Розроблення уніфікованої спектрально-кореляційної фазової математичної моделі для однохідної спільної оцінки часу затримки та напряму приходу шумових широкосмугових сигналів

The object of this study is the delay and direction of arrival (DOA) of wideband signals in additive noise observed by a synchronized two-channel receiving system. The problem is that most wideband delay and DOA estimation methods rely on multi-stage or iterative procedures – such as peak search, covariance and eigen-decomposition, angular scanning, or sparse recovery – which increase computational load and restrict real-time applicability. The essence of the obtained results is a unified spectral-correlation model that enables joint estimation of delay and DOA directly from the phase of the complex cross-spectrum. The delay is determined from the linear phase slope with respect to frequency, while the DOA is obtained from a deterministic phase offset defined by the known antenna baseline. This formulation eliminates iterative refinement and angular scanning and reduces computational complexity to O(Nlog2N). Numerical experiments with linear frequency-modulated signals in Gaussian noise demonstrate predictable accuracy behavior: with increasing signal-to-noise ratio, the DOA root-mean-square error decreases from 0.15–0.18° to 0.03–0.04°, while the delay error decreases from tens of nanoseconds to a few nanoseconds. These results are explained by the deterministic relationship between time shifts, spatial delays, and phase behavior in the frequency domain under stationary noise conditions. The proposed model can be applied in practice for synchronized two-channel wideband reception with known geometry and fixed antenna baseline in real-time passive monitoring and direction-finding systems.