## 2 Signal Model and Problem Formulation
### ISAC Signal Model with OFDM Waveform  
OFDM waveform is generally used to transmit signals in a 5G communication system. Considering the millimeter waveband in 5G, the OFDM communication symbols can be shared for sensing. In other words, an ISAC system can be constructed based on the 5G communication system. Analytically, the transmitted OFDM signal can be expressed as  
\[s(t)=\sum_{m=0}^{N_{s}-1}\sum_{n=0}^{N_{c}-1}a_{n,m}e^{|2\pi f_{c}t}e^{|2\pi n \Delta f(t-mT_{s})}\text{rect}(\frac{t-mT_{s}}{T_{s}}), \tag{1}\]  
where \(N_{s}\) denotes the number of OFDM symbols composing the communication frame, \(N_{c}\) denotes the total amount of subcarries, \(a_{n,m}\) denotes the \(n\)th complex modulation information of the \(m\)th OFDM symbol, \(n\Delta f\) is the \(n\)th individual subcarrier frequency, \(f_{c}\) denotes the carrier frequency, \(T_{s}=T_{cp}+T\) denotes the total OFDM symbol duration composed of an elementary symbol duration \(T=1/\Delta f\) and a CP duration \(T_{cp}\), and \(\text{rect}(t/T_{s})\) represents a rectangular window of duration \(T_{s}\). Considering an object in the range \(r\) with velocity \(v\), the received baseband signal can be described as  
\[ \begin{split} y(t)&=y_{t}(t)+y_{c}(t)+y_{n}(t)\\ &=\sigma\sum_{m=0}^{N_{s}-1}e^{|2\pi f_{d}mT_{s}}\sum_{n=0}^{N_{ s}-1}a_{n,m}e^{|2\pi n\Delta f(t-mT_{s})}e^{-|2\pi n\Delta f_{\tau}}\text{rect}( \frac{t-mT_{s}-\tau}{T_{s}})+y_{c}(t)+y_{n}(t),\end{split}  \tag{2}\]  
where \(y_{t}(t)\) denotes the object reflection signal, \(y_{c}(t)\) denotes the clutter due to the reflection of ground, green belts, and wind, etc., \(y_{n}(t)\) denotes the receiver thermal noise, \(\sigma\) denotes attenuation and phase shift occurring due to the propagation and scattering process, and \(f_{d}\) and \(\tau\) denote the Doppler shift and the round-trip propagation time of the object, respectively. They can be described by  
\[f_{d}=2 vf_{c}/c, \tag{3}\]  
with \(c\) being the speed of light, and  
\[\tau=2r/c. \tag{4}\]In order to describe the signal processing conveniently, we only rewrite the object reflection signal here. Then, according to (2), the \(m\)th received OFDM symbol with removing CP can be rewritten as  
\[y_{tm}(t)=\sigma e^{|2\pi f_{d}mT_{s}}\sum_{n=0}^{N_{c}-1}a_{n,m}e^{|2\pi m\Delta f (t-mT_{s})}e^{-|2\pi m\Delta f\tau}. \tag{5}\]  
Obviously, we can separate subcarrier of the \(m\)th OFDM symbol by exploiting DFT with sampling interval \(1/N_{c}\Delta f\). Specifically, the modulation information can be removed by an element-wise complex division to \(a_{n,m}\). Then, the \(m\)th received OFDM symbols can be described as a vector  
\[\mathbf{y}_{tm}=\sigma e^{|2\pi f_{d}mT_{s}}\Big{[}1,e^{-|2\pi\Delta f\tau},\dots, e^{-|2\pi(N_{c}-1)\Delta f\tau}\Big{]}. \tag{6}\]  
We perform the same operation on each received OFDM symbol in (2), and can obtain the received matrix expressed as  
\[\mathbf{Y} =\Big{[}\mathbf{y}_{l0}^{\mathrm{T}},\mathbf{y}_{l1}^{\mathrm{T}},\dots, \mathbf{y}_{l(N_{l}-1)}^{\mathrm{T}}\Big{]}^{\mathrm{T}}+\mathbf{Y}_{c}+\mathbf{Y}_{n} \tag{7}\] \[=\sigma\mathbf{b}_{fd}\otimes\mathbf{b}_{\tau}+\mathbf{Y}_{c}+\mathbf{Y}_{n},\]  
where \(\mathbf{Y}_{c}\) denotes the clutter matrix, \(\mathbf{Y}_{n}\) denotes the thermal noise matrix, \(\mathbf{b}_{fd}\) denotes the Doppler dimension vector, and \(\mathbf{b}_{\tau}\) denotes the range dimension vector, and they are expressed as  
\[\mathbf{b}_{fd}=\Big{[}1,e^{|2\pi f_{d}T_{s}},\dots,e^{|2\pi f_{d}(N_{c}-1)T_{s}} \Big{]}^{\mathrm{T}}, \tag{8}\]  
and  
\[\mathbf{b}_{\tau}=\Big{[}1,e^{-j2\pi\Delta f\tau},\dots,e^{-j2\pi(N_{c}-1)\Delta f \tau}\Big{]}. \tag{9}\]  
According to (9), we can conveniently evaluate \(\tau\) by computing the \(N_{c}\)-points inverse discrete Fourier transform (IDFT) of (7) along each row of \(\mathbf{Y}\). In a similar way, the determination of \(f_{d}\) can be solved by applying a \(N_{s}\)-points DFT along each column of \(\mathbf{Y}\).