# Removal of Cochannel Interference Using Probabilistic Latent Component Analysis in Passive Bistatic Radar

Authors: Tao Ying; Xuebao Wang; Wei Tian; Cheng Zhou
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper examines the problem of cancellation of cochannel interference (CCI) present in the same frequency channel as the signal of interest, which may bring a reduction in the performance of target detection, in passive bistatic radar. We propose a novel approach based on probabilistic latent component analysis for CCI removal. The highlight is that removing CCI is considered as reconstruction, and extraction of Doppler-shifted and time-delayed replicas of the reference signal exploited fully as training data. The results of the simulation show that the developed method is effective.


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## Body

## 1. Introduction

Recently, there has been a growing interest in passive bistatic radar (PBR) exploiting illuminators of opportunity [1–5]. Without the need for a dedicated transmitter’s deployment and operation, PBR systems are significantly less expensive to implement and operate than their conventional counterparts. Due to their bistatic or multistatic configuration, PBR systems provide RCS advantages to counter stealth targets. Also, PBR systems have several advantages over conventional monostatic radars, including smaller in size, needing no additional frequency channel allocation since no dedicated transmitter is used, and immune to antiradiation missiles. Further, employing PBR systems in a multistatic configuration provides spatial diversity similar to active MIMO radar systems [6].Many typical illuminators have already been employed for target detection in passive radars including FM radio [7, 8], Digital Audio Broadcast (DAB) [9], Digital Video Broadcast (DVB) [10–13], satellite signal [14], Universal Mobile Telecommunications System (UMTS) [15, 16], and Global System for Mobile communication (GSM) [17, 18]. In addition, several studies have looked at the use of one potential illuminator of opportunity, rapidly growing in coverage, which is related to wireless local area networks (LAN) [19–22]. Zhao et al. [23] employ the radio in HF band (3–30 MHz) to achieve wide-area moving target detection and ocean remote sensing. In this paper, we will consider DVB-T transmitters as illuminators of opportunity. They have ubiquitous spatial coverage, are permanent in time, and have a thumbtack-like function due to the noise-like behavior of the orthogonal frequency division multiplexing (OFDM) modulation used. It should be noted that the developed approach in this paper can be applied to any transmission of opportunity.The weak signal reflected from the target is received by PBR’s surveillance channel with a reference channel using a directive antenna steered towards the exploited illuminator of opportunity. Usually, the target echoes may be masked by the small fraction of the direct signal received by the side lobe of the antenna of the surveillance channel, strong clutter, or multipath echoes, all of which can be considered as Doppler-shifted and time-delayed replicas of the reference signal from the transmitter of opportunity, as well as injected with CCI from other nuisance illuminating sources such as intended jamming and involuntary disturbance. Indeed, weak target echoes may be masked by the other echoes from other targets of a higher level in large range-Doppler separations. Therefore, it is very significant to remove the disturbance for target detection, and there are increasing studies focusing on the solution to this problem. Kulpa and Czekała [24] present a method of adaptive removal of strong echoes from the received signal for the case when strong echoes shadow a distant, weak target echo. Raout et al. [12] achieve space-time clutter rejection and target passive detection using the amplitude and phase estimation method. An algorithm exploiting multichannel adaptive filters for sea clutter cancellation in PBR is proposed in [25]. Colone et al. [26] examine the problem of cancellation of direct signal, multipath, and clutter echoes in PBR and propose a multistage processing algorithm for disturbance removal and target detection. However, most of the above literature studies mainly focus on the cancellation of the direct signal, strong clutter or multipath echoes, and the echoes from other vital targets, without taking into account the problem of removal of CCI in a PBR due to spectrum congestion in the frequency range from VHF to 6 GHz in the complex electromagnetic environment. CCI from other transmitters may enter the surveillance channel from the main lobe or side lobe of the receiving antenna and degrade the performance of target detection of PBR systems. To better play a role in the future war, it is equally crucial for PBR systems to preprocess signals and achieve CCI’s cancellation from other nuisance transmitters. Therefore, we focus on the solution to this problem.In recent years, the latent variable model has been a subject of research. Scholars have proposed a variety of latent variable models [27–29]. Probabilistic latent component analysis (PLCA) is discussed and applied to the separation of sounds from single-channel mixtures in [30] and radar signal detection in [31]. Likewise, the cancellation of CCI can be considered as the separation of the desired signal from the surveillance channel mixtures in a PBR. Moreover, the reference signal can be used as training data for PLCA.Consequently, in order to make full utilization of the reference signal without increasing the complexity of the receiving system, PLCA is employed for the removal of CCI. First, in this paper, the problem of disturbance removal in the complex electromagnetic environment is proposed. Below, the PLCA method for signal preprocessing to achieve the cancellation of CCI is developed, which explicitly models the collected signal as a mixture of marginal distribution products. The estimation of the most appropriate marginal distributions, performed using the expectation-maximization algorithm, is employed selectively to reconstruct the desired signal, and eventually, the CCI suppression is achieved.The paper is organized as follows. Section2 describes the addressed problem. PLCA is introduced, and the PLCA method for the removal of CCI is developed in Section 3. Section 4 gives the simulation results. Finally, Section 5 provides the conclusion.


