Introduction

Passive bistatic radar (PBR) has gained significant attention in recent years due to its low cost, resilience to jamming, and ability to detect stealth targets1,2,3,4,5. Unlike conventional active radar, PBR exploits illuminators of opportunity (IOs)–such as digital video broadcast-terrestrial6, frequency-modulated (FM) radio7,8, cellular base stations9,10,11, and satellite signals12,13–as non-cooperative transmitters. A typical receiver comprises two channels: a reference channel that captures the direct-path signal from the illuminator, and a surveillance channel that receives echoes from potential targets. Target detection is generally achieved by clutter suppression across the two channels, followed by coherent processing techniques such as cross-correlation or matched filtering.

In ideal conditions, the reference channel is assumed to contain only the clean, undistorted direct-path signal. In practice, however, this assumption is often violated. Due to wide antenna beamwidths, complex clutter environments, and target dynamics, the reference signal may include unwanted components such as multipath clutter, interference, and even target echoes. Such contamination can severely degrade clutter suppression, reduce signal-to-noise ratio (SNR) in matched filtering, and introduce spurious peaks that interfere with subsequent detection and tracking.

Existing studies on reference-signal contamination can be broadly divided into three categories14,15,16,17,18. The first category addresses contamination caused by multipath clutter, which constitutes the majority of existing methods. These approaches aim to purify the contaminated reference signal by either reconstructing the clean signal or suppressing unwanted clutter components. For example, the authors investigated multipath-induced contamination in airborne PBR and proposed a clutter suppression algorithm based on sparse Bayesian learning, which demonstrated effective performance under reference contamination14. Notably, this method assumes a known clutter subspace, limiting its applicability when clutter statistics are partially unknown, thus motivating further research into adaptive or blind modeling. The second category focuses on contamination from target echoes. Although the presence of target-induced components in the reference signal typically does not severely degrade clutter suppression or matched filtering, it can produce ghost peaks in the detection output or introduce biases in direction-of-arrival (DOA) estimation, potentially impacting subsequent tracking stages. For instance, the authors reconstructed the embedded target echoes and subtracted them from the contaminated reference, effectively mitigating ghost peaks15. A robust DOA correction method was developed for FM radio-based passive radar to address target signal leakage in the reference channel16. The third category does not distinguish between contamination types but instead focuses on designing robust detectors. These methods apply advanced detection theory to improve performance under low SNR conditions in the reference channel. However, they may fail in heavily cluttered environments, where signal contamination is severe. For example, the authors formulated the detection task as a composite hypothesis test under noisy-reference conditions and proposed likelihood ratio test (LRT)-based and kernelized detectors17, including a threshold-adjustment strategy to mitigate false alarms at low direct-path SNR. Similarly, several generalized LRT variants were developed for detection with noisy reference signals, demonstrating significant performance improvements over conventional cross-correlation detectors in low-SNR conditions18.

According to this classification, the present work belongs to the second category, focusing on reference contamination by target echoes. While prior studies have examined its impact on DOA estimation, the effect on target detection has received limited attention. This paper fills this gap by providing both a theoretical analysis and a practical suppression strategy. Specifically, the mechanism behind the emergence of ghost peaks in the detection plane is theoretically derived, and a novel method for suppressing ghost peaks, based on multi-frame matching, is proposed to mitigate this phenomenon. The main contributions of this paper are summarized as follows:

  1. a)

    From a matched-filter perspective, we derive and explain the mechanism by which target echoes embedded in the reference channel generate ghost peaks in the detection plane.

  2. b)

    We propose a lightweight suppression strategy that leverages multi-frame temporal association and anchor-based masking. Unlike reconstruction-based approaches15, our method avoids explicit echo estimation, thereby improving robustness against noise and interference.

  3. c)

    The proposed approach requires no prior knowledge of clutter or environment statistics, in contrast to methods such as the work16. This enhances its generalizability across diverse scenarios.

Notation: Lowercase bold italic letters (e.g., \(\varvec{x}\)) denote vectors, uppercase bold italic letters (e.g., \(\varvec{D}\)) denote matrices. Scalars are represented by italic letters (e.g., m, M), while identifiers and special symbols are written in upright font (e.g., \(\textbf{I}\) for the identity matrix). Functions are expressed in italic font (e.g., \(\psi (\cdot )\)). The Hermitian transpose are denoted by \((\cdot )^{\textrm{H}}\).

Signal modeling

In a PBR system, the receiver consists of two distinct channels: a reference channel that captures the direct-path transmission from the IO, and a surveillance channel that receives echoes containing target returns, clutter, and noise. Target detection is typically achieved by coherent integration through cross-correlation between the reference and surveillance signals. Ideally, the reference channel provides a clean replica of the transmitted waveform. In practice, however, it may be corrupted by unintended target echoes due to the wide antenna beamwidth or angular variations of moving targets. To characterize this effect, we establish a unified discrete-time model for the reference and surveillance channels, with emphasis on target-induced contamination in the reference channel. Other types of corruption, such as clutter-induced contamination, are not considered in this work.

Let s[n] denote the baseband equivalent of the transmitted signal at discrete time index n, and let \(T_s\) represents the sampling interval. The received reference signal \(x_{\textrm{r}}[n]\) can then be modeled as

$$\begin{aligned} {x_{\mathrm{{r}}}}[n] = {\alpha _0}s[n] + \sum \limits _{k = 1}^{{N_{\mathrm{{t}}}}} {{\beta _k}s[n - \Delta {\tau _k}]} {e^{j2{{\uppi }}{f_{{{\textrm{d},k}}}}n{T_{\textrm{s}}}}} + {w_{\mathrm{{r}}}}[n], \end{aligned}$$
(1)

where \({\alpha _0} \in \mathbb {C}\) is the complex amplitude of the direct-path signal; the second term accounts for the \({N_{\textrm{t}}}\) target-induced components that leak into the reference channel, with \({\beta _k} \in \mathbb {C}\) denoting the complex amplitude, \(\Delta {\tau _k}\) the relative range-delay bin, and \({f_{{\textrm{d}},k}}\) Doppler shift bin of the k-th contaminating return; and \({w_{\textrm{r}}}[n]\) is additive white Gaussian noise.

