A Low-Complexity Near-Field Imaging Method for Multistatic Radar Systems Based on Receiver-Domain Decomposition

1. Introduction

Near-field radar imaging differs fundamentally from its far-field counterpart because the propagation delay depends nonlinearly on the candidate spatial location. In multistatic systems with sparse or diluted receiver layouts, this nonlinearity is compounded by the fact that each receiver observes a distinct bistatic geometry. As a result, the corresponding matched-filter image must typically be evaluated directly over the search volume, which is computationally demanding even in static-target scenarios.

Backprojection or direct matched filtering remains the standard accuracy benchmark because it preserves the exact geometric phase model. However, its computational burden scales linearly with the number of receivers and image voxels and becomes prohibitive in large-scale problems. FFT-based accelerations are powerful when the forward model admits a global Fourier structure, but near-field bistatic propagation generally does not satisfy the assumptions needed for such a reduction [1,2]. Fast backprojection methods and related approximations can reduce cost, but they may introduce artifacts or inherit structural assumptions that are not well matched to sparse near-field multistatic layouts [3].

Boag proposed a multilevel domain-decomposition framework for radar imaging in which the data are partitioned into subdomains, low-resolution images are formed, phase corrected, and then interpolated and aggregated into a fine-resolution result [4]. That framework was developed in a different setting, with decomposition performed in a structured data domain. In the present work, the same principle is adapted to a revised imaging problem which leverages the inherent structure of the sparse receiver geometry rather than relying on traditional regular angle-frequency grids.

Compared with brute-force backprojection, the proposed method aims to approximate the same near-field matched-filter image while reducing the number of full-resolution voxel evaluations. Compared with per-receiver FFT fusion, the present method does not impose a global Fourier interpretation on the spatial geometry; rather, it preserves the exact spatial matched-filter structure and uses interpolation only after explicit phase correction. Compared with Boag’s original frequency-angle decomposition, the present formulation partitions the receiver domain into subapertures, which is the natural decomposition axis for sparse near-field multistatic arrays.

1.1. Contributions

This work addresses the computational burden associated with near-field image formation in multistatic radar systems. The main contributions are:

1.

A receiver-domain decomposition framework for near-field multistatic radar imaging that partitions the receiver array into computationally manageable subapertures.

2.

A phase-corrected interpolation strategy that compensates for geometric phase variations before interpolation, thereby improving reconstruction accuracy while maintaining low computational complexity.

3.

A complexity and interpolation-error analysis that quantifies the trade-off between computational savings and reconstruction fidelity.

4.

A simulation-based evaluation demonstrating that the proposed method closely approximates the direct matched-filter image while significantly reducing computational cost.

5.

A discussion of the applicability of the proposed approach to distributed radar, sparse multistatic arrays, and emerging integrated sensing and communication (ISAC) systems.

1.2. Paper Organization

Section II reviews the relevant literature. Section III derives the radar model and matched-filter score for the stationary known-envelope case. Section IV develops the receiver-domain decomposition and phase-corrected interpolation method. Section V presents complexity and error analysis. Section VI provides numerical results, including plots that explicitly show how phase correction smooths the low-resolution block images. Section VII concludes the paper.

2. Related Work

Radar imaging has a long history in inverse scattering, synthetic aperture radar (SAR), and high-resolution radar cross-section imaging. Classical treatments establish that coherent image formation can be interpreted as a matched-filter or backprojection operation over a spatial reflectivity distribution [1,2]. These methods are attractive because they can accommodate arbitrary propagation geometries and near-field phase histories. However, their direct implementation becomes computationally expensive when the imaging region is volumetric, the spatial grid is dense, or the number of transmit–receive channels is large. This computational burden motivates fast image-formation methods such as fast backprojection, subaperture processing, and multilevel decomposition [3,4].

