Extended Cancelation Algorithm (ECA)

• ECA is a signal processing methodology for passive radar that cancels direct-path and clutter interference to estimate target delay and Doppler parameters.
• It employs a three-step process including IO waveform estimation, joint subspace interference cancellation, and delay–Doppler matched filtering, achieving near–ML performance.
• The algorithm meets the Cramér–Rao bound under high-SNR conditions and requires a significantly higher RC SNR than the interference-to-noise ratio for reliable target localization.

The Extended Cancelation Algorithm (ECA) is a methodology for target localization in passive radar systems, enabling receiver nodes (RNs) to estimate target delay and Doppler parameters by canceling interference from the direct path and stationary clutter. Exploiting a reference channel (RC) that directly samples the illuminator-of-opportunity (IO) signal, ECA achieves near–maximum-likelihood (ML) estimation of target parameters, with analytic guarantees on variance and bias under high signal-to-noise ratio (SNR) conditions. It is applicable to both monostatic and multistatic networks and is central to achieving statistically efficient target localization using passive radar (Viberg et al., 28 Jan 2026).

1. Signal Model and Problem Formulation

ECA operates in a multistatic passive radar network comprising $K$ geographically separated RNs and a single IO. Each RN acquires two digitized complex-baseband signals:

• Reference Channel (RC): $x_k(t) = a_k s_k(t) + n_k(t)$, where $s_k(t)$ is a delayed IO waveform (geometry determines $\tau_k^{\mathrm{RC}}$), $a_k$ is path-dependent amplitude, and $n_k(t) \sim \mathcal{CN}(0,\sigma_n^2)$ is additive white Gaussian noise.
• Surveillance Channel (SC): $$y_k(t) = b_k s_k(t) + S^c_k(t) + d_k s_k(t-\tau_k) e^{j\omega_k t} + e_k(t)$$, with $b_k$ the direct-path IO leakage, $S^c_k(t) = \sum_{l=1}^L c_{k,l} s_k(t - l\Delta T)$ modeling clutter as an $L$-tap FIR filter in $s_k$, $d_k$ the attenuated target echo, and $e_k(t) \sim \mathcal{CN}(0,\sigma_e^2)$.

The sampled data is collected as $N$-vectors, with Toeplitz matrices constructed for convolutional clutter modeling, and steering vectors encoding delay and Doppler shift.

2. ECA Processing Chain

The ECA pipeline comprises three core steps at each RN:

1. IO Waveform Estimation: The IO waveform in the SC model is substituted with the sampled RC, $\hat{\mathbf{s}}_k = \mathbf{x}_k$, assuming ${\rm SNR}_{\rm RC} \gg {\rm SNR}_{\rm SC}$. Clutter matrices $\mathbf{S}_k$ become $\mathbf{X}_k$.
2. Joint Subspace Interference Cancelation: The interference basis is constructed as $\mathbf{X}_{I,k} = [\mathbf{x}_k \; \mathbf{X}_k]$. An orthoprojector

$$\hat{\mathbf{\Pi}}_k^\perp = \mathbf{I} - \mathbf{X}_{I,k}\bigl(\mathbf{X}_{I,k}^H \mathbf{X}_{I,k}\bigr)^{-1} \mathbf{X}_{I,k}^H$$

projects $\mathbf{y}_k$ onto the subspace orthogonal to the direct-path and clutter components.

1. Delay–Doppler Matched Filtering: Construct estimated steering vectors $$\hat{\mathbf{a}}_k(\tau, \omega) = \mathbf{x}_k(\tau) \odot \mathbf{v}(\omega)$$ and compute the ECA spectrum:

$$P_k(\tau,\omega) = \frac{\bigl|\hat{\mathbf{a}}_k(\tau, \omega)^H \hat{\mathbf{\Pi}}_k^\perp \mathbf{y}_k\bigr|^2} {\hat{\mathbf{a}}_k(\tau, \omega)^H \hat{\mathbf{\Pi}}_k^\perp \hat{\mathbf{a}}_k(\tau, \omega)}.$$

The maximizer $(\hat{\tau}_k, \hat{\omega}_k)$ over a grid is the local target parameter estimate.

Subsequently, the central node fuses $\{\hat{\tau}_k, \hat{\omega}_k\}_{k=1}^K$ via bistatic-geometry relationships

$$\tau_k = \frac{\|u-r_k\| + \|u-r_{IO}\| - \|r_k - r_{IO}\|}{c}, \quad \omega_k = -\omega_c\,\dot{\tau}_k$$

and solves for target position $u$ and velocity $\dot{u}$ using nonlinear least-squares or global ML (Viberg et al., 28 Jan 2026).

3. Statistical Efficiency and Cramér–Rao Lower Bound

In the high-SNR regime ($\sigma_e^2 \to 0$, $\sigma_n^2 \to 0$), ECA yields asymptotically unbiased and Gaussian parameter estimates. Defining the unknown parameter vector $\boldsymbol{\theta} = [x, y, v_x, v_y]^T$ and letting $\mathbf{D}_k$ denote the $N \times 4$ Jacobian of the true steering vector, the following statistical results hold:

• Cramér–Rao Bound (CRB) with Perfect RC:

$${\rm CRB}_{\boldsymbol{\theta}} = \sigma_e^2\,\left[\frac{1}{2}\sum_{k=1}^K |d_k|^2\,\Re\left\{\mathbf{D}_k^H\,\widetilde{\mathbf{P}}_k\,\mathbf{D}_k\right\}\right]^{-1}$$

where $$\widetilde{\mathbf{P}}_k = \mathbf{\Pi}_k^\perp (\mathbf{I} - \mathbf{P}_k)\mathbf{\Pi}_k^\perp$$ and $\mathbf{P}_k$ projects onto the canceled subspace.

