(2) Detection statistics based on CCAF

To construct a detection using CCAF, we first need to do a cyclic autocorrelation of the reference signal *x*ˆ*IF <sup>i</sup>* (*nTs*) as

$$R\_{r\_l r\_l}^{a}(\tau) = \frac{1}{N} \sum\_{n=0}^{N-1} \mathfrak{X}\_i^{I F} \left( nT\_s + \tau/2 \right) \mathfrak{X}\_i^{I F} \left( nT\_s - \tau/2 \right)^\* e^{-j2\pi anT\_s}.\tag{15}$$

Then, the cyclic cross-correlation of the reference signal *x*ˆ*IF <sup>i</sup>* (*nTs*) and the echo signal *<sup>x</sup>*ˆ*IF <sup>s</sup>* (*nTs*) is

$$R\_{r,s}^{a-f}(\tau) = \frac{1}{N} \sum\_{n=0}^{N-1} \mathbf{x}\_s^{IF} \left( nT\_s + \tau/2 \right) \hat{\mathbf{x}}\_i^{IF} \left( nT\_s - \tau/2 \right)^\* e^{-j2\pi anT\_s},\tag{16}$$

where *τ* is the delay, *α* represents the cyclic frequency, and *N* stands for the number of sampling points.

The vectors at the cyclic frequencies *α* and *α* − *f* corresponding to the maximum peak values of *Rα riri* (*τ*) and *R<sup>α</sup> ris*(*τ*) are respectively extracted, and are recorded as *<sup>R</sup>α riri* (*τ*) and *Rα* −*f ris* (*τ*). The mutual fuzzy function processing is performed on these two vectors to obtain

$$\Psi\_i(u, f) = \sum\_{\tau=0}^{N-1} R\_{r\_i s}^{a'-f'}(\tau) R\_{r\_i r\_i}^{a'}(\tau - u)^\* e^{j\pi f \tau},\tag{17}$$

where Ψ*i*(*u*, *f*) represents the CCAF between the monitoring channel signal and the reference signal, *u* is the delay, and *f* is the Doppler shift.
