Sub-carrier processing

Taking a digital television signal based on cyclic prefix (CP-OFDM) as an example, the received signal in the useful duration at k-th sub-carrier can be expressed as [Reference Yi, Wan, Yi and Cao18]:

(2)$$\eqalign{S_k & = c_1Q_k^T + \sum\limits_{n = 2}^{M_c} {c_ne^{{-}j2\pi f_k\tau _n^c }} Q_k^T \cr & \quad + \sum\limits_{m = 1}^{M_t} {\alpha _me^{{-}j2\pi ( {\,f_k-f_m^D } ) \tau _m^t }} U_{k, m}^T + W_k} $$

where Q_k = [C ₀, k, C ₁, k,…, C_{L −1}, k]^T, L is the OFDM block number used as snapshots, C _1,k represent the normalized modulation code of the l-th OFDM block in k-th subcarrier, and

(3)$$U_{k, l}\approx [ {C_{0, k}, \;C_{1, k}e^{\,j2\pi f_l^D T_s}, \;\ldots , \;C_{L-1, k}e^{\,j2\pi f_l^D T_s( {L-1} ) }} ] ^T$$

where T_S is the duration of one OFDM block. It can be seen from (2) that the samples of the direct-path signal and multi-path clutters at one sub-carrier are coherent with each other, whereas target echoes are uncorrelated to the clutter signals due to phase rotation introduced by the Doppler frequency. Therefore, all the zero-Doppler signals as shown in the first term of the right-hand side of (4) are integrated into a single term:

(4)$$\eqalign{S_k = & \left[{c_1 + \sum\limits_{n = 2}^{M_c} {c_ne^{{-}j2\pi f_k\tau_n^c }} } \right]Q_k^T \cr & \quad + \sum\limits_{m = 1}^{M_t} {\alpha _me^{{-}j2\pi ( {\,f_k-f_m^D } ) \tau _m^t }} U_{k, m}^T + W_k} $$

Oblique projection

Oblique projection operator is an extension of the orthogonal projection operators, which has attracted much attention in the field of signal processing [Reference Behrens and Scharf20]. Oblique projection technology has been used to interference and clutter suppression [Reference Hou, Mao and Li25, Reference Yi, Ji, Kirubarajan, Xie and Hu26], direction of arrival estimation [Reference Boyer and Bouleux21, Reference Tannaka and Imai27], target detection [Reference Sun, Xie, Hao and Yi28], and polarization diversity technology [Reference Mao, Liu, Hou, Hong, Guo and Deng22, Reference Cao, Zhang, Liang, Wen, Jin and Zhang24, Reference Wang, Zhang and Cao29]. Assume that S is a complex matrix of size n × m having full column rank and, likewise, that J is a complex matrix of size n × k having full column rank. Assume further that subspace 〈S〉 and 〈J〉 are disjoint, which requires m + k ≤ n. The H represents conjugate transpose symbols. The well-known formula to build a projection with subspace 〈S〉 is

(5)$${\bf P}_S = {\bf S}( {\bf S}^H{\bf S}) ^{{-}1}{\bf S}^H$$

where (•)^H represents conjugate transposition. (•)⁻¹ represents inverse operation. The orthogonal projection with subspace 〈S〉^⊥ is given by

(6)$${\bf P}_S^\bot = {\bf I}-{\bf P}_S$$

Now examine the matrix (S J) that is composed from the columns of S and of J. Assume that m + k < n, and the column rank of (S J) is less than n. The orthogonal projection onto the linear subspace 〈S J〉 is

(7)$${\bf P}_{SJ} = \left[{\matrix{ {\bf S} & {\bf J} \cr } } \right]\left[{\matrix{ {{\bf S}^H{\bf S}} & {{\bf S}^H{\bf J}} \cr {{\bf J}^H{\bf S}} & {{\bf J}^H{\bf J}} \cr } } \right]^{{-}1}\left[{\matrix{ {{\bf S}^H} \cr {{\bf J}^H} \cr } } \right]$$

We may decompose this orthogonal projection as follows:

(8)$${\bf P}_{SJ} = {\bf E}_J + {\bf E}_S$$

(9)$${\bf E}_J = \left[{\matrix{ {\bf S} & 0 \cr } } \right]\left[{\matrix{ {{\bf S}^H{\bf S}} & {{\bf S}^H{\bf J}} \cr {{\bf J}^H{\bf S}} & {{\bf J}^H{\bf J}} \cr } } \right]^{{-}1}\left[{\matrix{ {{\bf S}^H} \cr {{\bf J}^H} \cr } } \right]$$

(10)$${\bf E}_S = \left[{\matrix{ 0 & {\bf J} \cr } } \right]\left[{\matrix{ {{\bf S}^H{\bf S}} & {{\bf S}^H{\bf J}} \cr {{\bf J}^H{\bf S}} & {{\bf J}^H{\bf J}} \cr } } \right]^{{-}1}\left[{\matrix{ {{\bf S}^H} \cr {{\bf J}^H} \cr } } \right]$$

Formulas (9) and (10) may be simplified as follows:

(11)$${\bf E}_J = {\bf S}( {\bf S}^H{\bf P}_J^\bot {\bf S}) ^{{-}1}{\bf S}^H{\bf P}_J^\bot $$

(12)$${\bf E}_S = {\bf J}( {\bf J}^H{\bf P}_S^\bot {\bf J}) ^{{-}1}{\bf J}^H{\bf P}_S^\bot $$

The subspaces 〈S〉 and 〈J〉 are disjoint, so

(13)$${\bf E}_J\left[{\matrix{ {\bf S} & {\bf J} \cr } } \right] = \left[{\matrix{ {\bf S} & 0 \cr } } \right]$$

(14)$${\bf E}_S\left[{\matrix{ {\bf S} & {\bf J} \cr } } \right] = \left[{\matrix{ 0 & {\bf J} \cr } } \right]$$

As shown in (13) and (14), the oblique projection operator E_J and E_S are orthogonal to the subspace 〈J〉 and 〈S〉, respectively. The subspace 〈S〉 and 〈J〉 are not required to be orthogonal.

Extensive cancelation algorithm

ECA is a clutter suppression algorithm based on interference subspace projection [Reference Wan, Yi, Zhao and Ke2]. The algorithm is to project the echo signals of the surveillance channel into the direct-path signal and its time delay spread space to eliminate clutter. Construct a subspace matrix X consisting of the reference signal as follows:

(15)$${\bf X} = \left[{\matrix{ {S_{ref}( 1 ) } & {S_{ref}( 2 ) } & \cdots & {S_{ref}( N ) } \cr } } \right]^T$$

where S_ref(n), n = 1, 2, …, N represent reference signal and (•)^T represents transposition.

The subspace projection method is used to suppress the direct-path signal and multi-path clutter in the surveillance channel. The orthogonal projection matrix can be represented by the subspace matrix X formed by the reference signal

(16)$${\bf P} = {\bf I}-{\bf X}( {{\bf X}^H{\bf X}} ) ^{{-}1}{\bf X}^H$$

Projecting the echo signal of the surveillance channel to the orthogonal subspace of the reference signal, the residual signal S _ECA in the surveillance channel can be expressed as follows:

(17)$$S_{ECA} = [ {{\bf I}-{\bf X}{( {{\bf X}^H{\bf X}} ) }^{{-}1}{\bf X}^H} ] S_{surv} = {\bf P}S_{surv}$$

After the orthogonal projection processing, the direct-path signal and clutter in the surveillance channel are canceled, and the target signal is retained.

Target suppression

It is assumed that the surveillance signal only contains direct-path signal, target signal, and Gaussian white noise, and the polarization state vectors of direct-path signal and target signal can be obtained by estimation, and are expressed as follows:

(19)$${\bf D} = \left[{\matrix{ {\cos \gamma_d} \cr {\sin \gamma_de^{\,j\varphi_d}} \cr } } \right]$$

(20)$${\bf T} = \left[{\matrix{ {\cos \gamma_t} \cr {\sin \gamma_te^{\,j\varphi_t}} \cr } } \right]$$

Using the method of oblique projection in Section “Theoretical basis,” the oblique projection operator can be express as

