# Robust Sampling Frequency Offset Estimation for OFDM over Frequency Selective Fading Channels
## 4 Proposed Estimation Scheme for DRM with Frequency Selective Fading
In the case of DRM channel model 1 [2], which corresponds to AWGN channel, one can write the correlation metric as  
\[\hat{R}_{l}(k_{m})\hat{R}_{l}^{*}(-k_{N_{p}/2+1-m})=E_{X}^{2}e^{ \beta 2\pi D_{l}\left(k_{m}+k_{N_{p}/2+1-m}\right)\Delta_{s}N_{u}/N}\left[1+\frac{ \left\{\mathcal{I}_{l}(k_{m})+\mathcal{Z}_{l}(k_{m})\right\}e^{-jm_{v}}}{E_{X}^ {2}}\right] \tag{26}\]  
whose argument can be factorized as \[\arg\left\{\hat{R}_{l}(k_{m})\hat{R}_{l}^{*}(-k_{N_{p}/2+1-m})\right\} =2\pi D_{t}\Delta_{s}N_{u}/N\left\{k_{m}+k_{N_{p}/2+1-m}\right\} \tag{27}\] \[\quad+\tan^{-1}\left\{\frac{\left[\left\{\mathcal{I}_{l}(k_{m})+ \mathcal{Z}_{l}(k_{m})\right\}e^{-jm_{v}}\right]^{Q}}{E_{X}^{2}}\right\}\] \[=2\pi D_{t}\hat{\Delta}_{s}N_{u}/N\left\{k_{m}+k_{N_{p}/2+1-m}\right\}\]  
Since the pilots are transmitted at a boosted power level, it is very likely that \(|\left[\left\{\mathcal{I}_{l}(k_{m})+\mathcal{Z}_{l}(k_{m})\right\}e^{-jm_{v} }\right]^{Q}|\ll E_{X}^{2}\) when SNR is high [24], hence the arctangent in (27) can be approximated to get  
\[\tan^{-1}\left\{\frac{\left[\left\{\mathcal{I}_{l}(k_{m})+\mathcal{Z}_{l}(k_ {m})\right\}e^{-jm_{v}}\right]^{Q}}{E_{X}^{2}}\right\}\approx\frac{\left[ \left\{\mathcal{I}_{l}(k_{m})+\mathcal{Z}_{l}(k_{m})\right\}e^{-jm_{v}} \right]^{Q}}{E_{X}^{2}} \tag{28}\]  
It is easily shown from (27) and (28) that we obtain  
\[2\pi D_{t}(\hat{\Delta}_{s}-\Delta_{s})N_{u}/N\approx\frac{\sum \limits_{m=1}^{N_{p}/2}\frac{\left[\left\{\mathcal{I}_{l}(k_{m})+\mathcal{Z}_{ l}(k_{m})\right\}e^{-jm_{v}}\right]^{Q}}{E_{X}^{2}}}{\sum\limits_{m=1}^{N_{p}/2} \left\{k_{m}+k_{N_{p}/2+1-m}\right\}} \tag{29}\]  
Since \(E\left[\left\{\mathcal{I}_{l}(k_{m})+\mathcal{Z}_{l}(k_{m})\right\}e^{-jm_{v} }\right]=0\), we arrive at  
\[E\left[\left|\hat{\Delta}_{s}-\Delta_{s}\right|^{2}\right]\approx\left(\frac{ N}{2\pi D_{t}N_{u}}\right)^{2}\frac{\sum\limits_{m=1}^{N_{p}/2}\frac{E\left[ \left|\left[\left\{\mathcal{I}_{l}(k_{m})+\mathcal{Z}_{l}(k_{m})\right\}e^{-jm _{v}}\right]^{Q}\right|^{2}\right]}{E_{X}^{4}}}{\left(\sum\limits_{m=1}^{N_{p}/ 2}\left\{k_{m}+k_{N_{p}/2+1-m}\right\}\right)^{2}} \tag{30}\]  
