Low-cost solutions for mobile passive radar based on multichannel DPCA and NULA configurations

Abstract

In this paper, we investigate low-cost solutions for enabling ground moving target indication applications with multichannel mobile passive radar systems. As known, in order to be competitive with their active counterparts, passive radars are typically characterized by severe constraints in terms of cost, complexity, and compactness, especially when installed on moving platforms. On the one hand, carrying out the computations onboard requires processing techniques as simple as possible. On the other hand, the need for lightweight and compact systems that can be installed on a moving platform requires using a limited number of receiving channels. To meet these requirements, we propose a series of nonadaptive detectors based on multichannel displaced phase center antennas, which allow suppressing the Doppler-spread clutter component without requiring computationally intensive space–time adaptive processing techniques. Moreover, we explore the use of nonuniformly spaced array configurations on receive, which represent a good alternative to conventional uniform linear arrays when a limited number of receiving channels can be implemented. The effectiveness of the proposed processing techniques and antenna design solutions is demonstrated via numerical analysis for the case of a DVB-T-based mobile passive radar system.

Introduction

In the last years, there has been significant interest in passive radar systems installed on moving platforms, as evidenced by several studies appeared in the open literature (e.g., see [1–13]). As is well known, passive radar is a receive-only system that allows to detect targets by parasitically exploiting existing transmitters as illuminators of opportunity. This principle of operation brings in a number of advantages including reduced complexity, low probability of intercept, and ease of deployment, [14, 15]. The use of passive radar onboard airborne or ground moving platforms provides enhanced strategic advantages by extending the functionalities of passive radar to applications such as synthetic aperture radar (SAR) imaging [2–4] or ground moving target indication (GMTI) [5–13].

Such systems pose stimulating challenges in terms of both system design and processing techniques. One of the major concerns is that the detection of slow-moving targets is severely degraded by the platform motion, which induces an angle-dependent Doppler component in the clutter signal. To tackle this issue, space–time clutter suppression techniques involving platform motion compensation must be exploited. Typically, the spatial information is provided by the $N$ elements of a multichannel receiving antenna, while the $M$ pulses in the coherent processing interval (CPI) serve as temporal information.

The easiest space–time clutter suppression technique is displaced phase center antennas (DPCAs), which requires an antenna with just two receiving elements [16]. The idea is to compensate the platform motion by subtracting the signals received by the two antenna elements at the time instants in which their phase centers occupy the same positions. The echoes from moving targets are preserved due to their own radial motion, while the contribution from stationary scatterers cancels out. Clearly, this approach requires the pulse repetition frequency (PRF) to be synchronized with the platform velocity, so that the second antenna occupies the position of the first one after an integer number of pulses.

A more advanced approach involving adaptive space–time clutter suppression was formerly introduced by Brennan and Reed in [17] back in 1973 and is usually referred to as space–time adaptive processing (STAP). In the following years, the research on STAP has drawn the attention of several experts in the field (e.g., see [18–20]). STAP is essentially a generalization of DPCA in which the pulses received by an arbitrary number of receiving elements are combined adaptively, based on the statistical properties of clutter scene. Thanks to its adaptivity, STAP is generally considered more robust than DPCA to nonideal conditions, such as rapid changes in the clutter scene or platform velocity perturbations. Moreover, the higher number of receiving elements provides additional degrees of freedom, which is beneficial to the detection of slow-moving targets. This is especially true when the antennas are positioned strategically, resorting to nonuniform sensor distributions.

However, we note that the improved performance of STAP is achieved at the expenses of a drastically higher computational load. As a matter of fact, STAP is based on the estimation and inversion of the clutter covariance matrix (CCM), which makes practical implementation challenging, especially when a high number of antennas and pulses is used. To tackle this issue, several reduced-order suboptimal STAP techniques have been proposed, which are usually referred to as partially adaptive STAP. An extensive review of fully adaptive STAP and its suboptimal variants can be found in [18].

Although STAP techniques are well-established and their effectiveness has been extensively demonstrated in the open literature, their high complexity does not make them best suited to mobile passive radar applications. Specifically, when dealing with passive radars installed on moving platforms, the designer should always recall that:

1. (1) the simplest processing schemes should generally be favored in order to guarantee that the required computations can be carried out onboard;

2. (2) key features such as weightlessness and compactness should be pursued, being crucial for the design of easily deployable systems, which can eventually be installed on a moving platform.

To properly address these requirements, in this paper, we work on two different but entwined fronts. On the one hand, the need for simple processing schemes suggests avoiding adaptive filtering techniques altogether, favoring nonadaptive ones. In this perspective, a multichannel DPCA-like solution might represent a convenient, undemanding alternative to reduced-order STAP approaches. On the other hand, the need to preserve ease of deployment fosters the use of lightweight and compact systems that in turn allow a limited number of receiving channels to be featured. To this end, we explore the use of radar receivers based on nonuniform linear array (NULA) configurations, as a way to reduce the number of receiving channels without compromising performance.

Preliminary results along the lines described above have been reported in our previous studies [11–13]. Specifically, in [11], we introduced a nonadaptive solution, referred to as “apodization approach,” in which we exploited three nonuniformly spaced receiving antennas installed on a moving platform to detect moving targets against clutter and noise.

In this paper, we significantly extend the results in [11] by deriving a series of low-complexity nonadaptive detectors based on different signal models, effectively implementing DPCA schemes possibly operating with any number of channels on receive. Furthermore, we characterize the performance of each detector in terms of false alarm probability and detection probability, via both theoretical and simulated analyses. Finally, the advantages and the limitations of nonuniform sensors spacings are investigated when exploiting the proposed detectors with different array layouts.

The remainder of this paper is organized as follows: in “System geometry and signal model”, we present the assumed system geometry and the signal model adopted. The multichannel DPCA detectors are introduced in “Nonadaptive multichannel DPCA”, and their target detection performance against clutter and noise are evaluated and compared. In “NULA configurations”, these detectors are tested considering both ULA (uniform linear array) and NULA configurations, to point out the advantages of nonuniform sensor spacings. In “Case study: passive radar system based on DVB-T”, the proposed NULA-based multichannel DPCA solutions are tested in a simulated environment, considering the case study of a passive radar system parasitically exploiting digital video broadcasting – terrestrial (DVB-T) transmissions as signals of opportunity. Finally, in “Conclusions”, we draw our conclusions.

System geometry and signal model

In this section, we introduce the assumed system geometry and signal model, constituting the foundation for the derivation of the multichannel DPCA detectors.

System geometry and reference scenario

The system geometry is sketched in Fig. 1. We consider a passive radar system parasitically exploiting continuous wave (CW) transmissions as signals of opportunity. The exploited transmitter (Tx) is assumed to be ground-based and stationary, while the radar receiver (Rx) is installed on a platform located at height ${H_p}$ and moving at a constant velocity ${v_p}$ along the direction of the $y$-axis. The Rx is based on an $N$-element linear array, also aligned along the $y$-axis (namely in side-looking configuration). The antenna indices are ordered such that $n = 0$ denotes the leading antenna and $n = N - 1$ denotes the trailing antenna. Let ${\boldsymbol{z}}$ be the vector describing the array layout:

(1)\begin{equation}{\boldsymbol{z}} = \left[ {{d_0},\,{d_1},\, \ldots ,\,{d_{N - 1}}\,} \right]\lambda ,\end{equation}

Figure 1. A sketch of the adopted system geometry.

where ${d_n}$ being the distance of the $n$th antenna from the leading antenna in $\lambda $ units (0 = d ₀ < d ₁ <…< d_{N −1}).

When CW transmissions are employed as waveforms of opportunity for passive radar application, a suboptimal batching strategy is usually employed to contain the computational load required for the evaluation of the cross-ambiguity function (e.g., see [1, 9]). Particularly, the reference and surveillance signals are fragmented into batches which are first separately range compressed and then jointly processed via a bank of Doppler filters. This allows emulating the fast-time/slow-time framework typical of a pulsed radar. We denote with $M$ the number of batches, or pulses, in each CPI and with $T$ the batch duration, which also corresponds to the equivalent pulse repetition time.

We assume that the observed scene is characterized by both stationary scatterers and moving targets, and we denote with ${\theta _0}$ the angle between the array end-fire and the receiver to scatterer line of sight. As known, the stationary scatterers contribute to the overall clutter return, which may hide the echoes from potential moving targets of interest.

Our aim is to exploit the described receiving-only system to detect slow-moving targets against clutter and noise. In the following, we will consider three simplifying assumptions, referred to as hypotheses H₁, H₂, and H₃ for future reference.

Hypothesis 1 (H₁): High clutter-to-noise power ratio (CNR). First, we will assume high CNR conditions. This is relatively common condition when dealing with mobile passive radars for short- and medium-range applications, for which the performance is usually clutter limited.