## 2. The Problem

The geometry of PBR is depicted schematically in Figure1. The reference and surveillance antennas are assumed to be colocated with the reference antenna steered toward the transmitter and the surveillance antenna pointed in the direction to be surveyed. The reference channel receives the direct signal from the transmitter, which is used to cancel the multipath at the surveillance channel and evaluate the delay-Doppler cross-correlation function (2D-CCF) between the surveillance and the reference signal.(1)χτ,v2=∫−∞+∞xts·t−τexpj2πvtdt2,where xt is the target echo signal, st is the reference signal, and v and τ are the Doppler frequency shift and the sample time-delay parameters, respectively.Figure 1
PBR geometry.In practice, the surveillance channel collects a mixture of Doppler-shifted and time-delayed replicas of the reference signal from the transmitter of opportunity, involving target echoes, direct signal, strong clutter, or multipath echoes, and the CCI signals from other nuisance illuminators. Different approaches have been proposed to cope with direct signal and multipath [12, 24, 26, 32]. However, with respect to the cancellation of direct signal and multipath, CCI rejection requires different techniques for preprocessing before matched filtering. Otherwise, detecting targets’ performance and capabilities, incredibly weak targets, will degrade a lot in a PBR. The PLCA method for signal preprocessing to remove the CCI in a PBR will be discussed in what follows.


## 3. The Probabilistic Latent Component Analysis Method

### 3.1. Probabilistic Latent Component Analysis

Probabilistic latent component analysis (PLCA) is a straightforward extension of probabilistic latent semantic indexing (PLSI) [33] which deals with an arbitrary number of dimensions. The basic model is defined as follows [30]:(2)Px=∑zPz∏j=1NPxj|z,where Px denotes a distribution over the N-dimensional random variable x and xj is j’th dimension. z is a latent variable, and Pxj|z is one-dimensional distribution. Effectively this model represents a mixture of marginal distribution products to approximate an N-dimensional distribution. Subsequent processing is to discover the most appropriate marginal distributions. To perform the estimation of the marginals Pxj|z, the EM algorithm is considered.(i)
E step: as seen from (2), Pxj|z is independent of each other:(3)Px|z=∏j=1NPxj|z.In this step, the posterior probability of the latent variablez is estimated as follows:(4)Pz|x=Pz∏j=1NPxj|z∑z′Pz′∏j=1NPxj|z′.(ii)
M step: a newz distribution is obtained as follows:(5)Pz=∫PxPz|xdx.Define(6)P∗xj|z=∫⋯∫PxPz|xdxk,∀k≠j.A new and more accurate estimation ofz is obtained as follows:(7)Pxj|z=P∗xj|zPz.A converging solution for the marginalsPxj|z along each dimension j and the latent variable priors Pz will be obtained by repeating the above steps in an alternating manner multiple times. It should be noted that Px and z are both discrete in practical application.


