A Two-stage Approach to Estimate CFO and Channel with One-bit ADCs

A Two-stage Approach to Estimate CFO and Channel with One-bit ADCs

Abstract

In this paper, we propose a two-stage approach to estimate the carrier frequency offset (CFO) and channel with one-bit analog-to-digital converters (ADCs). Firstly, a simple metric which is only a function of the CFO is proposed, and the CFO is estimated via solving the one-dimensional optimization problem. Secondly, the generalized approximate message passing (GAMP) algorithm combined with expectation maximization (EM) method is utilized to estimate the channel. In order to provide a benchmark of our proposed algorithm in terms of the CFO estimation, the corresponding Cramér-Rao bound (CRB) is derived. Furthermore, numerical results demonstrate the effectiveness of the proposed approach when applied to the general Gaussian channel and mmWave channel.

CFO, channel estimation, millimeter wave system, one-bit quantization

I Introduction

To provide a high-speed data rate in celluar systems, the mmWave multiple input multiple output (MIMO) system has been proposed as the key technology of the fifth generation (5G) cellular system [1]. Because of the larger bandwidths that accompany mmWave, the cost and power consumption are huge due to high precision (e.g., 10-12 bits) analog-to-digital converters (ADCs) [2]. As a result, a low precision (e.g., 1-4 bits) ADC is employed to relieve this ADC bottleneck [1, 3, 4]. However, as low precision quantization is severely nonlinear, traditional algorithms designed for high precision systems can not be applied directly because of significant performance degradation. As a consequence, new signal processing algorithms dealing with channel estimation and transmit precoding have been proposed, which work well in systems with low precision ADCs [5, 6, 7]. For the channel estimation in mmWave systems, it can be regarded as one-bit compressed sensing (CS) problems [8, 9], as the mmWave MIMO channel is approximately sparse in angle domain [10]. Therefore, many CS-based algorithms have been proposed to estimate the mmWave MIMO channel. In [11, 12], a modified expectation maximization (EM) algorithm and approximate message passing (AMP) algorithms are utilized to solve the channel estimation problem in mmWave MIMO systems.

In practice, the carrier frequencies between the local oscillators at the TX and the RX can be mismatched, which results in carrier frequency offset (CFO) impairing the phase of the channel measurements in systems. One approach to dealing with the above problem is to correct the CFO before channel estimation, which is impractical because the mmWave systems always work at low SNR prior to channel estimation [14]. As a result, several works have studied the joint CFO and channel estimation [14, 13]. In [13], a generalized AMP (GAMP) algorithm to jointly estimate the CFO and channel in mmWave narrowband systems with one-bit ADCs is developed. It utilizes a lifting technique which increases the problem’s dimension. In [14], an algorithm called PBiGAMP is proposed to jointly estimate CFO and wideband channel, which has a much lower computational complexity.

In this letter, we propose a two-stage approach to estimate the CFO and channel with one-bit ADCs. Firstly, we utilize Bussgang decomposition theorem which transforms the non-linear model into a linear model [15], and the CFO is estimated via solving the one-dimensional optimization problem. Secondly, by fixing the CFO with the estimated CFO, we apply the GAMP-EM algorithm [16, 17] to estimate the channel. Besides, the CRB is also derived for evaluating the performance of our algorithm in terms of CFO estimation. One appealing advantage of the proposed method is that both the CFO and channel can be estimated accurately without increasing the problem’s dimension. Numerical results show the effectiveness of the proposed two-stage approach, i.e., the estimation performance degradation in terms of the CFO and channel of the proposed method is marginal, compared to the benchmarks such as the CRB and the CFO-known (oracle) algorithm.

Notation: Let lowercase boldface letters like denote a vector and capital boldface letters like denote a matrix. For a vector , let denote a matrix whose diagonal is composed of . For a square matrix , let denote a vector whose elements are the diagonal elements of . Let and denote the real and imaginary part of , and and denote the real and imaginary part of . represents the th element of . We use the notation and to represent the transpose and conjugate transpose of . denotes a vector obtained by stacking all the columns of . The symbol denotes the kronecker product.

Ii Algorithm

In this section, the problem model and algorithm are introduced. Consider a MIMO system with one-bit ADCs and let denote the CFO. For a training block , the observation obtained at ADCs is

(1)

where is the channel matrix, is an element-wise quantization function given by with being the signum function, is the Vandermonde vector given by and is the additive white Gaussian noise with . We aim to estimate the CFO and channel based on the observation and the training block .

