*3.1. Signal-to-Interference Ratio*

In the proposed method, the signal-to-interference ratio (SIR) changes with the shifting of *k*. At the sensing receiver, strength of an echo depends on several factors such as antenna gain, round trip distance from the target, carrier frequency, and radar cross-section, as mentioned in (6). If we define *P*Rx1 as the received power of *y*1(*k*), *P*Rx2 for *y*2(*k*), and *P*Rx3 for *y*3(*k*), then the SIR during the shifting of *k* is as below.

• At the beginning, echoes within *y*1(*k*) are detected under the collective ISI caused by *y*3(*k*) and *y*2(*k*), and we can define the SIR during this process as

$$\text{SIR}\_1 = \frac{P\_{\text{Rx}\_1}}{a\_2(\not{k})P\_{\text{Rx}\_2} + P\_{\text{Rx}\_3}},\tag{30}$$

where ´ *<sup>k</sup>* indicates the shift in *<sup>k</sup>* and 0 <sup>≤</sup> *<sup>α</sup>*2(´ *k*) < 1 defines the part of *P*Rx2 appearing as interference. *α*2(´ *k*) = 0 indicates that there are no echoes to form *y*2(*k*).

• At the second stage, when ´ *k* is beyond the *N*cp and within *Ns*, echoes that form *y*2(*k*) are detected and a part of *y*1(*k*) causes ISI, which increases with ´ *k*. The SIR can be defined as

$$\text{SIR}\_2 = \frac{P\_{\text{Rx}\_2}}{\alpha\_1(\acute{k})P\_{\text{Rx}\_1} + P\_{\text{Rx}\_3}},\tag{31}$$

where 0 <sup>≤</sup> *<sup>α</sup>*1(´ *k*) < 1 is used to account for the ISI caused by part of *P*Rx1 . • Similarly, when we detect echoes in *y*3(*k*), the SIR is

$$\text{SIR}\_3 = \frac{P\_{\text{Rx}\_3}}{P\_{\text{Rx}\_1} + \alpha\_2(\not k)P\_{\text{Rx}\_2}},\tag{32}$$

where *P*Rx1 appears as ISI because at this stage the element-wise division is performed by the previous OFDM symbol *Sq*−1.

Here, it is important to mention that *P*Rx1 > *P*Rx2 > *P*Rx3 (assuming same radar crosssection for different targets associated with echoes) because of the FSPL difference between echoes that form *y*1(*k*), *y*2(*k*) and *y*3(*k*). Therefore, SIR1 > SIR2 > SIR3, which clearly indicates that detection of echoes in *y*2(*k*) and *y*3(*k*) is not possible without the sufficient processing gain obtained through the IFFT/FFT operation during sensing. Usually, the Doppler estimation requires large interval (compared to the OFDM symbol duration) of the waveform; therefore, large number of OFDM symbols can provide sufficient processing gain for the echoes to overcome ISI.