The surveillance signal \({x_{\textrm{s}}}[n]\), received by a spatially separated antenna pointing toward the target region, can be modeled as

$$\begin{aligned} x_{\textrm{s}}[n] = \sum _{i=1}^{N_{\textrm{c}}} \alpha _i \, s[n - \Delta \hat{\tau }_i] + \sum _{k=1}^{N_{\textrm{t}}^{\prime }} \beta _k^{\prime } \, s[n - \Delta \tilde{\tau }_k] \, e^{j 2 \uppi f_{\textrm{d},k}^{\prime } n T_s} + w_{\textrm{s}}[n], \end{aligned}$$
(2)

where the first summation models the multipath clutter in the surveillance channel, with \({N_{\textrm{c}}}\) clutter components, each characterized by a complex amplitude \({\alpha _i}\) and a relative range-delay bin \(\Delta {\hat{\tau } _i}\) the second summation represents \(N_{\mathrm{{t}}}^{\prime }\) desired target echoes to be detected, with each echo having amplitude \(\beta _k^{\prime }\), relative range-delay bin \(\Delta {\tilde{\tau } _k}\), and Doppler shift bin \(f_{\mathrm{{d,}}k}^{\prime }\); and \({w_{\textrm{s}}}[n]\) denotes additive noise in the surveillance channel.

All received signal components are modeled with respect to the direct-path signal s[n], providing a unified temporal and spatial reference for subsequent analysis. It should be emphasized that the target-induced components in the reference and surveillance channels are not necessarily associated with the same physical scatterers; their respective delays and Doppler shifts may differ due to distinct propagation paths. Compared with conventional passive radar models, the formulation here explicitly incorporates an additional target-induced term in the reference channel. Although this contamination usually has much lower energy than the direct-path signal and only marginally affects the channel SNR, it can nevertheless introduce distortions and ambiguities in the matched-filter output. Accurately characterizing this effect is therefore essential for ensuring reliable target detection in practical passive radar systems.

Ghost peaks mitigation under reference signal contamination

In a PBR, reliable target detection relies on clutter suppression and coherent integration. The extended cancellation algorithm (ECA)19,20,21,22,23 suppresses stationary and slow-moving clutter by projecting the surveillance signal onto the orthogonal subspace of the contaminated reference, thereby enabling matched filtering to enhance weak target returns.

This section introduces the proposed method for mitigating ghost peaks caused by target-induced contamination in the reference channel. As shown in Fig. 1, the overall procedure consists of four stages: (i) signal acquisition and modeling, (ii) clutter suppression via ECA, (iii) matched filtering and CFAR detection, and (iv) multi-frame suppression of ghost peaks. Unlike conventional approaches, the first stage explicitly incorporates target-induced contamination in the reference signal, providing the theoretical basis for explaining ghost peak formation. The second and third stages follow standard PBR processing, while the final stage constitutes the main contribution of this work by introducing a multi-frame strategy to identify suspicious Doppler bins and suppress spurious detections.

Fig. 1
figure 1

Overall flow of the proposed ghost peaks mitigation method. The pipeline includes signal acquisition, clutter suppression via ECA, matched filtering, CFAR detection, and multi-frame matching. The bottom part illustrates the decision logic and masking strategies used to suppress ghost peaks while preserving true target detections.

Full size image

Mechanism of ghost peaks generation

Let \({{\varvec{x}}_{\textrm{r}}} \in \mathbb {C}^{N \times 1}\) and \({{\varvec{x}}_{\textrm{s}}} \in \mathbb {C}^{N \times 1}\) denote the reference and surveillance signal vectors, respectively, collected over a coherent processing interval (CPI) of N samples. According to the signal model mentioned earlier, they can be expressed as

$$\begin{aligned} {{\varvec{x}}_{\mathrm{{r}}}}= & {\varvec{x}}_{\mathrm{{r}}}^{\mathrm{{d}}} + {\varvec{x}}_{\mathrm{{r}}}^{\mathrm{{t}}} + {{\varvec{w}}_{\mathrm{{r}}}}, \end{aligned}$$
(3)
$$\begin{aligned} {{\varvec{x}}_{\mathrm{{s}}}}= & {\varvec{x}}_{\mathrm{{s}}}^{\mathrm{{c}}} + {\varvec{x}}_{\mathrm{{s}}}^{\mathrm{{t}}} + {{\varvec{w}}_{\mathrm{{s}}}}, \end{aligned}$$
(4)

where \({\varvec{x}}_{\mathrm{{r}}}^{\mathrm{{d}}}\) represents the direct-path component in the reference channel, \({\varvec{x}}_{\mathrm{{s}}}^{\mathrm{{c}}}\) denotes clutter contributions in the surveillance channel,\({\varvec{x}}_{\mathrm{{r}}}^{\mathrm{{t}}}\) and \({\varvec{x}}_{\mathrm{{s}}}^{\mathrm{{t}}}\) correspond to target-induced returns embedded in the reference and surveillance channels, respectively, and \({{\varvec{w}}_{\mathrm{{r}}}}\), \({{\varvec{w}}_{\mathrm{{s}}}}\) are additive white Gaussian noise vectors.

