*Appendix A.2. Proof of Lemma 2*

First, the reference signal *x*ˆ*IF <sup>i</sup>* (*nTs*) is cyclically autocorrelated by

$$\begin{split} R\_{r\_l r\_l}^a(\boldsymbol{\pi}) &= \frac{1}{N} \sum\_{n=0}^{N-1} \boldsymbol{\hat{x}}\_i^{IF} (n T\_s + \boldsymbol{\tau}/2) \boldsymbol{\hat{x}}\_i^{IF} (n T\_s - \boldsymbol{\tau}/2)^\* e^{-j2\pi anT\_s} \\ &= \boldsymbol{\beta}\_i^2 R\_{p\_i p\_i}^a(\boldsymbol{\tau}), \end{split} \tag{A10}$$

where *R<sup>α</sup> riri* (*τ*) is the cyclic autocorrelation of two reference signals, *α* is the cyclic frequency, and *R<sup>α</sup> pi pi* (*τ*) stands for the cyclic autocorrelation normalized by the amplitudes of the two reference signals, which can be expressed as

$$R\_{p, \eta}^{a} \left( \tau \right) = \frac{1}{N} \sum\_{n=0}^{N-1} \left[ \mathbb{C}\_{l} \left( nT\_{s} + \tau/2 \right) \cdot D\_{l} \left( nT\_{s} + \tau/2 \right) \right] \cdot \left[ \mathbb{C}\_{l} \left( nT\_{s} - \tau/2 \right) \cdot D\_{l} \left( nT\_{s} - \tau/2 \right) \right]^{\*} e^{-j \left[ 2 \pi a nT\_{s} \right]}. \tag{A11}$$

*Remote Sens.* **2020**, *12*, 263

Then, the reference signal *x*ˆ*IF <sup>i</sup>* (*nTs*) and the echo signal *<sup>x</sup>IF <sup>s</sup>* (*nTs*) are subjected to cyclic cross-correlation processing under the assumption of *H*1. In addition, using the uncorrelated properties of the baseband GPS reference signal pseudo-random code to obtain

$$\begin{split} R\_{r,z}^{a-f}(\mathbf{r}) &= \frac{1}{N} \sum\_{n=0}^{N-1} \pounds\_{i}^{\text{IF}}(nT\_{s} + \tau/2) \mathbf{z}\_{s}^{\text{IF}}(nT\_{s} - \tau/2)^{\*} \mathbf{e}^{-j2\pi(a-f)nT\_{s}} \\ &= \frac{1}{N} \sum\_{n=0}^{N-1} \left[ \beta\_{i} p\_{i}(nT\_{s}) \right] \cdot \left[ \sum\_{\substack{j=1 \\ i \neq j}}^{M} a\_{j} p\_{j}(nT\_{s} - \tau\_{j}) \mathbf{e}^{j2\pi f\_{i}nT\_{s}} + \omega(nT\_{s}) \right]^{\*} \mathbf{e}^{-j2\pi(a-f)nT\_{s}} \\ &= \frac{1}{N} \sum\_{n=0}^{N-1} \left[ \beta\_{i} p\_{i}(nT\_{s}) \right] \cdot \left[ a\_{i} p\_{i}(nT\_{s} - \tau\_{i}) \mathbf{e}^{j2\pi f\_{i}nT\_{s}} + \omega(nT\_{s}) \right]^{\*} \mathbf{e}^{-j2\pi(a-f)nT\_{s}} \\ &= \beta\_{i} a\_{i} \mathbf{e}^{-j\pi f\_{i}\tau\_{i}} \mathbf{e}^{-j\pi(a-f+f\_{i})\tau\_{i}} \mathbf{R}\_{p\_{i}p\_{i}}^{a-f+f\_{i}}(\mathbf{r} - \tau\_{i}) + \mathbf{N}^{\text{u}}(\tau), \end{split} \tag{A12}$$

where *τ<sup>i</sup>* is the time delay of the echo relative to the direct wave, *fdi* is the Doppler shift of the echo relative to the direct wave, and *Nα*(*τ*) is the cyclic cross-correlation of the reference signal *x*ˆ*IF <sup>i</sup>* (*nTs*) and the monitoring channel noise *ω*(*nTs*), which can be expressed as follows:

$$N^{a}(\tau) = \frac{1}{N} \sum\_{n=0}^{N-1} \pounds\_{i}^{IF} \left( nT\_{s} + \tau/2 \right) \omega \left( nT\_{s} - \tau/2 \right)^{\*} e^{-j2\pi \left( n - f \right) nT\_{s}}.\tag{A13}$$

