用高级、科学、符合IEEE顶级期刊的语言，不改变下文意思的翻译下面的内容，结果用latex给我： ...

latex
This paper takes the aforementioned issue as its point of departure.  
When the number of targets increases, the vectorized Ziv–Zakharov bound (ZZB) evaluated numerically suffers from a severe \emph{curse of dimensionality} because the vector search governed by the maximization constraint becomes intractable, and the resulting bound is no longer trackable.  
Our objective is therefore to derive a closed-form expression that can faithfully characterize the ZZB for a multi-target system, thereby enabling optimized waveform design and \emph{resolving the Doppler ambiguity}.

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An intriguing observation is that the perceptual limit of the entire multi-target sensing system can be fully captured by examining the resolvability between the two \emph{closest} targets in the parameter space.  
This statement is rigorously justified by the analysis given in \textbf{Response~2.4}.

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Although directly obtaining the globally optimal vector that satisfies the maximization constraint of the vectorized ZZB is formidable, we establish a theoretical link between the numerical \emph{locally} optimal solution of the vectorized ZZB and the \emph{Doppler-difference ZZB} proposed herein.  
The connection is revealed by finely discretizing the feasible set of the parameters to be estimated and exhaustively enumerating $\xi$, thereby yielding a relaxed lower bound.  
While this bound is conservative, it nonetheless exposes the fundamental performance law of the system and admits a clear physical interpretation.

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We numerically evaluate the vectorized ZZB over the same parameter interval as that in Fig.~1, i.e.\ $h\!\in\![0,1]$ in~\eqref{vector ZZB}, assuming the unknown parameters are uniformly distributed over the domain and fixing the target amplitude ratio to $s\!=\!1$.  
First, the dual-target case is considered.  
The bounds associated with the \emph{unit} and \emph{opposite} projection vectors,
$$
  \mathbf{a}_{\text{unit}} \;=\; \frac{1}{\sqrt{2}}\bigl[1,\,1\bigr]^{\!\mathsf{T}}, 
  \qquad
  \mathbf{a}_{\text{oppo}} \;=\; \frac{1}{\sqrt{2}}\bigl[-1,\,1\bigr]^{\!\mathsf{T}},
$$
respectively constrain the mean-square errors of the estimates of the parameter \emph{sum} and \emph{difference}, the latter being directly pertinent to our goal.

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To verify the conclusion of \textbf{Response~2.4}—namely, that the overall perceptual limit of a multi-target system can be determined solely from the resolvability of the two closest targets—we further construct a four-target scenario with a uniform projection vector
$$
  \mathbf{a}'_{\text{unit}} \;=\; \frac{1}{\sqrt{2}}\bigl[1,\,1,\,1,\,1\bigr]^{\!\mathsf{T}},
$$
and compute its vectorized ZZB numerically.  
The resulting curves are compared with the analytical Doppler-difference ZZB derived herein under the assumption of equal target amplitudes (see Fig.~9).

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The numerically obtained vectorized ZZB for the four-target system manifests a high degree of consistency with that of the dual-target case, empirically confirming that the system’s perceptual limit is governed by the resolution between adjacent targets.  
Moreover, in the high-SNR regime, the proposed Doppler-difference ZZB coincides with the numerical vectorized solution, whereas the slight deviation observed at low SNR originates from the Taylor approximation of the zeroth-order modified Bessel function.  
Nevertheless, as a closed-form lower bound, the Doppler-difference ZZB accurately characterizes the sensing capability of the system and, in contrast to the non-trackable numerical vectorized ZZB, offers a tractable basis for waveform design aimed at resolving Doppler ambiguity.