(21)$${\bf E}_{op} = {\bf D}( {{\bf D}^H{\bf P}_T^\bot {\bf D}} ) ^{{-}1}{\bf D}^H{\bf P}_T^\bot $$

(22)$${\bf E}_{op}\left[{\matrix{ {\bf D} & {\bf T} \cr } } \right] = \left[{\matrix{ {\bf D} & 0 \cr } } \right]$$

In (22), the target signal is suppressed. But under actual conditions, the energy of target signal is very low, and its polarization state vector is difficult to estimate. In order to solve this problem, blind adaptive oblique projection technology is used for reference [Reference Behrens and Scharf20–Reference Cao, Zhang, Liang, Wen, Jin and Zhang24].

We define the surveillance signal covariance matrix as below:

(23)$${\bf R}_{{\rm sur}v} = {\bf R}_{DT} + \sigma ^2{\bf {\rm I}}{\rm} = \left[{\matrix{ {\bf D} & {\bf T} \cr } } \right]\left[{\matrix{ {R_D} & 0 \cr 0 & {R_T} \cr } } \right]\left[{\matrix{ {\bf D} & {\bf T} \cr } } \right]^H + \sigma ^2{\bf {\rm I}}$$

where ${\bf R}_{{\rm sur}v} = {\bf S}_{surv}{\bf S}_{surv}^H$, ${\bf S}_{surv} = {\bf D}s_D( t ) + {\bf T}s_T( t ) + {\bf W}_{surv}$, s _D(t) = cd(t), d(t) is the signal from transmitter, c represents the amplitude of direct-path signal; $s_T( t ) = \alpha d( t-\tau ) e^{j2\pi f_{}^D t}$, α represents the amplitude of target, τ represents the delay with respect to the direct-path signal, $f_{}^D$ represents the Doppler frequency of the target; ${\bf W}_{surv}$ $\left({{\bf W}_{surv} = \left[{\matrix{ {w_H} & {w_V} \cr } } \right]} \right)$ represents the Gaussian white noise of H-polarized channel and V-polarized channel; ${\bf R}_{DT} = [ {{\bf D}s_D( t ) + {\bf T}s_T( t ) } ] [ {{\bf D}s_D( t ) + {\bf T}s_T( t ) } ] ^H$, R_D (R _D = [s _D(t)][s _D(t)]^H) is the eigenvalue of R_DT related to the direct-path signal. R_T (R _T = [s _T(t)][s _T(t)]^H) is the eigenvalue of R_DT related to the target. σ ² represents noise variance. I represents the identity matrix.

Then, we use the method proposed in [Reference Coon, Sandell, Beach and Mcgeehan30, Reference Boumard31] to estimate the noise variance σ ², and get R_DT as shown below:

(24)$${\bf R}_{DT} = {\bf R}_{surv}-\sigma ^2{\bf {\rm I}} = \left[{\matrix{ {\bf D} & {\bf T} \cr } } \right]\left[{\matrix{ {R_D} & 0 \cr 0 & {R_T} \cr } } \right]\left[{\matrix{ {\bf D} & {\bf T} \cr } } \right]^H$$

According to the principle of blind adaptive oblique projection, the modified oblique projection operator as follows:

(25)$${\bf E}_{b-op}^{} = {\bf D}( {{\bf D}^H{\bf R}_{DT}^ + {\bf D}} ) ^{{-}1}{\bf D}^H{\bf R}_{DT}^ + $$

The symbol (•)⁺ represents the pseudo-inverse. We define

(26)$${\bf A} = \left[{\matrix{ {\bf D} & {\bf T} \cr } } \right]$$

(27)$${\bf B} = \left[{\matrix{ {R_D} & 0 \cr 0 & {R_T} \cr } } \right]$$

Then, we can obtain the pseudo-inverse of matrix A:

