lobe performance are the integrated sidelobe level (ISL) ratio and the peak sidelobe (PSL) ratio. The ISL compares the total energy of the sidelobe region with that of the mainlobe, whereas the PSL assesses the peak amplitude of the maximum sidelobe in relation to the peak of the mainlobe. Their mathematical representations are as follows:

$$\mathrm{ISL}={\displaystyle \frac{{\int }_{\Delta t}^T{\left|r_a\left(\tau \right)\right|}^{2}d\tau }{{\int }_0^{\Delta t}{\left|r_a\left(\tau \right)\right|}^{2}d\tau }}$$

(12)

and

$$\mathrm{PSL}={\displaystyle \frac{max{\left\{{\left|r_a\left(\tau \right)\right|}^2\right\}}_{\Delta t}^T}{max{\left\{{\left|r_a\left(\tau \right)\right|}^2\right\}}_0^{\Delta t}}}={\displaystyle \frac{max{\left\{{\left|r_a\left(\tau \right)\right|}^2\right\}}_{\Delta t}^T}{{\left|r_a\left(0\right)\right|}^2}}$$

(13)

where the mainlobe values of the autocorrelation function lie within the range $\left[-\Delta t,\Delta t\right]$. The preliminary simplified expressions of the ISL and PSL, as given in Equations (12) and (13), are derived by fully utilizing the conjugate symmetry property of the autocorrelation function.

A lower ISL and PSL value generally indicates better performance of a radar waveform. By reducing the ISL, the overall sidelobe response of the waveform is decreased, as it lessens the average energy in the sidelobes. In contrast, lowering the PSL often results in flatter sidelobes, which helps alleviate the issue of weak targets being overwhelmed by stronger sidelobes.

In general, optimizing waveforms to reduce the PSL and ISL values is more straightforward for functions that are convex, linear, and continuous. Equations (12) and (13) do not possess these properties, especially Equation (13), because the operator $max\left\{·\right\}$ prevents differentiation in the traditional way. Therefore, we can rewrite Equation (13) as follows:

$$\mathrm{PSL}=\underset{p\to \infty }{lim}{\left({\displaystyle \frac{{\int }_{\Delta t}^T{\left|r_a\left(\tau \right)\right|}^{p}d\tau }{{\int }_0^{\Delta t}{\left|r_a\left(\tau \right)\right|}^{p}d\tau }}\right)}^{2/p}$$

(14)

Equation (15) utilizes the fact that the infinity norm is equal to the operator $max\left\{·\right\}$. Therefore, Equations (12) and (14) can be uniformly written as follows:

$$\mathrm{GISL}={\left({\displaystyle \frac{{\int }_{\Delta t}^T{\left|r_a\left(\tau \right)\right|}^{p}d\tau }{{\int }_0^{\Delta t}{\left|r_a\left(\tau \right)\right|}^{p}d\tau }}\right)}^{2/p}$$

(15)

where $2<p<\infty$. When $p=2$, the GISL metric in Equation (15) becomes the ISL metric in Equation (12). Similarly, when $p\to \infty$, the GISL metric in Equation (15) approaches the PSL metric in Equation (14).

To mitigate spectral spreading, a frequency-domain template defines the mean square error (MSE) between the specified and the measured waveform spectra. This MSE is quantified as the frequency template error (FTE) [19], which serves to further refine the optimization procedure. The aim is to decrease the PSL and curtail the expansion of the out-of-band spectrum, a necessity stemming from the interplay between autocorrelation and power spectral density. The FTE metric is delineated as follows:

$$\mathrm{FTE}={\displaystyle \frac{1}{f_H-f_L}}{\int }_{f_L}^{f_H}{\left|S\left(f,\mathbf{x}\right)-U\left(f\right)\right|}^{2}df$$

(16)

where $U\left(f\right)$ represents the spectral template and $S\left(f\right)$ denotes the spectrum of the waveform under examination. The integration bounds, $f_L$ and $f_H$, correspond to the minimum and maximum frequencies of demand, encompassing the out-of-band spectral roll-off. The spectral template is established based on the spectrum of the initial PCFM waveform, which is identical to the LFM waveform, to ensure that the spectral content does not diverge significantly from a predefined shape.

