Delayed Kalman Filter (DKF) Explained

• Delayed Kalman Filter (DKF) is a formulation that addresses estimation problems where measurements depend on both current and prior states.
• It improves efficiency by avoiding state augmentation and employing a generalized gain that directly accounts for delayed-state correlations.
• DKF achieves optimality equivalent to stochastic cloning while reducing arithmetic and memory costs under many practical conditions.

The delayed Kalman filter (DKF) denotes a class of Kalman-filter formulations designed for estimation problems in which measurements depend on prior states rather than only on the current state. In the delayed-state setting, the central issue is that measurements such as odometry, $\Delta$-range GNSS, and relative visual-inertial observations depend on both $x_j$ and $x_k$ with $j<k$, so the usual hidden-Markov assumption is violated because $x_j$ and $x_k$ are correlated through process noise. A formal derivation shows that a properly derived delayed-state Kalman filter yields exactly the same state and covariance update as stochastic cloning (SC), but without explicit state augmentation; in a separate line of work, the acronym DKF is also used for an anti-delay distributed Kalman filter fusion algorithm for vehicle-borne sensor networks with time-varying transmission delays (Mina et al., 28 Aug 2025, Yu et al., 2022).

1. Delayed-state estimation problem

Standard Kalman filtering assumes that at time $k$ the measurement has the form

$y_k = H_k x_k + v_k,$

and depends only on the current state $x_k$. In many navigation problems, however, the measurement depends on a prior state $x_j$, $x_j$0. A generic linear delayed-state model is

$x_j$1

Because $x_j$2 and $x_j$3 are correlated through process noise, the usual conditional-independence assumption

$x_j$4

is violated. The required correction is to account for the correlation between the measurement and the prior state error (Mina et al., 28 Aug 2025).

This formulation is especially relevant when the sensor reports a relative change between states over time. The data identify odometry as a prominent example, and also list $x_j$5-range GNSS and relative visual-inertial measurements as delayed-state cases. A plausible implication is that the DKF is best understood not as an ad hoc workaround for latency, but as a generalized linear-Gaussian estimator for measurements with explicit temporal coupling.

2. Generalized delayed-state Kalman-filter formulation

The propagation step is identical to the standard Kalman filter. With $x_j$6 and $x_j$7 denoting the predicted state and covariance at time $x_j$8, given data through time $x_j$9,

$x_k$0

$x_k$1

where

$x_k$2

The delayed-state derivation introduces the composite matrices

$x_k$3

$x_k$4

$x_k$5

$x_k$6

The innovation is

$x_k$7

Since the innovation depends on process noise $x_k$8 as well as on $x_k$9, its covariance becomes

$j<k$0

The generalized Kalman gain that accounts for the cross-covariance $j<k$1 is

$j<k$2

The corresponding update is

$j<k$3

and

$j<k$4

The significance of this formulation is precise: the filter remains within Kalman-filter theory, but the innovation statistics and gain are modified so that delayed-state correlations are handled directly rather than absorbed through state augmentation (Mina et al., 28 Aug 2025).

3. Equivalence to stochastic cloning

Stochastic cloning augments the state vector by cloning $j<k$5 alongside $j<k$6, runs a standard KF in $j<k$7 dimensions, and then discards the clone. Mina et al. prove by induction that the bottom-block of the SC gain and covariance update match exactly the DKF gain $j<k$8 and posterior covariance $j<k$9.

The proof uses the SC-augmented Kalman gain together with the backward-propagation identity

$x_j$0

After algebraic rearrangement, the bottom-$x_j$1 rows give

$x_j$2

which is identical to the generalized DKF gain. A parallel block-algebra argument shows that the SC covariance update reduces to the DKF covariance update as well (Mina et al., 28 Aug 2025).

This result addresses a specific misconception identified in the source material: Kalman-filter variants are often taken to be inherently unable to handle correlated delayed-state measurements. The delayed-state formulation shows that this is not a limitation of Kalman-filter theory itself, but of the standard hidden-Markov specialization. In the linear-Gaussian case, DKF and SC yield the same optimal posterior. This suggests that the principal distinction between the two approaches is computational organization rather than estimation quality.

