# Wideband Spectrum Sensing Based on Reconfigurable Filter Bank in Cognitive Radio
## 3 Design of Reconfigurable P-DFTFB
### Design of Analysis P-DFTFB  
The conventional FB is structurally made in a lattice network or a tree cascade. Note that the number of divided sub-bands may increase to sharply put on more computational complexity of both structures, so it is not suitable for the large number of sub-bands. With a parallel structure, the polyphase uniform DFT modulation counts on an analysis FB that is modulated by the prototype low-pass filter, featuring a simple computational structure with lower complexity. The polyphase decimation is there to quicken the subsequent filtering and FFT operation. Having equipped with a P-DFTFB, wideband spectrum sensing technology exhibits attractive frequency characteristics being capable of eliminating the influence of aliasing. Its implementation structure is so efficient to be able to greatly fascinate the processing course of the whole system. This is truly remarkable and therefore widely welcomed in wideband spectrum sensing technology. Assume that the wideband input signal is \(r(n)\), the number of sub-bands to be divided is \(M\), and the prototype filter \(h(n)\) is a low-pass FIR filter, with \(N\) in length, as \(h(n)=\{h(0),h(1),\cdots,h(N-1)\}\). \(N\) is an integer multiple of \(M\), which is an integer power of two, i.e., \(L=N/M\) and \(L\) is an integer. The polyphase structure of the filter acts to get less computational burden and hardware complexity. The prototype filter \(h(n)\) has a transfer function \(H(z)\), which is defined as:  
\[ \begin{array}{ll}H(z)&=\sum\limits_{n=0}^{N-1}h(n)z^{-n}\\ &=h_{0}+h_{M}z^{-M}+\cdots+h_{(L-1)M}z^{-(L-1)M}\\ &+h_{1}z^{-1}+h_{M+1}z^{-(M+1)}+\cdots+h_{(L-1)M+1}z^{-(L-1)M-1}\\ &\vdots\\ &+h_{M-1}z^{-(M-1)}+h_{2M-1}z^{-(2M-1)}+\cdots+h_{LM-1}z^{-(LM-1)}\end{array}  \tag{1}\]  
Let \(H(z)\) be expressed by polyphase decomposed form with \(M\) polyphase branches as:  
\[ \begin{array}{ll}H(z)&=z^{0}(h_{0}+h_{M}z^{-M}+\cdots+h_{(L-1)M}z^{-(L-1)M}) \\ &+z^{-1}(h_{1}+h_{M+1}z^{-M}+\cdots+h_{(L-1)M+1}z^{-(L-1)M})\\ &\vdots\\ &+z^{-(M-1)}(h_{M-1}+h_{2M-1}z^{-M}+\cdots+h_{LM-1}z^{-(L-1)M})\\ &=\sum\limits_{m=0}^{M-1}z^{-m}\sum\limits_{l=0}^{L-1}h(Ml+m)z^{-Ml}\end{array}  \tag{2}\]  
where \(m=0,1,\cdots,M-1\) and \(l=0,1,\cdots,L-1\). For \(h(Ml+m)\) are the polyphase branches of \(h(n)\), represented by \(h_{m}(l)\), then  
\[H_{m}(z^{M})=\sum\limits_{l=0}^{L-1}h(Ml+m)z^{-Ml} \tag{3}\]  
Therefore, \(H(z)\) is expressed as:  
\[H(z)=\sum\limits_{m=0}^{M-1}H_{m}(z^{M})z^{-m} \tag{4}\]  
According to the definition of DFT, Equation (4) is a DFT transform of \(H_{m}(z^{M})\). So, the filter is implemented by a polyphase system followed by DFT modulator. The precondition such as center frequency is unnecessarily known in advance for input wideband signal \(r(n)\) when it is decomposed into \(M\) sub-band signals, and the sampling rate of each sub-band is reduced to \(1/M\) of the original. At the same time, the prototype filter coefficient \(h(n)\) is divided into \(M\) polyphase branches as \(\mathrm{h}_{m}(n)\). To keep the data rate in a lower level with less complexity, the prototype filter coefficient \(h(n)\) is decimated with \(M\) times. It is known that the equivalenttransformation principle of the network structure should be followed in the multisampling rate system, so the signal delay of \(M\) samples is given a \(M\)-times decimation, which is equivalent to decimate \(M\) times and then delay one sample. The sub-band division block diagram can be obtained for the P-DFTFB as shown in Figure 3. The output of each sub-band is \(Y_{m}(l)\), if \(M\) is an integer power of 2, an FFT operation may be performed to supersede the DFT operation. Meanwhile, the P-DFTFB from critical decimation (the decimation times is equal to the number of divided sub-bands) satisfied the reconstruction condition, so it does not need to demonstrate the reconfigurability of the FB, and the integrated FB will not be taken into account in the FB design.