Closed-Form Successive Relative Transfer Function Vector Estimation based on Blind Oblique Projection Incorporating Noise Whitening

Relative transfer functions (RTFs) of sound sources play a crucial role in beamforming, enabling effective noise and interference suppression. This paper addresses the challenge of online estimating the RTF vectors of multiple sound sources in noisy and reverberant environments, for the specific scenario where sources activate successively. While the RTF vector of the first source can be estimated straightforwardly, the main challenge arises in estimating the RTF vectors of subsequent sources during segments where multiple sources are simultaneously active. The blind oblique projection (BOP) method has been proposed to estimate the RTF vector of a newly activating source by optimally blocking this source. However, this method faces several limitations: high computational complexity due to its reliance on iterative gradient descent optimization, the introduction of random additional vectors, which can negatively impact performance, and the assumption of high signal-to-noise ratio (SNR). To overcome these limitations, in this paper we propose three extensions to the BOP method. First, we derive a closed-form solution for optimizing the BOP cost function, significantly reducing computational complexity. Second, we introduce orthogonal additional vectors instead of random vectors, enhancing RTF vector estimation accuracy. Third, we incorporate noise handling techniques inspired by covariance subtraction and whitening, increasing robustness in low SNR conditions. To provide a frame-by-frame estimate of the source activity pattern, required by both the conventional BOP method and the proposed method, we propose a spatial-coherence-based online source counting method. Simulations are performed with real-world reverberant noisy recordings featuring 3 successively activating speakers, with and without a-priori knowledge of the source activity pattern.