On the Spatial Ambiguity Function

The ambiguity function of a spatial signal in the “signal spatial frequency displacement vs antenna linear displacement” coordinates is considered in this paper; in case of a monochromatic signal with known carrier frequency, spatial signal vector angular displacement is considered as the second coordinate. The analysis is made taking into account an external (reradiated or thermal) background which contributes considerable peculiarities in the properties of the spatial ambiguity function. Accuracy of measuring the signal linear or angular displacement and resolution are determined. Examples for the Fresnel and Fraunhofer zones are given.

💡 Research Summary

The paper extends the classic ambiguity function (AF), traditionally defined in the time‑frequency domain, to the spatial domain by introducing a two‑dimensional coordinate system consisting of spatial frequency displacement (Δk) and linear antenna displacement (Δx). For a monochromatic source with known carrier frequency, the received field can be written as s(r)=A e^{j(k·r−ωt)}. When the antenna array is shifted by Δx, the received voltage acquires an additional phase factor e^{jΔk·Δx}, where Δk represents the change in the wave‑vector component parallel to the aperture. The spatial AF is defined as

 χ(Δx,Δk)=∫ S(r) S⁎(r+Δx) e^{−jΔk·r} dr,

where S(r)=s(r)+B(r) is the total field composed of the desired signal s(r) and an external background B(r) (reradiated or thermal noise). The background is modeled as a zero‑mean stochastic process with spatial autocorrelation C_B(Δr)=⟨B(r)B⁎(r+Δr)⟩. Consequently, χ splits into three contributions: signal‑signal, signal‑background cross‑terms, and background‑background. The latter raises the floor of the AF, limiting the smallest detectable displacement.

Assuming a white Gaussian background (C_B(Δr)=σ_B² δ(Δr)), the effective width of χ becomes √(σ_S²+σ_B²), where σ_S is the intrinsic spectral width of the signal. This demonstrates that background power directly broadens the ambiguity surface and degrades both resolution and measurement accuracy.

Resolution analysis shows that linear displacement resolution is limited by the maximum supported spatial‑frequency bandwidth Δk_max, giving a Rayleigh‑type criterion Δx_min≈1/Δk_max. Angular (or wave‑front tilt) resolution follows the classic aperture formula Δθ_min≈λ/(2π D), with D the aperture diameter. In the presence of background noise, the signal‑to‑noise ratio (SNR) drops, and the Cramér‑Rao lower bound (CRLB) for unbiased estimators becomes

 CRLB(Δx) = 1 /