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 [Reference Palmer, Cristallini and Kuschel1, Reference Wojaczek, Colone, Cristallini and Lombardo9]). 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.

(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).

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 [Reference Lombardo21] and Lombardo et al. [Reference Lombardo, Colone and Pastina22], 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.

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. [Reference Quirini, Blasone, Colone and Lombardo11] 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.