**5. Moving Aerial Target Detection Performance Analysis**

In order to better design the decision threshold of the detector and evaluate the detection performance of the two detection quantities, this section analyzes the probability distributions of Λ(*L*, *V*) and Ω(*L*, *V*), respectively.

For convenience, the reference baseband signal *x*ˆ*IF <sup>i</sup>* (*t*) of the GPS satellite is first discretized and expressed as

$$\pounds\_{i}^{IF}\left(nT\_{s}\right) = \beta\_{i} p\_{i}\left(nT\_{s}\right),\tag{26}$$

where *β<sup>i</sup>* is the amplitude of the baseband reference signal, *pi* (*nTs*) is the baseband reference signal after the amplitude normalized, and *Ts* stands for the sampling period.

The signal *xIF <sup>s</sup>* (*t*) of the monitoring channel is discretized and then expressed as

$$\mathbf{x}\_s^{IF} \left( nT\_s \right) = \sum\_{j=1}^{M} a\_j p\_j \left( nT\_s - \tau\_j \right) \exp \left( j2\pi f\_{d\_j} nT\_s \right) + \omega \left( nT\_s \right), \tag{27}$$

where *α<sup>j</sup>* is the amplitude of the echo signal, *pj nTs* − *τ<sup>j</sup>* ! exp  *j*2*π fdj nTs* is the amplitude-normalized echo signal, *ω* (*n*) is the complex Gaussian noise obeying the *N* 0, *σ*<sup>2</sup> *w* ! distribution, and *N*(·) is the Gaussian distribution.

The binary hypothesis of echo signal detection is: assuming that *H*<sup>1</sup> is the target existence, the signal *x <sup>s</sup>* (*nTs*) of the monitoring channel contains the echo signal *<sup>α</sup><sup>i</sup> pi* (*nTs* <sup>−</sup> *<sup>τ</sup>i*) exp *j*2*π fdi nTs* ! corresponding to a certain reference signal *x*ˆ*IF <sup>i</sup>* (*nTs*). Assuming that *H*<sup>0</sup> indicates that the target does not exist, the signal *x <sup>s</sup>* (*nTs*) of the monitoring channel does not contain any echo signal corresponding to *x*ˆ*IF <sup>i</sup>* (*nTs*), which is expressed as follows:

$$\begin{cases} \begin{array}{c} H\_0: \mathbf{x}\_s^{IF}(nT\_s) = \omega \left(nT\_s\right) \\ H\_1: \mathbf{x}\_s^{IF}\left(nT\_s\right) = \sum\_{j=1 \atop j \neq i}^{M} a\_j p\_j \left(nT\_s - \tau\_j\right) \exp\left(j2\pi f\_{d\_i} nT\_s\right) + \omega \left(nT\_s\right) \end{array} \tag{28}$$

*5.1. Performance Analysis of Detection Based on CAF*

**Lemma 1.** *The distribution of the detection statistic under the H*<sup>1</sup> *hypothesis is*

$$\mathcal{S}(\mathbb{S}\_{i}(\tau,f)|H\_{1}) \sim \text{CN}\left(\left\{a\_{i}^{\*}\beta\_{i}\chi\_{pp}\left(\tau - n\_{\tau\prime}f - f\_{d}\right)\right\}, \quad 2\text{N}\beta\_{i}^{2}\sigma\_{\omega}^{2}\right),\tag{29}$$

*where CN*(.) *represents the complex Gaussian process.*

**Proof.** See Appendix A.1.

As with the analysis under the *H*<sup>1</sup> hypothesis, it can be concluded that the detection statistic distribution under the *H*<sup>0</sup> hypothesis is

$$(S\_i(\tau, f)|H\_0) \sim \text{CN}\left(0, 2N\beta^2 \sigma\_\omega^2\right). \tag{30}$$

Using the data fusion technique of multiple GPS satellites, according to the cumulative nature of the Gaussian distribution, the distribution of the final detection statistic Λ (*Rr*, *V*) is

$$\Lambda(\Lambda|H\_0) \sim \text{CN}\left(0, \sum\_{i=1}^{M} 2N\beta\_i^2 \sigma\_{\omega}^2\right),\tag{31}$$

$$\mathbb{V}\left(\Lambda|H\_1\right) \sim \text{CN}\left(\sum\_{i=1}^{M} \left\{a\_i^\* \beta\_i \chi\_{pp}\left(\pi - n\_{\tau\prime}f - f\_d\right)\right\}, \sum\_{i=1}^{M} 2N\beta\_i^2 \sigma\_\omega^2\right). \tag{32}$$

The false alarm probability of detecting weak echo signals from different GPS satellite sources by Equations (28) and (31) is given by

$$P\_{FA} = \int\_{\lambda}^{\infty} f\left(\Lambda | H\_0\right) d\Lambda = \exp\left(-\frac{\lambda}{\sum\_{i=1}^{M} 2N\beta\_i^2 \sigma\_\omega^2}\right),\tag{33}$$

where *f* (Λ|*H*0) is the probability density function of (Λ|*H*0).

From Equation (33), it can be concluded that the adaptive detection threshold *λ* is

$$\lambda = -\sum\_{i=1}^{M} 2N\beta\_i^2 \sigma\_\omega^2 \ln\left(P\_{FA}\right). \tag{34}$$

According to Equationss (A2), (32), and (34), the detection probability can be obtained by using signal detection theory:

$$P\_D = \int\_{\lambda}^{\infty} f(\Lambda | H\_1) d\Lambda = Q\_m \left( \sqrt{\frac{\left| \sum\_{i=1}^M (\beta\_i \alpha\_i N) \right|^2}{\sum\_{i=1}^M N 2 \beta\_i^2 \sigma\_\omega^2}}, \sqrt{\frac{\lambda}{\sum\_{i=1}^M N 2 \beta\_i^2 \sigma\_\omega^2}} \right), \tag{35}$$

where *Qm*(·, ·) is the Marcum Q function, and *f* (Λ|*H*1) is the probability density function of (Λ|*H*1).

From Equation (35), it can be seen that the theoretical detection probability of the multi-star weak echo joint detection based on the CCA detection quantity construction method is related to the parameters such as the monitoring channel noise, the number of sampling points *N*, the number of satellites, and the false alarm probability. It can be seen that the detection probability is proportional to the number of satellites *M*, that is, the detection probability increases with the number of satellites. It is theoretically proved that the weak echo combined detection of multiple GPS satellites has a higher detection probability than the weak echo detection of a single GPS satellite, but the method does not reflect the noise suppression performance.
