Mono-bit millimeter-wave channel estimation: Bayesian and adaptive quantization frameworks

Abstract

To achieve a more reasonable cost, employment of a mono-bit quantization structure is essential in wide bandwidth millimeter-wave communications (mmW). However, in mmW channel estimation, one of the critical challenges of mono-bit quantization in each branch of the antenna is the loss of data amplitude. The latest researches offer several approaches to estimate the channel in mono-bit single and multiple antenna structures. Such methods not consider the recovery of channel amplitude and consequently are limited to finding direction. Moreover, they do not provide sparsity into account. Thus, two new schemes are introduced based on a non-adaptive Bayesian algorithm and adaptive sigma–delta modulation (SDM) to tackle these issues. In both methods, we exploit the sparsity of mmWC channel in the angle domain. To extract the sparse-based Laplacian density in case that the sparsity level is unknown, we propose a Bayesian algorithm which employs a hierarchical chain composed of Gaussian and exponential prior. Furthermore, in case that the sparsity level of an application is known or can be estimated, the adaptive mono-bit quantization schemes is proposed which retrieves the amplitude of the sparse channel by the aid of SDM and CS tools. Simulation results show that the adaptive mono-bit estimator outperforms the non-adaptive Bayesian approach comparing the mean square error as an index. Also, the proposed adaptive method, especially when a sufficient number of one-bit measurements are available, can asymptotically converge into optimum full-bit observations with lower computational complexity.

Appendices

Appendix 1

We know that

$$E\left[ {{\text{z}}_{i} |{\text{r}}_{i} = 1,{\mathbf{h}}_{v}^{(t)} } \right] = \frac{{\int\limits_{0}^{\infty } {z_{i} \exp ( - (z_{i} - {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} )^{2} /2\sigma^{2} )dz_{i} } }}{{\int\limits_{0}^{\infty } {\exp ( - (z_{i} - {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} )^{2} /2\sigma^{2} )dz_{i} } }}$$

(54)

and

$$E\left[ {{\text{z}}_{i} |{\text{r}}_{i} = - 1,{\mathbf{h}}_{v}^{(t)} } \right] = \frac{{\int\limits_{ - \infty }^{0} {z_{i} \exp ( - (z_{i} - {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} )^{2} /2\sigma^{2} )dz_{i} } }}{{\int\limits_{ - \infty }^{0} {\exp ( - (z_{i} - {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} )^{2} /2\sigma^{2} )dz_{i} } }}$$

(55)

which (55) can be computed in closed-form. From nominator of (54) we have

$$\begin{aligned} & \int\limits_{0}^{\infty } {z_{i} \exp ( - (z_{i} - {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} )^{2} /2\sigma^{2} )dz_{i} } \\ & \quad = \int\limits_{0}^{\infty } {(z_{i} - {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} + {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} )\exp ( - (z_{i} - {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} )^{2} /2\sigma^{2} )dz_{i} } \\ & \quad = \int\limits_{0}^{\infty } {(z_{i} - {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} )\exp ( - (z_{i} - {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} )^{2} /2\sigma^{2} ) + {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} \exp ( - (z_{i} - {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} )^{2} /2\sigma^{2} )dz_{i} } \\ & \quad = \int\limits_{{ - {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} }}^{\infty } {s\exp ( - (s)^{2} /2\sigma^{2} )ds_{i} + {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} \int\limits_{{ - {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} }}^{\infty } {\exp ( - (s)^{2} /2\sigma^{2} )ds_{i} } } \\ & \quad = - \sigma^{2} \exp ( - (s)^{2} /2\sigma^{2} )|_{{ - {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} }}^{\infty } + {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} \int\limits_{{ - {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} }}^{\infty } {\exp ( - (s)^{2} /2\sigma^{2} )ds_{i} } \\ & \quad = \sigma^{2} \exp ( - ({\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} )^{2} /2\sigma^{2} ) + {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} \sqrt {2\pi \sigma^{2} } \varPhi ({\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} ) \\ \end{aligned}$$

(56)

