Reduced Bussgang Kalman Filter for 1-Bit Data

• The paper introduces RBKF, a variant that reduces computational cost by projecting high-dimensional 1-bit measurements prior to a Bussgang-based Kalman update.
• It leverages adaptive thresholding and Bussgang linearization to cut complexity from O(n³) to O((a·n)³), making it suitable for large-scale ADC applications.
• RBKF supports both linear and nonlinear state-space models using EKF-style linearization, aligning with broader strategies for Kalman filter optimization.

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" (Jung et al., 23 Jul 2025), 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" (Raitoharju et al., 2015).

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 $x_0$ (Jung et al., 23 Jul 2025). 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 (Jung et al., 23 Jul 2025).

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 $E[z_k \mid r_{1:k-1}] = 0$ (Jung et al., 23 Jul 2025). 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 (Jung et al., 23 Jul 2025).

RBKF itself is a model-based method. The formulation assumes that $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$0, $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$1, $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$2, and $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$3 are known, possibly with EKF-style linearization for nonlinear models (Jung et al., 23 Jul 2025). 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 (Jung et al., 23 Jul 2025).

2. Bussgang linearization under 1-bit quantization

The filter is built on the Bussgang decomposition for Gaussian inputs. If $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$4 and $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$5 is applied elementwise, then the cross-covariance is

$u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$6

with $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$7 (Jung et al., 23 Jul 2025). The corresponding equivalent linearization is

$u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$8

where $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$9 is uncorrelated with $w_k \sim N(0,Q_k)$0, so $w_k \sim N(0,Q_k)$1 (Jung et al., 23 Jul 2025).

For the 1-bit output covariance, the method uses the arcsine law. Writing

$w_k \sim N(0,Q_k)$2

the correlation matrix of $w_k \sim N(0,Q_k)$3, one has

$w_k \sim N(0,Q_k)$4

with the $w_k \sim N(0,Q_k)$5 operator applied elementwise to entries in $w_k \sim N(0,Q_k)$6 (Jung et al., 23 Jul 2025).

Within the Kalman filtering context, these identities are applied to the dithered residual

$w_k \sim N(0,Q_k)$7

For linear $w_k \sim N(0,Q_k)$8, or for a Jacobian-based residual in the nonlinear case,

$w_k \sim N(0,Q_k)$9

and the quantized observation is approximated as

$v_k \sim N(0,R_k)$0

with

$v_k \sim N(0,R_k)$1

where $v_k \sim N(0,R_k)$2 is uncorrelated with $v_k \sim N(0,R_k)$3 (Jung et al., 23 Jul 2025). The effective linear measurement model is then expressed on the state deviation $v_k \sim N(0,R_k)$4 as

$v_k \sim N(0,R_k)$5

with

$v_k \sim N(0,R_k)$6

(Jung et al., 23 Jul 2025).

This construction is directly related to the statistical linearization viewpoint surveyed in (Raitoharju et al., 2015), where a Jacobian-like matrix is written as

$v_k \sim N(0,R_k)$7

and explicitly noted to be the Bussgang-type gain for memoryless nonlinearities when inputs are Gaussian (Raitoharju et al., 2015). 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:

$v_k \sim N(0,R_k)$8

$v_k \sim N(0,R_k)$9

$x_0$0

$x_0$1

$x_0$2

and

$x_0$3

(Jung et al., 23 Jul 2025).

Using the Bussgang approximation, BA-KF forms the output covariance

$x_0$4

or equivalently defines

$x_0$5

(Jung et al., 23 Jul 2025). The Kalman-like gain is

$x_0$6

and the update is

$x_0$7

$x_0$8

(Jung et al., 23 Jul 2025). Because adaptive dithering enforces $x_0$9, no centering term is needed in the innovation (Jung et al., 23 Jul 2025).

RBKF modifies only the measurement side of this construction. It reduces the measurement dimension before the Bussgang update by a fixed projection

$\tau_k$0

for example by averaging $\tau_k$1 replicated ADCs per feature:

$\tau_k$2

(Jung et al., 23 Jul 2025). The projected covariance and effective measurement matrix become

$\tau_k$3

$\tau_k$4

(Jung et al., 23 Jul 2025). The reduced update is then

$\tau_k$5

$\tau_k$6

$\tau_k$7

(Jung et al., 23 Jul 2025).

The reduction is therefore not a modification of the Bussgang constants themselves, but a projection of the quantized observation, its covariance, and the effective linearized measurement operator.

