Appendix
In this Appendix, we complete the derivation which was begun in [4, 9] to approve the outcome (time offset), which is a biased result.
According to Bouder et al. [4], Bouder et al. [9] and EVD of correlation matrix, we get:
$$\begin{aligned} \lambda _i =\alpha _i^2 =\left\{ {\begin{array}{ll} \left\Vert {\vec {h}_0 } \right\Vert^{2}+\sigma _n^2 =\left( {1+\rho \frac{T_s -t_0 }{T_e }} \right)\sigma _n^2&\quad i=1 \\ \left\Vert {\vec {h}_1 } \right\Vert^{2}+\sigma _n^2 =\left( {1+\rho \frac{t_0 }{T_e }} \right)\sigma _n^2&\quad i=2 \\ \sigma _n^2&3\le i\le M \\ \end{array}} \right.\nonumber \\ \end{aligned}$$
(34)
where the parameters’ definitions were given in Bouder et al. [4] and Bouder et al. [9].
Then the estimation of time offset is:
$$\begin{aligned} t_0 =\frac{\lambda _2 -\lambda _3 }{\lambda _1 +\lambda _2 -2\lambda _3 }\times N,\;\;\hat{{t}}_0 =\frac{\hat{{\lambda }}_2 -\hat{{\lambda }}_3 }{\hat{{\lambda }}_1 +\hat{{\lambda }}_2 -2\hat{{\lambda }}_3 }\times N\nonumber \\ \end{aligned}$$
(35)
According to Bouder et al. [4]:
$$\begin{aligned} E\left( {\hat{{\lambda }}_k } \right)=\lambda _k \left( {1+\frac{1}{M} {\mathop {\mathop {\sum \limits _{i=1}}\limits _{i\ne k}}^{N}} {\frac{\lambda _i }{\lambda _k -\lambda _i }} } \right) \end{aligned}$$
(36)
According to (35), we will get:
$$\begin{aligned} \hat{{t}}_0 -t_0&= \frac{1}{N}\left( {\frac{\hat{{\lambda }}_2 -\hat{{\lambda }}_3 }{\hat{{\lambda }}_1 +\hat{{\lambda }}_2 -2\hat{{\lambda }}_3 }-\frac{\lambda _2 -\lambda _3 }{\lambda _1 +\lambda _2 -2\lambda _3 }} \right) \nonumber \\&= \frac{1}{N}\left( {\frac{\left( {\hat{{\lambda }}_2 -\hat{{\lambda }}_3 } \right)\left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)-\left( {\lambda _2 -\lambda _3 } \right)\left( {\hat{{\lambda }}_1 +\hat{{\lambda }}_2 -2\hat{{\lambda }}_3 } \right)}{\left( {\hat{{\lambda }}_1 +\hat{{\lambda }}_2 -2\hat{{\lambda }}_3 } \right)\left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)}} \right) \nonumber \\&= \frac{\left( {\hat{{\lambda }}_2 -\hat{{\lambda }}_3 } \right)\left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)-\left( {\lambda _2 -\lambda _3 } \right)\left( {\hat{{\lambda }}_1 +\hat{{\lambda }}_2 -2\hat{{\lambda }}_3 } \right)}{N\times \left( {\hat{{\lambda }}_1 +\hat{{\lambda }}_2 -2\hat{{\lambda }}_3 } \right)\times \left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)} \nonumber \\&= \frac{\left( {\hat{{\lambda }}_2 \lambda _1 +\hat{{\lambda }}_2 \lambda _2 -2\hat{{\lambda }}_2 \lambda _3 -\hat{{\lambda }}_3 \lambda _1 -\hat{{\lambda }}_3 \lambda _2 +2\hat{{\lambda }}_3 \lambda _3 } \right)-\left( {\lambda _2 \hat{{\lambda }}_1 +\lambda _2 \hat{{\lambda }}_2 -2\lambda _2 \hat{{\lambda }}_3 -\lambda _3 \hat{{\lambda }}_1 -\lambda _3 \hat{{\lambda }}_2 +2\lambda _3 \hat{{\lambda }}_3 } \right)}{N\times \left( {\hat{{\lambda }}_1 +\hat{{\lambda }}_2 -2\hat{{\lambda }}_3 } \right)\times \left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)} \nonumber \\&= \frac{\hat{{\lambda }}_2 \times \left( {\lambda _1 -2\lambda _3 +\lambda _3 } \right)+\hat{{\lambda }}_3 \times \left( {2\lambda _2 -\lambda _1 -\lambda _2 } \right)-\hat{{\lambda }}_1 \times \left( {\lambda _2 -\lambda _3 } \right)}{N\times \left( {\hat{{\lambda }}_1 +\hat{{\lambda }}_2 -2\hat{{\lambda }}_3 } \right)\times \left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)}\nonumber \\&= \frac{\hat{{\lambda }}_2 \times \left( {\lambda _1 -\lambda _3 } \right)-\hat{{\lambda }}_3 \times \left( {\lambda _1 -\lambda _2 } \right)-\hat{{\lambda }}_1 \times \left( {\lambda _2 -\lambda _3 } \right)}{N\times \left( {\hat{{\lambda }}_1 +\hat{{\lambda }}_2 -2\hat{{\lambda }}_3 } \right)\times \left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)} \end{aligned}$$
(37)
If and only if:
$$\begin{aligned}&E\left( {\hat{{t}}_0 -t_0 } \right) \nonumber \\&\quad = E\left( {N\!\times \! \frac{\hat{{\lambda }}_2 \!\times \! \left( {\lambda _1 \!-\!\lambda _3 } \right)\!-\!\hat{{\lambda }}_3 \!\times \! \left( {\lambda _1 \!-\!\lambda _2 } \right)\!-\!\hat{{\lambda }}_1 \!\times \! \left( {\lambda _2 \!-\!\lambda _3 } \right)}{\left( {\hat{{\lambda }}_1 \!+\!\hat{{\lambda }}_2\!-\!2\hat{{\lambda }}_3 } \right)\!\times \! \left( {\lambda _1 \!+\!\lambda _2 \!-\!2\lambda _3 } \right)}} \right)\!=\!0\nonumber \\ \end{aligned}$$
(38)
\(\hat{{t}}_0 \) is unbiased estimator of \(t_0 \). Equation (38) will be
$$\begin{aligned}&E\left( {\hat{{t}}_0 -t_0 } \right)\nonumber \\&\quad = E\left( {\frac{N\!\times \! \left( {\hat{{\lambda }}_2 \!\times \! \left( {\lambda _1 \!-\!\lambda _3 } \right)\!-\!\hat{{\lambda }}_3 \!\times \! \left( {\lambda _1 \!-\!\lambda _2 } \right)\!-\!\hat{{\lambda }}_1 \!\times \! \left( {\lambda _2 \!-\!\lambda _3 } \right)} \right)}{\left( {\hat{{\lambda }}_1 \!+\!\hat{{\lambda }}_2 \!-\!2\hat{{\lambda }}_3 } \right)\!\times \! \left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)}} \right) \nonumber \\&\quad = \frac{N}{\left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)}\nonumber \\&\qquad \times E\left( {\frac{\hat{{\lambda }}_2 \!\times \! \left( {\lambda _1 \!-\!\lambda _3 } \right)\!-\!\hat{{\lambda }}_3 \!\times \! \left( {\lambda _1 \!-\!\lambda _2 } \right)\!-\!\hat{{\lambda }}_1 \!\times \! \left( {\lambda _2 \!-\!\lambda _3 } \right)}{\left( {\hat{{\lambda }}_1 \!+\!\hat{{\lambda }}_2 \!-\!2\hat{{\lambda }}_3 } \right)}} \right)\nonumber \\ \end{aligned}$$
(39)
And the Taylor series expansion of \(1/{\left( {\hat{{\lambda }}_1 +\hat{{\lambda }}_2 -2\hat{{\lambda }}_3 } \right)}\) is:
$$\begin{aligned}&\frac{1}{\left( {\hat{{\lambda }}_1 +\hat{{\lambda }}_2 -2\hat{{\lambda }}_3 } \right)} = \frac{1}{\left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)+\left( {\tilde{\lambda }_1 +\tilde{\lambda }_2 -2\tilde{\lambda }_3 } \right)} \nonumber \\&\quad \approx \frac{1}{\left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)}\!-\!\frac{1}{\left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)^{2}}\left( {\tilde{\lambda }_1 \!+\!\tilde{\lambda }_2 \!-\!2\tilde{\lambda }_3 } \right)\nonumber \\&\qquad +\,o\left( {\left( {\tilde{\lambda }_1 +\tilde{\lambda }_2 -2\tilde{\lambda }_3 } \right)^{2}} \right) \end{aligned}$$
(40)
Where \(\hat{{\lambda }}_i =\lambda _i +\tilde{\lambda }_i ,\;i=1,2,3\), then:
$$\begin{aligned}&\left( {\lambda _2 +\tilde{\lambda }_2 } \right)\times \left( {\lambda _1 -\lambda _3 } \right)-\left( {\lambda _3 +\tilde{\lambda }_3 } \right)\times \left( {\lambda _1 -\lambda _2 } \right) \nonumber \\&\qquad - \left( {\lambda _1 +\tilde{\lambda }_1 } \right)\times \left( {\lambda _2 -\lambda _3 } \right) \nonumber \\&\quad =\lambda _2 \times \left( {\lambda _1 \!-\!\lambda _3 } \right)+\tilde{\lambda }_2 \times \left( {\lambda _1 \!-\!\lambda _3 } \right)\!-\!\lambda _3 \times \left( {\lambda _1 -\lambda _2 } \right)\!-\!\tilde{\lambda }_3 \nonumber \\&\qquad \times \left( {\lambda _1 -\lambda _2 } \right)-\lambda _1 \times \left( {\lambda _2 -\lambda _3 } \right)-\tilde{\lambda }_1 \times \left( {\lambda _2 -\lambda _3 } \right) \nonumber \\&\quad =\tilde{\lambda }_2 \times \left( {\lambda _1 -\lambda _3 } \right)-\tilde{\lambda }_3 \times \left( {\lambda _1 -\lambda _2 } \right)-\tilde{\lambda }_1 \times \left( {\lambda _2 -\lambda _3 } \right)\nonumber \\ \end{aligned}$$
(41)
At the same time, multiplying (41) by (40) gives:
$$\begin{aligned}&\left( {\tilde{\lambda }_2 \times \left( {\lambda _1 -\lambda _3 } \right)\!-\!\tilde{\lambda }_3 \times \left( {\lambda _1 -\lambda _2 } \right)\!-\!\tilde{\lambda }_1 \times \left( {\lambda _2 -\lambda _3 } \right)} \right)\nonumber \\&\qquad \times \ldots \left( \frac{1}{\left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)}-\frac{1}{\left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)^{2}}\right.\nonumber \\&\quad \quad \left.\times \left( {\tilde{\lambda }_1 +\tilde{\lambda }_2 \!-\!2\tilde{\lambda }_3 } \right)+\,o\left( {\left( {\tilde{\lambda }_1 +\tilde{\lambda }_2 -2\tilde{\lambda }_3 } \right)^{2}} \right) \right) \nonumber \\&\quad =\left( {\tilde{\lambda }_2 \times \left( {\lambda _1 -\lambda _3 } \right)-\tilde{\lambda }_3 \times \left( {\lambda _1 -\lambda _2 } \right)-\tilde{\lambda }_1 \times \left( {\lambda _2 -\lambda _3 } \right)} \right)\nonumber \\&\quad \quad \times \frac{1}{\left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)} \nonumber \\&\quad \quad -\left( {\tilde{\lambda }_2 \times \left( {\lambda _1 -\lambda _3 } \right)\!-\!\tilde{\lambda }_3 \times \left( {\lambda _1 -\lambda _2 } \right)\!-\!\tilde{\lambda }_1 \times \left( {\lambda _2 -\lambda _3 } \right)} \right)\nonumber \\&\quad \quad \times \frac{1}{\left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)^{2}}\left( {\tilde{\lambda }_1 +\tilde{\lambda }_2 -2\tilde{\lambda }_3 } \right) \nonumber \\&\quad \quad +\left( {\tilde{\lambda }_2 \times \left( {\lambda _1 -\lambda _3 } \right)-\tilde{\lambda }_3 \!\times \! \left( {\lambda _1 -\lambda _2 } \right)-\tilde{\lambda }_1 \!\times \! \left( {\lambda _2 -\lambda _3 } \right)} \right)\nonumber \\&\quad \quad \times \, o\left( {\left( {\tilde{\lambda }_1 +\tilde{\lambda }_2 -2\tilde{\lambda }_3 } \right)^{2}} \right) \end{aligned}$$
(42)
In other word, if and only if
$$\begin{aligned} E\left\{ {\begin{array}{l} \left( {\tilde{\lambda }_2 \times \left( {\lambda _1 -\lambda _3 } \right)-\tilde{\lambda }_3 \times \left( {\lambda _1 -\lambda _2 } \right)-\tilde{\lambda }_1 \times \left( {\lambda _2 -\lambda _3 } \right)} \right) \times \frac{1}{\left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)} \\ -\left( {\tilde{\lambda }_2 \times \left( {\lambda _1 -\lambda _3 } \right)-\tilde{\lambda }_3 \times \left( {\lambda _1 -\lambda _2 } \right)-\tilde{\lambda }_1 \times \left( {\lambda _2 -\lambda _3 } \right)} \right) \times \frac{1}{\left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)^{2}}\left( {\tilde{\lambda }_1 +\tilde{\lambda }_2 -2\tilde{\lambda }_3 } \right)\\ +\left( {\tilde{\lambda }_2 \times \left( {\lambda _1 -\lambda _3 } \right)-\tilde{\lambda }_3 \times \left( {\lambda _1 -\lambda _2 } \right)-\tilde{\lambda }_1 \times \left( {\lambda _2 -\lambda _3 } \right)} \right) \times o\left( {\left( {\tilde{\lambda }_1 +\tilde{\lambda }_2 -2\tilde{\lambda }_3 } \right)^{2}} \right) \\ \end{array}} \right\} =0 \end{aligned}$$
(43)
\(\hat{{t}}_0 \) is unbiased estimator of \(t_0 \). And it is easy to approve that \(\hat{{t}}_0 \) is unbiased estimator, when \(\lambda _1 =\lambda _2 \), and
$$\begin{aligned}&E\left( {\tilde{\lambda }_2 \times \left( {\lambda _1 -\lambda _3 } \right)-\tilde{\lambda }_3 \times \left( {\lambda _1 -\lambda _2 } \right)-\tilde{\lambda }_1 \times \left( {\lambda _2 -\lambda _3 } \right)} \right) \nonumber \\&\quad =E\left( {\hat{{\lambda }}_2 } \right)\times \left( {\lambda _1 -\lambda _3 } \right)\!-\!E\left( {\hat{{\lambda }}_3 } \right)\times \left( {\lambda _1 -\lambda _2 } \right)\!-\!E\left( {\hat{{\lambda }}_1 } \right)\nonumber \\&\quad \quad \times \left( {\lambda _2 -\lambda _3 } \right)=0 \end{aligned}$$
(44)
While \(\lambda _1 \ne \lambda _2 \), according to (36), we will obtain:
$$\begin{aligned} E\left( {\hat{{\lambda }}_1 } \right)&= \lambda _1 \left( {1+\frac{1}{M} {\mathop {\mathop {\sum \limits _{i=1}}\limits _{i\ne 1}}^{N}} {\frac{\lambda _i }{\lambda _1 -\lambda _i }} } \right),\nonumber \\ E\left( {\hat{{\lambda }}_2 } \right)&= \lambda _2 \left( {1+\frac{1}{M} {\mathop {\mathop {\sum \limits _{i=1}}\limits _{i\ne 2}}^{N}} {\frac{\lambda _i }{\lambda _2 -\lambda _i }} } \right), \nonumber \\ E\left( {\hat{{\lambda }}_3 } \right)&= \lambda _3 \left( {1+\frac{1}{M} {\mathop {\mathop {\sum \limits _{i=1}}\limits _{i\ne 3}}^{2}} {\frac{\lambda _i }{\lambda _3 -\lambda _i }} } \right)\nonumber \\&= \lambda _3 \left( {1+\frac{1}{M}\left( {\frac{\lambda _1 }{\lambda _3 -\lambda _1 }+\frac{\lambda _2 }{\lambda _3 -\lambda _2 }} \right)} \right) \end{aligned}$$
(45)
So:
$$\begin{aligned}&E\left( {\hat{{\lambda }}_2 } \right)\times \left( {\lambda _1 -\lambda _3 } \right) = \lambda _2 \left( {1+\frac{1}{M} {\mathop {\mathop {\sum \limits _{i=1}}\limits _{i\ne 2}}^{N}} {\frac{\lambda _i }{\lambda _2 -\lambda _i }} } \right)\nonumber \\&\quad \times \left( {\lambda _1\!-\!\lambda _3 } \right)\!=\!\lambda _2 \times \left( {\lambda _1 \!-\!\lambda _3 } \right)\!+\!\frac{1}{M} {\mathop {\mathop {\sum \limits _{i=1}}\limits _{i\ne 2}}^{N}} {\frac{\lambda _i \!\times \! \lambda _2 \!\times \! \left( {\lambda _1 \!-\!\lambda _3 } \right)}{\lambda _2 \!-\!\lambda _i }} , \nonumber \\&E\left( {\hat{{\lambda }}_3 } \right)\times \left( {\lambda _1 -\lambda _2 } \right) = \lambda _3 \left( {1+\frac{1}{M} {\mathop {\mathop {\sum \limits _{j=1}}\limits _{j\ne 3}}^{N}} {\frac{\lambda _i }{\lambda _3 -\lambda _i }} } \right)\nonumber \\&\quad \times \left( {\lambda _1\!-\!\lambda _2 } \right)\!=\!\lambda _3 \times \left( {\lambda _1 \!-\!\lambda _2 } \right)\!+\!\frac{1}{M} {\mathop {\mathop {\sum \limits _{j=1}}\limits _{j\ne 3}}^{2}} {\frac{\lambda _j \!