Therefore, the key to designing waveforms with high similarity and low sidelobes can be represented by the following mathematical model:

$$min{\left(\mathrm{FTE}\right)}^{\sigma }{\left(\mathrm{GISL}\right)}^{1-\sigma }$$

(17)

where $0\le \sigma \le 1$, which is a user parameter that determines the ratio between detection performance and similarity.

3. Problem Optimization

In this section, computational efficiency is improved by discretizing Equation (17) during the optimization process. To address this problem, an optimization algorithm based on the heavy-ball gradient descent method is proposed.

3.1. Discretizing the Objective Function

To carry out numerical optimization of the objective function concerning the parameter vector $\mathbf{x}$ for a specified p, Equations (10) and (15) have to be discretized following the approach delineated in Equation (8), as expounded in Section 2. By doing so, the autocorrelation in Equation (10) is transformed into its discretized version:

$$r_a\left[l\right]=\sum _{m=1}^{M}s\left[m\right]s^{*}\left[m+l\right]=r_a^{*}\left[-l\right]$$

(18)

where $s\left[m\right]=0$ when $m\le 0$ or $m>M$. Exploiting the Fourier relationship between autocorrelation and spectral density, the vectorized version of Equation (18) is expressed as follows:

$$\mathbf{r}={\mathbf{F}}^{\mathrm{H}}\left[\left(\mathbf{F}\overline{\mathbf{s}}\right)\odot {\left(\mathbf{F}\overline{\mathbf{s}}\right)}^{*}\right]$$

(19)

where $\mathbf{F}$ is the $(2M-1)\times (2M-1)$ discrete Fourier transform (DFT) matrix and ${\mathbf{F}}^{\mathrm{H}}$ is its inverse DFT matrix. ⊙ denotes the Hadamard product. $\overline{\mathbf{s}}$ is the zero-padding form of the discrete waveform $\mathbf{s}$, which has the same length$-M$ sampling with the corresponding autocorrelation function in Equation (18):

$$\overline{\mathbf{s}}={\left[{\mathbf{s}}^{\mathrm{T}}{0}_{1\times \left(M-1\right)}\right]}^{\mathrm{T}}$$

(20)

According to Equation (19), the discrete form of the GISL can be expressed as

$$J_p={\displaystyle \frac{{\parallel {\mathbf{w}}_{\mathrm{SL}}\odot \mathbf{r}\parallel }_p^2}{{\parallel {\mathbf{w}}_{\mathrm{ML}}\odot \mathbf{r}\parallel }_p^2}}$$

(21)

where ${\parallel ·\parallel }_p$ denotes the discrete $p-$norm, while ${\mathbf{w}}_{\mathrm{SL}}$ and ${\mathbf{w}}_{\mathrm{ML}}$ are vectors of length $(2M-1)$, consisting solely of 0 and 1, which isolate the sidelobe and mainlobe regions of $\mathbf{r}$ respectively. The mainlobe segment of $\mathbf{r}$ is isolated by the non-zero central elements of ${\mathbf{w}}_{\mathrm{ML}}$ and is precisely defined by the 3-dB bandwidth oversampling factor K, resulting in the central $(2K-1)$ samples of $\mathbf{r}$ which are classified as the mainlobe. The authors of [21,32] demonstrated that this method effectively manages the desired bandwidth in the optimization process.