To capture the temporal structure of the reference signal and model correlated clutter interference, a range-delay dictionary matrix \(\varvec{D} \in \mathbb {C}^{N \times L}\) is constructed by stacking L delayed versions of \({{\varvec{x}}_{\mathrm{{r}}}}\). Each column corresponds to an integer range-delay bin of the reference signal:

$$\begin{aligned} {\varvec{D}} = \left[ {{\varvec{x}}_{\mathrm{{r}}}^0,{\varvec{x}}_{\mathrm{{r}}}^1,...,{\varvec{x}}_{\mathrm{{r}}}^{L - 1}} \right] , \end{aligned}$$
(5)

where \({\varvec{x}}_{\mathrm{{r}}}^l\) denotes the l-sample range delayed version of \({\varvec{x}}_{\mathrm{{r}}}\), appropriately zero-padded to preserve the dimension N. The parameter L is referred to as the order of clutter suppression, which controls the dimensionality of the clutter subspace and determines the degree of freedom available for cancelling correlated clutter. The columns of \({\varvec{D}}\) span the subspace associated with static and slowly-varying clutter components, which can be effectively suppressed by projecting the surveillance signal onto the orthogonal complement of this subspace.

To further illustrate the effect of target-induced contamination in the reference channel, the range-delay dictionary can be decomposed according to the components of \({{\varvec{x}}_{\mathrm{{r}}}}\):

$$\begin{aligned} {\varvec{D}} = {{\varvec{D}}^{\mathrm{{d}}}} + {{\varvec{D}}^{\mathrm{{t}}}}, \end{aligned}$$
(6)

where \({{\varvec{D}}^{\mathrm{{d}}}}\) and \({{\varvec{D}}^{\mathrm{{t}}}}\) are constructed from the delayed versions of the direct-path component \({\varvec{x}}_{\mathrm{{r}}}^{\mathrm{{d}}}\) and the target-induced component \({\varvec{x}}_{\mathrm{{r}}}^{\mathrm{{t}}}\), respectively. Each submatrix consists of L delayed versions of the corresponding signal component. The noise term is typically not included in the dictionary construction, since white Gaussian noise does not contribute significantly to the correlated clutter subspace.

Once the full range-delay matrix \({\varvec{D}}\) is obtained, the ECA projects the surveillance signal onto the orthogonal complement of the clutter subspace spanned by \({\varvec{D}}\). The orthogonal projection matrix is defined as:

$$\begin{aligned} {{\varvec{P}}_ \bot } = \mathbf{{I}} - {\varvec{D}}{({{\varvec{D}}^{\mathrm{{H}}}}{\varvec{D}})^{ - 1}}{{\varvec{D}}^{\mathrm{{H}}}}, \end{aligned}$$
(7)

and the clutter-suppressed surveillance signal is given by

$$\begin{aligned} {{\varvec{\tilde{x}}}_{\mathrm{{s}}}} = {{\varvec{P}}_ \bot }{{\varvec{x}}_{\mathrm{{s}}}}. \end{aligned}$$
(8)

Substituting the decomposition of \({\varvec{D}}\) into this expression and rearranging terms yields:

$$\begin{aligned} {{\varvec{\tilde{x}}}_{\mathrm{{s}}}} = {{\varvec{x}}_{\mathrm{{s}}}} - {\varvec{Dw}}. \end{aligned}$$
(9)

where

$$\begin{aligned} {\varvec{w}} = {({{\varvec{D}}^{\mathrm{{H}}}}{\varvec{D}})^{ - 1}}{{\varvec{D}}^{\mathrm{{H}}}}{{\varvec{x}}_{\mathrm{{s}}}}, \end{aligned}$$
(10)

represents the least-squares weight vector used in the ECA projection process. Intuitively, \({\varvec{Dw}}\) is the estimated clutter component reconstructed from the range-delay dictionary, and \({{\varvec{\tilde{x}}}_{\mathrm{{s}}}}\) is the residual signal after cancellation.

With the clutter components effectively suppressed by the ECA, the residual signal in the surveillance channel primarily consists of target returns and additive noise. To enhance the detection of weak moving targets, a matched-filter is employed to coherently accumulate the signal energy across time. For a hypothesized target with range-delay bin \(\Delta \tau\) and Doppler shift \({f_{\mathrm{{d}}}}\), the matched-filter output24,25 is expressed as a cross-ambiguity function:

$$\begin{aligned} {{\psi (}}\Delta \tau ,{f_{\mathrm{{d}}}}\mathrm{{) = }}\sum \limits _{n = 0}^{N - 1} {{{\tilde{x}}_{\mathrm{{s}}}}} [n]x_{\mathrm{{r}}}^*[n - \Delta \tau ]{e^{ - j2{{\uppi }}{f_{\mathrm{{d}}}}n{T_{\mathrm{{s}}}}}}, \end{aligned}$$
(11)

where \({\tilde{x}_{\mathrm{{s}}}}[n]\) denotes the clutter-suppressed residual signal at the n-th time index. Equivalently, the matched-filter can be expressed in vector form as:

$$\begin{aligned} {{\psi (}}\Delta \tau ,{f_{\mathrm{{d}}}}\mathrm{{) = }}{\varvec{x}}_{\mathrm{{shift}}}^{\mathrm{{H}}}{{\varvec{\tilde{x}}}_{\mathrm{{s}}}}, \end{aligned}$$
(12)

where \({{\varvec{x}}_{\mathrm{{shift}}}}\) is the delay- and Doppler-shifted version of the reference signal. Substituting (9) into this expression gives:

$$\begin{aligned} {{\psi (}}\Delta \tau ,{f_{\mathrm{{d}}}}\mathrm{{) = }}{\varvec{x}}_{\mathrm{{shift}}}^{\mathrm{{H}}}\left( {{{\varvec{x}}_{\mathrm{{s}}}} - {\varvec{Dw}}} \right) . \end{aligned}$$
(13)

