## 4 NBI Suppression Algorithm
### Cyclic Cascaded High Order Complex-Coefficient Matrix Adaptive Notch Filter  
In the linear cascaded high order complex-coefficient adaptive notch filter, the tracking performance will degrade, since the input of the current section of the adaptive notch filter unit is the output of the previous one. Moreover, the convergence rate will decrease. To overcome the above shortcomings, we propose a cyclic cascaded framework, which is depicted in Figure 4. A \(K\)-order cyclic cascaded matrix adaptive notch filter is composed of the \(K\times K\) basic one order IIR notch filter units. Apparently, \(K\times K\) adaptive coefficients should be calculated for each iteration in a \(K\)-order cyclic cascaded high order matrix adaptive notch filter. In fact, only \(K\) active adaptive weight coefficients in the left column as depicted under the solid line in Figure 4 should be determined according to the weight coefficient iteration algorithm. Meanwhile, the updated complex value weight coefficients are copied to the other corresponding adaptive notch filter units at each iteration, which reduced the algorithm complexity greatly. For a \(k\) order CCHOMANF, only \(12k\) multipliers and \(5k\) adders are used to fulfil the update of the adaptive weight coefficients. Commonly, the number of in-band narrow band interferences with different central frequencies is not more than 10 in practical ultra-wide band wireless sensor networks. In other words, the order of CCHOMANF is not very large, so we can set \(3\leq k<10\), which makes the computation complexity of the proposed narrow band interference suppression algorithm reasonable. To simplify the discussion, we define adaptive notch filter cell NF\({}_{k}\) (\(k=1,2,...,K\)) as the \(k\)-th active basic adaptive notch filter unit, which is depicted in left column in Figure 4. The transfer function of the \(k\)-th active adaptive notch filter is given by:  
\[H_{k}(z)=\frac{1-h_{k}z^{-1}}{1-\rho h_{k}z^{-1}}. \tag{40}\]  
Hence, the transfer function of cyclic cascaded high order matrix adaptive notch filter can be described as:  
\[H(z)=\frac{1}{K}\sum_{m=1}^{K}\prod_{k=1}^{K}H_{k}(z), \tag{41}\]  
Figure 4: The framework of cyclic cascaded high order complex-coefficient matrix adaptive notch filter based on one order complex-coefficient adaptive notch filter units. where \(H_{k}(z)\) designates the \(k\)-th one order complex-coefficient IIR adaptive notch filter unit. The normalized frequency response of \(K\) order CCHOMANF is depicted in Figure 5. In theory, a \(K\) order CCHOMANF can suppress \(K\) narrow band interferences. Substantially, in the cyclic cascaded high order matrix adaptive notch filter, the entire complex plane is divided into \(K\) circulator sections with each section being monitored by a basic one-order adaptive notch filter unit. The specific number of cascaded units is dependent on the practical application situation. In this place, we note that the order number \(K\) of CCHOMANF does not need to be adjusted in different application scenarios, even if the number of narrow band interferences is variable. Normally, we consider \(5\leq K\leq 10\) (let the order number \(K\) be lager than the number of narrow band interferences in different frequencies). When \(K\) is more than the number of narrow band interferences \(N_{\text{NBIs}}\), \(K-N_{\text{NBIs}}\) redundant adaptive notch filter units will present. The redundant complex-coefficient adaptive notch filter cells will vibrate around the central frequencies of narrow band interferences close to them. Therefore, the proposed CCHOMANF still runs soundly, even though the order of the CCHOMANF \(K\) is not equal to the number of narrow band interferences. It is worth noting that, in the initialization stage, we set \(h_{k}(1)\) as follows: \(h_{1}(1)=1,h_{2}(1)=cos(2\pi/K)+jsin(2\pi/K),h_{k}(1)=cos(2(k-1)\pi/K)+jsin(2( k-1)\pi/K)\), which guarantees that different adaptive notch filter units converge to different narrow band interferences close to them, rather than some certain narrow band interference.