Spatial Ambiguity Function (SAF)

• Spatial Ambiguity Function is a metric that characterizes an array’s ability to distinguish spatial positions and directions by comparing shifted aperture fields.
• It captures the inner-product response between displaced replicas of the aperture field, revealing couplings between linear and angular resolutions, especially in near-field regimes.
• SAF analysis informs practical trade-offs in array design by addressing aperture geometry, aliasing constraints, and Cramér–Rao bounds for localization accuracy.

The Spatial Ambiguity Function (SAF) quantifies the capability of an array (aperture) to distinguish different spatial configurations of a monochromatic signal source, subject to shifts in spatial position and direction of arrival. It is the spatial analog of the classical Doppler–delay ambiguity function, underlying the resolution limits and coupling between linear displacement and spatial frequency (or angle), including effects of aperture geometry, radiating backgrounds, and array sampling. The SAF is foundational in the analysis and design of sensor arrays for localization, synthetic aperture radar, and large-scale multi-antenna systems, particularly as next-generation arrays operate in near-field regimes where simple far-field approximations are inadequate (Yuryev et al., 2014, Monnoyer et al., 9 Dec 2025).

1. Mathematical Definition and Fundamental Properties

The classical definition of the SAF begins with a two-dimensional aperture with complex field-weight distribution $h(\mathbf r)$, $\mathbf r \in \mathbb R^2$, and received spatial signal $$H(\mathbf r) = \int_{\mathbb R^2} A(\boldsymbol\eta) e^{-j\,\boldsymbol\eta \cdot \mathbf r} d^2\eta$$, with $A(\boldsymbol\eta)$ the spatial frequency (directional) spectrum and $\lambda$ the wavelength. To analyze the effect of spatial (linear) and directional (spatial-frequency) perturbations, the function

$$\Xi(\Delta\mathbf x,\, \Delta\boldsymbol\eta) = \int_R h(\mathbf r + \tfrac{1}{2} \Delta\mathbf x)\, h^*(\mathbf r - \tfrac{1}{2} \Delta\mathbf x)\, e^{-j\,\Delta\boldsymbol\eta \cdot \mathbf r}\, d^2r,$$

is introduced, normalized such that $\Xi(\mathbf 0, \mathbf 0) = 1$. In one spatial dimension: $$\Xi(\Delta x,\; \Delta k) = \int_{-L/2}^{+L/2} h(x + \tfrac{\Delta x}{2})\, h^*(x - \tfrac{\Delta x}{2})\, e^{-j \Delta k x}\, dx.$$ This function characterizes the inner-product response between two shifted replicas of the aperture field, modulated by a relative phase gradient. The SAF quantifies the array’s ability to discriminate between a source at $(x, k)$ and one at $(x+\Delta x, k+\Delta k)$ in position and spatial frequency (Yuryev et al., 2014).

2. Far-Field and Near-Field Forms

In the far-field (Fraunhofer) regime, the SAF simplifies by identifying $$\Delta\boldsymbol{\eta} = (2\pi/\lambda) \Delta\boldsymbol{\theta}$$, with $\Delta\boldsymbol{\theta}$ the angular shift: $$\Xi(\Delta x, \Delta\theta) = \int_R h(r + \tfrac{\Delta x}{2})\, h^*(r - \tfrac{\Delta x}{2})\, e^{-j (2\pi/\lambda)\, \Delta\theta\, r} dr.$$ This form admits classical "sinc" behavior for uniform illumination: $$\Xi_{\mathrm{aper}}(\Delta x, \Delta k) = \frac{ \sin[\frac{1}{2}(L\Delta k - \Delta x)] }{ \frac{1}{2}(L\Delta k - \Delta x) },$$ yielding main-lobe widths $\Delta x \approx 2\pi / L$, $\Delta k \approx 2\pi / L$.