Near-field radar imaging has received renewed attention because of applications in millimeter-wave security screening, nondestructive testing, automotive sensing, indoor mapping, biomedical imaging, and distributed sensing systems. In contrast to far-field imaging, the near-field case cannot generally rely on planar-wave approximations. The spherical-wave phase curvature must be retained, and the propagation delay varies nonlinearly with both the sensor position and the candidate target location. These effects are particularly important in sparse MIMO and multistatic radar arrays, where each transmitter–receiver pair observes the scene from a different bistatic geometry [5,6,7]. As a result, near-field multistatic imaging is often formulated using exact or approximate matched filtering, backprojection, range migration, or tomographic reconstruction methods.

Multistatic and distributed radar systems provide spatial diversity that can improve localization accuracy, robustness, and imaging resolution. Widely separated transmitters and receivers observe the target under different aspect angles, which can be beneficial for resolving ambiguities and reducing shadowing effects [5,6]. At the same time, this diversity increases the dimensionality of the data and complicates coherent image formation. Recent work on near-field multistatic SAR has shown that combining measurements from multiple sensors can improve image quality, but it also introduces strong, position-dependent sidelobe structures that must be controlled during reconstruction [8]. Similar issues arise in multichannel ground-penetrating radar and distributed MIMO radar, where multiview and multifrequency data increase the available information while also increasing data volume and computational cost [9,10].

A central challenge in these systems is therefore the trade-off between imaging accuracy and computational complexity. Direct matched filtering or backprojection evaluates the hypothesized propagation phase for every sensor channel and every image voxel. This approach is accurate and flexible, but its cost scales linearly with the number of voxels and the number of measurements. Several acceleration strategies have been proposed, including fast backprojection, Fourier-domain reconstruction, subaperture processing, interpolation-based methods, GPU acceleration, compressed sensing, and learned reconstruction [3,4,7,11,12]. These approaches differ in the assumptions they make about the array geometry, sampling pattern, scene sparsity, and acceptable reconstruction error.

Interpolation is especially relevant for reducing the cost of coherent image formation. In complex-valued radar data, however, direct interpolation can introduce phase errors because the phase varies rapidly with range and sensor position. Recent work has therefore emphasized phase-aware interpolation, in which known propagation phase terms are removed or controlled before interpolation and restored afterward [11]. This idea is closely related to the approach adopted in the present work: the rapidly varying geometric phase is compensated before interpolation so that the remaining quantity is smoother and can be approximated on a coarser grid.

The domain-decomposition approach of Boag [4] provides an important foundation for reducing the cost of radar image formation. In that framework, the imaging problem is divided into smaller subproblems, enabling multilevel interpolation and reducing the number of expensive direct evaluations. The present work follows the same general philosophy but applies it to a different setting: sparse near-field multistatic radar imaging. In this case, the decomposition is performed over receiver subsets, and the interpolation must account for bistatic near-field phase variation. Therefore, the present method is not simply a direct implementation of the original domain-decomposition algorithm; rather, it adapts the decomposition principle to receiver-domain processing and combines it with phase-corrected interpolation for multistatic geometries.

Recent developments in integrated sensing and communication (ISAC) further motivate low-complexity imaging algorithms. ISAC systems are expected to reuse communication infrastructure and waveforms for sensing tasks in beyond-5G and 6G networks [13]. Such systems may involve distributed nodes, sparse apertures, limited computational resources, and real-time processing constraints. These trends increase the need for image-formation methods that retain the geometric flexibility of matched filtering while reducing computational cost. The proposed receiver-domain decomposition and phase-corrected interpolation method is intended to address this need by approximating the direct near-field multistatic matched-filter image with substantially fewer full-resolution evaluations.

The proposed method is inspired by the domain-decomposition philosophy introduced by Boag. However, several important differences distinguish the present formulation.

First, the original framework was developed for a structured imaging setting in which the decomposition was performed in a different data domain. In contrast, the present work operates directly on sparse multistatic receiver geometries and performs decomposition over receiver subsets.

Second, the proposed method introduces an explicit phase-compensation stage that is tailored to near-field multistatic propagation. This compensation substantially reduces local phase oscillations and enables accurate interpolation of the resulting subimages.