The discrete forms of $S(f,\mathbf{x})$ and $U\left(f\right)$ in Equation (16) can be obtained through the DFT of their time-domain discrete forms:

$$\begin{array}{c}\mathbf{S}=\mathbf{Fs}\\ \mathbf{U}=\mathbf{Fu}\end{array}$$

(22)

Since the interesting frequencies correspond to the bandwidths of $S\left(f\right)$ and $U\left(f\right)$, both of which are B, the integration interval for Equation (16) falls within B. And due to the fact that the square of vector modulus can be expressed as the product of a vector and its conjugate symmetric vector, the discrete form of the FTE can be expressed as

$$\begin{array}{cc}\hfill D& ={\displaystyle \frac{1}{f_H-f_L}}\sum _{m=1}^M{|\mathbf{Fs}-\mathbf{Fu}|}^2\hfill \\ \hfill & ={\displaystyle \frac{1}{f_H-f_L}}{(\mathbf{Fs}-\mathbf{Fu})}^{\mathrm{H}}(\mathbf{Fs}-\mathbf{Fu})\hfill \end{array}$$

(23)

According to Equations (21) and (23), the discrete form of the objective function in Equation (17) can be expressed as

$$Y_p=D^{\sigma }{J}_p^{(1-\sigma )}$$

(24)

3.2. Gradient Descent

Calculate the gradient of the objective function’s discrete representation about the vector $\mathbf{x}$, as presented in Equation (24). The gradient operator is a vector of length$-N$:

$${\nabla }_{\mathbf{x}}={\left[{\displaystyle \frac{\partial }{\partial {\alpha }_1}},{\displaystyle \frac{\partial }{\partial {\alpha }_2}},\cdots ,{\displaystyle \frac{\partial }{\partial {\alpha }_N}}\right]}^{\mathrm{T}}$$

(25)

By applying Equation (25) to Equation (24), as shown in the derivation process in Appendix A, the gradient of $Y_p$ can be obtained as

$$\begin{array}{cc}\hfill {\nabla }_{\mathbf{x}}{Y}_p=& {\left[{\displaystyle \frac{\partial Y_p}{\partial {\alpha }_1}},{\displaystyle \frac{\partial Y_p}{\partial {\alpha }_2}},\cdots ,{\displaystyle \frac{\partial Y_p}{\partial {\alpha }_N}}\right]}^{\mathrm{T}}\hfill \\ \hfill =& {\displaystyle \frac{-2\sigma }{f_H-f_L}}{D}^{\sigma -1}{J}_p^{(1-\sigma )}{\mathbf{B}}^{\mathrm{T}}\times \Im \left\{\left({\left(\mathbf{s}-\mathbf{u}\right)}^{*}\odot \mathbf{s}\right)\right\}+4D^{\sigma }(1-\sigma )J_p^{(1-\sigma )}{\overline{\mathbf{B}}}^{\mathrm{T}}\hfill \\ \hfill & \times \Im \left\{{\overline{\mathbf{s}}}^{*}\odot \left({\mathbf{F}}^{\mathrm{H}}\left[\mathbf{F}\left(\left[{\displaystyle \frac{{\mathbf{w}}_{\mathrm{SL}}}{{\mathbf{w}}_{\mathrm{SL}}^{\mathrm{T}}{\left|\mathbf{r}\right|}^p}}-{\displaystyle \frac{{\mathbf{w}}_{\mathrm{ML}}}{{\mathbf{w}}_{\mathrm{ML}}^{\mathrm{T}}{\left|\mathbf{r}\right|}^p}}\right]\odot {\left|\mathbf{r}\right|}^{(p-2)}\odot \mathbf{r}\right)\odot \left(\mathbf{F}\overline{\mathbf{s}}\right)\right]\right)\right\}\hfill \end{array}$$

(26)

where

$$\overline{\mathbf{B}}=\left[{\mathbf{B}}^{\mathrm{T}}{\mathbf{0}}_{N\times \left(M-1\right)}\right]$$

(27)

The operation $\Im \left\{·\right\}$ extracts the imaginary component of each element, while $\left|·\right|$ computes the magnitude of the elements within the vector. $\mathbf{B}$ contains N discrete basic functions as defined in Equation (8) for waveform construction. In Equation (27), $\overline{\mathbf{B}}$ is obtained by applying the same zero-padding method in Equation (20) to $\mathbf{B}$.