In the near-field (Fresnel) regime, phase curvature is significant: $$\Xi_{\mathrm{aper}}^{(\mathrm{near})}(\Delta x, \Delta k) = \int_{-L/2}^{L/2} \exp \left\{ -j \left[ \Delta k\, x - \tfrac{k}{2D} \Delta x x \right] \right\} dx.$$ The resulting SAF peak is no longer at $(0,0)$ but is displaced along $\Delta k \approx (k/2D) \Delta x$, reflecting the coupling between displacement and spatial frequency. The lobe widths scale as $\Delta x \sim \sqrt{2\pi D/k}$, $\Delta k \sim \sqrt{k/2\pi D}$ (Yuryev et al., 2014, Monnoyer et al., 9 Dec 2025).

3. Role of Extended and Reradiated Backgrounds

If the spatial scene reradiates or scatters with random, spatially-extended amplitude pattern $A(\boldsymbol\eta)$, the measured SAF manifests as a product of (i) the point-target SAF and (ii) the background-induced field’s autocorrelation,

$$\langle |\Xi| \rangle = \Xi_{\mathrm{aper}}(\Delta x, \Delta k) \times K_b(\Delta x),$$

where $$K_b(\Delta x) = \int E[A(\eta) A^*(\eta)] e^{-j \Delta k \eta}\, d\eta$$ or, in the spatial domain, $$K_b(\Delta x) \approx \int h_b(x + \tfrac{\Delta x}{2}) h_b^*(x - \tfrac{\Delta x}{2}) dx$$ for average background-induced field $h_b(x)$. This causes the overall SAF volume to shrink relative to the ideal value, and modifies the resolvable widths: the array’s ability to distinguish linear or angular displacements depends jointly on the aperture and on the spatial statistics of the background (Yuryev et al., 2014).

4. Resolution, Cramér–Rao Bounds, and Estimate Coupling

Resolution in $\Delta x$ and $\Delta k$ (or angle) is analytically derived from the second-order derivatives of $|\Xi|$ at the origin: $$(\delta x)^2 = -\left.\left( \frac{1}{\partial^2 |\Xi| / \partial (\Delta x)^2} \right) \right|_{(0,0)},\quad (\delta k)^2 = -\left.\left( \frac{1}{\partial^2 |\Xi| / \partial (\Delta k)^2} \right) \right|_{(0,0)}.$$ Given received SNR $q$, the Cramér–Rao lower bounds on unbiased estimation error variances are

$$\mathrm{Var}(\hat x) \approx \frac{1}{q} (\delta x)^2,\qquad \mathrm{Var}(\hat k) \approx \frac{1}{q} (\delta k)^2.$$

There is also a cross-covariance

$$\mathrm{Cov}(\hat x, \hat k) = -\frac{1}{q} \frac{\partial^2 |\Xi| / \partial (\Delta x)\, \partial (\Delta k)} { (\partial^2 |\Xi| / \partial (\Delta x)^2 )(\partial^2 |\Xi| / \partial (\Delta k)^2 )}.$$

Closed-form results for variance and correlation coefficients can be expressed in terms of the root-mean-square (rms) widths $\sigma_x$ (aperture field), $\sigma_k$ (aperture spectrum), and $\sigma_{k,b}$ (background spectrum width): $$\mathrm{Var}(\hat x) \approx \frac{1}{q} \frac{1}{4\pi^2 (\sigma_k^2 + \sigma_{k,b}^2)}, \quad \mathrm{Var}(\hat k) \approx \frac{1}{q} \frac{1}{4\pi^2 \sigma_x^2},$$ and coupling coefficient $$r_{x,k} = -\sigma_{k,b}^2 / \sqrt{(\sigma_k^2 + \sigma_{k,b}^2)\sigma_x^2}$$ (Yuryev et al., 2014).

5. Discrete Array Sampling, Aliasing, and Grating Lobes

When the aperture is discretely sampled, the SAF must account for the induced aliasing structure. For a grid $G$ with element spacing $\Delta$, the discrete SAF reads: $S(r, r') = \sum_{p_n \in G} g(p_n; r, r'),$ where $g(p; r, r')$ is the matched-filter kernel between putative source locations $r$ and $r'$.

