## 4 Doppler Frequency Estimation in the Presence of Noise  
In real scenarios, we have to consider the influence of noise, which can be modeled as an Additive White Gaussian Noise (AWGN), as commonly done in the literature. Then the received signal model becomes  
\[\widetilde{y}(nT_{s}) =y(nT_{s})+\eta(nT_{s})\] \[=Ac(nT_{s}-\tau)s_{b}(nT_{s}-\tau)d(nT_{s}-\tau)\] \[\cos\left(2\pi(f_{IF}+f_{d})nT_{s}+\varphi\right)+\eta(nT_{s}) \tag{34}\]  
where \(\eta(nT_{s})\) represents the white Gaussian noise normally distributed with zero mean and variance \(\sigma_{IF}^{2}\), related to the power spectral density \(S_{N}(f)=N_{0}/2\) of the analogue noise by the well-known formula  
\[\sigma_{IF}^{2}=E\left\{\eta^{2}(nT_{s})\right\}=\frac{N_{0}f_{s}}{2} \tag{35}\]  
valid when the transfer function of the equivalent front-end filter is assumed flat over the whole digitization bandwidth \((-f_{s}/2,f_{s}/2)\). Without considering the data-transition, the in-phase and quadrature components of the CAF for the local parameters \(\overline{f}_{d}\) and \(\overline{\tau}\) can be written as  
\[ \begin{split} I(\overline{\tau},\overline{f}_{d})& =\frac{1}{N}\sum_{n=0}^{N-1}\left(y(nT_{s})+\eta(nT_{s})\right)C (nT_{s}-\overline{\tau})\cos\left(2\pi(f_{IF}+\overline{f}_{d})\right)\\ Q(\overline{\tau},\overline{f}_{d})&=\frac{1}{N} \sum_{n=0}^{N-1}\left(y(nT_{s})+\eta(nT_{s})\right)C(nT_{s}-\overline{\tau}) \sin\left(2\pi(f_{IF}+\overline{f}_{d})\right)\end{split}  \tag{36}\]  
which, based on Equation (19) (and omitting the subscript \(i\)), becomes  
\[ \begin{split} I(\overline{\tau},\overline{f}_{d})& =\mathrm{Real}\left\{S(\overline{\tau},\overline{f}_{d})\right\}+N_{I}\\ Q(\overline{\tau},\overline{f}_{d})&=\mathrm{Im }\left\{S(\overline{\tau},\overline{f}_{d})\right\}+N_{Q}\end{split}  \tag{37}\]  
Figure 7: Cumulative distribution of the frequency errors (comparison between the conventional and methods R-3 and C-3). where Real\(\{\}\) and Img\(\{\}\) mean respectively real and imaginary part of a complex value, and  
\[ \begin{split} N_{I}&=\frac{1}{N}\sum_{n=0}^{N-1}\eta( nT_{s})C(nT_{s}-\overline{\tau})\cos\left(2\pi(f_{IF}+\overline{f}_{d})\right)\\ N_{Q}&=\frac{1}{N}\sum_{n=0}^{N-1}\eta(nT_{s})C( nT_{s}-\overline{\tau})\sin\left(2\pi(f_{IF}+\overline{f}_{d})\right)\end{split}  \tag{38}\]  
According to Reference [21], we know that \(N_{I}\) and \(N_{Q}\) are still white noise processes with variance \(\mathrm{var}(N_{I})=\mathrm{var}(N_{Q})=\sigma_{IF}^{2}/(2N)\), and the envelope of \(\widetilde{S}(\overline{\tau},\overline{f}_{d})=I+jQ\) is  
\[\widetilde{\widetilde{S}}(\overline{\tau},\overline{f}_{d})^{2} = I(\overline{\tau},\overline{f}_{d})^{2}+Q(\overline{\tau}, \overline{f}_{d})^{2} \tag{39}\] \[= \overline{S}(\overline{\tau},\overline{f}_{d})^{2}+2\mathrm{Real }\left\{S(\overline{\tau},\overline{f}_{d})\right\}N_{I}+2\mathrm{Img}\left\{S (\overline{\tau},\overline{f}_{d})\right\}N_{Q}+N_{I}^{2}+N_{Q}^{2}\] \[= \overline{S}(\overline{\tau},\overline{f}_{d})^{2}+w_{I}+w_{Q}\]  
where the term \(w_{I}+w_{Q}=w\) is a random process with mean \(E\{w\}=E\{N_{I}^{2}+N_{Q}^{2}\}=\sigma_{IF}^{2}/N\) including all the noise contributions. At this point, working as in Equations (27) or (31), we can develop two new estimators in the presence of noise, that is  
\[\widehat{f}_{d}=\frac{\overline{f}_{A}\widetilde{\overline{S}}_{A}+\overline{ f}_{B}\widetilde{\overline{S}}_{B}+\overline{f}_{C}\widetilde{\overline{S}}_{C}}{ \widetilde{\overline{S}}_{A}+\widetilde{\overline{S}}_{B}+\overline{\overline {S}}_{C}} \tag{40}\]  
and  
\[\widehat{f}_{d,complex}=\mathrm{Real}\left\{\frac{\overline{f}_{C}\widetilde{S }_{C}+\overline{f}_{A}\widetilde{S}_{A}e^{-j\pi f_{sp}T_{d}}+\overline{f}_{B} \widetilde{S}_{B}e^{-j2\pi f_{sp}T_{d}}}{\widetilde{S}_{C}+\widetilde{S}_{A}e ^{-j\pi f_{sp}T_{d}}+\widetilde{S}_{B}e^{-j2\pi f_{sp}T_{d}}}\right\} \tag{41}\]  
(for simplicity the Doppler frequency step is as in Equation (33)). Hereafter, we refer to Equation (40) as algorithm \"R\({}_{n}\)-3\" and to Equation (41) as algorithm \"C\({}_{n}\)-3\".