Third, the formulation is derived from the exact multistatic matched-filter model and is therefore directly applicable to sparse distributed radar configurations.

Consequently, the proposed approach should be viewed as an adaptation and extension of the domain-decomposition principle to near-field multistatic imaging rather than a direct implementation of the original framework.

3. Radar Model and Matched-Filter Derivation

Consider a transmitter located at ${\mathbf{p}}_{tx}\in {\mathbb{R}}^3$ and a set of M receivers at locations ${\mathbf{q}}_m\in {\mathbb{R}}^3$, $m=1,\dots ,M$. See Figure 1. We assume a stationary point target at position $\mathbf{p}\in {\mathbb{R}}^3$. The bistatic delay for the mth receiver is

$${\tau }_m\left(\mathbf{p}\right)={\displaystyle \frac{\parallel \mathbf{p}-{\mathbf{p}}_{tx}\parallel +\parallel \mathbf{p}-{\mathbf{q}}_m\parallel }{c}},$$

(1)

where c is the propagation speed and $\parallel \mathbf{x}\parallel$ is the $L_2$ norm of the vector $\mathbf{x}$.

Let $g_m(n,\mathbf{p})$ denote the n-th sample of the known signal envelope at the m-th receiver for a target located at $\mathbf{p}$, $f_c$ the carrier frequency, and ${\alpha }_m\left(\mathbf{p}\right)$ a known (up to a multiplicative factor) propagation related complex amplitude. The received data at receiver m and sample index n are modeled as

$$x_m\left[n\right]={\alpha }_m\left(\mathbf{p}\right)g_m(n,\mathbf{p})e^{-j2\pi f_c{\tau }_m\left(\mathbf{p}\right)}+w_m\left[n\right],$$

(2)

where $w_m\left[n\right]$ denotes additive complex Gaussian noise. Define the vectors

$$\mathbf{x}\left[n\right]={\left[\begin{array}{ccc}{x}_1\left[n\right]& \dots & x_M\left[n\right]\end{array}\right]}^T.$$

(3)

$$\mathbf{a}\left(\mathbf{p}\right)={\left[\begin{array}{ccc}{e}^{-j2\pi f_c{\tau }_1\left(\mathbf{p}\right)}& \dots & e^{-j2\pi f_c{\tau }_M\left(\mathbf{p}\right)}\end{array}\right]}^T.$$

(4)

$$\mathbf{w}\left[n\right]={\left[\begin{array}{ccc}{w}_1\left[n\right]& \dots & w_M\left[n\right]\end{array}\right]}^T.$$

(5)

Further, define

$$d(n,\mathbf{p})={\alpha }_m\left(\mathbf{p}\right)g_m(n,\mathbf{p})$$

(6)

Using these vectors, the observed signal equation becomes,

$$\mathbf{x}\left[n\right]=\mathbf{a}\left(\mathbf{p}\right)d(n,\mathbf{p})+\mathbf{w}\left[n\right],$$

(7)

A matched filter for this signal is

$$Q\left(\widehat{\mathbf{p}}\right)={\left|{\mathbf{a}}^H\left(\widehat{\mathbf{p}}\right){\displaystyle \sum _{n=1}^N}\overline{d}(n,\mathbf{p})\mathbf{x}\left[n\right]\right|}^2.$$

(8)

where the overline denotes complex conjugation. Let the measurements be organized into a matrix $\mathbf{X}\in {\mathbb{C}}^{M\times N}$ whose $(m,n)$th element is $x_m\left[n\right]$. Because the signal envelope is known, the time dimension can be compressed into a single vector

$$\mathbf{y}=\mathbf{X}\overline{\mathbf{d}},$$

(9)

where $\mathbf{d}={[d(0,\mathbf{p}),d(1,\mathbf{p}),\dots ,d(N-1,\mathbf{p})]}^T$.

