Pilot Optimization for Structured Compressive Sensing Based Channel Estimation in Large-Scale MIMO Systems with Superimposed Pilot Pattern

Abstract

Compressive sensing (CS) has attracted much attention in wireless communications due to its ability to attain acceptable channel estimates with a small number of pilots. To further reduce the pilot overhead in multi-input multi-output (MIMO) systems, CS-based channel estimation may employ superimposed pilot pattern. Previous works on superimposed pilot design generally allocate pilots randomly, which may give ill-posed measurement matrices. In this paper, we focus on deterministic pilot allocation for large-scale MIMO systems with superimposed pilot pattern to improve the performance of structured CS based channel estimation. By exploiting the spatial common sparsity and the error bound of block sparse reconstruction, a new criterion is firstly proposed to optimize the pilots in the Hadamard space. The proposed criterion makes full use of the information about the principal angles across the blocks in the measurement matrix, which can enhance the average recovery ability and exclude the worst pilots simultaneously. Secondly, a genetic algorithm is proposed to minimize the merit factor of the proposed criterion efficiently. Simulation results show that the proposed optimized pilots outperform the random pilots in terms of mean-squared error by about 3 dB. Moreover, the proposed criterion is more likely to achieve better measurement matrices than the traditional criteria.

Appendices

Appendix 1: Proof of ${\user2{v}}({\varvec{\Phi}}) = 0$

From Eq. (5), the i-th column of the l-th normalized block ${\tilde{\varvec{\Phi}}}_{l}$ is given by

$${\tilde{\varvec{\Phi}}}_{l,i} = \sqrt {N}{\mathcal{D}\left( {{\mathbf{f}}_{l} } \right)} \frac{{{\mathbf{u}}_{i} }}{{\left\| {{\mathbf{u}}_{i} } \right\|_{2} }}$$

(17)

By substituting (17) into Eq. (8), $v({\varvec{\Phi}})$ can be rewritten as

$$v({\varvec{\Phi}}) = \mathop {\hbox{max} }\limits_{l} \mathop {max}\limits_{i \ne j} \left| {N\frac{{{\mathbf{u}}_{i}^{\text{H}} }}{{\left\| {\left. {{\mathbf{u}}_{i} } \right\|} \right._{2} }}{\mathcal{D}}\left( {{\mathbf{f}}_{l}^{*} } \right){\mathcal{D}}({\mathbf{f}}_{l} )\frac{{{\mathbf{u}}_{j} }}{{\left\| {\left. {{\mathbf{u}}_{j} } \right\|} \right._{2} }}} \right| = \mathop {max}\limits_{i \ne j} \left| {\frac{{{\mathbf{u}}_{i}^{H} }}{{\left\| {\left. {{\mathbf{u}}_{i} } \right\|} \right._{2} }} \cdot \frac{{{\mathbf{u}}_{j} }}{{\left\| {{\mathbf{u}}_{j} } \right\|_{2} }}} \right|$$

(18)

As shown in (18), $v({\varvec{\Phi}})$ is determined by the normalized pilot vectors. If we take columns of a Hadamard matrix as pilot vectors, we can have $v({\varvec{\Phi}}) = 0$. It is because of the orthogonality between different columns, i.e., ${\mathbf{u}}_{i}^{H} {\mathbf{u}}_{j} = {\mathbf{0}},\forall i \ne j$. And $v({\varvec{\Phi}}) = 0$ indicates that the measurement matrix ${\varvec{\Phi}}$ is composed of orthogonal blocks.

