## 4 Proposed FFT Nulling with Fine Frequency Shift  
Based on the above results, which demonstrate that interference cancellation is perfectly achieved when the NBI frequency aligns with an FFT grid frequency, we propose a method to align the NBI frequency with an FFT grid. Figure 4: Spectrum before cancellation (**left**) and after cancellation (**right**) when \(f_{I}=50.4/256T\). Figure 3: Spectrum before cancellation (**left**) and after cancellation (**right**) when \(f_{I}=50/256T\). Figure 2: Block diagram of interference cancellation. First, we accurately estimate the NBI frequency. After identifying the interference frequency, the interference frequency can be expressed as  
\[f_{1}=(m+\beta)/NT \tag{9}\]  
where \(m\) is an integer and \(\beta\) is a value in \((-0.5,+0.5)\). The received signal's frequency is then shifted by \(f_{F}=-\beta/NT\). Through this process, the NBI frequency becomes \(f_{F}^{{}^{\prime}}=m/NT\), allowing for perfect interference cancellation using the technique described in Section 3. The block diagram for the proposed frequency shift method is shown in Figure 5.  
The fine frequency value, \(f_{F}\), is obtained using an \(NP\)-point FFT, where \(P\) is an integer greater than 1. Through this process, a spectrum oversampled by a factor of \(P\) is obtained. Since the length of the input signal is \(N\), the FFT output is  
\[Y_{NP}(k)=\sum_{n=0}^{N-1}y(n)exp\left(-j\frac{2\pi kn}{NP}\right),\ k=0, \cdots,NP-1 \tag{10}\]  
Then, the position of the NBI in \(Y_{NP}(k)\) is determined. For convenience, let us assume that the existence of a tone interference is known. The position of the interference can be identified as follows:  
\[k_{I}=argmax|Y_{NP}(k)|^{2} \tag{11}\]  
The parameter \(P\) determines the level of oversampling for identifying the NBI frequency. While a higher value of \(P\) allows for more precise estimation of the interference frequency, it also increases computational complexity, as the range of \(k_{I}\) to be searched in (11) expands to \(NP\). This highlights a trade-off between the accuracy of NBI frequency estimation and computational efficiency, which should be carefully considered in practical implementations. Considering that the spectrum is oversampled by a factor of \(P\), the values of \(m\) and \(\beta\) in Equation (9) can be found as follows:  
\[ \begin{split} m&=\left\lceil k_{I}/P\right\rceil\\ \beta&=\left(k_{I}-P_{m}\right)/P\end{split}  \tag{12}\]  
The larger the value of \(P\), the more precise the frequency estimation becomes, reducing the discrepancy between the actual NBI frequency and the estimated value. However, this improvement in precision comes at the cost of increased computational complexity, as a higher \(P\) requires more computation. Once the fine frequency is identified, the received signal's frequency is shifted in the opposite direction:  
\[y^{\prime}(n)=y(n)\exp\left(-j2\pi\frac{\beta}{N}n\right) \tag{13}\]  
Figure 5: Proposed fine frequency shift. Through this process, the interference frequency is aligned with the FFT grid frequency. Following that, interference cancellation is performed using the method shown in Figure 2. First, the FFT of the received signal is taken, and then the NBI frequency is nulled:  
\[Y^{\prime}(k)=\sum_{n=0}^{N-1}y^{\prime}(n)exp\left(-j\frac{2\pi kn}{N}\right), \ k=0,\cdots,N-1 \tag{14}\]  
\[Y\left(k^{\prime}\right)=0 \tag{15}\]  
where  
\[k^{\prime}=argmax|Y\left(k\right)|^{2},\ \ k=0,\cdots,N-1. \tag{16}\]  
Since the NBI frequency and the grid frequency are aligned at this point, the interference will be effectively canceled by single FFT grid nulling, regardless of the specific interference frequency. Let us check the example in Figure 4 again.