---
title: Delayed Kalman Filter (DKF) Explained
url: https://www.emergentmind.com/topics/delayed-kalman-filter-dkf
type: topic
---

# Delayed Kalman Filter (DKF) Explained

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 [2508.21260; 2209.12666].

## 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 $x_j$, $j<k$. A generic linear delayed-state model is
$$
y_k = [H_{j,k}\;\;H_{k,k}]
\begin{bmatrix}
x_j^T & x_k^T
\end{bmatrix}^T + v_k
\equiv H_{j,k}x_j + H_{k,k}x_k + v_k,\qquad v_k\sim\mathcal N(0,R_k).
$$
Because $x_j$ and $x_k$ are correlated through process noise, the usual conditional-independence assumption
$$
P(y_k|x_0\ldots x_j\ldots x_k)=P(y_k|x_k)
$$
is violated. The required correction is to account for the correlation between the measurement and the prior state error [2508.21260].

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 $\Delta$-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 $\hat x_{k|k-1}$ and $P_{k|k-1}$ denoting the predicted state and covariance at time $k$, given data through time $j<k$,
$$
\hat x_{k|k-1} = \Phi_{k,j}\hat x_{j|j} + B_k u_k,
$$
$$
P_{k|k-1} = \Phi_{k,j}P_{j|j}\Phi_{k,j}^T + S_k,
$$
where
$$
S_k \triangleq G_k Q_k G_k^T.
$$

The delayed-state derivation introduces the composite matrices
$$
J_k \triangleq H_{j,k}\Phi_{j,k},
$$
$$
H_k \triangleq H_{j,k}\Phi_{j,k} + H_{k,k},
$$
$$
N_k \triangleq J_k S_k,
$$
$$
\bar R_k \triangleq J_k S_k J_k^T + R_k.
$$
The innovation is
$$
r_k \triangleq y_k - H_k\hat x_{k|k-1}.
$$
Since the innovation depends on process noise $w_k$ as well as on $\hat x_{k|k-1}$, its covariance becomes
$$
S_r = H_kP_{k|k-1}H_k^T - N_kH_k^T - H_kN_k^T + \bar R_k.
$$

The generalized Kalman gain that accounts for the cross-covariance $N_k$ is
$$
K_k = (P_{k|k-1}H_k^T - N_k^T)
\bigl[H_kP_{k|k-1}H_k^T - N_kH_k^T - H_kN_k^T + \bar R_k\bigr]^{-1}.
$$
The corresponding update is
$$
\hat x_{k|k} = \hat x_{k|k-1} + K_k r_k,
$$
and
$$
P_{k|k} = (I-K_kH_k)P_{k|k-1}(I-K_kH_k)^T
+ (I-K_kH_k)N_k^T K_k^T
+ K_kN_k(I-K_kH_k)^T
+ K_k\bar R_kK_k^T.
$$

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 [2508.21260].

## 3. Equivalence to stochastic cloning

Stochastic cloning augments the state vector by cloning $x_j$ alongside $x_k$, runs a standard KF in $2n$ 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 $K_k$ and posterior covariance $P_{k|k}$.

The proof uses the SC-augmented Kalman gain together with the backward-propagation identity
$$
P_j = \Phi_{j,k}P_k\Phi_{j,k}^T - \Phi_{j,k}S_k\Phi_{j,k}^T.
$$
After algebraic rearrangement, the bottom-$n$ rows give
$$
K_k = (P H_k^T - N_k^T)\cdot[\ldots]^{-1},
$$
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 [2508.21260].

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 $2n\times 2n$ augmented covariance. The paper attributes the efficiency gain to never building or inverting that augmented covariance and instead exploiting the sparsity of $J_k$, $H_k$, and $N_k$ [2508.21260].

| Quantity | SC | DKF |
|---|---:|---:|
| Arithmetic cost | $O(32\,n^3 + 0.67\,m^3 + 24\,m\,n^2 + 12\,m^2\,n)$ flops | $O(6.67\,n^3 + 0.67\,m^3 + 12\,m\,n^2 + 10\,m^2\,n)$ flops |
| Memory | $O(80\,n^2 + 20\,m^2 + 40\,m\,n)$ floats | $O(60\,n^2 + 36\,m^2 + 36\,m\,n)$ floats |