### 3.2. The Probabilistic Latent Component Analysis Method

Actually, we can consider the removal of CCI as reconstruction and extraction of Doppler-shifted and time-delayed replicas of the reference signal from the mixture received in the surveillance channel. Moreover, the reference channel offers us a source of training data for PLCA without increasing the complexity of the receiving system. As a consequence, PLCA is modified and employed for the removal of CCI. The PLCA method for the removal of CCI explicitly models the collected signal as a mixture of marginal distribution products. The estimation of the most appropriate marginal distributions, performed using the expectation-maximization algorithm, is employed selectively to reconstruct and extract the desired signal. Eventually, the disturbance suppression is achieved.To begin with, perform the short-time Fourier transform of a signalst:(8)STFTst,f=∫−∞∞sτγ∗τ−te−j2πfτdτ.The spectrogram ofs(t) can be treated as the distribution of signal energy over time and frequency domain, P^f,t=absSTFTst,f.Proper normalization will make the spectrogram of signal a distribution of energy across the time-frequency planePf,t. Adopting this view allows us to employ PLCA directly for analysis of the signal. The 2D model of PLCA is defined as follows:(9)Pf,t=∑zPzPf|zPt|z,where Pf,t is a magnitude spectrogram of signal and Pf|zand Pt|zare frequency marginals and time marginals, respectively. The marginals will contain a dictionary of spectra which can best describe the signal represented by the input spectrogram. The application of PLCA on the spectrogram of a drum loop and its corresponding marginals are shown in Figure 2. Reconstruction of different frequency marginals and time marginals will lead to extraction of different components of the drum loop.Figure 2
Application of PLCA on the spectrogram of a drum loop.Notice that these marginals can be used as a model of a class of acoustic signals. Likewise, we can model the signal received by the surveillance channel as these marginals. Next, we will apply the PLCA to the CCI removal. The total collected signal in the surveillance channel can be given by(10)xt=Asurvdt+∑m=1Mamdt−τmej2πfdmt+∑i=1Ncidt−τci+∑k=1Kjkt+nsurvt,where dt is the complex envelope of the direct signal (a delayed replica of the transmitted signal from the illuminator of opportunity); Asurvis the complex amplitude of the direct signal received via the side/back lobe of the surveillance; am, τm, and fⅆm are the complex amplitude, the delay (with respect to the direct signal), and the Doppler frequency of the mth target (m=1,…,M); ci and τci are the complex amplitude and the delay (with respect to the direct signal) of the ith stationary ground scatter (i=1,…,N); jkt is the kth CCI contribution from other nuisance illuminators (j=1,…,K); and nsurvt is the thermal noise contribution at the surveillance antenna.Further, we define(11)St=Asurvdt+∑m=1Mamdt−τmej2πfdmt+∑i=1Ncidt−τci+nsurvt,(12)Jt=∑k=1Kjkt.After substitution of (11) and (12) into (10), we have(13)xt==St+Jt.The complex envelope of the signal at the reference channel is(14)sreft=Arefdt+nreft,where Arefis the complex amplitude and nreft is the thermal noise contribution at the reference antenna. Since the direct signal is received by the main lobe of the reference antenna, for simplification of analysis it is assumed that the reference signal is free of multipath. It is also noted that the effect of multipath on the reference channel has been analyzed in [34] where a potential algorithm for its removal has been introduced.The distributionPxf,t of signal energy across time-frequency plane will be obtained by performing the short-time Fourier transform of xt. The relevance between sreft and St implies that the data from the reference channel can be utilized as training data. The posterior probability of the latent variable z is estimated as follows:(15)Psz|f,t=PszPsf|zPst|z∑z′Pz′Psf|z′Pst|z′.We can estimate the frequency marginalsPsf|z to reconstruct St by performing the EM algorithm mentioned above. According to (11), (12), and (13), the collected signal in the surveillance channel can be treated as a mixture of Doppler-shifted and time-delayed replicas of the reference signal from the transmitter of opportunity and the CCI signals from other nuisance illuminating sources. Therefore, the removal of CCI can be considered as the equivalent of reconstruction and extraction of Doppler-shifted and time-delayed replicas of the reference signal from the mixture. Thus, our objective is to discover a dictionary of PJf|z and PJt|z to describe the source of CCI. Simultaneously, the solution of the time marginals Pst|z will be used to reconstruct St.To solvePJf|z, PJt|z, and Pst|z, extend the frequency marginals of the training data(16)Pxf|z=PSf|zPJf|z.For a fixedz,(17)Pz|f,t=PzPxf|zPxt|z∑z′Pz′Pxf|z′Pxt|z′.The new latent variable prior is obtained as follows:(18)Pz=∫∫Pxf,tPz|f,tdfdt.Define(19)Px∗f|z=∫Pxf,tPz|f,tdt,Px∗t|z=∫Pxf,tPz|f,tdf.Compute the new marginals(20)Pxf|z=Px∗f|zPz=Psf|zPJf|z,(21)Pxt|z=Px∗t|zPz=Pst|zPJt|z.Repeating (17), (18), (20), and (21) in an alternating manner multiple times yields a converging solution for PJf|z, PJt|z, and Pst|z. Reconstruction of the signal is obtained as follows:(22)Psf,t=∑zPsf|zPzPst|z.As a consequence, the CCI from other nuisance illuminators will be removed.