Hypothesis 2 (H₂): Perfect DPCA condition. As is well known, the DPCA techniques require the PRF to be synchronized with the platform velocity so that the $n$th antenna occupies the position of the $\left( {n - 1} \right)$th after an integer number ${K_n}$ of pulses. This is known as perfect DPCA condition. Denoting with ${\Delta }{T_n}$ the time interval required for the $n$th antenna to occupy the position of the $\left( {n - 1} \right)$th, we have

(2)\begin{equation}\Delta T_n=\frac{d_n-d_{n-1}}{v_p}.\end{equation}

Therefore, the perfect DPCA condition is satisfied if each of the time intervals ${\Delta }{T_n}$ in equation (2) is an integer multiple of the batch duration, namely,

(3)\begin{align}&\Delta T_n=\frac{d_n-d_{n-1}}{v_p}{}=K_n \cdot T,\nonumber\\ & K_n\in\mathbb{N},{}n=1,\dots,N-1.\end{align}

The implications of equation (3) are clearly visible in Fig. 2, which shows the positions consecutively occupied by the $N = 3$ antenna elements in different time instants. Denoting with $p$ the index of a generic position, we note that each antenna takes the $p$th position in an instant corresponding to the starting time of one of the $M$ pulses. For now, we only consider uniform spacings between the sensors (i.e., ${d_n} - {d_{n - 1}} = d,\,\,\forall n$), so that all the delays ${\Delta }{T_n}$ are equal (i.e., ${\Delta }{T_n} = {K_n} \cdot T = K \cdot T,\,\forall n$). In Fig. 2, we assumed $K = 2$.

Figure 2. The set of positions consequently occupied by different antenna elements in different time instants. The ULA configuration ${\boldsymbol{z}} = \left[ {0\,0.5\,1.0} \right]\lambda $ has been used. We assumed $K = 2$. The perfect DPCA condition guarantees perfect alignment between the $N$ antennas.

In the reminder of this paper, we assume that the perfect DPCA condition is always satisfied, meaning that the array inter-element distances, the platform velocity, and the batch duration are constrained through equation (3). Note that satisfying this condition is relatively straightforward when CW signals are fragmented into batches. As a matter of fact, in such cases, the equivalent PRF corresponds to the inverse of the batch duration and can be chosen so as to satisfy equation (3).

Hypothesis 3 (H₃): Absence of internal clutter motion (ICM). In the following, we will assume the absence of ICM, meaning that the clutter scatterers have no (or negligible) intrinsic radial velocity. On the one hand, this hypothesis allows guaranteeing that the Doppler shift of an echo from a stationary scatterer only depends on the platform motion and can thus be univocally determined if the direction of arrival of the scatterer is known. On the other hand, the absence of ICM allows guaranteeing that the clutter returns do not fluctuate from pulse to pulse so that different antenna elements occupying a given position at different times will observe exactly the same stationary scene.

Signal model

Let ${\boldsymbol{x}}$ denote the space–time data vector relative to the $l$th range bin. This vector can be written as the sum of three contributions:

(4)\begin{equation}{\boldsymbol{x}} = {\boldsymbol{t}} + {\boldsymbol{c}} + {\boldsymbol{n}}.\end{equation}

Particularly, ${\boldsymbol{t}}$ is a space–time column vector collecting the echoes from the moving target, ${\boldsymbol{c}}$ is a column vector collecting the returns from the stationary scatterers located at the $l$th range bin, and ${\boldsymbol{n}}$ is a column vector collecting additive white Gaussian noise samples. We also denote with ${\boldsymbol{d}} = {\boldsymbol{c}} + {\boldsymbol{n}}$ the overall disturbance signal.

In the following, we resort to the data selection strategy adopted in the studies by Lombardo [21] and Lombardo et al. [22], which is designed to remove the CPI border effect due to sensor misalignments. The idea is to only integrate the pulses collected from the positions sequentially occupied by each of the $N$ sensors. This corresponds to discarding the first and the last pulses received by the antenna elements at the borders of the array. Denoting with $P$ the total number of positions sequentially occupied, the position index $p = 0, \ldots ,P - 1$ can be written in terms of the space and time indices, $n$ and $m$, respectively, as follows:

(5)\begin{equation}p = m - \frac{{{d_n}}}{{{v_p}T}}.\end{equation}

Figure 2 shows a graphical representation of the described data selection strategy. The discarded pulses are colored in light blue, while the ones used for integration are colored in dark blue. If the perfect DPCA condition is satisfied, this strategy allows guaranteeing that the integrated pulses are received by the $N$ antennas from the same set of $P$ positions.

Based on the adopted selection strategy, the data vector ${\boldsymbol{x\,}}\left( {NP \times 1} \right)$ in equation (4), which collects the observations of the $N$ antennas from the $P$ positions, can be written as follows:

(6)\begin{equation}{\boldsymbol{x}} = {\left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{x}}_0}}&{{{\boldsymbol{x}}_1}}& \ldots &{{{\boldsymbol{x}}_{P - 1}}} \end{array}} \right]^T}\end{equation}

where ${{\boldsymbol{x}}_p}\,\left( {p = 0, \ldots ,P - 1} \right)$ is an $\left( {N \times 1} \right)$ column vector collecting the pulses received by the $N$ antennas from the $p$th position.

Based on the aforementioned data selection and assuming a perfect DPCA condition (H₂), the target contribution ${\boldsymbol{t}}$ in equation (4) can be written as follows:

(7)\begin{equation}{\boldsymbol{t}} = {\boldsymbol{\,}}{{\boldsymbol{s}}_{pos}}\left( {{\theta _0},{v_b}} \right) \otimes {{\boldsymbol{s}}_a}\left( {{v_b}} \right)\end{equation}

where

• – “$ \otimes $” denotes the Kronecker product;

• – the vector ${{\boldsymbol{s}}_{pos}}\left( {{\theta _0},{v_b}} \right)$ $\left( {P \times 1} \right)$ collects the $P$ spatial phase shifts between the $p$th position and the first one

(8)\begin{equation}{{\boldsymbol{s}}_{pos}}\left( {{\,f_d}} \right) = {\left\{ {\exp \left( {\,j2\pi {f_d}\left( {{\theta _0},{v_b}} \right)pT} \right)} \right\}_{p = 0 \ldots P - 1}},\end{equation}

with ${\theta _0}$ being the target direction of arrival and ${f_d}\left( {{\theta _0},{v_b}} \right)$ being the bistatic Doppler frequency of the considered target. In turn, the Doppler frequency can be decomposed into two contributions:

(9)\begin{equation}f_d=\frac{v_{p}\,{\it{cos{}}}\,\theta_0{}}\lambda-\frac{v_b}\lambda,\end{equation}

where ${v_p}{cos}{\theta _0}$ is the radial component of the platform velocity and ${v_b}$ is the intrinsic radial component of the observed target;

• – the vector ${{\boldsymbol{s}}_a}\left( {{v_b}} \right)$ $\left( {N \times 1} \right)$ collects the amplitudes and phases of the target’s echo at the N antennas from the generic position and depends on the propagation loss, the array layout, and the target velocity. In “Nonadaptive multichannel DPCA”, we will consider different models for vector ${{\boldsymbol{s}}_a}\left( {{v_b}} \right)$ depending on the assumptions made on the observed target, and this will allow us to derive different detection schemes.

The disturbance components $\textstyle\boldsymbol c\;(NP\times1)$ and $\boldsymbol n\;(NP\times1)$ are modeled as statistically independent Gaussian random variables with zero mean and covariance matrices ${{\mathbf{Q}}_{\text{c}}}$ and ${{\mathbf{Q}}_{\text{n}}}$, respectively. Specifically, the covariance matrix for thermal noise is simply written as ${{\mathbf{Q}}_{\text{n}}} = \sigma _n^2{{\mathbf{I}}_{NP}}$, where $\sigma _n^2 = \frac{1}{{NP}}E\left\{ {{{\boldsymbol{n}}^H}{\boldsymbol{n}}} \right\}$ is the noise power level at the generic antenna element, and we assumed identical receiving channels.

To derive a closed-form expression for the CCM ${{\mathbf{Q}}_{\text{c}}}$, we proceed as follows. First, note that the overall clutter return ${\boldsymbol{c}}$ from the $l$th range cell can be modeled as the superposition of the clutter returns from the stationary scatterers located in the angular sector ${{\Theta }} = \left[ { - \pi ,\pi } \right]$:

(10)\begin{equation}{\boldsymbol{c}} = \mathop{\smallint \limits_{ - \pi }^\pi}{A_c}\left( \theta \right){\boldsymbol{s}}\left( {{\theta _0},{v_b}} \right)d\theta \end{equation}

where ${A_c}\left( \theta \right)$ is the complex amplitude of the return from the stationary scatterer located at the $l$th range bin and at angle $\theta $ and ${\boldsymbol{s}}\left( \theta \right)$ is the corresponding space–time steering vector that can be written as follows:

(11)\begin{equation}{\boldsymbol{s}}\left( \theta \right) = {\boldsymbol{\,}}{{\boldsymbol{s}}_{pos}}\left( {\theta ,0} \right) \otimes {\left[ {1,\,1, \ldots ,1} \right]^T}\end{equation}

since, dealing with the stationary scene $\left( {{v_b} = 0} \right)\,$and having assumed the absence of ICM (H₃), the clutter echoes collected by the $N$ antennas from a given position $p$ are identical.