The matched-filter score is then:

$$Q\left(\widehat{\mathbf{p}}\right)={\left|{\mathbf{a}}^H\left(\widehat{\mathbf{p}}\right)\mathbf{y}\right|}^2.$$

(10)

The direct image is formed by evaluating (10) over all candidate voxels. As shown in the sequel in our setting, the size of a typical voxel is about 10 cm in each of the three dimensions (x,y,z). Thus, to cover a volume of a cubic km we need ${10}^{12}$ voxels. This motivates our search for computation load reduction.

4. Receiver-Domain Approximation

4.1. Receiver Partition

Let the receiver set be partitioned into K disjoint subsets,

$$\{1,\dots ,M\}={\displaystyle \bigcup _{k=1}^K}{\mathcal{S}}_k.$$

(11)

Correspondingly, define block steering vectors ${\mathbf{a}}_k\left(\mathbf{p}\right)$ and block-compressed data ${\mathbf{y}}_k$. Then the complex matched field is

$$G\left(\mathbf{p}\right)={\mathbf{a}}^H\left(\mathbf{p}\right)\mathbf{y}={\displaystyle \sum _{k=1}^K}{G}_k\left(\mathbf{p}\right),G_k\left(\mathbf{p}\right)={\mathbf{a}}_k^H\left(\mathbf{p}\right){\mathbf{y}}_k.$$

(12)

The score is $Q\left(\mathbf{p}\right)={\left|G\left(\mathbf{p}\right)\right|}^2$.

4.2. Coarse-Grid Subimages

Instead of evaluating each block field on the fine grid ${\Omega }_f$, we compute it on a coarse grid ${\Omega }_c$: The fine grid may contain approximately eight samples per resolution cell, whereas the coarse grid contains approximately one sample per resolution cell.

$$G_k\left({\mathbf{p}}_c\right),{\mathbf{p}}_c\in {\Omega }_c.$$

(13)

4.3. Reference Phase Correction

Let ${\overline{\mathbf{q}}}_k$ denote the centroid of the receiver subset ${\mathcal{S}}_k$. Define the reference delay

$${\tau }_k^{\mathrm{ref}}\left(\mathbf{p}\right)={\displaystyle \frac{\parallel \mathbf{p}-{\mathbf{p}}_{tx}\parallel +\parallel \mathbf{p}-{\overline{\mathbf{q}}}_k\parallel }{c}},$$

(14)

and the corresponding reference phase

$${\varphi }_k^{\mathrm{ref}}\left(\mathbf{p}\right)=2\pi f_c{\tau }_k^{\mathrm{ref}}\left(\mathbf{p}\right).$$

(15)

The compensated coarse block image is

$${\tilde{G}}_k\left({\mathbf{p}}_c\right)=G_k\left({\mathbf{p}}_c\right)e^{-j{\varphi }_k^{\mathrm{ref}}\left({\mathbf{p}}_c\right)}.$$

(16)

This removes the dominant oscillatory component and is the key step that enables stable interpolation.

4.4. Interpolation and Rephasing

Let ${\mathcal{I}}_c^f$ denote trilinear interpolation from the coarse to the fine grid. The reconstructed block field on the fine grid is

$${\widehat{G}}_k\left(\mathbf{p}\right)={\mathcal{I}}_c^f\left({\tilde{G}}_k\right)\left(\mathbf{p}\right)e^{j{\varphi }_k^{\mathrm{ref}}\left(\mathbf{p}\right)}.$$

(17)

The full approximation is then:

$$\widehat{Q}\left(\mathbf{p}\right)={\left|{\displaystyle \sum _{k=1}^K}{\widehat{G}}_k\left(\mathbf{p}\right)\right|}^2.$$

(18)

5. Complexity and Error Analysis

5.1. Computational Complexity

Let $P=N_x{N}_y{N}_z$ denote the number of fine-grid voxels. Direct evaluation of (10) requires

$$T_{\mathrm{dir}}=\Theta \left(MP\right).$$

(19)

If the coarse-grid decimation factors are $({\gamma }_x,{\gamma }_y,{\gamma }_z)$, the coarse grid contains approximately

$$P_c\approx {\displaystyle \frac{P}{{\gamma }_x{\gamma }_y{\gamma }_z}}.$$

(20)

The approximation requires evaluating all receiver blocks on the coarse grid plus interpolation on the fine grid:

$$T_{\mathrm{app}}=\Theta \left(M{P}_c+P\right).$$

(21)

Thus the potential gain appears only when the reduction in coarse-grid evaluations outweighs the interpolation overhead.