## 3.1. Probabilistic Latent Component Analysis

Probabilistic latent component analysis (PLCA) is a straightforward extension of probabilistic latent semantic indexing (PLSI) [33] which deals with an arbitrary number of dimensions. The basic model is defined as follows [30]:(2)Px=∑zPz∏j=1NPxj|z,where Px denotes a distribution over the N-dimensional random variable x and xj is j’th dimension. z is a latent variable, and Pxj|z is one-dimensional distribution. Effectively this model represents a mixture of marginal distribution products to approximate an N-dimensional distribution. Subsequent processing is to discover the most appropriate marginal distributions. To perform the estimation of the marginals Pxj|z, the EM algorithm is considered.(i)
E step: as seen from (2), Pxj|z is independent of each other:(3)Px|z=∏j=1NPxj|z.In this step, the posterior probability of the latent variablez is estimated as follows:(4)Pz|x=Pz∏j=1NPxj|z∑z′Pz′∏j=1NPxj|z′.(ii)
M step: a newz distribution is obtained as follows:(5)Pz=∫PxPz|xdx.Define(6)P∗xj|z=∫⋯∫PxPz|xdxk,∀k≠j.A new and more accurate estimation ofz is obtained as follows:(7)Pxj|z=P∗xj|zPz.A converging solution for the marginalsPxj|z along each dimension j and the latent variable priors Pz will be obtained by repeating the above steps in an alternating manner multiple times. It should be noted that Px and z are both discrete in practical application.