In the dominant $n^3$ term, DKF is roughly $4\times$ cheaper. For large $n$ and moderate $m$, DKF also saves memory. The source also notes a qualification: for very high-dimensional measurements $(m\gg n)$, 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 $\tau=1$, let
$$
x_k = [p_k, v_k]^T,
$$
and suppose a wheel-odometry measurement
$$
y_k = p_k - p_{k-1} + v_k.
$$
Then
$$
H_{j,k}=[-I\;\;0],\qquad H_{k,k}=[I\;\;0].
$$
If $\Phi$ is identity on position,
$$
J_k = H_{j,k}\Phi_{j,k} = -I,\qquad
H_k = H_{j,k}\Phi + H_{k,k} = 0,
$$
$$
N_k = J_kS_k = -S_k,\qquad
\bar R_k = J_kS_kJ_k^T + R_k = S_k + R_k.
$$

One iteration is summarized by the following sequence:

1. Predict $\hat x_{k|k-1}, P_{k|k-1}$.
2. Compute $J_k, H_k, N_k, \bar R_k$.
3. Form the innovation $r_k = y_k - H_k\hat x_{k|k-1}$.
4. Compute
   $$
   S_r = H_kPH_k^T - N_kH_k^T - H_kN_k^T + \bar R_k.
   $$
5. Compute
   $$
   K_k = (PH_k^T - N_k^T)S_r^{-1}.
   $$
6. Update $\hat x_{k|k}=\hat x_{k|k-1}+K_kr_k$, and update $P_{k|k}$ using the generalized covariance expression above.

In this simple case $H_k=0$, so the formulas collapse to a “difference measurement” KF [2508.21260]. 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 $n$, and generalizes immediately to any fixed delay $\tau$ or multi-state measurement. The stated limitations are that the algebraic complexity of deriving $J_k$, $H_k$, and $N_k$ may intimidate some implementers, and that for very high-dimensional measurements $(m\gg n)$ SC’s memory sometimes is smaller [2508.21260].

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 $d^t_{ij}(k)\in\{0,1,\dots,d_t\}$; each packet carries a time-stamp $s$; each node maintains a length-$L$ buffer; and the filter operates in information form with a consensus sub-routine over delayed neighbor data. The update is expressed through
$$
[P_k^i(s)]^{-1} = [P_k^i(s|s-1)]^{-1} + \Omega_i(d_g),
$$
$$
x_k^i(s)=P_k^i(s)\Bigl\{[P_k^i(s|s-1)]^{-1}x_k^i(s|s-1)+\Theta_i(d_g)\Bigr\},
$$
where $\Theta_i(d_g)$ and $\Omega_i(d_g)$ collect delayed neighbor information after $d_g$ consensus rounds. Under a connected tree or strongly connected digraph of diameter $d_g$, the information-consensus update converges exactly in $d_g$ hops, and with the $d_t+1$ buffer-indexed stamps the total iterations per time $k$ is at most $(d_t+1)d_g$ [2209.12666].

That distributed formulation further introduces an optimal global fusion
$$
x_k^f=\sum_{i=1}^n \Gamma_i x_k^i(k),
$$
with
$$
\Gamma=\begin{bmatrix}\Gamma_1&\Gamma_2&\cdots&\Gamma_n\end{bmatrix}
=\Xi^{-1}e(e^T\Xi^{-1}e)^{-1},
$$
and
$$
P_k^f=(e^T\Xi^{-1}e)^{-1}.
$$
Its simulations use $n=12$ sensors, delays $d^t_{ij}(k)\in\{0,1,2,3,4\}$ with $d_t=4$, and report that for $d_t=3$ the position, velocity, and acceleration RMSE of Algo 1 is consistently lower than the compared delayed distributed KF by about $20$–$30\%$; in a mobile-car trajectory tracking experiment with three 24\,GHz FMCW radar units, it reduces position error by $\sim 25\%$ relative to the compared method while nearly matching a centralized KF [2209.12666].

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.

Source: https://www.emergentmind.com/topics/delayed-kalman-filter-dkf