\times \! \lambda _3 \!\times \! \left( {\lambda _1 \!-\!\lambda _2 } \right)}{\lambda _3 \!-\!\lambda _j }} , \nonumber \\&E\left( {\hat{{\lambda }}_1 } \right)\times \left( {\lambda _2 -\lambda _3 } \right) = \lambda _1 \left( {1+\frac{1}{M} {\mathop {\mathop {\sum \limits _{n=1}}\limits _{n\ne 1}}^{N}} {\frac{\lambda _i }{\lambda _1 -\lambda _i }} } \right)\nonumber \\&\quad \times \left( {\lambda _2 \!-\!\lambda _3 } \right)\!=\!\lambda _1 \!\times \! \left( {\lambda _2 \!-\!\lambda _3 } \right)\!+\!\frac{1}{M} {\mathop {\mathop {\sum \limits _{n=2}}\limits _{n\ne 1}}^{N}} {\frac{\lambda _n \!\times \! \lambda _1 \!\times \! \left( {\lambda _2 \!-\!\lambda _3 } \right)}{\lambda _1 \!-\!\lambda _n }}\nonumber \\ \end{aligned}$$
(46)
And (44) will be:
$$\begin{aligned}&\frac{1}{M} {\mathop {\mathop {\sum \limits _{i\!=\!1}}\limits _{i\ne 2}}^{N}} {\frac{\lambda _i \times \lambda _2 \times \left( {\lambda _1 \!-\!\lambda _3 } \right)}{\lambda _2 \!-\!\lambda _i }} \!-\!\frac{1}{M} {\mathop {\mathop {\sum \limits _{j\!=\!1}}\limits _{j\ne 3}}^{N}} {\frac{\lambda _j \times \lambda _3 \times \left( {\lambda _1 \!-\!\lambda _2 } \right)}{\lambda _3 -\lambda _j }}\nonumber \\&\quad \quad -\frac{1}{M} {\mathop {\mathop {\sum \limits _{n=2}}\limits _{n\ne 1}}^{N}} {\frac{\lambda _n \times \lambda _1 \times \left( {\lambda _2 -\lambda _3 } \right)}{\lambda _1 -\lambda _n }} \nonumber \\&\quad =\frac{1}{M}\left( {\mathop {\mathop {\sum \limits _{i=1}}\limits _{i\ne 2}}^{N}} {\frac{\lambda _i \!\times \! \lambda _2 \!\times \! \left( {\lambda _1 \!-\!\lambda _3 } \right)}{\lambda _2 \!-\!\lambda _i }} \!-\!{\mathop {\mathop {\sum \limits _{n=2}}\limits _{n\ne 1}}^{N}} {\frac{\lambda _n \!\times \! \lambda _1 \!\times \! \left( {\lambda _2 \!-\!\lambda _3 } \right)}{\lambda _1 \!-\!\lambda _n }} \right. \nonumber \\&\quad \quad \left.-\left( {\frac{\lambda _1 }{\lambda _3 -\lambda _1 }+\frac{\lambda _2 }{\lambda _3 -\lambda _2 }} \right)\times \lambda _3 \times \left( {\lambda _1 -\lambda _2 } \right) \right) \end{aligned}$$
(47)
where:
$$\begin{aligned}&{\mathop {\mathop {\sum \limits _{i=1}}\limits _{i\ne 2}}^{N}} {\frac{\lambda _i \times \lambda _2 \times \left( {\lambda _1 -\lambda _3 } \right)}{\lambda _2 -\lambda _i }} \nonumber \\&\quad =\frac{\lambda _1 \times \lambda _2 \times \left( {\lambda _1 -\lambda _3 } \right)}{\lambda _2 -\lambda _1 }+\frac{\lambda _3 \times \lambda _2 \times \left( {\lambda _1 -\lambda _3 } \right)}{\lambda _2 -\lambda _3 }\nonumber \\&\qquad +\frac{\lambda _4 \times \lambda _2 \times \left( {\lambda _1 -\lambda _3 } \right)}{\lambda _2 -\lambda _4 }+\cdots +\frac{\lambda _N \times \lambda _2 \times \left( {\lambda _1 -\lambda _3 } \right)}{\lambda _2 -\lambda _N }\nonumber \\ \end{aligned}$$
(48)
At the same time:\(\lambda _2 -\lambda _3 =\lambda _2 -\lambda _4 =\cdots =\lambda _2 -\lambda _N \). Then (48) will be:
$$\begin{aligned}&{\mathop {\mathop {\sum \limits _{i=1}}\limits _{i\ne 2}}^{N}} {\frac{\lambda _i \times \lambda _2 \times \left( {\lambda _1 -\lambda _3 } \right)}{\lambda _2 -\lambda _i }} =\frac{\lambda _1 \times \lambda _2 \times \left( {\lambda _1 -\lambda _3 } \right)}{\lambda _2 -\lambda _1 }\nonumber \\&\quad +\left( {N-2} \right)\times \frac{\lambda _3 \times \lambda _2 \times \left( {\lambda _1 -\lambda _3 } \right)}{\lambda _2 -\lambda _3 } \end{aligned}$$