4. Computational reduction and relation to generic KF optimization

The explicit motivation for RBKF is that the dominant cost in BA-KF is inverting $\tau_k$8, which is $\tau_k$9. This is particularly restrictive when many 1-bit ADCs are used and often $$r_k = Q(u_k-\tau_k), \qquad Q(\cdot)=\mathrm{sign}(\cdot)$$0 (Jung et al., 23 Jul 2025). The reported per-step complexity is

• BA-KF: $$r_k = Q(u_k-\tau_k), \qquad Q(\cdot)=\mathrm{sign}(\cdot)$$1,
• RBKF: $$r_k = Q(u_k-\tau_k), \qquad Q(\cdot)=\mathrm{sign}(\cdot)$$2 plus the projection cost $$r_k = Q(u_k-\tau_k), \qquad Q(\cdot)=\mathrm{sign}(\cdot)$$3, with dominant term $$r_k = Q(u_k-\tau_k), \qquad Q(\cdot)=\mathrm{sign}(\cdot)$$4 (Jung et al., 23 Jul 2025).

Memory is also reduced: storing $$r_k = Q(u_k-\tau_k), \qquad Q(\cdot)=\mathrm{sign}(\cdot)$$5 and its factors changes from $$r_k = Q(u_k-\tau_k), \qquad Q(\cdot)=\mathrm{sign}(\cdot)$$6 to $$r_k = Q(u_k-\tau_k), \qquad Q(\cdot)=\mathrm{sign}(\cdot)$$7 (Jung et al., 23 Jul 2025). In applications with replicated measurements, the projection can be chosen as block-averaging,

$$r_k = Q(u_k-\tau_k), \qquad Q(\cdot)=\mathrm{sign}(\cdot)$$8

when $$r_k = Q(u_k-\tau_k), \qquad Q(\cdot)=\mathrm{sign}(\cdot)$$9 and $\tau_k = \hat y_{k|k-1} = h(\hat x_{k|k-1}),$0 is the number of physical features (Jung et al., 23 Jul 2025).

This reduction mechanism is closely aligned with the general principles in (Raitoharju et al., 2015). That tutorial emphasizes sequential measurement processing for diagonal or block-diagonal $\tau_k = \hat y_{k|k-1} = h(\hat x_{k|k-1}),$1, grouped or partitioned measurement updates, low-rank updates via the Woodbury identity, square-root filters, and partially linear or conditionally linear reductions as standard routes for lowering Kalman-filter complexity (Raitoharju et al., 2015). In that framework, RBKF can be interpreted as a measurement-side dimension-reduction strategy specialized to Bussgang-effective observation models. A plausible implication is that RBKF belongs to the same structural-optimization class as grouped measurement updates and reduced-dimensional measurement transforms, although the 2025 formulation presents it directly as a projection of $\tau_k = \hat y_{k|k-1} = h(\hat x_{k|k-1}),$2, $\tau_k = \hat y_{k|k-1} = h(\hat x_{k|k-1}),$3, and $\tau_k = \hat y_{k|k-1} = h(\hat x_{k|k-1}),$4 rather than through the generic notation $\tau_k = \hat y_{k|k-1} = h(\hat x_{k|k-1}),$5 used in (Raitoharju et al., 2015).

The broader tutorial also notes that exploitation of the structure of the problem can lead to improved estimation accuracy while reducing the computational load (Raitoharju et al., 2015). For RBKF, the article’s own empirical claims are more specific: when projections average multiple i.i.d. replicas with identical $\tau_k = \hat y_{k|k-1} = h(\hat x_{k|k-1}),$6, information loss can be negligible, whereas heterogeneous noise or spatially varying statistics cause mild performance loss (Jung et al., 23 Jul 2025).

5. Nonlinear integration, numerical conditioning, and implementation

RBKF supports nonlinear $\tau_k = \hat y_{k|k-1} = h(\hat x_{k|k-1}),$7 and $\tau_k = \hat y_{k|k-1} = h(\hat x_{k|k-1}),$8 through EKF linearizations. The state Jacobian is

$\tau_k = \hat y_{k|k-1} = h(\hat x_{k|k-1}),$9

and the measurement Jacobian is

$z_k := u_k-\tau_k$0

after which the method proceeds through $z_k := u_k-\tau_k$1, the Bussgang mapping, and the RBKF update (Jung et al., 23 Jul 2025). A UKF variant is stated to be possible by plugging sigma-point-based $z_k := u_k-\tau_k$2, but the reported implementation uses EKF (Jung et al., 23 Jul 2025).

The implementation guidance emphasizes regularization and conditioning. The recommended stabilized quantities are

$z_k := u_k-\tau_k$3

followed by safe inverse square root evaluation; the normalized covariance

$z_k := u_k-\tau_k$4

is clipped entrywise to $z_k := u_k-\tau_k$5 before applying $z_k := u_k-\tau_k$6; $z_k := u_k-\tau_k$7 is symmetrized and regularized by small diagonal loading; and $z_k := u_k-\tau_k$8 is regularized by $z_k := u_k-\tau_k$9 if needed (Jung et al., 23 Jul 2025). The update covariance $E[z_k \mid r_{1:k-1}] = 0$0 is then symmetrized and, if necessary, adjusted by tiny diagonal loading to enforce positive semidefiniteness (Jung et al., 23 Jul 2025).

