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你现在是无线通信方向的院士，我是大学老师。正在写一篇推导并改善CE-OFDMISAC中感知性能的文章...

Created on: January 8, 2026

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Question

CE_OFDM_ShiJingCheng (3).pdf

你现在是无线通信方向的院士，我是大学老师。正在写一篇推导并改善CE-OFDMISAC中感知性能的文章，准备投稿到IEEE TWC。附件是我么的呢初稿。请详细、仔细看看现在文章内容上面有没有写的有问题或者可以改善的地方。还有就是，如果需要删除一些内容，什么比较适合删除？我们先一起讨论下

Thought

Preparing critique in Chinese and reviewing details

I found the reference to Table II and other sections like "Finally, Sec. VII." Great! Now I need to provide a detailed critique in Chinese, covering major conceptual issues, math consistency, clarity, and missing information. I’ll ensure we avoid overstating academic credentials and correct the "cos h" typesetting. Also, I'll address notation duplication and assumptions in the model.

Reviewing assumptions and consistency details

In Assumption A, they use "s" with length N, but later refer to "Nd" in CE-OFDM. I should align these terms by defining Nd as the data subcarriers in the random symbol model. Also, the zero-padding vector "z_p" should be complex, though using real zeros is fine but unusual. I'll revise accordingly.

Clarifying pulse shaping, circulant properties, and PAPR issues

The matrix P is circulant with its first column as p, meaning the samples correspond to pulse offsets. However, pulse shaping is linear, not circular, so I need to address this approximation. For PAPR, row norms of circulant matrices are connected to pulse energy, and I’ll need to adjust the bound since P may not be unitary.

Clarifying PAPR calculations and pulse shaping assumptions

There's a subtle issue regarding constant modulus and the interdependence of xup entries. While they compute PAPR, I realize that row energy in pulse shaping could vary because, in linear convolution, rows may involve multiple symbols. This means the row energy might exceed 1, potentially up to (1+α).

Addressing pulse shaping and spectrum confusion

There seems to be confusion in their approach: using circulant matrix for pulse shaping but then treating it as linear in certain contexts like ACF and PAPR bounds. The folding criterion for the spectrum is unclear too, with the discrepancy between full and half-spectrum. Additionally, there's a mix-up between Nyquist pulse shaping being RRC or RC, and that requires clarification.

Clarifying pulse shaping and filter inconsistencies

The treatment of pulse shapes and filters is inconsistent: they refer to p(t) as Nyquist pulse while using RRC, but root-raised cosine (RRC) is indeed Nyquist, as it satisfies the ISI-free criteria. They also mishandle frequency response and notation. Additionally, some formulas for symbol duration and shift matrices require fixes to align with definitions properly.

Addressing scaling, Fourier convention, and assumptions

There are scaling issues with their treatment of matrices, especially around the diagonalization of J_k. I need to unify Fourier conventions—either use unitary or unnormalized consistently. Additionally, the paper assumes some variances that should be checked, and it’s unclear in some parts regarding element indexing and approximations for finite constellations. I think error bounds or more discussion would help clarify these points.

Fixing mathematical model inconsistencies and clarifying contributions

There’s an inconsistency in the symbol model where the symbol lengths should match between sections. I'll also check the approximation for the characteristic function, as factors could be off by a factor of 2. There are still missing steps, especially regarding Lindeberg conditions. Additionally, the paper seems to lack clarity in terms of the contribution and novelty, optimization, and table details.

Suggestions for organizing and clarifying the paper

I need to restructure the introduction, clearly state the novelty of CE-OFDM for power-constrained ISAC, and highlight its differences from normal OFDM. For clarity, I should fix ambiguities in the time-domain and symbol index definitions and update the system model. Pulse shaping should be clearly presented, and I'll ensure to unify DFT normalization. Additionally, I need to avoid unsupported claims in the PAPR section and align the sensing metric with the contributions.

Suggestions for improving and trimming content

To improve the structure, I should clarify the use of 'mainlobe normalization' and explain coherent integration's variance reduction. There are also phase rotation and optimization constraints that need fixing, alongside clarifying receiver compensation. For simulations, it's important to include default parameters and explore additional metrics, like RMSE. Lastly, trimming sections like SE vs BER, correcting formula errors, and reducing algorithmic details should be a focus based on the user's request. The highest priority issues include section numbering, DFT normalization, and pulse spectrum errors.

Thought for 4m 43s

Answer

先说一句避免误会：我不是“院士”，但我可以按 IEEE TWC 审稿人的口径，把你们这版初稿里可能会被卡住的硬伤、推导/建模一致性问题、以及可以增强或删减的内容逐条挑出来，方便我们接下来逐项讨论修改。

1) 目前最容易被审稿人直接指出的“硬伤/不一致”

1.1 文章结构编号错误

你们在“文章组织结构”里写“Finally, Sec. VII concludes the paper”，但正文结论是 Sec. VI，没有 Sec. VII。这个会显得不够仔细，建议立刻改掉。 (CE_OFDM_ShiJingCheng (3).pdf)

1.2 Table II 在文中被引用但实际缺失

数值部分开头写“default settings in Table II are used”，但 PDF 里并没有 Table II（至少当前稿里读不到）。这属于“可复现性信息缺失”的典型扣分点。 (CE_OFDM_ShiJingCheng (3).pdf)

1.3 DFT 归一化/尺度存在明显自相矛盾

你们说 FNF_N 是“normalized DFT matrix”，但紧接着 IDFT 展开式用了 1/N1/N 系数（这对应“非归一化 IDFT”）。 (CE_OFDM_ShiJingCheng (3).pdf)
这个问题会连带影响：CNC_N 的取值、能量归一化、以及后面很多推导里的常数因子（例如你们在 ACF 推导里出现的 LN\sqrt{LN} 等）。建议：全文统一一种 DFT 约定（unitary 或 1/N 版）并把所有相关公式整体对齐。

1.4 脉冲频域表达式疑似排版/公式错误（“cos h”）

在脉冲谱 P(f)P(f) 的分段式里出现了 “cos h …” 这种写法，很像 LaTeX/排版错误（容易被认为公式不严谨）。 (CE_OFDM_ShiJingCheng (3).pdf)
同时，你们全文多处称用 RRC pulse shaping，但这里给的又像 RC/RRC 的混合写法。建议明确：

  • 你们的 p(t)p(t) 到底是 RC 还是 RRC？
  • 如果是 RRC，频域幅度应是 RC\sqrt{\text{RC}} 形式；如果是 RC，则应是标准 0.5(1+cos⁡(⋅))0.5(1+\cos(\cdot)) 形式。
    这个点不改，TWC 很容易被抓住。

1.5 “相位旋转透明”与“优化变量维度”存在冲突（非常关键）

你们证明“symbol-wise phase rotation 对通信解调完全透明”，核心条件是 θ[n]=θn1\theta[n]=\theta_n\mathbf{1}（整块 OFDM 符号同相位偏置），这样 DFT 到有效子载波上为 0。 (CE_OFDM_ShiJingCheng (3).pdf)
但在你们的联合优化问题里，约束写的是 θn∈(0,2π],n=1,…,N\theta_n\in(0,2\pi], n=1,\dots,N，这看起来是在优化 一个 OFDM 符号内部每个采样点的相位序列。 (CE_OFDM_ShiJingCheng (3).pdf)

这两者逻辑上冲突：

  • 若 θ\theta 是逐采样点变化，一般不会对通信“透明”，除非你们额外说明通信端能已知并补偿该序列；
  • 若 θ\theta 必须保持“symbol-wise 常相位”才透明，那么优化自由度就几乎没了（无法塑形 ACF）。

这是我认为当前稿件最需要优先统一的地方：请你们决定 相位设计到底是“跨符号的块相位”还是“符号内序列相位”，以及通信端是否补偿。否则后面所有“在不牺牲通信性能的前提下优化感知”的结论都会被质疑。

1.6 PAPR 上界推导很可能不够严谨/甚至尺度不自洽

你们给了脉冲成形后 PAPR 上界：先得到 max⁡∥Pi∥2\max\|P_i\|^2，再进一步给出 ((2+α)/2)2((2+\alpha)/2)^2。 (CE_OFDM_ShiJingCheng (3).pdf)
但这里存在两个潜在问题会被审稿人追问：

  1. 你们把脉冲成形写成矩阵 PP（且后面 ACF 推导把它当 circulant 来对角化），那每行范数与“线性卷积+跨符号叠加”的直觉并不完全一致；
  2. ((2+α)/2)2((2+\alpha)/2)^2 这个“roughly 1+α symbol intervals” 的论证目前偏口语化，缺少明确假设（RRC/RC、截断长度、采样率 L、脉冲能量归一化方式等）。

建议：要么把这个上界推导补严谨（给出明确的离散脉冲长度/截断方式和可验证的 bound），要么干脆删除这条“漂亮但危险”的上界，改为“给出 max⁡∥Pi∥2\max\|P_i\|^2 的一般界 + 由仿真 CCDF 说明其很低”。

1.7 Shannon 容量公式写法明显有问题

你们写了 C=Blog⁡2(1+CBEbN0)C = B\log_2(1 + \frac{C}{B}\frac{E_b}{N_0})，右边又出现 C/BC/B（等式两边含同一个未知量），这在论文里会非常刺眼。 (CE_OFDM_ShiJingCheng (3).pdf)
如果你们想画“SE vs Eb/N0E_b/N_0” 的 Shannon 限，应写成标准的

CB=log⁡2(1+SNR),SNR=EbN0⋅RB\frac{C}{B}=\log_2(1+\text{SNR}),\quad \text{SNR}=\frac{E_b}{N_0}\cdot \frac{R}{B}

类似关系，并把“目标 BER 下所需 Eb/N0E_b/N_0”与“理论容量界”区分清楚。

2) 建模/符号层面：建议系统性梳理的点（能显著提升可读性）

2.1 “n” 的含义在文中容易混用

你们在系统模型里说“Let n serve as the index for time-domain symbols”， (CE_OFDM_ShiJingCheng (3).pdf)
但后面又用 n 做采样点/DFT 相关索引、又讨论 coherent integration（跨符号累积）。建议统一：

  • 用 mm 表示 符号内采样点 m=0,…,N−1m=0,\dots,N-1
  • 用 ℓ\ell 表示 OFDM 符号编号（coherent integration 的维度）
  • 用 kk 表示频域子载波
    这样可避免审稿人读着读着“索引语义漂移”。

2.2 “Random Symbol Model”里 s 的维度与后文 CE-OFDM 映射不一致

这里把 ss 写成长度 NN 的 i.i.d. 符号向量 (CE_OFDM_ShiJingCheng (3).pdf) ，
但 CE-OFDM 映射又用 NdN_d 个符号并构造 Hermitian 对称的 vv。 (CE_OFDM_ShiJingCheng (3).pdf)
建议在“随机符号模型”一开始就用 NdN_d（或明确 Nd=N/2−1N_d=N/2-1 的关系）。

3) “内容上可以加强”的方向（对 TWC 友好）

3.1 把“你们真正的理论贡献”讲得更“可审稿”

你们在引言贡献点里说“分离 pulse-dependent 与 phase-dependent 项” (CE_OFDM_ShiJingCheng (3).pdf) ，这是很好的卖点。建议在 Sec. III 里更“显式地”告诉读者：

  • Ik(g)I_k(g) 对应“均值项/脉冲塑形决定的底噪形状”，
  • Sk(g,θ)S_k(g,\theta) 对应“随机性引起的方差项/相位相关可压制”，
    并给一段直观解释“为什么相位只能压方差、不能改变均值形状”，这样审稿人更容易认可你们的“结构性发现”。

3.2 对“圆周相关/CP/最大无模糊距离”的假设写清楚

你们 ACF 用的是 circular shift matrix（JkJ_k），适用前提通常是“CP 足够长、感知延迟落在 CP 保护范围内”等，否则 radar 更自然的是 aperiodic correlation。你们需要把这个适用条件写在系统模型里，否则会被问“这个 ACF 指标是否对应真实 ranging sidelobe”。

4) 如果要删内容，哪些最适合删/移到附录？

我建议按“风险/性价比”排序：

4.1 优先删或大幅收缩：PAPR 解析上界那段“((2+α)/2)2((2+\alpha)/2)^2”

原因：目前论证不够严谨且和后续“circulant 建模”容易互相牵扯，属于“写了反而可能扣分”的内容。保留通用界 max⁡∥Pi∥2\max\|P_i\|^2 + 用 Fig.2 的 CCDF 说话会更稳。 (CE_OFDM_ShiJingCheng (3).pdf)

4.2 建议删/重写：SE–BER 与 Shannon limit 这段（至少先修公式）

因为容量公式当前是明显错误写法，且这一段如果不是你们核心贡献（你们核心是感知 ACF 与联合设计），可以缩成“1 段 + 1 图”甚至移到附录。 (CE_OFDM_ShiJingCheng (3).pdf)

4.3 算法细节可移附录：Algorithm 1 的逐行更新公式

你们算法流程非常长（PSO + RAdam 的混合），主文建议保留“思想 + 伪代码框架 + 复杂度量级 + 收敛/停止准则”，把推导细节/超参数表移附录或补充材料（TWC 审稿人一般不反对，但会反感正文太“工程实现说明书”）。 (CE_OFDM_ShiJingCheng (3).pdf)

4.4 背景铺垫可以更短

引言里关于 ISAC 大背景的文献堆叠可以再压缩，把空间留给“CE-OFDM 在随机信号下 ACF 机制不同于 OFDM/SC”的洞察，这样更 TWC。

5) 我建议我们下一步先“对齐”的三件事（决定后再继续细抠推导）

  1. θ\theta 到底是“符号内序列”还是“跨符号块相位”？通信端是否补偿？（决定你们 Proposition 5 与优化问题 (59) 是否成立/如何改） (CE_OFDM_ShiJingCheng (3).pdf) (CE_OFDM_ShiJingCheng (3).pdf)
  2. DFT 归一化全文统一（否则常数会到处漂）。 (CE_OFDM_ShiJingCheng (3).pdf)
  3. 脉冲成形模型：线性卷积 vs circular 近似、RC vs RRC、以及 P(f)P(f) 正确表达。 (CE_OFDM_ShiJingCheng (3).pdf) (CE_OFDM_ShiJingCheng (3).pdf)

如果你同意，我们可以先从第 1 点（θ\theta 的定义与“通信透明性”）开始讨论：
你们理想中是希望 在一个 OFDM 符号内部自由设计 θ[m]\theta[m] 来压 sidelobe，还是只想做 跨符号的 θℓ\theta_\ell（每个符号一个常相位）？这会决定后面整个优化框架怎么落地。

Question

我们的相位是在时域上的，一个不是一个符号内部一个theta，而是符号从频域转换到时域之后加上的theta

Thought

Clarifying phase vector and communication model

I need to explain to the user that the phase is a time-domain vector after IDFT, confirming equations (8)-(9). I’ll also highlight the conflict with Proposition 5 in Section IV-A, showing where they need adjustments. In their model, θ doesn't vanish with per-sample design, so subtraction at receiver might be needed.

