## Appendix A  
The amplitude \(A\) in Equation (13) can be expressed as  
\[A=\sigma_{n}\sqrt{2(c/n_{0})T_{coh}}\] (A1)  
Figure 15: Navigation solution errors of the simplified EUTC integration simulation: (**a**) attitude errors; (**b**) velocity errors; and (**c**) position errors.  
where \(\sigma_{n}\) is the standard deviation of noise terms in (13), \(c/n_{0}\) is the carrier-to-noise ratio in Hz, and \(T_{coh}\) is the coherent integration time.  
Substituting Equations (13) and (A1) into Equation (7) in [27], it is possible to restate the narrowband power as  
\[\begin{split} P_{n}=\left(\sum\limits_{n=1}^{M}I_{p}(n)\right)^{ 2}+\left(\sum\limits_{n=1}^{M}Q_{p}(n)\right)^{2}\\ =M^{2}A^{2}\text{sinc}^{2}(\pi f_{e}T_{coh})+\left(\sum\limits_{n =1}^{M}w_{lp,n}\right)^{2}+\left(\sum\limits_{u=1}^{M}w_{Qp,n}\right)^{2}\\ +2 MA\text{sinc}(\pi f_{e}T_{coh})\cos(\phi_{e})\sum\limits_{n=1}^ {M}w_{lp,n}+2 MA\text{sinc}(\pi f_{e}T_{coh})\sin(\phi_{e})\sum\limits_{n=1}^{M }w_{Qp,n}\end{split}\] (A2)  
Similarly, the wideband power can be expressed as  
\[\begin{split} P_{w}=\sum\limits_{n=1}^{M}\left(I_{p}^{2}(n)+Q_{p} ^{2}(n)\right)\\ =MA^{2}\text{sinc}^{2}(\pi f_{e}T_{coh})+\sum\limits_{n=1}^{M}w_ {lp,n}^{2}+\sum\limits_{n=1}^{M}w_{Qp,n}^{2}\\ +2A\text{sinc}(\pi f_{e}T_{coh})\cos(\phi_{e})\sum\limits_{n=1}^ {M}w_{lp,n}+2A\text{sinc}(\pi f_{e}T_{coh})\sin(\phi_{e})\sum\limits_{n=1}^{M }w_{Qp,n}\end{split}\] (A3)  
In order to calculate the mean and variance of \(P_{n}\) and \(P_{w}\), we introduce the following variables:  
\[\left\{\begin{array}{l}X_{I}=\sum\limits_{n=1}^{M}w_{lp,n}\\ X_{Q}=\sum\limits_{n=1}^{M}w_{Qp,n}\end{array}\right.\] (A4)  
where \(X_{I}\) and \(X_{Q}\) are normal random variables with mean 0 and variance \(M\sigma_{n}^{2}\).  
The expectation of \(X_{I}^{2}\) and \(X_{Q}^{2}\) is  
\[\text{E}\left(X_{I}^{2}\right)=\text{E}\left(X_{Q}^{2}\right)=M\sigma_{n}^{2}.\] (A5)  
In addition, the following variables are defined:  
\[\left\{\begin{array}{l}Y_{I}=\sum\limits_{n=1}^{M}w_{lp,n}^{2}\\ Y_{Q}=\sum\limits_{n=1}^{M}w_{Qp,n}^{2}\end{array}\right..\] (A6)  
The expectation of \(Y_{I}\) and \(Y_{Q}\) is  
\[\text{E}\left(Y_{I}\right)=\text{E}\left(Y_{Q}\right)=M\sigma_{n}^{2}.\] (A7)  
In addition,  
\[\begin{split}\text{E}\left(w_{lp,n}^{3}\right)=\text{E}\left(w_{ Qp,n}^{3}\right)=0\\ \text{E}\left(w_{lp,n}^{4}\right)=\text{E}\left(w_{Qp,n}^{4}\right)=3\sigma_{n }^{4}\end{array}.\] (A8)  
Based on Equations (A4)-(A8), the mean and variance of \(P_{n}\) are derived as follows:  
\[\begin{split}\text{E}\left(P_{n}\right)&=2M^{2}\sigma_{n} ^{2}(c/n_{0})T_{coh}\text{sinc}^{2}(\pi f_{e}T_{coh})+2M\sigma_{n}^{2}\\ \sigma^{2}(P_{n})&=4M^{2}\sigma_{n}^{4}\left[2M(c/n_{0})T_{coh}\text{sinc} ^{2}(\pi f_{e}T_{coh})+1\right]\end{split}.\end{split}\] (A9)Similarly, the mean and variance of \(P_{w}\) are obtained as follows:  
\[\begin{array}{l}\mathrm{E}(P_{w})=2M\sigma_{n}^{2}(c/n_{0})T_{col}\mathrm{sinc}^ {2}(\pi f_{e}T_{col})+2M\sigma_{n}^{2}\\ \sigma^{2}(P_{w})=8M\sigma_{n}^{4}(c/n_{0})T_{col}\mathrm{sinc}^{2}(\pi f_{e}T_ {col})+4M\sigma_{n}^{4}\end{array}. \tag{10}\]  
The covariance of \(P_{n}\) and \(P_{w}\) is  
\[\mathrm{cov}(P_{n},P_{w})=4M\sigma_{n}^{4}\Big{[}2M(c/n_{0})T_{col}\mathrm{sinc }^{2}(\pi f_{e}T_{col})+1\Big{]}. \tag{11}\]  
Using Equation (12) in [27], the expectation of the narrow-to-wide power ratio can be expressed as  
\[\mathrm{E}\Big{(}\frac{P_{n}}{P_{w}}\Big{)}\approx\frac{\mathrm{E}(P_{n})}{ \mathrm{E}(P_{w})}-\frac{\mathrm{cov}(P_{n},P_{w})}{\mathrm{E}^{2}(P_{w})}+ \frac{\mathrm{E}(P_{n})\sigma^{2}(P_{w})}{\mathrm{E}^{3}(P_{w})}. \tag{12}\]  
Substituting Equations (9)-(11) into (12), when the frequency error cannot be ignored, the expectation of the power ratio is approximated as  
\[\mathrm{E}\Big{(}\frac{P_{n}}{P_{w}}\Big{)}\approx\frac{M\Big{(}c/n_{0}\tau \mathrm{sinc}^{2}(\pi f_{e}\tau)+1\Big{)}}{M+c/n_{0}\tau\mathrm{sinc}^{2}(\pi f _{e}T_{col})}. \tag{13}\]