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A clean signal reconstruction approach for coherently combining multiple radars
EURASIP Journal on Advances in Signal Processing volume 2018, Article number: 47 (2018)
Abstract
Distributed radars have the potential to combine coherently for achieving a high signaltonoise ratio (SNR) while maintaining a moderate antenna size. The key to coherently combining multiple radars is obtaining accurate coherent parameters (CPs), which are used to adjust the transmitting/receiving time and phase of each radar. One approach for CP estimation is to transmit orthogonal waveforms. However, ideally, orthogonal waveforms occupying the same frequency band may not be found in practice. Crosscorrelation energy leakage exists between nonorthogonal waveforms, which seriously impairs the accurate acquisition of CPs. To solve this problem, we propose a clean signal reconstruction approach for CP estimation. This approach reconstructs clean echoes by gradually stripping out the crosscorrelation energy leakage with a reconstructioneliminationreconstruction framework. And CPs are obtained from these reconstructed clean echoes. Since the majority of crosscorrection energy leakages are eliminated, enhanced CP estimation performance can be achieved. Verified simulations are designed for a dual radar scenario. Results show that the proposed approach significantly improves the performance of CP estimation while reducing the SNR requirement for coherently combining multiple radars.
1 Introduction
A single radar has a theoretical maximum signaltonoise ratio (SNR) for a given size target at a given range, which directly affects radar’s ability to detect, track, and identify the target [1]. There are mainly two solutions to raise SNR. One is to develop highly sensitive radar with a large antenna. However, such large antenna systems are costly and not easily transportable. Alternatively, by jointly processing the transmitted and received signals from several colocated small independent radars, it is possible to combine them into a coherently functioning system for a significantly enhanced SNR [1–4]. When N radars are coherently combined, an N^{3} SNR gain over a single radar can be acquired, i.e., full coherence is achieved on both transmitting and receiving [5].
Coherently combining multiple radars is challenging, especially wideband radars. Signal processing in terms of timing and phase adjustments should be carried out to calibrate the incoherence induced by disparate propagation paths and synchronization errors. If relative positions of the radars are known to a fraction of a wavelength, accurate time delay or phase corrections can be implemented in accordance with this prior knowledge [2, 6]. However, measuring accurate relative positions is difficult and subject to increasing errors as the separation of radar antennas increases, if the radars are not relatively fixed [1]. Furthermore, only knowing the relative separation between the radars is not sufficient for multiple radars’ coherent combination, since it does not account for the internal electrical difference caused by synchronization errors. Hence, an additional calibration process is needed.
Another approach relies upon periodically separating the monostatic and bistatic echoes at each radar [3]. The monostatic echo corresponds to its transmitted radar signal, while the bistatic echo corresponds to a transmitted radar signal from the other radar. By comparing the relative time and phase relationship between these returns, coherence parameters (CPs), i.e., fine time and phase correcting values, can be estimated. These parameters are then used to adjust the transmitting/receiving time and phase of each radar for obtaining the full coherence SNR gain. A key feature of this approach is that relative positions of radar are not critical and coherently combining multiple radars is achieved via targetbased calibration. Inspired by such idea, relevant works are springing up, including CPs’ CramerRao bound (CRB) and the theoretical performance bounds analyze [7–9], scheme for dual radar coherent combination based on steppedfrequency signal [10–12], and dual radar coherent combination laboratory and field experiments [13, 14].
To separate the monostatic and bistatic returns, two different ways have been reported: keeping these echoes separated in the time domain, i.e., using timedivision multiplexing (TDM) technique [1] or using orthogonal waveforms [2, 3]. To note, the scattering response of the target is commonly timevarying and using TDM techniques will undermine the coherence among separated echoes. Hence, the latter approach is a better choice. When orthogonal waveforms are transmitted to estimate CPs, the monostatic and bistatic echoes can be separated by matched filtering. Conceptually, these orthogonal waveforms should have the same operating bandwidth and center frequency. Otherwise, CP estimates may be subject to additional errors due to the frequencyselective target scattering response. Aside from the common spectral requirement, the waveforms must be as nearly orthogonal as possible. However, it is hard to find ideally orthogonal waveforms occupying the same frequency band in practice. Crosscorrelation energy leakage between nonideally ones cannot be ignored, which will affect the accurate acquisition of CPs.
In this paper, to the best of our knowledge, the influence of nonorthogonal waveforms on CP estimation is analysed in detail for the first time. Theoretical analyses demonstrate that the crosscorrelation energy leakage stemmed from waveform nonorthogonality will introduce additional estimation errors and problems. Since the crosscorrelation energy leakage is the dominant reason for impairing CP estimates, a CP estimation approach based on clean signal reconstruction is proposed. The basic idea of this approach is to reconstruct the clean signal that is not or rarely interfered by crosscorrelation energy leakage. Once the clean single is reconstructed, improved CP estimates can be obtained via these reconstructed clean signals. And the effectiveness of our proposed approach is verified by a dual radar coherent combination example transmitting up and downchirp waveforms to estimate CPs.
The remainder of this paper is organized as follows. Section 2 introduces the principle of coherently combining multiple radars. Section 3 presents the CP estimation framework and discusses the influence of nonorthogonal waveforms on CP acquisition. The clean signal reconstruction approach for estimating CPs and validated simulations are detailed in Sections 4 and 5. Conclusions are given in Section 6.
2 Principle of coherently combining multiple radars
As radars are distributed, theoretically, each radartransmitted signal cannot superimpose coherently at the target due to disparate propagation paths from radars to the target and synchronization errors (The difference of target scattering characteristics among different radars can be approximately neglected, since radars are assumed to be nearly colocated compared to the target detection range). For the same reason, incoherence also exists among the echoes received by radars. And such incoherence among signals can be characterized as time and phase misalignments. Hence, to coherently combine multiple radars, i.e., achieve both coherentontransmit and coherentonreceive, signal processing concerning time and phase adjustments should be carried out to calibrate the incoherence induced by the above reasons. Put it more concretely, if the signal transmitted by radar 1 (reference radar) arrives at the target after the signal transmitted by radar 2, proper time and phase adjustments for radar 2 are needed to slow down and retard its transmitting time and phase appropriately.
Let \(\phantom {\dot {i}\!}s(t  {\kappa _{k}}){e^{j2\pi {f_{c}}({t  {\kappa _{k}}}) + j{\varphi _{k}}}}\) denote the transmitted signal of radar k, s(t) is the common baseband signal of all radars, κ_{k} is the time synchronization error of radar k compared to the reference clock, and φ_{k} is the initial phase of radar k. Then, these signals propagated to the target can be expressed as
where, f_{c} is the carrier frequency, τ_{k}=R_{k}/c, R_{k} is the range from radar k to the target (for the multiscatterer target model case, R_{k} denotes the range from radar k to the target scattering centre), and c is the speed of light.
Apparently, from (1), the arrival time and phase of these signals are misaligned. To enhance the radio frequency (RF) electric field impinging upon the target, i.e., achieve coherentontransmit, fine time and phase adjustments are necessary. Without loss of generality, if radar 1 is the reference radar, it is essential to adjust the transmitting time and phase of other radars. And the adjusted signals can be expressed as \(s\left ({t  {\kappa _{k}} + \Delta \tau _{k}^{t}}\right){e^{j2\pi {f_{c}}({t  {\kappa _{k}}}) + j{\varphi _{k}}}}{e^{ j\Delta \phi _{k}^{t}}}\), k=2,⋯,N, where
