Open AccessTechnical Note

An Optimized Diffuse Kalman Filter for Frequency and Phase Synchronization in Distributed Radar Networks

School of Electronic and Information Engineering, Beihang University, Beijing 100191, China

Hangzhou Innovation Institute, Beihang University, Hangzhou 310025, China

Author to whom correspondence should be addressed.

Remote Sens. 2025, 17(3), 497; https://doi.org/10.3390/rs17030497

Submission received: 13 December 2024 / Revised: 17 January 2025 / Accepted: 28 January 2025 / Published: 31 January 2025

(This article belongs to the Special Issue Array and Signal Processing for Radar)

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Abstract

Distributed radar networks have emerged as a key technology in remote sensing and surveillance due to their high transmission power and robustness against node failures. When performing coherent beamforming with multiple radars, frequency and phase deviations introduced by independent oscillators lead to a decrease in transmission power. This paper proposes an optimized diffuse Kalman filter (ODKF) for the frequency and phase synchronization. Specifically, each radar locally estimates its frequency and phase, then shares this information with neighboring nodes, which are used for incremental update and diffusion update to adjust local estimates. To further reduce synchronization errors, we incorporate a self-feedback strategy in the diffusion step, in which each node balances its own estimate with neighbor information by optimizing the diagonal weights in the diffusion matrix. Numerical simulations demonstrate the superior performance of the proposed method in terms of mean squared deviation (MSD) and convergence speed.

Keywords:

distributed radar networks; diffuse Kalman filter; frequency and phase synchronization; optimization

1. Introduction

Unmanned aerial vehicles (UAVs) equipped with radar systems have gained significant interest due to their low cost and mobility in surveillance and remote sensing applications [1,2]. By coordinating at the wavelength level, distributed UAV-borne radars can transmit high-power signals, with power proportional to the square of the number of antennas [3]. Compared to radar deployed on a single platform, this coherent distributed architecture offers higher angular resolution, robustness to system failure and flexibility in spatial positioning [4]. However, achieving wavelength-level wireless synchronization in practical scenarios is challenging, due to radio frequency (RF) signals being generated by independent local oscillators. The drift of these oscillators together with environmental disturbances introduces frequency and phase deviations, which degrade system coherence and result in reduced transmitted gain [5].

The fundamental idea for synchronization in distributed radar networks is selecting a reference signal to calibrate the other nodes. This reference signal can be generated from within the network, such as a particular node designated as the master clock [6], or obtained from an external source, such as a global positioning system (GPS) signal [7]. After receiving the reference signal, each node locally calculates the relative deviations in clock and carrier frequency and then aligns its local phase-locked loops. However, this centralizedstrategy is sensitive to the stability of the reference signal. The drift or instability of the reference clock can propagate to the slave clocks through the network, which degrades the synchronization accuracy [8]. Moreover, external GPS signals are susceptible to jamming and spoofing attacks, which causes it to be unable to provide a reliable reference signal [9].

In contrast, a decentralized strategy demonstrates robustness against node failures [10], in which each node exchanges synchronous signals and applies the same consensus algorithm to iteratively reach global phase and frequency alignment across the network [11,12]. Prager et al. [13] proposed a distributed two-way time transfer (TWTT) algorithm to estimate relative clock and RF carrier phase offsets, followed by compensation and transmission synchronization. Experimental results in [13] showed a time synchronization accuracy of 50 ps and a phase synchronization accuracy of

λ / 28

. The author in [14] demonstrates a software-defined frequency synchronization technique using digital radar transceivers, achieving a standard deviation of drift estimation below 10 parts per billion (ppb) and an average carrier frequency offset of less than 30 Hz at 3.07 GHz. Kenney et al. [15] extended this to analyze the clock drift estimation performance under platform motion, as well as with single-tone and two-tone synchronous signals.

To further improve frequency and phase state estimation accuracy and reduce residual phase errors, Rashid et al. in [16,17] proposed decentralized frequency and phase consensus (DFPC) and integrated the Kalman filter into synchronization period, in which each node shares the observations and covariance with their neighbors to update synchronization parameters. These algorithms depend on a Metropolis matrix [18] for weighting both estimates and error covariances. However, they do not fully exploit the underlying properties of radar network topology, such as sparsity or cluster structure. Moreover, UAV formation flying suffers from jittering effects due to random airflow and body vibrations. These motion-induced jitters cause more severe synchronization mismatches at each time interval, and degrade the coherent beamforming gain [19]. This requires the synchronization method to achieve the desired precision within a shorter synchronization period.

In this paper, an optimized diffuse Kalman filter (ODKF) for frequency and phase synchronization in distributed radar networks is proposed, in which each node estimates and shares its local frequency and phase, followed by incremental update and diffusion update. Further based on the network topology, we integrate a self-feedback strategy into the fastest convergence to adjust the network weights. Simulations show that the proposed method significantly reduces the mean square deviation (MSD) across the network and accelerates the synchronization process.