Furthermore, substituting (4) and (6) into the above equation and rearranging yields:

$$\begin{aligned} \psi (\Delta \tau , f_{\textrm{d}})&= \psi _{\textrm{clutter}}(\Delta \tau , f_{\textrm{d}}) + \psi _{\textrm{target}}(\Delta \tau , f_{\textrm{d}}) + \psi _{\textrm{noise}}(\Delta \tau , f_{\textrm{d}}) - \psi _{\textrm{ghost}}(\Delta \tau , f_{\textrm{d}}) \nonumber \\&= \varvec{x}_{\textrm{shift}}^{\textrm{H}} \left( \varvec{x}_{\textrm{s}}^{\textrm{c}} - \varvec{D}^{\textrm{d}}\varvec{w} \right) + \varvec{x}_{\textrm{shift}}^{\textrm{H}} \varvec{x}_{\textrm{s}}^{\textrm{t}} + \varvec{x}_{\textrm{shift}}^{\textrm{H}} \varvec{w}_{\textrm{s}} - \varvec{x}_{\textrm{shift}}^{\textrm{H}} \varvec{D}^{\textrm{t}}\varvec{w}. \end{aligned}$$
(14)

As seen from the above decomposition, the matched-filter output consists of four components: residual clutter, the true target return from the surveillance channel, additive noise, and ghost interference. The first three correspond to conventional outputs in passive radar detection, whereas the last term arises exclusively from target-induced contamination in the reference channel.

By combining (1), (3), (5), and (11), the ghost component can be expanded as:

$$\begin{aligned} {{{\psi }}_{\mathrm{{ghost}}}}\mathrm{{(}}\Delta \tau ,{f_{\mathrm{{d}}}}\mathrm{{)}} = \sum \limits _{n = 0}^{N - 1} {\left( {\sum \limits _{k = 1}^{{N_{\mathrm{{t}}}}} {\sum \limits _{l = 0}^{L - 1} {{\beta _k}w[l]s[n - \Delta {\tau _k} - l]} {e^{j2\mathrm{{\pi }}{f_{\mathrm{{d,}}k}}n{T_{\mathrm{{s}}}}}}} } \right) } x_{\mathrm{{r}}}^*[n - \Delta \tau ]{e^{ - j2\mathrm{{\uppi }}{f_{\mathrm{{d}}}}n{T_{\mathrm{{s}}}}}}, \end{aligned}$$
(15)

where w[l] denotes the l-th element of the least-squares weight vector in the ECA projection, \({\beta _k}\) and \({f_{\mathrm{{d,}}k}}\) are the amplitude and Doppler of the k-th target echo, and \({x_{\mathrm{{r}}}}[n]\) is the contaminated reference signal.

Since clutter components in both channels typically exhibit much higher power than either target returns or noise, the direct-path signal dominates the matched-filter reference. Thus, \({x_{\mathrm{{r}}}}[n]\) in the above expression can be approximated by the direct-path term defined in (1). Substituting this approximation yields:

$$\begin{aligned} {{{\psi }}_{\mathrm{{ghost}}}}\mathrm{{(}}\Delta \tau ,{f_{\mathrm{{d}}}}\mathrm{{)}} = \sum \limits _{n = 0}^{N - 1} {\left( {\sum \limits _{k = 1}^{{N_{\mathrm{{t}}}}} {\sum \limits _{l = 0}^{L - 1} {{\beta _k}w[l]s[n - \Delta {\tau _k} - l]} {e^{j2{{\uppi }}{f_{\mathrm{{d,}}k}}n{T_{\mathrm{{s}}}}}}} } \right) } x_{\mathrm{{r}}}^*[n - \Delta \tau ]{e^{ - j2{{\uppi }}{f_{\mathrm{{d}}}}n{T_{\mathrm{{s}}}}}}. \end{aligned}$$
(16)

In the above expression, the weight vector w[l], obtained via least-squares estimation, characterizes how each delayed replica of the direct-path signal contributes to reconstructing multipath clutter in the surveillance channel. In practice, only a small subset of delay indices yield significant coefficients, corresponding to dominant clutter paths, whereas the remaining coefficients are negligible and have little effect on clutter reconstruction. Specifically, the significance of each coefficient in the weight vector \({\varvec{w}}\) can be characterized by a thresholding rule:

$$\begin{aligned} |w[l]| = {\left\{ \begin{array}{ll} \;\;\ge \dfrac{\kappa }{L}\sum _{i=0}^{L-1} |w[i]|, & l \in \mathscr {G}_c \\ \;\;< \dfrac{\kappa }{L}\sum _{i=0}^{L-1} |w[i]|, & l \notin \mathscr {G}_c \end{array}\right. }, \end{aligned}$$
(17)

where \(\kappa \ge 1\) is a scaling factor and \({\mathscr {G}_c} = \left\{ {\Delta {{\hat{\tau } }_1},\Delta {{\hat{\tau } }_2}, \cdots ,\Delta {{\hat{\tau } }_{{N_c}}}} \right\}\) denotes the set of delay indices corresponding to the dominant multipath clutter components in the surveillance channel. Substituting this representation into (16), the ghost term can be approximated as:

$$\begin{aligned} & \psi _{\textrm{ghost}}(\Delta \tau , f_{\textrm{d}}) \approx \sum _{n = 0}^{N - 1} \left( \sum _{k = 1}^{N_{\textrm{t}}} \sum _{i = 1}^{N_c} \beta _k w[\Delta \hat{\tau }_i] s[n - \Delta \tau _k - \Delta \hat{\tau }_i] e^{j2\pi f_{\textrm{d},k} n T_{\textrm{s}}} \right) \alpha _0 s^{*}\nonumber \\ & \quad [n - \Delta \tau ]SSSSSS e^{-j2\pi f_{\textrm{d}} n T_{\textrm{s}}}. \end{aligned}$$
(18)

From this expression, it is evident that spurious peaks arise in the matched-filter output whenever the following conditions are approximately satisfied for some target-clutter delay pair:

$$\begin{aligned} \Delta \tau = \Delta {\tau _k} + \Delta {\hat{\tau } _i}, {f_{\mathrm{{d}}}} = {f_{\mathrm{{d}},k}}, \end{aligned}$$
(19)

with \(k = 1,2, \cdots ,{N_{\mathrm{{t}}}}\) and \(i = 1,2, \cdots ,{N_{\mathrm{{c}}}}\).