## 3.2. The Probabilistic Latent Component Analysis Method

Actually, we can consider the removal of CCI as reconstruction and extraction of Doppler-shifted and time-delayed replicas of the reference signal from the mixture received in the surveillance channel. Moreover, the reference channel offers us a source of training data for PLCA without increasing the complexity of the receiving system. As a consequence, PLCA is modified and employed for the removal of CCI. The PLCA method for the removal of CCI explicitly models the collected signal as a mixture of marginal distribution products. The estimation of the most appropriate marginal distributions, performed using the expectation-maximization algorithm, is employed selectively to reconstruct and extract the desired signal. Eventually, the disturbance suppression is achieved.To begin with, perform the short-time Fourier transform of a signalst:(8)STFTst,f=∫−∞∞sτγ∗τ−te−j2πfτdτ.The spectrogram ofs(t) can be treated as the distribution of signal energy over time and frequency domain, P^f,t=absSTFTst,f.Proper normalization will make the spectrogram of signal a distribution of energy across the time-frequency planePf,t. Adopting this view allows us to employ PLCA directly for analysis of the signal. The 2D model of PLCA is defined as follows:(9)Pf,t=∑zPzPf|zPt|z,where Pf,t is a magnitude spectrogram of signal and Pf|zand Pt|zare frequency marginals and time marginals, respectively. The marginals will contain a dictionary of spectra which can best describe the signal represented by the input spectrogram. The application of PLCA on the spectrogram of a drum loop and its corresponding marginals are shown in Figure 2. Reconstruction of different frequency marginals and time marginals will lead to extraction of different components of the drum loop.Figure 2
Application of PLCA on the spectrogram of a drum loop.Notice that these marginals can be used as a model of a class of acoustic signals. Likewise, we can model the signal received by the surveillance channel as these marginals. Next, we will apply the PLCA to the CCI removal. The total collected signal in the surveillance channel can be given by(10)xt=Asurvdt+∑m=1Mamdt−τmej2πfdmt+∑i=1Ncidt−τci+∑k=1Kjkt+nsurvt,where dt is the complex envelope of the direct signal (a delayed replica of the transmitted signal from the illuminator of opportunity); Asurvis the complex amplitude of the direct signal received via the side/back lobe of the surveillance; am, τm, and fⅆm are the complex amplitude, the delay (with respect to the direct signal), and the Doppler frequency of the mth target (m=1,…,M); ci and τci are the complex amplitude and the delay (with respect to the direct signal) of the ith stationary ground scatter (i=1,…,N); jkt is the kth CCI contribution from other nuisance illuminators (j=1,…,K); and nsurvt is the thermal noise contribution at the surveillance antenna.Further, we define(11)St=Asurvdt+∑m=1Mamdt−τmej2πfdmt+∑i=1Ncidt−τci+nsurvt,(12)Jt=∑k=1Kjkt.After substitution of (11) and (12) into (10), we have(13)xt==St+Jt.The complex envelope of the signal at the reference channel is(14)sreft=Arefdt+nreft,where Arefis the complex amplitude and nreft is the thermal noise contribution at the reference antenna. Since the direct signal is received by the main lobe of the reference antenna, for simplification of analysis it is assumed that the reference signal is free of multipath. It is also noted that the effect of multipath on the reference channel has been analyzed in [34] where a potential algorithm for its removal has been introduced.The distributionPxf,t of signal energy across time-frequency plane will be obtained by performing the short-time Fourier transform of xt. The relevance between sreft and St implies that the data from the reference channel can be utilized as training data. The posterior probability of the latent variable z is estimated as follows:(15)Psz|f,t=PszPsf|zPst|z∑z′Pz′Psf|z′Pst|z′.We can estimate the frequency marginalsPsf|z to reconstruct St by performing the EM algorithm mentioned above. According to (11), (12), and (13), the collected signal in the surveillance channel can be treated as a mixture of Doppler-shifted and time-delayed replicas of the reference signal from the transmitter of opportunity and the CCI signals from other nuisance illuminating sources. Therefore, the removal of CCI can be considered as the equivalent of reconstruction and extraction of Doppler-shifted and time-delayed replicas of the reference signal from the mixture. Thus, our objective is to discover a dictionary of PJf|z and PJt|z to describe the source of CCI. Simultaneously, the solution of the time marginals Pst|z will be used to reconstruct St.To solvePJf|z, PJt|z, and Pst|z, extend the frequency marginals of the training data(16)Pxf|z=PSf|zPJf|z.For a fixedz,(17)Pz|f,t=PzPxf|zPxt|z∑z′Pz′Pxf|z′Pxt|z′.The new latent variable prior is obtained as follows:(18)Pz=∫∫Pxf,tPz|f,tdfdt.Define(19)Px∗f|z=∫Pxf,tPz|f,tdt,Px∗t|z=∫Pxf,tPz|f,tdf.Compute the new marginals(20)Pxf|z=Px∗f|zPz=Psf|zPJf|z,(21)Pxt|z=Px∗t|zPz=Pst|zPJt|z.Repeating (17), (18), (20), and (21) in an alternating manner multiple times yields a converging solution for PJf|z, PJt|z, and Pst|z. Reconstruction of the signal is obtained as follows:(22)Psf,t=∑zPsf|zPzPst|z.As a consequence, the CCI from other nuisance illuminators will be removed.


