METHOD FOR ESTIMATING BIT ERROR PROBABILITY USING ERROR RATE RATIO OF FRAME SYNCHRONIZATION WORD

Abstract

The present invention relates to a method for estimating a bit error probability using an error rate ratio of a frame synchronization word, to lower computational complexity such that the method can be implemented in a relatively simple and economical way at a high computational speed. The method includes the steps of: a) defining error rate ratios of frame synchronization words; b) setting a weighted least squares cost function with weights greater than or equal to 0 for the bit error probability using the error rate ratios of the frame synchronization words set in the step a); c) obtaining an estimated bit error probability value that minimizes the cost function set in the step b); and d) sequentially obtaining the weights so that a mean squared error of the estimated bit error probability value obtained in the step c) becomes small.

Claims

1. A method for estimating a bit error probability using an error rate ratio of a frame synchronization word, the method comprising the steps of: a) defining error rate ratios of frame synchronization words; b) setting a weighted least squares (WLS) cost function with weights greater than or equal to 0 for the bit error probability using the error rate ratios of the frame synchronization words set in the step a); c) obtaining an estimated bit error probability value that minimizes the cost function set in the step b); and d) sequentially obtaining the weights so that a mean squared error of the estimated bit error probability value obtained in the step c) becomes small.

2. The method of claim 1, wherein in the step a), the error rate ratios .sub.0, .sub.1, . . . , and .sub.K1 of the frame synchronization words are defined as shown in the following Equation 1: $\begin{array}{cc}{}_k=\frac{n_{k+1}}{n_k}& \left[\mathrm{Equation}1\right]\end{array}$ where n.sub.k denotes the number of frames each having a frame synchronization word in which the number of bit errors is k, and K denotes a bit error allowable tolerance of a frame synchronization word having a length of M bits.

3. The method of claim 2, wherein in the step b), the WLS cost function (p.sub.e) with the weights w.sub.0, . . . , and w.sub.k1 as shown in the following Equation 4 is set using the error rate ratios set through the Equation 1: $\begin{array}{cc}f\left(p_e\right)=\underset{k=0}{\overset{K-1}{.Math.}}{w}_k{\left(\frac{{}_k}{c_k}-\right)}^2& \left[\mathrm{Equation}4\right]\end{array}$ where a constant c.sub.k is defined by the following Equation 3-2, and is defined by the following Equation 3-3, $\begin{array}{cc}{c}_k=\frac{M-k}{k+1}& \left[\mathrm{Equation}3-2\right]\end{array}$ $\begin{array}{cc}=\frac{p_e}{1-p_e},p_e=\frac{}{1+}.& \left[\mathrm{Equation}3-3\right]\end{array}$

4. The method of claim 3, wherein in the step c), values of and p.sub.e that minimize the cost function (p.sub.e) are obtained through the following Equations 5-1 and 5-2: $\begin{array}{cc}\hat{}=\frac{\underset{k=0}{\overset{K-1}{.Math.}}{w}_k{}_k/c_k}{\underset{k=0}{\overset{K-1}{.Math.}}{w}_k}& \left[\mathrm{Equation}5-1\right]\end{array}$ where {circumflex over ()} denotes an estimated value of the value of that minimizes (p.sub.e) of the Equation 4, $\begin{array}{cc}=\frac{\hat{}}{\hat{}+1}=\frac{\underset{k=0}{\overset{K-1}{.Math.}}{w}_k{}_k/c_k}{\underset{k=0}{\overset{K-1}{.Math.}}{w}_k{}_k/c_k+w_k}.& \left[\mathrm{Equation}5-2\right]\end{array}$

5. The method of claim 4, wherein in the step d), the weights w.sub.0, . . . , and w.sub.K1 are sequentially obtained so that the mean squared error of becomes small.

6. The method of claim 5, wherein in the step d), the bit error probability is estimated in a different manner depending on a value of K, which denotes a bit error allowable tolerance of a frame synchronization word, while w.sub.0 is set to 1.