2. Problem Formulation

The transmission coherence in distributed radar networks requires precise frequency and phase synchronization. Figure 1 illustrates a schematic example of the wireless synchronization process among N distributed radars, with

N = 5

for clarity. In Figure 1a, the dashed arrows represent the synchronization links, where the radars exchange synchronized signals to calibrate their frequencies and phases. The solid lines indicate each radar’s transmitted waveforms propagating toward a far field point target. After compensating for frequency and phase deviations, as depicted in Figure 1b, the transmitted waveforms become coherently aligned at the target.

The inconsistency of the initial phase and frequency of each transmitted signal will degrade system coherence. To evaluate how synchronization performance affects the system coherence, we begin by analyzing the total accumulated energy emitted by all nodes at angle

θ

. Assuming radar nodes are perfectly synchronized, the accumulated signal is

s_{i} (t) = C \sum_{n = 1}^{N} ω_{n} \cdot e^{j 2 π f_{D, n} t} \cdot e^{j 2 π (f_{c} \cdot t - d_{n} sin (θ) / λ)},

(1)

where C is a constant representing the amplitude of the signal, and

f_{c}

denotes the carrier frequency.

d_{n}

is the distance of the n-node from the coordinate origin, and

λ

represents the wavelength. For the moving platform, the radial velocity

v_{n}

introduces a Doppler frequency shift given by

f_{D, n} = 2 v_{n} / λ

. To coherently steer the beam towards a target at

θ

, each node n is applied with a phase shift

ω_{n} = exp (j 2 π d_{n} sin (θ) / λ)

Due to independent oscillators and unpredictable environmental factors, there are drifts in the clock, frequency, and phase of the signals transmitted by the UAV-borne radars. Here, we assume the time deviations have been sufficiently compensated or remain negligible compared to wavelength-level mismatch [13,20]. Consequently, the primary sources of coherence degradation are frequency and phase deviations. The actual accumulated signal at target is

\begin{matrix} s (t) = C \sum_{n = 1}^{N} & ω_{n} \cdot e^{j 2 π f_{D, n} t} \cdot e^{j 2 π (f_{c} + δ f_{n}) t} \cdot e^{j ϕ_{n}} \cdot e^{- 2 π (d_{n} + δ d_{n}) sin (θ + δ θ_{n}) / λ}, \end{matrix}

(2)

where

δ f_{n}

is the frequency drift, primarily caused by temperature changes or radio frequency device aging. Assuming that

δ f_{n}

is relatively small,

|δ f_{n}| ≪ f_{c}

, so that it does not significantly affect the wavelength associated with the Doppler frequency. The constant

ϕ_{n}

represents the phase deviation, caused by random initial oscillator phases and variations in signal travel time across the circuit board.

δ d_{n}

indicates positional error due to inaccuracies in GPS or inertial navigation calculations, and

δ θ_{n}

is the angular pointing error. We set

C = 1

, and the (2) can be further simplified into

s (t) = \sum_{n = 1}^{N} e^{j (2 π (f_{c} + δ f_{n} + f_{D, n}) t + φ_{n})},

(3)

where

φ_{n}

is the total phase deviation of the n-th node, which includes random initial oscillator phase errors, position errors, and beam steering errors. In the subsequent analysis, we model

φ_{n}

as a random variable.

φ_{n} = ϕ_{n} - 2 π ((d_{n} + δ d_{n}) sin (θ + δ θ_{n}) - d_{n} sin (θ)) / λ .

(4)

The total energy at

θ

is time-varying due to factors such as frequency jitter, time-varying Doppler effects, and coupled phase deviations, resulting in random characteristics of the instantaneous beamforming gain. The instantaneous gain at a specific moment fails to describe the statistical properties [21]. We employ the expected beamforming gain to smooth out the uncertainties caused by time-varying deviations and to quantify the statistical characteristics of this inconsistency.

\begin{matrix} G (t) & = E [{|s (t)|}^{2}] = E [\sum_{n = 1}^{N} \sum_{m = 1}^{N} e^{j 2 π ((δ f_{n} - δ f_{m}) + (f_{D, n} - f_{D, m})) t} \cdot e^{j (φ_{n} - φ_{m})}] . \end{matrix}

(5)

The beamforming gain

G (t)

depends on the statistical properties of the frequency

δ f_{n}

δ f_{m}

f_{D, n}

f_{D, m}

and phase deviations

φ_{n}

φ_{m}

. We assume that the frequency and phase deviations are independent and Gaussian distributed, with

δ f_{n}

∼

N (0, σ_{f}^{2})

and

φ_{n}

∼

N (0, σ_{φ}^{2})

. In practice, UAV platform jitter induces random motion fluctuations at each node, which leads to approximately Gaussian distributed Doppler frequency among nodes,

f_{D, n}

∼

N (0, σ_{f_{D}}^{2})

[22]. In order to further obtain the specific form of

G (t)

in (5), we separate the double sum into two categories: case (i)

n = m

and case (ii)

n \neq m

. For case (i), the gain is a constant

G (t) = \sum_{n = 1}^{N} 1 = N

. For case (ii) under the independent Gaussian distribution assumption, the expectation of the product can be factorized into a product of expectations.