Square-root filtering is recommended for improved numerical robustness (Jung et al., 23 Jul 2025). This recommendation is consistent with the general survey in (Raitoharju et al., 2015), where square-root filters based on Cholesky or QR are described as numerically favorable, replacing inversions with triangular solves. The same tutorial also recommends the Joseph-stabilized covariance update

$E[z_k \mid r_{1:k-1}] = 0$1

as preferred for stability (Raitoharju et al., 2015). The RBKF formulas reported in (Jung et al., 23 Jul 2025) use the simpler covariance update $E[z_k \mid r_{1:k-1}] = 0$2. This suggests that, in implementations where numerical robustness is critical, the generic stabilization guidance from (Raitoharju et al., 2015) is relevant even though the 2025 paper presents the reduced method in compact form.

The practical notes on the projection $E[z_k \mid r_{1:k-1}] = 0$3 are also explicit. Block-averaging is presented as a good default when there are $E[z_k \mid r_{1:k-1}] = 0$4 replicas per physical measurement; it preserves scale and keeps $E[z_k \mid r_{1:k-1}] = 0$5 well-conditioned. For heterogeneous noise, more sophisticated $E[z_k \mid r_{1:k-1}] = 0$6, such as weighted averaging, can improve performance at modest extra cost (Jung et al., 23 Jul 2025).

6. Empirical behavior, comparative position, and limitations

The reported experiments cover the Lorenz-Attractor model and the Michigan NCLT dataset (Jung et al., 23 Jul 2025). On the Lorenz attractor with a single 1-bit ADC per feature and no reduction, the following mean-squared error results are reported:

• EKF (ideal measurements): $E[z_k \mid r_{1:k-1}] = 0$7 dB,
• KalmanNet (ideal): $E[z_k \mid r_{1:k-1}] = 0$8 dB,
• EKF with 1-bit: $E[z_k \mid r_{1:k-1}] = 0$9 dB,
• KalmanNet with 1-bit: $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$00 dB,
• BKF with 1-bit: $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$01 dB,
• BKNet with 1-bit: $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$02 dB (Jung et al., 23 Jul 2025).

For multiple 1-bit ADCs per feature, the comparison between RBKF and BKF is more directly informative. With identical noise across ADCs, RBKF matches BKF MSE across noise levels, while inference time drops dramatically as the number of ADCs increases; an example given is 128 ADCs per feature, where BKF is approximately $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$03 s and RBKF approximately $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$04 s on the same setup (Jung et al., 23 Jul 2025). With heterogeneous noise across ADCs, where variance is uniform in $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$05 dB, the reported Lorenz results are:

• EKF (ideal): $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$06 dB,
• BKF for $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$07 ADCs per feature: $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$08 dB,
• RBKF for the same cases: $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$09 dB (Jung et al., 23 Jul 2025).

On Michigan NCLT with partial model knowledge and a single ADC per feature, the reported values are:

• EKF (ideal): $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$10 dB,
• KalmanNet (ideal): $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$11 dB,
• with 1-bit, EKF: $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$12 dB,
• KalmanNet: $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$13 dB,
• BKF: $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$14 dB,
• BKNet: $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$15 dB (Jung et al., 23 Jul 2025). These results are used in the source to distinguish the intended operating regimes: RBKF when models are known and efficiency is needed, BA-KF when measurement dimension is modest and runtime is not critical, and BKNet when $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$16, $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$17, $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$18, and $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$19 are partially known or mismatched (Jung et al., 23 Jul 2025).

The limitations are stated in model and distributional terms. Degradation is expected under strongly heterogeneous noise among replicas, correlations not preserved by simple averaging, severe model mismatch in $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$20, $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$21, $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$22, and $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$23, and non-Gaussian pre-quantization inputs, since Bussgang is second-order optimal for Gaussian inputs (Jung et al., 23 Jul 2025). The method is also sensitive to the centering step: dithering that re-centers the residual is described as critical (Jung et al., 23 Jul 2025). More generally, (Raitoharju et al., 2015) cautions that partitioning and structural reductions must be balanced against numerical conditioning and linearization drift, especially in strongly nonlinear regimes.

In summary, RBKF is a reduced-dimensional Bussgang-linearized Kalman update for 1-bit observations. Its defining ingredients are adaptive thresholding $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$24, the Bussgang coefficient $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$25, the arcsine-law covariance $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$26, and a fixed projection $u_k = y_k = h(x_k) + v_k \in \mathbb{R}^n,$27 that compresses the quantized measurement prior to inversion (Jung et al., 23 Jul 2025). Within the broader theory of Kalman filter complexity reduction, it is an instance of exploiting observation structure to replace a high-dimensional update by a reduced one, while retaining recursive state estimation under the same underlying model assumptions (Raitoharju et al., 2015).