*3.3. DPI and MPI Suppression Based on ECA*

After the reference signal separation and reconstruction, the DPI and MPI of multiple satellites can be suppressed. Different from the DPI and MPI suppression processes of a single GPS satellite, this section suppresses the DPI and MPI brought by multiple GPSs in the surveillance channel based on the extensive cancellation algorithm (ECA).

Firstly, the ECA uses multiple GPS reference signals *x*ˆ1(*t*), *x*ˆ2(*t*)... *x*ˆ*M*(*t*) to construct a delay spread matrix **X**ref of multi-satellite signals, which can be expressed as

$$\mathbf{X}^{\text{ref}} = \begin{bmatrix} \mathbf{X}\_1^{\text{ref}} & \mathbf{X}\_2^{\text{ref}} & \dots \mathbf{X}\_i^{\text{ref}} \dots & \mathbf{X}\_M^{\text{ref}} \end{bmatrix} , \tag{7}$$

where **X**ref *<sup>i</sup>* (*i* ∈ (1, *M*)) is the reconstructed matrix of the reconstructed ith reference signal *x*ˆ*i*(*t*) through different delays, and the extension matrix **X**ref *<sup>i</sup>* is an element in the matrix **<sup>X</sup>**ref and can be expressed as

$$\mathbf{X}\_{i}^{\text{ref}} = \begin{bmatrix} \pounds\_{i}(0) & \left| \pounds\_{i}(N-1) \right| & \dots & \pounds\_{i}(N-K) \\ \pounds\_{i}(1) & \left| \pounds\_{i}(0) \right| & \dots & \pounds\_{i}(N-K+1) \\ \dots & \left| \pounds\_{i}(N-2) \right| & \dots & \dots \\ \pounds\_{i}(N-1) & \pounds\_{i}(N-2) & \dots & \pounds\_{i}(N-K-1) \end{bmatrix} \tag{8}$$

where *N* is the number of sampling points, and *K* is the maximum delay, which can be obtained by dividing the maximum detection distance by the speed of light(*K* = *<sup>R</sup>*max *<sup>c</sup>* ). DPI and MPI can be expressed as

$$DMPI = \sum\_{i=0}^{W\_1} \omega\_{1\_i} \mathbf{x}\_1(t - \tau\_{1\_i}) + \sum\_{i=0}^{W\_2} \omega\_{2\_i} \mathbf{x}\_2(t - \tau\_{2\_i}) + \dots \\ \dots + \sum\_{i=0}^{W\_M} \omega\_{M\_i} \mathbf{x}\_M(t - \tau\_{M\_i}). \tag{9}$$

Then, adjust the value of *ε* = [*ε*0, *ε*<sup>1</sup> ... *ε***N**−1] **<sup>T</sup>** to make *ε***X**ref approach the direct wave and multipath interference. The problem is transformed into the following problem:

$$\min \left\| \mathbf{x}\_s(t) - \varepsilon \mathbf{X}^{\mathrm{ref}} \right\|^2. \tag{10}$$

The solution of Equation (10) uses the least squares criterion [27–29]; then, Equation (10) is equivalent to the following:

$$\frac{\partial \left( \left\| \mathbf{x}\_{\delta}(t) - \varepsilon \mathbf{X}^{\mathrm{ref}} \right\|^{2} \right)}{\partial(\varepsilon)} = 0. \tag{11}$$

Thus, *ε* = **X**ref *<sup>H</sup>* **X**ref−<sup>1</sup> **<sup>X</sup>**ref*xs*(*t*) is obtained, where  **X**ref *<sup>H</sup>* is the transpose of **Xref**, and the signal in the monitoring channel after interference suppression is expressed as

$$\mathbf{x}\_s'(t) = \mathbf{x}\_s(t) - DMPI = \mathbf{x}\_s(t) - \varepsilon \mathbf{X}^{\mathrm{ref}} = \mathbf{x}\_s(t) - \mathbf{X}^{\mathrm{ref}} \left( \left( \mathbf{X}^{\mathrm{ref}} \right)^H \mathbf{X}^{\mathrm{ref}} \right)^{-1} \mathbf{X}^{\mathrm{ref}} \mathbf{x}\_s(t), \tag{12}$$

where *x <sup>s</sup>*(*t*) only contains the echo signal and noise *ns*(*t*).

The proposed algorithm does not need to know the gain value, and the DMPI can be directly solved by the proposed algorithm. In order to verify the DPI and MPI suppression algorithms based on signal separation and reconstruction proposed in this paper, the specific parameter settings are shown in Table 3 as follows: the reference channel contains noise, five GPS signals; the monitoring channel contains noise, three GPS echoes, and DPIs corresponding to five reference channel GPS signals. Firstly, the reference signal separation and reconstruction algorithms are used to separate and reconstruct the reference signals 1, 2, 3, 4, and 5 of the reference channel. Then, the purified reference signal is used together with the suppression algorithm proposed in this section to perform DPI suppression on the signal of the monitoring channel. Finally, the signal of the monitoring channel after the suppression and the reference signal are subjected to cross ambiguity function (CAF) processing [30–32], and the DPI suppression effect is judged by observing whether there is a peak corresponding to the echo on the delay-Doppler spectrum.

Figure 10 shows the time-frequency two-dimensional correlation of un-suppressed DPI of multiple GPS satellites. It can be seen that the echo generated by the target is completely submerged in the peak generated by DPI. Figure 11 shows the comparison of the monitoring channel signals before and after the DPI and MPI suppression methods proposed in this paper. It can be seen that the amplitude of the monitoring channel signal decreases after interference suppression, which proves that strong DPI interference has been effectively suppressed.


**Table 3.** Parameter setting.

**Figure 10.** Time-frequency two-dimensional correlation graph with unsuppressed interference.

**Figure 11.** Monitoring channel signal before and after interference suppression.

Figure 12 shows the CCF of the reference and echo signals after DPI interference suppression. It can be seen that the peak of the echo signal is clearly highlighted after the DPI interference suppression. Figure 12 proves that the interference suppression scheme proposed in this paper can effectively suppress the DPI of the monitoring channel when the reference channel SNR is as low as −15 dB, and the reference channel has multiple GPS signals with similar power.

**Figure 12.** Time-frequency two-dimensional graph after direct wave multipath interference suppression.

This section uses the multiple GPS reference signals separated and reconstructed to construct the delay spread matrix **Xref** of the multiple satellite signals, and then finds the optimal weight *ε* based on the least squares criterion. Then, let *ε***X**ref approach multipath and then subtract *ε***X**ref from the monitoring channel to get the monitoring channel signal after DPI and MPI suppression.