6. Interpolation Error Analysis

The accuracy of the proposed receiver-domain decomposition method depends critically on the interpolation of low-resolution subimages. In this section, we quantify the approximation error introduced by interpolating phase-compensated block images from a coarse spatial grid to a fine grid.

6.1. Problem Formulation

Recall that the coherent block image associated with receiver subset ${\mathcal{S}}_k$ is given by

$$G_k\left(\mathbf{p}\right)={\mathbf{a}}_k^H\left(\mathbf{p}\right){\mathbf{y}}_k,$$

(22)

where ${\mathbf{a}}_k\left(\mathbf{p}\right)$ is the steering vector restricted to the receivers in subset ${\mathcal{S}}_k$, and ${\mathbf{y}}_k$ is the corresponding compressed data vector.

To enable interpolation, we introduce a reference phase

$${\varphi }_k^{\mathrm{ref}}\left(\mathbf{p}\right)=2\pi f_c{\tau }_k^{\mathrm{ref}}\left(\mathbf{p}\right),$$

(23)

where ${\tau }_k^{\mathrm{ref}}\left(\mathbf{p}\right)$ is defined using the centroid of the receiver subset. The phase-compensated field is then

$${\tilde{G}}_k\left(\mathbf{p}\right)=G_k\left(\mathbf{p}\right)e^{-j{\varphi }_k^{\mathrm{ref}}\left(\mathbf{p}\right)}.$$

(24)

The key idea is that ${\tilde{G}}_k\left(\mathbf{p}\right)$ exhibits significantly reduced oscillatory behavior compared to $G_k\left(\mathbf{p}\right)$, making it suitable for interpolation.

6.2. Interpolation Approximation

Let ${\Omega }_f$ denote the fine grid and ${\Omega }_c\subset {\Omega }_f$ a coarse grid with spacings $(c_x,c_y,c_z)$. Let ${\tilde{G}}_k^{\left(c\right)}$ denote samples of ${\tilde{G}}_k$ on ${\Omega }_c$. The interpolated approximation from the coarse grid $\left(c\right)$ to the fine grid $\left(f\right)$ is

$${\widehat{G}}_k\left(\mathbf{p}\right)={\mathcal{I}}_c^f\left({\tilde{G}}_k^{\left(c\right)}\right)\left(\mathbf{p}\right)e^{j{\varphi }_k^{\mathrm{ref}}\left(\mathbf{p}\right)},$$

(25)

where ${\mathcal{I}}_c^f$ denotes trilinear interpolation.

6.3. Error Bound

We now quantify the error introduced by this approximation.

Proposition 1.

Assume that the phase-compensated field ${\tilde{G}}_k\left(\mathbf{p}\right)$ belongs to $C^2(\Omega )$, and that all second-order partial derivatives are uniformly bounded:

$$\left|{\displaystyle \frac{{\partial }^2{\tilde{G}}_k}{\partial x_i\partial x_j}}\left(\mathbf{p}\right)\right|\le M_k,\forall \mathbf{p}\in \Omega .$$

(26)

Then the interpolation error satisfies

$$\left|{\tilde{G}}_k\left(\mathbf{p}\right)-{\mathcal{I}}_c^f\left({\tilde{G}}_k^{\left(c\right)}\right)\left(\mathbf{p}\right)\right|\le C_k\left(c_x^2+c_y^2+c_z^2\right),$$

(27)

for all $\mathbf{p}\in \Omega$, where $C_k$ depends on the derivative bounds.

