Training size optimization with reduced complexity in cell-free massive MIMO system

Abstract

Training sequence is used in multiple antenna systems to estimate channel state information and mitigate channel distortion between transmitter and receiver. However, the training sequence or pilot must be limited to a certain size in order to reduce the impact of overhead loss due to limited channel coherence length in mobile users. In this paper, we proposed to use training size optimization in cell-free massive MIMO system. In addition, we proposed and compared the performance of different training size optimization algorithms, namely exhaustive search optimization, bisection optimization and min–max optimization, with each method has different level of calculation complexities. The results showed that in general, all of the 3 training length optimization methods improved the downlink rate compared to the conventional pilot length method. We also showed that the training optimization methods are more effective when the coherence length is small or the number of users is very large. In the case of large number of users or small coherence length, the exhaustive search has the best median downlink rate, followed closely by min–max optimum and finally the bisection method. Even though the exhaustive search optimization has the best downlink rate, we showed that the proposed reduce optimization complexity methods has significantly less calculation complexity. In addition, the median downlink rate performance of min–max optimization method is only slightly less than that of the exhaustive search method for various number of users and coherence length.

Appendices

Appendix 1: Channel estimation

In order to transmit data from APs to the desired UEs using spatial multiplexing, it is necessary to acquire channel information between the transmitter and receiver through estimation. In our system model, the channel estimation is done at each AP using received uplink pilot from the UEs during the training phase. We use minimum mean square error (MMSE) channel estimation in order to obtain the lower bound ergodic downlink rate [18]. Even though not every UE is connected to every AP, this does not mean that APs can avoid receiving signal from non-connected UEs during the training phase. This is because an AP will receive signal from any UE as long as the large scale fading coefficient between the AP and UE is not zero. Since AP can receive pilot from any UEs, this may create interference during the training phase. As a results, pilot signal received at AP $l$ from $K$ UEs in the cell can be express as

$$\varvec{Y}_{l} = \sqrt {P_{\text{p}} } \mathop \sum \limits_{k = 1}^{K} \phi_{k} \varvec{g}_{lk} + \varvec{N}_{l}$$

(14)

where

• $\varvec{Y}_{l}$ is $\tau \times M$ received pilot signal at AP $l$,

• $P_{\text{p}}$ is uplink pilot power,

• ${{\varvec\phi }}_{k}$ is $\tau \times 1$ pilot from UE $k$,

• $\varvec{N}_{l}$ is $\tau \times M$ noise signal at AP $l$ with each element has ${\mathcal{C}\mathcal{N}}\left( {0, 1} \right)$ distribution.

To estimate the channel for UE $k$, we obtain channel projection of the received pilot signal, (14), with the actual pilot signal from UE $k$, $\phi_{k}$, as follows [19]

$${\check{\varvec{y}}}_{lk} = {{\varvec\phi }}_{k}^{H} \varvec{Y}_{l} = \sqrt {P_{\text{p}} } \mathop \sum \limits_{{k^{\prime} = 1}}^{K} {{\varvec\phi }}_{k}^{H} \phi_{{k^{\prime}}} \varvec{g}_{{lk^{\prime}}} + {{\varvec\phi }}_{k}^{H} \varvec{N}_{l}.$$

(15)

Using (15), we can obtain MMSE channel estimate as [4]

$$\hat{\varvec{g}}_{lk} = \theta_{lk} {\check{\varvec{y}}}_{lk},$$

(16)

where

$$\theta_{lk} = \frac{{\sqrt {P_{\text{p}} } \tau \beta_{lk} }}{{P_{\text{p}} \mathop \sum \nolimits_{{k^{\prime} = 1}}^{K} \left| {{{\varvec\phi }}_{{k^{\prime}}}^{H} {{\varvec\phi }}_{k} } \right|^{2} \beta_{{lk^{\prime}}} + \tau }}.$$

(17)