7. The method of claim 6, wherein in the step d), when a value of n.sub.K+1 which denotes the number of frames each having a frame synchronization word in which the number of bit errors exceeds the allowable tolerance K, is not 0, and the value of K, which denotes a bit error allowable tolerance of a frame synchronization word, is 1, the estimated bit error probability value is obtained through the following Equation 6:
c.sub.0=M,.sub.0=n.sub.1/n.sub.0
=.sub.0/.sub.0,=/(+1).[Equation 6]

8. The method of claim 6, wherein in the step d), when a value of n.sub.K+1 which denotes the number of frames each having a frame synchronization word in which the number of bit errors exceeds the allowable tolerance K, is not 0, and the value of K, which denotes a bit error allowable tolerance of a frame synchronization word, is 2, the estimated bit error probability value is obtained through the following Equation 7: $\begin{array}{cc}{c}_0=M,c_1=\left(M-1\right)/2,{}_0=n_1/n_0,{}_1=n_2/n_1={}_0/c_0,=\frac{c_1\widehat{{}_0}\left(c_0\widehat{{}_0}+2\right)}{2c_1\widehat{{}_0}+1}=\frac{{}_0/c_0+{}_1/c_1}{1+\widehat{w_1}},=\widehat{{}_1}/\left(\widehat{{}_1}+1\right)& \left[\mathrm{Equation}7\right]\end{array}$ where, when the number n.sub.1 of frames of which frame synchronization words each have 1 bit error is 0, the error rate ratio ${}_1=\frac{n_2}{n_1}$ is not defined, and the weight is set to 0 and .sub.1/c.sub.1 is calculated as 0 in the Equation 7.

9. The method of claim 6, wherein in the step d), when a value of n.sub.K+1 which denotes the number of frames each having a frame synchronization word in which the number of bit errors exceeds the allowable tolerance K, is not 0, and the value of K, which denotes a bit error allowable tolerance of a frame synchronization word, is 3, the estimated bit error probability value is obtained through the following Equation 8: $\begin{array}{cc}{c}_0=M,c_1=\left(M-1\right)/2,c_2=\left(M-2\right)/3,{}_0=n_1/n_0,{}_1=n_2/n_1,{}_2=n_3/n_2={}_0/c_0,=\frac{c_1\widehat{{}_0}\left(c_0\widehat{{}_0}+2\right)}{2c_1\widehat{{}_0}+1}=\frac{{}_0/c_0+\widehat{w_1}{}_1/c_1}{1+\widehat{w_1}},=\frac{c_2\widehat{{}_1}\left\{c_0{c}_1\widehat{{}_1^2}+c_1{\widehat{{}_1}\left(1-\widehat{w_1}\right)}^2+2\widehat{w_1^2}+\widehat{w_1}\right\}}{\left(c_2\widehat{{}_1}+1\right)+\left(2c_2\widehat{{}_1}+1\right)\widehat{w_1}}=\frac{{}_0/c_0+\widehat{w_1}{}_1/c_1+\widehat{w_2}{}_2/c_2}{1+\widehat{w_1}+\widehat{w_2}},=\widehat{{}_2}/\left(\widehat{{}_2}+1\right)& \left[\mathrm{Equation}8\right]\end{array}$ where, when the number n.sub.1 of frames of which frame synchronization words each have 1 bit error is 0, the error rate ratio ${}_1=\frac{n_2}{n_1}$ is not defined, and the weight is set to 0 and .sub.1/c.sub.1 is calculated as 0 in the Equation 8, and when the number n.sub.2 of frames of which frame synchronization words each have 2 bit errors is 0, the error rate ratio ${}_2=\frac{n_3}{n_2}$ is not defined, and the weight is set to 0 and .sub.2/c.sub.2 is calculated as 0 in the Equation 8.

Description

DESCRIPTION OF DRAWINGS

[0034] FIG. 1 is a flowchart of a method for estimating a bit error probability using an error rate ratio of a frame synchronization word according to an embodiment of the present invention.

[0035] FIG. 2 is a graph showing a mean squared error obtained by an estimator according to the above-described Equation 6 using an error rate ratio .sub.0=n.sub.1/n.sub.0 and a mean squared error obtained by a maximum likelihood estimation method, when K, which denotes a bit error allowable tolerance of a frame synchronization word, is 1.

[0036] FIG. 3 is a graph showing a mean squared error obtained by an estimator according to the above-described Equation 7 using error rate ratios .sub.0=n.sub.1/n.sub.0 and .sub.1=n.sub.2/n.sub.1 and a mean squared error obtained by a maximum likelihood estimation method, when K, which denotes a bit error allowable tolerance of a frame synchronization word, is 2.