are the time and phase correcting values (transmit CPs) of radar k. To note, as modern radars are assumed to be used, in which digital waveform generators allow independent control of time and phase.
Let \(\phantom {\dot {i}\!}u(t){e^{j2\pi {f_{c}}t}}\) denote the reflected signal at the target, then this reflected signal received by radar l after downconversion can be expressed as (assuming each radar can realize coherent reception)
For the same reason, to achieve coherentonreceive, appropriate time and phase shifts are also needed for the echo received by other radars except radar 1. And the adjusting echo can be expressed as \(u \left (t  {\tau _{l}} + {\kappa _{l}} + \Delta \tau _{l}^{r} \right){e^{j2\pi f_{c} \tau _{l} + j2\pi {f_{c}}{\kappa _{l}}  j{\varphi _{l}}}}{e^{ j\Delta \phi _{l}^{r}}}\), l=2,⋯,N, where
denote the time and phase correcting values (receive CPs) of radar l. Note that the superscript t is the abbreviation for “transmit,” while the superscript r is the abbreviation for “receive.” Apparently, \(\Delta \tau _{1}^{t}=\Delta \tau _{1}^{r}=\Delta \phi _{1}^{t}=\Delta \phi _{1}^{r}\), the remaining CPs to be estimated are \(\left \{\Delta \tau _{k}^{t},\Delta \tau _{l}^{r},\Delta \phi _{k}^{t},\Delta \phi _{l}^{r}\right \},k,l=2,\cdots,N\). For convenience, \(\left \{\Delta \tau _{k}^{t},\Delta \tau _{l}^{r}\right \}\) and \(\left \{\Delta \phi _{k}^{t},\Delta \phi _{l}^{r}\right \}\) are collectively called time and phase CPs.
3 Problem formulation
3.1 Signal model and CP estimation framework
Obviously, as the existence of synchronization errors, transmit and receive CPs of same radar are not identical. To estimate CPs, signals received by each radar have to be separated into monostatic and bistatic echoes firstly. It is the reason why orthogonal waveforms are transmitted. Assuming there is a target with Qfixed nonscintillating scatterers in the space, in the state of each radar transmitting orthogonal waveforms, the signal received by radar l after downconversion can be expressed as (approximation is done based on the above assumption that radars are nearly colocated)
where \(\bar \xi ^{q}\) denotes the scattering coefficient of scatterer q; τ_{lk}=τ_{k}+τ_{l}+κ_{k}−κ_{l} is the propagation time from radar k, to target scattering center, to radar l; Δτ^{q} is the corresponding twoway propagation delay from scatterer q to target scattering center in the radar line of sight (LOS); s_{k}(t) denotes the orthogonal waveform transmitted by radar k; w_{l}(t) denotes the noise introduced during radar l reception, which is assumed to be a zeromean, complex white Gaussian random process; \({s^{\prime }}(t)= \sum _{q=1}^{Q}\bar {\xi }^{q} s_{k}(t\Delta \tau ^{q})e^{j2\pi f_{c}\Delta \tau ^{q}}\); and \(\phantom {\dot {i}\!}\xi = e^{j2\pi f_{c} \tau _{11}}\).
By matched filtering (MF), the N received signals {r_{l}(t),l=1,⋯,N} can be separated into N^{2} echoes. And each separated echo characterizes a distinct propagation path from radar k to radar l, which can be formulated as
where \(A^{\prime }_{k}(t) = \sum _{q=1}^{Q}\bar {\xi }^{q} A_{k}(t\Delta \tau ^{q})e^{j2\pi {f_{c}}\Delta \tau ^{q}}\); \(C^{\prime }_{k^{\prime }k}(t) = \sum _{q=1}^{Q}\bar {\xi }^{q} C_{k^{\prime }k}(t\Delta \tau ^{q})e^{j2\pi {f_{c}}\Delta \tau ^{q}}\); A_{k}(t) is the autocorrelation function of s_{k}(t); \(C_{ {kk}^{\prime }}(t)\) is the crosscorrelation function between \(s_{k^{\prime }}(t)\) and s_{k}(t), i.e., the crosscorrelation energy leakage term; and \(w_{lk}^{\text {MF}}(t)=w_{l}(t)\otimes s_{k}^{*}(t)\), ⊗ is the convolution operation, and (·)^{∗} is the complex conjugate operation.
After separation, echo pair needs to be carefully selected to estimate CPs. For the dual radar case, the separated echo set is {y_{11}(t),y_{12}(t),y_{21}(t),y_{22}(t)}. Comparing y_{11}(t) with y_{12}(t), their receiving paths are identical, while transmitting paths are not. Hence, by comparing the time and phase differences between this echo pair, transmit CPs \(\left \{\Delta \tau _{2}^{t},\Delta \phi _{2}^{t}\right \}\) can be estimated. Meanwhile, comparing the echo pair {y_{21}(t),y_{22}(t)}, same transmit CPs can be also estimated. The ultimate transmit CP estimates will be the fusion of these two estimates. And the corresponding CP estimation framework is shown in Fig. 1, which can be easily extended to receive CP estimation and Nradar case.
On this basis, a CP estimator called crosscorrelation (CC) processing is proposed [3, 10]. In this estimator, the selected echo pair is crosscorrelated. The peak of the crosscorrelation output defines the corresponding time CPs, and the phase at the peak of the crosscorrelation output defined the corresponding phase CPs. This processing can be briefly formulated as follows
where \(\Delta \hat {\tau }_{k,l}^{t}\) and \(\Delta \hat {\tau }_{l,k}^{r}\) denote the peaks of echo pair crosscorrelation outputs \(x_{k,l}^{t} (\tau) = \int \limits _{T_{\mathrm {G}}}y_{lk}(t)y_{l1}^{*}(t\tau)dt\) and \(x_{l,k}^{r}(\tau) =\int \limits _{T_{\mathrm {G}}}y_{lk}(t)y_{1k}^{*}(t\tau)dt\), T_{G} is the time span of the radar receive range gate (see Fig. 1), and arg{·} denotes the phase of a complex argument.
3.2 Influence of transmitting nonorthogonal waveforms on CP estimation
To note, CC is proposed on the assumption of transmitting ideally orthogonal waveforms. In practice, ideally orthogonal waveforms occupying same frequency band are not existent. In this case, CP estimation performance of CC may be impaired. And detailed analyses are illustrated in the following.
3.2.1 Receive CP estimation
Assuming SNR tends to infinity, from (6), \(x_{l,k}^{r}(\tau)\) can be derived as (ξ^{2} is omitted)
where \(f_{k}(t) = {A^{\prime }}_{k}\left (t\tau _{11}\Delta \tau _{k}^{t}\right)e^{j\Delta \phi _{k}^{t}} + \sum _{\substack {{k^{\prime }}=1\\ {k^{\prime }}\neq k}}^{N}{C^{\prime }}_{k^{\prime }k}\left (t\tau _{11}\Delta \tau _{k^{\prime }}^{t}\right)e^{j\Delta \phi _{k^{\prime }}^{t}}\).
Obviously, the peak of \(x_{l,k}^{r}(\tau)\) is \(\Delta \tau _{l}^{r}\), i.e., \(\Delta \hat \tau _{l,k}^{r} = \Delta \tau _{l}^{r}\). By substituting \(\Delta \hat \tau _{l,k}^{r}\) into (8), we have \(x_{lk}^{r}\left (\Delta \hat \tau _{k,l}^{r}\right) = M_{k}^{r} e^{j\Delta \phi _{l}^{r}}\), where \(M_{k}^{r} \in \mathbb {R}\) denotes the effective estimation energy. Then, according to (7), we have
3.2.2 Transmit CP estimation
Under the same assumption, \(x_{k,l}^{t}(\tau)\) can be derived as(ξ^{2} is omitted)
where g_{k}(t,τ) is the term related to crosscorrelation energy leakage (for more details, see in Appendix A).
Divide (10) into two parts: the term with a peak at \(\Delta \tau _{k}^{t}\) and the disturbance
where T_{RP} denotes the time span of the target range profile (RP) without “blank” margin (see Fig. 1). And in this time span, A^{′}_{k}(t)≈A^{′}_{1}(t).
Generally, the amplitude of A_{k}(t) in the mainlobe region is far greater than that of A_{k}(t) in the sidelobe region and \({C_{k^{\prime }k}}\left (t \right)\). Even if the disturbance exists, the peak of \({x_{k,l}^{t}\left (\tau \right)}\) will lie in the neighborhood of point \(\Delta \tau _{k}^{t}\), that is
To note, \(\varepsilon _{k}^{\max } \geqslant 0\), whose value is determined by the selection of orthogonal waveforms (A_{k}(t) and \({C_{k^{\prime }k}}\left (t \right)\)) and the geometric arrangement of radars \(\left (\Delta \tau _{k}^{t}{\text {, }}k = 2, \cdots,N\right)\).
By substituting (12) into (11), we have
where \(E_{k}^{t} \!<\! \int \limits _{T_{\text {RP}}}\!{A^{\prime }}_{k}\!\left (t  \tau _{11} \Delta \tau _{k}^{t}\right)\left [{A^{\prime }}_{k}\left (t\tau _{11}\Delta \tau _{k}^{t}\right)\right ]^{*}dt\in \mathbb {R}\) denotes the effective estimation energy, \(F_{k}^{t}\left ({{\varepsilon _{k}}} \right) \in \mathbb {C}\) denotes the disturbance caused by peak position deviation, and \(J_{k}^{t}(\varepsilon _{k}) = \int \limits _{T_{\mathrm {G}}  T_{\text {RP}}}{A^{\prime }}_{k} \left (t \tau _{11}  \Delta \tau _{k}^{t}\right)\left [{A^{\prime }}_{1}\left (t\tau _{11}\Delta \tau _{k}^{t}  \varepsilon _{k}\right)\right ]^{*}dt+\int \limits _{T_{\mathrm {G}}}g_{k}\left (t,\Delta \tau _{k}^{t} + {\varepsilon _{k}}\right)dt\in \mathbb {C}\) denotes the disturbance introduced by crosscorrelation energy leakage and the mismatch between A^{′}_{1}(t) and A^{′}_{k}(t).
Combining (7), (12), and (13), we have
Throughout the above analyses, transmitting nonorthogonal waveforms seriously affect the estimation performance of CC. And such influences can be outlined as follows:

(1)
When SNR →∞, using CC, the estimated receive CPs are unbiased (see (9)), while the estimated transmit CPs are not (see (14)). And their estimated biases are determined by the selection of orthogonal waveforms and the geometric arrangement of radars;

(2)
Since T_{G}>T_{RP}, it is easy to verify that \(M_{k}^{r}>E_{k}^{t}\), which implies that estimating transmit CPs is more noisesensitive than estimating receive CPs.
4 Methods
As analyzed in Section 3, crosscorrelation energy leakages between nonorthogonal waveforms seriously impair the accurate acquisition of CPs. If such energy leakages can be eliminated before CP estimation, i.e., reconstruct clean signals and use them to estimate CPs, the impacts introduced by nonorthogonal waveforms will disappear. Inspired by this, a new approach for estimating CPs is proposed in this section. In the following, we first introduce the clean signal reconstruction framework. Then, an additional process called prematch processing is elaborated as an enhancement of our proposed approach. Finally, we show the overall CP estimation process.
4.1 Clean signal reconstruction
From (6), since crosscorrelation energy leakages in the separated echo originate from the presence of echoes of other waveforms in the mixed reception signal, by eliminating these echoes from the initial reception signal, a clean separated echo can be reconstructed. Specifically, take the clean reconstruction of y_{lN}(t) as an example, crosscorrelation energy leakages introduced from the echoes of {s_{1}(t),⋯,s_{N−1}(t)} in y_{lN}(t) can be suppressed by reconstructing and eliminating these echoes from the initially mixed reception signal r_{l}(t). Under this consideration, a clean signal reconstruction approach is designed with a reconstructioneliminationreconstruction (RER) framework, which can be illustrated as follows:

1.
Reconstruction (the former one). MF is implemented on r_{l}(t) in the frequency domain to separate the echo of s_{1}(t). The separated echo, i.e., the spectrum of y_{l1}(t), will be
$$ {\selectfont{\begin{array}{ll} Y_{l1} (f) &= R_{l}(f)\cdot S_{1}^{*}(f)\\ &= \xi\! \sum\limits_{q = 1}^{Q} \bar \!{\xi}^{q}e^{ j2\pi f\Delta\tau^{q}}\!e^{j2\pi f_{c}\Delta\tau^{q}}\! \!\cdot \! S_{1}(f)^{2} e^{ j2\pi f\left({{\tau_{11}} + \Delta \tau_{1}^{t} + \Delta \tau_{l}^{r}} \right)}\! e^{j\Delta \phi_{1}^{t} + j\Delta \phi_{l}^{r}}\\ & \quad + S_{1}^{*}(f)\left[\sum\limits_{k^{\prime}=2}^{N}V_{lk^{\prime}}(f)+W_{l}(f)\right] \end{array}}} $$(15)where f∈[−B/2,B/2], B is the bandwidth of the orthogonal waveforms, \(V_{lk^{\prime }}(f) = \xi \sum \limits _{q = 1}^{Q}\bar \xi ^{q} e^{j2\pi f\Delta \tau ^{q}}e^{ j2\pi {f_{c}}\Delta \tau ^{q}}\cdot S_{k^{\prime }}(f)e^{ j2\pi f\left (\tau _{11}+ \Delta \tau _{k^{\prime }}^{t} + \Delta \tau _{l}^{r}\right)}e^{j\Delta \phi _{k^{\prime }}^{t} + j\Delta \phi _{l}^{r}}\) is the crosscorrelation energy leakage term, R_{l}(f), S_{k}(f), and W_{l}(f) are the spectrum of r_{l}(t), s_{k}(t), and w_{l}(t), respectively.
At high frequencies, according to the geometrical diffraction theory (GDT), the radar backscatter from a target can be accurately represented by an allpole model [15–17]. Let X_{l1}(f)=Y_{l1}(f)/S_{1}(f)^{2} (remove the effect of waveform spectral envelop on solving allpole model). After discretization, X_{l1}(f) can be formulated as
$$ \begin{aligned} {}X_{l1}(f_{m}) &= \underbrace{\sum_{q=1}^{Q}d_{l1}^{q}\left(p_{l1}^{q}\right)^{m}}_{\text{allpole model}}\\ & \quad +\frac{S_{1}^{*}(m)}{S_{1}(m)^{2}}\left[\sum_{k^{\prime}=2}^{N}V_{lk^{\prime}}(m)+W_{l}(m)\right] \end{aligned} $$(16)where \(d_{l1}^{q} = \xi \bar {\xi }_{q} e^{j\pi B\left (\tau _{11}+\Delta \tau _{1}^{t} + \Delta \tau _{l}^{r} + \Delta \tau ^{q}\right)}e^{ j2\pi {f_{c}}\Delta \tau ^{q}}e^{j\Delta \phi _{1}^{t} + j\Delta \phi _{l}^{r}}\) and \(p_{l1}^{q} = e^{j2\pi \Delta f\left (\tau _{11} + \Delta \tau _{1}^{t} + \Delta \tau _{l}^{r} + \Delta \tau ^{q}\right)}\) are the poles and their amplitudes, f_{m}=−B/2+mΔf, m=1,⋯,M−1, Δf=B/M. By solving the parametric model in (16), Y_{l1}(f) can be reconstructed as
$$ \hat Y_{l1}^{c}({{f_{m}}}) = \sum\limits_{q = 1}^{{{\hat Q}_{l1}}} {\hat d_{l1}^{q}{{\left({\hat p_{l1}^{q}} \right)}^{m}}} {\left {{S_{1}}\left({{f_{m}}} \right)} \right^{2}} $$(17)where \(\hat d_{l1}^{q}\), \(\hat p_{l1}^{q}\), and \(\hat Q_{l1}^{q}\) are the solved parameters (more details in Appendix B);

