Bussgang-Aided Kalman Filter

• The paper introduces a Bussgang-aided Kalman filter that transforms highly quantized 1-bit measurements into an effective linear Gaussian model for state estimation.
• It presents a reduced-complexity variant that employs linear projection to average ADC outputs, significantly reducing computational cost while retaining accuracy.
• The approach is extended via Bussgang-aided KalmanNet, a deep learning architecture that compensates for model mismatches and improves real-world performance.

Searching arXiv for the specified paper and closely related context papers. A Bussgang-aided Kalman filter is a Kalman-type state estimator that has been modified to work with severely quantized measurements, specifically 1-bit observations, by using the Bussgang decomposition to construct an effective linear observation model inside the filter. In the formulation developed in "State Estimation with 1-Bit Observations and Imperfect Models: Bussgang Meets Kalman in Neural Networks," the method addresses state estimation under 1-bit quantization, introduces a reduced-complexity variant, and extends the framework to partially known models through a hybrid neural architecture termed the Bussgang-aided KalmanNet (Jung et al., 23 Jul 2025).

1. Problem formulation and motivation

The starting point is a general discrete-time nonlinear state-space model with ideal observations,

$$\mathbf{x}_{t} = \mathbf{f}(\mathbf{x}_{t-1}) + \mathbf{w}_{t}, \quad \mathbf{w}_t \sim \mathcal{N}(0,\mathbf{Q}_t),$$

$$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$

with $\mathbf{f}:\mathbb{R}^m\to\mathbb{R}^m$, $\mathbf{h}:\mathbb{R}^m\to\mathbb{R}^n$, and Gaussian process and measurement noises. The standard objective is

$$\hat{\mathbf{x}}_t = \arg\min_{\hat{\mathbf{x}}_t}\mathbb{E}\!\left[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{y}_1,\ldots,\mathbf{y}_t\right].$$

The 1-bit setting replaces the ideal observation with an element-wise sign quantizer,

$\mathbf{r}_t = \mathbf{Q}(\mathbf{y}_t),$

with components

$$\mathbf{r}_t[i]= \begin{cases} +1, & \mathbf{y}_t[i] > 0,\ -1, & \mathbf{y}_t[i] \le 0. \end{cases}$$

The estimation target becomes

$$\min_{\hat{\mathbf{x}}_t}\mathbb{E}\bigl[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_t\bigr].$$

This setting is difficult for standard Kalman filtering, the extended Kalman filter, and direct KalmanNet application because the observation relation $$\mathbf{x}_t \mapsto \mathbf{r}_t=\mathbf{Q}(\mathbf{h}(\mathbf{x}_t)+\mathbf{v}_t)$$ is strongly nonlinear and discontinuous; the sign function is non-differentiable; Kalman filtering relies on Gaussianity and linear observation models to propagate Gaussian posteriors; and 1-bit quantization induces highly non-Gaussian likelihoods. The core idea of the Bussgang-aided Kalman filter is therefore to approximate the nonlinear observation with a statistically equivalent linear model so that a Kalman-like update becomes possible even with 1-bit observations (Jung et al., 23 Jul 2025).

2. Bussgang decomposition and dithered observation centering

For a zero-mean Gaussian vector $\mathbf{y}\in\mathbb{R}^n$ with covariance $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$0, passed through a memoryless nonlinearity $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$1 to produce $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$2, Bussgang’s theorem states that the cross-correlation between input and output is proportional to the input covariance,

$$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$3

for some matrix $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$4 depending only on the nonlinearity and the input statistics. For 1-bit sign quantization with zero-mean Gaussian input, the Bussgang coefficient matrix is diagonal,

$$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$5

This yields the approximate linear model

$$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$6

where $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$7 is a zero-mean distortion term that is uncorrelated with $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$8 and is approximated as Gaussian for tractability.