[0037] FIG. 4 is a graph showing a mean squared error obtained by an estimator according to the above-described Equation 8 using error rate ratios .sub.0=n.sub.1/n.sub.0, and .sub.1=n.sub.2/n.sub.1, and a mean squared error obtained by a maximum likelihood estimation method, when K, which denotes a bit error allowable tolerance of a frame synchronization word, is 3.

BEST MODE

[0038] FIG. 1 is a flowchart of a method for estimating a bit error probability using an error rate ratio of a frame synchronization word according to an embodiment of the present invention. The present invention may be implemented by a kind of data processing device, and the data processing device may set K, which denotes a bit error allowable tolerance of a frame synchronization word having a length of M bits. The data processing device performs frame synchronization to figure out the numbers n.sub.0, . . . n.sub.K of frames of which frame synchronization words each have 0 bit error, . . . , and K bit errors, among received N frames. In addition, the data processing device figure outs the number n.sub.K+1 of frames each having a frame synchronization word in which the number of bit errors exceeds K, and n.sub.0+n.sub.1 . . . +n.sub.K+1=N.

[0039] The method for estimating a bit error probability using an error rate ratio of a frame synchronization word according to an embodiment of the present invention may include steps a), b), c), and d).

[0040] In step a), the error rate ratios, .sub.0, .sub.1, . . . , .sub.K1 synchronization words are defined as shown in the following Equation 1:

[00017]$\begin{array}{cc}{}_k=\frac{n_{k+1}}{n_k}& \left[\mathrm{Equation}1\right]\end{array}$ [0041] (where n.sub.k denotes the number of frames each having a frame synchronization word in which the number of bit errors is k, and K denotes a bit error allowable tolerance of a frame synchronization word having a length of M bits).

[0042] In step b), a weighted least squares (WLS) cost function with weights greater than or equal to 0 for the bit error probability is set using the error rate ratios set in step a). More specifically, in step b), the WLS cost function (p.sub.e) with the weights w.sub.0, . . . , and w.sub.k1 as shown in the following Equation 4 is set using the error rate ratios set through the above-described Equation 1:

[00018]$\begin{array}{cc}f\left(p_e\right)=\underset{k=0}{\overset{K-1}{.Math.}}{w}_k{\left(\frac{{}_k}{c_k}-\right)}^2& \left[\mathrm{Equation}4\right]\end{array}$ [0043] (where a constant c.sub.k is defined by the following Equation 3-2, and is defined by the following Equation 3-3),

[00019]$\begin{array}{cc}{c}_k=\frac{M-k}{k+1}& \left[\mathrm{Equation}3-2\right]\end{array}$$\begin{array}{cc}=\frac{p_e}{1-p_e},p_e=\frac{}{1+}.& \left[\mathrm{Equation}3-3\right]\end{array}$

[0044] In step c), an estimated bit error probability value that minimizes the cost function set in step b) is obtained. More specifically, in step c), values of and p.sub.e that minimize the cost function (p.sub.e) are obtained through the following Equations 5-1 and 5-2:

[00020]$\begin{array}{cc}\hat{}=\frac{\underset{k=0}{\stackrel{K-1}{.Math.}}{w}_k{}_k/c_k}{\underset{k=0}{\stackrel{K-1}{.Math.}}{w}_k}& \left[\mathrm{Equation}5-1\right]\end{array}$ [0045] (where {circumflex over ()} denotes an estimated value of the value of that minimizes (p.sub.e) of the above-described Equation 4),

[00021]$\begin{array}{cc}=\frac{\hat{}}{\hat{}+1}=\frac{\underset{k=0}{\stackrel{K-1}{.Math.}}{w}_k{}_k/c_k}{\underset{k=0}{\stackrel{K-1}{.Math.}}{w}_k{}_k/c_k+w_k}.& \left[\mathrm{Equation}5-2\right]\end{array}$

[0046] Here, Equation 5-2 may be derived through Equation 3-3 and Equation 5-1.

[0047] In step d), the weights w.sub.0, . . . , w.sub.K1 and are sequentially obtained so that the mean squared error of e becomes small. More specifically, in step d), the bit error probability is estimated in a different manner depending on a value of K, which denotes a bit error allowable tolerance of a frame synchronization word, while w.sub.0 is set to 1. This is because of the above-described Equation 5-2 does not change even if the weights w.sub.0, . . . , w.sub.K1 are multiplied by any positive number in common, and thus, does not lose generality.