(49)
Similarly
$$\begin{aligned}&{\mathop {\mathop {\sum \limits _{n=2}}\limits _{n\ne 1}}^{N}} {\frac{\lambda _n \times \lambda _1 \times \left( {\lambda _2 -\lambda _3 } \right)}{\lambda _1 -\lambda _n }} =\frac{\lambda _2 \times \lambda _1 \times \left( {\lambda _2 -\lambda _3 } \right)}{\lambda _1 -\lambda _2 }\nonumber \\&\quad +\left( {N-2} \right)\times \frac{\lambda _3 \times \lambda _1 \times \left( {\lambda _2 -\lambda _3 } \right)}{\lambda _1 -\lambda _3 } \end{aligned}$$
(50)
At last, (47) will be:
$$\begin{aligned}&\frac{1}{M}\left( {\mathop {\mathop {\sum \limits _{i=1}}\limits _{i\ne 2}}^{N}} {\frac{\lambda _i \times \lambda _2 \times \left( {\lambda _1 -\lambda _3 } \right)}{\lambda _2 -\lambda _i }} -{\mathop {\mathop {\sum \limits _{n=2}}\limits _{n\ne 1}}^{N}} {\frac{\lambda _n \times \lambda _1 \times \left( {\lambda _2 -\lambda _3 } \right)}{\lambda _1 -\lambda _n }} \right. \nonumber \\&\quad \quad \left.-\left( {\frac{\lambda _1 }{\lambda _3 -\lambda _1 }+\frac{\lambda _2 }{\lambda _3 -\lambda _2 }} \right)\times \lambda _3 \times \left( {\lambda _1 -\lambda _2 } \right) \right) \nonumber \\&\quad =\frac{1}{M}\!\left( {\begin{array}{l} \frac{\lambda _1 \times \lambda _2 \times \left( {\lambda _1 -\lambda _3 } \right)}{\lambda _2 -\lambda _1 }\!+\!\left( {N\!-\!2} \right)\!\times \! \frac{\lambda _3 \times \lambda _2 \times \left( {\lambda _1 -\lambda _3 } \right)}{\lambda _2 -\lambda _3 }\!-\!\frac{\lambda _2 \times \lambda _1 \times \left( {\lambda _2 -\lambda _3 } \right)}{\lambda _1 -\lambda _2 } \\ -\left( {N\!-\!2} \right)\!\times \! \frac{\lambda _3 \times \lambda _1 \times \left( {\lambda _2 -\lambda _3 } \right)}{\lambda _1 -\lambda _3 }\!-\!\left( {\frac{\lambda _1 }{\lambda _3 -\lambda _1 }\!+\!\frac{\lambda _2 }{\lambda _3 -\lambda _2 }} \right)\!\times \! \lambda _3 \!\times \! \left( {\lambda _1 \!-\!\lambda _2 } \right) \\ \end{array}} \right) \nonumber \\&\quad =\frac{1}{M}\left( \frac{\lambda _1 \times \lambda _2 \times \left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)}{\lambda _2 -\lambda _1 }+\left( {N-2} \right)\right.\nonumber \\&\qquad \times \left( {\frac{\lambda _3 \times \lambda _2 \times \left( {\lambda _1 -\lambda _3 } \right)}{\lambda _2 -\lambda _3 }-\frac{\lambda _3 \times \lambda _1 \times \left( {\lambda _2 -\lambda _3 } \right)}{\lambda _1 -\lambda _3 }} \right)\nonumber \\&\qquad \left.-\left( {\frac{\lambda _1 }{\lambda _3 -\lambda _1 }+\frac{\lambda _2 }{\lambda _3 -\lambda _2 }} \right)\times \lambda _3 \times \left( {\lambda _1 -\lambda _2 } \right) \right) \nonumber \\&\quad =\frac{1}{M}\left( \frac{\lambda _1 \times \lambda _2 \times \left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)}{\lambda _2 -\lambda _1 }+\left( {N-2} \right)\right.\nonumber \\&\qquad \times \left( {\frac{\lambda _3 \times \lambda _2 \times \left( {\lambda _1 -\lambda _3 } \right)^{2}-\lambda _3 \times \lambda _1 \times \left( {\lambda _2 -\lambda _3 } \right)^{2}}{\left( {\lambda _2 -\lambda _3 } \right)\left( {\lambda _1 -\lambda _3 } \right)}} \right)\nonumber \\&\qquad \left.