The zero-mean condition is central. If the input to the quantizer is non-zero mean, the corresponding linear minimum mean-square error form involves $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$9, $\mathbf{f}:\mathbb{R}^m\to\mathbb{R}^m$0, and $\mathbf{f}:\mathbb{R}^m\to\mathbb{R}^m$1 through analytically and computationally expensive expressions. To avoid this, and to maximize information content of 1-bit measurements, the framework introduces dithering by subtracting a predicted mean from the quantizer input. With

$\mathbf{f}:\mathbb{R}^m\to\mathbb{R}^m$2

the dithered signal is

$\mathbf{f}:\mathbb{R}^m\to\mathbb{R}^m$3

and the quantized observation becomes

$\mathbf{f}:\mathbb{R}^m\to\mathbb{R}^m$4

By construction,

$\mathbf{f}:\mathbb{R}^m\to\mathbb{R}^m$5

Under Gaussian assumptions, $\mathbf{f}:\mathbb{R}^m\to\mathbb{R}^m$6 has covariance

$\mathbf{f}:\mathbb{R}^m\to\mathbb{R}^m$7

where $\mathbf{f}:\mathbb{R}^m\to\mathbb{R}^m$8 is the Jacobian of $\mathbf{f}:\mathbb{R}^m\to\mathbb{R}^m$9 at $\mathbf{h}:\mathbb{R}^m\to\mathbb{R}^n$0. Applying Bussgang to $\mathbf{h}:\mathbb{R}^m\to\mathbb{R}^n$1 gives

$\mathbf{h}:\mathbb{R}^m\to\mathbb{R}^n$2

with

$\mathbf{h}:\mathbb{R}^m\to\mathbb{R}^n$3

The covariance of the 1-bit output is available in closed form through Price’s theorem,

$\mathbf{h}:\mathbb{R}^m\to\mathbb{R}^n$4

where $\mathbf{h}:\mathbb{R}^m\to\mathbb{R}^n$5. The resulting construction furnishes an effective linear Gaussian observation model in terms of second-order statistics, which is the analytical basis of the Bussgang-aided Kalman filter (Jung et al., 23 Jul 2025).

3. Bussgang-aided Kalman filter recursion

The Bussgang-aided Kalman filter is essentially an extended Kalman filter for the state evolution together with a Bussgang-based linearization of the 1-bit observation. The system model is assumed known, including $\mathbf{h}:\mathbb{R}^m\to\mathbb{R}^n$6, $\mathbf{h}:\mathbb{R}^m\to\mathbb{R}^n$7, $\mathbf{h}:\mathbb{R}^m\to\mathbb{R}^n$8, and $\mathbf{h}:\mathbb{R}^m\to\mathbb{R}^n$9.

Given the posterior pair $$\hat{\mathbf{x}}_t = \arg\min_{\hat{\mathbf{x}}_t}\mathbb{E}\!\left[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{y}_1,\ldots,\mathbf{y}_t\right].$$0, the prediction step is

$$\hat{\mathbf{x}}_t = \arg\min_{\hat{\mathbf{x}}_t}\mathbb{E}\!\left[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{y}_1,\ldots,\mathbf{y}_t\right].$$1

$$\hat{\mathbf{x}}_t = \arg\min_{\hat{\mathbf{x}}_t}\mathbb{E}\!\left[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{y}_1,\ldots,\mathbf{y}_t\right].$$2

where $$\hat{\mathbf{x}}_t = \arg\min_{\hat{\mathbf{x}}_t}\mathbb{E}\!\left[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{y}_1,\ldots,\mathbf{y}_t\right].$$3 is the Jacobian of $$\hat{\mathbf{x}}_t = \arg\min_{\hat{\mathbf{x}}_t}\mathbb{E}\!\left[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{y}_1,\ldots,\mathbf{y}_t\right].$$4 at $$\hat{\mathbf{x}}_t = \arg\min_{\hat{\mathbf{x}}_t}\mathbb{E}\!\left[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{y}_1,\ldots,\mathbf{y}_t\right].$$5. The predicted measurement and its covariance are

$$\hat{\mathbf{x}}_t = \arg\min_{\hat{\mathbf{x}}_t}\mathbb{E}\!\left[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{y}_1,\ldots,\mathbf{y}_t\right].$$6

$$\hat{\mathbf{x}}_t = \arg\min_{\hat{\mathbf{x}}_t}\mathbb{E}\!\left[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{y}_1,\ldots,\mathbf{y}_t\right].$$7

Using the dithered 1-bit observation, the effective linearized observation model is treated as

$$\hat{\mathbf{x}}_t = \arg\min_{\hat{\mathbf{x}}_t}\mathbb{E}\!\left[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{y}_1,\ldots,\mathbf{y}_t\right].$$8

