## 3 Digital SI Cancellation Using Adaptive Filter  
In the proposed prototype system, an AF is adopted for the SI cancellation in a digital domain. AF is generally utilized to deal with the interference in a system where a kind of reference signal that is related to the interference signal can be known. In an IBFD system, due to the fact that the SI comes from the DL signals, which can be known at BS, AF is a suitable technology for digital SI cancellation. The structure of the AF-based SI cancellation method in an IBFD system is provided in Figure 8. The basic structure of the AF-based cancellation method is an \(N_{\text{taps}}\)-tap adjustable finite impulse response (FIR) filter with \(N_{\text{taps}}\)-tap delay line and \(N_{\text{taps}}\) adjustable coefficients. At time index \(n\), the interfered received signal \(v(n)\) contains two parts: the desired received UL signal \(d(n)\) and SI signal \(s^{\prime}(n)\), i.e., \(v(n)=d(n)+s(n)^{\prime}\). The SI signal \(s^{\prime}(n)\) comes from the DL transmitted signal \(s(n)\), i.e., \(s^{\prime}(n)=f(s(n))\), and \(s\) is used as a reference signal to be fed into the adjustable FIR filter. The adjustable FIR filter outputs signal \(s^{\prime\prime}(n)\) as a trained version of \(s(n)\), i.e., \(s^{\prime\prime}(n)=g(s(n),\mathbf{h}_{n})\), where \(\mathbf{h}_{n}\) is a vector consists of the adjustable FIR filter coefficients at time index \(n\), which can be expressed as  
\[\mathbf{h}_{n}=[h_{n},h_{n-1},\cdots,h_{n-N_{\text{taps}}+1}]^{\text{T}}, \tag{1}\]  
where \(h_{j}\) represents the adjustable coefficient of the \(j\)-th tap of the adjustable FIR filter. Then, the total filter output \(y(n)\) is obtained from  
\[y(n)=v(n)-s^{\prime\prime}(n), \tag{2}\]  
where \(y(n)\) is the clean version, i.e., the SI canceled version, of the RX signal. From the description above, the target of the digital SI cancellation is to find best filter coefficients so that \(g(s(n),\mathbf{h}_{n})=f(s)\), which means that SI is completely canceled in the optimal case \(s^{\prime\prime}(n)=s^{\prime}(n)\). That is, the objective of the digital SI cancellation is to find an optimal filter coefficient vector that  
\[\mathbf{h}^{*}=\underset{\mathbf{h}_{n}}{\min}\big{|}\big{|}s^{\prime\prime}( n)-s^{\prime}(n)\big{|}\big{|}, \tag{3}\]  
Figure 7: Measured results of the antennas used in system experiment, return loss (S11) for each antenna, and self-interference (S21) between two antennas. The TX/RX antenna configuration is parallel (side-by-side), distance (_d_) is 150 mm. To optimize the optimization problem in (3), firstly, we re-formulate two expressions in the SI cancellation model at time index \(n\) in Figure 8 in vector form as  
\[\mathbf{v}(n) =[v(n),v(n-1),\cdots,v(n-N_{\text{taps}}+1)]^{\text{T}}, \tag{4}\] \[\mathbf{s}(n) =[s(n),s(n-1),\cdots,s(n-N_{\text{taps}}+1)]^{\text{T}}, \tag{5}\]  
where \(\mathbf{v}(n)\) and \(\mathbf{s}(n)\) represent the vector form of the interfered RX (UL) signal \(v(n)\) and TX (DL) signal \(s(n)\), respectively. Then, the output of the FIR structure of AF, i.e., \(s^{\prime\prime}(n)\), can be expressed as  
\[s^{\prime\prime}(n)=\mathbf{h}_{n}^{\text{H}}\mathbf{s}(n). \tag{6}\]  
where \(s^{\prime\prime}(n)\) means the trained version of the SI source \(s(n)\). Then, the AF SI cancellation can be operated by subtracting the result of (6) from the interfered RX signal \(v(n)\) as in (2). From the expression in (2), it can be known that the AF output \(y(n)\) is actually the difference between RX signal \(v(n)\) and FIR output \(s^{\prime\prime}(n)\), and can be reformulated as  
\[y(n)=d(n)+\big{(}s^{\prime}(n)-s^{\prime\prime}(n)\big{)}. \tag{7}\]  
Because the UL-desired signal \(d(n)\) is uncorrelated to either \(s^{\prime}(n)\) or \(s^{\prime\prime}(n)\), solving the optimization problem in (3) is equivalent to the minimization of (7).