Channel Estimation for mmWave Massive MIMO Systems With Mixed-ADC Architecture

Millimeter wave (mmWave) communications are widely preferred due to the rich bandwidth and potentially huge spectrum resources. Nowadays, mixed-ADC architecture combined with mmWave massive MIMO has become a communication mainstream, which can effectively solve the issue of high total power consumption and cost of base station (BS) circuits. However, the channel estimation problem for mmWave massive MIMO systems with mixed-ADC architecture has not been studied yet. In this paper, we develop the sparse channel estimation method on this framework. Specifically, by exploiting the sparsity of mmWave channels, the beamspace channel estimation problem can be transformed into a sparse matrix recovery problem, the channel parameters are recovered using compressive sensing (CS) techniques. Simulation results show that the algorithms quantized by the mixed-ADC outperforms the low-resolution ADC, and the best performance can be achieved when the low-resolution ADC in the mixed-ADC architecture reaches five-bit.

the number of ADCs and radio frequency (RF) chains in the system can be effectively reduced thus achieving a hardware cost balance. The second solution is to use low-resolution ADCs instead of high-resolution ADCs for quantization [5], [6], [7], which can directly reduce the power loss and hardware cost of the system, but quantization noise is an unavoidable problem.
Channel estimation has been a popular research topic in the field of mmWave communication, there have been numerous channel estimation algorithms during this period. From the perspective of the types of beamforming architectures, the channel estimation schemes for mmWave systems can be divided into hybrid beamforming structure equipped with the ideal state (high-resolution) ADC for quantization [8], [9], [10], [11] or fully digital beamforming structure equipped with low-resolution ADC for quantization [12], [13], [14]. For the full digital beamforming structure, that is, each antenna in the system is equipped with a corresponding RF chain, which allows flexible and high quality generation of multiple independently controllable beams, yet the number of antennas is usually limited considering the hardware cost. The hybrid beamforming structure is composed of a low-dimensional digital beamforming device with fewer RF chains and an analog beamforming device with an analog phase shifter, which is mostly used in the scenarios with a large amount of antennas, and solves the problem of high transmission loss in the millimeter band. In addition, considering the impact of quantization noise on signal transmission when equipped with a low-resolution ADC, in the references [15], [16], [17], [18], the authors proposed a mixed-ADC architecture. Specifically, the antennas is divided into two parts, most of the antennas are equipped with the low-resolution ADCs and a few antennas are equipped with the high-resolution ADCs, which effectively reduces the impact of quantization noise. This hybrid architecture makes economic sense because it significantly reduce hardware costs and power consumption during this period.
Due to the sparsity of millimeter-wave channel, compressed sensing technology naturally becomes a channel estimation method [19], [20], [21], [22], [23]. For mmWave massive MIMO systems, the authors in [19], [20] proposed the OMP algorithm for signal recovery based on compressed sensing. The work of [21] proposed an efficient CS channel estimation method, which is based on time-domain and frequency-domain methods [22], [23]. The authors proposed a CS-based channel estimation scheme for full-dimensional lens arrays [24], and the experiments demonstrated the effectiveness of the scheme. Since the neural network can build nonlinear complex relationship models and has strong learning ability. It is become a popular method using artificial intelligence to estimate channel and to detect signal, and so on [25]. The authors proposed a super-resolution channel estimation scheme that uses iterative weighting to make the estimated arrival/departure angle to be the optimal solution [26]. The authors proposed a better real-time CSI feedback architecture by extending the CSI sensing and recovery network [27]. Meanwhile, the deep learning (DL) technology has also made rapid development and has been applied in various fields, including communications field. The authors of [28] combined signal recovery iterative algorithms and DL techniques to design a learned denoisingbased approximate message passing (LDAMP) network. In addition, the work proposed a DL based mmWave channel estimation and tracking algorithm [29], which reduces the system overhead. However, all these studied were based on high-resolution ADCs, which bring about the potentially excessive power dissipation.
The intelligent reflective surface (IRS) is a hot topic of interest in communications in recent years. IRS consists of a low-complexity control circuit and a large number of passive reflective units, that can reconfigure the propagation environment by intelligently adjusting the amplitude and phase of the incident signals. For the channel estimation problem of IRS-assisted communication, the researchers proposed a channel estimation scheme based on cell on/off state control [30], [31], which can enable the user's reflected channel to be estimated and not suffer the interference from the reflected signals of other IRS elements. However, the cost is expensive to implement a large number of element on/off switches, as it requires individual control of each IRS element. Considering the system overhead problem, some researchers have proposed cascaded channel estimation schemes. The authors of [32] proposed a deep denoising neural network-assisted CS wideband channel estimation method to reduce the training overhead. The authors proposed a twostage algorithm based on sparse matrix decomposition and perfect matrix [33]. In [34], the authors proposed a joint multi-user channel estimation scheme that is based on a two-step process.
To the best of our knowledge, there are more studies in the literatures on the performance analysis of communication systems with mixed-ADC architecture, while the channel estimation problem has not been studied yet, so this paper can is cable to realize the effect of mixed ADC on channel estimation. In this paper, we propose an sparse channel estimation algorithms based the sparsity of the channel. Under the low-resolution quantization, we reconstruct the sparse channel by using orthogonal matching pursuit (OMP) technique. Simulation results show that the OMP algorithm outperforms the least squares (LS) under the same configuration conditions, and the algorithm with the mixed-ADC architecture outperforms the low-resolution ADC.
Notation: We provide explanatory notes for some symbols in the paper, where bold letters indicate vectors and bold uppercase letters indicate matrices. For any matrix B, vec(B) method means that the input matrix B is arranged in columns to form a column vector, and B i (or B(i)) represents the i th entry of B. B T , B H , B represent B's transpose, conjugate transpose, complex conjugate, respectively. In addition, diag(·) means to extract the diagonal elements of the square matrix and create a column vector, I is the unit matrix, • denotes the Khatri-Rao product operator, ⊗ denotes the Kronecker product, and · 2 represents the Euclidean norm.