By defining ${1_N} = {\left[ {1,\,1, \ldots ,1} \right]^T}$, the expression of the CCM ${{\mathbf{Q}}_{\text{c}}}$ can be developed as follows:

(12)\begin{equation}\begin{gathered} {{\mathbf{Q}}_{\text{c}}} = E\left\{ {{\boldsymbol{c}}{{\boldsymbol{c}}^H}} \right\} = \hfill \\ \;\;\;\;\; = \mathop {\smallint \limits_{ - \pi }^ \pi} \mathop {\smallint \limits_{ - \pi }^ \pi} E\left\{ {{A_c}\left( \theta \right){A_c}^*\left( {\theta {^{\prime}}} \right)} \right\}{\boldsymbol{s}}\left( \theta \right){{\boldsymbol{s}}^H}\left( \theta \right)d\theta d\theta {^{\prime}} = \hfill \\ \;\;\;\;\; = \mathop {\smallint \limits_{ - \pi }^ \pi} \sigma _c^2\left( \theta \right)\left( {{{\boldsymbol{s}}_{pos}}\left( {\theta ,0} \right) \otimes {1_N}} \right)\left( {{\boldsymbol{s}}_{pos}^H\left( {\theta ,0} \right) \otimes {1_N}} \right)d\theta \hfill \\ \end{gathered} \end{equation}

where we assumed that the amplitudes of the returns from different scatterers are independent, i.e.,

(13)\begin{equation}E\left\{ {{A_c}\left( \theta \right){A_c}^*\left( {\theta '} \right)} \right\} = \left\{ {\begin{array}{*{20}{c}} {0,\,\,\,\,\,\,\,\,\,\,for\,\theta \ne \theta '} \\ {\sigma _c^2\left( \theta \right),\,\,\,for\,\theta = \theta '}. \end{array}} \right.\end{equation}

Eventually, we obtain

(14)\begin{equation}\begin{gathered} \,\,\,\,\,\,\,{{\mathbf{Q}}_c} = \mathop {\smallint \limits_{ - \pi }^ \pi} \sigma _c^2\left( \theta \right)\left[ {{{\boldsymbol{s}}_{pos}}\left( {\theta ,0} \right){\boldsymbol{s}}_{pos}^H\left( {\theta ,0} \right)} \right]d\theta \otimes \left[ {{1_N}1_N^H} \right] = \hfill \\ \;\;\;\;\;\;\;\;\; = {{\mathbf{K}}_{\text{P}}} \otimes {{\mathbf{K}}_N}. \hfill \\ \end{gathered} \end{equation}

In the last expression of equation (14), we successfully decoupled the spatial and the temporal components of the CCM. Specifically, ${{\mathbf{K}}_{\text{P}}} = \mathop {\smallint \limits_{ - \pi }^\pi} \sigma _c^2\left( \theta \right)\left[ {{{\boldsymbol{s}}_{pos}}\left( {\theta ,0} \right){\boldsymbol{s}}_{pos}^H\left( {\theta ,0} \right)} \right]d\theta $ is a $P \times P$ cross-correlation matrix of the clutter returns observed by a fixed antenna from the $P$ positions, while ${{\mathbf{K}}_N} = \left[ {{1_N}1_N^H} \right]{\boldsymbol{\,}}$is a rank-one $N \times N$ cross-correlation matrix of the clutter returns observed by the $N$ antennas from a fixed position, which are perfectly correlated due to the adopted H₂ and H₃ hypotheses.

Hence, the overall disturbance covariance matrix (DCM) can be written as follows:

(15)\begin{equation}\mathbf Q=\sigma_n^2{\mathbf I}_{NP}+{\mathbf Q}_\text{c}\boldsymbol.\end{equation}

Finally, by recalling the following matrix inversion lemma:

\begin{equation*}{\mathbf{B}} = {\mathbf{A}} + {\mathbf{UV}}\end{equation*}

(16)\begin{equation}{{\mathbf{B}}^{-1}} = {{\mathbf{A}}^{-1}} - {{\mathbf{A}}^{-1}}{\mathbf{U}}{\left({\mathbf{I}} + {\mathbf{V}}{{\mathbf{A}}^{-1}}{\mathbf{U}}\right)^{-1}}{\mathbf{V}}{{\mathbf{A}}^{-1}}\end{equation}

we obtain a closed-form expression also for the inverse DCM:

(17)\begin{equation}\mathbf Q^{-1}=\frac1{\sigma_n^2}{\{{{\mathbf I}_{NP}-{({\lbrack\sigma_n^2\mathbf K_P^{-1}+N{\mathbf I}_P\rbrack^{-1}\otimes\lbrack}1_N1_N^H\rbrack})}\}}.\end{equation}

Under the hypothesis H₁ of high CNR, we have $\sigma _n^2{\mathbf{K}}_P^{-1} \to 0$ as $CNR \to + \infty $, and the inverse DCM expression in equation (17) further simplifies into

(18)\begin{equation}\begin{gathered} \,\,\,\,\,\,\,\,\,{{\mathbf{Q}}^{-1}} = \frac{1}{{\sigma _n^2}}\left\{ {{{\mathbf{I}}_{NP}} - \left( {{{\mathbf{I}}_P} \otimes \frac{1}{N}\left[ {{1_N}1_N^H} \right]} \right)} \right\} = \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\, = \frac{1}{{\sigma _n^2}}\left\{ {{{\mathbf{I}}_P} \otimes \left( {{{\mathbf{I}}_N} - \frac{1}{N}\left[ {{1_N}1_N^H} \right]} \right)} \right\} = \frac{1}{{\sigma _n^2}}\left\{ {{{\mathbf{I}}_P} \otimes {{\boldsymbol{\Pi }}_N}} \right\}. \hfill \\ \end{gathered} \end{equation}

where ${{{\boldsymbol\Pi }}_N} = \left( {{{\mathbf{I}}_N} - \frac{1}{N}\left[ {{1_N}1_N^H} \right]} \right)$. Of course, ${{\mathbf{Q}}^{-1}}$ is a tight approximation of the true DCM only if the hypotheses H₁–H₃ are satisfied.

Nonadaptive multichannel DPCA detectors

To derive the optimal detector based on the assumed signal model, we resort to the likelihood ratio test (LRT):

(19)\begin{equation}L{(\boldsymbol x)}=\frac{e^{{(\boldsymbol x-\boldsymbol t)}^H\mathbf Q^{-1}{(\boldsymbol x-\boldsymbol t)}}}{e^{\boldsymbol x^H\mathbf Q^{-1}\boldsymbol x}} \gt rless\gamma,\end{equation}

where $\gamma $ is an appropriate threshold. By evaluating the log-likelihood ${\Lambda }\left( {\boldsymbol{x}} \right) = \log \left( {L\left( {\boldsymbol{x}} \right)} \right)$, we obtain

(20)\begin{equation}\Lambda{(\boldsymbol x)}=-\boldsymbol x^H\mathbf Q^{-1}\boldsymbol t-\boldsymbol t^H\mathbf Q^{-1}\boldsymbol x+\boldsymbol t^H\boldsymbol Q^{-1}\boldsymbol t\boldsymbol.\end{equation}

Substituting equations (7) and (18) in equation (20) and omitting the dependence of the target component ${\boldsymbol{t}}$ on ${\theta _0}$ and ${v_b}$ for ease of notation, we obtain

(21)\begin{equation}\begin{gathered} {\boldsymbol{\,\,\,\,\,\,}}{\Lambda }\left( {\boldsymbol{x}} \right) = - {{\boldsymbol{x}}^H}\frac{1}{{\sigma _n^2}}\left( {{{\mathbf{I}}_P} \otimes {{\boldsymbol{\Pi }}_N}} \right)\left( {{{\boldsymbol{s}}_{pos}} \otimes {{\boldsymbol{s}}_a}} \right) \hfill \\ {\boldsymbol{\,}}\;\;\;\;\;\;\;\;\;\;\;\; - \left( {{\boldsymbol{s}}_{pos}^H \otimes {\boldsymbol{s}}_a^H} \right)\frac{1}{{\sigma _n^2}}\left( {{{\mathbf{I}}_P} \otimes {{\boldsymbol{\Pi }}_N}} \right){\boldsymbol{x}} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\; + \left( {{\boldsymbol{s}}_{pos}^H \otimes {\boldsymbol{s}}_a^H} \right)\frac{1}{{\sigma _n^2}}\left( {{{\mathbf{I}}_P} \otimes {{\boldsymbol{\Pi }}_N}} \right)\left( {{{\boldsymbol{s}}_{pos}} \otimes {{\boldsymbol{s}}_a}} \right). \hfill \\ \end{gathered} \end{equation}

