Bussgang-Aided KalmanNet

• The paper introduces a framework combining analytic Bussgang decomposition with deep recurrent neural networks to achieve high-accuracy state estimation under severe 1-bit quantization.
• It employs a Bussgang-aided Kalman filter and its reduced-complexity variant, significantly reducing computational cost without major RMSE performance loss.
• The architecture leverages GRU modules to adaptively learn Kalman gains, enhancing robustness against model mismatches and non-Gaussian quantization distortions.

Bussgang-aided KalmanNet is a state estimation framework designed for high-accuracy filtering in systems with 1-bit quantized observations and possibly imperfect process and measurement models. The approach combines analytic linearization based on the Bussgang decomposition with deep recurrent neural networks to mitigate the severe information loss induced by 1-bit quantization and to compensate for model mismatch. The algorithm extends traditional Kalman and hybrid filtering methodologies, providing robustness and computational efficiency in settings where quantization distortion and non-idealities are prominent (Jung et al., 23 Jul 2025).

1. System Model and 1-Bit Quantization

The foundation of Bussgang-aided KalmanNet is a discrete-time nonlinear state-space model: $$x_t = f(x_{t-1}) + w_t,\qquad w_t\sim\mathcal{N}(0,Q_t)$$

$$y_t = h(x_t) + v_t,\qquad v_t\sim\mathcal{N}(0,R_t)$$

where $x_t\in\mathbb{R}^m$ represents the latent state, and $y_t\in\mathbb{R}^n$ is the pre-quantized sensor measurement. Gaussian process noise $w_t$ and measurement noise $v_t$ are assumed mutually independent. Subsequently, the real-valued measurement is quantized by a 1-bit element-wise quantizer—possibly employing a dithering technique based on predictive centering: $z_t = y_t - \tau_t, \quad \tau_t = \hat y_{t|t-1}$

$$r_t = Q(z_t), \quad [Q(z)]_i = \begin{cases} +1,& z_i > 0 \ -1,& z_i \le 0 \end{cases}$$

The estimator aims to minimize mean squared state reconstruction error given binary observations: $$\hat x_{t|t} = \arg\min_{\hat x_t}\,\mathbb{E}\bigl[\|x_t-\hat x_t\|^2\,|\,r_{1},...,r_{t}\bigr]$$ 1-bit quantization, especially under non-dithered or asymmetric regimes, introduces significant distortion and non-Gaussianity, invalidating most classic Kalman-type solutions without substantial modification.

2. Bussgang Decomposition and Linear Surrogate

The Bussgang theorem is leveraged under the assumption that the quantizer input $z_t$ is (approximately) zero-mean Gaussian. The decomposition establishes a linear relationship between quantizer output and input through: $$y_t = h(x_t) + v_t,\qquad v_t\sim\mathcal{N}(0,R_t)$$0

$$y_t = h(x_t) + v_t,\qquad v_t\sim\mathcal{N}(0,R_t)$$1

yielding the surrogate linear model: $$y_t = h(x_t) + v_t,\qquad v_t\sim\mathcal{N}(0,R_t)$$2 Here, $$y_t = h(x_t) + v_t,\qquad v_t\sim\mathcal{N}(0,R_t)$$3 is an additive, zero-mean, uncorrelated distortion with covariance: $$y_t = h(x_t) + v_t,\qquad v_t\sim\mathcal{N}(0,R_t)$$4

$$y_t = h(x_t) + v_t,\qquad v_t\sim\mathcal{N}(0,R_t)$$5

This conversion enables 1-bit quantized observations to be processed using a Kalman-like linear update.

3. Bussgang-aided Kalman Filter and Reduced Complexity Variant

The Bussgang-aided Kalman Filter (BKF) propagates predictions as in the extended Kalman filter, but uses the linearized Bussgang measurement model for updates:

• Prediction:

  $$y_t = h(x_t) + v_t,\qquad v_t\sim\mathcal{N}(0,R_t)$$6

  $$y_t = h(x_t) + v_t,\qquad v_t\sim\mathcal{N}(0,R_t)$$7

• Update with the Bussgang gain:

  $$y_t = h(x_t) + v_t,\qquad v_t\sim\mathcal{N}(0,R_t)$$8

  $$y_t = h(x_t) + v_t,\qquad v_t\sim\mathcal{N}(0,R_t)$$9

  $x_t\in\mathbb{R}^m$0

The update treats $x_t\in\mathbb{R}^m$1 as a noisy, linear measurement.

