# Intrinsic Interference Cancellation Scheme for FBMC-OQAM Systems Based on Power Multiplexing
## 2 System Model  
### Intrinsic Interference of FBMC  
We denote the real-valued data symbol corresponding to the \(m\)th subcarrier of the \(n\)th FBMC symbol by \(a_{m,n}\), which is obtained by taking the real or imaginary parts of the complex QAM symbol. \(M\) represents the total number of subcarriers, and \(4T\) denotes the time duration of \(g(t)\), which is the impulse response of the PHYDYAS prototype filter [2; 22] with an overlapping coefficient equal to 4. Then the FBMC baseband signal transmitted in the time domain, as shown in Figure 1, can be expressed as [23]  
\[s(t)=\sum\limits_{n=0}^{+\infty}\sum\limits_{m=0}^{M-1}j^{m+n}a_{m,n}g\bigg{(} t-n\frac{T}{2}\bigg{)}e^{j2\pi\frac{\pi}{T}t}=\sum\limits_{n=0}^{+\infty} \sum\limits_{m=0}^{M-1}a_{m,n}g_{m,n}(t). \tag{1}\]  
As shown in Figure 1, the FBMC signal transmitted in the time domain adopts the \"overlap-add\" method, which can be fully explained by (1). \(\sum\limits_{m=0}^{M-1}a_{m,n}g_{m,n}(t)\) in (1) stands for the \(n\)th FBMC symbol, which has \(M\) subcarriers and a time duration of \(4T\), corresponding to a colored frame with \(nT/2\) time delay in Figure 1. \(s(t)=\sum\limits_{n=0}^{+\infty}\sum\limits_{m=0}^{M-1}a_{m,n}g_{m,n}(t)\) means all FBMC symbols overlap with respective time delays and then are added together to produce FBMC signals \(s(t)\), i.e., \"overlap-add\", corresponding to Figure 1. However, an overlap of several FBMC symbols results in an interference from the other 14 FBMC symbols to the FBMC symbol to be received and previous studies in the literature named it \"intrinsic interference\". Fortunately, the intrinsic interference can be eliminated completely in the AWGN channel with \"real-orthogonality\" expressed as follows [3]  
\[\Re\bigg{\{}\int g_{m,n}(t)g_{m_{0},n_{0}}^{*}(t)dt\bigg{\}}=\delta_{m,m_{0}} \delta_{n,n_{0}}, \tag{2}\]  
Figure 1: FBMC signal transmitted in the time domain with the “overlap-add” method. The red frame denotes the FBMC symbol to be received, and the blue frames stand for the FBMC symbols interfering with the red frame. where \(\delta_{m,n}\) denotes the Kronecker delta function. We express the received signal as \(y(t)=h(t)*s(t)+n(t)\), where \(h(t)\) denotes the channel impulse response and \(n(t)\) is the white noise. It is assumed that the time length of the prototype filter is much larger than the maximum delay of the channel, and the channel remains unchanged within the time length of the prototype filter. Then, the received signal is approximately expressed as  
\[y(t)\approx\sum_{n=0}^{+\infty}\sum_{m=0}^{M-1}H_{m,n}a_{m,n}g_{m,n}(t)+n(t), \tag{3}\]  
where \(H_{m,n}\) represents the channel frequency response (CFR) of the \(m\)th subcarrier of the \(n\)th FBMC symbol. In the receiver, data symbols without considering white noise are recovered by matched filtering, described as follows  
\[ \begin{split} r_{m_{0},n_{0}}&=\int y(t)g^{*}_{m_{0 },n_{0}}(t)dt\\ &=H_{m_{0},n_{0}}a_{m_{0},n_{0}}+\sum_{(m,n)\
eq(m_{0},n_{0})}H_{ m,n}a_{m,n}\int g_{m,n}(t)g^{*}_{m_{0},n_{0}}(t)dt,\\ &=H_{m_{0},n_{0}}a_{m_{0},n_{0}}+\sum_{(m,n)\
eq(m_{0},n_{0})}H_ {m,n}a_{m,n}\gamma^{m_{0},n_{0}}_{m,n}\end{split}  \tag{4}\]  
where \(\gamma^{m_{0},n_{0}}_{m,n}=\int g_{m,n}(t)g^{*}_{m_{0},n_{0}}(t)dt\) denotes the coefficient of intrinsic interference from the \(m\)th subcarrier of the \(n\)th FBMC symbol to the \(m_{0}\)th subcarrier of the \(n_{0}\)th FBMC symbol. As shown in Table 1, the coefficient \(\gamma^{m_{0},n_{0}}_{m,n}\) has a pure imaginary value. Assuming that channel changes in adjacent time-frequency points are not obvious and, thus, CFR of the time-frequency point \((m,n)\) near the time-frequency point \((m_{0},n_{0})\) meets the requirement \(H_{m,n}\approx H_{m_{0},n_{0}}\), Equation (4) can be simplified as  
\[ \begin{split} r_{m_{0},n_{0}}&=H_{m_{0},n_{0}} \left(a_{m_{0},n_{0}}+\sum_{(m,n)\
eq(m_{0},n_{0})}a_{m,n}\gamma^{m_{0},n_{0}} _{m,n}\right),\\ &=H_{m_{0},n_{0}}(a_{m_{0},n_{0}}+ju_{m_{0},n_{0}})\end{split}