The analogue of the Kalman gain is the Bussgang gain,

$$\hat{\mathbf{x}}_t = \arg\min_{\hat{\mathbf{x}}_t}\mathbb{E}\!\left[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{y}_1,\ldots,\mathbf{y}_t\right].$$9

The update equations are

$\mathbf{r}_t = \mathbf{Q}(\mathbf{y}_t),$0

$\mathbf{r}_t = \mathbf{Q}(\mathbf{y}_t),$1

Relative to the extended Kalman filter, the substitution is structural. The ordinary gain

$\mathbf{r}_t = \mathbf{Q}(\mathbf{y}_t),$2

is replaced by a gain that uses $\mathbf{r}_t = \mathbf{Q}(\mathbf{y}_t),$3 and the quantized-output covariance $\mathbf{r}_t = \mathbf{Q}(\mathbf{y}_t),$4. In this formulation, $\mathbf{r}_t = \mathbf{Q}(\mathbf{y}_t),$5 accounts for the scaling of the measurement residual due to quantization, while $\mathbf{r}_t = \mathbf{Q}(\mathbf{y}_t),$6 captures how much information remains in the sign measurements.

The validity conditions stated for the method are that the state and measurement models are known and differentiable so that $\mathbf{r}_t = \mathbf{Q}(\mathbf{y}_t),$7 and $\mathbf{r}_t = \mathbf{Q}(\mathbf{y}_t),$8 are available; the pre-quantized measurement $\mathbf{r}_t = \mathbf{Q}(\mathbf{y}_t),$9 is approximately Gaussian; the 1-bit quantizer is memoryless and element-wise; and the Bussgang distortion term is treated as zero-mean Gaussian and uncorrelated, enabling Kalman-like derivations (Jung et al., 23 Jul 2025).

4. Reduced-complexity formulation and the Bussgang-aided KalmanNet

The full Bussgang-aided Kalman filter requires inversion of the $$\mathbf{r}_t[i]= \begin{cases} +1, & \mathbf{y}_t[i] > 0,\ -1, & \mathbf{y}_t[i] \le 0. \end{cases}$$0 matrix $$\mathbf{r}_t[i]= \begin{cases} +1, & \mathbf{y}_t[i] > 0,\ -1, & \mathbf{y}_t[i] \le 0. \end{cases}$$1 at each step. When the number of 1-bit analog-to-digital converters per time step is large, the inversion cost is $$\mathbf{r}_t[i]= \begin{cases} +1, & \mathbf{y}_t[i] > 0,\ -1, & \mathbf{y}_t[i] \le 0. \end{cases}$$2. The reduced Bussgang-aided Kalman filter addresses this by reducing the dimension of the observation vector through a linear projection.

The reduced observation is defined as

$$\mathbf{r}_t[i]= \begin{cases} +1, & \mathbf{y}_t[i] > 0,\ -1, & \mathbf{y}_t[i] \le 0. \end{cases}$$3

where $$\mathbf{r}_t[i]= \begin{cases} +1, & \mathbf{y}_t[i] > 0,\ -1, & \mathbf{y}_t[i] \le 0. \end{cases}$$4 with $$\mathbf{r}_t[i]= \begin{cases} +1, & \mathbf{y}_t[i] > 0,\ -1, & \mathbf{y}_t[i] \le 0. \end{cases}$$5. The paper uses an averaging-based projection

$$\mathbf{r}_t[i]= \begin{cases} +1, & \mathbf{y}_t[i] > 0,\ -1, & \mathbf{y}_t[i] \le 0. \end{cases}$$6

which groups every $$\mathbf{r}_t[i]= \begin{cases} +1, & \mathbf{y}_t[i] > 0,\ -1, & \mathbf{y}_t[i] \le 0. \end{cases}$$7 analog-to-digital converters per measurement feature and averages their outputs. The measurement function is correspondingly adapted as

$$\mathbf{r}_t[i]= \begin{cases} +1, & \mathbf{y}_t[i] > 0,\ -1, & \mathbf{y}_t[i] \le 0. \end{cases}$$8

The reduced covariance is

$$\mathbf{r}_t[i]= \begin{cases} +1, & \mathbf{y}_t[i] > 0,\ -1, & \mathbf{y}_t[i] \le 0. \end{cases}$$9

and the reduced Bussgang gain is

$$\min_{\hat{\mathbf{x}}_t}\mathbb{E}\bigl[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_t\bigr].$$0

with $$\min_{\hat{\mathbf{x}}_t}\mathbb{E}\bigl[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_t\bigr].$$1. The update equations become

$$\min_{\hat{\mathbf{x}}_t}\mathbb{E}\bigl[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_t\bigr].$$2

$$\min_{\hat{\mathbf{x}}_t}\mathbb{E}\bigl[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_t\bigr].$$3

The inversion cost is therefore reduced to $$\min_{\hat{\mathbf{x}}_t}\mathbb{E}\bigl[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_t\bigr].$$4.