This alignment condition indicates that ghost peaks appear at the same Doppler frequency as the true target but at range bins shifted by clutter delays. Consequently, multiple spurious peaks may emerge, each corresponding to a different target-clutter combination. Motivated by this mechanism, the next section introduces a multi-frame suppression strategy to identify and eliminate such ghost responses, thereby enhancing target detection performance.

Ghost peaks suppression via multi-frame matching

Based on the analysis in (18) and (19), ghost peaks are not randomly distributed but instead exhibit a structured pattern in the range-Doppler domain. In particular, each ghost response tends to appear at a range-delay bin \(\Delta {\tau _k}\) equal to the sum of a true target range-delay bin and a clutter-induced delay \(\Delta {\hat{\tau } _i}\), while retaining the Doppler shift of the associated target. Consequently, ghost responses manifest as clusters of spurious peaks aligned along the Doppler axis at multiple range bins, thereby forming target-like trajectories in the range-Doppler map.

Furthermore, ghost peaks generally arise in the presence of true moving targets. They often coexist with genuine detections and may exhibit temporal continuity across consecutive frames, especially within Doppler-consistent bins. Unlike true targets, however, ghost responses do not correspond to physical motion but are structurally induced by contamination in the reference channel. Motivated by this structured behavior, a frame-based consistency analysis is proposed. The key idea is that persistent multi-frame peaks within the same Doppler bin are indicative of target-induced reference contamination. To address this, a multi-frame matching strategy is introduced to diagnose such cases and suppress ghost peaks by exploiting their temporal and spectral distribution patterns.

Let \({{{\psi }}_m}\mathrm{{(}}\Delta \tau ,{f_{\mathrm{{d}}}}\mathrm{{)}}\) denotes the matched-filter output at range-delay bin \(\Delta \tau\) and Doppler frequency bin \({f_{\mathrm{{d}}}}\) for the m-th frame. A detection strategy is introduced that consists of three processes: Stage 1 performs single-frame thresholding to identify candidate peaks, Stage 2 applies multi-frame consistency matching to retain persistent detections, and Stage 3 carries out Doppler-bin screening to flag channels potentially contaminated by ghost artifacts.

Stage 1 (single-frame detection): A binary indicator is defined based on a magnitude threshold \(\gamma\):

$$\begin{aligned} {\chi _m}\left( {\Delta \tau ,{f_{\mathrm{{d}}}}} \right) = \left\{ {\begin{array}{*{20}{c}} {1,}& {\left| {{{{\psi }}_m}\mathrm{{(}}\Delta \tau ,{f_{\mathrm{{d}}}}\mathrm{{)}}} \right|> \gamma }\\ {0,}& {\mathrm{{otherwise}}} \end{array}} \right. , \end{aligned}$$
(20)

where \({\chi _m}\left( {\Delta \tau ,{f_{\mathrm{{d}}}}} \right) = 1\) indicates that a peak exceeding the threshold is present in the m-th frame.

Stage 2 (multi-frame consistency matching): To exploit temporal persistence, a consistency score is computed across the past M−1 frames for the current frame M:

$$\begin{aligned} {S_M}\mathrm{{(}}\Delta \tau ,{f_{\mathrm{{d}}}}\mathrm{{) = }}\sum \limits _{m = 1}^{M - 1} {\sum \limits _{(\Delta {{\hat{\tau } }_m},{{\hat{f}}_{\mathrm{{d}},m}}) \in {\mathscr {D}_m}} {\chi \left( {\Delta {{\hat{\tau } }_m},{{\hat{f}}_{\mathrm{{d}},m}}} \right) {\mathbf{{1}}_{\{ \left| {\Delta \tau - \Delta {{\hat{\tau } }_m}} \right| \le \delta _{{\uptau }}^m\} }}{\mathbf{{1}}_{\{ \left| {{f_{\mathrm{{d}}}} - {{\hat{f}}_{\mathrm{{d}},m}}} \right| \le \delta _{\mathrm{{f}}}^m\} }}} } , \end{aligned}$$
(21)

where \({\mathscr {D}_m}\) denotes the set of detection points in the m-th frame, \(\mathbf{{1}}_{\{\cdot \}}\) denotes the indicator function that equals 1 if the condition is satisfied and 0 otherwise, and \(\delta _{{\uptau }}^m\) and \(\delta _{\mathrm{{f}}}^m\) denote the tolerance thresholds in the range and Doppler dimensions, respectively. The multi-frame consistency score \({S_M}\mathrm{{(}}\Delta \tau ,{f_{\mathrm{{d}}}}\mathrm{{)}}\) is obtained by accumulating all single-frame detections \(\chi \left( {\Delta {{\hat{\tau } }_m},{{\hat{f}}_{\mathrm{{d}},m}}} \right)\) that fall within a tolerance window \(\left( {\delta _{{\uptau }}^m,\delta _{\mathrm{{f}}}^m} \right)\) around the bin \(\mathrm{{(}}\Delta \tau ,{f_{\mathrm{{d}}}}\mathrm{{)}}\) across the past M−1 frames.

