A novel sequential algorithm for clutter and direct signal cancellation in passive bistatic radars
Abstract
Cancellation of clutter and multipath is an important problem in passive bistatic radars. Some important recent algorithms such as the ECA, the SCA and the ECA-B project the received signals onto a subspace orthogonal to both clutter and pre-detected target subspaces. In this paper, we generalize the SCA algorithm and propose a novel sequential algorithm for clutter and multipath cancellation in the passive radars. This proposed sequential cancellation batch (SCB) algorithm has lower complexity and requires less memory than the mentioned methods. The SCB algorithm can be employed for static and non-static clutter cancellation. The proposed algorithm is evaluated by computer simulation under practical FM radio signals. Simulation results reveal that the SCB provides an admissible performance with lower computational complexity.
Keywords
Passive radar Bistatic multipath Clutter cancellation1 Introduction
The reference antenna is adjusted to receive only the direct path of the signal from the transmitter, while the surveillance antenna receives signals from all directions which includes signals not only from the direct path from the FM station but also from the reflections produced by targets and clutters. Using the ambiguity function based on the matched filters [3, 9], the Range-Doppler targets and clutter are detectable.
Before computing the ambiguity function, there are some challenges that must be resolved. For example, the power of the direct path signal is significantly higher than the received power from targets, and the signal received from the target and clutter often go through multipath unknown channels. Various methods have been proposed to confront these problems. Some of them have considered the problem as a composite hypothesis test and have attempted to design sub-optimal detectors such as generalized likelihood ratio test for target detection in the presence of the interference [10, 11]. Some others have employed adaptive filters to estimate the clutter and direct path signal components in order to cancel them [12, 13]. However, an important class of methods is based on the projection of the received signal onto a subspace orthogonal to both the clutter and the pre-detected targets. The ECA, SCA and ECA-B are among these methods [14, 15, 16]. Recently, a version of ECA (ECA-S) has been proposed in [17].
In this paper, we propose a novel algorithm for clutter and multipath cancellation for the passive radars by generalization of some recent algorithms which we call as the sequential cancellation batch (SCB) algorithm. Our simulation results show that the proposed SCB outperforms or performs as good as the mentioned state-of-the-art methods, depending on the conditions. Furthermore, the proposed SCB requires lesser memory than these existing state-of-the-art methods. Our simulations show that after clutter and direct signal cancellation using the SCB algorithm, weak targets likely are not detectable. Hence, in this paper, we use CLEAN algorithm [18, 19, 20] for weak target detection. Although in this paper, we concentrate on the use of commercial FM radio signals, it should be noted that the proposed method (SCB) can be applied to any transmission of opportunity, such as GSM transmissions, DAB or DVB-T and satellites. Indeed, the choice of FM transmissions arguably results in waveforms with the worst ambiguity properties for target detection.
The paper is organized as follows. Section 2 presents the signal model and ambiguity function. Section 3 introduces the ECA and SCA algorithms and describes the proposed SCB technique, and in Section 4, three tests are introduced for comparison of the performance of algorithms. Finally, Section 5 is our conclusions.
Notations: Throughout this paper, we use boldface lower case and capital letters to denote vector and matrix, respectively. We use \(\mathcal {O}(.)\) as the complexity order of algorithms. diag(.,…,.) denotes diagonal matrix containing the elements on the main diameter. 0_{N×R} is an N×R zero matrix and I_{N} is an N×N identical matrix. Also (.)^{T}, (.)^{∗} and (.)^{H} stand for the transpose, conjugate and Hermitian of a matrix or vector, respectively. The operator ⌊.⌋ denotes the integer part (or floor) of a number.
2 Signal model and ambiguity function
where d(t) is the direct transmitted signal that is multiplied by the complex amplitude A_{sur}. The variables a_{m}, τ_{m} and f_{dm} are the complex amplitude, delay and Doppler frequency of the mth target signal (m=1,…,N_{T}), respectively, that is N_{T} is the number of targets. c_{i}(t) and τ_{ci} are the complex amplitude function and delay of the ith clutter (i=1,…,N_{C}), that is N_{C} is the number of clutters. All delays are calculated with respect to the direct signal. n_{sur}(t) is the thermal noise contribution at the receiver antenna.
where A_{ref} is a complex amplitude and n_{ref}(t) is the thermal noise contribution at the reference antenna.
where s_{sur}[ i] and s_{ref}[ i] denote s_{sur}(t_{i}) and s_{ref}(t_{i}), respectively. Consider that the discrete delay l corresponds to the delay T[ l]=lT_{s} and R is maximum delay bin of clutter. Similarly, the discrete Doppler frequency bin p, corresponds to the Doppler frequency f_{d}[ p]=p/(NT_{s}) and P is maximum Doppler bin of clutter.