4. Computational and memory characteristics

The reported arithmetic and memory costs distinguish the delayed-state DKF from SC primarily through the avoidance of a $x_j$3 augmented covariance. The paper attributes the efficiency gain to never building or inverting that augmented covariance and instead exploiting the sparsity of $x_j$4, $x_j$5, and $x_j$6 (Mina et al., 28 Aug 2025).

Quantity	SC	DKF
Arithmetic cost	$x_j$7 flops	$x_j$8 flops
Memory	$x_j$9 floats	$x_k$0 floats

In the dominant $x_k$1 term, DKF is roughly $x_k$2 cheaper. For large $x_k$3 and moderate $x_k$4, DKF also saves memory. The source also notes a qualification: for very high-dimensional measurements $x_k$5, SC’s memory sometimes is smaller. The comparison therefore does not reduce to a universal ordering; it is parameter-regime dependent, even though the state-augmentation-free formulation is generally emphasized as the more efficient one.

5. Canonical odometry example

For a one-step delay $x_k$6, let

$x_k$7

and suppose a wheel-odometry measurement

$x_k$8

Then

$x_k$9

If $k$0 is identity on position,

$k$1

$k$2

One iteration is summarized by the following sequence:

1. Predict $k$3.
2. Compute $k$4.
3. Form the innovation $k$5.
4. Compute

$k$6

1. Compute

$k$7

2. Update $k$8, and update $k$9 using the generalized covariance expression above.

In this simple case $y_k = H_k x_k + v_k,$0, so the formulas collapse to a “difference measurement” KF (Mina et al., 28 Aug 2025). The example is important because it makes explicit that delayed-state measurements need not be rare or exotic; they arise directly from common relative-motion sensors.

6. Advantages, limitations, and related delayed-network formulations

For delayed-state measurements, the stated advantages of DKF are that it yields the exact same optimal posterior as SC in the linear-Gaussian case, avoids explicit state augmentation, reduces flop-count and memory especially for large $y_k = H_k x_k + v_k,$1, and generalizes immediately to any fixed delay $y_k = H_k x_k + v_k,$2 or multi-state measurement. The stated limitations are that the algebraic complexity of deriving $y_k = H_k x_k + v_k,$3, $y_k = H_k x_k + v_k,$4, and $y_k = H_k x_k + v_k,$5 may intimidate some implementers, and that for very high-dimensional measurements $y_k = H_k x_k + v_k,$6 SC’s memory sometimes is smaller (Mina et al., 28 Aug 2025).

The acronym DKF is also used in a distinct but related context: the “anti-delay distributed Kalman filter (DKF) with finite-time convergence” for vehicle-borne sensor networks with time-varying transmission delays. In that setting, each inter-sensor transmission may suffer a different delay $y_k = H_k x_k + v_k,$7; each packet carries a time-stamp $y_k = H_k x_k + v_k,$8; each node maintains a length-$y_k = H_k x_k + v_k,$9 buffer; and the filter operates in information form with a consensus sub-routine over delayed neighbor data. The update is expressed through

$x_k$0

$x_k$1

where $x_k$2 and $x_k$3 collect delayed neighbor information after $x_k$4 consensus rounds. Under a connected tree or strongly connected digraph of diameter $x_k$5, the information-consensus update converges exactly in $x_k$6 hops, and with the $x_k$7 buffer-indexed stamps the total iterations per time $x_k$8 is at most $x_k$9 (Yu et al., 2022).

That distributed formulation further introduces an optimal global fusion

$x_j$0

with

$x_j$1

and

$x_j$2

Its simulations use $x_j$3 sensors, delays $x_j$4 with $x_j$5, and report that for $x_j$6 the position, velocity, and acceleration RMSE of Algo 1 is consistently lower than the compared delayed distributed KF by about $x_j$7–$x_j$8; in a mobile-car trajectory tracking experiment with three 24\,GHz FMCW radar units, it reduces position error by $x_j$9 relative to the compared method while nearly matching a centralized KF (Yu et al., 2022).

These two usages of DKF should not be conflated. One is a delayed-state single-filter formulation for correlated measurements involving prior states; the other is a distributed anti-delay fusion algorithm for networked sensing under transmission latency. Their common theme is explicit treatment of delay-induced correlation or asynchrony, but their state models, update structure, and intended applications are different.