2.
Elimination. Although the solved parameters in (17) are contaminated by \({V_{k^{\prime }}}\left (m \right),{k^{\prime }} = 2, \cdots,N\), \(\hat Y_{l1}^{c}({{f_{m}}})\) still reserves the majority energy of Y_{l1}(f_{m}). Generally speaking, \(\hat Y_{l1}^{c}({{f_{m}}})\) can be considered as an approximation of Y_{l1}(f_{m}) in the leastsquares sense. To note, the superscript c in \(\hat Y_{l1}^{c}({{f_{m}}})\) represents the word “contaminated.” And the echo of s_{1}(t) in r_{l}(t) can be reconstructed as \(\hat Y_{l1}^{c} (f_{m}){S_{1}(f_{m})}/{S_{1}(f_{m})^{2}}\). Then, by eliminating it from the initially received mixed signal, its introduced crosscorrelation energy leakage will be suppressed. This process is formulated as
$$ \left[Y_{l1}(f_{m})\hat Y_{l1}^{c} (f_{m})\right]\frac{S_{1}(f_{m})}{S_{1}(f_{m})^{2}} $$(18)where the product term S_{1}(f_{m})/S_{1}(f_{m})^{2} denotes the inverse MF processing contrary to multiplying \(S_{1}^{*}(f_{m})\).
By multiplying (18) with \(S_{2}^{*}(f_{m})\), the echo of s_{2}(t) can be separated
$$ {\tilde Y_{l2}}\left({{f_{m}}} \right) = \left[Y_{l1}(f_{m})\hat Y_{l1}^{c} (f_{m})\right]\frac{S_{1}(f_{m})}{S_{1}(f_{m})^{2}}S_{2}^{*}\left({{f_{m}}} \right)\cdot $$(19)To distinct with the separated echo Y_{l2}(f_{m}) via direct MF, this one is tilded. With the same process, the echo of s_{2}(t) in r_{l}(t) can be also reconstructed and eliminated. Doing this iterative reconstruction and elimination procedures until only the echo of s_{N}(t) remains in r_{l}(t), the elimination work is done;

