## 3 System Model
### OFDM-IM Communication Transmission Framework
\[\widetilde{\mathbf{X}}_{T}=\frac{U}{\sqrt{KG}}\text{IFFT}\Big{\{}\widetilde{ \mathbf{X}}_{F}\Big{\}}=\frac{1}{\sqrt{KG}}\mathbf{W}_{U}^{H}\widetilde{ \mathbf{X}}_{F}. \tag{18}\]  
\(\widetilde{\mathbf{X}}_{T}\) is the time-domain OFDM-IM block, \(\mathbf{W}_{U}\) is the FFT matrix and \(\mathbf{w}_{U}^{H}\mathbf{W}_{U}=U\mathbf{I}\mathbf{I}_{U}\), \(U/\sqrt{GK}\) is the normalization factor for the FFT, and \(E\big{\{}\widetilde{X}_{T}^{H}\widetilde{X}_{T}\big{\}}=U\). At the transmitter, a cyclic prefix (CP) with a length greater than the maximum channel delay must be added to \(\widetilde{\mathbf{X}}_{T}\) before transmission. At the receiver, after the signal passes through the wireless fading channel, it undergoes CP removal, FFT, and deinterleaving; the frequency domain data can be represented as  
\[\mathbf{y}_{F}=\mathbf{H}\mathbf{X}_{F}+\mathbf{n}_{F}, \tag{19}\]  
where \(\mathbf{y}_{F}=[y_{F}(1),y_{F}(2),\ldots,y_{F}(\mathbf{u}),\ldots y_{F}(U)]^{T}\), \(\mathbf{H}\) is a \(U\)-dimensional diagonal matrix with its diagonal elements representing the channel impulse response (CIR) coefficients of the multipath channel in the frequency domain, and \(\mathbf{n}_{F}\) denotes Gaussian noise in the frequency domain with a mean of 0 and variance of \(\sigma_{F}^{2}\). Based on the estimated CIR, channel equalization is applied to \(\mathbf{y}_{F}\), which can be represented as \(\hat{\mathbf{y}}_{F}\). To detect the positions of the active subcarriers in each subblock, the OFDM-IM signal separator divides \(\hat{\mathbf{y}}_{F}\) into \(G\) subblocks, with the \(\beta\)th subblock represented as  
\[\hat{\mathbf{y}}_{\beta}=\big{[}\hat{y}_{\beta}(1),\hat{y}_{\beta}(2),\ldots, \hat{y}_{\beta}(n),\ldots,\hat{y}_{\beta}(N)\big{]}^{T}, \tag{20}\]  
where \(\hat{\mathbf{y}}_{F}\) is passed to the log-likelihood ratio (LLR) detector to determine the positions of the active subcarriers. Subsequently, index inverse mapping and constellation symbol demodulation are performed, generating index information with a length of \(p_{1}\) bits and constellation information with a length of \(p_{2}\) bits. Finally, the bit combiner merges these into an information bit sequence of length \(B\). To connect the time-domain channel characteristics described by (1)-(11) with the frequency-domain representation in (20), we consider that, within the duration of one OFDM symbol \(T_{\text{sym}}\), the channel varies slowly. Hence, we assume that \(h_{TR}(\tau,t)\) is approximately time-invariant during one OFDM symbol, i.e., \(h_{TR}(\tau)\approx h_{TR}(\tau,t)\) for \(\text{t}\in\big{[}t_{0},t_{0}+T_{\text{sym}}\big{]}\). Sampling this time-invariant impulse response in the discrete-time domain (corresponding to \(U\) points for IFFT/FFT) yields a sequence \(h_{TR}[l],l=0,1,\ldots,L-1\)The DFT of the discrete-time impulse response \(h_{TR}[l]\), yielding the frequency-domain gains \(H_{k}\) for each subcarrier, is shown as  
\[H_{k}=\sum_{l=0}^{L-1}h_{TR}[l]e^{-j\frac{2\pi il}{l}},k=0,1,\ldots,U-1. \tag{21}\]  
Arranging \(H_{k}\) into the diagonal matrix \(\mathbf{H}=\text{diag}(H_{0},H_{1},\ldots,H_{U-1})\) corresponds directly to the frequency-domain model \(\mathbf{y}_{F}=\mathbf{H}\mathbf{X}_{F}+\mathbf{n}_{F}\) in (20).