E [e^{j 2 π ((δ f_{n} - δ f_{m}) + (f_{D, n} - f_{D, m})) t} \cdot e^{j (φ_{n} - φ_{m})}] = [E^{j 2 π δ f_{n} t}] \cdot [E^{- j 2 π δ f_{m} t}] \cdot \cdot \cdot [E^{- j φ_{m}}] .

(6)

According to the moment generating function for a random variable X∼

N (0, σ^{2})

[23], the expected value holds

E (e^{j X}) = \int_{- \infty}^{\infty} \frac{1}{σ \sqrt{2 π}} e^{- \frac{x^{2}}{2 σ^{2}}} e^{j x} d x = e^{- \frac{σ^{2}}{2}} .

(7)

Collecting all such terms for the difference

δ f_{n}

δ f_{m}

f_{D, n}

f_{D, m}

φ_{n}

and

φ_{m}

, we obtain an overall factor

e^{- 4 π^{2} t^{2} (σ_{f_{D}}^{2} + σ_{f}^{2}) - σ_{φ}^{2}}

. Since there are

N (N - 1)

terms in case (ii), summing them all, the expected value of statistical beamforming gain

G (t)

in (5) can be expressed as

G (t) = N + N (N - 1) e^{- 4 π^{2} t^{2} (σ_{f_{D}}^{2} + σ_{f}^{2}) - σ_{φ}^{2}} .

(8)

Equation (8) indicates that the system achieves its highest gain at the initial time

t = 0

, where the coherence among nodes is affected only by the phase deviation

σ_{φ}^{2}

. As time t increases, the normalized expected beamforming gain gradually decreases and eventually stabilizes at a completely uncorrelated state

{lim}_{t \to \infty} G (t) / N^{2} = 1 / N

. The rate of decline is determined by frequency variances

σ_{f_{D}}^{2}

and

σ_{f}^{2}

. Figure 2 simulates the radiation patterns of 5 antennas randomly arranged in the x-y plane, in which the angle

θ = 0

is designated as the statistical beamforming gain for evaluating the synchronization performance of the distributed radar networks. From Figure 2a,b), the normalized radiated energy is random and defocused as the total phase errors among multiple radars increase. More comparisons and analyses will be presented in the simulation section. It is necessary to synchronize frequency and phase with low variance to meet the requirements of high-power coherent beamforming.

3. Frequency and Phase Synchronization Method Based on Diffuse Kalman Filter

In distributed radar networks, the diffusion-based Kalman filter is a decentralized approach, which can achieve self-optimized estimation and prediction through information exchange among neighboring nodes in the network [24]. In this section, a DKF-based frequency and phase synchronization method for distributed radar networks is developed, and the diffusion matrix defined by the network topology is optimized to enhance synchronization speed and accuracy.

3.1. DKF-Based Frequency and Phase Synchronization

To mathematically model this distributed synchronization system, we represent it as an undirected graph

G = (N, E)

, where

N = {1, 2, \dots, N}

denotes the set of radar nodes, and

E \subseteq N \times N

represents the set of edges corresponding to wireless synchronization links. The undirected graph

G

is assumed to be connected to ensure the propagation of synchronization information among all radar nodes. For the n-th radar node at time k, the state vector to be synchronized includes the actual frequency and phase terms, defined as

x_{n} (k) = {[η_{n} (k), φ_{n} (k)]}^{T}

, where

η_{n} (k) = f_{c} + δ f_{n} (k) + f_{D, n} (k)

. Based on the analysis in Section 2, the state transition model can be expressed as

x_{n} (k) = x_{n} (k - 1) + w_{n} (k) n = 1, 2, \dots, N,

(9)

where

w_{n} (k) \in R^{2}

is a random variable representing the uncertainties introduced by oscillator drift and motion fluctuations, also known as process noise. We assume

w_{n}

∼

N (0, Q)

, and the covariance matrix

Q

w_{n} (k)

Q = [\begin{matrix} σ_{f}^{2} + σ_{f_{D}}^{2} & 2 π Δ t (σ_{f}^{2} + σ_{f_{D}}^{2}) \\ 2 π Δ t (σ_{f}^{2} + σ_{f_{D}}^{2}) & 4 π^{2} Δ t^{2} (σ_{f}^{2} + σ_{f_{D}}^{2}) + σ_{φ}^{2} \end{matrix}],