Revising Proposition 5 for clarity

I'll rewrite Proposition 5 to clarify that if the receiver de-rotates by subtracting θ, the Θ term is removed. For this, θ must be known and deterministic. I'll add a short proof showing this, but also note it may not hold under frequency-selective channels. To maintain sensing gains, option 1 is preferred, as option 2 conflicts with the optimization.

Aligning terminology and rewriting for clarity

I need to align the term "symbol-wise phase rotation" with "sample-wise deterministic phase sequence" or "time-domain phase rotation sequence" in the abstract and intro. I’ll clarify transparency after de-rotation at the CE-OFDM receiver. I also need to cite the relevant parts of the paper for consistency, including lines for the optimization and phase vector definitions.

Focusing on defining phase and proposing key edits

The user wants a focus on the phase definition and potential modifications. I will propose changing the notation for θ: defining it as a time-domain phase rotation sequence, applied after IDFT and before the phase modulator—maybe calling it "scrambling phase". I’ll provide structured edits for Section II-B, the abstract/intro, and Section IV-A to align with this change.

Addressing noise invariance and transparency assumptions

I'll add a note on noise invariance and clarify transparency: it only holds for symbol-wise constant phase. For AWGN, additive phase models work, but multipath requires a different approach with flat fading or equalization. I'll suggest simulations to justify stable performance. Also, I'll adjust Proposition 5.

Thought for 3m 5s

Answer

明白了：你们的相位确实是在时域上加的确定性相位序列——也就是先做 IDFT 得到 v~\tilde v，再在相位调制器里用 (2πhCNv~+θ)(2\pi h C_N \tilde v+\theta) 生成 CE-OFDM。这个在稿子里已经写得很清楚：θ∈RN×1\theta\in\mathbb{R}^{N\times 1} 是“deterministic phase vector”并且直接出现在(8)(9)里。(CE_OFDM_ShiJingCheng (3).pdf) (CE_OFDM_ShiJingCheng (3).pdf)

但这样一来，你们现在文稿里关于“通信透明”的表述就必须整体改一遍，因为当前版本混进了“symbol-wise phase rotation”的特殊情形（那是每个 OFDM 符号只加一个标量相位）：

  • 你们在 IV-A 的 Proposition 5 明确假设 θ[n]=θn1\theta[n]=\theta_n\mathbf{1}（symbol-wise），并据此证明 (52) 里的相位项 Θn,k\Theta_{n,k} 为 0。(CE_OFDM_ShiJingCheng (3).pdf)
  • 但你们真正要优化的是长度为 NN 的时域相位序列（(59) 里 θn∈(0,2π],n=1,…,N\theta_n\in(0,2\pi],n=1,\dots,N），这与 Proposition 5 的“symbol-wise”假设不一致。(CE_OFDM_ShiJingCheng (3).pdf)
  • 同时在 (52) 里也能看到：一般的时域序列 θ[n]\theta[n] 会产生 Θn,k=1NfkHJτ′θ[n]\Theta_{n,k}=\frac1N f_k^H J'_\tau \theta[n] 这一项，它只有在 θ[n]∝1\theta[n]\propto \mathbf{1} 时才必然为 0。(CE_OFDM_ShiJingCheng (3).pdf)

你们现在需要做的“统一口径”（建议选 A，不然后面感知增益也讲不圆）

A) 保留“时域相位序列（sample-wise）”作为核心设计自由度（推荐）

这条路能保住你们 (59) 的优化意义，也能保住“相位能压低 Sk(g,θ)S_k(g,\theta)”这条主线。(CE_OFDM_ShiJingCheng (3).pdf)

但代价是：“通信透明”不能再用 Proposition 5 那种“接收机完全不用改”的说法。你们需要把 IV-A 改成下面这种更严谨、审稿人更买账的表述：

  1. 把“透明性”的定义改为：接收端已知 θ\theta，先做去相位（de-rotation）再解调。
    你们现在已经采用了“接收相位向量模型”(51)：r~=hC~NFNHv+θ+ε\tilde r = h\tilde C_N F_N^H v + \theta + \varepsilon。(CE_OFDM_ShiJingCheng (3).pdf)
    那么最自然的处理就是先做
r~d≜r~−θ,\tilde r_d \triangleq \tilde r - \theta,

再送入 correlator/DFT 分支。这样 (52) 里单独的 Θn,k\Theta_{n,k} 项会被完全消掉，得到与未加相位时同分布的统计量（在你们的相位加性模型下，减去确定性 θ\theta 不改变噪声统计）。(CE_OFDM_ShiJingCheng (3).pdf)

  1. Proposition 5 建议改写成“Receiver-aided invariance”（把现在这条“symbol-wise 才为 0”的证明降级成 Remark/特殊情形）。
    也就是说：
  • 现在的 Proposition 5（θ[n]=θn1⇒Θn,k=0\theta[n]=\theta_n\mathbf{1}\Rightarrow\Theta_{n,k}=0）可以保留，但标题和叙述要改成“特殊情形：无需接收端去相位”。(CE_OFDM_ShiJingCheng (3).pdf)
  • 主结论应该变成：任意时域相位序列在“已知且同步”的条件下，通过 r~−θ\tilde r-\theta 可保证通信检测统计不变（对应你们想在摘要里强调的“不损通信”）。
  1. 摘要/引言里所有 “symbol-wise phase rotation” 都要替换
    摘要第 1 页就写了“controllable via symbol-wise phase rotation”并且说“proved that it does not impair communication performance”。如果你们走 A 方案，这两句要改成“time-domain deterministic phase sequence (known to the receiver) / receiver-aided de-rotation”。(CE_OFDM_ShiJingCheng (3).pdf)
    引言里“transparent to CE-OFDM demodulation”也建议加上 “after de-rotation” 这一句，否则审稿人会直接拿你们 (52)+(Proposition 5) 来质疑。(CE_OFDM_ShiJingCheng (3).pdf)

这一改法的好处：你们的“感知优化自由度”完全保住；坏处：需要你们在系统假设里明确“θ\theta 对通信接收端可知且同步”。

B) 如果坚持“接收机完全不改也透明”

那就只能用你们现在 Proposition 5 的 symbol-wise 情形（θ[n]=θn1\theta[n]=\theta_n\mathbf{1}）。(CE_OFDM_ShiJingCheng (3).pdf)
但这个选择会带来一个致命后果：这种“全采样同相位”的旋转几乎不会改变单个符号内部的 ACF 结构，你们 (59) 这种按 n=1,…,Nn=1,\dots,N 优化的意义也会大幅下降甚至不成立。(CE_OFDM_ShiJingCheng (3).pdf)
所以从你们整篇“靠 θ\theta 压 Sk(g,θ)S_k(g,\theta)”的框架看，我不建议走 B。

我建议你们现在立刻改的 3 个具体位置（最省力、收益最大）

  1. Sec. II-B：在(8)后面加一句非常明确的话：
    “θ\theta is an NN-length time-domain (sample-wise) deterministic phase sequence applied after IDFT and before the nonlinear phase modulator.” (CE_OFDM_ShiJingCheng (3).pdf)

  2. Sec. IV-A：把现在 Proposition 5 的标题/定位改成“特殊情形”，并新增一个主命题：
    “Given known θ\theta, the receiver de-rotates r~d=r~−θ\tilde r_d=\tilde r-\theta before correlation/DFT, hence Θn,k\Theta_{n,k} is removed from (52).” (CE_OFDM_ShiJingCheng (3).pdf)

  3. Abstract + Introduction：把 “symbol-wise phase rotation” 全部替换成与你们真实模型一致的说法，并把“transparent”改成“transparent after de-rotation”。(CE_OFDM_ShiJingCheng (3).pdf) (CE_OFDM_ShiJingCheng (3).pdf)

如果你愿意，我们下一步就按 A 方案把 IV-A 这一节重写成“审稿人挑不出硬伤”的版本：

  • 先给出“θ\theta 已知/同步”的系统假设；
  • 再给一个 6–8 行的严谨证明把 (52) 的 Θn,k\Theta_{n,k} 消掉；
  • 最后把原 Proposition 5 作为 Remark（不需要 de-rotation 的特例）。

Question

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可是你看我附件中发你的 实际上我们在讨论的是这个东西。我看他输的没影响的。我们现在想按照他这种思路来写

Thought

Exploring effects of time-domain phase on communication

Okay, they're saying time-domain phase doesn’t affect communication. However, the excerpt only holds true for a constant phase. If θ(t) changes, we could adjust by having the receiver know and subtract the phase. θ(t) might also not interfere if it lies in the right space, like DC or Nyquist.

Analyzing phase shift effects on demodulation outputs

If DC and Nyquist frequencies are deactivated, θ components at these frequencies won’t affect demodulation outputs. If a timing shift (τ) is applied, it’s adjusted once delay is estimated, but for invariance, θ must be constant (DC). If τ=0, any oscillations orthogonal to active subcarriers won't interfere.

Exploring the impact of sample-wise θ on communication

If we use sample-wise θ, claiming "no impact" is tricky unless we assume de-rotation or treat the phase as known pseudo-noise. In the case of CE-OFDM, the receiver can subtract the phase before correlating to remove the impact. They didn't explicitly say they want no receiver modifications, so we can guide them with the derivation structure. We could cite precise sections from their PDF for clarity.

Addressing citation format with file parsing

Okay, we have the parsed text, but I need to find the line numbers for proper citations. The current file search output has parsed pages without line markers. I think I need to use a different approach to retrieve line numbers or gather snippets with them. This isn't quite matching up to the citation format I need.

Planning the response with citations and details

The file's citation request is simple, but the inclusion of line numbers is trickier. I could reference the file with (website) for simplicity. However, to be precise, I could use mclick to get the correct chunk with line numbers, or extract lines using Python, but filecite seems necessary for formal citations. I'll focus on clarifying the user's design approach with phase rotation.

Explaining phase demodulation and de-rotation

The user can apply the reference by recognizing that deterministic, known θ can be treated as an additive phase. By design, θ’s DFT support can be placed in unused carriers, ensuring it doesn’t impact communication. For arbitrary θ, the receiver needs to subtract it for "no impairment" before demodulation.

Clarifying θ design and its impact on transmission

It seems θ is computed as symbol-wise constant during integration, with zero-mean basis functions. To match the reference, you can treat θ as constant within symbols, but vary between symbols during optimization. This adjustment implies using sample-wise design and requires de-rotation in the receiver for performance consistency.

Linking reference to user’s paper

The user’s reference shows that θ is a scalar per block, matching their Eq (55), and the DFT property causes the integral to vanish. For performance analysis, I recommend focusing on the message without diving into BER evaluation. I’d also suggest modifying the excerpt for discrete-time phase demod, but avoid excessive analysis.

Strengthening proposal with SNR phase demod analysis

I can propose incorporating a high SNR phase demodulation approximation to justify the additive phase model, reinforcing Section IV-A. The current Eq (51) oversimplifies things—I'll suggest adding a paragraph explaining phase demod with noise, proposing de-rotation if θ affects communication. The receiver model needs some tweaks in Fig.1.

Crafting response and suggestions for de-rotation

We can suggest adding "de-rotation" after phase demodulation. The key options are: (a) subtract θ after phase demodulation, requiring θ to be known, (b) constrain θ, or (c) limit θ spectrum. I'll provide a concise response to the user with a small question/interjection for clarity, keeping explanations clear and minimal.

Thought for 1m 44s

Answer

你截图里那段推导（相位解调器 + 一组相关器）确实能得到“额外相位不影响解调输出”，但它成立的关键前提是：

  • 加进去的是 每个符号区间内的常数相位 θi\theta_i（不是随 tt 变化的序列）；
  • 相关器基函数 qk(t)q_k(t) 满足 零均值：∫0TBqk(t) dt=0\int_0^{T_B} q_k(t)\,dt=0（DCT/DFT 子载波基一般满足，DC 分量除外）。

因此 Ψi,k=1TB∫θiqk(⋅) dt=θi⋅0=0\Psi_{i,k}=\frac{1}{T_B}\int \theta_i q_k(\cdot)\,dt=\theta_i \cdot 0=0。你说的“他输的没影响”，本质就是“常相位落在 DC 子空间，而解调分支把 DC 扔掉了”。

1) 这和你们稿子里现有 Proposition 5 是同一个逻辑，但只覆盖“symbol-wise 常相位”

你们现在稿子里的通信部分已经写成了同样的结构：把相关器输出分解成信号项、相位项、噪声项，并证明在 symbol-wise 情况下相位项为 0（靠的也是 fkH1=0f_k^H\mathbf{1}=0 这个“零均值/去 DC”性质）。(CE_OFDM_ShiJingCheng (3).pdf)

2) 但你们真正要做的相位设计是“时域序列 θ\theta”：一般情况下不会自动为 0

你们系统模型里 θ=[θ1,…,θN]T\theta=[\theta_1,\ldots,\theta_N]^T 是在 IDFT 后、相位调制前加上的 时域采样级相位序列（sample-wise）。(CE_OFDM_ShiJingCheng (3).pdf)

这种情况下，通信解调分支里那一项本质上就是“θ\theta 的 DFT 投影/相关投影”：

  • 除非 θ\theta 恰好是常向量（只含 DC），或者只落在你们未使用的频点子空间（DC / Nyquist 等），否则它一般会泄漏到有效子载波上，不能像截图那样自动消失。

所以：如果你们坚持用“序列级 θ\theta”来优化感知，就不能再用截图那种结论直接写成“对通信完全透明（无需处理）”。

3) 想“按他那种思路写”，最稳的写法是：把“透明性”改成 receiver-aided de-rotation

也就是保留你们现在(类似截图的)“相关器输出分解”写法，但把结论改成：

只要接收端已知 θ\theta，在进入 CE-OFDM 解调相关器/DFT 之前，先在相位域做去旋转（de-rotation），则通信统计与 BER 不变。

在你们的相位域模型里（你们已经写了“接收相位 = 有效相位 + θ\theta + 噪声”这种形式），证明会非常短：

  • 相位解调输出：r~=(data term)+θ+ε\tilde r = \text{(data term)} + \theta + \varepsilon (CE_OFDM_ShiJingCheng (3).pdf)
  • 去旋转：r~d≜r~−θ=(data term)+ε\tilde r_d \triangleq \tilde r - \theta = \text{(data term)} + \varepsilon
  • 再送入相关器/DFT：所有分支输出就回到“无 θ\theta”的情形，因此 BER 统计不变。

这样写的好处：

  • 你们仍然可以优化 sample-wise θ\theta（自由度保住，感知优化才有意义）；
  • “不牺牲通信”的论断变得严谨，不会被审稿人用你们现有 Proposition 5 的前提去卡。

同时，把现在的 Proposition 5（symbol-wise 常相位自动消失）降级成一个 **Remark / Corollary（无需显式 de-rotation 的特例）**即可。(CE_OFDM_ShiJingCheng (3).pdf)

4) 具体落到你们稿子里，建议你们这样改（最省改动）

  1. Sec. IV-A 标题与主结论改口径：
    从“phase rotation transparent”改成“known deterministic phase sequence is removable by de-rotation, hence communication-invariant”。

  2. 在 Fig.1 的 Comm Rx 流程里（Phase de-modulation 后）加一个小方块：** De-rotation by known θ\theta**。(CE_OFDM_ShiJingCheng (3).pdf)

  3. 把 Abstract/Introduction 里所有 “symbol-wise phase rotation” 改成与你们真实模型一致的表述：
    “time-domain deterministic phase sequence (known to Rx)”。

如果你们同意走这条写法，我可以按你给的截图结构，帮你把 IV-A 整段“像教科书/经典 CE-OFDM 论文那样”的推导叙述整理成可直接替换进正文的版本（含：连续时间相位解调近似→相关器输出分解→de-rotation 不变性→symbol-wise 特例）。

Question

thompson-thesis-ce-ofdm-final.pdf

我把他的文章发你，我感觉他不是常相位呢？你来仔细学习下。因为我看他有下标i，是第i个采样点的相位啊

Thought

Clarifying θ_i as a memory term

I found that in Thompson's thesis, θ_i is used as a memory term for a constant phase within each block, not the phase at an i-th sample point. As explained in equation (3.12), θ_i represents the memory term for the ith OFDM symbol interval. I’ll provide citations for the explanation in Chinese.