3 Clutter and direct signal cancellation
In this section, first we introduce two known algorithms ECA and SCA for clutter and direct signal cancellation in passive bistatic radars. Then the proposed algorithm is presented.
3.1 Extensive cancellation algorithm (ECA)
The computational complexity of the ECA algorithm is \(\mathcal {O}(NM^{2}+M^{3})\). This complexity is high because the estimation of vector θ requires the inversion of the matrix H^{H}H with dimensions M×M.
3.2 Sequential cancellation algorithm (SCA)
Aiming at reducing the computational load of the ECA algorithm described in Section 3.1, a sequential solution algorithm has been offered in [14] for clutter and direct signal cancellation, called SCA.
- Start with initial equations as:$$\begin{array}{@{}rcl@{}} \mathbf{P}_{M}=\mathbf{I}_{N}, \end{array} $$(10)$$\begin{array}{@{}rcl@{}} \bar{\mathbf{s}}_{\text{sur}}^{(M)}=\mathbf{s}_{\text{sur}}. \end{array} $$(11)
- Then, the output vector of SCA algorithm is obtained by implementing the below recursive equations for i=M,…,2,1 respectively:$$\begin{array}{@{}rcl@{}} \bar{\mathbf{x}}_{j}^{(i)}=\mathbf{P}_{i}\mathbf{x}_{j}\quad for \quad j=0, 1, \dots, i-1, \end{array} $$(12)$$\begin{array}{@{}rcl@{}} \mathbf{Q}_{i}=\left[\mathbf{I}_{N}-\frac{\bar{\mathbf{x}}_{{i-1}}^{({i})}{\bar{\mathbf{x}}_{{i-1}}^{({i})\text{H}}}}{{{\bar{\mathbf{x}}_{{i-1}}^{({i})\text{H}}}}\bar{\mathbf{x}}_{{i-1}}^{({i})}}\right], \end{array} $$(13)$$\begin{array}{@{}rcl@{}} \mathbf{P}_{{i-1}}=Q_{i}P_{i} \end{array} $$(14)
- In each step of the above equations, the received signal can be improved one level as:$$ \mathbf{s}_{\text{sur}}^{({i-1})}=\mathbf{P}_{{i-1}}\mathbf{s}_{\text{sur}}=\mathbf{Q}_{i}{\mathbf{s}_{\text{sur}}^{({i})}}. $$(15)
- After finishing the above loop, by using the final projection matrix P_{0}, the output vector s_{SCA} is obtained as follows:$$ \mathbf{s}_{\text{SCA}}=\mathbf{s}_{\text{sur}}^{({0})}=\mathbf{P}_{{0}}\mathbf{s}_{\text{sur}}. $$(16)
Almost all steps of a SCA algorithm have been shown in this figure. It is possible to limit the computational of the cancellation algorithm by arresting it after stage S (S<M). The computational complexity of the SCA algorithm limited to S stage is \(\mathcal {O}(NMS)\), which can be significantly smaller than the computational cost of the corresponding complete ECA algorithm.
3.3 Sequential cancellation batch (SCB) algorithm
In order to improve the cancellation performance with a limited computational load, a modification of the SCA is proposed which is called SCB. The received signal at the surveillance antenna is divided into sections with length T_{B}. If the entire length of the surveillance antenna signal is T_{int}, the total number of samples of the signal at the antenna will be N=⌊T_{int}f_{s}⌋, where f_{s} is the sampling frequency. The signal is divided into b packets with N_{B}=⌊N/b⌋ available samples. First, the SCA algorithm is applied to each of these packets distinctly. The output of the SCA algorithm on each packet is a vector removed of the clutter and direct signal. Then, the main cleaned vector is obtained from the union of these sub-vectors. Finally, the main vector can be used for computing and plotting the ambiguity diagram and target detection.
where ξ(τ_{d},f_{d}) is the ambiguity function of \( s_{\text {sur}}^{j}(t)\) at position (τ_{d},f_{d}) and η is a small value selected between zero and one in our simulations.
The computational complexity of the SCB algorithm in each batch is \(\mathcal {O}(N_{B}MS)\). This means that the SCB algorithm requires lesser memory than the ECA and SCA algorithms. When the SCB algorithm is run on b batches, the computational complexity will be \(\mathcal {O}({bN}_{B}MS)\) which is equal to the SCA algorithm because bN_{B} equals N.