• Excess Variance from RC Noise:

$$\mathbf{Q} = 2\,\sum_{k=1}^K\frac{|d_k|^2}{|a_k|^2}\,\sigma_n^2\,\Re\left\{\mathbf{D}_k^H\,\widetilde{\mathbf{P}}_k\,\mathbf{Z}_k\,\mathbf{J}_k\mathbf{J}_k^T\mathbf{Z}_k^H\,\widetilde{\mathbf{P}}_k\,\mathbf{D}_k\right\}$$

with $\mathbf{Z}_k\mathbf{J}_k$ characterizing the steering estimation and residual interference from noisy RC.

• Asymptotic Covariance:

$$\operatorname{Cov}(\hat{\boldsymbol{\theta}}) \approx {\rm CRB}_{\boldsymbol{\theta}} + ({\rm CRB}_{\boldsymbol{\theta}})\,\mathbf{Q}\,({\rm CRB}_{\boldsymbol{\theta}})$$

When $\sigma_n^2$ is sufficiently small, the excess variance term becomes negligible and the ECA estimator achieves the CRB.

4. SNR Requirements and Sufficient Conditions

A sufficient scalar condition ensures that the excess variance from RC noise is negligible: $$(L+1)\,\frac{|b_k|^2 + |d_k|^2 + \|\mathbf{c}_k\|^2}{\sigma_e^2} \ll \frac{|a_k|^2}{\sigma_n^2} \quad \forall k$$ or, in the notation of interference-to-noise and SNR ratios: $${\rm INR}_{SC} = (L + 1)\frac{|b_k|^2 + |d_k|^2 + \|\mathbf{c}_k\|^2}{\sigma_e^2} \ll {\rm SNR}_{RC} = \frac{|a_k|^2}{\sigma_n^2}$$ The RC SNR must comfortably exceed the SC total interference-to-noise ratio (from direct path, clutter, and steering mismatch) for the ECA to deliver statistically efficient (CRB-achieving) estimates.

5. Simulation Results and Empirical Validation

Monte Carlo simulations corroborate the derived statistical results:

• For $K=1$ bistatic configurations, the measured mean-squared error (MSE) of $(\hat{\tau}, \hat{\omega})$ matches the “perfect-RC” CRB down to a SC SNR threshold (e.g., $-25$ dB), below which the ML search fails or biases arise.
• Varying RC SNR at fixed SC SNR (e.g., $15$ dB) demonstrates excess variance prediction; when RC SNR exceeds $\sim95$ dB, the estimator variance reaches the CRB.
• Frame-based (batch) processing, even with sparse sampling and numerous batches, shows negligible performance loss compared with full integration, whereas naive concatenation of separate batches incurs significant degradation in variance.
• In a 3-node network ($K=3$), empirical RMSE for position and velocity estimation agrees with the matrix covariance formula within the high-SNR regime.
• In dynamic (tracking) scenarios, position and velocity estimates at each scan remain within their respective $95\%$ CRB ellipses.

6. Practical Design Considerations and Limitations

The integration time $T = N\Delta T$ sets the tradeoff between delay (range) resolution, $\Delta\tau \propto 1/B$, and Doppler (velocity) resolution, $\Delta\omega \propto 1/T$. Longer $T$ reduces estimator variance but can introduce range migration if $\omega_k T \gtrsim 1$, mitigated via extended steering models or Keystone formatting.

RC link budget design must satisfy ${\rm SNR}_{RC} \gg {\rm INR}_{SC}$, achievable by employing directive antennas, low-noise amplifiers, and sufficient IO illumination. Clutter model order $L$ should be minimized, including only significant delay bins, to reduce both computational cost and the excess interference penalty.

For feasibility studies, one computes $$\sum_k |d_k|^2\,\Re\left\{\mathbf{D}_k^H \widetilde{\mathbf{P}}_k \mathbf{D}_k\right\}$$ from realistic terrain and target cross-section models, evaluates the CRB at design SNRs, and verifies that $$\frac{|a_k|^2}{\sigma_n^2} \gtrsim 10\,\frac{|b_k|^2 + |d_k|^2 + \|\mathbf{c}_k\|^2}{\sigma_e^2}$$ for a $10\times$ margin.

While ECA scales to multistatic networks via incoherent fusion, its computational burden for grid-based delay-Doppler search remains significant. Frame-based and approximate decoupling strategies somewhat alleviate this with minor losses in resolution. Limitations appear when RC SNR does not substantially exceed SC interference, or when range migration is unaddressed in extended-interval processing.

7. Summary of Capabilities and Constraints

ECA constitutes a near-ML approach for passive radar delay/Doppler estimation, based on RC-driven subtraction of direct path and clutter. When RC SNR is adequately superior, the algorithm achieves the CRB for target localization and velocity variance. It is well-suited for multistatic extension, though at the cost of increased search complexity and sensitivity to RC SNR. Proper balancing of RC front-end quality, integration time, clutter model complexity, and grid resolution is essential to meet performance guarantees under the analytically derived CRB and SNR conditions (Viberg et al., 28 Jan 2026).