# Null Subcarrier Index Modulation in OFDM Systems for 6G and Beyond
## 4 Proposed NSC-OFDM-IM
### Receiver Design and Detection
#### 4.1.1 Log-Likelihood Ratio Processing  
The next step is to find out the index of the inactive carrier location within each complex subgroup. As we explained at the transmitter side, the inactive carrier location is left as zero, and no carrier frequency is generated at this frequency location. Hence, at the receiver side, this allows us to design a very flexible and easily applicable detection mechanism with lower computational complexity. Figure 5: The proposed NSC-OFDM-IM receiver. In order to obtain the null carrier indices, the frequency domain signal is again processed separately, and thus, the log-likelihood ratio detection mechanism is applied to the complex signal vector. The LLR detector in the proposed NSC-OFDM-IM scheme provides the logarithm of the ratio of the a posteriori probabilities of the frequency domain signal vector, and it works by considering all frequency locations, regardless of whether they are active or inactive, along the frequency axis. The ratio of these a posteriori probabilities is given by,  
\[\lambda(\beta)=ln\frac{\sum\limits_{\kappa=0}^{M}P(X(\beta)=s_{ \kappa}|y_{F}(\beta)}{P(X(\beta)=0|y_{F}(\beta)}\quad\beta=1,2,\ldots,N \tag{17}\]  
As a simplified representation of this ratio, by using the Bayesian theorem, the \(\lambda(\beta)\) can be expressed for BPSK-modulated NSC-OFDM-IM as below:  
\[\lambda(\beta)=max(a,b)+ln(1+exp(-|b-a|))+\frac{|y_{F}(\beta)|^{2}}{N_{0,F}} \tag{18}\]  
where \(a=-|y_{F}(\beta)-h_{F}(\beta)|^{2}/N_{0,F}\) and \(b=-|y_{F}(\beta)+h_{F}(\beta)|^{2}/N_{0,F}\). \(N_{0,F}\) denotes the noise variance in the frequency domain, and it can be expressed as,  
\[N_{0,F}=(\frac{N_{used}}{N_{FFT}})N_{0,T} \tag{19}\]  
where \(N_{used}\) denotes the number of used active subcarriers in the frequency spectrum. After applying the log-likelihood ratio to the frequency signal vector at the output of FFT processing, the LLR vector with a length of N is achieved, and now, it is ready for subgroup separation. The size of the LLR subgroups is the same as the complex subgroups. The main objective of constructing these subgroups in the LLR domain is to find out the null carrier index within each subgroup so that the index modulated data are extracted from the received signal. The coefficients in the LLR vector are given by,  
\[\tilde{X}=\{\tilde{X}(1),\tilde{X}(2),\ldots,\tilde{X}(N)\} \tag{20}\]  
At this stage, we have two types of subgroups with the same size \(\tilde{M}\), which are complex subgroups for complex data extraction, and LLR subgroups to be used for index data extraction. These subgroups are processed separately in parallel form. For example, the first null carrier index is obtained from the first LLR subgroup, and in the meantime, by excluding this null carrier index, the active subcarriers are extracted from the first complex subgroup. As we designed at the transmitter side, each complex subgroup contains one null subcarrier, which is used for conveying the index data, and \((\tilde{M}-1)\) active subcarriers, which are used for complex data transmission. With this in mind, after the LLR calculation and subgroup separation in the LLR domain, the values in each LLR subgroup can be interpreted as the likelihood ratio, with a low value, which indicates that the observed result is unlikely to occur, which means that the occurrence of the null subcarrier at this location is highly possible, and with the remaining values, which can be interpreted as the occurrence of the active subcarriers. The interpretation of LLR values in each subgroup allows us to easily extract the null carrier location among the \(\tilde{M}\) carrier locations. To do so, each subgroup is evaluated separately in terms of its minimum real value, which indicates the null carrier position in the complex subgroup.