(10)

where

Δ t

is the time interval. The first diagonal entry

σ_{f}^{2} + σ_{f_{D}}^{2}

Q

quantifies both the local oscillator instability and platform motion-induced Doppler uncertainties. The term

4 π^{2} Δ t^{2} (σ_{f}^{2} + σ_{f_{D}}^{2}) + σ_{φ}^{2}

represents accumulated phase drift over time. The off-diagonal terms capture the coupling between frequency and phase over time

Δ t

The true phase and frequency states of each node need to be estimated, which can be achieved using frequency estimation methods such as fast Fourier transform spectral analysis method, as well as phase estimation methods, including the quadratic least squares [13,15]. We assume that the observation vector of the n-th node is

y_{n} (k) = {[{\hat{η}}_{n} (k), {\hat{φ}}_{n} (k)]}^{T}

, and the observation model can be expressed as

y_{n} (k) = x_{n} (k) + v_{n} n = 1, 2, \dots, N,

(11)

where the estimation errors are independent and modeled as

v_{n}

∼

N (0, R)

, where the covariance matrix

R

is a diagonal matrix with

ε_{η}^{2}

and

ε_{φ}^{2}

on the diagonal.

After estimating and broadcasting the local frequency and phase

y_{n}

, the DKF processing involves two main steps: an incremental update for data sharing and a diffusion update for estimate sharing [25].

Similar to the measurement update in a traditional Kalman filter, the incremental update refines each node’s estimate by incorporating not just its own observations but also the measurement information from its neighbors

N_{n} = {j \in N | (n, j) \in E}

. The incremental update at node n follows the standard Kalman measurement update structure, and combines the observations using a weighted aggregation to refine its own estimate. The covariance and observation are incrementally incorporated using the following processing:

P_{n, k | k} = {({(P_{n, k | k - 1})}^{- 1} + \sum_{j \in N_{n}} R_{j, k}^{- 1})}^{- 1},

(12)

{\hat{x}}_{n, k | k} = {\hat{x}}_{n, k |k - 1} + P_{n, k |k} (\sum_{j \in N_{n}} R_{j, k}^{- 1} (y_{j, k} - {\hat{x}}_{n, k |k - 1})),

(13)

where

{\hat{x}}_{n, k |k - 1}

is the predicted state vector of node n at time k; this is similar to the Kalman filter’s prior estimate, based on data up to time

k - 1

{\hat{x}}_{n, k |k}

is an intermediate estimate at node n after incorporating measurement information.

P_{n, k | k - 1}

is the predicted covariance matrix corresponding to

{\hat{x}}_{n, k |k - 1}

, and

P_{n, k | k}

is the updated covariance after fusing measurement data.

y_{j, k}

and

R_{j, k}

are the observed data and covariance matrix from neighbor j at time k, respectively.

In the incremental update process, the measurements remain distributed across various nodes instead of being centralized as in the traditional Kalman filter. An additional step diffusion update is needed to exchange intermediate estimates among neighbors and achieve global consistency throughout the network. During the diffusion update, each node n computes a new estimate by combining its own updated state estimate

{\hat{x}}_{n, k | k}

with those received from its neighbors

{\hat{x}}_{j, k | k}

. The core in the diffusion update is linear weighting of the estimates between node n and its neighbors,

{\tilde{x}}_{n, k | k} = \sum_{j \in N_{n}} c_{j, n} {\hat{x}}_{n, k | k} s . t . \sum_{j = 1}^{N} c_{j, n} = 1,

(14)

where

c_{j, n}

is the

(j, n)

element of diffusion matrix

C

. Specifically, if nodes j and n can transmit synchronization signals to each other, the corresponding element in the matrix

C

is a non-zero weighted value. Otherwise,

C_{j, n} = 0

, indicating that there is no exchanged information.

Figure 3 illustrates the basic process of the diffusion Kalman filter. In the left figure (incremental update), taking node n as an example, node n first receives the measurement data

y_{l} (k)

and

y_{m} (k)

from node l and node m, then performs steps (12) and (13) to update its local state estimate. Next, in the right figure (diffusion update), node n obtains the incremental updates

{\hat{x}}_{l, k | k}

and

{\hat{x}}_{m, k | k}

from its neighbors l and m, then applies step (14) to fuse these updates and achieve a globally consistent estimate. Algorithm 1 illustrates the implementation process of the frequency and phase synchronization method based on DKF. The initial state

{\hat{x}}_{n, 0 | - 1}

and covariance matrix

P_{n, 0 |- 1}

of each node are set to

0

and

Π_{0}

, where

Π_{0}

can be set relatively large, indicating high uncertainty about the initial state. The state prediction step after the diffusion update is consistent with that of the standard Kalman filter; here, the state transfer matrix is the identity matrix.

