# An Improved Blind Recognition Algorithm of Frame Parameters Based on Self-Correlation
## 4 Improved Frame Parameters Recognizing Algorithm Based on Self-Correlation  
The algorithm described in the previous section uses the 38 criterion to determine the threshold during the decision phase. Actually, if the number of rows of the sampling matrix \(Y^{*}\) is too low or the bit error rate (BER) is high, this decision method might lead to low accuracy. In this section, the peak-to-average ratio (PAR) and 38-criterion are combined to search for the length of the frame. Afterwards, the frame length value obtained by the search is substituted into the self-correlation algorithm described in the previous section to obtain the synchronization word sequence. The definition of the PAR is as shown in Equation (20). Suppose array \(X\) contains n elements, then:  
\[R=\frac{X_{\max}}{\frac{1}{n}\sum\limits_{i=0}^{n-1}X_{i}} \tag{20}\]  
where \(R\) is PAR, \(X_{i}(0\ \leq\ i\ \leq\ n)\) is any element in array \(X\). \(X_{\max}\) is the maximum value in the array.  
While searching for the frame length value, it needs to first apply the upper limit value \(\delta_{up}=\ u+3\sigma\) for coarse filtering and record the estimated frame length values and corresponding \(Y_{sum}^{*}\) arrays. Afterwards, for the filtered \(Y_{sum}^{*}\) arrays, it needs to calculate the peak-to-average ratio R of each array and determine the frame length corresponding to the array \(Y_{sum}^{*}\) with the largest R value as a reasonable frame length estimate. It can be shown from the simulation results that the correctness of frame length recognition and error-resistance performance can be greatly improved by applying this kind of judgment method.  
After obtaining the credible frame length estimation value, it is substituted into the algorithm in the previous section. When determining the 0 and 1, the up and down threshold are in the form of \(u\pm\alpha\sigma\), where \(\alpha\) is a constant slightly less than 3. Because the peak value is not as high as the ideal case when the bit error rate is high, if \(\alpha\) takes 3, it will cause the problem of missing the \"1\". However, the impact on the \"0\" decision is not obvious. The recommended value of \(\alpha\) is 2.7.  
As is shown in Figure 2, the \(\alpha\) of up and down thresholds are all 3. Figure 2a shows the result of \(Y_{sum}^{*}\) when the frame length is correctly detected at bit error rate (BER) = 0.08. Figure 2b is the partial enlargement of Figure 2a at the frame header. There are two \"1\"s below the up threshold, which leads to the missing judgement. In Figure 3, \(\alpha\) is 2.7, no missing judgement occurs when the receiving bit stream is unchanged, indicating that the fault tolerance is enhanced. It needs to be noticed that for Figures 2-4, all the coordinates on y-axis have been normalized while x-axis represents bit numbers.  
Figure 3: **(a)** This is the figure of \(Y_{sum}^{*}\) with BER = 0.08, \(\alpha\) = 2.7. **(b)** This is the partial enlargement of Figure 3a at the frame header. There are no “1”s below the up threshold, indicating that all “1”s are determined correctly.  
Figure 2: **(a)** This is the figure of \(Y_{sum}^{*}\) with BER = 0.08, \(\alpha\) = 3. **(b)** This is the partial enlargement of Figure 2a at the frame header. There are two “1”s below the up threshold, causing the missing judgement.