The covariance of $\hat{\varvec{g}}_{lk}$ can be written as [4]

$${\mathbb{E}}\left[ {\hat{\varvec{g}}_{lk}^{H} \hat{\varvec{g}}_{lk} } \right] = \frac{{P_{\text{p}} \tau^{2} \beta_{lk}^{2} }}{{P_{\text{p}} \mathop \sum \nolimits_{{k^{\prime} = 1}}^{K} \left| {{{\varvec\phi }}_{{k^{\prime}}}^{H} {{\varvec\phi }}_{k} } \right|^{2} \beta_{{lk^{\prime}}} + \tau }}\varvec{I}_{M} = \gamma_{lk} \varvec{I}_{M},$$

(18)

where $\gamma_{lk} = \frac{{P_{p} \tau^{2} \beta_{lk}^{2} }}{{P_{p} \mathop \sum \nolimits_{{k^{\prime} = 1}}^{K} \left| {{{\varvec\phi }}_{{k^{\prime}}}^{H} {{\varvec\phi }}_{k} } \right|^{2} \beta_{{lk^{\prime}}} + \tau }}.$

The error for MMSE channel estimate, $\tilde{\varvec{g}}_{lk} = \varvec{g}_{lk} - \hat{\varvec{g}}_{lk}$, is uncorrelated with $\hat{\varvec{g}}_{lk}$ and has the following covariance

$${\mathbb{E}}\left[ {\tilde{\varvec{g}}_{lk}^{H} \tilde{\varvec{g}}_{lk} } \right] = {\mathbb{E}}\left[ {\varvec{g}_{lk}^{H} \varvec{g}_{lk} } \right] - {\mathbb{E}}\left[ {\hat{\varvec{g}}_{lk}^{H} \hat{\varvec{g}}_{lk} } \right] = \left( {\beta_{lk} - \gamma_{lk} } \right)\varvec{I}_{M}.$$

(19)

By using (18) and (19), along with Chi-squared moment properties [31], we obtain the following identities

$${\mathbb{E}}\left[ {{||\hat{\varvec{g}}}_{lk}||^{2} } \right] = M\gamma_{lk},$$

(20)

$${\mathbb{E}}\left[ {||\hat{\varvec{g}}_{lk}||^{4} } \right] = \left( {M + 1} \right)\gamma_{lk}^{2},$$

(21)

$${\mathbb{E}}\left[ {{||\hat{\varvec{g}}}_{{lk^{\prime}}}||^{2} \hat{||\varvec{g}}_{lk}||^{2} } \right] = M^{2} \gamma_{{lk^{\prime}}} \gamma_{lk} \quad {\text{for}}\quad k^{\prime} \ne k,$$

(22)

$${\mathbb{E}}\left[ {\left| {{|\tilde{\varvec{g}}}_{lk} \hat{\varvec{g}}_{lk}^{H} } \right|^{2} } \right] = \left( {\beta_{lk} - \gamma_{lk} } \right)\gamma_{lk} M.$$

(23)

Appendix 2: Closed form ergodic lower bound downlink rate

We can express the lower bound downlink rate for UE $k$ as

$$R_{k} = \left( { \frac{{T_{\text{coh}} - \tau }}{{T_{\text{coh}} }}} \right)\log_{2} \left( {1 + \frac{{S_{k} }}{{N_{k} }}} \right),$$

(24)

where $S_{k}$ is the effective average desired power while $N_{k}$ is the average interference and noise power. Based from lower bound ergodic downlink rate theorem [20], we can write $S_{k}$ as

$$S_{k} = \left| {{\mathcal{D}}_{k} } \right|^{2}$$

Using ${\mathcal{D}}_{k}$ from (6), $S_{k}$ can be written as [14]

$$S_{k} = \left| {\sqrt {P_{\text{d}} } \sum\limits_{{l \in {\mathcal{L}}_{k} }} {\rho_{lk}^{1/2} {\mathbb{E}}} \left[ {\varvec{g}_{lk} \hat{\varvec{g}}_{lk}^{H} } \right]} \right|^{2}$$

(25)

$$= P_{\text{d}} M^{2} \left| {\mathop \sum \limits_{{l \in {\mathcal{L}}_{k} }}^{{}} \rho_{lk}^{1/2} \gamma_{lk} } \right|^{2}.$$

(26)