Stage 3 (Doppler-bin screening): an aggregated activity indicator is defined for each Doppler bin by summing the multi-frame consistency scores across all range-delay bins:

$$\begin{aligned} \mathscr {C} ({f_{\mathrm{{d}}}}) = \sum \limits _{\Delta \tau } {S\mathrm{{(}}\Delta \tau ,{f_{\mathrm{{d}}}}\mathrm{{)}}}. \end{aligned}$$
(22)

This function counts the number of thresholded detections within a given Doppler bin that consistently match across frames within predefined range-Doppler gates. If the accumulated count exceeds a threshold \(\eta\), the Doppler bin is flagged as potentially contaminated by ghost artifacts. This criterion serves as a coarse screening mechanism to identify Doppler channels likely affected by reference-induced interference, thereby enabling targeted suppression in subsequent processing stages.

Based on this mechanism, after performing consistency matching across M frames, suppose that a Doppler bin \({f_{\mathrm{{d}}}} = f_{\mathrm{{d}}}^\dag\) is identified as contaminated. According to (19), the corresponding suspected target peaks within this Doppler bin can be expressed as:

$$\begin{aligned} \Delta \tau = \Delta {\tau ^\dag } + \Delta {\hat{\tau } _i}, i = 1,2, \cdots ,{N_{\mathrm{{c}}}}, \end{aligned}$$
(23)

here, \(\Delta {\tau ^\dag }\) denotes the range-delay bin of a true target present in the reference channel (and possibly also in the surveillance channel), while \(\Delta {\hat{\tau } _i}\) corresponds to clutter-related delays introduced by multipath propagation. This additive alignment gives rise to spurious peaks that resemble true target responses through coherent integration. To suppress such ghost peaks, the first step is to identify the true target peak among the set of suspected detections. Based on (23), three possible cases are considered:

The first case considers whether there exists a range-delay bin \(\Delta {\hat{\tau } _i} = 0\), which implies that the multipath clutter in the surveillance channel contains a direct-path component. In practice, this condition can be inferred from the magnitude of the corresponding ECA weight: a significantly large coefficient indicates that the direct-path signal is also present in the surveillance channel. Under this condition, the earliest observed peak necessarily corresponds to the true target, i.e.,

$$\begin{aligned} \Delta {\tau _{\mathrm{{target}}}} = \Delta {\tau ^\dag }. \end{aligned}$$
(24)

whereas all subsequent peaks are identified as ghost responses, i.e.,

$$\begin{aligned} \Delta {\tau _{\mathrm{{ghost}}}} = \Delta {\tau ^\dag } + \Delta {\hat{\tau } _i}, i = 2, \cdots ,{N_{\mathrm{{c}}}}. \end{aligned}$$
(25)

To suppress ghost peaks while retaining the true target response, a binary masking strategy is employed. Specifically, a mask vector of length L is defined as

$$\begin{aligned} {\varvec{m}} = \left[ {1,0, \cdots ,0} \right] . \end{aligned}$$
(26)

such that only the first element is retained. In the range-Doppler plane, for a Doppler bin identified via multi-frame consistency analysis, the first stable peak is selected as the anchor detection. A logical AND operation is then performed between the peak vector and the predefined mask, thereby retaining the earliest consistent detection while suppressing subsequent ghost peaks aligned along the same Doppler bin.

The second case arises when the direct-path signal is absent from the surveillance channel, i.e., \(\Delta {\hat{\tau } _i}> 0\) and the target contaminating the reference signal does not appear in the surveillance channel. In this case, the observed range-delay bins satisfy \(\Delta \tau> \Delta {\tau ^\dag }\), which implies that all detections within the contaminated Doppler bin correspond exclusively to ghost responses:

$$\begin{aligned} \Delta {\tau _{\mathrm{{ghost}}}} = \Delta {\tau ^\dag } + \Delta {\hat{\tau } _i}, \Delta {\hat{\tau } _i}> 0, i = 1,2, \cdots ,{N_{\mathrm{{c}}}}. \end{aligned}$$
(27)

Accordingly, a suppression strategy similar to that in the first scenario is applied, but with the binary mask vector defined as

$$\begin{aligned} {\varvec{m}} = \left[ {0,0, \cdots ,0} \right] . \end{aligned}$$
(28)

thereby eliminating all peaks across the L range-delay bins within the affected Doppler bin.

The third case similar to the second in that \(\Delta {\hat{\tau } _i}> 0\). However, in this scenario, the target contaminating the reference signal is also present in the surveillance channel. Consequently, within the Doppler bin \({f_{\mathrm{{d}}}} = f_{\mathrm{{d}}}^\dag\), at least one range-delay bin corresponds to a true target peak satisfying (23), while the remaining peaks with \(\Delta {\hat{\tau } _i}> 0\) are regarded as ghost responses satisfying (27). The suppression strategy therefore follows the same principle as in the first scenario: the true target is preserved, and shifted ghost peaks are eliminated through binary masking.

A practical issue arises in distinguishing the true target from ghost responses prior to masking. To this end, an anchor selection criterion is introduced. Let \(\Delta {\tau _{\mathrm{{1st}}}}\) and \(\Delta {\tau _{\mathrm{{2nd}}}}\) denote the range-delay bins of the first and second detected peaks, respectively. The first peak is accepted as a reliable anchor if

$$\begin{aligned} \left| {\Delta {\tau _{\mathrm{{2nd}}}} - \Delta {\tau _{\mathrm{{1st}}}}} \right| \approx \Delta {\hat{\tau } _1} + \delta , \end{aligned}$$
(29)

where \(\Delta {\hat{\tau } _1}\) is the clutter-related range-delay bin derived from the ECA, and \(\delta\) accounts for tolerance due to range-bin quantization or motion-induced variation. This condition ensures that the earliest detection corresponds to the true target, enabling reliable binary masking in the subsequent suppression step.