Defining the $\left( {N \times 1} \right)$ column vector ${\boldsymbol{\chi }} = \left\{ {{\boldsymbol{s}}_{pos}^H \otimes {{\mathbf{I}}_N}} \right\}{\boldsymbol{x}}$, we can rewrite equation (21) as follows:

(22)\begin{equation}{\Lambda }\left( {\boldsymbol{x}} \right) = - \frac{1}{{\sigma _n^2}}{{\boldsymbol{\chi }}^H}{{{\boldsymbol\Pi }}_N}{{\boldsymbol{s}}_a} - \frac{1}{{\sigma _n^2}}{\boldsymbol{s}}_a^H{{{\boldsymbol\Pi }}_N}{\boldsymbol{\chi }} + P{\boldsymbol{s}}_a^H{{{\boldsymbol\Pi }}_N}{{\boldsymbol{s}}_a}.\end{equation}

Note that ${\boldsymbol{\chi }}$ represents the Doppler bin of the DFT (discrete fourier transform) of ${\boldsymbol{x}}$, corresponding to the Doppler frequency ${f_d}$ for which the target is sought. Finally, defining the column vector ${\boldsymbol{y}}\,\left( {N \times 1} \right)$ as

(23)\begin{equation}{\boldsymbol{y}} = \frac{1}{{\sigma _n^2}}{{\boldsymbol{\Pi }}_N}{\boldsymbol{\chi }},\end{equation}

the log-likelihood in equation (22) can be rewritten as follows:

(24)\begin{equation}{\Lambda }\left( {\boldsymbol{x}} \right) = - {{\boldsymbol{y}}^H}{{\boldsymbol{s}}_a} - {\boldsymbol{s}}_a^H{\boldsymbol{y}} + P{\boldsymbol{s}}_a^H{{{\boldsymbol\Pi }}_N}{{\boldsymbol{s}}_a}.\end{equation}

Note that each element of ${\boldsymbol{y}}$ can be interpreted as the output of a multichannel DPCA. To show this, let us rewrite equation (23) for the case of $N = 2$ antennas:

(25)\begin{equation}{\boldsymbol{y}} = \frac{1}{{\sigma _n^2}}\frac{1}{2}\left[ {\begin{array}{*{20}{c}} 1&{-1} \\ {-1}&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\chi \left[ 0 \right]} \\ {\chi \left[ 1 \right]} \end{array}} \right] = \frac{1}{{2\sigma _n^2}}\left[ {\begin{array}{*{20}{c}} {\chi \left[ 0 \right] - \chi \left[ 1 \right]} \\ { - \chi \left[ 0 \right] + \chi \left[ 1 \right]} \end{array}} \right].\end{equation}

As apparent, ${\boldsymbol{y}}$ collects two equal and opposite values, corresponding to the output of a conventional DPCA. In turn, the rows of ${{{\boldsymbol\Pi }}_{N = 2}}$ contain the two possible permutations of the weights of a standard single canceler [23], namely ${\boldsymbol{v}} = \left[ {1, - 1} \right]$. In other words, conventional DPCA can be seen as the counterpart of the single canceler for moving radar systems. While DPCA virtually compensates for the platform motion by subtracting the signals received by the two antennas at the time instants in which their phase centers occupy the same positions, a single canceler takes the difference between two temporally displaced replicas of the received signal, resorting to a tapped delay line (TDL) with $N = 2$ taps and weight coefficients ${\boldsymbol{v}} = \left[ {1, - 1} \right]$. In either case, the clutter returns from stationary scatterers cancel out, while the echoes from moving targets, shifted in phase due to their own radial motion, are preserved.

In general, for $N \geq 2$, the $n$th element of ${\boldsymbol{y}}$ is a linear combination of the $N$ elements of ${\boldsymbol{\chi }}$, with coefficients given by the $n$th row of ${{{\boldsymbol\Pi }}_N}$. Particularly, each row of ${{{\boldsymbol\Pi }}_N}$ contains one of the $N$ permutations of the weights in the vector ${\boldsymbol{v}} = \left[ {\frac{{N - 1}}{N},\, - \frac{1}{N}{1_{N - 1}}} \right]$. Note that the vector ${\boldsymbol{v}}$ is such that

(26)\begin{equation}\mathop {\sum \limits_{i = 1}^N} v\left[ i \right] = 0.\end{equation}

This is a fundamental property of the weights of any TDL-based clutter canceler, which guarantees that the filter response will show a notch at zero bistatic velocity, enabling the suppression of the clutter component. Incidentally, we note that the specific order in which the weights are applied does not affect clutter cancelation capabilities, while it may affect the target response, depending on its radial velocity.

Based on the foregoing considerations and observing that the log-likelihood function in equation (24) depends on unknown parameters, we resort to the generalized LRT, in which the log-likelihood is maximized with respect to the unknowns. Depending on the model adopted for the target contribution, this approach allows us to derive a series of nonadaptive multichannel DPCA-like detectors suitable for low-cost radar systems.

Fully coherent detector

To derive this first detector, we assume that the target contribution in equation (7) is determined up to a multiplicative constant given by the deterministic but unknown complex target amplitude $A$, i.e., ${{\boldsymbol{s}}_a} = $ $A{{\boldsymbol{s}}_{del}}$, where

(27)\begin{equation}{{\boldsymbol{s}}_{del}}\left( {{v_b}} \right) = {\left\{ {\exp \left( {j\frac{{2 \pi }}{\lambda }\frac{{{v_b}}}{{{v_p}}}{d_n}} \right)} \right\}_{n = 0, \cdots ,N - 1}}.\end{equation}

is fully specified and depends on the antenna element spacings ${d_n}$ and on the ratio ${v_b}/{v_p}$ between the target bistatic velocity and the platform velocity.

In this case, maximizing equation (24) with respect to the unknown parameter $A$, we have

(28)\begin{equation}\mathop {\max }\limits_A \left\{ {\Lambda } \right\} = - {{\boldsymbol{y}}^H}\tilde A{{\boldsymbol{s}}_{del}} - {\tilde A^*}{\boldsymbol{s}}_{del}^H{\boldsymbol{y}} + P{\tilde A^*}{\boldsymbol{s}}_{del}^H{{\boldsymbol{\Pi }}_N}\tilde A{{\boldsymbol{s}}_{del}},\end{equation}

which is maximum for $\tilde A = \frac{{{\boldsymbol{s}}_{del}^H{\boldsymbol{y}}}}{{P{\boldsymbol{s}}_{del}^H{{{\boldsymbol\Pi }}_N}{{\boldsymbol{s}}_{del}}}}$, yielding

(29)\begin{equation}\mathop {\max }\limits_A \left\{ {\Lambda } \right\} = \frac{{{{\left| {{\boldsymbol{s}}_{del}^H{\boldsymbol{y}}} \right|}^2}}}{{P{\boldsymbol{s}}_{del}^H{{{\boldsymbol\Pi }}_N}{{\boldsymbol{s}}_{del}}}}.\end{equation}

By selecting an appropriate threshold value $T$ to guarantee the desired false alarm probability, we obtain the following detector:

(30)\begin{equation}{\vert\boldsymbol s_{del}^H\boldsymbol y\vert}^2 \gt rless T,\end{equation}

where the denominator of equation (29) has been included in the threshold $T$, being a constant value. In the following, this will be referred to as fully coherent (FC) detector as both the spatial and the temporal phase information are exploited.

Based on equations (23), (27), and (30), the FC detector can be interpreted as the cascade of the following operations:

1. 1. Coherent integration along the position domain by evaluating DFT of the data vector ${\boldsymbol{x}}$.

2. 2. Suppression of the clutter contributions within the generic Doppler bin ${\boldsymbol{\chi }}$ of the Fourier-transformed data vector by evaluating ${\boldsymbol{y}} = \frac{1}{{\sigma _n^2}}{{{\boldsymbol\Pi }}_N}{\boldsymbol{\chi }}$.

3. 3. Coherent integration along the antenna domain by multiplying ${\boldsymbol{y}}$ by the temporal steering vector ${\boldsymbol{s}}_{del}^H\left( {{v_b}} \right)$.

Based on the observations at the end of the previous section, the FC detector can also be interpreted as the coherent integration along the antenna domain of the different outputs of $N$ multichannel DPCAs, collected in the vector ${\boldsymbol{y}}$, each obtained considering a different permutation of the weights ${\boldsymbol{v}}$ in the $n$th row of ${{{\boldsymbol\Pi }}_N}$.