Proposing clearer notation and phase explanation

I’ll suggest better notation for our paper, like using ℓ for the block index and n for the sample index. Also, I’ll clarify that θ_i is a random offset based on memory or channel, but constant within the block. I’ll provide citations from sections 3.12 and 3.22 to explain this phase offset.

Clarifying phase and block/sample indexing

The user thinks "i" refers to the sample index, but in the thesis, there are two different uses of "i". In equations like r[i] for discrete-time samples, "i" is the sample index. However, in phase expressions, it represents the block index. I'll cite sections 3.12, 4.18, and 4.20 to explain.

Adapting phase design based on thesis approach

I’ll cite the relevant sections of the thesis to define r[i] and explain the phase offset θ. We could adapt the approach by restricting θ to be constant within each block, as suggested in our CE-OFDM ISAC paper, which would eliminate sample-wise handling. However, if sample-wise is needed, we would require additional receiver de-rotation for the design.

Clarifying design approach and proposing alternatives

We can propose adding a constant per block with a shaping sequence in the null space or reserved subcarriers, but the user wants a full-time sequence. Instead, we might suggest subtracting the known sequence before DFT. In the thesis, the phase unwrapper ignores phase offsets thanks to the correlator, but this doesn’t apply for sample-wise θ, which would require adjustments to the receiver.

Clarifying system model and θ definitions

In your paper, you're currently defining θ as a vector of length N. However, Proposition 5 uses θ[n] = θ_n 1, meaning it's symbol-wise and consistent with Thompson's approach. To follow his model, you should rewrite θ as scalar per OFDM symbol. Alternatively, you can keep it per-sample but show how the receiver removes it. The key is that θ_i in Thompson's thesis represents a memory term constant within a block, not as a per-sample index.

Thought for 1m 7s

Answer

你这个疑惑非常正常：Thompson 的符号 ii 在论文里被“复用”了，所以看起来像“第 ii 个采样点”。但你截图那一套推导里，θi\theta_i 其实不是“采样点相位”，而是第 ii 个 CE-OFDM block（等价于第 ii 个 OFDM 符号区间）内的常相位/记忆项——它在一个 block 内是常数，只是在 block 之间随 ii 变化（而且还可能由记忆/连续相位机制递推产生）。

1) Thompson 文中 θi\theta_i 到底是什么？

1.1 在信号定义里，ii 明确是“第 ii 个 block”

他在 CE-OFDM 相位信号定义里写得非常清楚：

  • “during the iith block”，并且给出
ϕ(t)=θi+2πhCN∑k=1NIi,kqk(t−iTB),iTB≤t<(i+1)TB\phi(t)=\theta_i + 2\pi h C_N\sum_{k=1}^N I_{i,k}q_k(t-iT_B),\quad iT_B\le t < (i+1)T_B

同时还直接说 θi\theta_i is a memory term（记忆项，用于相位连续）(thompson-thesis-ce-ofdm-final.pdf)。

所以这里的 θi\theta_i 是块索引 ii 对应的“块内常数相位偏置/记忆项”。

1.2 θi\theta_i 还是“递推的记忆项”，不是采样点相位

后面他为了保证连续相位，给了递推关系（(3.22)）：

θi≡θi−1+K∑k=1N[⋯ ]\theta_i \equiv \theta_{i-1} + K\sum_{k=1}^N[\cdots]

并解释 “Notice that θi\theta_i depends on θi−1\theta_{i-1} …”，也就是它是跨 block 演化的记忆项(thompson-thesis-ce-ofdm-final.pdf)。

这进一步说明：θi\theta_i 的下标 ii 是块序号，不是“采样点序号”。

2) 你觉得“像采样点”的原因：他同一章前面也用 r[i]r[i] 表示离散采样

在 4.1 开头，他把连续时间接收信号采样得到离散序列：

r[i]=s[i]+n[i], i=0,1,2,…r[i]=s[i]+n[i],\ i=0,1,2,\dots

这里的 ii 确实是采样点索引(thompson-thesis-ce-ofdm-final.pdf)。

但到了 4.1.1 “Performance Analysis”，他又回到连续时间分析，并在 (4.20) 里写：

arg⁡[r(t)]=θi+⋯ ,iTB≤t<(i+1)TB\arg[r(t)] = \theta_i + \cdots,\quad iT_B\le t<(i+1)T_B

此时的 ii 又变成了块索引(thompson-thesis-ce-ofdm-final.pdf)。

结论：Thompson 在同一章里把 ii 既当“采样点索引”又当“block 索引”，导致你现在的直觉冲突。

3) 他为什么能证明“没影响通信性能”？——因为偏置是“块内常数”，相关器基函数零均值

你截图里核心就是：

Ψi,k=1TB∫iTB(i+1)TBθiqk(t−iTB) dt\Psi_{i,k}=\frac{1}{T_B}\int_{iT_B}^{(i+1)T_B}\theta_i q_k(t-iT_B)\,dt

然后用 ∫0TBqk(t)dt=0\int_0^{T_B}q_k(t)dt=0 得到 Ψi,k=0\Psi_{i,k}=0(thompson-thesis-ce-ofdm-final.pdf)。

在他 4.1.2 里还把信道相位偏置 ϕ0\phi_0 加进去，变成 [θi+ϕ0][\theta_i+\phi_0]，同样被零均值相关器“吃掉”，所以也为 0(thompson-thesis-ce-ofdm-final.pdf)。

这套逻辑只对“块内常相位”成立。如果是“随时间变化的相位序列 θ(t)\theta(t)”或“采样级 θ[n]\theta[n]”一般不会被 ∫qk=0\int q_k = 0 自动消掉。

4) 这和你们稿子的关系：你们现在的 Proposition 5 本质上就是 Thompson 这一条（symbol-wise 常相位）

你们稿子里 Proposition 5 的前提写得非常像 Thompson：symbol-wise phase rotation，θ[n]=θn1\theta[n]=\theta_n\mathbf{1}，并用 fkH1=0f_k^H\mathbf{1}=0 得到 Θn,k=0\Theta_{n,k}=0(CE_OFDM_ShiJingCheng (3).pdf)。

所以：

  • 如果你们想“按 Thompson 的思路写通信不受影响”，那就要把你们的相位设计明确成：
    每个 OFDM 符号（或每个 transmission slot）一个标量相位 θℓ\theta_\ell，对该符号内所有时域采样点都一样。这依然是“IDFT 后加到时域上”，但它是“块内常数”，不是“逐采样点自由变化”。

  • 如果你们坚持要做“逐采样点的 θ[n]\theta[n]”并拿它来压 ACF sidelobe，那 Thompson/Proposition 5 这条“天然透明”就不能直接用；你们需要在通信接收机里加一个“已知 θ\theta 的去旋转/去除”步骤（否则一般会泄漏到有效子载波）。

5) 我建议你先定一个写作口径（两条路选其一，逻辑就会非常干净）

路线 A（完全沿用 Thompson 口径，写起来最顺）
把你们的“相位设计变量”改成 symbol-wise 标量 θℓ\theta_\ell（块内常数），并把系统模型/优化维度都对齐到这个设定。

路线 B（保留 sample-wise θ[n]\theta[n] 的大自由度）
在通信部分把“透明”改成：“接收端已知并同步 θ[n]\theta[n]，在相位解调后先做去除再 FFT/相关”，这样才能严谨。

如果你愿意，我们下一步可以直接把你们论文 Sec. IV-A 按“Thompson 风格”重写成一段可以直接替换的版本（并顺手把你们现在符号里 n/k/ℓn/k/\ell 的索引统一，避免再出现他那种 ii 复用引起的歧义）。你更倾向走 路线 A 还是 路线 B？

Question

你觉得哪个方案合理，对于提升感知性能而言。我们主要是要提升感知性能。但我觉得是不是差不多？因为CE-OFDM到最后都是放到一个圆上去做感知，只是在圆上的分布不同？

Thought

Proposing explanation for sensing advantage

We have the necessary definitions now. For sensing, a sample-wise phase vector is critical because it affects the E|r_k|^2 decomposition, where p̃(θ) influences leakage. With a symbol-wise constant, p̃(θ) only affects DC, providing no sensing advantage. We'll bundle this with an intuitive explanation, linking to source formulas. But the user is asking specifically which scheme is "reasonably" better for sensing performance.

Expounding on sensing strategy and phase continuity

To enhance sensing, we recommend using a sample-wise phase vector (θ). For communication, a receiver de-rotation step is necessary, coupled with treating Thompson's block constant θ_i as a memory term for phase continuity. This ensures the autocorrelation function (ACF) isn't shaped. We note Thompson's methods and explain communication mismatch if sample-wise θ is used but not corrected for by de-rotation.

Clarifying phase difference and sampling impact

We'll explain that constant envelope mapping means points on a circle, but temporal order matters. The autocorrelation function (ACF) depends on phase differences like exp(jϕ_{p,q}). A change in θ alters cancellations. It’s not just a histogram; adjacency patterns and phase differences matter. We can suggest a low-dimensional, piecewise constant θ for partial communication invariance. For overhead, θ can be generated using LFSR or embedded in a preamble. We’ll emphasize the difference between constant symbol-based θ vs. sample-wise adjustments.

Crafting a concise response in Chinese

For the user’s inquiry, I’ll structure a clear, not too verbose answer in Chinese:

  • Explain two types of "θ";
  • Discuss which improves sensing;
  • Relate it to CE-OFDM on the circle;
  • Show how to stay consistent with your paper.

Additionally, I’ll mention that using a constant "

Thought for 4m 30s

Answer

我觉得为了“提升感知性能”，两种方案并不“差不多”，差别反而很关键——主要看你们的 θ\theta 是不是在一个 CE-OFDM 符号内部随采样点变化。

1) Thompson 那套证明里，θi\theta_i 本质是“块内常相位（memory term）”，它不会帮你改善 ACF 旁瓣

Thompson 在推导里写的是：在第 ii 个 block 区间 iTB≤t<(i+1)TBiT_B\le t<(i+1)T_B，接收相位里有一个加性项 θi\theta_i（block-wise 常数） (thompson-thesis-ce-ofdm-final.pdf)。
后面相关器输出里出现 Ψi,k=1TB∫θiqk(⋅) dt\Psi_{i,k}=\frac{1}{T_B}\int \theta_i q_k(\cdot)\,dt，并用 ∫0TBqk(t)dt=0⇒Ψi,k=0\int_0^{T_B}q_k(t)dt=0\Rightarrow \Psi_{i,k}=0 来说明它不影响解调 (thompson-thesis-ce-ofdm-final.pdf)。

关键点：这种“块内常相位”对同一个符号内部的自相关基本是“整体旋转”，在 xHJkxx^H J_k x 里会被相消掉（因为相关用的是样本间相位差），因此几乎不可能成为你们“压旁瓣”的自由度。

2) 你们要做的“提升感知”，真正有用的是 sample-wise 的 θ=[θ1,…,θN]T\theta=[\theta_1,\dots,\theta_N]^T

你们稿子里系统模型明确把

x=A~exp⁡{j(2πhCNv~+θ)},x=\tilde A\exp\{j(2\pi hC_N\tilde v+\theta)\},

并且直接说明 θ=[θ1,…,θN]T\theta=[\theta_1,\dots,\theta_N]^T 是“to be optimized for sensing” (CE_OFDM_ShiJingCheng (3).pdf)。这就是 sample-wise（符号内逐采样点） 的相位序列。

更重要的是，你们理论分解把

E{∣rk∣2}=Ik(g)+Sk(g,θ)\mathbb E\{|r_k|^2\}=I_k(g)+S_k(g,\theta)

写成“脉冲决定的 pedestal + 相位可控的 leakage” (CE_OFDM_ShiJingCheng (3).pdf)，并且明确指出 structured leakage 只通过相位功率谱 p~(θ)\tilde p(\theta) 进入，因此“amenable to optimization” (CE_OFDM_ShiJingCheng (3).pdf)。这意味着：** 如果 θ\theta 不是一个“有形状的序列”，而只是块内常数，那么 p~(θ)\tilde p(\theta) 退化，旁瓣可控性会大幅消失。**

另外你们的相位优化价值在“相干积累不够长”时更明显，因为你们也证明了相干积累 LcL_c 会把 Sk(g,θ)S_k(g,\theta) 压到 1/Lc1/L_c，最后极限是 Ik(g)I_k(g) 这个结构下界 (CE_OFDM_ShiJingCheng (3).pdf)。所以：

  • LcL_c 很大：相位优化收益会被 1/Lc1/L_c 稀释，主要靠 pulse shaping 决定下界；
  • LcL_c 中小：相位优化（尤其是 sample-wise）更值钱。

3) 你说“都在圆上，只是分布不同，所以差不多吗？”——不太对，关键不是“落点分布”，而是“相位差结构”

CE-OFDM 确实常包络，样本都在单位圆附近；但匹配滤波/ACF 看的是

rk=x~HJkx~r_k=\tilde x^H J_k\tilde x

这种移位内积 (CE_OFDM_ShiJingCheng (3).pdf)，本质受控于大量的“样本对的相位差”。你们在展开 ∣rk∣2|r_k|^2 时，系数里显式出现了相位差组合项 ejφ~p,q,p′,q′e^{j\tilde\varphi_{p,q,p',q'}} (CE_OFDM_ShiJingCheng (3).pdf)——这说明决定旁瓣的是“相位差的耦合结构”，不是“圆上的直方图”。
块内常相位只会给所有样本加同一个角度，相位差不变 ⇒ ACF 统计基本不变；而 sample-wise θ[n]\theta[n] 会改变相位差结构 ⇒ 这才是你们能优化 Sk(g,θ)S_k(g,\theta) 的根源。