Algorithm 1 DKF-based frequency and phase synchronization.

Initialize the state vector ${\hat{x}}_{n, 0 |- 1} = 0$ and covariance matrix $P_{n, 0 |- 1} = Π_{0}$ according to the state model (9) and observation model (11).
for $k = 1 : K$ do
for $n = 1 : N$ do
Incremental Update:
“collecte local covariance and observation information”
$S_{n, k} = \sum_{j \in N_{n}} R_{j, k}^{- 1}$
$q_{n, k} = \sum_{j \in N_{n}} R_{j, k}^{- 1} y_{n} (k)$
   “update covariance and observation”
    $P_{n, k | k} = {({(P_{n, k | k - 1})}^{- 1} + S_{n, k})}^{- 1}$
    ${\hat{x}}_{n, k | k} = {\hat{x}}_{n, k |k - 1} + P_{n, k |k} (q_{n, k} - S_{n, k} {\hat{x}}_{n, k |k - 1})$
   Diffusion Update:
    ${\tilde{x}}_{n, k | k} = \sum_{j \in N_{n}} c_{j, n} {\hat{x}}_{n, k | k}$
    ${\hat{x}}_{n, k + 1 | k} = {\tilde{x}}_{n, k | k}$
    $P_{n, k + 1 | k} = P_{n, k | k} + Q$
end for
end for

Next, we will analyze the properties of the diffusion matrix

C

in detail and design an optimized diffusion strategy to accelerate iteration and improve frequency and phase synchronization accuracy.

3.2. ODKF Optimization for Diffusion Matrix

In a distributed radar network, the number of neighbors

N_{n}

and reliability of each node is different. By adjusting the weights and optimizing the diffusion matrix

C

, the MSD across the network can be reduced, and the convergence speed

μ (C)

can be accelerated. The diffusion matrix

C

satisfies

C 1 = 1, C = C^{T}, c_{j, n} \geq 0, and c_{j, n} = 0 if j \notin N_{n},

(15)

where

1

represents an

N \times 1

column vector of all ones, which means

C

is a row-stochastic matrix. The symmetry means the weight assigned between any two nodes i and j is the same. According to the Perron–Frobenius theorem [26], the matrix

C

has the uniquely largest eigenvalue equal to 1, and all of its eigenvalues are real.

1 = λ_{1} (C) \geq λ_{2} (C) \geq \dots \geq λ_{N} (C) \geq - 1 .

(16)

The convergence speed

μ (C)

is determined by the second-largest eigenvalue in magnitude [27],

μ (C) = max_{i = 2, \dots, N} | λ_{i} (C) | = max \{λ_{2} (C), - λ_{N} (C)\} .

(17)

Another important metric is MSD, which is used for evaluating convergence accuracy. After each node performs a diffusion update, the frequency and phase will gradually converge to a consensus state. This converged value is typically the average of the initial states of all nodes

\bar{x}

, but it is not necessarily zero. The MSD is defined as

{MSD}_{k} = lim_{k \to \infty} \frac{1}{N} \sum_{n = 1}^{N} E {∥x_{n} - \bar{x}∥}^{2} .

(18)

As the number of wireless synchronization links increases, the additional edges provide more paths for information exchange. So the consensus convergence speed is fast and

μ (C)

is relatively small. However, adding more edges does not enhance radar cooperation, and it comes at the cost of higher MSD in estimation performance. This phenomenon is similar to the idea that “too much information and communication can make a network of agents less intelligent” [24]. To balance this trade-off in a distributed radar network topology, we introduce a self-feedback weight,

trace (C)

, which ensures that each node prioritizes the weights of its neighboring nodes over its own information during the diffusion update. The objective function can be expressed as

\begin{matrix} \min μ (C) + γ \cdot t r a c e (C) \\ s . t . C 1 = 1, C \geq 0, C = C^{T} \\ c_{i, j} = 0 if (i, j) \notin E, \end{matrix}

(19)

where

γ

is a regularization parameter that controls the convergence rate and synchronization accuracy.

μ (C)

is proven to be a convex function, as both

λ_{2} (C)

and

λ_{N} (C)

are the supremum of a family of quadratic forms

λ_{n} (C) = sup {u^{T} C u |{∥u∥}_{2} \leq 1}

[27], and

μ (C)

can be transformed into the spectral norm restricted to the subspace

1^{⊥}

. The objective function (19) is equivalent to

\begin{matrix} \min {∥C - \frac{1}{N} 1 1^{T}∥}_{2} + γ \cdot t r a c e (C) \\ s . t . C 1 = 1, C \geq 0, C = C^{T} \\ c_{i, j} = 0 if (i, j) \notin E . \end{matrix}

(20)

For the problem (20), the squared Frobenius norm and the trace operator are convex. In the constraints,

C 1 = 1

C = C^{T}

, and

c_{i, j} = 0 if (i, j) \notin E

are linear constraints, and

C \geq 0

is a convex constraint. Therefore, problem (20) is convex and can be directly solved using CVX in MATLAB R2022b. The optimized diffusion matrix

C_{o}

is computed after establishing the wireless synchronization links. By replacing

C_{o}

into the diffusion update step of Algorithm 1, fast and accurate frequency and phase synchronization can be achieved.