Proof. 

The result follows from standard interpolation error bounds for trilinear interpolation applied independently to the real and imaginary parts of ${\tilde{G}}_k\left(\mathbf{p}\right)$. Since ${\tilde{G}}_k$ is twice continuously differentiable, the interpolation error is proportional to the second derivatives and scales quadratically with the grid spacing. □

Since multiplication by the unit-modulus phase factor $e^{j{\varphi }_k^{\mathrm{ref}}\left(\mathbf{p}\right)}$ does not affect the magnitude of the error, the same bound applies to the reconstructed field:

$$|G_k\left(\mathbf{p}\right)-{\widehat{G}}_k\left(\mathbf{p}\right)|\le C_k(c_x^2+c_y^2+c_z^2).$$

(28)

Summing over all receiver blocks yields the global bound

$$|G\left(\mathbf{p}\right)-\widehat{G}\left(\mathbf{p}\right)|\le \left({\displaystyle \sum _{k=1}^K}{C}_k\right)(c_x^2+c_y^2+c_z^2).$$

(29)

6.4. Role of Phase Compensation

The validity of the above bound depends critically on the smoothness of the compensated field ${\tilde{G}}_k\left(\mathbf{p}\right)$. Without phase compensation, the block field is

$$G_k\left(\mathbf{p}\right)={\mathbf{a}}_k^H\left(\mathbf{p}\right){\mathbf{y}}_k=\sum _{m\in {\mathcal{S}}_k}{e}^{j2\pi f_c{\tau }_m\left(\mathbf{p}\right)}{y}_m,$$

(30)

which generally exhibits rapid oscillation with respect to $\mathbf{p}$ due to the carrier-phase term. After reference-phase compensation, the block field becomes

$${\tilde{G}}_k\left(\mathbf{p}\right)=G_k\left(\mathbf{p}\right)e^{-j{\varphi }_k^{\mathrm{ref}}\left(\mathbf{p}\right)}=\sum _{m\in {\mathcal{S}}_k}{y}_m{e}^{j2\pi f_c\left({\tau }_m\left(\mathbf{p}\right)-{\tau }_k^{\mathrm{ref}}\left(\mathbf{p}\right)\right)},$$

(31)

so that the dominant oscillatory component is removed and the residual field varies more slowly with $\mathbf{p}$. This reduction in oscillation directly reduces the constants $C_k$ in the interpolation error bound, enabling accurate interpolation on a coarse grid.

6.5. Practical Implications

The error bound highlights three key design considerations:

• Grid spacing: The approximation error scales quadratically with the coarse-grid spacing.
• Receiver partitioning: Smaller receiver blocks improve the accuracy of the reference phase model and reduce residual oscillation.
• Phase compensation: Proper phase correction is essential; without it, interpolation becomes unreliable due to rapid phase variation.

These observations are consistent with the numerical results, which show that phase compensation significantly smooths the low-resolution block images and improves interpolation accuracy.

7. Numerical Results

7.1. Simulation Setup

The numerical experiments use a single transmitter at the origin and $M=80$ receivers uniformly distributed on a circular ring of radius 400 m in the plane $z=0$. The target is stationary and located at

$${\mathbf{p}}_{★}={[200,300,500]}^T\mathrm{m}.$$

(32)

The carrier frequency is 2 GHz, and the known signal envelope is used for time compression.

7.2. Direct Versus Approximation Runtime

For a grid of $201\times 201\times 21=848,421$ voxels, the measured runtime of the direct matched filter was

$$T_{\mathrm{dir}}=9.142\mathrm{s},$$

(33)

while the approximation required

$$T_{\mathrm{app}}=3.986\mathrm{s}.$$

(34)

In this toy example, the speed up is a factor of 2.293. Both methods recover the correct target position.

