# Intrinsic Interference Cancellation Scheme for FBMC-OQAM Systems Based on Power Multiplexing
## 3 Description of the Proposed Scheme
### Power Multiplexing and Intrinsic Interference Cancellation in Transmitter  
Denote OQAM symbols around the pilot by vector \(\mathbf{a}=[a_{1}\;a_{2}\;\cdots\;a_{K}]^{T}\) and represent coded symbols as \(\mathbf{d}=[d_{1}\;d_{2}\;\cdots\;d_{K}]^{T}\), which are obtained by  
\[\mathbf{d}=\mathbf{Ga}, \tag{9}\]  
where \(\mathbf{G}\) is a power multiplexing coding matrix of size \(K\times K\), which is calculated by  
\[\mathbf{G}=\mu_{1} \begin{bmatrix}1&0&0&0&0\\ c&1&0&0&0\\ c&0&1&0&0\\ \vdots&0&0&\ddots&0\\ c&0&0&0&1\end{bmatrix} , \tag{10}\]  
\[\mu_{1}=\sqrt{\frac{K}{(1+c^{2})K-c^{2}}}, \tag{11}\]  
The estimate of \(\mathbf{s}\) after equalization can be expressed as \(\mathbf{\hat{s}}=\mathbf{s}+\mathbf{n}\), where \(\mathbf{\hat{s}}\) stands for the estimate of \(\mathbf{s}\) and \(\mathbf{n}\) is a vector of independent and identically distributed (i.i.d.) \(\mathcal{CN}(\mathbf{0},\sigma^{2})\) noise. The estimate of \(\mathbf{d}_{1,\ N}\) is calculated by  
\[\mathbf{\hat{d}}_{1,\ N}=\mathbf{Q}^{T}\mathbf{\hat{s}}= \begin{bmatrix}0& \mathbf{d}_{2,\ N}^{T}\end{bmatrix} ^{T}+\mathbf{\hat{n}}, \tag{18}\]  
where \(\mathbf{d}_{2,\ N}= \begin{bmatrix}d_{2}&\cdots&d_{N}\end{bmatrix} ^{T}\), \(\mathbf{\hat{n}}=\mathbf{Q}^{T}\mathbf{n}\), \(\mathbf{\hat{d}}_{1,\ N}= \begin{bmatrix}0&\hat{d}_{2}&\cdots&\hat{d}_{N}\end{bmatrix} ^ {T}\) is an estimate of \(\mathbf{d}_{1,\ N}\), and it is easy to know that \(\hat{d}_{k}=a_{k}+ca_{1}+\bar{n}_{k}\), \((k=2,3,\cdots,K)\) according to (9) and (18). The successive interference cancellation algorithm described in Figure 2 is used to recover \(a_{k}\) from \(d_{k}\). The first step of this algorithm regards \(ca_{1}+\bar{n}_{k}\) as additive noise to \(a_{k}\) and adopts detection algorithms to recover \(a_{k}\) from \(d_{k}\) for \(k=2,3,\cdots,K\). It is meaningful to note that the maximum likelihood (ML) detection algorithm is a good choice in this situation. It is known that M points in a constellation diagram need to be compared if we choose M-order quadrature amplitude modulation (QAM) in traditional multicarrier communication systems, such as OFDM. Fortunately, the ML detection algorithm only needs to compare \(\sqrt{M}\) points for M-order QAM in FBMC because data symbols in FBMC are real-valued symbols extracted from real or imaginary parts of complex QAM symbols. The difference between OFDM and FBMC in an M-order QAM constellation diagram is described in Figure 3.  
Supposing that \(a_{k}\) is perfectly recovered from \(d_{k}\), as shown in Figure 2 for \(k=2,3,\cdots,K\), i.e., \(\mathbf{\hat{a}}_{2,\ K}=\mathbf{a}_{2,\ K}= \begin{bmatrix}a_{2}&\cdots&a_{K }\end{bmatrix} ^{T}\), the next step of successive interference cancellation algorithm is to recover \(a_{1}\) from \(a_{k}\) and \(d_{k}\) for \(k=2,3,\cdots,K\). We represent \(\mathbf{\tilde{a}}_{1}\) as follows:  
\[\mathbf{\tilde{a}}_{1}=\mathbf{d}_{2,\ K}-\mathbf{a}_{2,\ K}= \begin{bmatrix} ca_{1}+\bar{n}_{2}\\ ca_{1}+\bar{n}_{3}\\ \vdots\\ ca_{1}+\bar{n}_{K}\end{bmatrix} , \tag{19}\]  
Figure 3: Constellation diagrams of M-order QAM in OFDM (**left**) and FBMC (**right**). Figure 2: Successive interference cancellation algorithm. and the estimate of \(a_{1}\) is calculated by the following step:  
\[\hat{a}_{1}=\frac{\mathbf{a}_{1}^{T}\mathbf{e}_{K-1}}{c(K-1)}=a_{1}+\bar{n}, \tag{20}\]  
where \(\mathbf{e}_{K-1}\) is a unit vector of size \((K-1)\times 1\), and \(\bar{n}\) is a noise of \(\mathcal{CN}\big{(}0,\ \sigma^{2}/\big{\{}c^{2}(K-1)\big{\}}\ \big{)}\). It is obvious that increasing the values of \(c\) and \(K\) is beneficial to the signal-to-noise ratio (SNR) of \(\hat{a}_{1}\), but it also means more inaccurate signal detection results of \(a_{k},\ (k=2,\cdots,K)\) and higher computational complexity. As a result, the values of \(c\) and \(K\) should be assigned properly to achieve a compromise among these factors.