The normal pdf and normal CDF in above are defined as ${\rm N}({\mathbf{W}}_{i}^{H} {\hat{\mathbf{h}}}_{v}^{(t)} |0,\sigma^{2} ) \triangleq {1 \mathord{\left/ {\vphantom {1 {\sqrt {2\pi \sigma^{2} } }}} \right. \kern-0pt} {\sqrt {2\pi \sigma^{2} } }}\exp - ({\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} )^{2} /2\sigma^{2} )$ and $\varPhi ({\mathbf{W}}_{i}^{H} {\hat{\mathbf{h}}}_{v}^{(t)} ) \triangleq \frac{1}{{\sqrt {2\pi \sigma^{2} } }}\int\limits_{{ - {\mathbf{W}}_{i}^{H} {\hat{\mathbf{h}}}_{v}^{(t)} }}^{\infty } {\exp ( - \frac{{(s)^{2} }}{{2\sigma^{2} }}} )ds$ respectively. With a few manipulations, we have

$$E\left[ {{\text{z}}_{i} |{\text{r}}_{i} ,{\mathbf{h}}_{v}^{(t)} } \right] = {\mathbf{W}}_{i}^{{^{H} }} {\mathbf{h}}_{v}^{(t)} + {\text{r}}_{i} \sigma^{2} \frac{{{\rm N}({\mathbf{W}}_{i}^{H} {\hat{\mathbf{h}}}_{v}^{(t)} |0,\sigma^{2} )}}{{\varPhi ({\text{r}}_{i} {\mathbf{W}}_{i}^{H} {\hat{\mathbf{h}}}_{v}^{(t)} )}}$$

(57)

Also, the expectation of $\alpha_{i}^{ - 1}$ is obtained by simple manipulation as follows

$$\begin{aligned} \vartheta_{i} \equiv E\left[ {\alpha_{i}^{ - 1} |{\mathbf{r}},{\hat{\mathbf{h}}}_{v}^{(t)} } \right] & = \frac{{\int\limits_{0}^{\infty } {\alpha_{i}^{ - 1} p({\hat{\mathbf{h}}}_{i,v}^{(t)} |\alpha_{i} )p(\alpha_{i} )d\alpha_{i} } }}{{\int\limits_{0}^{\infty } {p({\hat{\mathbf{h}}}_{i,v}^{(t)} |\alpha_{i} )p(\alpha_{i} )d\alpha_{i} } }} = \frac{{\int\limits_{0}^{\infty } {\alpha_{i}^{ - 1} \frac{1}{{\sqrt {2\pi \alpha_{i} } }}\exp ( - {\raise0.7ex\hbox{${{\text{h}}_{i,v}^{2} }$} \!\mathord{\left/ {\vphantom {{{\text{h}}_{i,v}^{2} } {2\alpha_{i} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${2\alpha_{i} }$}})\frac{1}{{\alpha_{i} }}d\alpha_{i} } }}{{\int\limits_{0}^{\infty } {\frac{1}{{\sqrt {2\pi \alpha_{i} } }}\exp ( - {\raise0.7ex\hbox{${{\text{h}}_{i,v}^{2} }$} \!\mathord{\left/ {\vphantom {{{\text{h}}_{i,v}^{2} } {2\alpha_{i} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${2\alpha_{i} }$}})\frac{1}{{\alpha_{i} }}d\alpha_{i} } }} \\ & = \frac{{4\int\limits_{0}^{\infty } {\frac{{t^{2} }}{{h_{i,v}^{3} }}\exp ( - t^{2} )dt} }}{{2\int\limits_{0}^{\infty } {\exp ( - h_{i,v}^{2} t^{2} /2)dt} }} = \frac{{\frac{4}{{h_{i,v}^{3} }}\sqrt {\frac{\pi }{2}} }}{{\sqrt {\frac{\pi }{2}} \frac{1}{{h_{i,v} }}}} = \frac{1}{{h_{i,v}^{2} }} \\ \end{aligned}$$

(58)

Appendix 2

The normal CDF is given by by

$$F_{1} (x) = \frac{1}{{\sigma \sqrt {2\pi } }}\int\limits_{ - \infty }^{x} {\exp ( - \frac{{t^{2} }}{{2\sigma^{2} }})} dt$$

(59)

For having the normal complementary CDF (CCDF), we use $F_{1} ( - x)$. Thus, it can be obtained

$$F_{2} (x) = F_{1} ( - x) = \frac{1}{{\sigma \sqrt {2\pi } }}\int\limits_{ - \infty }^{ - x} {\exp ( - \frac{{t^{2} }}{{2\sigma^{2} }})} dt$$

