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Table 4 Summary of pilot training-based channel estimation schemes

From: Application of cell-free massive MIMO in 5G and beyond 5G wireless networks: a survey

Scheme Estimate of dkl Mean and covariance of estimate Mean and covariance of estimate error
LS \( {\hat{d}}_{kl}^{LS}=\frac{1}{\sqrt{p_k{\tau}_p}}{\boldsymbol{X}}_{t_kl}^{pilot} \) \( \mathbbm{E}\left\{{\hat{d}}_{kl}^{LS}\right\}=0 \) \( \mathbbm{E}\left\{{\tilde{d}}_{kl}^{LS}\right\}=0 \)
\( \mathbb{C}\left\{{\hat{d}}_{kl}^{LS}\right\}= \) \( {\sum}_{i\in {S}_k}\frac{p_i}{p_k}{\boldsymbol{R}}_{il}+\frac{\sigma_{ul}^{{}^2}}{p_k{\tau}_p}{\mathbf{I}}_N \) \( \mathbb{C}\left\{{\hat{d}}_{kl}^{LS}\right\}=\mathbb{C}\left\{{\hat{d}}_{kl}^{LS}\right\}-{\boldsymbol{R}}_{kl} \)
MMSE \( {\hat{d}}_{kl}^{MMSE}=\sqrt{\tau_p{p}_k}{\boldsymbol{R}}_{kl}{\varPsi}_{t_kl}^{-1}{\boldsymbol{X}}_{t_kl}^{pilot} \) \( \mathbbm{E}\left\{{\hat{d}}_{kl}^{MMSE}\right\}=0 \) \( \mathbbm{E}\left\{{\tilde{d}}_{kl}^{MMSE}\right\}=0 \)
\( \mathbb{C}\left\{{\hat{d}}_{kl}^{MMSE}\right\}={\tau}_p{p}_k{\boldsymbol{R}}_{kl}{\varPsi}_{t_kl}^{-1}{\boldsymbol{R}}_{kl} \) \( \mathbb{C}\left\{{\tilde{d}}_{kl}^{MMSE}\right\}={\boldsymbol{R}}_{kl}-\mathbb{C}\left\{{\hat{d}}_{kl}^{MMSE}\right\} \)
EW-MMSE \( {\left[{\hat{d}}_{kl}^{EW- MMSE}\right]}_n=\frac{\sqrt{p_k{\tau}_p}\ {\left[{\boldsymbol{R}}_{kl}\right]}_{nn}}{\sum_{i\in {S}_k}{p}_i{\tau}_p\ {\left[{\boldsymbol{R}}_{kl}\right]}_{nn}+{\sigma}_{ul}^{{}^2}}\times {\left[{\boldsymbol{X}}_{t_kl}^{pilot}\right]}_n \) \( \mathbbm{E}\left\{{\left[{\hat{d}}_{kl}^{EW- MMSE}\right]}_n\right\}=0 \) \( \mathbbm{E}\left\{{\left[\tilde{d}_{kl}^{EW- MMSE}\right]}_n\right\}=0 \)
\( \mathbb{C}\left\{{\left[{\hat{d}}_{kl}^{EW- MMSE}\right]}_n\right\}=\frac{p_k{\tau}_p{\left({\left[{\boldsymbol{R}}_{kl}\right]}_{nn}\right)}^2}{\sum_{i\in {S}_k}{p}_i{\tau}_p\ {\left[{\boldsymbol{R}}_{kl}\right]}_{nn}+{\sigma}_{ul}^{{}^2}} \) \( \mathbb{C}\left\{{\left[{\tilde{d}}_{kl}^{EW- MMSE}\right]}_n\right\}={\left[{\boldsymbol{R}}_{kl}\right]}_{nn}-\mathbb{C}\left\{{\left[{\hat{d}}_{kl}^{EW- MMSE}\right]}_n\right\} \)
PA-MMSE \( {\hat{d}}_{kl}^{PA- MMSE}={\overline{d}}_{kl}{e}^{j{\varphi}_{kl}}+ \) \( \sqrt{p_k}{\boldsymbol{R}}_{kl}\times {\varPsi}_{t_kl}^{-1}\left({\boldsymbol{X}}_{t_kl}^{pilot}-{\overline{z}}_{t_kl}\right) \) \( \mathbbm{E}\left\{{\hat{d}}_{kl}^{PA- MMSE}|{\varphi}_{kl}\right\}={\overline{d}}_{kl}{e}^{j{\varphi}_{kl}} \) \( \mathbbm{E}\left\{{\tilde{d}}_{kl}^{PA- MMSE}\right\}=0 \)
\( \mathbb{C}\left\{{\hat{d}}_{kl}^{PA- MMSE}|{\varphi}_{kl}\right\}={p}_k{\tau}_p{\boldsymbol{R}}_{kl}{\varPsi}_{t_kl}^{-1}{\boldsymbol{R}}_{kl} \) \( \mathbb{C}\left\{{\tilde{d}}_{kl}^{PA- MMSE}\right\}={\boldsymbol{R}}_{kl}-{p}_k{\tau}_p{\boldsymbol{R}}_{kl}{\varPsi}_{t_kl}^{-1}{\boldsymbol{R}}_{kl} \)
LMMSE \( {\hat{d}}_{kl}^{LMMSE}=\sqrt{p_k}{\overset{\acute{\mkern6mu}}{\boldsymbol{R}}}_{kl}{\left({\overset{\acute{\mkern6mu}}{\varPsi}}_{t_kl}\right)}^{-1}{\boldsymbol{X}}_{t_kl}^{pilot} \) \( \mathbbm{E}\left\{{\hat{d}}_{kl}^{LMMSE}\right\}=0 \) \( \mathbbm{E}\left\{{\tilde{d}}_{kl}^{LMMSE}\right\}=0 \)
\( \mathbb{C}\left\{{\hat{d}}_{kl}^{LMMSE}\right\}={p}_k{\tau}_p{\overset{\acute{\mkern6mu}}{\boldsymbol{R}}}_{kl}{\left({\overset{\acute{\mkern6mu}}{\varPsi}}_{t_kl}\right)}^{-1}{\overset{\acute{\mkern6mu}}{\boldsymbol{R}}}_{kl} \) \( \mathbb{C}\left\{{\tilde{d}}_{kl}^{LMMSE}\right\}={\overset{\acute{\mkern6mu}}{\boldsymbol{R}}}_{kl}-\mathbb{C}\left\{{\hat{d}}_{kl}^{LMMSE}\right\} \)