(28)$${\bf A}^ + { = } \left[{\matrix{ {{( {{\bf D}^H{\bf P}_T^\bot {\bf D}} ) }^{{-}1}{\bf D}^H{\bf P}_T^\bot } \cr {{( {{\bf T}^H{\bf P}_D^\bot {\bf T}} ) }^{{-}1}{\bf T}^H{\bf P}_D^\bot } \cr } } \right]$$

where ${\bf P}_T^\bot$ represents the orthogonal projection matrix of T. ${\bf P}_D^\bot$ represents the orthogonal projection matrix of D. The covariance matrix of surveillance channel signal without noise can be simplified as

(29)$${\bf R}_{DT}^ + { = } ( {{\bf AB}{\bf A}^H} ) ^ + $$

In order to remove the target signal and extract the reference signal, we need

(30)$${\bf E}_{b-op}^{} \left[{\matrix{ {\bf D} & {\bf T} \cr } } \right] = \left[{\matrix{ {\bf D} & 0 \cr } } \right]$$

We can plug (28) and (29) into (25), and get

(31)$$\eqalign{& {\bf E}_{b-op} = {\bf D}({\bf D}^H{\bf R}_{DT}^ + {\bf D})^{-1}{\bf D}^H({\bf AB}{\bf A}^H)^ + \cr & \quad \quad \quad = \;{\bf D}({\bf D}^H{\bf R}_{DT}^ + {\bf D})^{-1}({\bf A}^ + {\bf D})^H{\bf B}^{-1}{\bf A}^ + \cr & \quad = {\bf D}({\bf D}^H{\bf R}_{DT}^ + {\bf D})^{-1}\left[ {\matrix{ {{({\bf D}^H{\bf P}_T^\bot {\bf D})}^{-1}{\bf D}^H{\bf P}_T^\bot {\bf D}} \hfill \cr {{({\bf T}^H{\bf P}_D^\bot {\bf T})}^{-1}{\bf T}^H{\bf P}_D^\bot {\bf D}} \hfill \cr } } \right]^H \cr & \quad {\bf B}^{-1}\left[ {\matrix{ {{({\bf D}^H{\bf P}_T^\bot {\bf D})}^{-1}{\bf D}^H{\bf P}_T^\bot } \hfill \cr {{({\bf T}^H{\bf P}_D^\bot {\bf T})}^{-1}{\bf T}^H{\bf P}_D^\bot } \hfill \cr } } \right] \cr & \quad = {\bf D}({\bf D}^H{\bf R}_{DT}^ + {\bf D})^{-1}\left( {\left[ {\matrix{ {\bf I} \hfill \cr 0 \hfill \cr } } \right]} \right)^H{\bf B}^{-1}\left[ {\matrix{ {{({\bf D}^H{\bf P}_T^\bot {\bf D})}^{-1}{\bf D}^H{\bf P}_T^\bot } \hfill \cr {{({\bf T}^H{\bf P}_D^\bot {\bf T})}^{-1}{\bf T}^H{\bf P}_D^\bot } \hfill \cr } } \right]} $$

Then, combined with (30) and (31), we can obtain

(32)$$\eqalign{{\bf E}_{b-op}^{} {\bf T} & = {\bf D}({\bf D}^{H}{\bf R}_{DT}^{+} {\bf D} ) ^{-1} \left({\left[{\matrix{{\bf I} \cr 0 \cr}}\right]}\right)^{H}{\bf B}^{-1} \left[{\matrix{{{({{\bf D}^H{\bf P}_T^\bot {\bf D}})}^{-1}{\bf D}^H{\bf P}_T^\bot {\bf T}} \cr {{( {{\bf T}^H{\bf P}_D^\bot {\bf T}} ) }^{{-}1}{\bf T}^H{\bf P}_D^\bot {\bf T}} \cr }}\right] \cr & \quad \quad \quad \quad \quad \quad \quad \quad = {\bf D}({{\bf D}^H{\bf R}_{DT}^ + {\bf D}} ) ^{{-}1} \left({ \left[{ \matrix{ {\bf I} \cr 0 \cr }}\right]}\right)^H {\bf B}^{{-}1} \left[{ \matrix{ 0 \cr {\bf I} \cr } } \right] \cr & = {\bf D}({{\bf D}^H{\bf R}_{DT}^ + {\bf D}} )^{{-}1} \left({ \left[{ \matrix{ {\bf I} \cr 0 \cr } } \right]} \right)^H \left[{ \matrix{{{( {R_D} ) }^{{-}1}} & 0 \cr 0 & {{( {R_T} ) }^{{-}1}} \cr } } \right] \left[{ \matrix{ 0 \cr {\bf I} \cr } } \right] = 0}$$