To address imperfect or unknown models, the framework introduces the Bussgang-aided KalmanNet, a model-based deep learning architecture that retains the recursive prediction-correction structure, uses the reduced Bussgang-aided Kalman filter as its skeleton, and replaces the analytical reduced Bussgang gain by a learned Bussgang gain $$\min_{\hat{\mathbf{x}}_t}\mathbb{E}\bigl[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_t\bigr].$$5 produced by a gated recurrent unit architecture:

$$\min_{\hat{\mathbf{x}}_t}\mathbb{E}\bigl[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_t\bigr].$$6

The architecture uses multiple gated recurrent units—$$\min_{\hat{\mathbf{x}}_t}\mathbb{E}\bigl[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_t\bigr].$$7-GRU, $$\min_{\hat{\mathbf{x}}_t}\mathbb{E}\bigl[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_t\bigr].$$8-GRU, and $$\min_{\hat{\mathbf{x}}_t}\mathbb{E}\bigl[\|\mathbf{x}_t-\hat{\mathbf{x}}_t\|^2 \mid \mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_t\bigr].$$9-GRU—together with fully connected layers. The design mimics the analytical structure of the reduced Bussgang gain and processes temporal features derived from past and current state estimates, differences in state estimates, reduced quantized observations, and their temporal differences. The $$\mathbf{x}_t \mapsto \mathbf{r}_t=\mathbf{Q}(\mathbf{h}(\mathbf{x}_t)+\mathbf{v}_t)$$0-GRU learns an effective process noise covariance from state variations over time; the $$\mathbf{x}_t \mapsto \mathbf{r}_t=\mathbf{Q}(\mathbf{h}(\mathbf{x}_t)+\mathbf{v}_t)$$1-GRU emulates the recursive covariance update without knowing the true model; and the $$\mathbf{x}_t \mapsto \mathbf{r}_t=\mathbf{Q}(\mathbf{h}(\mathbf{x}_t)+\mathbf{v}_t)$$2-GRU learns the combined factor $$\mathbf{x}_t \mapsto \mathbf{r}_t=\mathbf{Q}(\mathbf{h}(\mathbf{x}_t)+\mathbf{v}_t)$$3. Fully connected layers combine the outputs to produce $$\mathbf{x}_t \mapsto \mathbf{r}_t=\mathbf{Q}(\mathbf{h}(\mathbf{x}_t)+\mathbf{v}_t)$$4 and update an internal estimate of $$\mathbf{x}_t \mapsto \mathbf{r}_t=\mathbf{Q}(\mathbf{h}(\mathbf{x}_t)+\mathbf{v}_t)$$5 for the next time step. The same dithering idea is used in the neural architecture: a measurement prediction is computed from the predicted state and subtracted before 1-bit quantization, centering the quantizer input and maximizing information flow through the 1-bit analog-to-digital converter (Jung et al., 23 Jul 2025).

5. Training objective and treatment of model mismatch

The Bussgang-aided KalmanNet is trained offline on sequences of ground-truth states and 1-bit observations. The dataset is

$$\mathbf{x}_t \mapsto \mathbf{r}_t=\mathbf{Q}(\mathbf{h}(\mathbf{x}_t)+\mathbf{v}_t)$$6

with

$$\mathbf{x}_t \mapsto \mathbf{r}_t=\mathbf{Q}(\mathbf{h}(\mathbf{x}_t)+\mathbf{v}_t)$$7

For sequence $$\mathbf{x}_t \mapsto \mathbf{r}_t=\mathbf{Q}(\mathbf{h}(\mathbf{x}_t)+\mathbf{v}_t)$$8, the loss is

$$\mathbf{x}_t \mapsto \mathbf{r}_t=\mathbf{Q}(\mathbf{h}(\mathbf{x}_t)+\mathbf{v}_t)$$9

and the mini-batch loss is

$\mathbf{y}\in\mathbb{R}^n$0

For the instantaneous loss

$\mathbf{y}\in\mathbb{R}^n$1

with $\mathbf{y}\in\mathbb{R}^n$2, the gradient with respect to the gain mapping is

$\mathbf{y}\in\mathbb{R}^n$3

Parameters $\mathbf{y}\in\mathbb{R}^n$4 are updated via Adam.