4) 我对“哪个方案更合理”的建议（面向你们的目标：提升感知）

结论：若论文主目标是“显著提升感知性能”，应当选 sample-wise θ\theta（你们现在模型其实也是这么设的） (CE_OFDM_ShiJingCheng (3).pdf)。
Thompson 那种 block-wise memory term 更适合“保证相位连续/便于通信分析”，但对你们要压 ACF 旁瓣帮助非常有限。

但有个必须同步解决的写作一致性问题：你们通信“不损失”的证明目前是按 symbol-wise（块内常相位）去证的（例如用 θ[n]=θn1\theta[n]=\theta_n\mathbf 1 进而 Θn,k=0\Theta_{n,k}=0 那条逻辑） (CE_OFDM_ShiJingCheng (3).pdf)。如果正文又把 θ\theta 当作 [θ1,…,θN]T[\theta_1,\dots,\theta_N]^T 来优化感知 (CE_OFDM_ShiJingCheng (3).pdf)，审稿人很容易抓住：** 你证明“透明”的前提和你优化用的变量不是同一个。**

换句话说：“不改接收机且通信完全透明” 与 “靠 sample-wise θ\theta 明显提升感知” 通常不能同时无条件成立。

5) 下一步我们怎么推进讨论

你们想保留 sample-wise θ\theta 来提升感知，我建议通信部分用更通用的口径：

  • 要么引入 receiver-aided de-rotation（接收端已知 θ\theta 后去除） 来保证 BER 不变；
  • 要么把“通信完全不受影响”限定为一个 特例/Remark：当 θ\theta 为块内常数时，回到 Thompson/你们 Proposition 5 的结论。

你先告诉我你们更倾向哪种写法：

  1. 保持“无需改接收机”（那就要接受相位自由度大幅收缩，感知收益可能变小）；
  2. 允许接收端已知并去除 θ\theta（更符合你们现在“优化 θ\theta 提升感知”的主线）。

Question

我们还是改成sample-wise theta吧，那我觉得证明通信性能不变这个就没啥用了，可以不写。但是我们前面的理论推导的公式需要调整么？按照这个思路，我们一起看看文中什么地方需要调整

Thought

Outlining changes and clarifying indexing

I’ll take the approach of deleting or modifying specific sections like the "symbol-wise phase rotation" mention in the Abstract, and we need to adjust for clarity around notation in Proposition 5. I'll recommend renaming the indexing variable "n" to make it clearer. Also, I'll update terminology to reflect sample-wise phase shaping and make the sample indexing consistent across sections.

Refining theoretical statements and focusing on sensing

For the ACF derivation, it’s important to ensure no assumptions about constant θ. I’ll suggest removing references to “symbol-wise” and reframing communication-focused statements. In the design framework, we should shift the emphasis toward sensing, but keep a brief mention of communication for completeness, noting that changes in θ would impact performance.

Proposing a minimal adjustment for clarity

It seems like proposing a new assumption about the communication receiver knowing θ could reduce complexity without overloading the draft. But, as the user mentioned, the proof might not be needed. We can still add a brief remark to address the reviewer’s potential questions. We’ll ensure the objective and derived functions stay valid for sample-wise.

Tracking required citations and formula adjustments

I need to gather citations for key sections, starting with the introduction (lines 44-47), Proposition 5 (lines 64-116), and the conclusion (lines 30-39). The system model also defines sample-wise theta as a vector in lines 88-96. Most formulas don’t need adjustments beyond removing the symbol-wise assumption. I’ll double-check DFT normalization.

Addressing review and terminology changes

I’ll highlight that equations (42)-(47) are already general with a sample-wise approach, so no modification is needed. I'll mention removing any occurrences of θ[n]=θ_n 1 in derivations and clarify the system model’s use of "n" for time-domain symbols. There are key notes on terminology, notation, and figure adjustments.

Optimizing communication terminology and adjustments

Start by confirming the decision for sample-wise phase sequence. Most ACF theory already uses vector theta, so no big changes. Communication-related adjustments and wording are a priority. I'll recommend keeping assumptions or some discussion for reviewer feedback. I'll provide specific replacements, citations, and restructuring ideas, including renaming Section IV-A and updating notation.

Thought for 3m 18s

Answer

好的，既然我们把 θ\theta 明确为 sample-wise（一个CE-OFDM符号内逐采样点的时域相位序列），那你们论文里需要调整的地方，主要不在前面“ACF期望推导”的核心公式，而集中在通信“透明/不影响”相关的表述、命题、以及若干符号一致性。

下面我按“必须改 / 建议改”给一份逐段定位清单（都对应你们稿子里的具体段落/行号）。

A. 结论先讲清：前面理论推导公式要不要动？

你们的 ACF 期望推导、以及“structured leakage 只通过相位谱 p~(θ)\tilde p(\theta) 进入、可优化”的主结论，本质上就是把 θ\theta 当成一个长度 NN 的设计向量来做的（这正是 sample-wise）。例如你们明确写了 p~(θ)=FHu(θ)⊙[FHu(θ)]\*\tilde p(\theta)=F^H u(\theta)\odot[F^H u(\theta)]^\* 是“designable phase vector θ\theta”的频域功率谱 (CE_OFDM_ShiJingCheng (3).pdf)，并据此讨论 Sk(g,θ)S_k(g,\theta) 可优化 (CE_OFDM_ShiJingCheng (3).pdf)。
所以：ACF 推导主链条大体不需要为 sample-wise 再重推。

需要动的是：你们文中仍然有几处把相位说成 symbol-wise 或者声称 “对通信解调透明/不影响通信性能”，这些在 sample-wise 设定下会变成逻辑硬伤。

B. 必须修改/删除的部分（否则会被审稿人直接抓住）

B1) 摘要：两句话与 sample-wise 冲突

摘要里写：

  • “leakage term controllable via symbol-wise phase rotation”
  • “and proved that it does not impair communication performance” (CE_OFDM_ShiJingCheng (3).pdf)

如果你们决定走 sample-wise，这里至少要改成：

  • 把 symbol-wise 改为 time-domain sample-wise phase sequence / phase shaping；
  • 把“proved does not impair communication performance”删掉或改成“communication impact is discussed / evaluated numerically / can be compensated if θ\theta is known …”（后面我给你们推荐写法）。

B2) 引言：关于“透明不损通信”的论断需要改口径

引言中有一句非常关键的“背书”：

“introducing a deterministic phase vector can be made transparent to CE-OFDM demodulation, so … does not compromise communication performance” (CE_OFDM_ShiJingCheng (3).pdf)

这句话在你们当前稿子里是靠 **Sec. IV-A 的 Proposition 5（symbol-wise 特例）**撑起来的。你们如果删掉 IV-A 或改成 sample-wise，这里必须同步改，否则前后不一致。

B3) Sec. IV-A 整节（Proposition 5）与 sample-wise 设定直接冲突 —— 建议整节删或重写成“讨论”

你们现在 Sec. IV-A 标题就是“Communication Performance Invariant to Phase Rotation”，并且 Proposition 5 明确假设 symbol-wise：

  • “applies phase rotation in a symbol-wise manner” (CE_OFDM_ShiJingCheng (3).pdf)
  • “θ[n]=θn1\theta[n]=\theta_n\mathbf{1}” (CE_OFDM_ShiJingCheng (3).pdf)
    并由此推出 Θn,k=0\Theta_{n,k}=0 (CE_OFDM_ShiJingCheng (3).pdf)，最后得出“completely transparent… does not alter BER statistics” (CE_OFDM_ShiJingCheng (3).pdf)。

但 sample-wise 的 θ=[θ1,…,θN]T\theta=[\theta_1,\dots,\theta_N]^T（你们系统模型里其实已经这么写了） (CE_OFDM_ShiJingCheng (3).pdf) 时，一般 Θn,k\Theta_{n,k} 不会自动为 0（你们(52)里也把它单列出来了） (CE_OFDM_ShiJingCheng (3).pdf)。
所以这节如果保留原样，会被直接判为“假设不一致”。

建议：如果你们决定“不写证明通信不变”，那最干净的做法是：

  • 删除 Sec. IV-A 整节；
  • 同时把摘要/引言/结论里所有“透明/不损通信/证明不变”的句子删掉或改成更弱的说法。

B4) Sec. IV 开头那句“确保通信性能”要同步改

你们在提出优化策略时写：

“enhance … sensing capability while ensuring the communication performance …” (CE_OFDM_ShiJingCheng (3).pdf)

如果通信不变不再证明、且 sample-wise 可能影响解调，这句话必须改成更稳的版本，例如：

  • “while maintaining low-PAPR; the communication impact is evaluated in Sec. V”
    或
  • “assuming θ\theta is known to the communication receiver (pre-compensated), communication performance can be preserved”
    （两种写法选其一，取决于你们想不想给接收端加一行“去相位”的前处理。）

B5) 结论：与“通信约束/不牺牲通信”有关的句子要弱化或删除

结论里有：

  • “Subject to power-efficiency and communication-performance constraints …” (CE_OFDM_ShiJingCheng (3).pdf)
  • “without compromising communication performance …” (CE_OFDM_ShiJingCheng (3).pdf)

如果你们不再证明通信不变，这些也要同步改口径（否则审稿人会问：你约束了什么？怎么保证的？）。

C. 建议改的部分（提高一致性与可读性，审稿体验会好很多）

C1) 把“phase rotation”统一改叫“sample-wise phase shaping/sequence”

你们系统模型里 θ\theta 已经是向量形式 (CE_OFDM_ShiJingCheng (3).pdf)，优化里也是 θn∈(0,2π]\theta_n\in(0,2\pi] (CE_OFDM_ShiJingCheng (3).pdf)。
那正文里凡是出现 “symbol-wise phase rotation” 的地方都建议统一成：

  • “time-domain sample-wise phase sequence θ\theta”
  • 或 “phase shaping vector θ\theta”
    这样不会再和 Thompson 那种“block-wise memory term”混淆。

C2) 索引符号建议统一（你们现在的 n 很容易两种含义混用）

系统模型里写 “Let n serve as the index for time-domain symbols” (CE_OFDM_ShiJingCheng (3).pdf)，但优化里又用 n=1,…,Nn=1,\dots,N 表示 θ\theta 的分量 (CE_OFDM_ShiJingCheng (3).pdf)；而通信节里又出现 “n-th symbol” (CE_OFDM_ShiJingCheng (3).pdf)。
既然你们要删 IV-A，这个混用会少一大块，但我仍建议你们把：

  • 符号内采样点：用 m=0,…,N−1m=0,\dots,N-1
  • OFDM 符号/slot：用 ℓ\ell
    这样后面任何涉及“coherent integration over slots” (CE_OFDM_ShiJingCheng (3).pdf) 都会更清晰。

D. 一个“现实建议”：即便删证明，也别完全不提通信影响

你们可以不做 Proposition 5 那种“严格零影响”的证明（sample-wise 下确实很难“零影响且无需改接收机”），但最好留 3–5 行解释，否则 TWC 审稿人几乎一定会问：

你优化 θ\theta 压旁瓣，会不会把通信 BER/SE 打坏？你怎么处理？

最省事、最稳的一种写法是（不需要长推导）：

  • 说明 θ\theta 是确定性已知序列（离线设计/预共享/低速控制信令下发）；
  • 通信端在进入 FFT/相关器前做一次 de-rotation / subtract known θ\theta（你们相位域模型里其实就是一行减法），随后按常规 CE-OFDM 解调；
  • 然后在 Sec. V 给一张 BER 曲线支撑“几乎不变”。

这样你们仍能在引言里保留“不会明显牺牲通信”的叙述，但口径会从“天然透明”变成“可补偿/可保持”。

如果你愿意，我们下一步可以按这个清单一起逐段改：我建议先从 摘要 + 引言 + Sec. IV-A 删除/改写 + 结论 这四块下手（这些是“前后不一致”的源头）。你想把 Sec. IV-A 整节删除，还是保留一个 短的 communication impact discussion（不做 Proposition）？

Question

我们在system model里面把theta的定义修改完善一下：while θ=[θ1,θ2,…,θN]T∈RN×1\boldsymbol{\theta}=[\theta_1,\theta_2,\dots,\theta_{N}]^T\in\mathbb{R}^{N\times1} denotes the deterministic phase vector to be optimized for sensing, which is known to the communication receiver, thus communication performance could be preserved.
主要想说 θ 是确定性已知序列，是离线设计的。然后我们再看看通信端在进入 FFT/相关器前做一次 de-rotation / subtract known
θ
θ（你们相位域模型里其实就是一行减法），随后按常规 CE-OFDM 解调；这句话加在哪里，我想加在下面的部分：
For the communication receiver under the additive white Gaussian noise (AWGN) condition, the received signal after cyclic prefix (CP) removal could be expressed as:
\begin{equation}
\mathbf{y}_{\mathrm{c}} = \sqrt{\rho}\mathbf{x}+\mathbf{z},
\label{commsignal}
\end{equation}
where ρ\rho is the signal-to-noise ratio (SNR), and z∼CN(0,IN)\mathbf{z}\sim\mathcal{C}\mathcal{N}(\mathbf{0},\mathbf{I}_N) denotes the white Gaussian noise vector with zero mean and unit variance.
另外，结论部分关于theta和通信性能的表述也进行对应的修改：
In this paper, we consider power-limited ISAC scenarios and conduct a systematic comparison of multiple communication-centric candidate modulation schemes, with particular emphasis on CE-OFDM, which exhibits a 0-dB PAPR. To investigate how random data-bearing waveforms in CE-OFDM can simultaneously support communication and sensing, we aim to reshape the statistical characteristics of the ACFs so as to improve sensing capability. Under a Nyquist pulse-shaping framework, and accounting for the nonlinear mapping intrinsic to CE-OFDM, we derive a closed-form expression for the expectation of the squared ACF of the resulting random ISAC signal. The derived result reveals a decomposition into a squared
mean term dictated by pulse shaping and a variance term induced by the waveform randomness, which could be mitigated via phase rotation. Moreover, by incorporating a coherent integration analysis, we establish key statistical laws that yield a computable objective function for subsequent structured waveform design.

Building upon these analytical findings, we propose a joint optimization framework that couples phase rotation and pulse shaping. Subject to power-efficiency and communication-performance constraints, the proposed design suppresses correlation sidelobes over a prescribed delay interval, thereby enhancing range resolvability. Numerical results demonstrate that the proposed joint design significantly reduces the average squared-ACF sidelobe level of CE-OFDM within the delay region of interest, while preserving its low-PAPR advantage and maintaining stable communication performance. These observations substantiate CE-OFDM as a compelling solution for ISAC system design in power-constrained scenarios, with the proposed optimization methodology effectively enhancing distance sensing capabilities without compromising communication performance or sacrificing inherent power efficiency advantages.