Generally, gradient-based methods function by iteratively adjusting the optimized parameters to minimize the objective function with each iteration. This arrangement can be expressed during the ith iteration as

$${\mathbf{x}}_i={\mathbf{x}}_{i-1}+{\mu }_i{\mathbf{q}}_i$$

(28)

where ${\mathbf{q}}_i$ represents the pursuit direction and ${\mu }_i$ denotes the step length for descent at the present. The pursuit direction ${\mathbf{q}}_i$ depends on the new gradient ${\nabla }_{\mathbf{x}}{Y}_p\left({\mathbf{x}}_{i-1}\right)$ and incorporates previous pursuit directions ${\mathbf{q}}_{i-1}$. The expression is as follows:

$${\mathbf{q}}_i=\left\{\begin{array}{cc}-{\nabla }_{\mathbf{x}}{Y}_p\left({\mathbf{x}}_{i-1}\right)\hfill & i=0\hfill \\ -{\nabla }_{\mathbf{x}}{Y}_p\left({\mathbf{x}}_{i-1}\right)+\beta {\mathbf{q}}_{i-1}\hfill & i\ne 0\hfill \end{array}\right.$$

(29)

where the gradient is a function of ${\mathbf{s}}_{i-1}$ (${\mathbf{x}}_{i-1}$) derived from the last iteration.

The gradient descent method mentioned in Equation (29) refers to the heavy-ball gradient descent method [33] for which $0<\beta <1$. Compared to other gradient descent methods, the heavy-ball gradient descent algorithm suppresses sudden changes in the pursuit direction by relying on gradient “inertia”. It strikes a wonderful balance between fast convergence and simple computation. Algorithm 1 shows the various steps of the algorithm. If ${\mathbf{q}}_i$ does not align with the descent direction of ${\nabla }_{\mathbf{x}}{Y}_p\left({\mathbf{x}}_{i-1}\right)$ (as indicated in Step 6), then the pursuit direction is redefined as the new gradient (refer to Step 7), effectively disregarding previous gradients.

The new step length $\mu$ is ascertained using a straightforward backtracking method, incorporating a “adequate decrease” parameter c in Step 9. Additionally, to address the issue of the step size $\mu$ being too large or too small, we set a step size reduction parameter ${\rho }_{\mathrm{down}}$ slightly less than 1 in Step 10 and a step size increase parameter ${\rho }_{\mathrm{up}}$ slightly greater than 1 in Step 12. Steps 6 and 7 ensure descent and trigger Step 10 when necessary to ensure the optimization remains within the local minimum vicinity of the waveform initialization.

Algorithm 1: Gradient optimization of $Y_p$.
Step	Operation
1:	Input: $K,BT,L,N=L\left(BT\right),p,I,\beta ,\mu ,{\rho }_{\mathrm{up}},{\rho }_{\mathrm{down}},c,\sigma$
2:	Initialize: ${\mathbf{x}}_0,{\mathbf{q}}_0=0_{N\times 1},i=1$
3:	Repeat
4:	Evaluate: $Y_p\left({\mathbf{x}}_{i-1}\right)$ and ${\nabla }_{\mathbf{x}}{Y}_p\left({\mathbf{x}}_{i-1}\right)$ using Equations (24) and (26)
5:	${\mathbf{q}}_i=-{\nabla }_{\mathbf{x}}{Y}_p\left({\mathbf{x}}_{i-1}\right)+\beta {\mathbf{q}}_{i-1}$
6:	If ${\left({\nabla }_{\mathbf{x}}{Y}_p\left({\mathbf{x}}_{i-1}\right)\right)}^{\mathrm{T}}{\mathbf{q}}_i\ge 0$
7:	${\mathbf{q}}_i=-{\nabla }_{\mathbf{x}}{Y}_p\left({\mathbf{x}}_{i-1}\right)$
8:	End (If)
9:	While $Y_p\left({\mathbf{x}}_{i-1}+\mu {\mathbf{q}}_i\right)>Y_p\left({\mathbf{x}}_{i-1}\right)+c\mu {\left({\nabla }_{\mathbf{x}}{Y}_p\left({\mathbf{x}}_{i-1}\right)\right)}^{\mathrm{T}}{\mathbf{q}}_i$
10:	$\mu ={\rho }_{\mathrm{down}}\mu$
11:	End (While)
12:	$\mu ={\rho }_{\mathrm{up}}\mu$
13:	${\mathbf{x}}_i={\mathbf{x}}_{i-1}+\mu {\mathbf{q}}_i$
14:	Until i = I
15:	Output: x