As shown in (32), the target signal is completely removed. However, in the actual working process of the SDPPR, it will inevitably be affected by the complex and variable electromagnetic environment. The electromagnetic wave received by the radar will exhibit a partially polarized wave characteristic. As shown in (1), clutter will cause random modulation of amplitude on the direct-path signal [Reference Yi, Wan, Yi and Cao18].

The composite clutter signal that contains both direct-path signal and random scattered clutter signals is regarded as a random amplitude modulation [Reference Yi, Wan, Yi and Cao18, Reference Behrens and Scharf20]. In the presence of clutter, the reception signal model is shown as below:

(33)$$\eqalign{{\bf S} = & \left[{\matrix{ {\cos \gamma_d} \cr {\sin \gamma_de^{\,j\varphi_d}} \cr } } \right]c_1d( t) + \sum\limits_{n = 2}^{M_C} {\left[{\matrix{ {\cos \gamma_c^n } \cr {\sin \gamma_c^n e^{\,j\varphi_c^n }} \cr } } \right]c_nd( t-\tau _n) } \cr & + \sum\limits_{m = 1}^{M_t} {\left[{\matrix{ {\cos \gamma_t^m } \cr {\sin \gamma_t^m e^{\,j\varphi_t^m }} \cr } } \right]\alpha _md( t-\tau _t) e^{\,j2\pi f^Dt}} + \left[{\matrix{ {w_h} \cr {w_v} \cr } } \right]} $$

where γ _d and ϕ _d represent polarization angle and polarization phase difference of the direct-path signal, respectively. $\gamma _c^n$ and $\varphi _c^n$ represent polarization angle and polarization phase difference of clutter, respectively. $\gamma _t^m$ and $\varphi _t^m$ represent polarization angle and polarization phase difference of target, respectively.

According to (4), we rewrite (33) and get

(34)$$\eqalign{{\bf S} = & \left[{\matrix{ {( {1 + \beta_h} ) \cos \gamma_d} \cr {( {1 + \beta_v} ) \sin \gamma_de^{\,j\varphi_d}} \cr } } \right]\varepsilon d( t) \cr & + \sum\limits_{m = 1}^{M_t} {\left[{\matrix{ {\cos \gamma_t^m } \cr {\sin \gamma_t^m e^{\,j\varphi_t^m }} \cr } } \right]\alpha _md( t-\tau _t) e^{\,j2\pi f^Dt}} + \left[{\matrix{ {w_h} \cr {w_v} \cr } } \right]} $$

where ε represents the amplitude of the composite clutter signal, β_h represents amplitude modulation factor of H-polarized channel. β_v represents channel amplitude modulation factor of V-polarized. β_h and β_v follow normal distribution.

In order to avoid the influence of amplitude modulation in (34), we will use sub-carrier processing technology. Convert (34) into sub-carrier expression

(35)$$\eqalign{{\bf S}_k = & \left[{\matrix{ {c_{1, h} + \sum\limits_{n = 2}^{M_c} {c_{n, h}e^{{-}j2\pi f_k\tau_n^{} }} } \cr {c_{1, v} + \sum\limits_{n = 2}^{M_c} {c_{n, v}e^{{-}j2\pi f_k\tau_n^{} }} } \cr } } \right]Q_k^T \cr & + \left[{\matrix{ {\sum\limits_{m = 1}^{M_t} {\alpha_{m, h}e^{{-}j2\pi ( {\,f_k-f_m^D } ) \tau_m^{} }} } \cr {\sum\limits_{m = 1}^{M_t} {\alpha_{m, v}e^{{-}j2\pi ( {\,f_k-f_m^D } ) \tau_m^{} }} } \cr } } \right]U_{k, m}^T + \left[{\matrix{ {W_{k, h}} \cr {W_{k, v}} \cr } } \right]} $$