At the beginning, we reformulate model (1) to a real-valued form. Utilizing the property , it can be transformed to a vector form firstly as

(2)

where , , , and . By defining

(3a)
(3b)

we obtain a real-valued equivalent model

(4)

where and .

Ii-a CFO Estimation

Before performing the channel estimation, we estimate the CFO at first. We assume that the prior distribution of follows and we use the method proposed in [15] to linearize the model as

(5)

where is the linearization matrix and is a residual error vector consisting of noise and linearization artifacts. According to [15], is calculated as

(6)

where

(7)

To estimate the CFO, we maximize the expected energy (taken with respect to ) of the output of the matched filtering of the observation , which can be expressed as

(8)

Assuming and omitting the constant coefficient, (8) can be simplified as

(9)

Furthermore, for an independent and identically distributed (iid) QPSK training block , the optimization problem (9) can be simplified further. First, we rewrite as

(10)

From equation (6), we extract the diagonal elements of and using the property , we obtain

Recall that

where denotes the th column of . The diagonal elements of are

(11)

where

(12)

For an iid QPSK training block which takes values in , is equal to . Therefore, is simplified as , and is simplified as . As a result, the optimization problem (9) is further simplified as

(13)

To solve the problem (9) or (13), we adopt two steps [18, 19]: Detection and Refinement.

Detection: The Detection step includes coarse detection and refined detection. Firstly, we solve the optimization problem (9) or (13) and obtain a coarse estimate of by restricting it to a discrete set denoted by . Secondly, we implement a refined detection over the frequencies around . We solve the same problem again, but restrict to the discrete set this time, and finally update as . We found that and work well for a large number of problems. Due to page limitations, we refer interested readers to the supplementary materials for more details about the parameters and .

Refinement: Numerical results show that problem (9) or (13) is locally concave around the global optimum. As a result, the estimate given by the Detection step is used as an initial point and the gradient descent algortihm is performed to refine the estimate as .

Furthermore, in order to evaluate the performance of the proposed approach for CFO estimation, the CRB of CFO (21) is derived in Appendix.

Ii-B Channel Estimation

The channel estimation problem can be transformed to a general model

(14)

and the GAMP algorithm can be directly applied with the prior . If some nuisance parameters of the prior are unknown, GAMP-EM can be used to jointly learn the nuisance parameters and estimate [17]. The implementation details about the GAMP-EM algorithm to model (14) can be found in [20]. In our channel estimation problem, we set and consider two kinds of channel: The general Gaussian channel and the mmWave channel. And more information about how to transform the corresponding problem model of two different channels to model (14) are provided below.

General Gaussian Channel

For the general Gaussian channel, the channel matrix follows a zero-mean Gaussian distribution, i.e., with being unknown. We apply the GAMP-EM algorithm directly on . Therefore, the corresponding and in model (14) are and in the model (4), respectively. For the denoising step in the GAMP algorithm, we denoise the noisy signal with the prior of being Gaussian.

mmWave Channel

A narrowband mmWave channel can be modeled by a ray-based model [13]. For a propagation environment having clusters and rays in the th cluster, the channel matrix is described as

Here, , and are the complex gain, angle-of-arrival and angle-of-departure of the th ray in the th cluster, respectively. and denote the carrier wavelength and antenna spacing.

For the mmWave MIMO channel, its beamspace representation of channel matrix is

(16)

where and are unitary Discrete Fourier Transform matrices. Since the mmWave MIMO channel is approximately sparse in angle domain, in (16) is a sparse matrix. We assume that follows the Bernoulli-Gaussian distribution and apply the GAMP-EM algorithm on instead of , where and . In this case, we can obtain a model similar to model (2), where and . Then follow the similar steps through transforming model (2) to (4), a real-valued equivalent model is obtained. Therefore, for the GAMP-EM algorithm, the corresponding and in model (14) are and , respectively.

Iii Numerical Simulation

In this section, the performance of the proposed algorithm is evaluated by applying to both the general Gaussian channel and mmWave channel, which is measured by the mean square error (MSE) of CFO estimate and the debiased MSE (dMSE) of channel estimate (). Meanwhile, the CRB of CFO (21) is also plotted. In our simulations, the MSE of CFO estimate is denoted by and the dMSE of channel estimate is denoted by for the Gaussian channel and for the mmWave channel, where is . An iid QPSK training block is used in our experiments and the system parameters are set as follows: , , . We consider a symbol rate and a CFO kHz, and we choose , where the we choose is maximally off the grids in Detection step. We set to satisfy , where . All the results are averaged over Monte Carlo (MC) trials.