$N_{k}$ is the interference power which can be obtained from the average power of the total received signal, $x_{k}$, minus the effective desired signal at UE $k$, ${\mathcal{D}}_{k}$, as follows

$$N_{k} = {\mathbb{E}}\left[ {\left| {x_{k} - {\mathcal{D}}_{k} d_{k} } \right|^{2} } \right]$$

(27)

From (16), we can write $N_{k}$ as

$$N_{k} = {\mathbb{E}}\left[ {\left| {{\mathcal{C}}_{k} d_{k} + \mathop \sum \limits_{k' \ne k}^{K} {\mathcal{M}}_{{kk^{\prime}}} d_{{k^{\prime}}} + n_{k} } \right|^{2} } \right].$$

(28)

Since the downlink data $d_{k}$ is not correlated for different value of $k$, then ${\mathcal{M}}_{{kk^{\prime}}}$ is not correlated for different $k^{\prime}$. This also means ${\mathcal{C}}_{k}$ is not correlated with ${\mathcal{M}}_{{kk^{\prime}}}$. Hence, $N_{k}$ becomes

$$N_{k} = {\mathbb{E}}\left[ {\left| {{\mathcal{C}}_{k} } \right|^{2} } \right] + \mathop \sum \limits_{k' \ne k}^{K} {\mathbb{E}}\left[ {\left| {{\mathcal{M}}_{{kk^{\prime}}} } \right|^{2} } \right] + 1.$$

(29)

Substituting (15) and (16) into ${\mathcal{C}}_{k}$ in (6), and applying expectation properties from (20) to (23), we can solve ${\mathbb{E}}\left[ {\left| {{\mathcal{C}}_{k} } \right|^{2} } \right]$ in (29) and get [14]

$${\mathbb{E}}\left[ {\left| {{\mathcal{C}}_{k} } \right|^{2} } \right] = P_{\text{d}} \mathop \sum \limits_{{l \in {\mathcal{L}}_{k} }}^{{}} \rho_{lk} \left( {\left( {M^{2} + M} \right)\gamma_{lk}^{2} + M\gamma_{lk} \left( {\beta_{lk} - \gamma_{lk} } \right) - M^{2} \gamma_{lk}^{2} } \right) = P_{\text{d}} \mathop \sum \limits_{{l \in {\mathcal{L}}_{k} }}^{{}} \rho_{lk} M\gamma_{lk} \beta_{lk}.$$

(30)

Substituting (15) and (16) into ${\mathcal{M}}_{{kk^{\prime}}}$ in (6), and applying expectation properties from (20) to (23), the expectation variable ${\mathbb{E}}\left[ {\left| {{\mathcal{M}}_{{kk^{\prime}}} } \right|^{2} } \right]$ in (29) can be solved as [14]

$${\mathbb{E}}\left[ {\left| {{\mathcal{M}}_{{kk^{\prime}}} } \right|^{2} } \right] = P_{\text{p}} P_{\text{d}} \left( {\left| {\phi_{k}^{H} \phi_{{k^{\prime}}} } \right|^{2} \left( {\mathop \sum \limits_{{l \in {\mathcal{L}}_{k} }}^{{}} \mathop \sum \limits_{{l_{1} \in {\mathcal{L}}_{k} }}^{{}} \rho_{{lk^{\prime}}}^{1/2} \theta_{{lk^{\prime}}} \rho_{{l_{1} k^{\prime}}}^{1/2} \theta_{{l_{1} k^{\prime}}} \beta_{lk} \beta_{{l_{1} k}} M^{2} } \right) + \mathop \sum \limits_{{l \in {\mathcal{L}}_{k} }}^{{}} \mathop \sum \limits_{i = 1}^{K} \rho_{{lk^{\prime}}} \theta_{{lk^{\prime}}}^{2} \beta_{lk} \beta_{li} M\left| {\phi_{i}^{H} \phi_{{k^{\prime}}} } \right|^{2} } \right) + P_{\text{d}} \tau \mathop \sum \limits_{{l \in {\mathcal{L}}_{k} }}^{{}} \rho_{{lk^{\prime}}} \theta_{{lk^{\prime}}}^{2} \beta_{lk} M$$

(31)

Substituting (26), (29), (30) and (31) into (24), the ergodic lower bound downlink rate becomes (7).