where c _1,h and c _1,v represent the complex amplitude of direct-path signal in the H-polarized channel and V-polarized channel, respectively. c_{n ,h} and c_{n ,v} represent the complex amplitude of the n-th stationary multi-path clutters in the H-polarized channel and V-polarized channel, respectively. τ_n represents the delay of the n-th stationary multi-path clutters. α_{m ,h} and α_{m ,v} represent the complex amplitude of target in the H-polarized channel and V-polarized channel, respectively. τ_m and $f_m^D$ are the delay and the Doppler frequency of m-th target, respectively. M_c and M_t represent the number of multi-path clutter and target, respectively. W_{k ,h} and W_{k ,v} represents white Gaussian noise in the H-polarized channel and V-polarized channel, respectively.

$\left[{c_{1, h} + \sum\nolimits_{n = 2}^{M_c} {c_{n, h}e^{{-}j2\pi f_k\tau_n^{} }} } \right]$ and $\left[{c_{1, v} + \sum\nolimits_{n = 2}^{M_c} {c_{n, v}e^{{-}j2\pi f_k\tau_n^{} }} } \right]$ could be viewed as the composite weight of clutters in the H-polarized and V-polarized channels, and in each sub-carrier the composite weight is constant.

Considering single target, (35) may be simplified as

(36)$${\bf S}_k = {\bf D}_k^{cs} \varepsilon _kQ_k^T + {\bf T}_k^t \alpha _ke^{{-}j2\pi ( {\,f_k-f^D} ) \tau ^t}U_k^T + \left[{\matrix{ {W_{k, h}} \cr {W_{k, v}} \cr } } \right]$$

${\bf D}_k^{cs} = \left[{\matrix{ {\cos \gamma_k^c } \cr {\sin \gamma_k^c e^{j\varphi_k^c }} \cr } } \right]$ represents polarization vector of composite clutter signal in the k-th sub-carrier. $\gamma _k^c$ and $\varphi _k^c$ represent polarization angle and polarization phase difference of composite clutter signal. ${\bf T}_k^t = \left[{\matrix{ {\cos \gamma_k^t } \cr {\sin \gamma_k^t e^{j\varphi_k^t }} \cr } } \right]$ represents polarization vector of target in the k-th sub-carrier. $\gamma _k^t$ and $\varphi _k^t$represent polarization angle and polarization phase difference of target. $\varepsilon _k$ and α _k represent the amplitude of the composite clutter signal and target in the k-th sub-carrier.

The blind adaptive oblique projection operator in each sub-carrier can be expressed as follows:

(37)$${\bf E}_k^{b-op} = {\bf D}_k^{cs} ( {{( {\bf D}_k^{cs} ) }^H{( {\bf R}_k^{DT} ) }^ + {\bf D}_k^{cs} } ) ^{{-}1}( {\bf D}_k^{cs} ) ^H( {\bf R}_k^{DT} ) ^ + $$

(38)$${\bf R}_k^{DT} = ( {{\bf A}_k{\bf B}_k{\bf A}_k^H } ) ^ + $$

where ${\bf A}_k = \left[{\matrix{ {{\bf D}_k^{DT} } & {{\bf T}_k^t } \cr } } \right]$ and ${\bf B}_k = \left[{\matrix{ {R_{k, D}} & 0 \cr 0 & {R_{k, T}} \cr } } \right]$.

Combining (36) and (37), we can effectively suppress the target signal:

(39)$$\eqalign{{\bf S}_k^{b-op} & = {\bf {\rm E}}_k^{b-op} {\bf S}_k = {\bf D}_k^{cs} ( {{( {\bf D}_k^{cs} ) }^H{( {\bf R}_k^{DT} ) }^ + {\bf D}_k^{cs} } ) ^{{-}1}( {\bf D}_k^{cs} ) ^H( {\bf R}_k^{DT} ) ^ + {\bf S}_k \cr& \quad= {\bf D}_k^{cs} \varepsilon _kQ_k^T + \left[{\matrix{ {\tilde{W}_k^h } \cr {\tilde{W}_k^v } \cr } } \right]} $$

where $\tilde{W}_k^h = E_k^{b-op} W_k^h$ and $\tilde{W}_k^v = E_k^{b-op} W_k^v$. As shown in (39), the composite clutter signal is extracted and the target signal is suppressed.

