# A Carrier Estimation Method Based on MLE and KF for Weak GNSS Signals
## 3 MLE Discriminator
### Cramer-Rao Bound (CRB)  
The CRB expresses a lower bound on the variance of estimators of a deterministic parameter. The derivation method of CRB is mature and can be found in [13]. Based on the work in [13], the discrete form of the CRB of the proposed maximum likelihood (ML) discriminator is derived in this section. The multiple-parameter CRB for any unbiased estimate of a generic, real-valued parameter vector \(\theta\) means the covariance matrix of the estimates \(C(\theta)\) is bounded as  
\[C(\hat{\theta})>J^{-1}(\theta) \tag{15}\]  
where \(J(\theta)\) is commonly referred to as the Fisher information matrix (FIM), whose inverse is the CRB matrix. With \(\Lambda(\theta)\) being the log-likelihood function, the FIM elements are defined by:  
\[[J(\theta)]_{u,\varphi}=-E\bigg{\{}\frac{\partial^{2}\Lambda(\theta)}{ \partial\theta_{u}\partial\theta_{\varphi}}\bigg{\}} \tag{16}\]  
where \(u\), \(v\) denotes the index of the parameters.  
The second-order partial derivative in (16) can be calculated as,  
\[\left\{\begin{array}{l}\frac{\partial^{2}\Lambda}{\partial f^{2}}=-\frac{4}{ \sigma^{2}}A\pi^{2}T^{2}\sum\limits_{n=0}^{N-1}n^{2}(R(r)\cdot\cos(\alpha)+I(r) \cdot\sin(\alpha))\\ \frac{\partial^{2}\Lambda}{\partial\varphi^{2}}=-\frac{A}{\sigma^{2}}\sum \limits_{n=0}^{N-1}(R(r)\cdot\cos(\alpha)+I(r)\cdot\sin(\alpha))\\ \frac{\partial^{2}\Lambda}{\partial f\partial\varphi}=\frac{\partial^{2} \Lambda}{\partial\varphi\partial f}=-\frac{2\pi TA}{\sigma^{2}}\sum\limits_{n= 0}^{N-1}n(R(r)\cdot\cos(\alpha)+I(r)\cdot\sin(\alpha))\end{array}\right. \tag{17}\]  
Substituting (17) to (16), the FIM can be obtained.  
\[J(\theta)=\frac{A^{2}}{\sigma^{2}}\left[\begin{array}{cc}N&\pi TN(N-1)\\ \pi TN(N-1)&2\pi^{2}T^{2}(N-1)N(2N-1)/3\end{array}\right] \tag{18}\]  
Then CRB of \(f\) and \(\varphi\) can be expressed as,  
\[\left\{\begin{array}{l}CR_{f}=1/\left[J(\theta)\right]_{1,1}=3\sigma^{2}/2A^ {2}\pi^{2}T^{2}(N-1)N(2N-1)\\ CR_{\varphi}=1/\left[J(\theta)\right]_{2,2}=\sigma^{2}/A^{2}N\end{array}\right. \tag{19}\]  
Similarly, the CRB of \(A\) is derived separately as,  
\[CR_{A}=-\bigg{/}E\bigg{\{}\frac{\partial^{2}\Lambda(\theta)}{\partial A^{2}} \bigg{\}}=\sigma^{2}/N \tag{20}\]  
Equations (19) and (20) determine the lower bound on the variance of \(f\), \(\varphi\) and \(A\). As explained in Section 2, \(A^{2}\) and \(2\sigma^{2}\) can be seen as the signal power and noise power of \(r[n]\), respectively. \(A^{2}/2\sigma^{2}\) denotes the SNR of \(r[n]\) which is proportional to the integration time \(T\) when the RF signal power and sampling rate are fixed. Therefore, the CRB of \(f\) and \(\varphi\) can be seen as a constant multiplied by the term \(1/T^{3}(N-1)N(2N-1)\) and \(1/TN\), respectively. It iswhere \(C_{N0}\) is the \(C/N_{0}\) of IF signal expressed in the form of dB, and \(BW\) is the noise bandwidth of the receiver. The second term on the right side converts SNR to \(C/N_{0}\) and the third term on the right side denotes the coherent integration gain.  
Figure 4 shows the CRB of \(f\) and \(\varphi\) with respect to \(C/N_{0}\) of IF signal. It can be seen that the CRB increases exponentially with the \(C/N_{0}\) decreases. Strategy 1 can obtain lower CRB of \(f\) than strategy 2. However, for the CRB of \(\varphi\), there is no difference.