To assess the target detection performance of the FC detector, we define the signal-to-clutter-plus-noise power ratio (SCNR) as follows:

(31)\begin{equation}SCN{R_{FC}} = \frac{{P_s^{out}}}{{P_d^{out}}} = \frac{{{{\left| {{\boldsymbol{w}}_{FC}^H{\boldsymbol{t}}} \right|}^2}}}{{E\left\{ {{{\left| {{\boldsymbol{w}}_{FC}^H{\boldsymbol{d}}} \right|}^2}} \right\}}} = \frac{{{A^2}{{\left| {{\boldsymbol{w}}_{FC}^H\left( {{{\boldsymbol{s}}_{pos}} \otimes {{\boldsymbol{s}}_{del}}} \right)} \right|}^2}}}{{{\boldsymbol{w}}_{FC}^H\widetilde {\mathbf{Q}}{{\boldsymbol{w}}_{FC}}}}\end{equation}

where $P_s^{out}$ is the output power of the useful signal, $P_d^{out} = \sigma {_{c^{2^{out}}}} + \sigma {_n^{2^{out}}}$ is the output power of the disturbance component, $\widetilde {\mathbf{Q}} = E\left\{ {{\boldsymbol{d}}{{\boldsymbol{d}}^H}} \right\}$ denotes the true DCM, and ${{\boldsymbol{w}}_{FC}}$ is an $\left( {NP \times 1} \right)$ column vector collecting the complex-valued coefficients, which represent the sequence of operations performed by the FC detector, namely,

(32)\begin{equation}{{\boldsymbol{w}}_{FC}} = \frac{1}{{\sigma _n^2}}\left\{ {{{\boldsymbol{s}}_{pos}} \otimes {{\mathbf{I}}_N}} \right\}\left( {{{\mathbf{I}}_N} - \frac{1}{N}\left[ {{1_N}1_N^H} \right]} \right){{\boldsymbol{s}}_{del}}.\end{equation}

Substituting the expression of ${{\boldsymbol{w}}_{FC}}$ in equation (31), we obtain

(33)\begin{equation}SCN{R_{FC}} = \frac{{{{\left| A \right|}^2}NP}}{{\sigma _n^2}}\left[ {1 - {{\left| {\frac{1}{N}\mathop {\sum \limits_{n = 0}^{N - 1}} \exp \left( {j\frac{{2 \pi }}{\lambda }\frac{{{v_b}}}{{{v_p}}}{d_n}} \right)} \right|}^2}} \right].\end{equation}

Clearly, the last expression in equation (33) holds only when ${\mathbf{Q}} = \widetilde {\mathbf{Q}}$, namely if hypotheses H₁–H₃ are satisfied.

Assuming a Swerling 0 (SW0) model for the target amplitudes, we can evaluate the detection performance in terms of probability of false alarm (${P_{fa}}$) and probability of detection (${P_d}$), using the following well-known expressions:

(34)\begin{equation}{P_{fa}} = \exp \left( { - \frac{{{T^2}}}{{\sigma _N^2}}} \right)\end{equation}

(35)\begin{equation}{P_d} = Q\left( {\sqrt {2\,SCN{R_{FC}}} ,\sqrt {2{T^2}} } \right)\end{equation}

where $T$ is the threshold value and $Q\left( {a,b} \right)$ is the Marcum function.

In conclusion, when H₁–H₃ are satisfied, the FC detector achieves optimal performance without involving any estimation process. However, it requires both the steering vectors ${{\boldsymbol{s}}_{pos}}\left( {{\theta _0},{v_b}} \right)$ and ${{\boldsymbol{s}}_{del}}\left( {{v_b}} \right)$ to be completely determined. Hence, implementing the FC detector involves a two-dimensional exhaustive search along the angle–velocity axes. In the following subsections, we introduce two additional detectors that do not rely on the phase information in ${{\boldsymbol{s}}_{del}}\left( {{v_b}} \right)$, thereby reducing the processing complexity.

Partially coherent detector

Note that the two-dimensional exhaustive search can be circumvented by performing noncoherent integration along the antenna domain. In fact, we observe that the steering vector ${{\boldsymbol{s}}_{pos}}\left( {{\theta _0},{v_b}} \right)$ depends on both ${\theta _0}$ and ${v_b}$. However, equation (8) shows that ${{\boldsymbol{s}}_{pos}}\left( {{\theta _0},{v_b}} \right)$ actually depends only on the target Doppler frequency ${f_d}\left( {{\theta _0},{v_b}} \right)$. Therefore, the phase information in ${{\boldsymbol{s}}_{pos}}\left( {{\theta _0},{v_b}} \right)$ can be exploited by resorting to a bank of filters tuned to different Doppler frequencies. By neglecting the phase information in ${{\boldsymbol{s}}_{del}}\left( {{v_b}} \right)$, we only need a one-dimensional exhaustive search along the target Doppler frequency axis.

Based on this observation, in the following, we introduce the partially coherent (PC) detector, which only exploits the phase information along the position domain. Recalling that only a limited number of receiving channels should be used to preserve ease of deployment, the PC detector should not undergo significant losses compared to the FC one.

To derive the PC detector, we modify the model adopted in “System geometry and reference scenario” for ${{\boldsymbol{s}}_a}$ to reflect the limited knowledge on the target velocity. To this purpose, ${{\boldsymbol{s}}_a} = \left[ {{A_0}, \ldots ,{A_{N - 1}}} \right]$ is modeled as a vector of deterministic but unknown complex amplitudes. Maximizing equation (24) with respect to ${{\boldsymbol{s}}_a}$, we have

(36)\begin{equation}\mathop {\max }\limits_{{A_{\text{o}}},{A_1}, \ldots ,{A_{N - 1}}} \left\{ { - {{\boldsymbol{y}}^H}{{\boldsymbol{s}}_a} - {\boldsymbol{s}}_a^H{\boldsymbol{y}} + P{\boldsymbol{s}}_a^H{{{\boldsymbol\Pi }}_N}{{\boldsymbol{s}}_a}} \right\}\end{equation}

whose maximum is given by

(37)\begin{equation}\mathop {\max }\limits_{{A_0},{A_1}, \ldots ,{A_{N - 1}}} \left\{ {\Lambda } \right\} = \frac{1}{P}{{\boldsymbol{y}}^H}{\boldsymbol{y}}.\end{equation}

By selecting an appropriate threshold value $T$ to guarantee the desired false alarm probability, we obtain the following detector:

(38)\begin{equation}{\vert\boldsymbol y\vert}^2 \gt rless T\end{equation}

where $P$ has been included in the threshold $T$.

Since the PC detector involves nonlinear operations, its performance cannot be evaluated in terms of the SCNR, as the superposition principle does not hold, and the effects of the different signal components cannot be analyzed separately. Therefore, we directly report the expressions for the achieved ${P_{fa}}$ and the ${P_d}$:

(39)\begin{equation}{P_{fa}} = \frac{1}{{\left( {N - 2} \right)!}}\gamma \left( {\frac{{\left( {N - 1} \right)T}}{{N\sigma _N^2}},N - 1} \right)\end{equation}

(40)\begin{equation}{P_d} = Q\left( {\sqrt {\frac{{\left( {N - 1} \right)2\xi }}{{N\sigma _N^2}}} ,\sqrt {\frac{{\left( {N - 1} \right)2{T^2}}}{{N\sigma _N^2}}} ,N - 1} \right)\end{equation}

where $\gamma \left( {\frac{T}{\alpha },N - 1} \right)$ is the upper incomplete gamma function, $T$ is the threshold value, $\alpha = \frac{{N - 1}}{N}$ is a scale factor, and $\xi $ is a parameter depending on the input power of the target signal. The detailed derivation of these expressions can be found in Appendix A.

Apodized partially coherent detector

Based on the foregoing considerations, the apodization approach introduced in the study by Quirini et al. [11] appears to be a suboptimal variant of the PC detector. Therein, the maximum between the absolute square values of the $N$ DPCA outputs collected in ${\boldsymbol{y}}$ was evaluated, thus obtaining

(41)\begin{equation}\max {{(\boldsymbol y}^{\boldsymbol\ast}\odot\boldsymbol y)} \gt rless T,\end{equation}

where $ \odot $ denotes the element-wise product. In the following, this detector will be referred to as apodization partially coherent (APC) detector.

As for the PC detector, the performance of the APC detector is evaluated in terms of ${P_{fa}}$ and ${P_d}$. In this case, we have we have the following upper bounds for the false alarm and detection probabilities:

(42)\begin{equation}{P_{fa}} \lesssim 1 - {\left( {1 - {e^{ - T/\sigma _N^2}}} \right)^N}\end{equation}

(43)\begin{equation}{P_d} \lt 1 - \mathop {\prod \limits_{k = 0}^{N - 1}} \left( {1 - Q\left( {\sqrt {\frac{{2\left( {\mu _k^R + \mu _k^I} \right)}}{{\sigma _N^2}}} ,\sqrt {\frac{{2T}}{{\sigma _N^2}}} } \right)} \right).\end{equation}

The detailed derivation of these expressions can be found in Appendix B.

Performance evaluation and comparison

In this final subsection, we validate the theoretical characterization of the performance of the FC, PC, and APC detectors via numerical analysis. To this end, the simulations are carried out assuming the validity of the hypotheses H₁–H₃ and generating the signal components according to the model outlined in “System geometry and signal model”. Hence, in this simulation, the conditions in which the three detectors have been defined are always satisfied. By ensuring this, we can effectively validate the theoretical performance of these detectors.

