## 2 System Model  
OFDM-IM is a sophisticated modulation technique that extends the traditional OFDM architecture. Unlike conventional OFDM, which utilizes all subcarriers to carry data, OFDM-IM conveys additional information by using both data symbols and the indices of active subcarriers. By selectively activating a subset of subcarriers, OFDM-IM enhances spectral efficiency and improves the system's resilience against interference and fading, leading to better BER performance. The signal transmission process of an OFDM-IM system is illustrated in Figure 1.  
There are \(N\) subcarriers in each of the sections \(S\) that make up the entire transmitted signal bandwidth. Therefore, the total number of transmitted subcarriers in this OFDM-IM system is \(N_{T}\), where \(N_{T}=SN\). Each section's signal processing system is independent from the others and operates separately. Of the \(N\) subcarriers, only \(A\) are active in each section, while the remaining \((N-A)\) subcarriers are deactivated (zero-padded). A total of \(m\) bits is transmitted per section in each transmission. These \(m\) bits consist of \(m_{1}\) bits sent by the modulated symbols and \(m_{2}\) bits sent by the active subcarriers' indices. In this case, \(m_{1}\) bits is represented as \(m_{1}=Alog_{2}M\) and \(m_{2}\) bits is represented as \(m_{2}=\lfloor log_{2}C(N,A)\rfloor\); hence, the total bits \(m\) can be expressed as follows:  
\[m=m_{1}+m_{2}=Alog_{2}M+\lfloor log_{2}C(N,A)\rfloor \tag{1}\]  
where \(M\) is the \(M\)-\(ary\) modulation size. The \(m_{1}\) bits are mapped to a set of \(A\) active subcarrier indices using combinatorial methods. As a result, the transmit vector \(\mathbf{x}=[x_{1},...,x_{N}]\) is formed by assigning \(A\) nonzero data symbols to the \(A\) active subcarriers based on the incoming \(m\) bits. The mapping from bits to symbols is denoted as \(\mathbf{x}=f_{OFDM-IM}(\mathbf{D_{i}})\), where \(\mathbf{D_{i}}\) represents the input sequence of \(m\) bits for each group.  
The received signal in the frequency domain is represented as follows:  
\[\mathbf{y}=\mathbf{h}\odot\mathbf{x}+\mathbf{w} \tag{2}\]  
where \(\mathbf{h}=[h_{1},\dots,h_{N}]\) denotes the Rayleigh fading channel and \(\boldsymbol{\hat{h}}{\sim}\mathcal{CN}(0,1)\). The term \(\mathbf{w}_{i}{\sim}\mathcal{CN}(0,\sigma)\) represents the additive white Gaussian noise (AWGN).  
For the imperfect CSI settings, we consider a system in which the receiver experiences channel estimation errors. Let \(\hat{h}(\alpha)\) denote the estimated channel, while the actual channel is modeled as follows:  
\[h(\alpha)=\hat{h}(\alpha)+e(\alpha) \tag{3}\]  
where the channel estimation error \(e(\alpha)\) follows a complex Gaussian distribution \(e(\alpha)\sim\mathcal{CN}(0,\epsilon^{2})\), where \(\epsilon^{2}\) represents the channel estimation error variance. The estimated channel \(\hat{h}(\alpha)\) is modeled as \(\hat{h}(\alpha)\sim\mathcal{CN}(0,1-\epsilon^{2})\). The CSI error variance \(\epsilon^{2}\) is inversely related to the average SNR by \(\bar{\gamma}=\frac{E_{a}^{2}}{\epsilon^{2}}\), where \({E_{a}}^{2}\) is the average energy of a transmitted \(M\)-\(ary\) symbol and \(\sigma^{2}\) is the noise variance.  
Figure 1: Generalized data transmission process for an OFDM-IM system.  
In this experiment, \(\epsilon\) is computed as \(\epsilon=\frac{1}{1+\text{SNR}}\), with \(\text{SNR}=10^{\frac{\text{SNR}_{B}}{10}}\). This relationship adjusts the error variance based on the average SNR, which can more accurately reflect the real-world conditions of imperfect CSI under Rayleigh fading.