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The bandwidth shortage has motivated the exploration of the millimeter wave (mmWave) frequency spectrum for future communication networks. To compensate for the severe propagation attenuation in the mmWave band, massive antenna arrays can be adopted at both the transmitter and receiver to provide large array gains via directional beamforming. To achieve such array gains, channel estimation (CE) with high resolution and low latency is of great importance for mmWave communications. However, classic super-resolution subspace CE methods such as multiple signal classification (MUSIC) and estimation of signal parameters via rotation invariant technique (ESPRIT) cannot be applied here due to RF chain constraints. In this paper, an enhanced CE algorithm is developed for the off-grid problem when quantizing the angles of mmWave channel in the spatial domain where off-grid problem refers to the scenario that angles do not lie on the quantization grids with high probability, and it results in power leakage and severe reduction of the CE performance. A new model is first proposed to formulate the off-grid problem. The new model divides the continuously-distributed angle into a quantized discrete grid part, referred to as the integral grid angle, and an offset part, termed fractional off-grid angle. Accordingly, an iterative off-grid turbo CE (IOTCE) algorithm is proposed to renew and upgrade the CE between the integral grid part and the fractional off-grid part under the Turbo principle. By fully exploiting the sparse structure of mmWave channels, the integral grid part is estimated by a soft-decoding based compressed sensing (CS) method called improved turbo compressed channel sensing (ITCCS). It iteratively updates the soft information between the linear minimum mean square error (LMMSE) estimator and the sparsity combiner. Monte Carlo simulations are presented to evaluate the performance of the proposed method, and the results show that it enhances the angle detection resolution greatly.

Thanks to the large bandwidth available at millimeter wave (mmWave) frequencies, mmWave communication technology has become a promising technology to meet the experientially increasing demands of future wireless networks [

Recently, many research efforts have been devoted to the CE for the mmWave systems [

In this paper, we study the off-grid problem and propose an enhanced method to improve the resolution of angle estimation in mmWave systems with massive antenna arrays and RF chain constraints. Firstly, we formulate the signal model to describe the off-grid problem in the angle quantization of the mmWave channel, which decomposes each continuously-distributed angle into the integral grid part and fractional off-grid part. Accordingly, an iterative off-grid turbo CE (IOTCE) algorithm is developed to renew and upgrade the CE between the integral grid and fractional off-grid angle information under the Turbo principle. By fully exploiting the sparse structure of mmWave channels, a soft-decoding based CS method, termed improved turbo compressed channel sensing (ITCCS), is developed to estimate the integral grid angle information, which iteratively updates the soft information between the linear minimum mean square error (LMMSE) estimator and the sparsity combiner. Simulation results show that the proposed CE method can achieve a higher resolution than the existing ones.

The rest of the paper is organized as follows. The mmWave system model is introduced in

According to [

In this paper, we also assume:

uniform linear arrays (ULAs) with half-wavelength spacing are deployed at both the BS and MS;

far-field scattering and block-fading are held, which means the signal waves arrive at different antennas with the same fading amplitudes but distinct phases and the channel fadings are kept constant during the CE precedure.

Under these assumptions, the single-path channel between the BS and MS can be expressed as:

The

In order to suppress the negative effect caused by the off-grid problem, a specific and operable virtual channel representation model is introduced. From this virtual model, an enhanced off-grid angle estimation model is developed.

The continuously-distributed AoA can be written as:

Accordingly, the channel matrix in Equation (2) can be reformulated as:

Then, the virtual representation model of the mmWave channel with continuously-distributed angle can be written as:

In order to facilitate the operation of the proposed CE algorithm, the codebook designed in [

In this way, the received signal at the MS can be written as:

Stacking

From Equation (17), the CE is accomplished by estimating the integral virtual channel, which, in turn, defines integral grid angle, and the fractional off-grid angle matrix, which stands for the fractional off-grid angle. As shown in

Equation (17) degenerates into a CS recovery algorithm with

Let

As illustrated in

Therefore, by fixing

By fixing the integral virtual channel

According to the proposed iteration, the IOTCE algorithm is illustrated in Algorithm 1, which presents a structure of a nested loop. In addition, the corresponding explanation of parameters in the IOTCE algorithm is listed in

Received signal

Sampling matrix

Estimate of

Estimation of

(1) Fixing

(1) Fixing

(2) Calculating

output the estimated value.

Then, and turn to Step 2.

In this section, we evaluate the performance of the proposed algorithm in the mmWave communication systems. The simulation system consists of a BS and an MS, and the same ULA with

The simulation results of the average angle estimation error (AAEE), defined as ^{−2} at SNR

The average probability of integral grid point estimation error (APIEE) performance, which is defined as ^{−2}, the algorithm with

The shortage of spectrum at present will be alleviated by advances in the mmWave communication systems, and directional precoding/beamforming with large antenna arrays appears to be inevitable to support longer outdoor links and to provide sufficient received signal power than before. In this paper, an enhanced CE algorithm is developed for the off-grid problem in the mmWave systems with massive antenna arrays and RF chain constraints. First, we developed an off-grid formulation to catch the quantization problem. Then, an iterative method, which depends on the developed off-grid formulation is proposed to renew and upgrade the CE between the integral and fraction angle information under the Turbo principle. Simulation results show the efficiency and high resolution of the proposed method. For future work, it would be more practical to consider the multi-paths channel with independently but not identically distributed paths, and develop an efficient method to calculate the Log Likelihood Ratio (LLR) information.

The work of Yuexing Peng was supported by the NSFC under grant 61171106, the National Key Scientific Instrument and Equipment Development Project under Grant 2013YQ20060706 and the 863 Program with grant 2014AA01A705. The work of Yonghui Li was supported by the ARC grants DP150104019, LP150100994, and FT120100487 as well as by funding from the Faculty of Engineering & Information Technologies, The University of Sydney, under the Faculty Research Cluster Program.

This article is the product of joint efforts. All the authors contributed to the conception, design and performance of the experiments, the analysis of the data and the writing of the paper. L. Han and Y. Peng conceived and designed the algorithm; P. Wang and Y. Li discussed and provided valuable comments on the algorithm design; L. Han performed the experiments and wrote the paper; Y. Peng, P. Wang and Y. Li revised the paper.

The authors declare no conflicts of interest.

Block diagram of structure with beamforming at the base station (BS) and combining at the mobile station (MS).

Flow chart of the iterative off-grid turbo channel estimation algorithm.

Flow chart of the improved turbo compressed channel sensing algorithm. FFT and IFFT denote the fast Fourier transform processing and the inverse transform processing, respectively. The modules

The average angle estimation error (AAEE) versus signal-to-noise ratio (SNR) for the proposed algorithm for different values of

The average probability of integral grid point estimation error (APIEE) versus signal-to-noise ratio (SNR) for the proposed algorithm for different values of

Explanation of parameters in the iterative off-grid turbo channel estimation algorithm.

Parameters | Explanation |
---|---|

the number of antennas at receiver | |

the integral virtual channel vector | |

the combining matrix at receiver | |

the fractional off-grid angle | |

the corresponding fractional off-grid angle |