4. Simulations and Performance Analysis

This section verifies the optimization effect of the proposed heavy-ball gradient descent algorithm on PCFM waveforms through numerical simulation experiments. The feasibility of the proposed heavy-ball gradient descent algorithm is validated through numerical simulations, analyzing the influence of parameters such as L and $\sigma$ on the sidelobe performance of PCFM waveforms and the similarity between PCFM and LFM waveforms.

4.1. Parameters Setting

The selection of waveform parameters and the setting of phase-difference codes have a significant impact on the final optimization results. The bandwidth $B=2$ MHz, carrier frequency $f_s=10$ GHz, and pulse width $T_c=20$ μs of LFM and PCFM waveforms are set. For the PCFM waveform, the main parameters are BT, K, and L. Evidently, a comprehensive exploration of the intricate interplay among the various parameters, implementation methods, and initial conditions is impractical. Therefore, we examine a subset of the design space to discern significant patterns. Specifically, we set $K=16$, which avoids spectral aliasing. According to $M=K\left(BT\right)$, M is set to 640. We set $L=1$ to ensure that the designed waveform has high similarity to LFM (the influence of L on waveform similarity will be discussed in detail later). According to $N=L\left(BT\right)$, N is set to 40.

The designed PCFM waveform needs to have high similarity to the initial LFM waveform, so it is necessary to select appropriate initial phase-difference codes ${\mathbf{x}}_0$. The authors of [34,35] studied P1, P2, P3, and P4 polyphase codes, which are step approximations of LFM pulse compression waveforms. Among them, the P4 code has the highest similarity to the LFM waveform. Here, we select the phase of the P4 code as the initial phase $\mathsf{\theta }=[{\theta }_0,{\theta }_1,\cdots ,{\theta }_N]$. The nth phase of the P4 code can be represented as

$${\theta }_n={\displaystyle \frac{\pi }{N}}{n}^2-\pi n,0\le n\le N$$

(30)

The initial phase changes ${\mathbf{x}}_0$ of the initial PCFM waveform can be represented as the changes between adjacent phases of the P4 code. The nth phase change ${\alpha }_n$ can be represented as:

$${\alpha }_n={\theta }_n-{\theta }_{n-1}={\displaystyle \frac{\pi }{N}}\left(2n-1\right)-\pi ,1\le n\le N$$

(31)

In the forthcoming results, setting $\beta =0.95$ based on experience provides a better balance between the inertia and response of the gradient for the objective function. To confine the optimization process to the local area surrounding the initialized waveform, the initial step size should be set to a small value $\mu ={10}^{-4}$. The increasing and decreasing step sizes should be set to ${\rho }_{\mathrm{up}}=1.01$ and ${\rho }_{\mathrm{down}}=0.9$ respectively, which effectively balances the fast convergence and simple computation of the algorithm. Ultimately, the sufficient decline parameter is set to $c={10}^{-2}$, and the maximum number of iterations is established at $I={10}^4$ based on empirical observations.