Reference signal purification

Performing IDFT on the data obtained in the previous step to convert the frequency domain signal into the time domain. At this point, the target signal has been suppressed, leaving only the composite signal containing the direct-path wave and zero-Doppler multi-path clutter with different time delays. In order to obtain a purer reference signal, we need to further suppress multi-path clutter. Therefore, we will study in detail the use of ECA to suppress multi-path clutter in this section.

Assume that the transmitted signal is V-polarized polarization, the energy of direct-path signal received by the V-polarized channel is much stronger than that of the H-polarized channel in general. Thus, we consider extracting the direct-path signal in the V-polarized channel and construct composite signal delay subspace matrix C_v as follows:

(40)$${\bf C}_v = \left[{\matrix{ 0 & {S_{sc}^v ( 1) } & {S_{sc}^v ( 2) } & \cdots & {S_{sc}^v ( N-1) } \cr 0 & 0 & {S_{sc}^v ( 1) } & \cdots & {S_{sc}^v ( N-2) } \cr \vdots & \vdots & \vdots & \ddots & \vdots \cr 0 & 0 & 0 & \cdots & {S_{sc}^v ( N-K + 1) } \cr } } \right]^T$$

where $S_{sc}^v ( n) \,\,\matrix{ {n = 1, \;2, \;\ldots , \;N} \cr }$ represents composite signal in the V-polarized channel. N represents the total data length. K represents the number of cancelation distance cells. In matrix C_v, the first row to the (K − 1)-th row represent delay of one range bin to (K − 1)-th range bin in sample signal, respectively.

According to the orthogonal characteristics of the subspace, the orthogonal projection operator can be obtained as

(41)$${\bf P}_{sc}^v = {\bf I}-{\bf C}_v( {{\bf C}_v^H {\bf C}_v} ) ^{{-}1}{\bf C}_v^H $$

The residual signal $S_{ECA, d}^v$ in the V-polarized channel of surveillance signal can be expressed as follows:

(42)$$S_{ECA, d}^v = [ {{\bf I}-{\bf C}_v{( {{\bf C}_v^H {\bf C}_v} ) }^{{-}1}{\bf C}_v^H } ] S_{sc}^v = {\bf P}_{sc}^v S_{sc}^v $$

where $S_{ECA, d}^v$ is the purified reference signal.

The signal processing flow of SDPPR by using the new method of reference extraction is shown in Fig. 2.

Fig. 2. Signal processing flow of combined processing method.

As shown in Fig. 2, the echo signal has been received in the surveillance channel by the single dual-polarization antenna. After the echo signal is processed by sub-carrier, blind adaptive oblique projection technology and ECA, the pure reference signal is obtained. Then, we can suppress the direct-path signal and multi-path clutters by employing signal processing method of [Reference Yi, Wan, Li and Leung32].

Experimental scheme

The method proposed in this paper is an effective alternative when the SDPPR system RSR fails. In order to compare the performance of the proposed method, a reference antenna was added on the SDPPR system.

We will obtain the reference signal in two ways. (1) The reference signal is extracted from the reference channel using RSR method. The direct-path signal with high SNR can be obtained by using reference antenna point to the transmitting station. (2) Reference signal extraction in SDPPR based on multi-processing (RSMP) method. When the direct-path signal SNR of SDPPR system surveillance channel is low, the reference signal cannot be reconstructed, and the radar system fails. The combined processing method proposed in this paper can effectively extract the reference signal and restore the working state of the radar system.

Experimental setup

The experimental system consists of reference antenna, surveillance antenna, multi-channel receiver, and upper computer. The surveillance antenna [Reference Cao, Gong, Yi, Wang and Wan33] is shown in Fig. 4. The whole antenna system is composed of one pair of dual-polarization antenna and one pair of single polarization antenna. Dual-polarization antenna is called surveillance antenna and used to receive surveillance signal. Single polarization antenna is used to receive reference signal and called reference antenna. The polarization isolation of surveillance antenna in the frequency band (710–718 MHz) is >20 dB. The gain of surveillance antenna is ~10 dB.

