Bussgang decomposition-based sparse channel estimation in wideband hybrid millimeter wave MIMO systems with finite-bit ADCs

doi:10.1016/j.dsp.2018.11.003

Digital Signal Processing

Volume 85, February 2019, Pages 29-40

https://doi.org/10.1016/j.dsp.2018.11.003 Get rights and content

Abstract

Both the hybrid architecture and low precision analog-to-digital converters (ADCs) are considered to alleviate the burden of high power cost and hardware implementation of millimeter wave (mmWave) communication system. Accordingly, the channel estimation issue in wideband mmWave system with finite-bit ADCs becomes even challenging. To address this issue, the non-linear mmWave channel estimation problem is reformulated into a linear sparse signal recovery problem by utilizing the Bussgang decomposition. Then, based on the equivalent linear sparse model, a Bussgang decomposition-based OMP (BD-OMP) algorithm is proposed to both exploit the inherent sparsity of wideband mmWave channel and alleviate the quantization error. Furthermore, we analyze that the actual noise of linear sparse model consists of combined noise and distortion noise, which is related to the number of quantization bits, antennas, and received signal power in a large scale regime. In addition, the terminal condition of BD-OMP algorithm is derived based on the residual difference of two consecutive iterations, where the expectation of residual difference is the variance of the actual noise. Simulation results demonstrate that the proposed approach can significantly reduce the training overhead for estimating wideband mmWave channel with finite-bit ADCs at the receiver.

Introduction

Millimeter wave (mmWave) communication system has been recognized as a key technology in future 5G cellular networks [1]. By employing large number of antennas, the corresponding beamforming gain can resist the huge path loss in mmWave frequency band, which also provides vast bandwidth to enable gigabit per second data transmission [2]. Unfortunately, the large antenna arrays and ultra high data rates result into the challenging channel estimation, and sophisticated hardware implementation [3]. To alleviate the power consumption and hardware cost, hybrid precoder/combiner architecture [4], [5], and low-bit resolution analog-to-digital converter (ADC) receivers [6], [7], are two promising solutions, which reduce the number of radio frequency (RF) chains and bits, respectively. Due to such hardware constraints, the channel estimation issue in mmWave frequency becomes even challenging.

For the hybrid analog/digital architecture-based mmWave systems, the channel estimation problem is mainly caused by both the high-dimensional channel matrix and low SNR before beamforming [8], [9]. To deal with this problem, the straightforward method is to perform beam searching [10], [11], [12], which is a closed-loop channel estimation method and consumes certain feedback overhead. Specifically, a hierarchical codebook consisted of low-resolution and high-resolution beams is deployed at both transmitter and receiver. Through exhaustive or divide-and-conquer search, the coarse best angle sector can be first found on the low-resolution codebook level. Then the best beam can be further determined via the high-resolution codebook level, which characterizes the angle of arrival and departure (AoA/AoD) of mmWave channel. However, this method requires a large amount of training and feedback overhead, which may be computationally prohibitive and limited by the size of beam codebook. On the other hand, channel sparsity can be also exploited to perform the channel estimation. [13] considered a hybrid precoding based mmWave system and exploited the sparse structure of narrowband mmWave channel in the angular domain [14]. In [15], an efficient open-loop channel estimator was proposed for hybrid mmWave MIMO system, where the training vectors are designed based on the total coherence, and the bounds of orthogonal matching pursuit (OMP)-based estimator are analytically derived. Since mmWave MIMO channel is represented under the basis that consists of steering vectors with the grid of angles of departure and arrival (AoD/AoA), the basis matches caused by grid points occur. [16] proposed a channel subspace matching pursuit algorithm by taking the multiple signal classification into account, so that the super resolution capability can be achieved, especially when the angular grid resolutions are relative high. However, the aforementioned solutions only focused on narrowband mmWave channels based on compressed sensing (CS) theory [17], [18], while the practical mmWave systems tend to be operated in wideband and frequency selective channels. Accordingly, the wideband channel estimators are developed in [19], [20], [21], [22], [23]. In [21], the wideband mmWave channel estimation problem was considered in both the time domain and frequency domain. The system constraints, including pulse shaping filter response and frame structure, are taken into consideration. Accordingly, the sparsifying dictionary was redefined for the sparse wideband mmWave channels, either in angular or delay domain. In [22], the received signal at the base station (BS) is expressed as a third-order tensor, and a CANDECOMP/PARAFAC decomposition-based method was proposed. The analysis demonstrates the uniqueness of the tensor decomposition even under the small size of tensor. Further, a CS based channel estimation strategy in the frequency domain was presented in [23], where the common sparsity among multiple subcarriers are exploited. The exploitation of common sparsity among frequency domain is also extended to the multi-user case in [24]. Nevertheless, a common assumption of the above mmWave channel estimation schemes is based on the perfect-precision ADCs at the receiver, which is costly and power consuming [25].