To this aim, for now we consider a phased array radar receiver ${\boldsymbol{z}} = \left[ {0\,0.5\,1.0} \right]\lambda $ with $N = 3$ antenna elements, characterized by uniform spacings of $\lambda /2$ between the sensors. We also assume that $P = 6$ positions are coherently integrated. Furthermore, we assume that the platform velocity ${v_p}$ is chosen so that the $n$th antenna occupies the position of the $n - 1$th after $K = 2$ pulses. With this choice of parameters, we precisely match the conditions sketched in Fig. 2. We also assume that the input signal-to-noise power ratio (SNR), defined as the ratio between the absolute square of complex target amplitude ${\left| A \right|^2}$ and the input noise power $\sigma _n^2$, is equal to $SN{R_{in}} = 0dB$.

The validity of the ${P_{fa}}$ and ${P_d}$ expressions shown in the previous subsection is verified by means of a Monte Carlo simulation with ${R_{MC}} = {10^6}$ runs. Specifically, the useful component of the received signal is generated as in equation (4), the noise component is modeled as a complex white Gaussian random variable, and the clutter component is modeled as a complex Gaussian random variable with covariance matrix ${{\mathbf{Q}}_c}$ as in equation (14). The sum of these contributions corresponds to an $\left( {NP \times 1} \right)$ column vector ${\boldsymbol{x}}$, which is the simulated data vector relative to the $l$th range bin. The decision variable can be obtained from the data vector ${\boldsymbol{x}}$ by using one of the three detectors defined in the previous subsections, namely the FC detector, the PC detector, or the APC detector.

First, we verify the derived ${P_{fa}}$ expressions by evaluating the theoretical and simulated values of ${P_{fa}}$ as a function of the threshold value, as shown in Fig. 3. As visible, all the derived theoretical expressions for ${P_{fa}}$, represented as solid curves, provide an accurate prediction of the simulated ${P_{fa}}$ values, represented as colored markers. Specifically, the upper bound derived for the ${P_{fa}}$ of the APC detector represents a good approximation for the actual ${P_{fa}}$ value.

Figure 3. Probability of false alarm achieved by the FC, PC, and APC detectors, as a function of the threshold value.

The simulated value for the ${P_d}$ can be estimated as the ratio between the values of $y$ overcoming the threshold value $T$ and the total number of Monte Carlo runs. For each detector, the threshold value $T$ is determined by inverting the relative ${P_{fa}}$ formula, namely equations (34), (39), and (42), respectively. In this case, a constant false alarm probability of ${P_{fa}} = {10^{ - 3}}$ has been assumed.

The theoretical and simulated values of ${P_d}$ are studied as a function of the ratio ${v_b}/{v_p}$ between the target bistatic velocity and the platform velocity. The ${P_d}$ values achieved by the different detectors are compared in Fig. 4. The red curve represents the FC detector, while the blue and the magenta ones represent the PC and the APC detectors, respectively. Furthermore, the solid lines represent the theoretical ${P_d}$ values, while the markers represent the simulated ones. The following observations are in order:

Figure 4. Probability of detection achieved by the FC, PC, and APC detectors, as a function of the ratio between target bistatic velocity ${v_b}$ and the platform velocity ${v_p}$.

• • Since the clutter signal has been generated according to the CCM model in equation (14), no model mismatches can possibly be present. Therefore, the FC detector achieves optimal performance, thus providing a useful reference for the other detectors.

• • The PC and the APC detectors achieve slightly degraded performances compared to the FC detector, due to the noncoherent integration along the antenna domain.

• • Overall, the PC detector achieves higher ${P_d}$ values than the APC one. This is visible by comparing not only the simulated values for ${P_d}$ but also the theoretical prediction of the PC with the upper bound of the APC.

• • Each of the three detectors shows a cancelation notch at ${v_b} = 0$, which enables clutter cancelation. This notch is also responsible for the partial attenuation affecting slow-moving targets.

• • For the considered set of parameters, the cancelation notch shows replicas located at $\frac{{{v_b}}}{{{v_p}}} = 2k,\,\,k \in \mathbb{Z}$, which in turn result in the appearance of blind velocities. As discussed in the next section, the width of the cancelation notch and the distance between blind velocities depend on the array design.

In conclusion, both PC and APC detectors suffer from the noncoherent integration in the antenna domain, reaching lower ${P_d}$ values compared to the FC detector. However, as mentioned earlier, these detectors do not require a two-dimensional exhaustive search on the bistatic target velocity ${v_b}$ and on the target direction of arrival ${\theta _0}$, possibly making them an attractive alternative when designing a low-cost mobile passive radar.

To conclude this section, we note that improved detection performance on slow-moving targets could be achieved by limiting the notch width, and that the set of unambiguous target velocities could be widened by increasing the distance between consecutive notch ambiguous replicas. As shown in the next section, these two competing needs sensibly depend on the receiving array design. When an ULA configuration is used, improved performance can be achieved only by increasing the number of sensors. This is not always feasible when striving for cost-containment, weightlessness, and compactness. Therefore, in the next section, we explore the use of NULA configurations, with the aim of realizing satisfactory trade-offs between the width of the cancelation notch and the distance between blind velocities, all without increasing the number of sensors.

NULA configurations

As mentioned at the end of the last section, target detection performance depends not only on the detector used but also on the array configuration adopted as radar receiver. Therefore, alongside the implementation of appropriate processing techniques, it is essential to pay careful attention to the design of the receiving array, which is crucial to preserve ease of deployment and compactness.

When designing a receiving array configuration, two competing needs should be considered:

1. 1. On the one hand, the global array length should be maximized. As a matter of fact, from Fig. 2, it is apparent that the global array length is directly related to the total observation time from the $p$th position. Maximizing the observation time results in an increased capability of distinguishing slow-moving targets from stationary scatterers, since the phase shift induced by the nonzero bistatic radial velocity of a slow-moving target is observed over a larger time span, ultimately resulting in a narrower cancelation notch.

2. 2. On the other hand, the inter-element distances should be minimized. As a matter of fact, reducing the distance between the sensors corresponds to minimizing the delay between the observations from the $p$th position. This results in a finer sampling grid in the time domain, which allows to increase the maximum unambiguous velocity, ultimately resulting in a larger distance between two replicas of the cancelation notch.

As apparent, when a ULA configuration is adopted, these requirements cannot be simultaneously pursued, unless a larger number of receiving elements is employed. However, when dealing with low-cost mobile radar systems, every additional sensor adversely affects the system compactness. Hence, our aim is to make the most of the limited number of available antennas by resorting to nonuniform sensor distributions. In contrast, as also studied in [11–13], NULA configurations allow reducing the notch width without simultaneously increasing the distance between the notch replicas, benefiting the overall detection performance.

As an illustrative example, we consider the NULA configuration ${\boldsymbol{z}} = \left[ {0\,0.5\,1.5} \right]\lambda $, with the same minimum element distance of 0.5$\lambda $ of the reference ULA array considered in “Nonadaptive multichannel DPCA”, and a larger maximum distance of 1.5$\lambda $. The set of $P = 6$ positions consecutively occupied by the $N = 3$ antenna elements is sketched in Fig. 5. Having assumed nonuniform sensor spacings, the time delay between the $n$th and the $n - 1$th sensors now depends on the antenna index $n$ and is given as follows:

Figure 5. The set of positions consequently occupied by different antenna elements in different time instants. The NULA configuration ${\boldsymbol{z}} = \left[ {0\,0.5\,1.5} \right]\lambda $ has been used. We assumed ${K_1} = 3$ and ${K_2} = 6$. The perfect DPCA condition guarantees perfect alignment between the $N$ antennas.

(44)\begin{equation}{\Delta }{T_n} = {K_n} \cdot T.\end{equation}

In Fig. 5 we assume ${K_1} = 3$ and ${K_2} = 6$. The positions outline clearly shows how the antenna design reflects in the time domain.

To better understand the role played by the inherent spatial diversity of NULA configurations, we note that each of the detectors introduced in the previous section represent a different way of combining the outputs of $N$ different DPCA filters, each obtained considering one of the possible permutations of the weight vector ${\boldsymbol{v}} = \left[ {\frac{{N - 1}}{N},\, - \frac{1}{N}{1_{N - 1}}} \right]$. Figure 6(a) and (b) shows these responses of the N = 3 DPCA filters (namely, absolute value of their outputs) as a function of the ratio ${v_b}/{v_p}$ between the target bistatic velocity and the platform velocity obtained using the reference three-element ULA of “Nonadaptive multichannel DPCA” ${\boldsymbol{z}} = \left[ {0\,0.5\,1.0} \right]\lambda $ and the three-element NULA ${\boldsymbol{z}} = \left[ {0\,0.5\,1.5} \right]\lambda $, respectively.