Thought

Citing system model and de-rotation insertion

Now that the conclusion has proper citations, I need to align the citations throughout, especially the system model definition. I'll use earlier citation markers (turn1file0) for consistency. When adding the de-rotation equation after equation (10), I'll reference that formula properly, placing citations outside LaTeX blocks to avoid issues.

Crafting suggestions for clarity and consistency

I want to refine their θ definition. I'd change "known to the communication receiver" to "pre-shared and synchronized," ensuring clarity. The AWGN aspect can be emphasized: element-wise de-rotation preserves noise distribution. After equation (10), I'd suggest adding equation (10a) with the appropriate references. Then, if the communication section IV-A stays, it should be slightly adjusted.

Proposing updates for clarity and alignment

I suggest keeping a short subsection on "Communication impact" with 2-3 paragraphs and a de-rotation equation. I'd remove the symbol-wise Proposition 5 or move it to a remark. I'll also revise the abstract, intro, and conclusion to align with these changes. We can update terms like "phase rotation" to "sample-wise phase shaping" for clarity, and add the de-rotation block to Fig 1. Additionally, we'll clarify the conditions for the frequency-selective channel.

Proposing de-rotation and fading adjustments

For the AWGN model (10), de-rotation works well, but for fading, it could be more applicable after equalization and simulation. I’ll suggest inserting the de-rotation explanation right after equation (10), before the demodulation discussion. The text will read: "Given θ is deterministic and known at Rx, we can compensate it via element-wise de-rotation..."

Proposing conclusion and system model updates

For the conclusion, I suggest wording it like this: "Subject to power-efficiency constraints and assuming the optimized deterministic phase sequence is available at the communication receiver for de-rotation/compensation, the proposed design..." Also, I’ll make sure to adjust the communication performance sections and refine mentions of system model constraints and θ.

Thought for 2m 16s

Answer

你这套改法（sample-wise θ\theta + θ\theta 为离线确定性序列且通信端已知）是合理的，而且不会动到你们前面“感知统计推导”的主链条；需要改的是通信相关表述/命题的口径，把原来“天然透明（symbol-wise 才成立）”统一换成“已知可补偿”。

下面按你给的两处文字，给你一个“放哪里、怎么写、还要同步改哪里”的清单。

1) System model 里 θ\theta 的定义：建议这样写更稳

你准备加的这句方向是对的（你们本来就把 θ\theta 当长度 NN 的设计向量来优化）(CE_OFDM_ShiJingCheng (3).pdf)。我建议把最后半句写得更“可审稿”（强调假设 + 补偿方式 + 适用条件），比如：

while θ=[θ1,…,θN]T∈RN×1\boldsymbol{\theta}=[\theta_1,\ldots,\theta_N]^T\in\mathbb{R}^{N\times1} denotes a deterministic sample-wise phase sequence optimized offline for sensing. We assume that θ\boldsymbol{\theta} is available at the communication receiver (e.g., pre-shared or signaled over a low-rate control channel), so that its effect can be compensated prior to standard CE-OFDM demodulation (especially under AWGN/flat-fading conditions).

这样写的好处是：你不会“硬承诺”在所有信道下都完全不影响通信，但也把“可保持通信性能”的逻辑讲通了。

2) “de-rotation / subtract known θ\theta”这句话放哪里最合适？

你想放在 (10) 之后，这个位置非常合适：因为 (9) 里 x\mathbf{x} 的确含有逐采样点的 ejθe^{j\boldsymbol{\theta}} 因子-L27，而 (10) 给的是 AWGN 下 CP removal 后的离散模型(CE_OFDM_ShiJingCheng (3).pdf)。

建议你在 (10) 后面紧跟一个“去相位”公式（比纯文字更有说服力）

例如新增一式（不妨编号为 (11)）：

yc(dr)≜yc⊙e−jθ=ρ A~exp⁡ ⁣(jh~FNHv)+z(dr),\mathbf{y}_{\mathrm{c}}^{(\mathrm{dr})} \triangleq \mathbf{y}_{\mathrm{c}} \odot e^{-j\boldsymbol{\theta}} = \sqrt{\rho}\,\tilde A \exp\!\big(j\tilde h\mathbf{F}_N^H\mathbf{v}\big) + \mathbf{z}^{(\mathrm{dr})},

其中 z(dr)=z⊙e−jθ∼CN(0,IN)\mathbf{z}^{(\mathrm{dr})}=\mathbf{z}\odot e^{-j\boldsymbol{\theta}}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_N)（圆对称复高斯乘单位模相位不变）。

然后接一句话就够了：

After de-rotation, the receiver can follow the conventional CE-OFDM demodulation chain.

这段刚好把你“已知可补偿 ⇒ 通信可保持”的论点闭环，而且完全贴合你们的 (9)(10) 模型-L27(CE_OFDM_ShiJingCheng (3).pdf)。

如果你们通信接收链更偏“相位域”（你们后面也写了类似 r~=⋯+θ+ε\tilde{\mathbf{r}}=\cdots+\boldsymbol{\theta}+\varepsilon 的形式）(CE_OFDM_ShiJingCheng (3).pdf)，那也可以写成“phase-demod 后做 r~−θ\tilde{\mathbf{r}}-\boldsymbol{\theta}”，本质等价。

3) 需要同步改的“冲突点”（改完才不会前后打架）

3.1 处理掉 “symbol-wise phase rotation / 通信天然透明”这条旧口径

你们引言/贡献里目前明确写了“transparent to CE-OFDM demodulation… does not compromise communication performance”(CE_OFDM_ShiJingCheng (3).pdf)，这其实是靠你们后面 symbol-wise 特例的 Proposition 5 在撑。

但你现在要走 sample-wise，所以必须把这些句子统一改成：

  • “θ\theta is deterministic and known at the receiver; its effect can be compensated (de-rotation)” 而不要再写“天然透明”。

3.2 Sec. IV-A（Proposition 5）要么删、要么改成“已知可补偿”

你们当前 Proposition 5 的假设是“symbol-wise phase rotation”且 θ[n]=θn1\theta[n]=\theta_n\mathbf{1} 才推出 Θn,k=0\Theta_{n,k}=0(CE_OFDM_ShiJingCheng (3).pdf)；这和你们现在的 sample-wise 设计目标会直接矛盾。

两种处理方式，二选一：

  1. 删掉整节 IV-A，并把所有“proved does not impair communication”改成“can be compensated / evaluated numerically”；
  2. 保留一小节但重写：把命题改成“任意 deterministic sample-wise θ\theta，只要 Rx 已知则通过 de-rotation 可恢复常规解调统计（AWGN 下严格成立）”，并把原来的 symbol-wise 当 Remark。

如果你结论里还要说“maintaining stable communication performance”，我更推荐 方案 2（哪怕只写 6–8 行），审稿人会更舒服。

4) 结论段落你这版怎么改最匹配（给你一个直接可替换版本）

你们当前结论里这几句需要同步改口径：
“Subject to power-efficiency and communication-performance constraints … maintaining stable communication performance … without compromising communication performance …”(CE_OFDM_ShiJingCheng (3).pdf)

建议改成下面这种（核心是把“通信约束”换成“已知可补偿假设 + 仿真验证”）：

  • 把 “communication-performance constraints” 改成
    “subject to power-efficiency constraints and assuming the optimized deterministic phase sequence is available for receiver-side de-rotation”；
  • 最后一句把 “without compromising communication performance” 改成
    “while maintaining stable BER performance after de-rotation (as validated numerically)”（你们 BER 图本来就在数值部分展示了）(CE_OFDM_ShiJingCheng (3).pdf)。

5) 这次改动对前面感知推导公式要不要“整体重推”？

不需要。因为你们的感知统计推导本来就是把 θ\theta 当 长度 NN 的设计向量来处理的（通过 p~(θ)\tilde p(\theta) 进入 Sk(g,θ)S_k(g,\theta) 的那条主线），这和 sample-wise 完全一致(thompson-thesis-ce-ofdm-final.pdf)。
真正需要动的，是把“symbol-wise 透明”相关的文字、命题、以及摘要/引言/结论里“绝对不影响通信”的措辞统一改成“已知可补偿”。

如果你愿意，我下一步可以按你们 PDF 的结构，把 (i) System model 的 θ\theta 定义 + (ii) (10) 后的 de-rotation 段落 + (iii) IV-A 的最短重写版本（或删节方案）+ (iv) 摘要/引言/结论对应替换句 全部给你整理成一组可直接粘贴的 LaTeX 文段。你更倾向 删掉 IV-A 还是 保留一个简短“de-rotation 命题/讨论”？

Question

把下面这一段公式10前后看看应该怎么修改：
For the communication receiver under the additive white Gaussian noise (AWGN) condition, the received signal after cyclic prefix (CP) removal and could be expressed as:
\begin{equation}
\mathbf{y}_{\mathrm{c}} = \sqrt{\rho}\mathbf{x}+\mathbf{z},
\label{commsignal}
\end{equation}
where ρ\rho is the signal-to-noise ratio (SNR), and z∼CN(0,IN)\mathbf{z}\sim\mathcal{C}\mathcal{N}(\mathbf{0},\mathbf{I}_N) denotes the white Gaussian noise vector with zero mean and unit variance.
加上After phase de-rotation, the receiver can follow the conventional CE-OFDM demodulation chain.

Answer

Crafting modified LaTeX and clarifying equation details

I found the relevant text with equation (10). Now I need to modify the LaTeX paragraph to include the de-rotation step and fix a bit of grammar. I'll adjust notation for better clarity and ensure consistency between the signal dimensions under AWGN. The tricky bit is deciding when to apply de-rotation — either before or after phase demodulation.

Clarifying phase de-rotation approach for the receiver

I’m considering the phase demodulation step in the receiver. The user wants to add a sentence about phase de-rotation after equation (10). The method of applying de-rotation on the complex samples before demodulation needs to be clear, along with the correct notation. I’ll also mention noise and the conventional demodulation chain.

Crafting revised text with de-rotation and phase demodulation

I’ll revise the text around equations (9) and (10) to include the de-rotation step for the communication receiver under AWGN. My goal is to write it smoothly, stating that after phase de-rotation, conventional CE-OFDM demodulation applies. I'll also ensure to incorporate the correct notation and equation detail consistency.

Question

把下面这一段公式10前后看看应该怎么修改：
For the communication receiver under the additive white Gaussian noise (AWGN) condition, the received signal after cyclic prefix (CP) removal and could be expressed as:
\begin{equation}
\mathbf{y}_{\mathrm{c}} = \sqrt{\rho}\mathbf{x}+\mathbf{z},
\label{commsignal}
\end{equation}
where ρ\rho is the signal-to-noise ratio (SNR), and z∼CN(0,IN)\mathbf{z}\sim\mathcal{C}\mathcal{N}(\mathbf{0},\mathbf{I}_N) denotes the white Gaussian noise vector with zero mean and unit variance.
加上After phase de-rotation, the receiver can follow the conventional CE-OFDM demodulation chain.

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Thought

Using equation from CE_OFDM for paragraph improvements

The search results show the relevant section from the CE_OFDM PDF (lines 66-74) with equation (10). I'll use this to refine the paragraph and introduce the updated equation for de-rotation. I'll correct the grammar and include the definition for yc(dr)y_c^{(dr)} or y~c\tilde{y}_c. The notations can be clarified based on equation (9) from the draft!

Describing phase de-rotation in CE-OFDM

In the standard CE-OFDM demodulation, we expect the signal xx without θ\theta, but phase de-rotation is necessary. I'll clarify that after phase de-rotation, the received signal follows the conventional baseband model. We can propose adding this statement and equation after equation (10).

Finalizing LaTeX and citations

I’ll mention that ⊙\odot indicates the Hadamard product, unless already defined earlier. We'll cite lines 66-74 from the relevant chunk and ensure the elementwise exponential definition is clear with exp⁡(−jθ)\exp(-j\boldsymbol{\theta}). Additionally, I'll note the noise’s circular symmetry. After that, I’ll finalize the LaTeX block and provide the citations outside the LaTeX code.

Thought for 1m 16s

Answer

你这段放在(10)后面加 de-rotation 很合适；同时把原句里多余的 “and” 去掉，并把“de-rotation”写成一条明确的向量运算 + 噪声统计不变，审稿人会更容易接受。你们原文的(10)位置在系统模型里这一段附近。(CE_OFDM_ShiJingCheng (3).pdf)

建议改成下面这样（直接可粘贴）：

latex
For the communication receiver under the additive white Gaussian noise (AWGN) condition, the received signal after cyclic prefix (CP) removal can be expressed as
\begin{equation}
    \mathbf{y}_{\mathrm{c}} = \sqrt{\rho}\mathbf{x}+\mathbf{z},
\label{commsignal}
\end{equation}
where $\rho$ is the signal-to-noise ratio (SNR), and $\mathbf{z}\sim\mathcal{C}\mathcal{N}(\mathbf{0},\mathbf{I}_N)$ denotes the white Gaussian noise vector with zero mean and unit variance.

Since the deterministic phase sequence $\boldsymbol{\theta}$ is known to the communication receiver, an element-wise phase de-rotation can be applied as
\begin{equation}
\mathbf{y}_{\mathrm{c}}^{(\mathrm{dr})} \triangleq \mathbf{y}_{\mathrm{c}} \odot \exp(-j\boldsymbol{\theta})
= \sqrt{\rho}\,\tilde A \exp\!\left(j\tilde h\,\mathbf{F}_N^{H}\mathbf{v}\right) + \mathbf{z}^{(\mathrm{dr})},
\label{eq:derotation}
\end{equation}
where $\odot$ denotes the Hadamard product and $\mathbf{z}^{(\mathrm{dr})} \triangleq \mathbf{z}\odot \exp(-j\boldsymbol{\theta}) \sim \mathcal{C}\mathcal{N}(\mathbf{0},\mathbf{I}_N)$.
After phase de-rotation, the receiver can follow the conventional CE-OFDM demodulation chain.

要点：

  • de-rotation 用 乘以 exp⁡(−jθ)\exp(-j\boldsymbol{\theta})（逐元素）更符合“相位去旋转”的直觉；
  • 顺手说明噪声仍是 CN(0,IN)\mathcal{CN}(\mathbf{0},\mathbf{I}_N)，这样你们后面说“通信性能可保持”更站得住。

Question

我们直接用语言表达，不用写公式。用1句话来说

Thought

Thought for 12s

Answer

Since the deterministic sample-wise phase sequence θ\boldsymbol{\theta} is known at the communication receiver, it can be compensated by performing an element-wise phase de-rotation on yc\mathbf{y}_{\mathrm{c}} prior to the subsequent demodulation (e.g., FFT/phase demodulation), after which the receiver can follow the conventional CE-OFDM demodulation chain. (CE_OFDM_ShiJingCheng (3).pdf)

Question

我们接下来要修改下面的section。现在我们计划把第一个subsection删掉，然后在这一章节讲一下后面优化问题中g的限制条件，或者看看应该改成写点什么别的，但最好别太长：

\section{Optimization of the Random CE-OFDM ACF}
In this section, we propose a joint phase rotation and pulse design optimization strategy for CE-OFDM signals, which could enhance the distance sensing capability while maintaining low-PAPR.