Applicability and limitations

In summary, the proposed method adaptively suppresses ghost peaks by combining mechanism analysis with multi-frame consistency matching. Doppler bins suspected of reference contamination are first identified, after which case-dependent anchor selection and binary masking are applied in the range–Doppler domain. This strategy enables effective suppression of spurious peaks while preserving genuine target detections.

The method is broadly applicable to both single- and multi-target scenarios, provided that the contaminating targets are independent. However, it is not well suited for formation targets, since multiple closely spaced echoes with identical Doppler velocities may simultaneously contaminate the reference channel, rendering the resulting responses indistinguishable. In addition, the multi-frame consistency check introduces extra computational cost and requires parameter tuning, which may hinder real-time deployment in highly dynamic environments.

Despite these limitations, the proposed approach substantially improves detection reliability under realistic operating conditions and provides a practical means of mitigating reference contamination in passive bistatic radar. The following section presents simulation results that validate its effectiveness.

Simulation results

Simulation setup

To validate the effectiveness of the proposed ghost peak mitigation method, numerical simulations were conducted using both measured and synthesized data. All simulations and processing-time evaluations were implemented in MATLAB R2021a on a desktop equipped with an Intel multi-core i7 processor running at 2.1 GHz.

A single moving target was assumed to travel from the reference antenna beam toward the surveillance antenna beam, such that its visibility in the two channels changed dynamically during the trajectory. Four representative visibility stages were considered. In Stage I, the target contaminates only the reference channel while the surveillance channel contains the direct-path, multipath clutter, and noise. In Stage II, the target remains in the reference channel while the surveillance channel contains only multipath clutter and noise without the direct-path. In Stage III, the target appears simultaneously in both the reference and surveillance channels, regardless of whether the direct-path is present. In Stage IV, the target is observed solely in the surveillance channel, representing the normal case without reference contamination. These stages emulate the practical process of a target gradually transitioning from the reference beam into the surveillance beam and are schematically illustrated in Fig. 2.

Fig. 2
figure 2

Illustration of the four-stage simulation scenario, showing target and direct-path visibility in the reference and surveillance channels.

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The simulation dataset was generated from a measured FM broadcast signal, which served as the clean direct-path reference. Multipath clutter was synthesized by applying random integer-sample delays to the measured waveform, whereas target echoes were constructed by imposing both an integer range-delay bin and a Doppler shift. The FM signal had a bandwidth of 200 kHz, and the CPI was set to 1 s, corresponding to N = 200,000 samples per CPI. The target trajectory spanned 50 s, equally divided into the four visibility stages described above. The key simulation parameters are summarized in Table 1.

Table 1 Key simulation parameters for the single-target four-stage scenario.
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Results and analysis

Based on the simulation setup described above, Fig. 3 presents the range-domain slice of the matched-filter output, representing the detection results of the conventional method for a single frame in each of the four stages. The conventional method refers to a target detection processing chain that includes three main modules: clutter suppression, matched filtering, and CFAR detection. This method serves as a baseline for comparison with the proposed approach, which incorporates additional steps for ghost peak suppression, as well as another reconstruction-based ghost peak suppression method15. It can be observed that only in Stage IV does the conventional detector yield the correct result, corresponding to the normal clean-reference case. In contrast, in Stages I and III, spurious peaks appear alongside the true target response. These ghost peaks are aligned with the true Doppler frequency but shifted in delay, making them difficult to distinguish from true detections. In Stage II, however, due to modulation by multipath clutter, all spurious peaks are ghost ones. As a result, multiple false alarms may occur in the range-Doppler plane, which not only degrade detection performance but also lead to erroneous track formation in subsequent data association and tracking stages. This confirms that when the reference signal is modulated or contaminated by target echoes, the standard detection procedure inevitably produces ghost peaks, thereby motivating the need for dedicated suppression strategies such as the one proposed in this work.

Fig. 3
figure 3

Range-domain side view of the matched-filter output obtained with the conventional method for a single frame in each of the four stages: (a) Stage I; (b) Stage II; (c) Stage III; (d) Stage IV.

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Building upon the single-frame observations in Fig. 3, it is also necessary to evaluate detection performance over the entire target trajectory as well as the computational cost. Fig. 4 provides a trajectory-level evaluation of the detection performance. Intuitively comparing the three methods, the proposed method demonstrates the best detection results. The conventional method and the reconstruction-based method each have their strengths: the conventional method shows a higher degree of overlap between the detected tracks and the true target trajectory, while the reconstruction-based method excels at suppressing false targets. Specifically, the results can be further interpreted through three representative phenomena rather than by strictly separating the four motion stages. First, compared with the ground-truth trajectory, the conventional method tends to produce multiple spurious tracks in the detection plane. While one of these tracks corresponds to the actual target, the others are false and may mislead subsequent tracking or association processes. The reconstruction-based method exhibits similar behavior.

Fig. 4
figure 4

Comparison of detection results across the target trajectory, presented together with the ground-truth trajectory: (a) conventional method, where multiple spurious and continuous tracks are observed; (b) reconstruction-based method, where multiple spurious and fragmented tracks are observed; (c) proposed method, which suppresses ghost peaks to preserve a single valid track.