Figure 6. Outputs of $N = 3$ multichannel DPCAs, obtained using the weighting coefficients given by the $N$ rows of ${{{\boldsymbol\Pi }}_N}$, and (a) the ULA ${\boldsymbol{z}} = \left[ {0\,0.5\,1.0} \right]\lambda $, (b) the NULA ${\boldsymbol{z}} = \left[ {0\,0.5\,1.5} \right]\lambda $.

Note that two of the three DPCA outputs in Fig. 6(a) are identical, due to the symmetry of the employed $\lambda /2$-ULA configuration, while the all three DPCA outputs in Fig. 6(b) are different. This is a result of the spatial diversity characterizing the employed NULA configuration: one of the three outputs responds better to the targets with low ${v_b}/{v_p}$ values, ultimately allowing to reduce the notch width, which is beneficial for the detection of slow-moving targets, while performing worse for the intermediate values; the other two outputs have a worse response to the targets with low ${v_b}/{v_p}$ values, but a higher response at the intermediate results. Their combination allows to reduce the notch, without simultaneously changing the distance between the notch replicas, which are responsible for blind velocities.

To provide a wider analysis of the performance improvement achievable by jointly exploiting the detectors in “Nonadaptive multichannel DPCA” with the NULA configurations, we consider for the comparison also an ULA array with maximum element distance of 1.5$\,\lambda $ and a NULA array with maximum element distance of $\lambda $ but shorter minimum element distance. Therefore, we compare the theoretical and simulated performance of the three detectors derived in the previous section, operating with the four different array configurations:

1. 1. The ULA ${{\boldsymbol{z}}_1} = \left[ {0\,0.50\,1.00} \right]\lambda $.

2. 2. The ULA ${{\boldsymbol{z}}_2} = \left[ {0\,0.75\,1.50} \right]\lambda $.

3. 3. The NULA ${{\boldsymbol{z}}_3} = \left[ {0\,0.35\,1.00} \right]\lambda $.

4. 4. The NULA ${{\boldsymbol{z}}_4} = \left[ {0\,0.50\,1.50} \right]\lambda $.

A Monte Carlo simulation with ${R_{MC}} = {10^6}$ runs is carried out as in “Nonadaptive multichannel DPCA”. A fixed false alarm probability of ${P_{fa}} = {10^{ - 3}}$ has been assumed, and an estimate of the ${P_d}$ value is obtained as the percentage of runs for which the threshold value $T$ is overcome.

Figure 7(a–d) shows the results obtained by the three different detectors using the four different array configurations. The following observations are in order:

• • As expected, an increased array total length results in a narrower cancelation notch. This is clearly visible by comparing longer array configurations (i.e., ${{\boldsymbol{z}}_2}$ or ${{\boldsymbol{z}}_4}$) to shorter ones (i.e., ${{\boldsymbol{z}}_1}$ or ${{\boldsymbol{z}}_3}$).

• • As also expected, a reduced minimum inter-element distance results in a larger distance between the notch replicas. Thus, for example, since configuration ${{\boldsymbol{z}}_3}$ has the minimum inter-element distance, it achieves the maximum distance between two adjacent notch replicas. Conversely, configuration ${{\boldsymbol{z}}_2}$ has the largest minimum inter-element distance, which results in the greater distance between two adjacent notch replicas.

• • Different trade-offs between the notch width and the distance between the blind velocities could be achieved by choosing the appropriate NULA design. This flexibility is paid in terms of ${P_d}$ fluctuations for the target velocities outside the clutter cancelation notch.

• • The correspondence between the theoretical and the simulated values of ${P_d}$ confirms the validity of the analysis.

• • The loss of the PC and APC detectors with respect to the FC one increases as the array deviates from the ULA configuration. Specifically, the PC performs almost as the FC when used together with an ULA. In contrast, when trying to leverage the NULA to either narrow the cancelation notch or increase the distance between notch replicas, the performance gap between the detection losses achieved by the PC and the FC significantly increases.

Figure 7. Probability of detection as a function of the ratio ${v_b}/{v_p}$ for the three proposed detectors. (a) ${{\boldsymbol{z}}_1} = \left[ {0\,0.50\,1.00} \right]\lambda $; (b) ${{\boldsymbol{z}}_2} = \left[ {0\,0.75\,1.50} \right]\lambda $; (c) ${{\boldsymbol{z}}_3} = \left[ {0\,0.35\,1.00} \right]\lambda $; (d) ${{\boldsymbol{z}}_4} = \left[ {0\,0.50\,1.50} \right]\lambda $.

In conclusion, when aiming at increasing the unambiguous velocity sector, the NULA configuration ${{\boldsymbol{z}}_3} = \left[ {0\,0.35\,1.00} \right]\lambda $ represents a better design choice with respect to the $\lambda /2 - $ULA ${{\boldsymbol{z}}_1} = \left[ {0\,0.50\,1.00} \right]\lambda $, as it allows to increase the distance between the adjacent notch replicas without compromising on the notch width. Conversely, when interested in the detection of particularly slow-moving targets, the NULA ${{\boldsymbol{z}}_4} = \left[ {0\,0.50\,1.50} \right]\lambda $ represents a better design choice with respect to the $3\lambda /2$-ULA ${{\boldsymbol{z}}_2} = \left[ {0\,0.75\,1.50} \right]\lambda $, as it allows to narrow down the cancelation notch without reducing the unambiguous velocity sector. In this case, it is likely that the performance loss of the PC detector (still limited but greater than for the ULA case) is acceptable for a low-cost sensor. However, if this is not the case, it is apparent that this loss can be avoided by implementing the full 2D filter bank required by the FC detector at the expense of providing the sensor with the necessary increased computational capability. An alternative would be using an ULA with four elements ($\left[ {0\,0.50\,\,1.0\,\,1.50} \right]\lambda $ or $\left[ {0\,0.33\,0.66\,1.00} \right]\lambda $) with a PC detector. This would trade processing hardware for antenna elements with their respective receiving channels, with the corresponding costs, mass, volume, and power absorptions.

Case study: passive radar system based on DVB-T

In the previous section, we always assumed that the DCM model is an accurate representation of the true clutter statistics, since the clutter signal has always been generated according to the statistics described by the CCM model in equation (14). In this section, we try to validate the above observations in a different simulated scenario in which the clutter signal is generated as the superposition of the returns from a grid of stationary scatterers rather than as a colored Gaussian sequence. Indeed, the proposed detectors are tested against a simulated data set, employing the same ULA and NULA configurations considered in “NULA configurations”. Differently from “Nonadaptive multichannel DPCA”, the simulated analysis focuses on a real-world scenario. Specifically, we consider a multichannel DVB-T-based passive radar installed on a ground moving platform.

The simulation setup recalls the one presented in [9] and [10]. An 8k mode DVB-T waveform of opportunity is used as a reference signal. The guard interval is equal to ${T_G} = {T_U}/4$, such that the overall duration of the orthogonal frequency-division multiplexing (OFDM) symbol is ${T_{OFDM}} = {T_U} + {T_G} = 1120\mu s$, being ${T_U} = 896\mu s$. This results in a CPI of about 0.57s, corresponding to 512 OFDM symbols. The carrier frequency has been set to ${f_C} = 690MHz$, which corresponds to the wavelength $\lambda \approx 0.4345m$. In this case, we assume that the reference signal is obtained by means of a decode/recode approach, so that a dedicated receiving channel for the reference signal is not needed.

The $N$ receiving channels are installed on a ground moving platform. The platform velocity has been set to ${v_p} = 6.4655m/s$ so that the perfect DPCA condition holds for each of the arrays ${{\boldsymbol{z}}_1}$, ${{\boldsymbol{z}}_2}$, ${{\boldsymbol{z}}_3}$, and ${{\boldsymbol{z}}_4}$ considered in the previous section. Specifically, for the chosen set of parameters, we have $\frac{\lambda }{{{v_p}T}} = 60$, i.e., the platform travels exactly one wavelength in 60 PRTs. This guarantees that the sub-wavelength inter-element distances in the chosen array configurations are also traveled in a time corresponding to integer multiple of PRTs.

As to the signal components, thermal noise has been generated as a zero mean, unitary variance, Gaussian-distributed complex random variable $\mathcal{N}\left( {0,\sigma _N^2} \right)$, and its power level has been deliberately set to unity (i.e., $\sigma _N^2 = 0dB$).

The four targets reported in Table 1 have been simulated:

Table 1. Simulated targets parameters

The target signal is obtained as the superposition of the signals backscattered from these targets. Due to the unitary noise scaling, the overall input power $P_S^{\left( {in} \right)}$ also corresponds to the input SNR, for which we assumed $\textstyle{\mathrm{SNR}}_{in}=-55$$\textstyle\mathrm{dB}$.

Finally, a clutter scene spanning the first $R_b=8$$km$, corresponding to ${N_R} = 1000$ range cell, and an angular sector ${{{\Theta }}_q} = \left[ {0,\pi} \right],\,\forall q$ has been simulated. The complex amplitudes of the scatterers are assumed to be independent and identically distributed, following a Gaussian probability density function. The absence of ICM is assumed. The clutter component is scaled in amplitude so that it has an assigned power level $P_C^{\left( {in} \right)}$at the input of each receiving channel. Due to the unitary noise scaling, the input power level of the clutter component also corresponds to the input CNR, for which we assumed $CNR_{in}=20$$\mathrm{dB}$.