4. Simulations

In this section, we simulate the relationship between the normalized statistical beamforming gain and frequency and phase deviations, which provides evaluation metrics for synchronization performance. Then, we conduct numerical simulations to evaluate the proposed ODKF method in three aspects: the (1) iterative estimation of frequency and phase deviations, (2) convergence accuracy and (3) convergence speed and complexity. All simulations are performed in MATLAB 2022b. Additionally, to further demonstrate the effectiveness of the ODKF algorithm, we compare it with two other algorithms in [16]: Decentralized Frequency and Phase Consensus (DFPC), a consensus-based synchronization method in which each node updates its frequency and phase by averaging those of its neighbors, and Kalman Filter-Based DFPC (KF-DFPC), a variant of DFPC incorporating a local Kalman filter at each node.

(1) Statistical beamforming gain: Figure 4 and Figure 5 evaluate the normalized statistical beamforming gain based on the analysis in Section 2. In Figure 4, as the variance of the phase mismatch among nodes increases, the distributed radars experience a more significant loss in statistical gain, particularly when the number of nodes is large. To meet the coherence gain requirement

G (t) / N^{2} > 0.9

in [10], the standard deviation of phase synchronization among nodes should be less than

27^{\circ}

20 . 9^{\circ}

19 . 8^{\circ}

19 . 1^{\circ}

and

18 . 7^{\circ}

with

N = 2, 5, 10, 20

and 100 nodes, respectively. In Figure 5, as time t increases, the normalized expected beamforming gain gradually decreases and eventually stabilizes at a completely uncorrelated state

{lim}_{t \to \infty} G (t) / N^{2} = 1 / N

. The rate of decline is determined by frequency variances

σ_{f_{D}}^{2}

and

σ_{f}^{2}

. Specifically, for

N = 20

with

σ_{f} = 100

Hz and

σ_{f_{D}} = 50

Hz, the required frequency synchronization period should be less than

0.5

ms. Once this period is exceeded, the beamforming gain will drop rapidly.

To further evaluate the synchronization performance of the proposed algorithm, we designed a scenario illustrated in Figure 6, where 20 UAV-borne radar nodes are randomly deployed in three-dimensional space. Each node in the graph represents a radar, and each edge denotes a wireless synchronization link for information exchange.

(2) Iterative estimation of frequency and phase deviations: Figure 7 and Figure 8 evaluate frequency and phase synchronization deviations over iterations. We compare the proposed method with the Metropolis-based DKF, DFPC, and KF-DFPC methods. The construction of the Metropolis-based diffusion matrix is

c_{i, j} = \{\begin{matrix} \frac{1}{max {d_{i}, d_{j}}} & if i \neq j, (i, j) \in E \\ 1 - \sum_{j \in N_{n}} c_{i, j} & if i = j \\ 0 & if (i, j) \notin E . \end{matrix}

where

d_{i}

is the number of neighbor nodes that are connected to the node i. The carrier frequency is set to

f_{c} = 1

GHz, and the frequency stability of the crystal oscillator is set to 0.1 parts per million (ppm), which is

σ_{f} = 100

Hz, and the standard deviation of the Doppler frequency is set to

σ_{f_{D}} = 2 v / λ = 50

Hz. The initial phase for each node is generated randomly by a uniform distribution with

σ_{φ} = π

. The noise covariance matrix is set to

Q_{0} = d i a g (σ_{f_{D}} + σ_{f}, σ_{φ})

, where

d i a g (\cdot)

denotes a column vector composed of the diagonal elements of a matrix. From Figure 7 and Figure 8, it can be observed that the Metropolis-based DKF method demonstrates low synchronization accuracy, with frequency deviations fluctuating around

10

Hz and phase error within

20

degrees. The DFPC algorithm, which relies on open-loop measurements and repeated consensus steps to reach a common frequency and phase, exhibits even slower convergence. Although KF-DFPC leverages local Kalman updates before consensus and thus outperforms DFPC, it still converges more slowly and with higher converged synchronization errors than ODKF. In contrast, the ODKF method achieves low frequency and phase synchronization deviations, with rapid convergence and minimal synchronization error within about five iterations.