\subsection{Communication Performance Invariant to Phase Rotation}
As observed from the analytical derivation of the CE-OFDM ACF in the section above, phase rotation can effectively enhance the system's sensing capability. However, due to the inherent framework of phase modulation, it is essential to first investigate the impact of extra phase on system communication performance before exploring specific optimization schemes.

Based on \eqref{signal} and \eqref{commsignal}, the received phase vector can be modeled as:
\begin{equation}
\tilde{\mathbf{r}}
= \tilde{h} C_N \mathbf{F}N^H \mathbf{v}+ \boldsymbol{\theta}+ \boldsymbol \varepsilon,
\label{eq:disc_model_phase}
\end{equation}
where ε∈RN×1\boldsymbol \varepsilon\in\mathbb{R}^{N\times 1} represents the phase noise. Then, the kk-th correlator for the nn-th symbol in the demodulator is given by:
\begin{align}
z{n,k}
&= \frac{1}{N}\mathbf f_k^{H}\mathbf J'{\tau}\tilde{\mathbf r}[n]
\notag\
&=\underbrace{\frac{1}{N}\mathbf f_k^{H}\mathbf J'{\tau}\tilde h C_N \mathbf{F}N^H\mathbf{v}[n]}{H_{n,k}}
+
\underbrace{\frac{1}{N}\mathbf f_k^{H}\mathbf J'{\tau}\boldsymbol{\theta}[n]}{\Theta_{n,k}}
+
\underbrace{\frac{1}{N}\mathbf f_k^{H}\mathbf J'{\tau}\boldsymbol{\varepsilon}[n]}{W_{n,k}},
\label{comm}
\end{align}
where zn,kz_{n,k} represents the kk-th branch output associated with the nn-th symbol, Hn,kH_{n,k} and Wn,kW_{n,k} represent the contributions of the desired signal and noise, respectively, while Θn,k\Theta_{n,k} captures the impact of phase rotation on the communication.

\begin{proposition}
For CE-OFDM ISAC system that turns off the DC component and applies phase rotation in a symbol-wise manner, the phase-rotation term in \eqref{comm} vanishes for any time shift τ\tau and any active subcarrier index kk:
\begin{equation}
\Theta_{n,k} = \frac{1}{N}\mathbf f_k^{H}\mathbf J'_{\tau}\boldsymbol{\theta}[n]
= 0, ; \forall;k\in{1,2,\dots,N-1},;\forall;\tau.
\label{eq:theta_zero_nonDC}
\end{equation}
\end{proposition}

\begin{IEEEproof}
Since Jτ′\mathbf J'_{\tau} is a cyclic-shift matrix, which satisfies:
\begin{equation}
\mathbf J'\tau \mathbf 1=\mathbf 1,\quad \forall;\tau.
\label{circ}
\end{equation}
Under symbol-wise phase rotation, the phase vector corresponding to the nn-th OFDM symbol takes the form:
\begin{equation}
\boldsymbol{\theta}[n] = \theta_n\mathbf 1,\quad \theta_n\in\mathbb{R}.
\label{time_domain}
\end{equation}
Hence,
\begin{equation}
\mathbf{J}'{\tau}\boldsymbol{\theta}[n]
= \theta_n \mathbf{J}'{\tau}\mathbf{1}
= \theta_n \mathbf{1}.
\end{equation}
Since fk\mathbf{f}_k denotes the kk-th column of the NN-point DFT matrix FN\mathbf{F}_N, it follows that:
\begin{equation}
\mathbf{f}k^{H}\mathbf{1}
= \sum{m=0}^{N-1} e^{j2\pi km/N}
= 0,\quad k\in{1,2,\dots,N-1}.
\end{equation}
Using the properties above, Θn,k\Theta_{n,k} can be further derived as:
\begin{align}
\Theta{n,k}
&= \frac{1}{N}\mathbf{f}k^{H}\mathbf{J}'{\tau}\boldsymbol{\theta}[n]
= \frac{\theta_n}{N}\mathbf{f}_k^{H}\mathbf{1}
= 0.
\end{align}
\end{IEEEproof}

Following the standard CE-OFDM setting, the symbol wise phase vector θ\boldsymbol{\theta} is completely transparent to the communication demodulator which does not alter Hn,kH_{n,k}, Wn,kW_{n,k}, or the BER statistics for any active subcarrier. Therefore, the phase rotation vector could be freely optimized across symbols to serve sensing objectives without degrading per-symbol detection or violating the PAPR constraint.

\subsection{Non-Convex Constrained Problem}
Our focus is the joint design of Nyquist pulse shaping and phase rotation for random CE-OFDM signal. This joint design naturally yields two coupled subproblems, namely, the pulse-shaping design in g\mathbf g and the phase-rotation design in θ\boldsymbol\theta, which will be handled in a structured manner in the sequel.

To ensure that the folded power-spectrum vector g\mathbf g corresponds to a band-limited Nyquist pulse with roll-off factor α\alpha, we enforce standard discrete spectral constraints. Specifically, the non-roll-off portions are fixed to 0/10/1 to preserve the band edges, and the roll-off part is constrained to be monotone to prevent spectral notches. The area constraint maintains a constant pulse energy, while the box constraint 0≤gn≤10\le g_n\le 1 ensures realizability. Since these constraints are linear in g\mathbf g, the pulse-shaping subproblem remains convex when θ\boldsymbol\theta is fixed.

Over a prescribed delay region of interest k∈[K1,K2]k\in[K_1,K_2], we adopt the sum of the average squared ACF after LcL_c coherent integrations, J(g,θ)J(\mathbf g,\boldsymbol{\theta}), as the performance metric and formulate the optimization problem as:
\begin{equation}
\begin{aligned}
\underset{\boldsymbol{\theta},\mathbf{g}}{\min}\quad
& J(\mathbf{g},\boldsymbol{\theta})\triangleq \sum_{k=K_1}^{K_2}\mathbb{E}{|\bar{R}{k}|^2}=\sum{k=K_1}^{K_2}\big[\mathcal{I}k(\mathbf g)+\frac{1}{L_c}\mathcal{S}k(\mathbf g,\boldsymbol\theta) \big] \
\text{s.t.}\quad
&\theta_n \in (0,2\pi], \quad n = 1,\dots,N,\
& g_n = 0,\quad n = 1,\dots,(N-N\alpha)/2, \
& g_n = 1,\quad n = (N+N\alpha)/2+1,\dots,N, \
& g{n+1}-g_n \ge 0, \quad n = 1,\dots,N-1, \
& \sum{n=1}^{N} g_n = \frac{N}{2}, \quad 0 \le g_n \le 1,\quad n = 1,\dots,N.
\end{aligned}
\label{opt_joint_pulse_phase}
\end{equation}
where Ik(g)\mathcal I_k(\mathbf g) and Sk(g,θ)\mathcal S_k(\mathbf g,\boldsymbol\theta) are given in \eqref{eq:Ik_def} and \eqref{eq:Sk_quad_form}.

For fixed θ\boldsymbol{\theta}, J(g,θ)J(\mathbf g,\boldsymbol\theta) reduces to a convex quadratic function of g\mathbf g over linear constraints, hence the pulse-design subproblem is a convex QP solvable by off-the-shelf QP solvers. In contrast, for fixed g\mathbf g, the dependence of Sk(g,θ)\mathcal S_k(\mathbf g,\boldsymbol\theta) on the phase-only vector θ\boldsymbol\theta renders the phase-design subproblem non-convex.

%For the CE-OFDM system, the intrinsic nonlinear phase modulation induces a highly non-convex dependence of the ACF on the phase parameters. In particular, the nonlinear mapping from the phase modulation to the ACF produces a strongly asymmetric response, while a larger modulation index amplifies fluctuations of the objective in the phase space. Moreover, the higher-order nonlinearity of the Bessel functions, combined with the multi-frequency oscillatory terms introduced by phase differences, generates multiple local extrema in the joint delay–phase domain. As a result, the ACF-based objective in \eqref{opt} exhibits pronounced non-convexity along essentially any parameter cross-section.

\subsection{Two-Stage Optimization Algorithm}
As discussed before, the joint problem in \eqref{opt_joint_pulse_phase} is convex in the folded power-spectrum vector g\mathbf g for fixed θ\boldsymbol{\theta}, but highly non-convex in θ\boldsymbol{\theta} for fixed g\mathbf g. This motivates a two-stage optimization strategy that first designs the Nyquist pulse via convex programming and then optimizes the CE-OFDM phase rotation by a dedicated non-convex algorithm.

\textbf{Stage 1 (pulse shaping design).}
We fix a nominal phase vector θ(0)\boldsymbol{\theta}^{(0)} and optimize g\mathbf g in \eqref{opt_joint_pulse_phase}.
With θ=θ(0)\boldsymbol{\theta}=\boldsymbol{\theta}^{(0)}, the objective can be written as:
\begin{equation}
\min_{\mathbf g}\ \sum_{k=K_1}^{K_2}\left[\mathcal I_k(\mathbf g)+\frac{1}{L_c}\mathcal S_k(\mathbf g,\boldsymbol\theta^{(0)})\right].
\end{equation}
Since both Ik(g)\mathcal I_k(\mathbf g) and Sk(g,θ(0))\mathcal S_k(\mathbf g,\boldsymbol\theta^{(0)}) are quadratic in g\mathbf g and the constraints are linear, the pulse-design subproblem is a convex quadratic program (QP), which we solve via CVX to obtain an optimized Nyquist pulse g⋆\mathbf g^\star.

\textbf{Stage 2 (phase rotation design).}
We keep g⋆\mathbf g^\star fixed and optimize the phase vector.
Note that ∑k=K1K2Ik(g⋆)\sum_{k=K_1}^{K_2}\mathcal I_k(\mathbf g^\star) is independent of θ\boldsymbol\theta. Then the phase design is equivalently cast as:
\begin{equation}
\min_{\boldsymbol\theta\in[0,2\pi)^N}\ \mathcal{F}(\boldsymbol{\theta})
\triangleq \sum_{k=K_1}^{K_2}\mathcal S_k(\mathbf g^\star,\boldsymbol\theta).
\label{eq:phase_obj_F}
\end{equation}
Due to the intrinsic nonlinear phase structure of CE-OFDM, F(θ)\mathcal{F}(\boldsymbol{\theta}) is highly oscillatory and multimodal over the phase torus, and purely gradient-based methods may be trapped in poor local minima.
We therefore employ a swarm-based scheme with adaptive-moment gradient refinement, combining global exploration with a Rectified-Adam (RAdam) update.

For clarity, the proposed algorithm can be interpreted as an enhanced version of particle swarm optimization (PSO), and the neighbor-set Ni\mathcal{N}_i for the ii-th particle at the tt-th iteration is defined by
\begin{equation}
\mathcal{N}_i=\left{j \mid \mathcal{F}_j^{(t-1)} < \mathcal{F}i^{(t-1)} - \Delta{th}\right},
\end{equation}
where Fj(t−1)\mathcal{F}_j^{(t-1)} denotes the fitness of particle jj at iteration t−1t-1, and Δth\Delta_{th} is the fitness difference threshold for selecting significantly superior neighbors.

Then, the velocity of particle ii is updated via
\begin{equation}
\boldsymbol{\omega}_i^{(t)}
=\zeta^{(t)} \boldsymbol{\omega}_i^{(t-1)}

  • c_1 \Delta\boldsymbol{\theta}_{\text{g}}^{(t)}
  • c_2 \Delta\boldsymbol{\theta}_{\text{p}}^{(t)}
  • c_3 \Delta\boldsymbol{\theta}{\text{a}}^{(t)},
    \end{equation}
    where
    \begin{equation}
    \Delta\boldsymbol{\theta}    {\text{g}}^{(t)}
    =r_1 \left(\boldsymbol{\theta}{\text{g}} \ominus \boldsymbol{\theta}i^{(t-1)}\right),
    \end{equation}
    \begin{equation}
    \Delta\boldsymbol{\theta}    {\text{p}}^{(t)}
    =r_2 \left(\boldsymbol{\theta}    {\text{p},i} \ominus \boldsymbol{\theta}i^{(t-1)}\right),
    \end{equation}
    \begin{equation}
    \Delta\boldsymbol{\theta}    {\text{a}}^{(t)}
    =\sum_{j\in\mathcal{N}i} \frac{\beta_0}{1+\gamma d{ij}^2}
    \left(\boldsymbol{\theta}_j^{(t-1)} \ominus \boldsymbol{\theta}_i^{(t-1)}\right),
    \end{equation}
    with ζ(t)\zeta^{(t)} the inertia weight, c1,c2,c3c_1,c_2,c_3 the acceleration coefficients, and θg\boldsymbol{\theta}_{\text{g}} and θp,i\boldsymbol{\theta}_{\text{p},i} representing the global and ii-th particle's historical optimal phase vectors, respectively.
    The phase operator ⊖\ominus is defined as a⊖b≜arg⁡ ⁣[ej(a−b)]\mathbf{a} \ominus \mathbf{b} \triangleq \arg\!\left[e^{j(\mathbf{a}-\mathbf{b})}\right], with γ\gamma the absorption coefficient and β0\beta_0 the maximum attraction intensity.
    Moreover, dij=1N∥θi⊖θj∥d_{ij} = \frac{1}{N}\|\boldsymbol{\theta}_i \ominus \boldsymbol{\theta}_j\| denotes the normalized phase difference.