Figure 8 shows a sketch of the employed processing scheme, similar to the one described in [9]. First, a suboptimal batching strategy is adopted to contain the computational load required for the evaluation of the range-Doppler maps. Specifically, the CW signal is segmented into batches, which are separately range compressed and Doppler processed along the equivalent slow time dimension. The duration of one batch is assumed to be equal to the duration of one OFDM symbol, so that $T = {T_{OFDM}}$. As discussed in “System geometry and signal model”, the duration of one batch also corresponds to the equivalent PRT.

Figure 8. Sketch of the adopted processing scheme.

The subsequent range compression stage exploits a reciprocal filter in place of a matched filter. This allows to equalize the variability associated to the information content of the waveform of opportunity, which is fundamental to guarantee effective clutter cancelation, as extensively discussed in [9].

After range compression, the received signals are appropriately delayed by ${\Delta }{T_n}$ before applying the data selection strategy described in “System geometry and signal model”. This allows us to obtain the space–time data vector ${\boldsymbol{x}}$, which collects the returns from the $N$ antenna from the $P$ positions for the $l$th range bin.

Note that the data vector ${\boldsymbol{x}}$ has not been generated according to the signal model in “System geometry and signal model”. However, assuming that the hypotheses H₁–H₃ still hold and that the sidelobes of the exploited waveform of opportunity are properly controlled, the FC, PC, and APC detectors should still achieve satisfactory target detection performance.

Consequently, we proceed to evaluate the effectiveness of the proposed detectors in the described real-world scenario, with a specific focus on the PC detector. However, we note that similar considerations apply to the FC and APC detectors as well. As to the array design, we consider the same ULA and NULA configurations employed in the previous section.

To assess the detection performance of the PC detector for different array configurations, we report the achieved test statistics on the bistatic range-Doppler maps. Specifically, we define a series of desired ${P_{fa}}$ values and we resort to equation (39) to evaluate the corresponding thresholds. Then, for each pixel in the map, we report the ${P_{fa}}$ value that corresponds to the minimum threshold required for that pixel to yield a detection. The results are reported as ${\log _{10}}{P_{fa}}$.

The described range-Doppler maps are reported in Fig. 9. Specifically, Fig. 9(a) and (b) is relative to the ULA configurations ${{\boldsymbol{z}}_1} = \left[ {0\,0.50\,1.00} \right]\lambda $ and ${{\boldsymbol{z}}_2} = \left[ {1.50\,0.75\,0} \right]\lambda $, while Fig. 9(c) and (d) are relative to the NULA configurations ${{\boldsymbol{z}}_3} = \left[ {0\,0.35,\,1.00} \right]\lambda $ and ${{\boldsymbol{z}}_4} = \left[ {0\,0.50\,1.50} \right]\lambda $. Recall that the shorter NULA ${{\boldsymbol{z}}_3}$ was intended to increase the spacing between the blind velocities, while the NULA ${{\boldsymbol{z}}_4}$ was intended to improve the detection performance against the slow targets. The following observations are in order:

• • Focusing on the slowest target, moving at $v_b^{\left( 1 \right)} = 1.5m/s$, we note that the longest array configurations, namely, the ULA ${{\boldsymbol{z}}_2}$ and the NULA ${{\boldsymbol{z}}_4}$, achieve the best detection performance, as evidenced by the lower nominal ${P_{fa}}$ value required to detect this target.

• • Target 2 is detected with satisfactory performance by most array configurations. Specifically, only the ULA ${{\boldsymbol{z}}_2}$ does not allow detecting this target, due to the presence of a notch replica in its proximity.

• • As for target 3, moving at $v_b^{\left( 3 \right)} = - 13m/s$, the detection performance sensibly depends on the width and positions of the notch replicas. On the one hand, the ULA ${{\boldsymbol{z}}_1}$ and the NULA ${{\boldsymbol{z}}_4}$ allow detecting this target with satisfactory performance. Conversely, both the ULA ${{\boldsymbol{z}}_2}$ and the NULA ${{\boldsymbol{z}}_3}$ have the first replica of the notch in its proximity. Despite this, the NULA ${{\boldsymbol{z}}_3}$ still achieves satisfactory detection performance against target 3, thanks to its narrower cancelation notch, which results from its higher global length compared to ${{\boldsymbol{z}}_2}$.

• • Finally, the target moving at $v_b^{\left( 4 \right)} = 16m/s$ is detected with satisfactory performance by all the array configurations, since none of the arrays shows any notch in its proximity.

Figure 9. Minimum ${P_{fa}}$ required to detect each bin using the PC detector: (a) ${{\boldsymbol{z}}_1} = \left[ {0\,0.50\,1.00} \right]\lambda $; (b) ${{\boldsymbol{z}}_2} = \left[ {0\,0.75\,1.50} \right]\lambda $; (c) ${{\boldsymbol{z}}_3} = \left[ {0\,0.35\,1.00} \right]\lambda $; and (d) ${{\boldsymbol{z}}_4} = \left[ {0\,0.50\,1.50} \right]\lambda $.

In conclusion, the DVB-T based simulations, obtained by generating the clutter echoes as the signal scattered by a large set of angularly distributed scatterers and using realistic continuous waveforms, confirms the results of the simplified theoretical analysis based on statistical clutter description and ideal waveforms. In particular, it showed that the PC detectors operates effectively against the simulated scenario with both ULA and NULA arrays. Moreover, this analysis confirms that ULA configurations are affected by generally larger cancelation notches and smaller unambiguous velocity sectors. This affects the detection performance on some of the targets of interest. Conversely, the flexibility of the nonuniform spatial sampling provided by NULA configurations allows to achieve overall better performance on the whole velocity sector of interest. Thus, the realistic simulation analysis confirms the results of “Case study: passive radar system based on DVB-T”.

Conclusions

In this paper, we studied the case of a passive radar system mounted on a moving platform. Specifically, we addressed the problem of clutter suppression, which is a particularly critical task, due to the spread in Doppler of the clutter returns, working on two different but entwined fronts.

On the one hand, based on the signal model derived in “System geometry and signal model”, we proposed three detection schemes, deriving from the generalized LRT: an FC detector, a PC detector, and an APC detector. Each of the detectors has been recognized to consist in a specific combination of the output of $N$ nonadaptive DPCA-like filters. Each of the $N$ DPCA-like filters has a clutter notch with different characteristics, and their combinations allow to have a good response for target moving at the different velocities. They are especially appealing for low-cost, lightweight, and compact systems that can be installed on a moving platform, allowing to detect moving targets against clutter and noise without requiring the estimation and inversion of the CCM of STAP techniques.

From the system designs perspective, we explored the use of NULA configurations at receive to trade-off the notch width and the spacing between its replicas, without increasing the number of employed sensors. The joint use of nonadaptive processing schemes and NULA configurations allowed to preserve the inherent advantages of passive radars, namely low costs, limited complexity, and ease of deployment.

The comparative performance analysis between ULA and NULA configurations shows that the PC detector, despite its significantly lower computational load, achieves performance very close to the FC detector, requiring intensive computations, when operating together with an ULA. Leveraging the NULA setup to either narrow down the clutter cancelation notch width or increase the spacing between blind velocities causes the PC scheme to operate with limited yet appreciable losses compared to the FC scheme. Hence, the combination of NULA and PC detector, which provides a sensor with the mostly reduced hardware and power absorption requirements adequate for a low-cost, lightweight sensor, must tolerate some limited detection losses. When this compromise is not acceptable, solutions involving slightly increased hardware or computation can be obtained by either combining an NULA configuration with the FC detector or a ULA configuration with more antenna elements with the PC detector.

Clearly, the target detection performance could be negatively affected by the presence of a model mismatch. However, the results obtained against the simulated scenario of a DVB-T-based passive radar provide encouraging results. As a matter of fact, as detailed in “Case study: passive radar system based on DVB-T”, the clutter signal is simulated as the superposition of the returns from stationary scatterers, and thus it does not rely on the signal model derived in “System geometry and signal model”. Despite this, each of the proposed processing schemes allows to achieve almost perfect clutter suppression.

Future research lines will include the experimental validation of the proposed processing techniques and design strategies in order to investigate their robustness in real-world scenarios. This in turn will require the removal of some of the assumptions made in developing the considered model. Specifically, the perfect DPCA condition, essential for achieving perfect clutter suppression, imposes a severe quantization constraint on the array inter-element distances. As a matter of fact, for a fixed platform velocity and when the batch duration is equal to the OFDM symbol duration, only a handful of NULA configurations allows to guarantee the perfect DPCA condition. Therefore, our future research endeavors will be focused on introducing a flexible batching strategy, independent of the OFDM symbol structure, to decouple the array design from the processing, thus unlocking the full potential of NULA configurations.