# An Improved Blind Recognition Algorithm of Frame Parameters Based on Self-Correlation
## 3 Frame Parameters Recognition Algorithm Based on Self-Correlation
In order to realize the synchronization word recognition in the non-cooperative communication background, the sampling data storage matrix \(Y\) is defined, and the multiplication results of the sampling data at corresponding positions of different moments in the sliding window are sequentially put into the storage matrix, as shown in Equation (14). \[Y=[ \begin{array}{cccc}y^{1}_{\tau_{m}}y^{1}_{\tau_{n}}&y^{2}_{\tau_{m}}y^{2 }_{\tau_{n}}&\cdots&y^{n}_{\tau_{m}}y^{n}_{\tau_{n}}\end{array} ] \tag{14}\]  
In Equation (14), \(n\) is the number of sampled signals in the sliding window  
When \(n\) equals to the frame length or an integer multiple of the frame length, assuming that the received bit stream error rate is 0, the columns corresponding to the frame synchronization word should be all 0 or all 1. The data in other columns should be distributed randomly. When n is not the frame length or an integer multiple of the frame length, elements of each column in the storage matrix should be randomly distributed. Meanwhile, there is almost no case of all \"0\"s or all \"1\"s.  
Considering the case of a data payload sequence, the probability that each element in the storage vector is 0 and 1 is:  
\[\left\{ \begin{array}{l}\mathrm{P}\big{(}y_{i}=0\big{)}=\!\frac{3}{4}\\ \mathrm{P}\Big{(}y_{i}=1\big{)}=\!\frac{1}{4}\end{array} \right.,\ (1\ \leq\ i\ \leq\ n) \tag{15}\]  
According to the knowledge of mathematical statistics [12], \(y_{i}\) obeys the binomial distribution, and its mathematical expectation and variance are:  
\[\left\{ \begin{array}{l}\mathrm{E}(y_{i})=\mathrm{P}\Big{(}y_{i}=1\big{)}= \!\frac{1}{4}\\ \mathrm{D}(y_{i})=\mathrm{P}(y_{i}=0)\times\mathrm{P}\big{(}y_{i}=1\big{)}=\! \frac{3}{16}\end{array} \right. \tag{16}\]  
In the synchronization word sequence and the data payload sequence, the probability distributions of 0 or 1 are different. Therefore, in the case where the starting time and the length of sliding time window are constant, the presence of synchronization word sequence can be detected by calculating the distributions of 0 and 1 corresponding to each position in the sliding time window. After multiple iterations of the search, the final conclusion is reached. The sampling matrix \(Y^{*}\) is defined as:  
\[ \begin{array}{l}Y^{*}=\left[\begin{array}{cccc}y^{1}_{\tau_{m}}y^{1}_{ \tau_{n}}&y^{2}_{\tau_{m}}y^{2}_{\tau_{n}}&\cdots&y^{n}_{\tau_{m}}y^{n}_{\tau_ {n}}\\ y^{1}_{(\tau_{n}+n)}y^{1}_{(\tau_{n}+n)}&y^{2}_{(\tau_{n}+n)}y^{2}_{(\tau_{n}+n) }&\cdots&y^{n}_{(\tau_{m}+n)}y^{(\tau_{n}+n)}\\ \vdots&\vdots&\ddots&\vdots\\ y^{1}_{(\tau_{m}+Nn-n)}y^{1}_{(\tau_{n}+Nn-n)}&y^{2}_{(\tau_{m}+Nn-n)}y^{2}_{( \tau_{n}+Nn-n)}&\cdots&y^{n}_{(\tau_{m}+Nn-n)}y^{n}_{(\tau_{n}+Nn-n)}\end{array}\right]\end{array}  \tag{17}\]  
In Equation (17), \(N\) is the number of matrix rows. By summing the data in each column of \(Y^{*}\):  
\[ \begin{array}{l}Y_{sum}{}^{*}\\ =\left[\begin{array}{cccc}\sum\limits_{i=0}^{N-1}y^{1}_{(\tau_{n}+n)}y^{1}_ {(\tau_{n}+n)}&\sum\limits_{i=0}^{N-1}y^{2}_{(\tau_{n}+ni)}y^{2}_{(\tau_{n}+n) }&\cdots&\sum\limits_{i=0}^{N-1}y^{n}_{(\tau_{n}+ni)}y^{n}_{(\tau_{n}+ni)}\\ =\left[\begin{array}{cccc}y^{1}_{1}&y^{2}_{2}&\cdots&y^{n}_{\tau}\end{array} \right]\end{array}\right]\end{array}  \tag{18}\]  
In Equation (18), \(y^{*}_{i}\) obeys the binomial distribution with the parameter \((N,\ p)\), where \(p=0.5\). Supposing that the random variable \(X_{n}\) obeys the binomial distribution with the parameter \((n,\ p)\) (\(0<p<1\), \(n\ \geq\ 1\)), when \(n\) is large enough, the binomial distribution has a normal distribution as its law of limit distribution according to the DeMoivre-Laplace central limit theorem. Thus, when the column \(y^{*}_{i}\) corresponds to the data payload sequence, taking a sufficient number of sliding window sample data, the probability distribution of \(y^{*}_{i}\) can be equivalent to that generated by obeying \(\mathrm{N}(Np,\ Np(1-p))\).