The formulation explicitly considers model mismatch. The cases described are inaccurate Taylor expansion order of a nonlinear state model in the Lorenz system, rotations of the state evolution function $\mathbf{y}\in\mathbb{R}^n$5, rotations of the measurement function $\mathbf{y}\in\mathbb{R}^n$6, and inaccurate noise covariances together with partially known dynamics in the Michigan NCLT dataset. The stated conclusion is that the model-based Bussgang-aided Kalman filter and its reduced variant assume the given model is correct and degrade badly when it is not, whereas the Bussgang-aided KalmanNet learns the effective gain directly from data and can compensate for such mismatches (Jung et al., 23 Jul 2025).

6. Empirical results on Lorenz and Michigan NCLT

The paper evaluates the Bussgang-aided Kalman filter, the reduced Bussgang-aided Kalman filter, and the Bussgang-aided KalmanNet on the Lorenz attractor and the Michigan NCLT dataset. Comparisons are made against the extended Kalman filter and KalmanNet under ideal and 1-bit observations.

For the Lorenz attractor, the continuous-time dynamics are described through the Jacobian

$\mathbf{y}\in\mathbb{R}^n$7

with $\mathbf{y}\in\mathbb{R}^n$8, $\mathbf{y}\in\mathbb{R}^n$9, and $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$00. A discrete-time approximation uses

$$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$01

and

$$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$02

In the basic Lorenz experiments, the measurement model is an identity measurement per state component.

Under single 1-bit analog-to-digital conversion per feature, signal-to-noise ratio $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$03 dB, and $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$04 dB, the reported mean-squared errors in dB are as follows.

Setting	Method	MSE
Ideal observation	EKF	-19.31 dB
Ideal observation	KalmanNet	-19.49 dB
1-bit observation	EKF	17.85 dB
1-bit observation	KalmanNet	12.95 dB
1-bit observation	BKF	-17.38 dB
1-bit observation	BKNet	-17.31 dB

For multiple analog-to-digital converters per measurement feature, the study considers 1, 8, 64, and 128 analog-to-digital converters under both identical and heterogeneous noise. In the identical-noise case, the mean-squared-error curves of the Bussgang-aided Kalman filter and the reduced Bussgang-aided Kalman filter are essentially identical at each analog-to-digital converter count, and for moderate noise with $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$05 dB and multiple analog-to-digital converters, the 1-bit Bussgang methods outperform the extended Kalman filter with ideal observations. Average inference time in the identical-noise case is reported as 0.601 s for ideal-observation extended Kalman filtering, 0.691 s to 5.428 s for the Bussgang-aided Kalman filter from 1 to 128 analog-to-digital converters, and approximately 0.72 s to 0.89 s for the reduced Bussgang-aided Kalman filter across all analog-to-digital converter counts.

In the heterogeneous-noise Lorenz case, where each analog-to-digital converter has independent noise drawn uniformly from $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$06 dB, the mean-squared errors in dB as a function of analog-to-digital converter count are reported as follows: ideal-observation extended Kalman filter, $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$07 dB; Bussgang-aided Kalman filter, $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$08, $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$09, $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$10, and $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$11 dB for 1, 8, 64, and 128 analog-to-digital converters; reduced Bussgang-aided Kalman filter, $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$12, $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$13, $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$14, and $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$15 dB. The reduced method is therefore very close to the full Bussgang-aided Kalman filter, with small degradation due to averaging.

For the neural variant with multiple analog-to-digital converters, the identical-noise experiments show that the Bussgang-aided KalmanNet error decreases as noise decreases, and except for 1 analog-to-digital converter per feature, the Bussgang-aided KalmanNet with 1-bit observations significantly outperforms KalmanNet with ideal observations. Its inference time remains almost constant, approximately 1.03 s to 1.05 s across analog-to-digital converter counts, compared with 1.014 s for KalmanNet. In the heterogeneous-noise case, KalmanNet with ideal observations achieves $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$16 dB, while the Bussgang-aided KalmanNet with 1, 8, 64, and 128 analog-to-digital converters achieves $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$17, $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$18, $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$19, and $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$20 dB.