Appendix 2: Proof of Proposition 1

The proof of proposition 1 consists of two parts which correspond to the proof of (12) and (13), respectively. We begin by denoting ${\mathbf{R}}_{lm} = {\tilde{\varvec{\Phi }}}_{l}^{H} {\tilde{\varvec{\Phi }}}_{m}$. From Eq. (5), ${\mathbf{R}}_{lm}$ can be formulated as

$${\mathbf{R}}_{lm} = N{\tilde{\varvec{\Pi }}}^{H} {\mathcal{D}}\left( {{\mathbf{f}}_{l}^{*} \odot {\mathbf{f}}_{m} } \right){\tilde{\varvec{\Pi }}}$$

(19)

where ${\tilde{\varvec{\Pi }}} = \left[ {\frac{{{\mathbf{u}}_{1} }}{{\left\| {{\mathbf{u}}_{1} } \right\|_{2} }},\frac{{{\mathbf{u}}_{2} }}{{\left\| {{\mathbf{u}}_{2} } \right\|_{2} }}, \ldots ,\frac{{{\mathbf{u}}_{{N_{t} }} }}{{\left\| {{\mathbf{u}}_{{N_{t} }} } \right\|_{2} }}} \right]$ is the normalized version of the pilot matrix ${\varvec{\Pi}} = [{\mathbf{u}}_{1} ,{\mathbf{u}}_{2} , \ldots ,{\mathbf{u}}_{{N_{t} }} ]$. Since the value of each pilot symbol is + 1 or − 1, every pilot vector has the same Frobenius norm, i.e., $\left\| {\left. {{\mathbf{u}}_{i} } \right\|} \right._{2} = \sqrt N ,1 \le i \le N_{t}$. Then (19) can be rewritten as

$${\mathbf{R}}_{lm} = {\varvec{\Pi}}^{H} {\mathcal{D}}({\mathbf{w}}_{lm} ){\varvec{\Pi}}$$

(20)

where ${\mathbf{w}}_{lm} = {\mathbf{f}}_{l}^{*} \odot {\mathbf{f}}_{m} = {\mathcal{D}}({\mathbf{f}}_{l}^{*} ){\mathbf{f}}_{m} = {\mathcal{D}}({\mathbf{f}}_{m} ){\mathbf{f}}_{l}^{*}$. By substituting (20) into Eq. (7), the interblock coherence $\mu_{B} ({\varvec{\Phi}})$ can be written as

$$\mu_{B} ({\varvec{\Phi}}) = \mathop {\hbox{max} }\limits_{l \ne m} \frac{1}{{N_{t} }}\rho \left( {{\varvec{\Pi}}^{\text{H} } {\mathcal{D}}({\mathbf{w}}_{lm} ){\varvec{\Pi}}} \right)$$

(21)

From Eq. (21), we can conclude that $\mu_{B} ({\varvec{\Phi}})$ is determined by ${\varvec{\Pi}}$. By using the definition of ${\mathbf{R}}_{lm}$, $\mu_{B}^{t} ({\varvec{\Phi}})$ in (10) can be rewritten as

$$\mu_{B}^{t} \left( {\varvec{\Phi}} \right) = \sum\limits_{l = 1}^{L - 1} {\sum\limits_{m \ne l} {\left\| {{\mathbf{R}}_{lm} } \right\|}_{F}^{2} = \sum\limits_{l} {\sum\limits_{m} {\left\| {{\mathbf{R}}_{lm} } \right\|}_{F}^{2} } - \sum\limits_{l = m} {\left\| {{\mathbf{R}}_{lm} } \right\|} _{F}^{2}}$$

(22)

To obtain $\mu_{B}^{t} ({\varvec{\Phi}})$, the term $\left\| {\left. {{\mathbf{R}}_{lm} } \right\|} \right._{F}^{2}$ is analyzed for two cases: the special case with l = m and the general case with any pair (l, m). For the special case with l = m, from Eq. (20), $\left\| {\left. {{\mathbf{R}}_{lm} } \right\|} \right._{F}^{2}$ can be written as

$$\left\| {{\mathbf{R}}_{lm} } \right\|_{F}^{2} = \left\| {{\varvec{\Pi}}^{\text{H}} {\mathcal{D}}\left( {{\mathbf{w}}_{{{l,l} }} } \right){\varvec{\Pi}}} \right\|_{F}^{2} = \left\| {\frac{1}{N}{\varvec{\Pi}}^{\text{H}} {\varvec{\Pi}}} \right\|_{F}^{2} \mathop = \limits^{{({\text{a}})}} N_{t}$$

(23)

where step (a) follows from ${\varvec{\Pi}}^{\text{H}} {\varvec{\Pi}} = N{\mathbf{I}}_{{N_{t} }}$ when ${\mathbf{u}}_{i}$ is a Hadamard column of dimension N.