(3) Convergence accuracy: Figure 9 and Figure 10 evaluate the convergence accuracy using MSD as the quantitative metric of average network performance. In both frequency and phase synchronization, we conduct 200 Monte Carlo simulations. The Metropolis-based DKF method performs the worst among the six methods, with high MSD values that remain nearly constant, indicating poor convergence and minimal improvement in synchronization accuracy. DFPC shows a moderate decrease in MSD initially but stabilizes at a relatively large residual error, while KF-DFPC achieves a drop to around

10^{- 12}

Hz and

10^{- 16}

degrees after about 55 and 68 iterations in frequency and phase synchronization, which demonstrates the benefit of local Kalman updates prior to consensus. In the ODKF method, a smaller

γ

means that each node relies more on its own observations rather than sharing information with other nodes. When

γ = 1

, the diagonal elements of the optimized matrix

C_{o}

are nearly zero, and the method achieves superior performance. ODKF with

γ = 1

can achieve

10^{- 12}

Hz and

10^{- 16}

degrees after about 14 and 18 iterations, which meets the coherence requirement of distributed radars.

(4) Convergence speed and complexity: Figure 11 evaluates the convergence speeds of various diffusion matrices, which is mainly determined by the second-largest eigenvalue. Figure 11 shows that the largest eigenvalue of diffusion matrices satisfying property (16) is always 1. The Metropolis strategy exhibits the slowest convergence rate, as its second-largest eigenvalue (in magnitude) is smaller. This conclusion is consistent with the detailed plots in Figure 9 and Figure 10, where the slope for the Metropolis-based DKF method is the lowest. The ODKF with

γ = 0

and

γ = 1

strategy exhibits a symmetric distribution of eigenvalues around zero, with many eigenvalues clustered near the critical values of

\pm 0.46

and

\pm 0.5

. This distribution shows a faster convergence ratio, as the second-largest eigenvalue is relatively small and the distribution is balanced. We evaluated the average time consumption per iteration for

N = 20

, at 0.13 ms, 0.04 ms, and 0.086 ms for the ODKF, DFPC, and KF-DFPC methods, respectively. Within a 0.5 ms synchronization period, ODKF reduces the frequency error to about 6.5 Hz and the phase error to about 25°, which corresponds to a coherent gain of approximately

G (t) / N^{2} > 0.86

. Limited by the current hardware capabilities, a faster CPU could potentially achieve a coherent gain exceeding 0.9. The computational complexity of DKF in each iteration can be viewed in two main parts: (i) local matrix operations, and (ii) neighbor-to-neighbor operations. Let N be the number of nodes, M the measurement dimension, P the state dimension, and

E

the average number of neighbors per node. In each incremental update, every node inverts both the covariance matrix and the measurement matrix, costing

O (M^{3} + P^{3})

. Additionally, each node performs neighbor-based linear combinations, costing

O (E \cdot P^{2})

. The total complexity across the network is

O (N \cdot (M^{3} + P^{3} + E \cdot P^{2}))

. The CVX-based matrix optimization step involves

N^{2}

variables, and can rely on interior-point solvers with generic complexity on the order of

O (N^{4})

O (N^{6})

. For a fixed network topology, computing (20) at every iteration is unnecessary. Optimization is performed once after establishing the network connections.

5. Conclusions

In this paper, an ODKF-based algorithm for frequency and phase synchronization is proposed. By integrating the Kalman filter and an optimized diffusion matrix, each node in the network exchanges observations and covariance, and iteratively synchronizes the frequency and phase deviations caused by independent oscillators. Simulations compared it with the KF-DFPC method and Metropolis-based DKF in terms of iterative estimation of frequency and phase deviations, and convergence accuracy and speed. The results demonstrate that the proposed method balances the trade-off between convergance ratio and MSD, which meet the requirement of coherence gain in distributed UAV-borne radars.

Author Contributions

X.G.: Conceptualization, Methodology, Software, Formal Analysis, Writing—Original Draft. J.W.: Conceptualization, Project Administration, Resources. B.Y.: Conceptualization, Formal Analysis, Validation, Writing—Review and Editing, Resources. J.S.: Conceptualization, Resources, Writing—Review and Editing. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China, Grant No. 62131001 and 62171029, in part by the “Ling Yan” plan Zhejiang Province Science and Technology Department (2023C01148), in part by the Aeronautical Science Foundation of China (2024M017051001).

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. (a) A schematic diagram of wireless synchronization for distributed UAV-borne radars, with

N = 5

as an example. Each colored line represents the transmitted waveform propagating toward a far field point target. (b) The transmitted signals are coherently superimposed at the point target.

Figure 1. (a) A schematic diagram of wireless synchronization for distributed UAV-borne radars, with

N = 5

as an example. Each colored line represents the transmitted waveform propagating toward a far field point target. (b) The transmitted signals are coherently superimposed at the point target.