3.
Reconstruction (the latter one). By multiplying the final eliminated signal with \(S_{N}^{*}(f_{m})\), we can obtain \(\tilde Y_{lN}(f_{m})\). Applying the same reconstruction work to \(\tilde Y_{lN}(f_{m})\), we have
$$ {\tilde X_{lN}}\left(f_{m} \right) = \sum\limits_{q = 1}^{{{Q}_{LN}}} {d_{lN}^{q}} {\left({p_{lN}^{q}} \right)^{m}} + \frac{{S_{N}^{*}\left(m \right)}}{{{{\left {{S_{N}}(m)} \right}^{2}}}}{\delta_{lN}}\left(m \right) $$(20)where \({\tilde X_{lN}(f_{m})}=\tilde Y_{lN}(f_{m})/S_{N}(f_{m})^{2}\) and δ_{lN}(m) contains the noise and eliminating residuals. Since the echo of {s_{1}(t),⋯,s_{N−1}(t)} has been eliminated from the mixed reception signal r_{l}(t), crosscorrelation energy leakage terms in (20) disappear. And the clean signal \(\hat Y_{1N}(f_{m})\) can be reconstructed by solving allpole in (20). Essentially, each time for solving allpole model can be seen as a filtering process, it reserves the signal part while eliminating the disturbance (crosscorrelation energy leakage, noise and elimination residuals). Therefore, with this additional reconstruction step, rather than direct MF, this disturbance can be further suppressed. Then, using the inverse fast Fourier transform (IFFT), its corresponding time domain signal \(\hat y_{1N}(t_{m})\) can be also recovered.
As mentioned above, we only present the clean reconstruction procedures related to \(\hat y_{1N}(t_{m})\). General reconstruction procedures are outlined as Algorithm 1. Notice that the clean signal cannot be reconstructed in one step and iterative reconstruction and elimination procedures are necessary. Such stepbystep reconstruction and elimination is the major feature of our proposed clean signal reconstruction approach. Particularly, for the dual radar case, the visualized RER framework is shown in Fig. 2. After reconstructing clean echoes \(\left \{\hat y_{lk}(t_{m}),l,k = 1, \cdots,N\right \}\), CPs will be estimated according to (7), i.e., CC method. For distinction, we refer to our proposed CP estimation approach as reconstructed crosscorrelation (RECC) processing.
4.2 Prematch processing
In practice, the number of poles Q is usually unknown. The Akaike Information Criterion (ALC) [18] and minimum description length (MDL) [19] are available to estimate Q. But Q estimated in (20) corresponding to different reconstructed clean echoes is not always identical (see Fig. 3a). This is especially true for real measurements characterized by a colored noise and interference. As poles contain the whole information of signal, mismatched poles may cause extra deviation from the true CPs.
Therefore, we propose a method to select the matched pole pairs from raw pole sets as the enhancement of RECC, called prematch processing. The selection principle is based on the assumption mentioned earlier; that is, every radar displays same target scattering characteristics. In other words, the relative phase between poles of each reconstructed clean echo is approximately fixed, in spite of disparate absolute pole phase, which can be traced in (16). Assuming pole A and pole B are the raw pole sets stemmed from two different reconstructed clean echoes, the matched pole pairs can be selected by:

1.
Quantify. Dividing the pole phase span of poles A and B into several bins, the length of bin γ should satisfy ε<γ<Δ_{min} to ensure that at most one pole existing in a bin. ε is the maximum estimated deviation of the pole phase and Δ_{min} is the minimum pole phase span of the raw pole sets (see Fig. 3a). And the bin amplitude equals the corresponding pole’s normalized amplitude. If no pole falls into a bin, the amplitude is zero (see Fig. 3b).