For the general case, $\left\| {\left. {{\mathbf{R}}_{lm} } \right\|} \right._{F}^{2}$ can be written as

$$\begin{aligned} \left\| {\left. {{\mathbf{R}}_{lm} } \right\|} \right._{F}^{2} &\mathop = \limits^{{\mathrm{(a)}}} \sum\limits_{i = 1}^{{N_{t} }} {{\mathbf{u}}_{i}^{H} {\mathcal{D}}^{H} \left( {{\mathbf{w}}_{lm} } \right)\left( {\sum\limits_{t = 1}^{{N_{t} }} {{\mathbf{u}}_{t} {\mathbf{u}}_{t}^{H} } } \right){\mathcal{D}}({\mathbf{w}}_{lm} ){\mathbf{u}}_{i} } \\ & \mathop = \limits^{{\mathrm{(b)}}} {\mathbf{w}}_{lm}^{H} \mathop {\underline{{\sum\limits_{i = 1}^{{N_{t} }} {\left( {\sum\limits_{t = 1}^{{N_{t} }} {{\mathbf{U}}_{i}^{H} {\mathbf{u}}_{t} {\mathbf{u}}_{t}^{H} {\mathbf{U}}_{i} } } \right)} }} }\limits_{{\mathbf{B}}} {\mathbf{w}}_{lm} \\ & \mathop = \limits^{{\mathrm{(c)}}} \text{S} \left\{ {\mathop {\underline{{\left( {\sum\limits_{i = 1}^{{N_{t} }} {\sum\limits_{t = 1}^{{N_{t} }} {\left( {{\mathbf{u}}_{t} {\mathbf{u}}_{t}^{H} } \right) \odot \left( {{\mathbf{u}}_{i}^{*} {\mathbf{u}}_{i}^{T} } \right)} } } \right)}} }\limits_{{\mathbf{B}}} \odot \left( {{\mathbf{w}}_{lm}^{*} {\mathbf{w}}_{lm}^{T} } \right)} \right\} \\ & \mathop = \limits^{{({\text{d}})}} \text{S} \left\{ {\mathop {\underline{{\left( {\left( {{\varvec{\Pi \Pi }}^{H} } \right) \odot \left( {{\varvec{\Pi \Pi }}^{H} } \right)^{*} } \right)}} }\limits_{{\mathbf{B}}} \odot \left( {{\mathbf{w}}_{lm}^{*} {\mathbf{w}}_{lm}^{T} } \right)} \right\} \\ \quad & = \text{S} \left\{ {{\mathbf{B}} \odot \left( {{\mathbf{w}}_{lm}^{*} {\mathbf{w}}_{lm}^{T} } \right)} \right\} \\ \end{aligned}$$

(24)

where step (a) follows from $\left\| {\left. {{\mathbf{R}}_{lm} } \right\|} \right._{F}^{2} = tr({\mathbf{R}}_{lm}^{H} {\mathbf{R}}_{lm} )$ and ${\mathbf{R}}_{lm}^{H} {\mathbf{R}}_{lm} =$ ${\varvec{\Pi}}^{H} {\mathcal{D}}^{H} ({\mathbf{w}}_{lm} )(\sum\nolimits_{t = 1}^{{N_{t} }} {{\mathbf{u}}_{t} {\mathbf{u}}_{t}^{H} } ){\mathcal{D}}({\mathbf{w}}_{lm} ){\varvec{\Pi}}$, step (b) utilizes the property of diagonal matrices, step (c) follows from ${\mathbf{x}}^{H} {\mathbf{Ay}} = \sum\nolimits_{i} {\sum\nolimits_{j} {x_{i}^{*} a_{ij} } } y_{j} = S\{ {\mathbf{A}} \odot ({\mathbf{x}}^{*} {\mathbf{y}}^{T} )\}$ and ${\mathcal{D}}^{H} ({\mathbf{x}}){\mathbf{A}}{\mathcal{D}}({\mathbf{y}}) = {\mathbf{A}} \odot ({\mathbf{x}}^{*} {\mathbf{y}}^{T} )$, step (d) follows from ${\mathbf{X}} \odot \left( {\sum\nolimits_{i} {{\mathbf{Y}}_{i} } } \right) = \sum\nolimits_{i} {({\mathbf{X}} \odot {\mathbf{Y}}_{i} )}$.