Equation (A13) obeys a Gaussian distribution with a variance of <sup>1</sup> *<sup>N</sup> <sup>β</sup>*<sup>2</sup> *i σ*2 *<sup>ω</sup>* and a mean of 0, which is denoted here by *<sup>N</sup>α*(*τ*). It can be seen that *<sup>R</sup>α*−*<sup>f</sup> ris* (*τ*) is obtained by delay and frequency offset of *Rα riri* (*τ*), and the delay and frequency offset of *<sup>R</sup>α*−*<sup>f</sup> ris* (*τ*) with respect to *<sup>R</sup><sup>α</sup> riri* (*τ*) is just the delay and frequency offset of the echo. Next, the column vectors corresponding to the cyclic frequencies of the maximum peaks of *R<sup>α</sup> riri* (*τ*) and *<sup>R</sup>α*−*<sup>f</sup> ris* (*τ*) are extracted, denoted respectively as *<sup>R</sup>α* −*f ris* (*τ*) and *<sup>R</sup>α riri* (*τ*), and subjected to mutual blur function processing to obtain

$$\begin{split} \Psi\_{i}(\boldsymbol{u},\boldsymbol{f}) &= \sum\_{\tau=0}^{N} R\_{r\_{i}\mathbf{r}}^{a'-f'}(\tau) R\_{r\_{i}\mathbf{r}\_{i}}^{a'}(\tau-\boldsymbol{u})^{\*} e^{j2\pi f\tau} \\ &= \beta\_{i}^{3} a\_{i} e^{-j\pi\boldsymbol{a}'\tau\_{i}} \sum\_{\tau=0}^{N} R\_{p\_{i}p\_{i}}^{a'-f+f\_{d\_{i}}}(\tau-\boldsymbol{\tau}\_{i}) R\_{p\_{i}p\_{i}}^{a'}(\tau-\boldsymbol{u})^{\*} e^{j2\pi(f-f\_{d\_{i}})\tau} \\ &+ \beta\_{i}^{2} \sum\_{\tau=0}^{N} N^{a'}(\tau) R\_{p\_{i}p\_{i}}^{a'}(\tau-\boldsymbol{u})^{\*} e^{j2\pi f\tau} \\ &= \Psi\_{i,R\_{i}}(\boldsymbol{u},f) + \Psi\_{NR\_{i}}(\boldsymbol{u},f), \end{split} \tag{A14}$$

where Ψ*R*,*Ri* (*u*, *<sup>f</sup>*) ≤ *<sup>β</sup>*<sup>3</sup> *i αi Rα pi pi* (*τ*) 2 *dτ*. When *u* = *τ<sup>i</sup>* and *f* = *fdi* ,

$$\Psi\_{R\_i R\_i}(\mu\_\prime f) = \beta\_i^3 a\_i \int \left| R\_{p\_i p\_i}^{a'}(\tau) \right|^2 d\tau,\tag{A15}$$

where Ψ*NRi* (*u*, *f*) is the noise term. According to the principle of constant false alarm detection, the size of the detection threshold is related to the false alarm probability. Therefore, the probability distribution of the detection quantity Ψ*i*(*u*, *f*) needs to be analyzed under the assumption of *H*1. The first term of Equation (A14) belongs to the determined detection amount and is expressed as

$$\Psi\_{R\_i R\_i}(u, f) = \beta\_i^3 a\_i e^{-j\pi a' \tau\_i} \sum\_{\tau=0}^N R\_{p\_i p\_i}^{a'-f+f\_{d\_i}}(\tau - \tau\_i) R\_{p\_i p\_i}^{a'}(\tau - u)^\* e^{j2\pi (f - f\_{d\_i})\tau}. \tag{A16}$$