Then, the gradient is computed as
\begin{equation}
\nabla\mathcal{F}i^{(t)}
= \nabla{\boldsymbol{\theta}i^{(t-1)}} \sum{k=K_1}^{K_2}
\mathcal S_k!\left(\mathbf g^\star,\boldsymbol{\theta}_i^{(t-1)}\right).
\label{eq:grad_F}
\end{equation}
The first-order momentum and second-order variance are updated by
\begin{equation}
\mathbf{m}_i^{(t)} = \beta_1 \mathbf{m}_i^{(t-1)} + (1-\beta_1)\nabla\mathcal{F}_i^{(t)},
\end{equation}
\begin{equation}
\mathbf{v}_i^{(t)} = \beta_2 \mathbf{v}_i^{(t-1)} + (1-\beta_2)\left(\nabla\mathcal{F}_i^{(t)}\right)^2,
\end{equation}
where β1,β2∈(0,1)\beta_1,\beta_2 \in (0,1) are the RAdam momentum coefficients.
The phase of particle ii is subsequently updated as
\begin{equation}
\tilde{\boldsymbol{\theta}}_i^{(t)}
= \boldsymbol{\theta}_i^{(t-1)}

  • \eta_{\text{b}}\boldsymbol{\omega}_i^{(t)}
  • \eta_{\text{r}}\tilde{\beta}\mathbf{m}_i^{(t)}/\left(\sqrt{\mathbf{v}_i^{(t)}} + \epsilon\right),
    \end{equation}
    where β~=1−β2/(1−β1)\tilde{\beta}=\sqrt{1-\beta_2}/(1-\beta_1), ϵ\epsilon is the numerical stability constant, and ηb\eta_{\text{b}} and ηr\eta_{\text{r}} denote the base and rectified learning rates, respectively.
    Finally, phase normalization is performed by
    \begin{equation}
    \boldsymbol{\theta}_i^{(t)} = \left[\tilde{\boldsymbol{\theta}}i^{(t)}\right]{2\pi}
    \triangleq \tilde{\boldsymbol{\theta}}_i^{(t)} - 2\pi\left\lfloor\frac{\tilde{\boldsymbol{\theta}}_i^{(t)}}{2\pi}\right\rfloor.
    \end{equation}

The complete workflow of the algorithm is summarized in Algorithm~\ref{alg:psf_RAmam}.

\begin{algorithm}[h]
\caption{The Two-Stage Optimization Algorithm.}
\label{alg:psf_RAmam}
\begin{algorithmic}[1]
\STATE \textbf{Input:} maximum iterations TT, population size II, threshold Δth\Delta_{th}, absorption γ\gamma, attraction β0\beta_0, acceleration c1,c2,c3c_1,c_2,c_3, momentum β1,β2\beta_1,\beta_2, learning rates ηb,ηr\eta_{\text{b}}, \eta_{\text{r}}, nominal phase vector θ(0)\boldsymbol{\theta}^{(0)};

\STATE \textbf{Stage 1 (CVX pulse optimization):} Fix θ=θ(0)\boldsymbol{\theta}=\boldsymbol{\theta}^{(0)} and solve \eqref{opt_joint_pulse_phase} w.r.t. g\mathbf g via CVX to obtain g⋆\mathbf g^\star;

\STATE \textbf{Stage 2 (phase design):} Fix g=g⋆\mathbf g=\mathbf g^\star, F(θ)≜J(g⋆,θ)\mathcal{F}(\boldsymbol{\theta}) \triangleq J(\mathbf g^\star,\boldsymbol{\theta});

\STATE Initialize: {θi(0)}\{\boldsymbol{\theta}_i^{(0)}\}, ωi(0)=0\boldsymbol{\omega}_i^{(0)} = \mathbf{0}, θp,i=θi(0)\boldsymbol{\theta}_{\text{p},i} = \boldsymbol{\theta}_i^{(0)}, θg=arg⁡min⁡Fi\boldsymbol{\theta}_{\text{g}} = \arg\min\mathcal{F}_i, mi(0)=0\mathbf{m}_i^{(0)} = \mathbf{0}, vi(0)=0\mathbf{v}_i^{(0)} = \mathbf{0};

\FOR{t=1t = 1 \textbf{to} TT}
\FOR{i=1i = 1 \textbf{to} II}
\STATE Compute Fi(t−1)=∑k=K1K2Sk(g⋆,θi(t−1))\mathcal{F}_i^{(t-1)} = \sum_{k=K_1}^{K_2} \mathcal S_k(\mathbf g^\star,\boldsymbol{\theta}_i^{(t-1)}).
\STATE Find Ni={j∣Fj(t−1)<Fi(t−1)−Δth}\mathcal{N}_i = \{ j \mid \mathcal{F}_j^{(t-1)} < \mathcal{F}_i^{(t-1)} - \Delta_{th} \}.
\STATE Compute ∇Fi(t)=∇θi(t−1)∑k=K1K2Sk(g⋆,θi(t−1))\nabla\mathcal{F}_i^{(t)} = \nabla_{\boldsymbol{\theta}_i^{(t-1)}}\sum_{k=K_1}^{K_2} \mathcal S_k(\mathbf g^\star,\boldsymbol{\theta}_i^{(t-1)}).
\STATE Update mi(t)=β1mi(t−1)+(1−β1)∇Fi(t)\mathbf{m}_i^{(t)} = \beta_1 \mathbf{m}_i^{(t-1)} + (1-\beta_1)\nabla\mathcal{F}_i^{(t)}.
\STATE Update vi(t)=β2vi(t−1)+(1−β2)(∇Fi(t))2\mathbf{v}_i^{(t)} = \beta_2 \mathbf{v}_i^{(t-1)} + (1-\beta_2)(\nabla\mathcal{F}_i^{(t)})^2.
\STATE Compute ωi(t)=ζ(t)ωi(t−1)+c1Δθg(t)+c2Δθp(t)+c3Δθa(t)\boldsymbol{\omega}_i^{(t)} = \zeta^{(t)} \boldsymbol{\omega}_i^{(t-1)} + c_1 \Delta\boldsymbol{\theta}_{\text{g}}^{(t)} + c_2 \Delta\boldsymbol{\theta}_{\text{p}}^{(t)} + c_3 \Delta\boldsymbol{\theta}_{\text{a}}^{(t)}.
\STATE Update θ~i(t)=θi(t−1)+ηbωi(t)−ηrβ~mi(t)/(vi(t)+ϵ)\tilde{\boldsymbol{\theta}}_i^{(t)} = \boldsymbol{\theta}_i^{(t-1)} + \eta_{\text{b}}\boldsymbol{\omega}_i^{(t)} - \eta_{\text{r}}\tilde{\beta}\mathbf{m}_i^{(t)}/(\sqrt{\mathbf{v}_i^{(t)}}+\epsilon).
\STATE Phase normalization θi(t)=[θ~i(t)]2π\boldsymbol{\theta}_i^{(t)} = [\tilde{\boldsymbol{\theta}}_i^{(t)}]_{2\pi}.
\ENDFOR
\STATE Update θg\boldsymbol{\theta}_{\text{g}} and {θp,i}\{\boldsymbol{\theta}_{\text{p},i}\}.
\STATE Update ζ(t+1)=ζmax−(ζmax−ζmin)[(t+1)/T]\zeta^{(t+1)} = \zeta_{\text{max}}-(\zeta_{\text{max}}-\zeta_{\text{min}})[(t+1)/T].
\ENDFOR
\STATE \textbf{Output:} Optimized parameters g⋆\mathbf g^\star and θg\boldsymbol{\theta}_{\text{g}};
\end{algorithmic}
\end{algorithm}

Thought

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Proposing edit for new subsection on g constraints

To replace the first subsection, I can suggest a short discussion on g constraints. It should focus on the folding power spectrum, symmetry, nulling, and roll-off, and include references to the CE_OFDM PDF for supporting citations. I'll propose titles like "Nyquist Pulse Constraint Set for g" or "Feasible Set of g."

Refining update with constraints and folding criteria

I’ll suggest including the folding criterion, which reduces the optimization design variables and motivates the constraints, and reference the necessary citations. I’ll also incorporate the squared spectrum definition and the de-rotation statement. The current section introduction could stay as is, just keeping focus on enhancing distance sensing with low-PAPR.

Proposing the revised subsection and adjustments

I’ll suggest removing IV-A and replacing it with a new subsection, maybe titled "Design constraints on folded spectrum g" or "Feasible set of g under Nyquist pulse shaping." The introduction can mention the assumption of deterministic θ, keeping communication stable. I'll keep it short and focused on g constraints for optimization.

Expanding and structuring the optimization section

I can suggest providing references to (30) and the nullification criterion. Then, in the optimization section, I'll note the simplified constraints like g_n=0 for the stopband, g_n=1 for the passband, monotonic roll-off, and energy constraints. I'll mention they're linear; hence it’s a convex QP, though it’s already in the non-convex constraints paragraph. Perhaps we can split the section:

  • Subsection A: “Feasible Set of Nyquist Pulse Spectrum”
  • Subsection B: “Non-Convex Constrained Problem”

Proposing section rewrite with g constraints

I'll propose adding the explanation of g constraints before the optimization, possibly under the "Non-Convex Constrained Problem" section to avoid duplication. We can rename the section to "Joint Phase-Sequence and Pulse-Shaping Optimization." The text will describe g constraints as the folded power spectrum, and include spectral constraints (monotonicity, energy summation, etc.), linear constraints preserving convexity. A concise, detailed edit plan and suggested replacement text in Chinese will be prepared. Citations will be included.

Proposing structure and modifications

I need to be mindful regarding citations and LaTeX code. I can't include citations inside code blocks, but I can append them as regular text outside of them. I'll present a new section structure with subsections, add suggested text (without citations in the code block), and also modify the section intro as requested. Here's the plan:

  1. Update the outline and subsection text.
  2. Provide suggested LaTeX for Subsection A.
  3. Edit introduction and paragraph in "Non-Convex Constrained Problem."

Thought for 1m 19s

Answer

可以的：删掉原来的“Communication Performance Invariant…”整段之后，我建议把这一节的第一个小节换成一个“g\mathbf g 的可行域/约束来源说明”，专门解释你们在(59)里对 g\mathbf g 那几条线性约束为什么合理、在物理上对应什么 Nyquist/roll-off 结构——写短一点，但让审稿人一眼看懂“不是拍脑袋加的约束”。你们文中其实已经有现成材料：g\mathbf g 的定义、folding-spectrum criterion、以及(59)里“0/1/单调/面积/盒约束”的动机句子。(CE_OFDM_ShiJingCheng (3).pdf) (CE_OFDM_ShiJingCheng (3).pdf)

推荐的新结构（替换掉原 Subsection 1）

把原来的

  • Communication Performance Invariant to Phase Rotation（整段删）

替换为类似下面这个短小节：

建议标题

  • Nyquist Pulse Feasible Set for g\mathbf g
    或更直白：
  • Pulse-Shaping Constraints on g\mathbf g

建议你们直接放进论文里的“短小节文字”（不长，但够解释清楚）

（你按需要微调符号/引用公式号即可）

latex
\subsection{Pulse-Shaping Constraints on $\mathbf g$}
Recall that $\mathbf g$ denotes the folded power-spectrum of the Nyquist shaping filter in the oversampled domain. 
Specifically, $\mathbf g$ is defined from the squared magnitude spectrum of the shaping pulse, and the Nyquist condition can be enforced via the folding-spectrum criterion, which relates the aliased spectral components by $g_{(L-1)N+n}=1-g_n$ for $n=1,\ldots,N$, while the intermediate components are nulled for $0\le \alpha <1$. 
As a result, it suffices to optimize the $N$-dimensional folded spectrum over the single-sided band.

To ensure that the resulting spectrum corresponds to a realizable band-limited Nyquist pulse with roll-off factor $\alpha$, we impose standard discrete spectral constraints: 
(i) the non-roll-off portions are fixed to $0/1$ to preserve the band edges; 
(ii) the roll-off portion is constrained to be monotone to avoid spectral notches; 
(iii) an area (energy) constraint is enforced to keep the pulse energy constant; and 
(iv) a box constraint $0\le g_n\le 1$ guarantees nonnegativity and realizability. 
These constraints are linear in $\mathbf g$, so the pulse-design subproblem remains convex when $\boldsymbol{\theta}$ is fixed.

这段的依据在你们文中已经出现过：

  • folding-spectrum criterion + 中间频段置零：(CE_OFDM_ShiJingCheng (3).pdf)
  • 你们对“0/1固定 + roll-off单调 + 面积约束 + 盒约束”的动机说明：(CE_OFDM_ShiJingCheng (3).pdf)
  • (59) 里具体约束形式（让小节和优化问题完全对齐）：(CE_OFDM_ShiJingCheng (3).pdf)

你们“Non-Convex Constrained Problem”小节建议顺手改两处（让重复更少）

因为上面新小节已经解释了 g\mathbf g 约束来源，你们原来的这段：

“To ensure that … we enforce standard discrete spectral constraints. Specifically, …”

就可以在 Non-Convex Constrained Problem 里缩成一句话，避免重复，例如：

“With the feasible set of g\mathbf g specified in the previous subsection, we now formulate the joint optimization…”

然后直接进入(59)。

一个小但重要的符号一致性提醒（建议你们顺手修）

你们前面定义 g\mathbf g 时把它写成了复向量（CLN×1\mathbb C^{LN\times 1}），但它其实是“平方幅度谱”应当是实非负，而(59)里也用 0≤gn≤10\le g_n\le 1。这个在审稿时经常被挑。建议把 g\mathbf g 的定义域统一成 RLN×1\mathbb R^{LN\times1} 或 R+LN×1\mathbb R_+^{LN\times1}。(CE_OFDM_ShiJingCheng (3).pdf) (CE_OFDM_ShiJingCheng (3).pdf)

如果你愿意，我下一步可以按你们现稿把该 section 的三处衔接句（section开头、Non-Convex开头、Two-Stage开头）一起改成完全顺滑、不重复、逻辑闭环的版本。

Question

我们第一个subsection里面，直接结合限制条件的公式来说吧，可以用quad并列相似限制条件。然后不要分点来说，要和我现在的风格一样，用段落来说

Thought

Reworking subsection with constraints and explanation

The user wants the first subsection edited to address constraints on gg and related concepts in paragraph form. I’ll integrate LaTeX with \quad to show constraints side-by-side, reference equation (59), and include citations. I'll clarify the difference between constraints for optimization (∑gn=N/2\sum g_n = N/2) versus normalization (∑gn=N\sum g_n = N).

Defining folded spectrum and constraints

I need to focus on the folded power-spectrum constraint, with gng_n representing the folded spectrum across n=1n = 1 to NN. The sum of gng_n should equal N/2N/2. I’ll clarify that gg is real, then write the subsection detailing the folded spectrum constraints, which include passband, stopband, and roll-off constraints. Let’s present them using LaTeX \quad as requested.

Finalizing the revised subsection

The user wants the first subsection rewritten with constraints presented in paragraph form, not bullet points. I’ll include relevant LaTeX code, like \quad to present constraints inline. I'll also reference the specified lines from the files and mention how these constraints lead to a convex set. I’ll also include the folding criterion and cite the necessary sections.

