Algorithm 2: Noise-Corrected Clustered-Sparse Channel Estimation Algorithm

Input: the received signal $\mathbf{y}$; the noise variance $\lambda$; the length of discreted paths L;

the maximum number of iterations $r_{max}$; the maximun number of discreted

paths in one cluster $L{C}_{max}$; the threshold for prunning small

hyperparameters ${\gamma }_{th}$; the threshold to stop the whole algorithm $ϵ$.

Initialize: The PB-CSBL algorithm is utilized to obtain the posterior mean ${\overline{\mu }}_{\mathbf{h}}\leftarrow {\mu }_{\mathbf{h}}$

and covariance matrix ${{\displaystyle \overline{\sum }}}_{\mathbf{h}}\leftarrow {\sum }_{\mathbf{h}}$; the noise vector $\lambda =\lambda {\mathbf{I}}_K$; the list of path

power ${\gamma }_{list}={\mathbf{1}}_L$; the iteration counter $r=0$.

Cluster Evolution:

1: for $d=1,2,\dots ,C$do

2:  for $i=1,2,\dots ,L_d$ do

3:   ${\gamma }_{d,i}\leftarrow \frac{1}{M}Tr\left[{\mathbf{B}}_d^{-1}\left({{\displaystyle \overline{\sum }}}_{\mathbf{h}}^{d,i}+{\overline{\mu }}_{\mathbf{h}}^{d,i}{\left({\overline{\mu }}_{\mathbf{h}}^{d,i}\right)}^H\right)\right]$, and update ${\gamma }_{list}$ with ${\gamma }_{d,i}$.

4:  end for

5:  $L_d^{\prime }\leftarrow min(L_d,L{C}_{max})$.

6:  ${\mathbf{B}}_d\leftarrow \frac{1}{L_d^{\prime }}\sum _{i=1}^{L_d^{\prime }}\frac{{{\displaystyle \overline{\sum }}}_{\mathbf{h}}^{d,i}+{\overline{\mu }}_{\mathbf{h}}^{d,i}{\left({\overline{\mu }}_{\mathbf{h}}^{d,i}\right)}^H}{{\gamma }_{d,i}}$ using the $L_d^{\prime }$ most significant continuous paths.

7:  ${\beta }_d\leftarrow \frac{{\alpha }_1}{{\alpha }_0}$, where ${\alpha }_1$ and ${\alpha }_0$ can be obtained through ${\mathbf{B}}_d$.

8: end for

Cluster Partition:

9: ${\mathbf{p}}^{\left(0\right)}\leftarrow$ FindIndex$(0<{\gamma }_{list}<{\gamma }_{th})$.

10: if${\mathbf{p}}^{\left(0\right)}$ is not empty then

11:  ${\gamma }_{list}^{\left(old\right)}\leftarrow {\gamma }_{list}$ and ${\gamma }_{list}\left[{\mathbf{p}}^{\left(0\right)}\right]\leftarrow \mathbf{0}$.

12:  $\mathbf{CLS}\leftarrow$ Split($\mathbf{CLS}$, ${\mathbf{p}}^{\left(0\right)}$), where $\mathbf{CLS}$ is splitted into C clusters according

to ${\mathbf{p}}^{\left(0\right)}$.

13: end if

14: for $d=1,2,\dots ,C$do

15:  Update $R_d$, ${\mathsf{\Gamma }}_d$ and ${\mathbf{B}}_d$ in $\mathbf{CLS}$ according to ${\mathbf{p}}^{\left(0\right)}$, ${\gamma }_{list}$ and ${\beta }_d$, respectively.

16: end for

Data Detection:

17: Refer to (36) and (37).

Noise Measurement:

18: Update the noise vector $\lambda$ according to (38).

Channel Estimation:

19: ${\sum }_{\mathbf{0}}\leftarrow diag\left({\mathsf{\Gamma }}_1\otimes {\mathbf{B}}_1,{\mathsf{\Gamma }}_2\otimes {\mathbf{B}}_2,\dots ,{\mathsf{\Gamma }}_C\otimes {\mathbf{B}}_C\right)$.

20: Obtain $\mathbf{U}$ according to (11).

21: ${\mathbf{p}}^{\left(1\right)}\leftarrow$ FindIndex$({\gamma }_{list}\ge {\gamma }_{th})$ and $\mathbf{U}\leftarrow \mathbf{U}[:,{\mathbf{p}}^{\left(1\right)}\otimes (1:M)]$.

22: Refer to (32) and (33).

Check stopping conditions:

23: $r\leftarrow r+1$.

24: return the sub-part of $\mathbf{Cluster}\mathbf{Evolution}$ until $r\ge r_{max}$ or $\parallel {\gamma }_{list}-{\gamma }_{list}^{\left(old\right)}{\parallel }_2^2<ϵ$.

Output: the pruned channel ${\overline{\mu }}_{\mathbf{h}}$ with the covariance matrix ${{\displaystyle \overline{\sum }}}_{\mathbf{h}}$, the cluster list $\mathbf{CLS}$,

the noise vector $\lambda$, and the transmitted data symbols ${\mathbf{x}}_{\mathbf{d}}^{\left(m\right)}$ for $m=1,2,\dots ,M$.