For the second term of Equation (A14), this term represents the mutual fuzzy function of the cyclic autocorrelation of the reference channel and the noise of the echo channel. If the cycle frequency is *α* , the term can be expressed as

$$\Psi\_{NR\_i}(u\_\prime f) = \beta\_i^2 \sum\_{\tau=0}^N \mathcal{N}^{a'}(\tau) R\_{p\_i p\_i}^{a'}(\tau - u)^\* e^{j2\pi f\tau},\tag{A17}$$

where *Nα* (*τ*) obeys a Gaussian distribution with a mean of 0 and a variance of <sup>1</sup> *<sup>N</sup> <sup>β</sup>*<sup>2</sup> *i σ*2 *<sup>ω</sup>* , and *<sup>R</sup>α pi pi* is a cyclic autocorrelation of the signal. Since the term is obtained by linear integral operation on noise, the term is still subject to Gaussian distribution, so the mean and variance can be used to characterize the probability distribution. The mean and variance are given by

$$E\left\{\Psi\_{NR\_i}(\mu,f)\right\} = 0,\tag{A18}$$

*Var*{*ψNRi* (*u*, *f*)} = *E ψNRi* (*u*, *f*) 2 − *E* " *ψNRi* (*u*, *f*) #2 = *E* %*β*2 *i N*−1 ∑ *τ*1=0 *N<sup>α</sup>* (*τ*<sup>1</sup> <sup>−</sup> *<sup>u</sup>*)*R<sup>α</sup> pi pi* (*τ*1)*ej<sup>π</sup> <sup>f</sup> <sup>τ</sup>*<sup>1</sup> & ∗ % *β*2 *i N*−1 ∑ *τ*2=0 *N<sup>α</sup>* (*τ*<sup>2</sup> <sup>−</sup> *<sup>u</sup>*)*R<sup>α</sup> pi pi* (*τ*2)*ej<sup>π</sup> <sup>f</sup> <sup>τ</sup>*<sup>2</sup> &∗\$ = *β*<sup>4</sup> *i N*−1 ∑ *τ*1=0 *N*−1 ∑ *τ*2=0 *E N<sup>α</sup>* (*τ*<sup>1</sup> <sup>−</sup> *<sup>u</sup>*)*R<sup>α</sup> pi pi* (*τ*1)*ej<sup>π</sup> <sup>f</sup> <sup>τ</sup>*<sup>1</sup> <sup>∗</sup> -*N*−<sup>1</sup> ∑ *τ*2=0 *N<sup>α</sup>* (*τ*<sup>2</sup> <sup>−</sup> *<sup>u</sup>*)*R<sup>α</sup> pi pi* (*τ*2)*ej<sup>π</sup> <sup>f</sup> <sup>τ</sup>*<sup>2</sup> .∗\$ = *β*<sup>4</sup> *i N*−1 ∑ *τ*1=0 *N*−1 ∑ *τ*2=0 *RNN*(*τ*<sup>1</sup> − *τ*2)*E* - *Rα pi pi* (*τ*1)*R<sup>α</sup>* ∗ *pi pi* (*τ*2)*ej<sup>π</sup> <sup>f</sup>*(*τ*1−*τ*2) = *β*<sup>4</sup> *i N*−1 ∑ *τ*1=0 *N*−1 ∑ *τ*2=0 1 *<sup>N</sup> <sup>β</sup>*<sup>2</sup> *i σ*2 *ωδ*(*τ*<sup>1</sup> − *τ*2)*E* - *Rα pi pi* (*τ*1)*R<sup>α</sup>* ∗ *pi pi* (*τ*2)*ej<sup>π</sup> <sup>f</sup>*(*τ*1−*τ*2) <sup>=</sup> *<sup>σ</sup>*<sup>2</sup> *<sup>ω</sup> β*<sup>6</sup> *i <sup>N</sup>* . (A19)

Therefore, the distribution of Ψ*NRi* (*u*, *f*) can be expressed as Ψ*NRi* (*u*, *<sup>f</sup>*) <sup>∼</sup> *CN* 0, *<sup>σ</sup>*<sup>2</sup> *<sup>ω</sup> β*<sup>6</sup> *i N* . It can be seen that the variance of the noise of the item is inversely proportional to the number of sampling points. Obviously, as the number of sampling points increases, the variance of the noise decreases, showing good noise suppression performance. Thus, the distribution of Ψ*i*(*u*, *f*) under the assumption of *H*<sup>1</sup> is given by

$$\Psi\_i\left((u\_\prime f)|H\_1\right) \sim \text{CN}\left(\Psi\_{\mathbb{R}\_i\mathbb{R}\_i}(u\_\prime f)\_\prime \frac{\sigma\_\omega^2 \beta\_i^6}{N}\right). \tag{A20}$$