Iii-a General Gaussian Channel

In this experiment, the MSE of CFO and the dMSE of channel are compared with the corresponding CRB and CFO-known algorithm, respectively. The results are presented in Fig. 1 and Fig. 2. In Fig. 1, the MSE of CFO decreases as the length of training block increases or the SNR increases. And the performance gap between the MSE and CRB is less than about dB. Fig. 2 presents the dMSE of channel with unknown CFO and known CFO respectively. It can be seen that the dMSE of channel decreases when or SNR increases. And the dMSE of channel with unknown CFO is close to that with known CFO.

Fig. 1: The MSE and CRB of the CFO versus the length of training block with different SNRs for the general Gaussian channel.
Fig. 2: The dMSE of the channel versus the length of training block with different SNRs for the general Gaussian channel.

Iii-B mmWave Channel

For the mmWave channel, we set the parameters of channel as follows: , and . We generate and from a laplacian distribution with an angle spread of 10 degrees [13]. The results of the experiment for mmWave channel are presented in Fig. 3 and Fig. 4.

From Fig. 3 and Fig. 4, we can see that the performance of our proposed approach for mmWave channel is similar to that for general Gaussian channel, which demonstrates that the proposed approach is effective for the mmWave channel.

Besides, the running time of the proposed approach to CFO and channel estimation in both the general Gaussian channel and mmWave channel is shown in TABLE I. The value in parenthesis represents the running time of the algorithm for mmWave channel.

Fig. 3: The MSE and CRB of the CFO versus the length of training block with different SNRs for the mmWave channel.
Fig. 4: The dMSE of the channel versus the length of training block with different SNRs for the mmWave channel.
Estimation
CFO est. 1.1(1.1) 4.5(4.5) 9.8(9.8)
Channel est. 0.1(0.3) 0.3(1.2) 0.6(2.2)
Total 1.2(1.3) 4.8(5.6) 10.4(12.0)
TABLE I: The averaged running time (seconds) of the proposed approach to estimate CFO and channel.

Iv Conclusion

We have designed a two-stage approach to estimate CFO and channel with one-bit ADCs, and derived the CRB of CFO. Numerical results demonstrate that the proposed approach works well for both the general Gaussian channel and mmWave channel, and the gap between the MSE and CRB of CFO is less than about 3dB. Compared to the CFO-known GAMP-EM algorithm, the performance degradation of the proposed approach is negligible.

V Appendix

In this part, the details about the calculation of the CRB of CFO are presented. First we start from the problem model (4) and to be more concretely, the two parts and of matrix are

where the th element of and are and , for . Let denote the th row of , the likelihood function is and the corresponding log-likelihood function is given by

(17)

By defining , we calculate

(18a)
(18b)
(18c)

where is the th row of and is

(19)

Here,

where the th element of is for . Let , the Fisher Information Matrix (FIM) is

(20)

where

and is a diagonal matrix with elements . The CRB is equal to . For the CFO, its CRB is

(21)

Vi Supplementary Material

In this supplementary material, the way to empirically choose the parameters (the number of grids in coarse detection) and (the number of grids in refined detection) is presented. For the convenience, the parameters of the numerical experiments in this material are set the same as that in our letter for the general Gaussian channel and SNR is chosen to be 10dB.

(a)
(b)
(c)
Fig. 5: The curve of the objective function with different . The red dash line represents the true CFO .

Let , Fig. 5 plots the curves of with different . From Fig. 5, we can see that there is an obvious main lobe and it is locally concave around the global optimum. Therefore, as long as is large enough, we can detect a value of on the main lobe which can make the gradient descent algorithm converge to the global optimum. However, in order to reduce the computational complexity, we choose reasonably which works well for the Detection step.

Fig. 6: The width of the main lobe versus the length of training block .

Through a large number of experiments, we found that the width of the main lobe depends mainly on . Fig. 6 presents the width of the main lobe with different . It can be seen that the width of the main lobe decreases when increases and the smallest width of the main lobe is about 0.025. Thus, should be at least larger than . However, we can see that there are many side lobes close to the main lobe, and meanwhile the main lobe in figures may also contain invisible side lobes, which may degrade the detection performance. Therefore, we design a two-step Detection: coarse detection and refined detection, and choose and for the simulations.

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