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By contrast, the proposed method effectively suppresses ghost peaks in most frames, thereby maintaining a single, continuous track aligned with the ground truth. Second, under the influence of multipath clutter, the conventional method fails more severely: all detections deviate from the true trajectory and form false targets, resulting in fragmented or incorrect tracks that undermine detection reliability. The reconstruction-based method has a slight advantage in suppressing false target formation. The proposed method, however, successfully identifies and rejects such spurious responses. Although this leads to intermittent detections, the retained track segments remain consistent with the actual trajectory, thus avoiding false-track formation. Finally, when the reference signal is clean and uncontaminated, all methods correctly detect a single trajectory coinciding with the ground truth. This confirms that the proposed approach introduces no performance loss under normal conditions while providing clear advantages in contaminated scenarios. Overall, the results demonstrate that the proposed method robustly suppresses ghost peaks and clutter-induced artifacts, ensuring reliable trajectory estimation across diverse operating environments.

To further quantify the overall performance differences among the three methods, in addition to the average processing time per frame, a new metric has been introduced: the average weighted score to measure the time-domain target detection performance. This metric combines two factors: the overlap rate between the detected target tracks and the true trajectory, and the false target suppression rate. Based on the detection results in Figs. 4, 5 illustrates a coordinate comparison of the three methods, with average processing time per frame on the x-axis and the weighted detection performance score on the y-axis. Clearly, the proposed method exhibits the best overall performance. From the perspective of the quantifiable detection performance dimension, the proposed method scores significantly higher than the other two methods. This is primarily due to the effective ghost peak suppression technique adopted in the proposed method, which stems from theoretical analysis. On the other hand, the reconstruction-based method performs less well, mainly because its adaptability to the specific scenario is weaker. Additionally, from the computational efficiency dimension, the proposed method and the conventional method have comparable computational complexities. However, the proposed method, due to the addition of lightweight ghost peak suppression, results in a slightly higher per-frame runtime. In contrast, the reconstruction-based method significantly increases its computational complexity due to the need to reconstruct the target echoes from the contaminated reference signal, subtract it from the original reference signal, and then execute the standard detection process again. This leads to the longest processing time. Overall, the proposed method only incurs a 1.25% increase in computational load compared to the conventional method, yet it significantly mitigates the ghost peak problem. This advantage stems from the design philosophy of the proposed strategy: rather than reconstructing the contaminated reference signal, it leverages offline analytical insights into the ghost peak mechanism to guide lightweight online suppression in the post-detection stage. This design achieves a favorable balance between suppression performance and processing efficiency, avoiding the impracticality often associated with reconstruction-based methods that emphasize accuracy at the expense of complexity. Moreover, this balance is crucial for real-time passive radar systems, where both detection reliability and processing efficiency directly impact subsequent tracking and situational awareness performance.

To further illustrate how the proposed method suppresses ghost peaks through the masking operation, one representative frame is selected from each of Stages I–III for detailed analysis, as shown in Figs. 6, 7, 8. In Stage I, after conventional detection, the Doppler bin containing the target is flagged as suspicious and exhibits multiple peaks, as shown in Fig. 6a, which the conventional method would all treat as true targets. In contrast, the proposed method exploits clutter-suppression weights to distinguish true and false responses: the first peak is identified as the true target and designated as the anchor, and the following \(L-1\) detections are set to zero regardless of amplitude, yielding the result in Fig. 6b.

Fig. 5
figure 5

Comparison of the average processing time per frame and weighted detection performance score for three methods.

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Fig. 6
figure 6

Processing of a representative frame in Stage I by the proposed method: (a) range profile of the suspicious Doppler bin with multiple peaks; (b) detection result after masking, where the true target is preserved and ghost peaks are suppressed.

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Here, L denotes the clutter suppression order, which ensures that the L consecutive detections starting from the anchor always cover the range-delay bins where ghost peaks occur, thereby effectively removing them. In Stage II, since the surveillance channel does not contain the direct-path signal, all peaks observed are shifted by multipath clutter and correspond to false targets. Although the first peak is still designated as the anchor, it is not treated as a true detection, and the masking operation zeros out L consecutive detections starting from the anchor, resulting in an empty output, as shown in Fig. 7. Finally, in Stage III, both the reference and surveillance channels contain the target signal, leading the conventional detector to produce multiple tracks—one true and others spurious—whereas the proposed method, by applying the same anchor-based masking strategy as in Stage I, suppresses ghost peaks while retaining the true target, as shown in Fig. 8. These representative examples demonstrate that the anchor-and-mask mechanism generalizes well across different contamination scenarios, providing effective suppression without explicit reference reconstruction, which in turn reduces complexity and enhances the suitability of the method for real-time passive radar applications.

Fig. 7
figure 7

Processing of a representative frame in Stage II by the proposed method: (a) range profile showing clutter-induced spurious peaks; (b) detection result after masking, where all peaks are rejected, yielding an empty output.

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Fig. 8
figure 8

Processing of a representative frame in Stage III by the proposed method: (a) range profile of the suspicious Doppler bin containing both true and false peaks; (b) detection result after masking, where only the true target is retained.

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Conclusion

This paper investigated the challenge of ghost peak generation in passive bistatic radar caused by target-induced contamination in the reference channel. Unlike clutter- or noise-related degradation, this type of contamination directly distorts the detection process by producing spurious peaks in the range-Doppler plane that are difficult to distinguish from true targets. To address this issue, a suppression strategy was proposed that combines multi-frame consistency analysis with an anchor-based masking operation. This approach avoids explicit reconstruction of the contaminated reference and instead performs lightweight post-detection suppression, ensuring that spurious responses are removed while genuine target detections are preserved. Simulation results based on measured FM signals confirmed that the proposed method effectively mitigates ghost peaks under different contamination scenarios, yielding reliable detection performance with only a minor computational overhead compared to conventional processing. These findings demonstrate that the method offers a favorable trade-off between suppression capability and efficiency, making it suitable for real-time passive radar applications. Future work will focus on extending the proposed approach to more complex scenarios involving multiple closely spaced or formation targets with identical Doppler frequency.