## 4. Simulation Results

In this section, the developed PLCA method’s performance is verified using DVB-T signals in 2k mode for simulations. The numerical values for the OFDM parameters are listed in Table1. The number of OFDM symbols is 30. Figures 3(a)–3(d) show the spectrum, 2D autoambiguity function, and the 2D autoambiguity function’s major cuts at zero range and zero Doppler. The DVB-T-modulated signal achieves a fair range resolution due to its high bandwidth. Besides, some side peaks in the 2D autoambiguity function of a DVB-T-modulated signal can be suppressed by the developed algorithm in [35], thus yielding a reduction in false alarms.Table 1
Numerical values for the OFDM parameters for the 2k mode.
Parameter2k modeSampling frequency64/7 MHzNumber of carriers1705Bandwidth7.61 MHzDuration of symbol part224μsDuration of guard interval14μsFigure 3
Spectrum and ambiguity function of DVB-T signal. (a) Spectrum; (b) 2D autoambiguity function; (c) zero range cut of autoambiguity function; (d) zero Doppler cut of autoambiguity function.
(a)(b)(c)(d)For emphasis on evaluating the performance and capabilities of CCI removal of the PLCA method, we simplify the scenario of simulation where two targets with the identical signal-to-noise ratio (SNR) of 5 dB are placed at (15 km, 600 Hz) and (18 km, −300 Hz), respectively, in the range-Doppler domain in the absence of direct signal and multipath clutter which can be cancelled by the examined approach in [32] in the subsequent processing.Assume that there are two nuisance sources, transmitting noise-amplitude-modulated signal and noise-frequency-modulated signal, respectively, and both the ratios of powers of target echo signal and CCI (SIR) are −30 dB. If we do not take measures to remove the disturbance in the first step using some method, the subsequent target detection will be affected severely. The comparison of Figures4 and 5 confirms this point. The two target echoes are entirely submerged in the background of CCI from other nuisance illuminators without removal.Figure 4
2D-CCF in the absence of CCI from other nuisance illuminators.Figure 5
2D-CCF in the presence of CCI from other nuisance illuminators.Now, the PLCA method is utilized for the removal of CCI from other nuisance illuminators. The reference signal is exploited as training data modeled using 20 marginals while the collected signal in the surveillance channel was modeled using 40 marginals. The number of iterations for the EM algorithm is 200. Figure6 shows the target echos, the corrupted signal in the surveillance channel, and the reconstruction and extraction of the desired signal by using the PLCA method, respectively, from top to bottom when SIR = −30 dB, SNR = 5 dB, and SNRre = 5 dB (the ratio of powers of the reference signal and noise). The 2D-CCF between the reconstructed signal by using the PLCA method and the reference signal indicates that the CCI from other nuisance illuminators is removed effectively, and the target echoes stand out in Figure 7.Figure 6
Target echos, corrupted signal, and reconstruction of the desired signal, respectively, from top to bottom with SIR = −30 dB, SNR = 5 dB, and SNRre = 5 dB.Figure 7
2D-CCF with disturbance removal using the PLCA method.For further evaluation of the proposed PLCA method’s performance in comparison with traditional methods, we employ the methods based on LMS adaptive filter and wavelet transform for CCI removal. Wavelet decomposition is performed at level 2 with a selection of heursure threshold and the orthogonal wavelet sym8 is adopted. We specify the order and step size of the LMS filter as 5 and 0.001, respectively. Besides, we define the signal-to-distortion ratio (SDR) as follows:(23)SDR=10log10st22st−s′t22,where st is the desired signal and s′t is the reconstruction of st. The comparison of CCI removal performance of the PLCA method regarding traditional methods, including LMS adaptive filter and wavelet transform, is depicted schematically in Figure 8. Obviously, the PLCA method is superior to the LMS method and the wavelet method. Moreover, the traditional methods can barely work to remove CCI and yield contribution to target detection, significantly when CCI exceeds target echoes over 20 dB in power. Figure 9 shows that a wide variety in SNR does not bring a significant change in SDR, while SDR changes sensitively with SNRre for the PLCA method. That is because the reference signal is exploited as training data for modeling and reconstruction of the desired signal in the corrupted signal in the surveillance channel. The purity of the reference signal is significant for the effectiveness of the PLCA method.Figure 8
Comparison of SDR versus SIR with different methods.Figure 9
SDR versus SNRre with different SNRs when SIR = −25 dB.


## 5. Conclusions

In this paper, the problem of removal of CCI from other nuisance illuminators was investigated in a PBR. The PLCA method was proposed and treated the removal of CCI as the reconstruction and extraction of Doppler-shifted and time-delayed replicas of the reference signal. Finally, some methods were compared by simulation, and it was demonstrated that the proposed PLCA method was effective and superior to the other methods.



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