7.3. Phase-Correction Smoothing Effect

The most important qualitative diagnostic is the low-resolution block image before and after phase correction. Figure 2 and Figure 3 show for a representative receiver block:

1.

magnitude and phase of the coarse block field before correction,

2.

magnitude and phase after multiplying by $e^{-j{\varphi }_k^{\mathrm{ref}}}$,

3.

one-dimensional cuts of the real and imaginary parts before and after correction.

These plots show that the uncompensated block image is highly oscillatory, while the corrected image varies much more slowly. This qualitative observation is confirmed by the phase-gradient-energy metric computed on the coarse block image. In the reported experiment, the phase-gradient energy decreased from

$$E_{\varphi ,\mathrm{before}}=3.15104$$

(35)

before correction to

$$E_{\varphi ,\mathrm{after}}=0.870591$$

(36)

after correction, corresponding to a reduction factor of

$${\displaystyle \frac{E_{\varphi ,\mathrm{before}}}{E_{\varphi ,\mathrm{after}}}}=3.619.$$

(37)

This confirms quantitatively that the reference-phase compensation substantially smooths the low-resolution block image and improves its suitability for interpolation. This is the central mechanism that makes interpolation plausible.

7.4. Score Cuts Along the Coordinate Axes

To evaluate the impact of the approximation on localization structure, one-dimensional score cuts are plotted along the x, y, and z axes while fixing the remaining coordinates at the true target location. The scores are normalized and shown in dB. Figure 4, Figure 5 and Figure 6 reveal whether the approximation preserves the peak location and the local curvature of the matched-filter surface.

7.5. Accuracy Measures

The approximation can be quantified using normalized RMS and maximum error relative to the direct image:

$${\Delta }_{\mathrm{rms}}={\displaystyle \frac{\parallel \widehat{Q}{-Q\parallel }_2}{{\parallel Q\parallel }_2}},{\Delta }_{max}={\displaystyle \frac{\parallel \widehat{Q}{-Q\parallel }_{\infty }}{{\parallel Q\parallel }_{\infty }}}.$$

(38)

In the reported experiment, the peak location was preserved exactly even when global image differences remained non-negligible, indicating that the approximation can remain useful for localization despite score-surface distortion away from the maximum.

7.6. Complexity–Accuracy Tradeoff

To summarize the tradeoff between computational efficiency and approximation fidelity, Fig. Figure 7 presents the measured speedup as a function of the normalized RMS error for the considered transmitter–receiver–target geometry. As expected, higher computational savings are accompanied by increased approximation error, illustrating the fundamental tradeoff between reduced evaluation time and interpolation accuracy.

8. Discussion

The results support several conclusions. First, in the static known-envelope case, time compression makes the direct matched filter particularly efficient, so an approximation is unlikely to be beneficial on very small 3D grids. Second, phase correction is essential; without it, the coarse block fields remain highly oscillatory and interpolation is ineffective. Third, the main value of the approximation idea in this setting lies not in small proof-of-concept problems but in larger-scale images where full-grid matched filtering becomes substantially more expensive.

The method is expected to degrade in at least three regimes. First, if the uncompensated or only weakly compensated block images remain highly oscillatory, the smoothness assumptions behind interpolation are violated. Second, overly aggressive coarse-grid spacing can remove local peak structure and distort the matched-filter surface. Third, strong extended-target interference or dense multipoint scattering may produce block fields whose behavior cannot be accurately represented by simple coarse-grid interpolation without adaptive refinement.

9. Conclusion

A receiver-domain-decomposition principle has been developed for near-field multistatic matched filtering with a known signal envelope and stationary target. The revised formulation reduces the problem to coherent spatial matched filtering of time-compressed data, which provides a clean baseline for evaluating phase-corrected interpolation. The study confirms that phase correction smooths low-resolution block images and is therefore the essential enabler for interpolation-based acceleration. At the same time, the experiments show that small static problems can favor direct matched filtering because the interpolation overhead dominates. The proposed framework is therefore best viewed as a scalable strategy for larger search spaces and more demanding configurations rather than as a universal replacement for direct computation.