-\left( {\frac{\lambda _1 }{\lambda _3 -\lambda _1 }+\frac{\lambda _2 }{\lambda _3 -\lambda _2 }} \right)\times \lambda _3 \times \left( {\lambda _1 -\lambda _2 } \right) \right) \nonumber \\&\quad =\frac{1}{M}\left( \frac{\lambda _1 \times \lambda _2 \times \left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)}{\lambda _2 -\lambda _1 }+\left( {N-2} \right)\right.\nonumber \\&\qquad \times \left( {\frac{\lambda _3 \times \left( {\lambda _2 \times \left( {\lambda _1 -\lambda _3 } \right)^{2}-\lambda _1 \times \left( {\lambda _2 -\lambda _3 } \right)^{2}} \right)}{\left( {\lambda _2 -\lambda _3 } \right)\left( {\lambda _1 -\lambda _3 } \right)}} \right)\nonumber \\&\qquad \left.-\left( {\frac{\lambda _1 }{\lambda _3 -\lambda _1 }+\frac{\lambda _2 }{\lambda _3 -\lambda _2 }} \right)\times \lambda _3 \times \left( {\lambda _1 -\lambda _2 } \right) \right) \end{aligned}$$
(51)
where
$$\begin{aligned}&\frac{\lambda _3 \times \left( {\lambda _2 \times \left( {\lambda _1 -\lambda _3 } \right)^{2}-\lambda _1 \times \left( {\lambda _2 -\lambda _3 } \right)^{2}} \right)}{\left( {\lambda _2 -\lambda _3 } \right)\left( {\lambda _1 -\lambda _3 } \right)} \\&\quad = \frac{\lambda _3 \!\times \! \left( {\lambda _2 \!\times \! \left( {\lambda _1^2 \!-\!2\lambda _1 \lambda _3 \!+\!\lambda _3^2 } \right)\!-\!\lambda _1 \!\times \! \left( {\lambda _2^2 \!-\!2\lambda _2 \lambda _3 \!+\!\lambda _3^2 } \right)} \right)}{\left( {\lambda _2 \!-\!\lambda _3 } \right)\left( {\lambda _1\!-\!\lambda _3 } \right)} \\&\quad \!=\! \frac{\lambda _3 \!\times \! \left(\! {\left( {\lambda _2 \lambda _1^2 \!-\!2\lambda _1 \lambda _2 \lambda _3 \!+\!\lambda _2 \lambda _3^2 } \right)\!-\!\left(\! {\lambda _1 \lambda _2^2 \!-\!2\lambda _1 \lambda _2 \lambda _3 \!+\!\lambda _1 \lambda _3^2 } \right)} \!\right)}{\left( {\lambda _2 \!-\!\lambda _3 } \right)\left( {\lambda _1\!-\!\lambda _3 } \right)} \\&\quad =\frac{\lambda _3 \times \left( {\lambda _1 \lambda _2 \left( {\lambda _1 -\lambda _2 } \right)-\lambda _3^2 \left( {\lambda _1 -\lambda _2 } \right)} \right)}{\left( {\lambda _2 -\lambda _3 } \right)\left( {\lambda _1 -\lambda _3 } \right)} \nonumber \\&\quad =\frac{\lambda _3 \times \left( {\lambda _1 -\lambda _2 } \right)\times \left( {\lambda _1 \lambda _2 -\lambda _3^2 } \right)}{\left( {\lambda _2 -\lambda _3 } \right)\left( {\lambda _1 -\lambda _3 } \right)} \end{aligned}$$
So (51) will be:
$$\begin{aligned}&\frac{1}{M}\left( \frac{\lambda _1 \times \lambda _2 \times \left( {\lambda _1 +\lambda _2 -2\lambda _3 } \right)}{\lambda _2 -\lambda _1 }+\left( {N-2} \right) \right.\nonumber \\&\quad \quad \times \left( {\frac{\lambda _3 \times \left( {\lambda _1 -\lambda _2 } \right)\times \left( {\lambda _1 \lambda _2 -\lambda _3^2 } \right)}{\left( {\lambda _2 -\lambda _3 } \right)\left( {\lambda _1 -\lambda _3 } \right)}} \right) \nonumber \\&\quad \quad \left.+\left( {\frac{\lambda _1 \!\times \! \left( {\lambda _2 \!-\!\lambda _3 } \right)\!+\!\lambda _2 \!\times \! \left( {\lambda _1 \!-\!\lambda _3 } \right)}{\left( {\lambda _1 \!-\!\lambda _3 } \right)\left( {\lambda _2\!-\!\lambda _3 } \right)}} \right)\!\times \! \lambda _3 \!\times \! \left( {\lambda _1\!-\!\lambda _2 } \right) \right)\nonumber \\ \end{aligned}$$