2.
Select. Find the maximum crosscorrelation output position of the two quantified sequences as the best match position (see Fig. 3c). Then, we retain the position coincident poles in both sequences as selected pole pairs (see Fig. 3d).
To conclude, combined the RER clean signal reconstruction framework with the prematching processing, the overall flow of the RECC is shown in Fig. 4.
5 Simulation results and discussion
To validate the effectiveness of our proposed approach, we design several simulations in dual radar coherently combining scenario. The dual radar, radar 1 and radar 2, are located at (−d/2,0) and (d/2,0) respectively, where d=7 m. We choose up and downchirp waveforms as the transmitted orthogonal waveform set to separate the monostatic and bistatic echoes and estimate CPs. Radar 1 acts as the reference radar and transmits an upchirp waveform, while radar 2 transmits a downchirp waveform. Assume that there is a target with five scatterers in the range direction, whose scattering center is located at (R_{0} cosθ,R_{0} sinθ), θ=60^{∘}, R_{0}=200 km, and R_{0}≫d. Other parameters used in the simulation are listed in Table 1.
5.1 CP estimation performance
According to the discussion in Section 2, in this dual radar coherently combining scenario, the CPs to be estimated are \(\left \{\Delta \tau _{2}^{t},\Delta \tau _{2}^{r},\Delta \phi _{2}^{t},\Delta \phi _{2}^{r}\right \}\). In simulations, the effect of noise on CP estimation is further considered. And the noise is added as complex white Gaussian random process, whose intensity is set based on the SNR of a single radar (\(\text {SNR}_{\text {MF}}^{\text {in}}\)). Meanwhile, we assume both radars have the same \(\text {SNR}_{\text {MF}}^{\text {in}}\). Besides, for evaluating the CP estimation performance, the root mean square error (RMSE) is introduced, which is defined as (the RMSE of \(\Delta \tau _{2}^{t}\) )
where M_{c} is the number of Monte Carlo (MC) runs, \(\Delta \tau _{2}^{t}(i)\) and \(\Delta \hat \tau _{2}^{t}(i)\) are the true and estimated CPs of run i. In our simulation, 200 MC runs are executed per each \(\text {SNR}_{\text {MF}}^{\text {in}}\) sampled value for each run to capture the average estimating performance.
For comparison, we compare the proposed RECC with the CC and the TDMCC. TDMCC is the case separating monostatic and bistatic echoes using TDM technique and estimating CPs via CC. To note, when TDM technique is applied, the monostatic and bistatic echoes are staggered in time. And no crosscorrelation energy leakages exist in separated echoes, which is equivalent to using ideally orthogonal waveforms. In this case, transmit CP estimation and receive CP estimation are identical. Figure 5 presents the RMSEs for all the three methods.
Noticed that the receive CPs estimated via CC are asymptotically unbiased and their RMSEs approach the CRB arbitrarily close in high \(\text {SNR}_{\text {MF}}^{\text {in}}\), while the transmit CPs estimated via CC are not, their RMSEs deviate away from the CRB. Also, at the same \(\text {SNR}_{\text {MF}}^{\text {in}}\), the RMSEs of estimated transmit CPs are always greater than those of estimated receive CPs when CC is used. The reason lies in the existence of crosscorrelation energy leakages introduced by nonideally orthogonal up and downchirp waveforms. It is these leakages that cause the difference in transmit and receive CP estimation. Simulation results coincide with the analyses in Section 3.2. Besides, such estimation problems are basically solved by RECC, whose performance is comparable to that of TDMCC, which is the obtainable optimal performance using CC under the current conditions. Simulation results verify the effectiveness of RECC.
To further illustrate the reason behind RECC for the enhancement of CP estimation performance, Fig. 6 compares the separated echoes from radar 1 reception before and after clean reconstruction, when SNRMFin = 34.77 dB. As up and downchirp waveforms are not ideally orthogonal, crosscorrelation energy leakages exist before reconstruction, which are presented as fluctuant sidelobes in Fig. 6 (see the black line). Once a clean signal is reconstructed, those sidelobes disappear (see the red line). And RP approximately remains the same, which implies that no information loss occurred. This is the reason why improved CP estimates can be obtained using RECC.
5.2 Dual radar coherent combination performance comparison
Next, to get a clear understanding of coherently combining benefits, we evaluate the performance of the dual radar system after coherent combination, whose incoherence is calibrated via the CP estimates obtained by RECC and CC, respectively. Once CPs are estimated, to realize constructive interference impinging upon the target, both radars should transmit identical waveform. In our simulation, the upchirp waveform is transmitted by both radars at this stage. Other parameters remain unchanged. Figure 7 compares the reference and combined signal, when SNRMFin = 18.77 dB. The reference signal is the echo transmitted and received by only radar 1, while the combined signal is the synthetic echo after dual radar coherent combining.
In Fig. 7a, the dual radar system is combined by RECC. Comparing the combined and reference signal, there is a marked drop in noise level for the combined signal. That is, the dual radar system obtains the SNR gain of coherentontransmit and coherentonreceive. To evaluate this SNR gain, 200 MC trials are carried out. And the calculated SNR gain is 8.37 dB, approaching the ideal level 9 dB (10 log2^{3}). By contrast, in Fig. 7b, the dual radar system is combined via CC. Compared to the reference signal, a slight noise level drop still exists in the combined signal. But RPs of the two signals no longer overlap. In this case, the SNR gain is also calculated, which is 4.52 dB. Apparently, under this situation, using RECC can obtain higher SNR gain than CC.
Furthermore, from these simulation results, it can be inferred that using the approach transmitting orthogonal waveforms and estimating CPs to coherently combine multiple radars is conditional. Once the SNR of single radar is too low, the mutual incoherence among radars cannot be calibrated via the inaccurate estimated CPs. In this case, coherently combining radars is in vain.
6 Conclusions
In this paper, we proposed a novel approach (RECC) for coherently combining multiple radars, based on a clean signal reconstruction framework (RER). This method mainly solves the impaired estimation performance of CPs resulting from transmitting nonorthogonal waveforms. Theoretical analyses and simulation results indicate that the influence of transmitting nonorthogonal waveforms on CP acquisition can be approximately ignored after employing RECC. In addition, further comparisons are carried out on dual radar coherent combination performance. Results show that, under the same conditions, using RECC can obtain higher coherent combination SNR gain than using CC.
Furthermore, it is worth to note that the application of the RER framework is not limited to reconstruct the clean signal from the signal mixed with up and downchirp waveforms and, for a signal mixed with general orthogonal waveforms, e.g., the orthogonal phasecoding waveforms, it is also effective. Meanwhile, in contrary to the perfect orthogonal waveform design, RER framework offers an alternative attempt for crosscorrelation energy leakage suppression, which is expected to be extended into much wider radar applications. Besides, when the target is moving, CPs estimated via the previous pulse will no longer adapt the pulse next. Therefore, how to predict the CPs to adapt the successive pulse will be an interesting subject for future research.
7 Appendix A: Expression of g _{k}(t,τ) in (10)
In this appendix, we give the detailed expression of g_{k}(t,τ). According to (6) and \(\Delta \tau _{1}^{t}= \Delta \phi _{1}^{t}=0\), we have
Since \(\int \limits _{T_{\mathrm {G}}}y_{lk}(t)y_{l1}^{*}(t\tau)dt = \int \limits _{T_{\mathrm {G}}}y_{lk}\left (t+\Delta \tau _{l}^{r}\right)y_{l1}^{*}\left (t+\Delta \tau _{l}^{r}\tau \right)dt\) holds, we can express \(x_{k,l}^{t}(\tau)\) in the following simplified form(ξ^{2} is omitted)
where
8 Appendix B: Procedures for solving allpole model
The allpole model in (16) can be solved with the following fourstep process:

1
Use the sampling data X_{l1}(m) to construct the Hankel matrix
$${{\mathbf{H}}_{l1}} = \left[ {\begin{array}{*{20}{c}} {{X_{l1}}\left(0 \right)}&{{X_{l1}}\left(1 \right)}& \cdots &{{X_{l1}}\left({M  L} \right)} \\ {{X_{l1}}\left(1 \right)}&{{X_{l1}}\left(2 \right)}& \cdots &{{X_{l1}}\left({M  L + 1} \right)} \\ \vdots & \vdots & \ddots & \vdots \\ {{X_{l1}}\left({L  1} \right)}&{{X_{l1}}\left(L \right)}& \cdots &{{X_{l1}}\left({M  1} \right)} \end{array}} \right] $$where L is the length of the correlation window length. Generally, L=⌈M/3⌉;

2
Applying the singularvalue decomposition (SVD) to express \({{\mathbf {H}}_{l1}} = {{\mathbf {U}}_{l1}}{{\mathbf {S}}_{l1}}{\mathbf {V}}_{l1}^{\mathrm {H}}\), and the model order Q_{l1} can be determined by the decomposed singularvalue matrice S_{l1};

3
According to the estimated \(\hat Q_{l1}\), U_{l1}, and V_{l1}, signal or noise subspace can be built. On the basis of these subspaces, the estimation of signal parameters via rotational invariance techniques (ESPRIT) [20] and the root multiple signal classification (rootMUSIC) [21] are available to estimate poles \(p_{l1}^{q}, q=1,\cdots,\hat Q_{l1}\);

4
Once the poles are known, the amplitude terms \(d_{l1}^{q}, q=1,\cdots,\hat Q_{l1}\) are estimated by fitting the allpole model to the data X_{l1}(m) using a linear leastsquares(LLS) or a nonlinear leastsquares (NLLS) algorithm.
Abbreviations
 ALC:

Akaike Information Criterion
 CC:

Crosscorrelation
 CPs:

Coherent parameters
 CRB:

CramerRao bound
 ESPRIT:

Estimation of signal parameters via rotational invariance techniques
 GDT:

Geometrical diffraction theory
 IFFT:

Inverse fast Fourier transform
 LLS:

Linear leastsquares
 LOS:

Line of sight
 MDL:

Minimum description length
 MF:

Matched filtering
 NLLS:

Nonlinear leastsquares
 RECC:

Reconstructed crosscorrelation
 RER:

Reconstructioneliminationreconstruction
 RF:

Radio frequency
 RP:

Range profile
 RMSE:

Root mean square error
 rootMUSIC:

Root multiple signal classification
 SNR:

Signaltonoise ratio
 SVD:

Singularvalue decomposition
 TDM:

Timedivision multiplexing
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Liu, X., Xu, Z., Liu, X. et al. A clean signal reconstruction approach for coherently combining multiple radars. EURASIP J. Adv. Signal Process. 2018, 47 (2018). https://doi.org/10.1186/s1363401805691
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DOI: https://doi.org/10.1186/s1363401805691