(60)

Furthermore, the complementary error function is defined

$$erfc(x) \triangleq \frac{2}{\sqrt \pi }\int\limits_{x}^{\infty } {\exp ( - t^{2} )} dt$$

(61)

By changing the variable $\frac{t}{\sigma \sqrt 2 }$ to $t^{\prime }$ into the (60) and some manipulations, we can get the relation between $erfc(.)$ and CCDF. We have,

$$F_{2} (x) = \frac{1}{2}erfc\left( {\frac{x}{\sigma \sqrt 2 }} \right)$$

(62)

By assumption of $\sigma = 1$, the first and the second derivative are obtained from (60) as

$$F_{2}^{\prime } (x) \triangleq f(x) = - \frac{1}{{\sqrt {2\pi } }}\exp \left( { - \frac{{x^{2} }}{2}} \right),\quad {\text{F}}_{2}^{{\prime \prime }} (x) \triangleq f^{\prime}(x) = \frac{x}{{\sqrt {2\pi } }}\exp \left( { - \frac{{x^{2} }}{2}} \right)$$

(63)

In above, the first derivative is pdf. By considering (36) and (38) and defining of ${\varvec{\upvarepsilon}}_{n} = {\varvec{\uptau}}_{n} - {\varvec{\upomega}}_{k - 1}$, we have,

$$L^{\prime}_{MLQ} ({\varvec{\upomega}}_{k - 1} ) = \sum\limits_{n = 1}^{k - 1} { - \left( {\frac{{1 + {\mathbf{r}}_{n} }}{2}} \right)\frac{{f({\varvec{\upvarepsilon}}_{n} )}}{{F{}_{\zeta }({\varvec{\upvarepsilon}}_{n} )}} + } \left( {\frac{{1 - {\mathbf{r}}_{n} }}{2}} \right)\frac{{f({\varvec{\upvarepsilon}}_{n} )}}{{1 - F{}_{\zeta }({\varvec{\upvarepsilon}}_{n} )}}$$

(64)

And

$$L_{MLQ}^{{\prime \prime }} ({\varvec{\upomega}}_{k - 1} ) = \frac{{\partial^{2} L_{MAQ} }}{{\partial {\varvec{\upomega}}_{k - 1}^{2} }} = \sum\limits_{n = 1}^{k - 1} {\left( {\frac{{1 + {\mathbf{r}}_{n} }}{2}} \right)\left[ {\frac{{f^{\prime}({\varvec{\upvarepsilon}}_{n} )}}{{F{}_{\zeta }({\varvec{\upvarepsilon}}_{n} )}} - \frac{{f^{2} ({\varvec{\upvarepsilon}}_{n} )}}{{F_{\zeta }^{2} ({\varvec{\upvarepsilon}}_{n} )}}} \right] + } \left( {\frac{{1 - {\mathbf{r}}_{n} }}{2}} \right)\left[ {\frac{{ - f^{\prime } ({\varvec{\upvarepsilon}}_{n} )}}{{1 - F{}_{\zeta }({\varvec{\upvarepsilon}}_{n} )}} - \frac{{f^{2} ({\varvec{\upvarepsilon}}_{n} )}}{{(1 - F{}_{\zeta }({\varvec{\upvarepsilon}}_{n} ))^{2} }}} \right]$$

(65)

Appendix 3

From the mono-bit received data model (2) and the estimated channel, we know that

$${\hat{\mathbf{r}}}(k)=sign\left( {{\hat{\mathbf{C}}}_{\text{r,RF}}^{\text{H}} {\mathbf{\hat{H}\hat{P}}}_{\text{t,RF}} {\hat{\mathbf{P}}}_{\text{t,BB}} {\mathbf{s}}(k){\mathbf{ + \hat{C}}}_{\text{r,RF}}^{\text{H}} {\mathbf{n}}(k)} \right)$$

(66)