An alternative solution is to deploy low-resolution ADCs to reduce the power consumption and hardware implementation [6], [7], [26], [27]. The side-effect is that the channel estimation with finite-bit ADCs is challenging due to the quantization of received signals. When only one-bit ADCs are employed at the receiver, at the low SNR region, the power penalty is approximately $\frac{π}{2}$ (1.96 dB) and the loss of MIMO capacity is not severe [28]. The one-bit channel estimation issue in massive MIMO system was considered in [29], where the nonlinear one-bit quantizer operation was reformulated as an equivalent linear system with the aid of Bussgang decomposition [30]. To deal with the quantization effects in mmWave channel estimation, [31] proposed a modified expectation maximization (EM) algorithm to exploit the sparsity of narrowband channel, where the channel estimation problem was formulated as a one-bit CS problem. In [32], by taking the hybrid beamforming into consideration, a variant generalized approximate message passing (GAMP) algorithm was proposed, which achieves better performance than least-squared (LS) method without quantization. These two methods exhibit high complexity, however, since many iterations are required to guarantee the convergence. Additionally, only narrowband mmWave channel was taken into account. In [33], the channel estimation issue in broadband mmWave system model was investigated by exploiting the angle-domain sparsity. Then, the GAMP-based methods [34], [35] were utilized for channel estimation under the condition of few-bit ADCs at the receiver. The training sequence were also designed to achieve the low complexity and low peak-to-average power ratio. However, the assumption that each antenna at the receiver has a RF chain is impractical in practical mmWave system. In [36], both hybrid beamforming architecture and low-resolution ADCs were considered into the mmWave system to reduce power consumption, where the channel estimation becomes challenging. In particular, the quantization error was treated as additive white Gaussian noise. As a result, a large estimation error was introduced.

In this paper, the wideband channel estimation is investigated in hybrid mmWave MIMO system with finite-bit ADCs. Compared with the previous work, the system constraints, including both the hybrid architectures and low-resolution ADCs, are simultaneously considered in wideband mmWave system. Specifically, the channel estimation is reformulated as a sparse recovery problem, where the sparsity of wideband frequency selective channel is exploited in both the angular and the delay domains. Then, with the aid of the Bussgang decomposition, which accurately characters the quantization effect, the non-linear channel estimation problem can be transformed into a linear sparse system. The actual noise consists of combined noise and distortion noise, which are caused by the hybrid combiner and quantization operation, respectively. In a large scale regime, the actual noise is related to the number of quantization bits, antennas, received signal power. Based on the equivalent linear sparse model, a Bussgang decomposition-based OMP (BD-OMP) algorithm is proposed to both exploit the inherent sparsity of wideband mmWave channel and alleviate the quantization error. Additionally, the terminal condition is derived, where the expectation of residual difference is the variance of the actual noise.