4.2. Results Analysis

Section 4.2.1, Section 4.2.2 and Section 4.2.3 analyze the impact of different parameters ($p,L,\sigma$) on the paper performance of the PSL and $I_{sim}$. In Section 4.2.4, a comparative analysis is conducted on the performance of different algorithms, demonstrating the superiority of the proposed algorithm.

4.2.1. Influence of p-Norm

From Equations (24) and (26), it can be observed that the p-norm only exists in $J_p$ of $Y_p$ and is independent of $D_p$. Therefore, $\sigma$ can be set to 0 when discussing the relationship between the p-norm and sidelobe performance. Indeed, the preliminary results appear to suggest that it can provides sufficient breadth for evaluating the GISL metric by setting the p-norm in the integer range from 2 to 20.

Figure 2 shows the normalized evolution curve of sidelobe performance and similarity with respect to the number of iterations, illustrating the normalized reduction of the PSL, ISL, and $I_{sim}$ relative to the initial value when $p=2,p=5,p=10$, and $p=20$. Each data point in Figure 2 represents the value at the new iteration count. Analyzing Figure 2a,b, it can be observed that with the increase in p, the convergence speed and degree of performance optimization of the PSL and ISL exhibit consistent trends. As p increases, the number of optimization iterations also increases. However, within 4000 iterations, the ISL, PSL, and $I_{sim}$ all converge to their optimal values. Figure 2a shows that the optimization effect of the ISL reaches its best when $p=5$. Figure 2b shows that the optimization effect of the PSL reaches its best when $p=20$. Figure 2b shows that the optimization effect of PSL reaches its best when $p=20$. However, Figure 2c shows that the similarity between the optimized PCFM and LFM waveforms is greater than 0.8 when p is only 2, which ensures that the designed PCFM is similar to LFM. Therefore, we will choose $p=2$ for the waveform design in the following steps.

4.2.2. Influence of L

In the previous discussion, we mentioned that the over-coding factor L was set to 1 to ensure waveform similarity. Here, we delve into the impact of L on sidelobe performance and waveform similarity.

Observing

Table 1. The PSL values of different waveforms under different over-coding factors.

	LFM (dB)	Initial PCFM (dB)	Optimized PCFM (dB)
$L=1$	−13.6011	−13.6016	−31.6202
$L=2$	−13.6011	−13.4988	−32.1821
$L=4$	−13.6011	−13.4733	−40.8512

, we can see that as L increases, the PSL of the initial PCFM remains constant, while the PSL of the optimized PCFM decreases. This is because the coding length $N=LBT$, and with each doubling of L, the code length N also doubles. Meanwhile, the codes of the intial PCFM are P4 codes whose encoding rule is fixed, so their PSL remains unchanged with the increase in code length. However, longer codes provide with greater freedom, which helps find better solutions, thus the PSL of the optimized PCFM decreases as L increases. Figure 3a shows the autocorrelation functions of LFM, the initial PCFM, and the optimized PCFM under different values of L. The relationship between the ISL value and L can be clearly observed in Figure 3a. Correspondingly, the bandwidth effectively becomes $LB$ due to the fact that the code element time is halved and the time-bandwidth product within a code element satisfies $B{T}_p=1$. As the distance resolution is inversely proportional to the bandwidth, increasing L improves the distance resolution. Figure 3b depicts a zoomed-in view of Figure 3a at a 3-dB bandwidth, illustrating that the optimization algorithm significantly enhances the distance resolution at lower values of L, while the improvement diminishes as L increases. However, the distance resolution is primarily determined by the signal bandwidth.

Observing

Table 2. The similarity $I_{sim}$ between different waveforms under different over-coding factors.

	${\mathit{I}}_{\mathit{sim}1}$ 1	${\mathit{I}}_{\mathit{sim}2}$ 2	${\mathit{I}}_{\mathit{sim}3}$ 3
$L=1$	0.9984	0.8369	0.8317
$L=2$	0.1905	0.2150	0.6552
$L=4$	0.1090	0.1055	0.4948

¹ The similarity between the initial PCFM and LFM. ² The similarity between the optimized PCFM and LFM. ³ The similarity between the initial PCFM and optimized PCFM.

