## 3 MLE Parameter Estimation Model
### MLE Cost Function  
Given a set of observations \(x\), the MLE of \(\theta\) is the \(\theta_{MLE}(x)\) at which the probability density function is maximized:  
\[\theta_{MLE}(x)=\underset{\theta}{\mathrm{argmax}}p(x\,|\,\theta) \tag{9}\]  
According to the above analysis, when the likelihood function \(p(x\,|\,\theta)\) is known, \(\theta_{MLE}(x)\) can be obtained by the following equation and the logarithmic form is given in Equation (11):  
\[\frac{\partial p(x\,|\,\theta)}{\partial\theta}\bigg{|}_{\theta=\theta_{MLE}}=0 \tag{10}\]  
\[\frac{\partial\ln p(x\,|\,\theta)}{\partial\theta}\bigg{|}_{\theta=\theta_{MLE }}=0 \tag{11}\]  
Since MLE is considered a valid parameter estimation algorithm in the tracking loop of the TC-OFDM receiver, we use MLE to effectively estimate the residual carrier frequency for weak TC-OFDM signals indoors. Meanwhile, the MLE produces the minimum variance of estimation error, when the statistical distribution of the estimated signal parameters in an uncertain interval is unknown [29]. When it comes to multi-parameter problems, the MLE of signal parameters is the estimation of signal parameters when the joint conditional probability density of a set of signal observations gets maximized. Since the coherent integration time in the algorithm is 1.6 ms which is shorter than the loop update time, we assume that the carrier is correctly tracked within such a short coherent integration time. Therefore, the parameters to be estimated can be regarded as unknown  
Figure 3: The principle of MLE. constants over the interval. Meanwhile, \(N\) is defined as the number of consecutive coherent integration results in this paper. The coherent integration results are adopted as observation sample parameters of MLE, which contains the signal amplitude, residual carrier and phase value in Equation (8). The value of \(N\) is selected to ensure that the parameters to be estimated remain constant over the entire interval, and \(N\) is closely related to the dynamic bandwidth of the receiver. The TC-OFDM system is for indoor positioning, which is mainly under low dynamic cases. In the low dynamic case, signal amplitude is the main change parameter. We use the MLE to estimate the signal amplitude, and other parameters (including residual carrier and phase) are little changed under such the low dynamic case. Also, the simulation and real test are used to determine the value of \(N\) in this research. Therefore, in order to obtain the MLE of the unknown \(A_{mp}\), \(\Delta f\) and \(\Delta\varphi\) in Equation (8), the joint probability density function of \(N\) consecutive coherent integration results is obtained according to [30]:  
\[p(Corr_{N}|A_{mp},\Delta f,\Delta\varphi)=\frac{1}{(2\pi\sigma^{2})^{N}}e^{(- \frac{1}{2\sigma^{2}}(Corr_{N}-\overset{\wedge}{Corr_{N}})W(Corr_{N}-\overset{ \wedge}{Corr_{N}})^{H})} \tag{12}\]  
where \(Corr_{N}=[Corr(0),\,Corr\,(1),\,\ldots\,\,\,\,\,,\,Corr\,(N-1)]\) represents \(N\) consecutive coherent integration results, \(\overset{\wedge}{Corr_{N}}\) is the estimated value of \(Corr_{N}\). \(W\) is the diagonal matrix which stands for the weight factor of \(Corr(n)\), superscript \(H\) denotes the matrix transpose and conjugate operations. The MLE minimum variance unbiased estimation can be achieved by obtaining the maximum value of (10) to get the values \(A_{mp}\), \(\Delta f\) and \(\Delta t\varphi\). The diagonal elements of the diagonal matrix \(W\) are set as \(1\) to obtain the log-likelihood cost function of Equation (12) as in [31; 32]:  
\[