Figure 2. Radiated normalized energy of 5 radars randomly arranged in the x-y plane, with the statistical beamforming gain at

θ = 0

(indicated by a red cross symbol), is designated as the evaluation metric for synchronization performance. (a) Pattern of synchronized radars. (b) Pattern of unsynchronized radars with

σ_{φ} = 20^{\circ}

phase errors.

Figure 2. Radiated normalized energy of 5 radars randomly arranged in the x-y plane, with the statistical beamforming gain at

θ = 0

(indicated by a red cross symbol), is designated as the evaluation metric for synchronization performance. (a) Pattern of synchronized radars. (b) Pattern of unsynchronized radars with

σ_{φ} = 20^{\circ}

phase errors.

Figure 3. Diagram of the diffusion Kalman filter.

Figure 4. Normalized statistical beamforming gain with phase variances

σ_{ϕ}

Figure 4. Normalized statistical beamforming gain with phase variances

σ_{ϕ}

Figure 5. Normalized statistical beamforming gain with frequency variances

σ_{f_{D}}, σ_{f}

and time t.

Figure 5. Normalized statistical beamforming gain with frequency variances

σ_{f_{D}}, σ_{f}

and time t.

Figure 6. The network topology consists of

N = 20

distributed radar nodes labeled 1–20, where the dotted lines represent the wireless synchronization links.

Figure 6. The network topology consists of

N = 20

distributed radar nodes labeled 1–20, where the dotted lines represent the wireless synchronization links.

Figure 7. Frequency synchronization deviations (in Hz) over 15 iterations for DFPC, KF-DFPC, Metropolis-based DKF, and ODKF methods, under conditions

σ_{f} = 100

Hz and

σ_{f_{D}} = 50

Hz. Each colored line represents the frequency deviation of a different node.

Figure 7. Frequency synchronization deviations (in Hz) over 15 iterations for DFPC, KF-DFPC, Metropolis-based DKF, and ODKF methods, under conditions

σ_{f} = 100

Hz and

σ_{f_{D}} = 50

Hz. Each colored line represents the frequency deviation of a different node.

Figure 8. Phase synchronization deviations (in degree) over 15 iterations for DFPC, KF-DFPC, Metropolis-based DKF, and ODKF methods, under condition

σ_{φ} = π

. Each colored line represents the phase deviation of a different node.

Figure 8. Phase synchronization deviations (in degree) over 15 iterations for DFPC, KF-DFPC, Metropolis-based DKF, and ODKF methods, under condition

σ_{φ} = π

. Each colored line represents the phase deviation of a different node.

Figure 9. Comparison of normalized MSD for frequency synchronization using DKF, DFPC, KF-DFPC, and FA-DKF with different

γ

value.

Figure 9. Comparison of normalized MSD for frequency synchronization using DKF, DFPC, KF-DFPC, and FA-DKF with different

γ

value.

Figure 10. Comparison of normalized MSD for phase synchronization using DKF, DFPC, KF-DFPC, and FA-DKF with different

γ

value.

Figure 10. Comparison of normalized MSD for phase synchronization using DKF, DFPC, KF-DFPC, and FA-DKF with different

γ

value.

Figure 11. Eigenvalue distributions of the diffusion matrix

C

. The red dotted line indicates the second largest eigenvalue.

Figure 11. Eigenvalue distributions of the diffusion matrix

C

. The red dotted line indicates the second largest eigenvalue.

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Geng, X.; Wang, J.; Yang, B.; Sun, J. An Optimized Diffuse Kalman Filter for Frequency and Phase Synchronization in Distributed Radar Networks. Remote Sens. 2025, 17, 497. https://doi.org/10.3390/rs17030497

AMA Style

Geng X, Wang J, Yang B, Sun J. An Optimized Diffuse Kalman Filter for Frequency and Phase Synchronization in Distributed Radar Networks. Remote Sensing. 2025; 17(3):497. https://doi.org/10.3390/rs17030497

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Geng, Xueyin, Jun Wang, Bin Yang, and Jinping Sun. 2025. "An Optimized Diffuse Kalman Filter for Frequency and Phase Synchronization in Distributed Radar Networks" Remote Sensing 17, no. 3: 497. https://doi.org/10.3390/rs17030497

APA Style

Geng, X., Wang, J., Yang, B., & Sun, J. (2025). An Optimized Diffuse Kalman Filter for Frequency and Phase Synchronization in Distributed Radar Networks. Remote Sensing, 17(3), 497. https://doi.org/10.3390/rs17030497

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An Optimized Diffuse Kalman Filter for Frequency and Phase Synchronization in Distributed Radar Networks

Abstract

1. Introduction

2. Problem Formulation

3. Frequency and Phase Synchronization Method Based on Diffuse Kalman Filter

3.1. DKF-Based Frequency and Phase Synchronization

3.2. ODKF Optimization for Diffusion Matrix

4. Simulations

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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