A close observation of (23) and (24) reveals that the variance of the denominator in (30) is calculated by  
\[\frac{E\left[\left|\left[\left\{\mathcal{I}_{l}(k_{m})+\mathcal{Z}_{l}(k_{m}) \right\}e^{-jm_{v}}\right]^{Q}\right|^{2}\right]}{E_{X}^{4}} =\left(\frac{\sigma_{Z}^{2}}{E_{X}}+\frac{\sigma_{Z}^{4}}{2E_{X}^ {2}}\right)\left[1+\frac{1}{4}\left(\frac{\sigma_{Z}^{2}}{E_{X}}+\frac{\sigma_ {Z}^{4}}{2E_{X}^{2}}\right)\right] \tag{31}\] \[\quad+\frac{\sigma_{I}^{2}}{E_{X}}\left(1+\frac{\sigma_{Z}^{2}}{ E_{X}}\right)\left(1+\frac{\sigma_{Z}^{2}}{E_{X}}+\frac{\sigma_{Z}^{4}}{2E_{X}^ {2}}\right)\]  
Substituting (31) to (30), the MSE expression of (25) can be calculated by  
\[\hat{R}_{l}(k_{m})\hat{R}_{l}^{*}(-k_{N_{p}/2+1-m})=E_{X}^{2}e^{ \beta 2\pi D_{l}\left(k_{m}+k_{N_{p}/2+1-m}\right)\Delta_{s}N_{u}/N}\left[1+\frac{ \left\{\mathcal{I}_{l}(k_{m})+\mathcal{Z}_{l}(k_{m})\right\}e^{-jm_{v}}}{E_{X}^ {2}}\right] \tag{26}\]0 \[\hat{R}_{l}(k_{m})\hat{R}_{l}^{*}(-k_{N_{p}/2+1-m})=E_{X}^{2}e^{ \beta 2\pi D_{l}\left(k_{m}+k_{N_{p}/2+1-m}\right)\Delta_{s}N_{u}/N}\left[1+\frac{ \left\{\mathcal{I}_{l}(k_{m})+\mathcal{Z}_{l}(k_{m})\right\}e^{-jm_{v}}}{E_{X}^ {2}}\right] \tag{26}\]1  
where  
\[\hat{R}_{l}(k_{m})\hat{R}_{l}^{*}(-k_{N_{p}/2+1-m})=E_{X}^{2}e^{ \beta 2\pi D_{l}\left(k_{m}+k_{N_{p}/2+1-m}\right)\Delta_{s}N_{u}/N}\left[1+\frac{ \left\{\mathcal{I}_{l}(k_{m})+\mathcal{Z}_{l}(k_{m})\right\}e^{-jm_{v}}}{E_{X}^ {2}}\right] \tag{26}\]2In the case of the uniform distribution of pilot tones, one can see that  
\[\hat{R}_{l}(k_{m})\hat{R}_{l}^{*}(-k_{N_{p}/2+1-m})=E_{X}^{2}e^{ \beta 2\pi D_{l}\left(k_{m}+k_{N_{p}/2+1-m}\right)\Delta_{s}N_{u}/N}\left[1+\frac{ \left\{\mathcal{I}_{l}(k_{m})+\mathcal{Z}_{l}(k_{m})\right\}e^{-jm_{v}}}{E_{X}^ {2}}\right] \tag{26}\]3  
which results in  
\[\hat{R}_{l}(k_{m})\hat{R}_{l}^{*}(-k_{N_{p}/2+1-m})=E_{X}^{2}e^{ \beta 2\pi D_{l}\left(k_{m}+k_{N_{p}/2+1-m}\right)\Delta_{s}N_{u}/N}\left[1+\frac{ \left\{\mathcal{I}_{l}(k_{m})+\mathcal{Z}_{l}(k_{m})\right\}e^{-jm_{v}}}{E_{X}^ {2}}\right] \tag{26}\]4  
It is worthwhile to note that all GRC patterns defined in each DRM mode satisfy (34) because the GRC is uniformly distributed in the DRM specification.