Notations: Vectors and matrices are written in lower-case and upper-case boldface, respectively; ${| \cdot |}_{c}$ denotes the cardinality of a set, while ${‖ \cdot ‖}_{0}$ and ${‖ \cdot ‖}_{2}$ denote the $l_{0}$ norm and $l_{2}$ norm, respectively. The matrix transpose, conjugate, conjugate transpose and inversion are denoted by ${(\cdot)}^{T}$ , ${(\cdot)}^{⁎}$ , ${(\cdot)}^{H}$ and ${(\cdot)}^{- 1}$ respectively; The $(i, j)$ -th entry of matrix A and i-th entry of vector a are denoted by ${[A]}_{i, j}$ and ${[a]}_{i}$ , respectively; I is an identity matrix; $E [\cdot]$ defines the expectation operation; $j = \sqrt{- 1}$ is an imaginary symbol; $sign (\cdot)$ denotes the sign function. ⊙ and ⊗ denote the Khatri–Rao product and Kronecker product, respectively. $diag (A)$ extracts the diagonal elements of matrix A as a vector, while $vec (A)$ denotes the vectorization operation for matrix A. trace is the trace operation; $d i a g (\cdot)$ is the operation to force the non-diagonal elements to zeros.

Section snippets

Channel model

To characterize the wideband and limited scattering property of mmWave channel, a typical geometric channel model with L multipaths is considered. Specifically, the channel of dth delay tap is expressed as $H_{d} = \sum_{l = 1}^{L} α_{l} p (d T_{s} - τ_{l}) a_{R} (ϕ_{l}) a_{T}^{H} (θ_{l}),$ where $α_{l}$ , $τ_{l}$ , $ϕ_{l}$ and $θ_{l}$ are the complex channel gain, time delay, angle of arrival and angle of departure, of the lth channel path, respectively. $p (t)$ is the raised-cosine filter at the time t, which incorporates the effect of pulse shaping of receiver. $a_{R} (ϕ_{l})$

Sparse formulation

To formulate the channel estimation problem, by removing the notation of summation, the received signal before quantization (8) is re-formulated as $r_{m} [n] = \sqrt{ρ} W_{m}^{H} H {\tilde{F}}_{m} {\tilde{s}}_{m}^{n} + e_{m} [n],$ where $H = [H_{0} H_{1} \dots H_{D - 1}] \in C^{N_{R} \times D N_{T}}$ is the concatenation matrix of the D delay channels, ${\tilde{F}}_{m} = I_{D} \otimes F_{m} \in C^{D N_{T} \times D N_{s}}$ , ${\tilde{s}}_{m}^{n} = {[s_{m} {[n]}^{T} s_{m} {[n - 1]}^{T} \dots s_{m} {[n - (D - 1)]}^{T}]}^{T} \in C^{D N_{s} \times 1}$ is the concatenated transmitted signal within D channel delay. To perform the channel estimation at the receiver side, we collect N received symbols in (8). The combined signal

Performance analysis

In this section, the performance analysis of the proposed scheme is provided, which includes analysis of terminal condition, and the complexity analysis.

Simulation results

In this section, several simulations are presented to investigate the performance of the proposed scheme. Unless otherwise indicated, the system parameters are set as $N_{T} = 32$ , $N_{R} = 16$ , $N_{RF}^{T} = 4$ , $N_{RF}^{R} = 4$ , $G_{T} = 64$ , $G_{R} = 32$ , $L = 2$ , $N_{q} = 6$ , $D = 4$ , $M = 40$ , $N = 16$ , system bandwidth $B = 500 MHz$ , and noise variance is computed via $σ^{2} = 10^{(- 174 + 10 \log B) / 10}$ . For the channel parameters, the raised-cosine filter $p (t)$ is modeled as $p (t) = {\begin{matrix} \frac{π}{4} sinc (\frac{1}{2 β}), & t = \pm \frac{T_{s}}{2 β} \\ sinc (\frac{t}{T_{s}}) \frac{\cos (\frac{π β t}{T_{s}})}{1 - {(\frac{2 β t}{T_{s}})}^{2}}, & t \neq \pm \frac{T_{s}}{2 β}, \end{matrix}$ where $T_{s}$ is the sampling time and β is

Conclusion

A Bussgang decomposition-based wideband channel estimation formulation has been proposed for hybrid mmWave MIMO system with finite-bit ADCs, where both the hybrid architectures and low-resolution ADCs, are simultaneously taken into consideration. Specifically, the non-linear mmWave channel estimation problem is reformulated into a linear sparse signal recovery problem by accurately capturing the quantization effect. Then, a BD-OMP algorithm has been proposed to both exploit the inherent

Acknowledgement

The authors would like to thank Dr. Jianhua Mo for the discussion of mmWave MIMO system with low-resolution ADCs. This work was supported in part by the Fundamental Research Funds for the Center Universities under Grant No. HIT.MKSTISP.2016 13, in part by the National Natural Science Foundation of China under Grant 61671176 and China Scholarship Council under Grant 201706120126.