Taking the spatial Fourier transform yields

$$G(\omega; r, r') = \int_{\mathbb R} g(p; r, r') e^{-j \omega p} dp;\quad S(r, r') = \sum_{m \in \mathbb Z} G\left(m \frac{2\pi}{\Delta}; r, r'\right).$$

Thus, discrete sampling folds the spatial spectrum $G(\omega)$ every $\pm 2\pi/\Delta$ and generates grating lobes wherever the support of $G(\omega)$ exceeds $[ -2\pi/\Delta, 2\pi/\Delta ]$ (Monnoyer et al., 9 Dec 2025).

The soft bandlimit

$$K(r, r') = \max_{p \in \text{array}} \left| \frac{d}{dp}[ \phi(p; r') - \phi(p; r)] \right|$$

quantifies the maximum local spatial frequency. The aliasing-free condition is $K(r, r') \leq 2\pi/\Delta$, defining the aliasing-free region (AFR) in which ambiguous grating lobes do not appear. For canonical arrays:

• For a Uniform Linear Array (ULA), the AFR is an "eye-shaped" region described in closed form in the infinite-aperture limit. Its width and height are set by the intersection of the bandlimit $K$ and the sampling interval $\Delta$.
• For a Uniform Circular Array (UCA), the AFR is a disk of radius $\rho \leq \lambda/(\Delta\Gamma(\theta_0))$, where $\Gamma(\theta_0)$ encodes the angular geometry (Monnoyer et al., 9 Dec 2025).

6. Trade-offs in Aperture Design, Thinning, and Aliasing

Increasing the array element spacing $\Delta$ narrows the AFR; grating lobes appear once the maximum local spatial frequency $K$ exceeds $2\pi/\Delta$. Enlarging aperture length or radius (increasing $L$ or $R_a$) raises the maximum frequency and further shrinks the AFR, even as it improves point-target resolution. The design of sparse (thinned) or extremely large-scale arrays requires careful balance between the desired resolution and the tolerable region of ambiguity-free operation (Monnoyer et al., 9 Dec 2025).

Schematic guidelines:

• To maximize resolution in $\Delta x$, use large $L$ or a scene with large background spectral support $\sigma_{k,b}$.
• To maximize angular resolution, use large $L$ (increasing $\sigma_k$) or ensure the scene provides high spatial position diversity ($\sigma_x$ small).
• For synthetic-aperture applications, the time-growing $L$ improves angular resolution only if the correlation function $K_b(\Delta x)$ remains stationary.
• In the near field, nontrivial coupling between linear and angular estimates requires either multi-frequency or multi-baseline measurements to separate effects.

7. Illustrative Examples and Comparison to Classical Far-Field AF

Canonical cases clarify essential SAF behaviors:

• In the Fraunhofer (far-field) limit, the spatial-frequency shift is constant; the SAF reduces to a one-dimensional sinc response in angle, with straight and periodic grating lobes appearing for $\Delta > \lambda/2$.
• In the Fresnel (near-field) regime, SAF mainlobe width is broadened and the peak shifts along the line $\Delta k = (k/2D)\Delta x$, reflecting linear–angular estimate coupling and chirp-like spectral properties.
• Thinned or sparse arrays manifest curved aliasing fronts in the SAF, as predicted quantitatively by the soft bandlimit $K$ and allegiance to the AFR (Monnoyer et al., 9 Dec 2025).

A summary table encapsulates key differences:

Regime	SAF Form	Grating Lobe Structure
Far-field	Sinc in angle (linear spatial freq.)	Straight, periodic; $\Delta > \lambda/2$
Near-field	Chirped/shifted-sinc, mainlobe at $\Delta k \sim \frac{k}{2D}\Delta x$	Curved, array- and geometry-dependent
Thinned array (NF)	Discrete-spectrum, aliasing controlled by $K \leq 2\pi/\Delta$	Multiple, non-uniform, eye/disk AFR

The transition from far- to near-field involves a shift from constant to locally-varying spatial frequency, with corresponding changes in resolution, ambiguity, and grating lobe morphology. By leveraging closed-form expressions for $K$ and the AFRs, one can predict and control these phenomena in array system design (Yuryev et al., 2014, Monnoyer et al., 9 Dec 2025).