## 4 The Improved Frequency Domain Equalization Algorithm with Reduced Cyclic Prefix
As shown in Figure 2, the error signal of the frequency domain equalizer at this time can be written into Equation (10) with the multiplication of matrixes representing the multiplication of the corresponding elements in these matrixes. \[E_{i}=X_{i}-pf_{i}+qb_{i}=X_{i}-\sum_{s=0}^{t\_i-1}Y_{i-s}Wf_{i}(s+1)+\sum_{i=1 }^{t\_b}X_{i-t}Wb_{i}(t) \tag{10}\]  
Following the minimum mean square error (MMSE) criterion; the least mean square (LMS) algorithm characterized by low computational complexity was used to adjust the frequency domain coefficient of the equalization filters [32]. The iteration of the frequency domain coefficients \(Wf_{i}\) and \(Wb_{i}\) of the OFDM-based frequency domain decision feedback equalizer was defined by Equation (11) with \(s=0,1\ldots(t\_i-1)\) and \(t=1,2\ldots t\_b\), respectively. Moreover, \(\mu_{i}(s)\) and \(\rho_{i}(t)\) represented the step length of \(Wf_{i}(s)\) and \(Wb_{i}(t)\) adjusted at the time \(i\), respectively. In order to further speed up the convergence of the equalization algorithm, the iteration step length of the LMS algorithm was also adjusted according to the MMSE criterion. The adjusted iteration was given by Equation (12). In  
Figure 2: OFDM-based frequency domain decision feedback adaptive equalization algorithm. the equation, \(W\!f^{\prime}_{i}\) (\(s\)) and \(W\!b^{\prime}_{i}(t)\) were the derivative of \(W\!f_{i}\)(\(s\)) and \(W\!b_{i}(t)\) at the step length \(\mu_{i}(s)\) and \(\rho_{i}(t)\), respectively. As derived from Equation (11), their relationship was defined by Equation (13)  
\[\left\{ \begin{array}{l}W\!f_{i+1}(s+1)=W\!f_{i}(s+1)+2\mu_{i}(s+1)E_{i}Y^{*}_ {i-s}\\ W\!b_{i+1}(t)=W\!b_{i}(t)-2\rho_{i}(t)E_{i}X^{*}_{i-t}\end{array} \right. \tag{11}\]  
\[\left\{ \begin{array}{l}\mu_{i+1}(s)=\mu_{i}(s)+\alpha\cdot\text{Re}[W\!f^{ \prime}_{i}(s+1)Y_{i-s}E^{*}_{i}]\\ \rho_{i+1}(t)=\rho_{i}(t)-\beta\cdot\text{Re}[W\!b^{\prime}_{i}(t)X_{i-t}E^{*} _{i}]\end{array} \right. \tag{12}\]  
\[\left\{ \begin{array}{l}W\!f^{\prime}_{i+1}(s+1)=W\!f^{\prime}_{i}(s+1)+2E_{i }Y^{*}_{i-s}-2\mu_{i}(s)W\!f^{\prime}_{i}(s+1)|Y_{i-s}|^{2}\\ W\!b^{\prime}_{i+1}(t)=W\!b^{\prime}_{i}(t)-2E_{i}X^{*}_{i-t}-2\rho_{i}(t)W\!b^{ \prime}_{i}(t)|X_{i-t}|^{2}\end{array} \right. \tag{13}\]  
According to the Equations (11)-(13), the addition operation is ignored, and each symbol equalization requires (t_i + t_b) * (7N) times multiplication. In terms of computation, the proposed frequency domain decision feedback equalization algorithm involves more filter parameters in the operation since it adopts a hierarchical feedback mechanism. In practical applications, communication signals utilize only \(N\) subcarriers (numbered from \(m_{0}\) to \(m_{0}+N-1\)) out of \(V/2\) available carriers [33]. It is therefore needed to compensate for the frequency selective fading on \(N\) subcarriers. The compensation for that on the other subcarriers can be ignored since it does not affect the final decoding. It means multiplying the frequency band range occupied by \(N\) subcarriers by a frequency domain rectangular window. In terms of time domain, it is equivalent to the convolution of the received signal and the signal given by Equation (14). Thus, an additional multipath interference is imposed on the received signal. The energy of the sampling function \(Sa(t)\) is primarily concentrated within the interval \([-3\pi,3\pi]\). Based on Equation (16), the length of additional multipath effect is only \(6V/N\) at this time, so that it has little influence on communication signals. The compensation for such effect is very easy to achieve with the equalization algorithm. Consequently, the simplified equalization algorithm needs only the calculation of error and filter parameters on \(N\) subcarriers based on Equations (11)-(13). \[I_{add}(n)=\frac{2\pi Nf_{s}}{V}Sa(\frac{\pi Nn}{V})\cos[\frac{\pi(2m_{0}+N)n }{V}] \tag{14}\]