Replacing the estimated baseband precoder ${\hat{\mathbf{P}}}_{\text{t,BB}}$ (47) into (66), we obtain

$${\hat{\mathbf{r}}}(k)=sign\left( {{\hat{\mathbf{C}}}_{\text{r,RF}}^{\text{H}} {\mathbf{\hat{H}\hat{P}}}_{\text{t,RF}} \sqrt {\frac{{N_{s} }}{{trace\left\{ {\left( {{\hat{\mathbf{H}}}_{eff} {\hat{\mathbf{H}}}_{eff}^{H} } \right)^{ - 1} } \right\}}}} {\hat{\mathbf{H}}}_{eff}^{H} \left( {{\hat{\mathbf{H}}}_{eff} {\hat{\mathbf{H}}}_{eff}^{H} } \right)^{ - 1} {\mathbf{s}}(k){\mathbf{ + \hat{C}}}_{\text{r,RF}}^{\text{H}} {\mathbf{n}}(k)} \right)$$

(67)

Using the definition of ${\hat{\mathbf{H}}}_{eff} \equiv {\hat{\mathbf{C}}}_{r,RF}^{H} {\mathbf{\hat{H}\hat{P}}}_{t,RF}$ and some manipulations, we have

$${\hat{\mathbf{r}}}(k) = sign\left( {\sqrt {\frac{{N_{s} }}{{trace\left\{ {\left( {{\hat{\mathbf{H}}}_{eff} {\hat{\mathbf{H}}}_{eff}^{H} } \right)^{ - 1} } \right\}}}} {\mathbf{s}}(k){\mathbf{ + \hat{C}}}_{\text{r,RF}}^{\text{H}} {\mathbf{n}}(k)} \right)$$

(68)

The relation (68) shows that the interference between multi streams is annihilated. Therefore, we have $2N_{s}$ real mono-bit data at the receiver. The SNR in each branch of the antenna for real received data is obtained as

$$SNR = E\left[ {\left\| {\sqrt {\frac{{N_{s} }}{{trace\left\{ {\left( {{\hat{\mathbf{H}}}_{eff} {\hat{\mathbf{H}}}_{eff}^{H} } \right)^{ - 1} } \right\}}}} {\mathbf{s}}(k)} \right\|^{2} } \right]/E\left[ {\left\| {{\hat{\mathbf{C}}}_{\text{r,RF}}^{\text{H}} {\mathbf{n}}(k)} \right\|^{2} } \right] = \frac{{P_{T} }}{{trace\left\{ {\left( {{\hat{\mathbf{H}}}_{eff} {\hat{\mathbf{H}}}_{eff}^{H} } \right)^{ - 1} } \right\}\sigma_{n}^{2} }}$$

(69)

Maximizing SNR means the minimization of $trace\left\{ {\left( {{\hat{\mathbf{H}}}_{eff} {\hat{\mathbf{H}}}_{eff}^{H} } \right)^{ - 1} } \right\}$ in (69). We utilize the singular value decomposition (SVD) over ${\hat{\mathbf{H}}}_{eff}$ as ${\hat{\mathbf{H}}}_{eff} = {\mathbf{\hat{U}\hat{\varGamma }\hat{V}}}^{H}$ that ${\hat{\mathbf{U}}}$ and ${\hat{\mathbf{V}}}$ are left and right singular vectors. Furthermore,${\hat{\mathbf{\varGamma }}}$ is a diagonal matrix consist of singular values in descending order. Consequently, maximization of SNR results as,

$$\hbox{max} (SNR) = \hbox{min} (\lambda_{1}^{ - 2} + \lambda_{2}^{ - 2} + \cdots + \lambda_{{N_{s} }}^{ - 2} )$$

(70)

where $\lambda_{i}$ is the ith entry of diagonal matrix ${\hat{\mathbf{\varGamma }}}$. Since $\lambda_{1}$ to $\lambda_{{N_{s} }}$ are the most significant singular values, The minimization term in (70) maximizes SNR certainly. Thus, it means that the best choice for ${\hat{\mathbf{P}}}_{\text{t,RF}}$ and ${\hat{\mathbf{C}}}_{\text{r,RF}}$ in term of ${\hat{\mathbf{H}}}_{eff} \equiv {\hat{\mathbf{C}}}_{r,RF}^{H} {\mathbf{\hat{H}\hat{P}}}_{t,RF}$ is $N_{s}$ singular vectors from ${\hat{\mathbf{V}}}$ and ${\hat{\mathbf{U}}}$, respectively. This choice makes $2N_{s}$ separated and parallel sub-channels.