According to (23) and (24), $\mu_{B}^{t} ({\varvec{\Phi}})$ in (22) can be written as

$$\mu_{B}^{t} ({\varvec{\Phi}}) = \sum\limits_{l,m} {S\left\{ {{\mathbf{B}} \odot \left( {{\mathbf{w}}_{lm}^{*} {\mathbf{w}}_{lm}^{T} } \right)} \right\}} - LN_{t}$$

(25)

Since ${\mathbf{w}}_{lm}^{*} {\mathbf{w}}_{lm}^{T} = {\mathcal{D}}({\mathbf{f}}_{m}^{*} ){\mathbf{f}}_{l} {\mathbf{f}}_{l}^{H} {\mathcal{D}}({\mathbf{f}}_{m} ) = ({\mathbf{f}}_{l} {\mathbf{f}}_{l}^{H} ) \odot ({\mathbf{f}}_{m} {\mathbf{f}}_{m}^{H} )^{*}$, the sum term on the right hand of (25) can be rewritten as

$$\begin{aligned} \sum\limits_{l,m} {\text{S} \left\{ {{\mathbf{B}} \odot \left( {{\mathbf{w}}_{lm}^{*} {\mathbf{w}}_{lm}^{T} } \right)} \right\}} & = \text{S} \left\{ {{\mathbf{B}} \odot \sum\limits_{l,m} {\left( {{\mathbf{f}}_{l} {\mathbf{f}}_{l}^{H} } \right) \odot \left( {{\mathbf{f}}_{m} {\mathbf{f}}_{m}^{H} } \right)^{*} } } \right\} \\ & = \text{S} \left\{ {{\mathbf{B}} \odot \left( {\sum\limits_{l} {{\mathbf{f}}_{l} {\mathbf{f}}_{l}^{H} } } \right) \odot \sum\limits_{m} {\left( {{\mathbf{f}}_{m} {\mathbf{f}}_{m}^{H} } \right)^{*} } } \right\} \\ & = \text{S} \left\{ {\mathop {\underline{{\left( {{\varvec{\Pi \Pi }}^{H} } \right) \odot \left( {{\varvec{\Pi \Pi }}^{H} } \right)^{*} }} }\limits_{{\mathbf{B}}} \odot {\mathbf{F}}_{s} {\mathbf{F}}_{s}^{H} \odot \left( {{\mathbf{F}}_{s} {\mathbf{F}}_{s}^{H} } \right)^{*} } \right\} \\ & = \left\| {\left( {{\varvec{\Pi \Pi }}^{H} } \right) \odot \left( {{\mathbf{F}}_{s} {\mathbf{F}}_{s}^{H} } \right)} \right\|_{F}^{2} \\ \end{aligned}$$

(26)

Finally, $\mu_{B}^{t} ({\varvec{\Phi}})$ can be written as

$$\mu_{B}^{t} ({\varvec{\Phi}}) = \sum\limits_{l \ne m} {\left\| {\left. {{\mathbf{R}}_{lm} } \right\|} \right._{F}^{2} } = \left\| {\left( {{\varvec{\Pi \Pi }}^{H} } \right) \odot \left( {{\mathbf{F}}_{s} {\mathbf{F}}_{s}^{H} } \right)} \right\|_{F}^{2} - LN_{t}$$

(27)