, we can see that as L increases, the similarity $I_{sim1}$ decreases sharply and becomes very low. This is not difficult to understand because the signal’s pulse width remains fixed, and the increase in L affects the bandwidth, expanding it from B to $LB$. At this point, the bandwidths of the LFM signal and the PCFM signal are no longer the same, as shown in Figure 4. The “Reference Line” in Figure 4 represents the amplitude attenuation of the LFM signal within the bandwidth B. When measuring the changes in bandwidth of various waveforms, the bandwidth B of the LFM signal is used as a reference, and the amplitude attenuation value serves as a comparison. Based on the frequency modulation slope $k=B/T$, the frequency modulation slope of the PCFM waveform will also increase, which makes them no longer similar. Thus, considering waveform similarity, we set L to 1 in the subsequent simulations.

Meanwhile, in Table 2, we calculate the similarity $I_{sim3}$. The results show that while the optimization algorithm improves the sidelobe performance of the waveform, it also decreases the similarity of the waveforms. Figure 5 shows the changes in bandwidth of the PCFM waveform before and after optimization when $L=1$. We can observe that the optimization process improves sidelobe performance by expanding the spectral content, which inevitably leads to a decrease in similarity between the initial PCFM and the optimized PCFM. Meanwhile, from Table 1 and Table 2, we should also note that the decrease in waveform similarity and the improvement in sidelobe performance are determined by the algorithm, both of which are independent of L.

4.2.3. Influence of σ

Through the previous simulation analysis, we learned that while the algorithm optimizes sidelobe performance, it also leads to spectral expansion, thereby reducing waveform similarity. In the following, we introduce the FTE to enhance spectral containment and ensure waveform similarity.

When $\sigma$ in the objective function (Equation (24)) is set to 0, the frequency template error (FTE) is introduced. Next, we will analyze the impact of the FTE on waveform similarity and sidelobe performance. In Figure 6a, we can observe that as $\sigma$ increases, the similarity curve between the optimized PCFM and LFM waveforms decreases, then increases, and finally converges at $\sigma =0.3$. The similarity becomes equal to $I_{sim1}$ (the similarity between the initial PCFM and LFM) (as shown in Table 2) when $\sigma =0.3$. Meanwhile, in Figure 6b, we can observe that the sidelobe performance of the PCFM waveform remains unchanged after gradient optimization when $\sigma =0.3$, which has the same ISL and PSL values with the initial PCFM (as shown in Table 2). So, we conclude the following: when $\sigma \ge 0.3$, the FTE essentially controls the gradient optimization process of the objective function. The optimized PCFM waveform becomes the same as the initial PCFM, and the sidelobe performance of PCFM is improved very little. Therefore, the value of $\sigma$ should be set within the range from 0 to 0.3 to reduce the ISL and PSL while improving $I_{sim}$.

Figure 7 shows the changes in bandwidth of the optimized PCFM relative to the LFM under different $\sigma$. The “Reference Line” represents the amplitude attenuation when the LFM bandwidth is B. We can observe that the bandwidth of the optimized PCFM changes when the amplitude attenuation reaches the “Reference Line”, indicating the occurrence of spectral expansion. The closer the $\sigma$ approaches to 0, the more severe spectral expansion becomes. Conversely, as the $\sigma$ approaches 0.3, the spectral expansion becomes smaller. It can be seen that the PCFM bandwidth is the same as the LFM bandwidth when $\sigma =0.3$. And there is no spectral expansion, which indicates that FTE effectively suppresses the occurrence of spectral expansion during the optimization process.

