--- title: Reduced Bussgang Kalman Filter for 1-Bit Data url: https://www.emergentmind.com/topics/reduced-bussgang-aided-kalman-filter type: topic --- # Reduced Bussgang Kalman Filter for 1-Bit Data Searching arXiv for the specified papers to ground the article in the cited literature. Reduced Bussgang-aided Kalman filter (RBKF) is a computationally efficient variant of the Bussgang-aided Kalman filter (BA-KF) for state estimation from 1-bit observations. It is designed for settings in which the observation dimension is large, often with many 1-bit analog-to-digital converters, so that the dominant computational cost of the Bussgang-based update is the inversion of a high-dimensional innovation covariance. RBKF preserves the Bussgang linearization and adaptive thresholding strategy of BA-KF, but reduces the measurement dimension by a fixed linear projection before the update. In the formulation reported in "State Estimation with 1-Bit Observations and Imperfect Models: Bussgang Meets Kalman in Neural Networks" [2507.17284], the method targets both linear and nonlinear state-space models, while its computational rationale aligns with the broader catalog of structure-exploiting Kalman filter optimizations surveyed in "On Computational Complexity Reduction Methods for Kalman Filter Extensions" [1512.03077]. ## 1. Problem formulation and scope RBKF is posed for a possibly nonlinear state-space model with state evolution $$ x_k = f(x_{k-1}) + w_k, $$ and pre-quantization observation $$ u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n, $$ where \(w_k \sim N(0,Q_k)\) and \(v_k \sim N(0,R_k)\), mutually independent, and independent of \(x_0\) [2507.17284]. The filter operates after 1-bit quantization, with the analog-to-digital converter applying the elementwise sign function with threshold \(\tau_k\): $$ r_k = Q(u_k-\tau_k), \qquad Q(\cdot)=\mathrm{sign}(\cdot) $$ applied coordinatewise [2507.17284]. A defining feature of the method is adaptive thresholding, also described as dithering. The threshold is set to the one-step-ahead prediction of the measurement, $$ \tau_k = \hat y_{k|k-1} = h(\hat x_{k|k-1}), $$ so that $$ z_k := u_k-\tau_k $$ is zero-centered in the conditional sense \(E[z_k \mid r_{1:k-1}] = 0\) [2507.17284]. This step is central because it restores the simple zero-mean Bussgang coefficients for 1-bit quantization and avoids the more expensive non-zero-mean formulas involving Gaussian integrals [2507.17284]. RBKF itself is a model-based method. The formulation assumes that \(f(\cdot)\), \(h(\cdot)\), \(Q_k\), and \(R_k\) are known, possibly with EKF-style linearization for nonlinear models [2507.17284]. Robustness to model mismatch is limited in comparison with the accompanying Bussgang-aided KalmanNet, which was introduced in the same work for partial model knowledge [2507.17284]. ## 2. Bussgang linearization under 1-bit quantization The filter is built on the Bussgang decomposition for Gaussian inputs. If \(u \sim N(0,C_u)\) and \(y=\mathrm{sign}(u)\) is applied elementwise, then the cross-covariance is $$ C_{yu}=E[yu^T]=\sqrt{\frac{2}{\pi}}\,D_u^{-1/2}C_u, $$ with \(D_u:=\mathrm{diag}(C_u)\) [2507.17284]. The corresponding equivalent linearization is $$ y \approx A u + e, \qquad A=C_{yu}C_u^{-1}=\sqrt{\frac{2}{\pi}}\,D_u^{-1/2}, $$ where \(e\) is uncorrelated with \(u\), so \(E[e\,u^T]=0\) [2507.17284]. For the 1-bit output covariance, the method uses the arcsine law. Writing $$ \rho_u=D_u^{-1/2}C_uD_u^{-1/2}, $$ the correlation matrix of \(u\), one has $$ C_{yy}=\frac{2}{\pi}\arcsin(\rho_u), $$ with the \(\arcsin\) operator applied elementwise to entries in \([-1,1]\) [2507.17284]. Within the Kalman filtering context, these identities are applied to the dithered residual $$ z_k=u_k-\hat y_{k|k-1}. $$ For linear \(h\), or for a Jacobian-based residual in the nonlinear case, $$ z_k = H_k(x_k-\hat x_{k|k-1}) + v_k, $$ and the quantized observation is approximated as $$ r_k = Q(z_k) \approx B_k z_k + \eta_k, $$ with $$ B_k=\sqrt{\frac{2}{\pi}}\,D_{z,k}^{-1/2}, \qquad D_{z,k}:=\mathrm{diag}(P_{k|k-1}), $$ where \(\eta_k\) is uncorrelated with \(z_k\) [2507.17284]. The effective linear measurement model is then expressed on the state deviation \(\delta x_k:=x_k-\hat x_{k|k-1}\) as $$ r_k \approx H_{b,k}\,\delta x_k + w_k^b, $$ with $$ H_{b,k}:=B_k H_k, \qquad w_k^b:=B_k v_k+\eta_k $$ [2507.17284]. This construction is directly related to the statistical linearization viewpoint surveyed in [1512.03077], where a Jacobian-like matrix is written as $$ J=P_{g(z)z}P_{zz}^{-1}, $$ and explicitly noted to be the Bussgang-type gain for memoryless nonlinearities when inputs are Gaussian [1512.03077]. This suggests that RBKF can be understood as a specialized, quantization-aware instance of the broader statistical-linearization family. ## 3. From BA-KF to the reduced formulation The parent BA-KF first performs prediction in EKF style for nonlinear models: $$ \hat x_{k|k-1}=f(\hat x_{k-1|k-1}), $$ $$ F_k=\frac{\partial f}{\partial x}\ \text{evaluated at}\ \hat x_{k-1|k-1}, $$ $$ \Sigma_{k|k-1}=F_k\Sigma_{k-1|k-1}F_k^T+Q_k, $$ $$ \hat y_{k|k-1}=h(\hat x_{k|k-1}), $$ $$ H_k=\frac{\partial h}{\partial x}\ \text{evaluated at}\ \hat x_{k|k-1}, $$ and $$ P_{k|k-1}=H_k\Sigma_{k|k-1}H_k^T+R_k $$ [2507.17284]. Using the Bussgang approximation, BA-KF forms the output covariance $$ S_k:=\mathrm{Cov}(r_k)=\frac{2}{\pi}\arcsin\!\left(D_{z,k}^{-1/2}P_{k|k-1}D_{z,k}^{-1/2}\right), $$ or equivalently defines $$ R_{b,k}:=S_k-H_{b,k}\Sigma_{k|k-1}H_{b,k}^T $$ [2507.17284]. The Kalman-like gain is $$ K_k:=\Sigma_{k|k-1}H_{b,k}^T S_k^{-1}, $$ and the update is $$ \hat x_{k|k}=\hat x_{k|k-1}+K_k r_k, $$ $$ \Sigma_{k|k}=(I-K_kH_{b,k})\Sigma_{k|k-1} $$ [2507.17284]. Because adaptive dithering enforces \(E[r_k]=0\), no centering term is needed in the innovation [2507.17284]. RBKF modifies only the measurement side of this construction. It reduces the measurement dimension before the Bussgang update by a fixed projection $$ A \in \mathbb{R}^{n_r \times n}, \qquad n_r = a n,\quad 0