## 3 Optimized Joint Signal Processing Schemes
### Sparsity Reconstruction-Based Improved Parameter Estimation
#### 3.1.1 Conventional Processing  
The conventional processing of the radar echo signal in the proposed integrated system is shown in Figure 4.  
Figure 4: Flowchart of the conventional processing. Figure 3: Flow chart of the joint signal processing schemes in the proposed system. Supposing that there are \(Q\) detectable targets in the application scenario, the distance between the radar transmitter and the \(q\)th target is \(R_{q}\) and the relative velocity of the \(q\)th target moving toward the radar is \(v_{q}\). It can be obtained that the time delay of the \(q\)th target is \(\tau_{q}=2R_{q}/c\); the Doppler frequency shift is \(f_{d}^{q}=-2f_{c}v_{q}/c\), where \(f_{c}\) is the carrier frequency; and \(c\) is the propagation velocity. For simplicity, we suppose that the direction of arrival at the receiving end of the integrated platform is accurately known, and the echo is processed in units of pulses. Then, we can perform digital beamforming; that is, use the aperture of the array antenna to form the receiving beam in the desired direction through digital signal processing so as to form the equivalent single-transmit waveform for the colocated MIMO radar at the receiver [4]. The echo signal of the \(p\)th pulse can be formulated as  
\[ \begin{split} y_{p}(t)&=\mathbf{a}_{r}^{H}(\theta_{r})[ y_{p,1}(t),y_{p,2}(t),\cdots,y_{p,N_{r}}(t)]^{\mathrm{T}}\\ &=N_{r}\sum_{q=1}^{Q}\beta_{q}e^{-2\pi f_{c}\tau_{q}}e^{2\pi f_{ d}^{q}\tau_{p}}\mathbf{a}_{t}^{T}(\theta_{t})\mathbf{s}_{p}(t-\tau_{q})+\omega_{p}(t) \end{split}  \tag{7}\]  
where \(\mathbf{a}_{r}(\theta_{r})\) denotes the receive steering vector, \(\beta_{q}\) is the scattering factor of the \(q\)th target, \(T_{p}\) is the pulse repetition period of the transmitted waveform, \(\mathbf{s}_{p}(t)\) is the transmitted signal vector of the \(p\)th pulse, \(\omega_{p}(t)\) is Gaussian noise, and \(\mathbf{a}_{t}(\theta_{t})\) indicates the transmit steering vector, and it can be written as  
\[\mathbf{a}_{t}(\theta_{t})=\left[1,e^{-2\pi d_{t}\sin\theta_{t}/\lambda},\cdots,e ^{-2\pi(N_{t}-1)d_{t}\sin\theta_{t}/\lambda}\right]^{\mathrm{T}} \tag{8}\]  
where \(d_{t}\) is the transmitting antenna spacing and \(\lambda=c/f_{c}\) is the signal wavelength. Then, the radar-received signal in (7) is sampled at the sampling frequency \(f_{s}=N_{c}\Delta f\), and the discrete format of the signal in (7) can be denoted as  
\[ \begin{split}\mathbf{y}_{p}=& N_{r}\sum_{q=1}^{Q}\beta_{q}e^{ 2\pi\gamma f_{d}^{q}\tau_{p}T_{r}}\mathbf{F}_{N_{c}}^{-1}\\ &\times\big{[}\mathbf{\psi}_{1}\big{(}\mathbf{s}_{p}\odot\tau_{q}\big{)},\mathbf{\psi}_{2}\big{(}\mathbf{s}_{p}\odot\tau_{q}\big{)},\cdots,\mathbf{\psi}_{N_{t}} \big{(}\mathbf{s}_{p}\odot\tau_{q}\big{)}\big{]}\\ &\times\mathbf{a}_{t}(\theta_{t})+\mathbf{\omega}_{p}\end{split}  \tag{9}\]  
where \(\mathbf{\psi}_{g}\) indicates the diagonal matrix of the \(g\)th channel set for realizing the subcarrier interleaving structure, \(\odot\) indicates the Hadamard product, \(\mathbf{s}_{p}\) indicates the frequency domain weight vector in the \(p\)th pulse, and \(\mathbf{F}_{N_{c}}^{-1}\) indicates an inverse discrete Fourier transform (IDFT) matrix, which can be written as  
\[\mathbf{F}_{N_{c}}^{-1}=\frac{1}{\sqrt{N_{c}}}\bigg{[}e^{2\pi\frac{(1-1)(j-1)}{N_ {c}}}\bigg{]}_{N_{c}\times N_{c}},i,j=1,2,\cdots,N_{c} \tag{10}\]  
In addition, \(\tau_{q}\) is the delay vector of the \(q\)th target, which is given by  
\[\mathbf{\tau}_{q}=\left[e^{-j2\pi f_{c}\tau_{q}},e^{-j2\pi(f_{c}+\Delta f)\tau_{ q}},\cdots,e^{-j2\pi(f_{c}+(N_{c}-1)\Delta f)\tau_{q}}\right]^{\mathrm{T}} \tag{11}\]  
To minimize the influence of the phase difference information of different channels on the signal processing results, we need to compensate for the received signals of different channels. However, the phase difference information of different channels is different, so it is necessary to separate the channels before compensation.