Ruoyu Zhang received the B.E. degree in communication engineering from Harbin Institute of Technology, Harbin, China, in 2014. He is currently pursuing the Ph.D. degree in electrical engineering from Harbin Institute of Technology. From Sept. 2017 to Aug. 2018, he was a visiting student in the Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, B.C., Canada. His research interests include compressed sensing, channel estimation, massive MIMO system.

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Jiayan Zhang received Ph.D. degree in communication engineering from Harbin Institute of Technology, Harbin, China, in 2008. He is currently an associate Professor with the Communication Research Center, School of Electronics and Information Engineering, Harbin Institute of Technology. His research interests include spread spectrum communication, high speed signal processing and channel coding.

Yulong Gao received Ph.D. degree in communication engineering from Harbin Institute of Technology, Harbin, China, in 2008. He is currently an associate Professor with the Communication Research Center, School of Electronics and Information Engineering, Harbin Institute of Technology. His research interests include cognitive radio, compressed sensing and statistical signal processing.

Honglin Zhao received the B.E. and Ph.D. degrees in communication engineering from Harbin Institute of Technology, Harbin, China, in 1991 and 1998, respectively. He is currently a Professor with the Communication Research Center, School of Electronics and Information Engineering, Harbin Institute of Technology. His research interests include spread spectrum communication, cognitive radio networks and wireless broadband networks.

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Bussgang decomposition-based sparse channel estimation in wideband hybrid millimeter wave MIMO systems with finite-bit ADCs☆

Abstract

Introduction

Section snippets

Channel model

Sparse formulation

Performance analysis

Simulation results

Conclusion

Acknowledgement

AEÜ, Int. J. Electron. Commun.

Signal Process.

Millimeter-wave cellular wireless networks: potentials and challenges

Proc. IEEE

An introduction to millimeter-wave mobile broadband systems

IEEE Commun. Mag.

Low RF-complexity technologies to enable millimeter-wave MIMO with large antenna array for 5G wireless communications

IEEE Commun. Mag.

Spatially sparse precoding in millimeter wave MIMO systems

IEEE Trans. Wirel. Commun.

Large-scale antenna systems with hybrid analog and digital beamforming for millimeter wave 5G

IEEE Commun. Mag.

Capacity analysis of one-bit quantized MIMO systems with transmitter channel state information

IEEE Trans. Signal Process.

Hybrid architectures with few-bit ADC receivers: achievable rates and energy-rate tradeoffs

IEEE Trans. Wirel. Commun.

An overview of signal processing techniques for millimeter wave MIMO systems

IEEE J. Sel. Top. Signal Process.

Beam codebook based beamforming protocol for multi-Gbps millimeter-wave WPAN systems

IEEE J. Sel. Areas Commun.

Millimeter wave beamforming for wireless backhaul and access in small cell networks

IEEE Trans. Commun.

Millimeter wave MIMO channel estimation using overlapped beam patterns and rate adaptation

IEEE Trans. Signal Process.

Channel estimation and hybrid precoding for millimeter wave cellular systems

IEEE J. Sel. Top. Signal Process.

Fundamentals of Wireless Communication

Channel estimation via orthogonal matching pursuit for hybrid MIMO systems in millimeter wave communications

IEEE Trans. Commun.

Efficient compressive channel estimation for millimeter-wave large-scale antenna systems

IEEE Trans. Signal Process.

Compressed channel sensing: a new approach to estimating sparse multipath channels

Proc. IEEE

Distributed compressed sensing aided sparse channel estimation in FDD massive MIMO system

IEEE Access

Channel estimation for millimeter-wave massive MIMO with hybrid precoding over frequency-selective fading channels

IEEE Commun. Lett.

Channel estimation for hybrid architecture-based wideband millimeter wave systems

IEEE J. Sel. Areas Commun.