[0048] Then, a method for estimating a bit error probability according to a value of K, which denotes a bit error allowable tolerance of a frame synchronization word, will be described in detail below. First, when a value of n.sub.K+1 which denotes the number of frames each having a frame synchronization word in which the number of bit errors exceeds the allowable tolerance K, is 0, an estimated bit error probability value is obtained according to Equation e for a maximum likelihood estimator described above in the section [Background Art] through the following Equation 9:

[00022]$\begin{array}{cc}=\frac{\underset{k=1}{\overset{K}{.Math.}}k{n}_k}{MN}.& \left[\mathrm{Equation}9\right]\end{array}$

[0049] When the value of n.sub.K+1 is not 0, and the value of K, which denotes a bit error allowable tolerance of a frame synchronization word, is 0, an estimated bit error probability value is obtained according to Equation f for a maximum likelihood estimator described above in the section [Background Art] through the following Equation 10:

=1(n.sub.0/N).sup.1/M[Equation 10]

[0050] When the value of K, which denotes an allowable error tolerance, is set to a value greater than 0 in order to improve the synchronizing performance of the data processing device, and the value of n.sub.K+1 is not 0, a maximum likelihood estimator is not given mathematically and it is required to obtain an estimated value through numerical analysis as described in the section [Background Art]. In this case, an estimated bit error probability value is obtained through simple computation using error rate ratios of frame synchronization words. The estimated bit error probability value is calculated in a different manner depending on the value of K, which denotes an allowable error tolerance of the frame synchronization word.

[0051] In addition, in step d), when the value of n.sub.K+1 is not 0 and the value of K is 1, a constant c.sub.0 and an error rate ratio .sub.0=n.sub.1/n.sub.0 of a frame synchronization word are calculated, and an estimated bit error probability value is obtained through the following

[0052] Equation 6:

c.sub.0=M,.sub.0=n.sub.1/n.sub.0

=.sub.0/c.sub.0,=/(+1).

[0053] In step d), when the value of n.sub.K+1 is not 0 and the value of K is 2, constants c.sub.0 and c.sub.1 and error rate ratios .sub.0=n.sub.1/n.sub.0 and .sub.1=n.sub.2/n.sub.1 of frame synchronization words are calculated, and an estimated bit error probability value is obtained through the above-described Equation 7:

[00023]$\begin{array}{cc}{c}_0=M,c_1=\left(M-1\right)/2,{}_0=n_1/n_0,{}_1-n_2/n_1={}_0/c_0,=\frac{c_1\widehat{{}_0}\left(c_0\widehat{{}_0}+2\right)}{2c_1\widehat{{}_0}+1}=\frac{{}_0/c_0+{}_1/c_1}{1+\widehat{w_1}},=\widehat{{}_1}/\left(\widehat{{}_1}+1\right)& \left[\mathrm{Equation}7\right]\end{array}$ [0054] (where, when the number n.sub.1 of frames of which frame synchronization words each have 1 bit error is 0, the error rate ratio

[00024]${}_1=\frac{n_2}{n_1}$ is not defined, and the weight is set to 0 and .sub.1/c.sub.1 is calculated as 0 in the above-described Equation 7).

[0055] In step d), when the value of n.sub.K+1 is not 0 and the value of K is 3, constants c.sub.0, c.sub.1, and c.sub.2, and error rate ratios .sub.0=n.sub.1/n.sub.0, .sub.1=n.sub.2/n.sub.2, and .sub.2=n.sub.3/n.sub.2 of frame synchronization words are calculated, and an estimated bit error probability value is obtained through the above-described Equation 8:

[00025]$\begin{array}{cc}{c}_0=M,c_1=\left(M-1\right)/2,c_2=\left(M-2\right)/3,{}_0=n_1/n_0,{}_1=n_2/n_1,{}_2=n_3/n_2={}_0/c_0,=\frac{c_1\widehat{{}_0}\left(c_0\widehat{{}_0}+2\right)}{2c_1\widehat{{}_0}+1}=\frac{{}_0/c_0+\widehat{w_1}{}_1/c_1}{1+\widehat{w_1}},\widehat{w_2}=\frac{c_2\widehat{{}_1}\left\{c_0{c}_1\widehat{{}_1^2}+c_1{\widehat{{}_1}\left(1-\widehat{w_1}\right)}^2+2\widehat{w_1^2}+\widehat{w_1}\right\}}{\left(c_2\widehat{{}_1}+1\right)+\left(2c_2\widehat{{}_1}+1\right)\widehat{w_1}}\widehat{{}_2}=\frac{{}_0/c_0+\widehat{w_1}{}_1/c_1+{}_2/c_2}{1+\widehat{w_1}+\widehat{w_2}},=/\left(\widehat{{}_2}+1\right)& \left[\mathrm{Equation}8\right]\end{array}$ [0056] (where, when the number n.sub.1 of frames of which frame synchronization words each have 1 bit error is 0, the error rate ratio

