## 2 Fundamentals  
### IM-OFDM Module  
The block diagram of the considered IM-OFDM system is presented in Figure 1. Let the total available subcarriers be \(M\) with subcarrier spacing \(\Delta f\) and symbol duration \(T_{s}\). For implementation convenience, these subcarriers are divided into \(G\) groups, each containing \(N_{g}=M/G\) subcarriers. Since each subblock undergoes identical procedures, it suffices to consider the \(\xi\)-th subblock (\(\xi\in\{1,\ldots,G\}\)). Within each group, there are \(N_{g}\) subcarriers with indices \((i_{1},\ldots,i_{n},\ldots,i_{N_{g}})\), where \(i_{n}\) is the index of the \(n\)-th subcarrier in each subblock. In the IM-OFDM system, \(k\) subcarriers in each subblock are activated to transmit modulated symbols, where the indices of the activated subcarriers are stored in the set \(I_{g}\), and the complex modulated symbol set is \(S\). An example look-up table with a subblock of \(N_{g}=4,k=2\) with symbols \(C_{1},C_{2}\) is shown in Figure 1. Assume the input bitstream of \(N\) bits is partitioned into \(N_{1}\) and \(N_{2}\) bits, which are given by  
\[N_{1}=\left\lfloor\log_{2}\binom{N_{g}}{k}\right\rfloor,N_{2}=k\log_{2}(Q),\]  
where \((^{n}_{k})\) denotes the binomial coefficient representing the number of \(k\)-element combinations from \(n\) elements, and \(Q\) is the modulation order defined by the constellation diagram. \(N_{1}\) bits determine indices of activated subcarriers, and the remaining \(N_{2}\) bits form the information sequence \(\mathbf{b}_{B}(n)=[b_{B}(1),\ldots,b_{B}(N_{2})]^{T}\), which is mapped via constellation modulation to generate the symbol sequence \(\mathbf{c}(n)=[c(1),\ldots,c(N_{c})]^{T}\), where \(c(n)\in S\). Subsequently, during the \(j\)-th OFDM symbol (\(j=1,\ldots,N_{s}\)), where \(N_{s}\) is the length of the OFDM symbol, its communication-oriented signal is expressed as  
\[\mathbf{x}_{c}(j)=[x_{c,1}(j),\ldots,x_{c,m}(j),\ldots,x_{c,M}(j)]^{T}, \tag{1}\]  
where \(x_{c,m}(j)\in S\) if \(m\in I_{g}\); otherwise, \(x_{c,m}(j)=0\). The total number of information bits per symbol is given by \(N_{1}+N_{2}\) bits. Different from conventional approaches to realizing sensing functionality, which insert communication and sensing signals into separate subcarriers or superimpose both onto index-modulated subcarriers [13], the sensing signal \(\mathbf{x}_{s}(j)=[x_{s,1}(j),\ldots,x_{s,M}(j)]^{T}\) in our  
Figure 1: The framework of the IM-OFDM system. considered scheme employs pseudo-random sequences with superior auto-correlation properties. The sequences satisfy \(\mathbb{E}[\mathbf{x}_{c}^{H}(j)\mathbf{x}_{s}(j)]=0\), indicating statistical orthogonality between communication and sensing signals. Through the power allocation factor \(\rho\), we linearly combine them to maintain consistent transmission power, yielding the transmitted signal \(\mathbf{x}_{F}(j)=\mathbf{x}_{s}(j)+\mathbf{x}_{c}(j)\), where \(\mathbf{x}_{F}(j)=[x_{F,1}(j),\ldots,x_{F,M}(j)]^{T}\), which can guarantee both communication and sensing functionalities. By performing inverse fast Fourier transform (IFFT) on the frequency-domain samples \(\mathbf{x}_{F}\), the time-domain OFDM symbol is given by  
\[\mathbf{x}(j)=\text{IFFT}\{\mathbf{x}_{F}(j)\}=[x_{1}(j),\cdots,x_{m}(j),\cdots,x_{M}(j )]^{T},(m=1,\ldots,M,j=1,\ldots,N_{s}) \tag{2}\]  
where \(x_{m}(j)\) is the element of the OFDM symbol matrix \(\mathbf{X}_{M\times N_{s}}\in\mathbb{C}^{M\times N_{s}}\). After IFFT operation, a cyclic prefix (CP) of length \(C\) is appended to the beginning of the OFDM symbol, given by  
\[\mathbf{x}_{\text{cp}}(j)=[x_{M-C+1}(j),\cdots,x_{M}(j),x_{1}(j),x_{2}(j),\cdots x _{M}(j)]^{T}, \tag{3}\]  
where \(\mathbf{x}_{\text{cp}}\in\mathbb{C}^{(M+C)\times N_{s}}\). Then, a parallel-to-serial (P/S) conversion is performed to sequentially concatenate IM-OFDM symbols, forming the transmitted signal \(\mathbf{x}_{\text{t}}\in\mathbb{C}^{N_{s}(M+C)\times 1}\).