Thought for 44s

Answer

可以。你们把原来的“Communication Performance Invariant …”删掉后，第一个小节我建议直接改成专门解释 g\mathbf g 的可行域/约束来源，并且把(59)里的约束“合在一条式子里”用 \quad 并列写出来，风格和你们现在一致（段落叙述 + 一条集中公式）。

下面给你一段可直接替换的 LaTeX（不分点、段落写法）：

latex
\subsection{Pulse-Shaping Constraints on $\mathbf g$}
Recall that $\mathbf g$ denotes the folded power-spectrum associated with the (normalized) Nyquist pulse shaping filter, which enters the expected squared-ACF expression through the diagonalized pulse-shaping operation. In particular, under the folding spectrum criterion, the aliased spectral components satisfy $g_{(L-1)N+n}=1-g_n$ for $n=1,\ldots,N$, while the intermediate components are nulled when $0\le \alpha <1$, so that it suffices to optimize the $N$-dimensional folded spectrum over the single-sided band. (CE_OFDM_ShiJingCheng (3).pdf)

To ensure that the optimized $\mathbf g$ corresponds to a realizable band-limited Nyquist pulse with roll-off factor $\alpha$, we directly impose the discrete spectral constraints used in \eqref{opt_joint_pulse_phase}, where the non-roll-off portions are fixed to $0/1$ to preserve the band edges, the roll-off part is constrained to be monotone to avoid spectral notches, and an area constraint is enforced to keep the pulse energy normalized, together with the box constraint for realizability. Specifically, these constraints take the form
\begin{equation}
g_n=0,\ n=1,\ldots,\frac{N-N\alpha}{2}
\quad
g_n=1,\ n=\frac{N+N\alpha}{2}+1,\ldots,N
\quad
g_{n+1}-g_n\ge 0,\ n=1,\ldots,N-1
\quad
\sum_{n=1}^{N} g_n=\frac{N}{2},\ 0\le g_n\le 1,
\label{eq:g_constraints}
\end{equation}
which are all linear in $\mathbf g$ and therefore keep the pulse-design subproblem convex for any fixed $\boldsymbol{\theta}$. (CE_OFDM_ShiJingCheng (3).pdf) (CE_OFDM_ShiJingCheng (3).pdf)

两个小提醒（顺手一起改掉会更一致）：

  1. 你们前面曾把 g\mathbf g 写成 CLN×1\mathbb{C}^{LN\times 1} 的“squared spectrum”，但它实际是非负实数谱，而且后面约束也写了 0≤gn≤10\le g_n\le 1。建议把定义域改成 RLN×1\mathbb{R}^{LN\times 1} 或 R+LN×1\mathbb{R}_+^{LN\times 1}。(CE_OFDM_ShiJingCheng (3).pdf)
  2. 你们在这里写“optimize the NN-dimensional folded spectrum”时，最好和(59)一致，明确“g=[g1,…,gN]\mathbf g=[g_1,\ldots,g_N]”是折叠后的那一段（避免读者以为还在优化 LNLN 维）。(CE_OFDM_ShiJingCheng (3).pdf)

如果你把你们 g\mathbf g 的定义那一段（在系统推导里引入 gg 的位置）也发我一下，我可以顺便把“LNLN 维→折叠成 NN 维→(59) 里只优化 NN 维”的叙述衔接得更丝滑，彻底消除维度歧义。

Question

Let g=[g1,g2,…,gLN]T∈CLN×1\mathbf{g} = [g_1,g_2,\dots,g_{LN}]^T \in \mathbb{C}^{LN\times1} be the squared spectrum of NP\sqrt{N}\mathbf{P}, which could be expressed as:
\begin{equation}
\mathbf{g} = N \mathbf{F}{LN} \mathbf{p} \odot \mathbf{F}{LN}^* \mathbf{p}^*.
\end{equation}

To mitigate ISI, Nyquist pulse shaping with a roll-off factor in the range 0≤α≤10 \leq \alpha \leq 1 is employed. Additionally, for the normalized pulse p\mathbf{p}, the property ∑n=1LNgn=N\sum_{n=1}^{LN} g_n = N holds.

Leveraging this observation alongside the characteristic f~LN−k+1=f~k+1∗\tilde{\mathbf{f}}_{LN-k+1} = \tilde{\mathbf{f}}_{k+1}^*, the ACF of CE-OFDM signal with pulse shaping can be derived as :
\begin{equation}
r_{k} = \sum_{n=1}^N g_n |\tilde{\mathbf{f}}^H\mathbf{x}|^2 e^{\frac{j 2 \pi k(n-1)}{LN}} + g_{(L-1)N + n} |\tilde{\mathbf{f}}_n^H\mathbf{x}|^2 e^{\frac{j 2 \pi k(n-N-1)}{LN}}.
\label{pulse1}
\end{equation}

To address the frequency-domain behavior of the Nyquist pulse, we invoke the folding spectrum criterion \cite{proakis2001digital}, which enforces the relation:
\begin{equation}\label{eq:nyquist_constraint}
g_{(L-1)N+n} = 1 - g_n, \quad n = 1, 2, \dots, N.
\end{equation}
Due to the adoption of a roll-off factor 0≤α<10 \leq \alpha < 1, all gng_n for n∈[N+1,(L−1)N]n \in [N+1, (L-1)N] are nullified.

Then, substitute the Nyquist pulse constraint in \eqref{eq:nyquist_constraint} and \eqref{signal} into \eqref{pulse1}, yields:
\begin{align}
r_k&=\sum_{n=1}^N\tilde{g}{n,k}\big|\mathbf{f}n\mathbf{x}\big|^2 e^{j\frac{2\pi k (n-1)}{N}}\notag\
&=\sum{n=1}^N\tilde{g}{n,k}\sum_{p=1}^N\sum_{q=1}^N\mathbf{f}n[p]\mathbf{f}n^*[q]e^{j(\theta_p-\theta_q)} e^{j[\tilde{h}\left(\tilde{\mathbf{v}}p-\tilde{\mathbf{v}}q\right)]} e^{j\frac{2\pi k (n-1)}{N}}\notag\
& = \frac{1}{N} \sum{n=1}^N\tilde{g}{n,k} \sum{p=1}^N\sum{q=1}^N e^{j\phi_{p,q}}e^{j[\tilde{h}(\mathbf{f}_p^H-\mathbf{f}_q^H)\mathbf{v}]} e^{-j\frac{2\pi(n-1)(p-q-k)}{N}},
\label{eq:rk_expand}
\end{align}
where g~n,k~=A~[gn+(1−gn)e−j2πk~L]\tilde{g}_{n,\tilde{k}} = \tilde{A}[g_n + (1 - g_n)e^{-\frac{j 2\pi \tilde{k}}{L}}], and ejϕp,q≜ej(θp−θq)e^{j\phi_{p,q}}\triangleq e^{j(\theta_p-\theta_q)}.

Question

g的定义：Let g=[g1,g2,…,gLN]T∈CLN×1\mathbf{g} = [g_1,g_2,\dots,g_{LN}]^T \in \mathbb{C}^{LN\times1} be the squared spectrum of NP\sqrt{N}\mathbf{P}, which could be expressed as:
\begin{equation}
\mathbf{g} = N \mathbf{F}{LN} \mathbf{p} \odot \mathbf{F}{LN}^* \mathbf{p}^*.
\end{equation}

To mitigate ISI, Nyquist pulse shaping with a roll-off factor in the range 0≤α≤10 \leq \alpha \leq 1 is employed. Additionally, for the normalized pulse p\mathbf{p}, the property ∑n=1LNgn=N\sum_{n=1}^{LN} g_n = N holds.

Leveraging this observation alongside the characteristic f~LN−k+1=f~k+1∗\tilde{\mathbf{f}}_{LN-k+1} = \tilde{\mathbf{f}}_{k+1}^*, the ACF of CE-OFDM signal with pulse shaping can be derived as :
\begin{equation}
r_{k} = \sum_{n=1}^N g_n |\tilde{\mathbf{f}}^H\mathbf{x}|^2 e^{\frac{j 2 \pi k(n-1)}{LN}} + g_{(L-1)N + n} |\tilde{\mathbf{f}}_n^H\mathbf{x}|^2 e^{\frac{j 2 \pi k(n-N-1)}{LN}}.
\label{pulse1}
\end{equation}

To address the frequency-domain behavior of the Nyquist pulse, we invoke the folding spectrum criterion \cite{proakis2001digital}, which enforces the relation:
\begin{equation}\label{eq:nyquist_constraint}
g_{(L-1)N+n} = 1 - g_n, \quad n = 1, 2, \dots, N.
\end{equation}
Due to the adoption of a roll-off factor 0≤α<10 \leq \alpha < 1, all gng_n for n∈[N+1,(L−1)N]n \in [N+1, (L-1)N] are nullified.

Then, substitute the Nyquist pulse constraint in \eqref{eq:nyquist_constraint} and \eqref{signal} into \eqref{pulse1}, yields:
\begin{align}
r_k&=\sum_{n=1}^N\tilde{g}{n,k}\big|\mathbf{f}n\mathbf{x}\big|^2 e^{j\frac{2\pi k (n-1)}{N}}\notag\
&=\sum{n=1}^N\tilde{g}{n,k}\sum_{p=1}^N\sum_{q=1}^N\mathbf{f}n[p]\mathbf{f}n^*[q]e^{j(\theta_p-\theta_q)} e^{j[\tilde{h}\left(\tilde{\mathbf{v}}p-\tilde{\mathbf{v}}q\right)]} e^{j\frac{2\pi k (n-1)}{N}}\notag\
& = \frac{1}{N} \sum{n=1}^N\tilde{g}{n,k} \sum{p=1}^N\sum{q=1}^N e^{j\phi_{p,q}}e^{j[\tilde{h}(\mathbf{f}_p^H-\mathbf{f}_q^H)\mathbf{v}]} e^{-j\frac{2\pi(n-1)(p-q-k)}{N}},
\label{eq:rk_expand}
\end{align}
where g~n,k~=A~[gn+(1−gn)e−j2πk~L]\tilde{g}_{n,\tilde{k}} = \tilde{A}[g_n + (1 - g_n)e^{-\frac{j 2\pi \tilde{k}}{L}}], and ejϕp,q≜ej(θp−θq)e^{j\phi_{p,q}}\triangleq e^{j(\theta_p-\theta_q)}.

Thought

Clarifying modifications to g definition and constraints

I need to propose changes for the g definition to prevent complex values, aligning with the new subsection style. I’ll also reconcile the sum constraint issue: currently, it's unclear whether sum_{n=1}^{LN} g_n should equal N or N/2. I think defining a folded vector g of length N will clarify this. If we set g_{(L-1)N+n}=1-g_n, the sum will be N. That should work!

Reevaluating sum constraints for energy distribution

Okay, so if the middle zeros hold, sum_{n=1}^{LN} g_n = N, but the sum over the first and last N terms always adds to N. So, their sum g_n = N/2 constraint seems to be ensuring symmetry, possibly for equal energy balance between positive and negative frequencies. I think it's adding to fix the energy distribution, but I'm not certain.

Clarifying folded vector and normalization

Okay, so it seems like their g_n might be single-sided, corresponding to the Nyquist pulse. The sum constraint of g_n = N/2 suggests they already assume the vector is folded. I need to propose clearer definitions: full spectrum g_full with LN entries, and the folded vector g with N entries, normalized appropriately. I’ll adjust the earlier statement to remove confusion but keep the scaling consistent.

Adjusting vector definition and normalization

I should change the earlier definition of g from being a complex vector to a real-valued vector, as we’re dealing with squared magnitudes. The modifications include adjusting the definition of g in the context of normalization—ensuring that the DFT (unitary or not) is handled correctly. If the DFT is unitary, then the sum g_n equals N.

Thought for 1m 35s

Answer

你们这段 g\mathbf g 的定义现在有两个“容易被审稿人挑”的点，需要顺手统一一下，否则后面(59)里把 g\mathbf g当成 NN 维变量会显得前后不一致：

  1. “squared spectrum” 应该是实非负，但你们写成了 CLN×1\mathbb C^{LN\times1}。你们(28)本身是模平方形式，所以 g∈R+LN×1\mathbf g\in\mathbb R_+^{LN\times1} 更合适。(CE_OFDM_ShiJingCheng (3).pdf)
  2. 你们前面把 g\mathbf g定义为 LNLN 维 (CE_OFDM_ShiJingCheng (3).pdf)，但后面折叠后推导/以及优化(59)里实际只在用并优化 NN 维 folded spectrum（约束也是对 n=1,…,Nn=1,\dots,N 写的）。(CE_OFDM_ShiJingCheng (3).pdf)

建议怎么改（最省改动、逻辑最顺）

A) 把“全谱”和“折叠谱”区分命名（推荐）

把你们现在这句

Let g=[g1,…,gLN]T∈CLN×1\mathbf g=[g_1,\dots,g_{LN}]^T\in\mathbb C^{LN\times1} be the squared spectrum …

改成下面这种口径（只要改符号和一句话解释，后文基本不用大改）：

  • 用 gfull∈R+LN×1\mathbf g_{\rm full}\in\mathbb R_+^{LN\times1} 表示 LNLN 点全谱（对应你们(28)）；
  • 再定义 g∈RN×1\mathbf g\in\mathbb R^{N\times1} 为 folded spectrum（对应你们后面一直在用、以及(59)优化的那个变量）。

你们原文(28)(29)(30)这段的结构是：先给全谱，再用 folding criterion把有效自由度压成 NN 维。(CE_OFDM_ShiJingCheng (3).pdf) 这样命名一改，(59)就自然接上了。(CE_OFDM_ShiJingCheng (3).pdf)

B) 直接把域从 C\mathbb C 改成 R+\mathbb R_+，并补一句“后续只优化 folded NN 维”

如果你们不想引入新符号，也至少把“C\mathbb C”改掉，并加一句：

“By the folding spectrum criterion … it suffices to optimize the NN-dimensional folded spectrum [g1,…,gN]T[g_1,\ldots,g_N]^T, and the remaining components are determined.”

另外：你们(29)里第一项的 ∣f~Hx∣2|\tilde{\mathbf f}^H\mathbf x|^2 很像漏了下标

在你们稿子对应位置，(29)第一项写成了 ∣f~Hx∣2|\tilde{\mathbf f}^H\mathbf x|^2，第二项才是 ∣f~nHx∣2|\tilde{\mathbf f}_n^H\mathbf x|^2。(CE_OFDM_ShiJingCheng (3).pdf)
从后面(31)推到 ∣fnx∣2|\mathbf f_n\mathbf x|^2 的形式看，第一项大概率也应该带 nn（否则维度/含义不一致）。这个小 typo 建议顺手修一下。

你们想写的新第一小节（直接用约束式并列、段落风格）怎么衔接(59)

你们(59)里的 g\mathbf g 约束写得很清楚了：固定 0/1 的非 roll-off 区、roll-off 单调、面积约束、盒约束。(CE_OFDM_ShiJingCheng (3).pdf)
所以新的 subsection 完全可以围绕这条式子来写（不分点、用段落解释“为什么这样约束能保证Nyquist/可实现”），并把那几条用 \quad 并列成一条集中公式——这就和你们现在的写作风格一致。

如果你愿意，我可以按你们现有的 LaTeX 结构，把删除原通信小节后这一章开头的三段（section引入 + 新subsection + “Non-Convex Constrained Problem”开头两句）改成完全无重复、无维度歧义的一版。

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