(52)
according to (36), set \(t_1 =T_s -t_0 \), then (52) will be:
$$\begin{aligned}&\frac{1}{M}\left( \frac{\lambda _1 \times \lambda _2 \times \left( {\lambda _1 +\lambda _{2} -2\lambda _3 } \right)}{\lambda _2 -\lambda _1 }+\left( {N-2} \right)\right.\nonumber \\&\quad \quad \times \left( {\frac{\lambda _3 \times \left( {\lambda _1 -\lambda _2 } \right)\times \left( {\lambda _1 \lambda _2 -\lambda _3^2 } \right)}{\left( {\lambda _2 -\lambda _3 } \right)\left( {\lambda _1 -\lambda _3 } \right)}} \right)\nonumber \\&\quad \quad \left.+\left( {\frac{\lambda _1 \times \left( {\lambda _2 -\lambda _3 } \right)+\lambda _2 \times \left( {\lambda _1 -\lambda _3 } \right)}{\left( {\lambda _1 -\lambda _3 } \right)\left( {\lambda _2 -\lambda _3 } \right)}} \right)\times \lambda _3 \times \left( {\lambda _1 -\lambda _2 } \right) \right) \\&\quad =\frac{1}{M}\!\left(\! {\begin{array}{l} \frac{\left( {1+\rho \frac{t_1 }{T_e }} \right)\times \left( {1+\rho \frac{t_0 }{T_e }} \right)\times N\rho }{\left( {\rho \frac{t_0 -t_1 }{T_e }} \right)}\!+\!\left( {N\!-\!2} \right)\!\times \! \left( {\frac{\left( {\rho \frac{t_1 -t_0 }{T_e }} \right)\times \left( {N\rho +\rho ^{2}\frac{t_1 t_0 }{T_e }} \right)}{\left( {\rho \frac{t_0 }{T_e }} \right)\left( {\rho \frac{t_1 }{T_e }} \right)}} \right) \\ \quad +\left( {\frac{\left( {1+\rho \frac{t_1 }{T_e }} \right)\times \left( {\rho \frac{t_0 }{T_e }} \right)+\left( {1+\rho \frac{t_0 }{T_e }} \right)\times \left( {\rho \frac{t_1 }{T_e }} \right)}{\left( {\rho \frac{t_0 }{T_e }} \right)\left( {\rho \frac{t_1 }{T_e }} \right)}} \right)\times \left( {\rho \frac{t_1 -t_0 }{T_e }} \right) \\ \end{array}} \!\!\right)\\&\quad \quad \times \, \sigma _n^4 \end{aligned}$$
If and only if:
$$\begin{aligned}&\frac{\left( {1+\rho \frac{t_1 }{T_e }} \right)\times \left( {1+\rho \frac{t_0 }{T_e }} \right)\times N\rho }{\left( {\rho \frac{t_1 -t_0 }{T_e }} \right)}=\ldots \left( {N-2} \right) \\&\quad \quad \times \left( {\frac{\left( {\rho \frac{t_1 -t_0 }{T_e }} \right)\times \left( {N\rho +\rho ^{2}\frac{t_1 }{T_e }\frac{t_0 }{T_e }} \right)}{\left( {\rho \frac{t_0 }{T_e }} \right)\left( {\rho \frac{t_1 }{T_e }} \right)}} \right)\\&\quad \quad +\left( {\frac{\left( {1+\rho \frac{t_1 }{T_e }} \right)\times \left( {\rho \frac{t_0 }{T_e }} \right)+\left( {1+\rho \frac{t_0 }{T_e }} \right)\times \left( {\rho \frac{t_1 }{T_e }} \right)}{\left( {\rho \frac{t_0 }{T_e }} \right)\left( {\rho \frac{t_1 }{T_e }} \right)}} \right)\\&\quad \quad \times \left( {\rho \frac{t_1 -t_0 }{T_e }} \right) \frac{\left( {1+N\rho \frac{t_1 }{T_e }+\rho ^{2}\frac{t_1 }{T_e }\frac{t_0 }{T_e }} \right)\times N}{\left( {\frac{t_1 -t_0 }{T_e }} \right)}=\left( {N-2} \right)\\&\quad \quad \times \left( {\frac{\left( {\frac{t_1 -t_0 }{T_e }} \right)\times \left( {N+\frac{t_1 }{T_e }\frac{t_0 }{T_e }} \right)}{\left( {\frac{t_0 }{T_e }} \right)\left( {\frac{t_1 }{T_e }} \right)}} \right)+\left( {\frac{N+2\rho \frac{t_1 }{T_e }\frac{t_0 }{T_e }}{\left( {\frac{t_0 }{T_e }} \right)\left( {\frac{t_1 }{T_e }} \right)}} \right)\\&\quad \quad \times \left( {\rho \frac{t_1 -t_0 }{T_e }} \right) \end{aligned}$$
\(\hat{{t}}_0 \) is unbiased estimator. And in practical application, it is hard to satisfy the condition.
