*4.2. Detection Statistics Construction with Multiple GPS Satellites*

Due to the different distribution positions of different GPS satellites, the peak coordinates of multiple GPS satellite echo detections *Si*(*τ*, *f*) and Ψ*i*(*u*, *f*) are also different. Thus, it is impossible to add a plurality of detection amounts to the fusion structure detection statistic. Aiming at this problem, this section unifies the detection peak coordinates of different satellites by coordinate transformation, which can superimpose the echo detection spectrum of several different GPS satellites to achieve the purpose of non-correlated cumulative enhanced signal-to-noise ratio. Thereby, a detection statistic with a higher SNR is constructed.

Figure 13 shows the geometry of the receiving system, where *θ* is the signal arrival angle, *δ* is the angle between the bistatic angle bisector and the speed *ν* of the aircraft, and *β* is the bistatic angle. It can be seen from the figure that the positions of different GPS satellites are different, so the delay *τ* and the Doppler shift *fd* corresponding to the peak values of the two-dimensional correlation between the different satellite reference signals and the monitoring channel echo signals are different. However, the common edge *Rr* and the velocity *ν* corresponding to different peak coordinates *τ* and *fd* are the same, so the detection spectrum can be converted from the delay-Doppler dimension to the distance velocity dimension so that the coordinate peaks are the same. Thereby, it is possible to accumulate different detection amounts. The relationship between *τ* and *Rr* and *fd* and *ν* is

$$\begin{cases} \begin{array}{c} R\_{\bar{r}} + R\_{\bar{t}} = L + c\tau, \\\ R\_{\bar{t}}^2 = R\_{\bar{r}}^2 + L^2 - 2R\_{\bar{r}}L\cos\theta. \end{array} \tag{18}$$

Solving Equation (18) can obtain:

$$R\_r = \frac{c^2 \tau^2 + 2Lc\tau}{2(L + c\tau - L\cos\theta)} = f(\tau),\tag{19}$$

$$f\_d = \frac{2v}{\lambda} \cos \delta \cos \frac{\beta}{2} = \frac{2v'}{\lambda} \cos \frac{\beta}{2} = g(v),\tag{20}$$

where *v* = *v* cos *δ*, and *v* represents the speed *v* of the target on the bisecting angle bisector, and *β* can be obtained from

$$\sin(\beta) = \frac{2R\_r \sin \theta \left(R\_r + c\tau - R\_r \cos \theta\right)}{R\_r^2 - 2R\_I \left(R\_I + c\tau\right) \cos R\_I + (L + c\tau)^2}.\tag{21}$$

**Figure 13.** The geometry of the receiving system.

The detection quantities *Si*(*τ*, *f*) and Ψ*i*(*u*, *f*) obtained by different methods are transformed by Equations (19) and (21) to obtain distance-velocity spectra *Si f* <sup>−</sup><sup>1</sup> (*Rr*), *g*(*v*) ! and Ψ*<sup>i</sup> f* <sup>−</sup><sup>1</sup> (*Rr*), *g*(*v*) ! , which are expressed as:

$$S\_{\rm i}\left(f^{-1}\left(\mathcal{R}\_{\rm r}\right),g\left(v\right)\right) = \sum\_{n=0}^{N} \mathbf{x}\_{\rm s}^{IF}\left(nT\_{\rm s}\right)\mathbf{\hat{x}}\_{\rm i}^{IF}\left(nT\_{\rm s} - f^{-1}\left(\mathcal{R}\_{\rm r}\right)\right)^{\*}e^{j2\pi\mathbf{g}\left(v\right)nT\_{\rm s}}\tag{22}$$

$$\Psi\_i \left( f^{-1} \left( R\_r \right), g(\upsilon) \right) = \sum\_{n=0}^N R\_{r\_i s}^{a'-f'} \left( n T\_s \right) R\_{r\_i r\_i}^{a'} \left( n T\_s - f^{-1} \left( R\_r \right) \right)^\* e^{j2 \pi g(\upsilon) n T\_s}. \tag{23}$$

At this time, the detection amount *Si f* <sup>−</sup><sup>1</sup> (*Rr*), *g*(*v*) ! or Ψ*<sup>i</sup> f* <sup>−</sup><sup>1</sup> (*Rr*), *g*(*v*) ! of the plurality of GPS satellites can be non-coherently superimposed in the distance-speed domain. The final superimposed detection statistics obtained by the two methods as

$$\Lambda\left(R\_{r}, V\right) = \sum\_{i=1}^{M} \mathbb{S}\_{i}\left(f^{-1}\left(R\_{r}\right), g(v)\right),\tag{24}$$

$$\Omega\left(R\_{\mathcal{I}}, V\right) = \sum\_{i=1}^{M} \Psi\_i\left(f^{-1}\left(R\_{\mathcal{I}}\right), \mathcal{g}\left(v\right)\right),\tag{25}$$

where *Rr* represents the distance from the target to the receiver and *V* represents the speed of the target. It can be seen from the above process that this paper uses the multiple GPS reference signals separated and reconstructed to calculate the detection quantity of different GPS satellites as the illumination source, which provides more favorable information for target detection. However, due to the different peak values of the detected quantities obtained by multiple GPSs, these detection quantities cannot be effectively fused, so this paper uses the coordinate fusion algorithm to combine them to obtain the final detection amount after peak enhancement.