4.2.4. Algorithm Comparison

Based on the previous analysis of p, L and $\sigma$, it is evident that to ensure high similarity between the designed PCFM waveform and LFM, the final parameters were set to $p=2$ and $L=1$. Additionally, to ensure that the designed PCFM waveform possesses lower sidelobes, $\sigma$ was set to 0. Therefore, with the parameters $p=2$, $L=1$ and $\sigma =0$, we conducted a comparative analysis of the performance achieved by different algorithms to demonstrate the superiority of the proposed algorithm.

The authors of [19] used the ISL, PSL, or FTE as the objective function and the greedy search algorithm (GSA) to traverse the $N\times N$ phase perturbation space to seek the optimal solution. The greedy search algorithm was improved by taking the optimal solution as the initial value and conducting the greedy search process again until it met the stopping condition, which is called the cyclic greedy search algorithm (CGSA). Figure 8 shows the autocorrelation function of optimized waveforms under different algorithms, and it can be observed that the gradient descent algorithm (GDA) exhibited the best optimization effect on decreasing sidelobes.

We calculated the values of the PSL and $I_{sim2}$ under different algorithms, as shown in

Table 3. Values of the PSL and $I_{sim2}$ under different algorithms.

	LFM	GSA	CGSA	GDA
PSL (dB)	−13.6011	−16.9854	−23.4733	−31.6202
$I_{sim2}$	1.0000	0.8453	0.5812	0.8371

. It is obvious that the GDA has not only has the lowest PSL but also has the same $I_{sim2}$ as the GSA. The proposed GDA can better suppress sidelobes while maintaining high similarity with LFM.

Section 3.2 indicates that the heavy-ball gradient descent algorithm suppresses sudden changes in the pursuit direction by relying on gradient “inertia”. The differences between the heavy-ball gradient descent algorithm and the ordinary gradient descent algorithm lie in whether Steps 9–12 in Table 1 are included. Steps 9–12 can adaptively adjust the step size when the gradient of descent is too large or too small. On the one hand, an excessive step size can lead to missing the optimal solution and unsatisfactory optimization results, as shown in Figure 9. On the other hand, a fixed step size leads to slow convergence speed and long computation time. The heavy-ball gradient descent algorithm can stop iterating after about 1000 iterations, but the ordinary gradient descent algorithm continues iterating until $I={10}^4$.

5. Conclusions

This paper analyzes the decrease in similarity between the PCFM waveform and the initial LFM waveform during the sidelobe optimization process and identifies it as a result of spectral expansion. We propose a mathematical optimization model that considers both the sidelobe optimization and similarity to LFM. We use the GISL and FTE as simultaneous objective functions in the optimization, controlling their relationship through parameters. The mathematical model is solved using the heavy-ball gradient descent method, which efficiently optimizes all waveform parameters. It can be executed using FFT operations and matrix/vector multiplication. The initial PCFM waveform is constructed based on LFM. The similarities between LFM and optimized PCFM, with and without the FTE in the objective function, are compared. The results show that the proposed method effectively mitigates the issue of spectral expansion.

Due to the heavy-ball gradient descent method used in this article, the choice of initial PCFM codes will affect the final similarity between LFM and the optimized PCFM. The P4 codes used in the article are the initial PCFM codes, but there may be better initial PCFM codes to improve the final similarity. Based on the results, there is a trade-off between sidelobe suppression and spectral containment and the value of $\sigma$ in the objective function determines the weight ratio. The limitation is that it cannot simultaneously achieve sidelobe suppression and spectral containment. Only one main optimization direction can be selected based on specific needs to determine the value of $\sigma$.

The exploration of mitigating spectral expansion in the optimization process of PCFM waveforms is still in its preliminary stage. On one hand, suppressing spectral expansion leads to a decrease in sidelobe optimization performance, so it is meaningful to decouple these two performance metrics and improve them simultaneously. On the other hand, this article only considers waveform design under a matched filtering system, while the design of PCFM waveforms for mismatched filtering systems is worth studying.