The Lorenz mismatch experiments further quantify the effect of imperfect models. When the true state dynamics are represented by a 5th-order Taylor expansion but the filter uses only $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$21, the Bussgang-aided Kalman filter yields $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$22 dB while the Bussgang-aided KalmanNet yields $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$23 dB, close to the $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$24 dB achieved with the full model. With a rotation of $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$25 by $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$26, the Bussgang-aided Kalman filter gives $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$27 dB whereas the Bussgang-aided KalmanNet gives $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$28 dB. With a rotation of $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$29 by $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$30, the Bussgang-aided Kalman filter gives $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$31 dB and the Bussgang-aided KalmanNet gives $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$32 dB (Jung et al., 23 Jul 2025).

For the Michigan NCLT dataset, the state is

$$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$33

with each axis modeled as a discrete Wiener-velocity process with transition matrix

$$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$34

process covariance

$$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$35

and two-dimensional block forms

$$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$36

Measurements are noisy velocities along each axis, with

$$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$37

and

$$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$38

The paper states that in practice $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$39 and $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$40 are not known exactly, so the experiments reflect partial model knowledge. With one 1-bit analog-to-digital converter per velocity measurement, the mean-squared errors in dB are: ideal-observation extended Kalman filter, $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$41 dB; ideal-observation KalmanNet, $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$42 dB; 1-bit extended Kalman filter, $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$43 dB; 1-bit KalmanNet, $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$44 dB; 1-bit Bussgang-aided Kalman filter, $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$45 dB; and 1-bit Bussgang-aided KalmanNet, $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$46 dB. The stated interpretation is that the extended Kalman filter and KalmanNet fail under 1-bit quantization, the Bussgang-aided Kalman filter improves but still suffers from model mismatch, and the Bussgang-aided KalmanNet under 1-bit observations and partial model knowledge achieves better performance than KalmanNet with ideal observations (Jung et al., 23 Jul 2025).

7. Scope, interpretation, and related distinctions

Within this framework, the term “Bussgang-aided Kalman filter” denotes a Kalman-type state estimator that uses Bussgang decomposition to transform 1-bit sign-quantized measurements into an effective linear Gaussian observation model, via a Bussgang gain and a quantized-output covariance, followed by prediction and update steps analogous to an extended Kalman filter and supported by a dithering scheme that maintains zero-mean input to the quantizer (Jung et al., 23 Jul 2025).

Three variants are distinguished. The full Bussgang-aided Kalman filter is statistically principled, requires no training, and is intended for settings with accurate model knowledge and moderate observation dimension. The reduced Bussgang-aided Kalman filter lowers complexity through projection while preserving almost all performance of the full method in the reported experiments. The Bussgang-aided KalmanNet keeps the recursive, model-based structure but replaces the analytical reduced Bussgang gain with a learned gain, allowing operation under partially known or mismatched models.

A common misconception is that conventional extended Kalman filtering or KalmanNet can simply be applied to 1-bit sign measurements. The reported Lorenz and NCLT results do not support that view: direct application under 1-bit observations gives large positive dB errors, whereas the Bussgang-based formulations explicitly capture quantization distortion through $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$47 and $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$48 and thereby recover accurate state estimation in cases where standard approaches degrade severely. Another possible misconception is that the reduced variant is merely a heuristic compression step. The derivation instead preserves the covariance structure under a linear projection and yields a reduced Bussgang gain with a stated inversion cost of $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$49 rather than $$\mathbf{y}_{t} = \mathbf{h}(\mathbf{x}_{t}) + \mathbf{v}_{t}, \quad \mathbf{v}_t \sim \mathcal{N}(0,\mathbf{R}_t),$$50.

The paper further states that the same Bussgang idea would generalize to other memoryless nonlinearities and to multi-bit low-resolution analog-to-digital converters, because for Gaussian inputs the Bussgang theorem still gives a linear relationship between input and output covariance. This suggests a broader methodological role for Bussgang-aided state estimation beyond the extreme 1-bit case, although the developed analysis and experiments focus specifically on 1-bit observations (Jung et al., 23 Jul 2025).