Fig. 4. Antenna system.

As shown in Fig. 5, the green sector area represents surveillance antenna beam direction. The blue sector area represents reference antenna beam direction. The yellow elliptical area represents ground and buildings which bring clutter. The detection target was Boeing 737 plane.

Fig. 5. Experimental geometry.

Experimental analysis

The SDPPR system needs to reconstruct the reference signal from the surveillance channel. In order to verify the effectiveness of the method proposed in this paper, the surveillance antenna of the SDPPR system was pointed at an angle of 120° with the transmitter, so as to weaken the influence of direct-path signal on the surveillance signal. In this scenario, we can obtain the surveillance signal spectrum as shown in Fig. 6(a), and the constellation diagram on surveillance signal as shown in Fig. 6(b). When using the reference antenna to collect data, the received signal spectrum is shown in Fig. 7(a), and the signal constellation diagram is shown in Fig. 7(b).

Fig. 6. Surveillance channel data: (a) signal frequency spectrum and (b) constellation diagram.

Fig. 7. Reference channel data: (a) signal frequency spectrum and (b) constellation diagram.

In the effective bandwidth, the signal spectrum amplitude of the reference channel is at least 10 dB stronger than that of the surveillance channel, which also means that the SNR of the direct-path signal in the reference channel is much stronger than that in the surveillance channel. Therefore, the reference channel signal constellation is more focused than the surveillance channel constellation. In this scenario, the reference signal cannot be reconstructed from the surveillance channel.

The experimental data were collected from a Boeing 737 in the radar surveillance area. After obtaining the reference signal, we perform time-domain filtering and matched filtering on the V-polarized channel and H-polarized channel data, respectively. When the reference signal is obtained from the reference channel using signal reconstruction, the range Doppler (RD) map of the H-polarized channel and the V-polarized channel are shown in Fig. 8. When using the method in this paper to obtain the reference signal from the surveillance channel, the RD map of the H-polarized channel and the V-polarized channel are shown in Fig. 9.

Fig. 8. RD maps in the 80-th frame in RSR: (a) RD map of V channel and (b) RD map of H channel.

Fig. 9. RD maps in the 80-th frame in RSMP: (a) RD map in V channel and (b) RD map in H channel.

It can be seen from Figs 8 and 9, whether it is the H-polarized channel or the V-polarized channel, the reference signal obtained based on the reconstruction method has a better clutter suppression effect. This is because the SNR of the reference signal extracted from the surveillance channel is lower than that obtained from the reference channel reconstruction. However, the target peak is still evident in Fig. 9.

The RD map data of the H-polarized and V-polarized channels were fused [Reference Yi, Zhu and Cao34], and then the cell-average constant false alarm rate detection was performed, and the detection results are shown in Fig. 10. The red dots represent the target detection points. The green line represents the target real track obtained from the automatic-dependent surveillance broadcast system.

Fig. 10. Result of target detection: (a) detection result in RSR and (b) detection result in RSMP.

The detection result by using the RSR method as shown in Fig. 10(a). The target detection points are very continuous between the 50th and 100th seconds. The detection result by using the RSMP method as shown in Fig. 10(b). Although the reference signal cannot be reconstructed from the surveillance channel, the detection points obtained by using the RSMP method are still satisfactory.

In the scenario set in this paper, the statistics of the target detection results are shown in Table 2.

Table 2. Target detection statistical result

In Table 2, we can clearly see that the best result is obtained by using the reference signal reconstruction from reference channel (RSR-RC) method, followed by the reference signal extraction in multi-processing from surveillance channel (RSMP-SC) method. The reference signal reconstruction from surveillance channel (RSR-SC) method cannot obtain reference signal, and the target detection fails. For the SDPPR system, there are only one surveillance antenna and no reference antenna, so the conventional method to obtain reference signal is signal reconstruction. However, in urban environments, due to factors such as terrain and interference, signal reconstruction may fail. The method proposed in this paper can solve the above problems and is an effective alternative.