[00026]${}_1=\frac{n_2}{n_1}$ is not defined, and the weight is set to 0 and .sub.1/c.sub.1 is calculated as 0 in the above-described Equation 8, and [0057] when the number n.sub.2 of frames of which frame synchronization words each have 2 bit errors is 0, the error rate ratio

[00027]${}_2=\frac{n_3}{n_2}$ is not defined, and the weight is set to 0 and .sub.2/c.sub.2 is calculated as 0 in the above-described Equation 8).

[0058] It is known in aeronautical telemetry that there are many cases where K is 2. Although a case where K is 4 or more is not described herein because there are hardly cases where K is 4 or more in aeronautical telemetry, the scope of the present invention can also be extended to the case where K is 4 or more.

[0059] The mean squared error performance of the estimator can be confirmed through Monte-Carlo simulation. In the simulation, a bit string length M of a frame synchronization word is 24 and the total number N of frames received by the data processing device is 10.sup.5. The simulator may figure out the numbers n.sub.0, . . . n.sub.K of frames of which frame synchronization words each have 0 bit error, . . . , and K bit errors. Also, the simulator may figure out the number n.sub.0, . . . n.sub.K+1 of frames each having a frame synchronization word in which the number of bit errors exceeds K.

[0060] FIG. 2 shows a mean squared error obtained by the estimator according to the above-described Equation 6 using an error rate ratio .sub.0=n.sub.1/n.sub.0 and a mean squared error obtained by the maximum likelihood estimation method, when K, which denotes a bit error allowable tolerance of a frame synchronization word, is 1. Here, the mean squared error is calculated according to the above-described Equation d. When n.sub.2 is 0, an estimated value is obtained according to the above-described Equation 9 in the maximum likelihood estimation method. When n.sub.2 is not 0, a value at which the likelihood function of the above-described Equation c becomes maximum is found through numerical analysis in the maximum likelihood estimation method.

[0061] FIG. 3 shows a mean squared error obtained by the estimator according to the above-described Equation 7 using error rate ratios .sub.0=n.sub.1/n.sub.0 and .sub.1=n.sub.2/n.sub.1 and a mean squared error obtained by the maximum likelihood estimation method, when K, which denotes a bit error allowable tolerance of a frame synchronization word, is 2. When n.sub.3 is 0, an estimated value is obtained according to the above-described Equation 9 in the maximum likelihood estimation method. When n.sub.3 is not 0, a value at which the likelihood function of the above-described Equation c becomes maximum is found through numerical analysis in the maximum likelihood estimation method.

[0062] FIG. 4 shows a mean squared error obtained by the estimator according to the above-described Equation 8 using error rate ratios .sub.0=n.sub.1/n.sub.0, .sub.1=n.sub.2/n.sub.1, and .sub.2=n.sub.3/n.sub.2, and a mean squared error obtained by the maximum likelihood estimation method, when K, which denotes a bit error allowable tolerance of a frame synchronization word, is 3. When n.sub.4 is 0, an estimated value is obtained according to the above-described Equation 9 in the maximum likelihood estimation method. When n.sub.4 is not 0, a value at which the likelihood function of the above-described Equation c becomes maximum is found through numerical analysis in the maximum likelihood estimation method.

[0063] Through FIGS. 2, 3, and 4, it can be confirmed that, when the value of K, which denotes a bit error allowable tolerance of a frame synchronization word, is 1, 2, or 3, there is no great difference between the mean squared error obtained by the conventionally used maximum likelihood estimator and the mean squared error obtained by the estimator using the error rate ratio.

[0064] The present invention is not limited to the above-described embodiment, and may be applied in a wide range. Various modifications may be made by any person having ordinary knowledge in the art to which the present invention pertains without departing from the gist of the present invention claimed in the claims.