We remind that the computational complexity of ECA-B method is \(\mathcal {O}\left (NM^{2}+M^{3}\right)\) similar to the ECA method; but its required memory is \(\mathcal {O}\left (N_{B}M^{2}+M^{3}\right)\) which is less than that of ECA. Anyway, both computational complexity and required memory of the proposed SCB method are considerably less than those of ECA-B method.
4 Results and discussion
Clutter echo parameters in scenario #1
Clutter | #1 | #2 | #3 | #4 | #5 | #6 | #7 | #8 | #9 |
---|---|---|---|---|---|---|---|---|---|
Delay (ms) | 0.05 | 0.1 | 0.15 | 0.2 | 0.25 | 0.1 | 0.17 | 0.22 | 0.25 |
Doppler (Hz) | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
CNR (dB) | 40 | 30 | 20 | 10 | 5 | 27 | 18 | 8 | 5 |
Target echo parameters in scenario #1
Target | #1 | #2 | #3 |
---|---|---|---|
Delay (ms) | 0.3 | 0.5 | 0.6 |
Doppler (Hz) | –50 | 100 | 50 |
SNR (dB) | 4 | 2 | –10 |
Selected parameters for simulation of the SCB algorithm
Observation time | T_{int} | 1 s |
---|---|---|
Sampling time | T_{s} | 0.005 ms |
Number of batch | b | 10 |
In the following, three tests are introduced for evaluation of SCB in comparison with ECA, SCA and ECA-B algorithms.
4.1 Evaluation using CA and TA tests
Clutter and target parameters in scenario #2 for calculation of CA and TA in SCB, ECA and SCA algorithms
Delay (ms) | Doppler (Hz) | SNR (dB) | |
---|---|---|---|
Clutter1 | 0.25 | 0 | 25 |
Clutter2 | 0.25 | Exponential Spectrum between −1 and 1 Hz | 25 |
Target | 0.6 | 100 | 5 |
4.2 Evaluation using CFAR target detection
In this section, in order to evaluate the proposed algorithm based on the target detection criteria, we use a CA-CFAR detector after clutter and direct signal cancellation using the mentioned algorithms. We use receiver operating characteristic (ROC) curves for detection performance comparison. In this manner, first, clutter and direct signal are removed by the SCB (or ECA and ECA-B) algorithm, and then targets are detected based on the output of ambiguity function and CA-CFAR detector. The detectors based on the ECA, ECA-B and SCB algorithms are called ECA-CA, ECA-B-CA and SCB-CA, respectively.
Clutter and target parameters in scenario #2 for calculation of CA and TA in SCB, ECA, ECA-B and SCA algorithms
Target T_{1} | Target T_{2} | |
---|---|---|
Delay (ms) | 0.1 | 0.9 |
Doppler (Hz) | 10 | 50 |
It is observed that the ROC of SCB and ECA-B is similar, and when the clutter has an exponential spectrum SNR is reduced to 4 dB in both methods. It is seen that by decreasing the number of batch in the SCB algorithm, the performance of the SCB-CA improves so that the ROC of SCB-CA with b = 10 is close to the ROC of ECA for both targets T_{1} and T_{2}. This means the SCB-CA detector performs similar to the ECA-CA detector if the number of batch is low. Consider that the SCB-CA has less computational complexity than that of ECA-CA. The ECA-CA and SCB-CA detectors degrade if Doppler frequency of target tends to be 0 Hz.
5 Conclusions
In this paper, the SCB algorithm is proposed for cancellation of static and non-static clutters as well as elimination of direct signal component in passive bistatic radars based on projections of the received signals onto a subspace orthogonal to the signal subspace of the clutter and the subspace of the previously detected targets. The SCB algorithm is first used for clutter and direct signal cancellation and detection of strong targets. To enhance the detection performance, the observation algorithm is then investigated and applied for detection of targets with weak signals. The simulation results revealed that the SCB algorithm performers well in the detection of targets compared with the state-of-the-art methods. The TA, CA and CFAR detection tests were used for comparing the SCB with the ECA, ECA-B and SCA algorithms. These tests showed that targets may hide in the ambiguity function when the number of batches increases. The SCB algorithm has lesser computational complexity than the ECA and ECA-B algorithms. Moreover, the proposed method requires lesser memory than these algorithms and the SCA method.
Notes
Competing interests
The authors declare that they have no competing interests.
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