For high-dimensional observations, an order-of-magnitude computational reduction is achieved using the reduced-complexity variant (rBKF). This introduces a projection matrix $x_t\in\mathbb{R}^m$2 (block averages, etc.), forming: $x_t\in\mathbb{R}^m$3

$x_t\in\mathbb{R}^m$4

All update equations use these projected quantities, reducing matrix inversion cost from $x_t\in\mathbb{R}^m$5 to $x_t\in\mathbb{R}^m$6.

4. Bussgang-aided KalmanNet Architecture

To address imperfect or partially known process/measurement models, Bussgang-aided KalmanNet (BKNet) parameterizes the gain computation as a learned function: $x_t\in\mathbb{R}^m$7

$x_t\in\mathbb{R}^m$8

The gain module $x_t\in\mathbb{R}^m$9 is realized by a sequence of three gated recurrent unit (GRU) blocks:

• GRU$y_t\in\mathbb{R}^n$0: captures dependency on process noise $y_t\in\mathbb{R}^n$1
• GRU$y_t\in\mathbb{R}^n$2: models prior state covariance $y_t\in\mathbb{R}^n$3
• GRU$y_t\in\mathbb{R}^n$4: encodes measurement covariance and quantizer nonlinearity

Inputs incorporate estimated state differences ($y_t\in\mathbb{R}^n$5), residual differences ($y_t\in\mathbb{R}^n$6), and matrix embeddings via fully-connected layers. A final fully-connected layer maps GRU outputs to the gain estimate. This modular, recurrent design preserves interpretability and low parameter count while enabling data-driven adaptation to unknown or non-Gaussian regimes.

5. Training Methodology and Pseudocode

BKNet employs a mean-square error loss with $y_t\in\mathbb{R}^n$7 regularization: $y_t\in\mathbb{R}^n$8 Training uses the Adam optimizer with a learning rate of $y_t\in\mathbb{R}^n$9, batch size $w_t$0, and regularization coefficient $w_t$1. Two types of data are used:

• Simulated: Lorenz attractor (sequence length 100, Gaussian noise, dithering quantization)
• Real-world: NCLT robot dataset (GPS/odometry, 6-dimensional state, 2-dimensional measurement, 1 Hz sampling; train/test on separate days)

Dithering centers the quantizer inputs at current predictions ($w_t$2), improving symmetry and Fisher information.

The update procedure follows the pseudocode:

$v_t$4 with $w_t$3 and $w_t$4.

6. Empirical Results and Performance

Key measured results:

Scenario	RMSE (dB)	Notes
Lorenz, EKF/KalmanNet (ideal)	$w_t$5
Lorenz, EKF/KalmanNet (1-bit)	fail ($w_t$612–18)	Unable to track accurately
Lorenz, BKF/rBKF (1-bit)	$w_t$7	Analytic Bussgang handling
Lorenz, BKNet (1-bit, mismatched models)	$w_t$8	Robust to model errors, Taylor expansions, rotational mismatches
NCLT, KalmanNet (ideal)	$w_t$9	Real robot data
NCLT, EKF/KalmanNet (1-bit)	fail	State estimation fails under strong quantization
NCLT, BKF (1-bit)	$v_t$0	High accuracy under 1-bit quantization
NCLT, BKNet (1-bit)	$v_t$1	Matches KalmanNet with ideal (unquantized) data

Reduced-complexity BKF with $v_t$2 ADCs at the sensor achieves a runtime decrease by a factor of 4–10 compared to the full BKF with negligible RMSE increase.

7. Theoretical Foundations and Interpretational Insights

Crucial to the performance of Bussgang-aided KalmanNet are several theoretical conditions:

• Centering quantizer inputs via dithering is critical, ensuring a computable diagonal structure for Bussgang gains and maximizing information extraction.
• For (near-)Gaussian residuals, the Bussgang decomposition ensures that the linear surrogate maintains second-order statistics of the original quantized system. Thus, Kalman recursions yield (approximately) optimal LMMSE state estimates.
• The residual noise $v_t$3 is approximated as Gaussian, with empirical justification in cases of moderate distortion or when large numbers of parallel 1-bit channels are aggregated.
• Within the BKNet extension, GRU modules flexibly adapt to biases, higher-order nonlinearities, and mismatches. The universal approximation capabilities of recurrent neural networks, under sufficient data and capacity, guarantee that the true time-varying gains can be learned.

Together, the Bussgang-aided Kalman filter and its neural extension KalmanNet form a principled, modular, and data-adaptive framework for nonlinear state estimation under severe quantization, robust to model errors and suitable for both simulated and real-world settings (Jung et al., 23 Jul 2025).