Channel Estimation for Finite Scatterers Massive Multi-User MIMO System

Abstract

In finite scattering propagation environment, when the number of scatterers is very small compared to the number of base station antennas and users in the cell and if the same scatterers are shared by all users, then the correlation among the users increases. Hence, the high-dimensional multi-user MIMO system is likely to have low-rank channel. To estimate the channel matrix, weighted nuclear norm minimization method is proposed in this paper. Iterative weighted singular value thresholding algorithm is used to solve the optimization problem. To recover the low-rank channel, a partial random Fourier matrix (PRFM) satisfying the restricted isometric property is adapted as the training matrix. The PRFM forces the iterative algorithm to converge in one iteration which reduces the huge computational complexity of the proposed channel estimation method. The mean square error and achievable sum rate are the criteria used to measure the performance of the proposed method. The results show that the proposed method outperforms the least square estimation method and the nuclear norm minimization method for various finite scatterers in different SNR levels.

Appendix: Convergence Analysis

The convergence of the proposed iterative algorithm is analysed by assuming the regularizer factor $\lambda =0$. Then the term $||\mathbf y -\varvec{\varPsi } \mathbf h ||_2^2$ in (6) can be solved iteratively using Landweber iterative method in matrix form as

$$\begin{aligned} \begin{aligned} \mathbf H _{i+1}=\mathbf H _{i} + \beta (\mathbf Y -\mathbf H _{i}\varvec{\varPhi })\varvec{\varPhi }^H \end{aligned} \end{aligned}$$

(17)

which can be rewritten as

$$\begin{aligned} \begin{aligned} \mathbf H _{i+1}=\mathbf H _{i}(\mathbf I -\beta \varvec{\varPhi }\varvec{\varPhi }^H) + \beta \mathbf Y \varvec{\varPhi }^H \end{aligned} \end{aligned}$$

(18)

Since $\mathbf H _0$ is initialized to be all zero matrix, obtain using the above recursion $\mathbf H _1$ to be

$$\begin{aligned} \begin{aligned} \mathbf H _{1}=\beta \mathbf H \varvec{\varPhi }^H \end{aligned} \end{aligned}$$

(19)

and

$$\begin{aligned} \begin{aligned} \mathbf H _{2}=\beta \mathbf Y \varvec{\varPhi }^H + (\mathbf I -\beta \varvec{\varPhi }\varvec{\varPhi }^H) \mathbf Y \varvec{\varPhi }^H \end{aligned} \end{aligned}$$

(20)

Rearranging the equation $\mathbf H _2$ in terms of $\mathbf H _1$

$$\begin{aligned} \begin{aligned} \mathbf H _{2}=\mathbf H _{1} + (\mathbf I - \beta \varvec{\varPhi }\varvec{\varPhi }^H) \mathbf Y \varvec{\varPhi }^H \end{aligned} \end{aligned}$$

(21)

Similarly, we obtain $\mathbf H _3$ as

$$\begin{aligned} \begin{aligned} \mathbf H _{3}=\mathbf H _{2} + (\mathbf I -\beta \varvec{\varPhi }\varvec{\varPhi }^H)^2 \mathbf Y \varvec{\varPhi }^H \end{aligned} \end{aligned}$$

(22)

In general, the iterative equation is written as

$$\begin{aligned} \begin{aligned} \mathbf H _{i}=\sum _{j=0}^{i-1}\beta \mathbf Y \varvec{\varPhi }^H(\mathbf I -\beta \varvec{\varPhi }\varvec{\varPhi }^H)^j \end{aligned} \end{aligned}$$

(23)

Using the expression for the sum of a geometric series, we obtain

$$\begin{aligned} \begin{aligned} \mathbf H _{i}= \beta \mathbf Y \varvec{\varPhi }^H [\mathbf I -(\mathbf I -\beta \varvec{\varPhi }\varvec{\varPhi }^H)]^{-1} [\mathbf I -(\mathbf I -\beta \varvec{\varPhi }\varvec{\varPhi }^H)^i] \end{aligned} \end{aligned}$$

(24)

For a partial DFT matrix $\varvec{\varPhi }\varvec{\varPhi }^H=\mathbf I$ and if we assume $\beta$ =1 then (9) converges to $\mathbf Y \varvec{\varPhi }^H$, i.e. in one iteration.