II. SYSTEM MODEL
We consider a mmWave massive MIMO system equipped with a hybrid analog and digital beamforming structure, where N t and N r antennas are provided at the transmitter and receiver terminals, separately. N s data streams are transmitted at both ends via N RF RF chains and satisfy N RF ≤ min(N r , N t ), as shown in Fig. 1. In contrast to the previous all-digital structure, we used a mixed ADC architecture at the receiver side. For the hybrid beamforming structure, we assume that the RF precoder (F RF ∈ C N t ×N RF ) and combiner (W RF ∈ C N r ×N RF ) are implemented with a network of phase shifters. At the m th time, the transmitter first preencodes the data symbol through the baseband preencoder F BB , and then uses the RF preencoder F RF to construct the transmission signal and transmit it to the receiver. To simplify the formula, we use a hybrid precoder F denote F RF F BB ∈ C N t ×N s , F BB represents the baseband processing matrix of the transmitter, and W denote W RF . For the received signal in the m th frame before quantization, we can express it as where ρ indicates the transmitting power, s ∈ C N s ×1 denotes the training symbol vector, satisfying E[ss H ] = 1 N s I. n m ∈ C N r ×1 is the noise vector, and H d ∈ C N r ×N t is the flat-fading MIMO channel.

A. MIXED-ADC ARCHITECTURE
We consider a mixed-ADC architecture equipped at the receiver side. Specifically, the antennas can be divided into two parts, in which the N 0 = N r antennas are attached to the high-resolution ADCs and the N 1 = (1 − )N r antennas are attached to the low-resolution ADCs, the coefficient represents the number of antennas attached to the highresolution ADC at the receiver as a percentage of the total number of antennas. For computational convenience, the antenna numbers N 0 and N 1 are set to integers. For the n th element of the received signal quantized in the m th frame, which can be expressed as where Q(·) denotes the quantization operator of the signal.
In (2), we assume that the signal via the high-resolution ADC is unaffected by quantization noise, so the overall received signals can be distinguished as where Y 0 is the received signals quantized by the highresolution ADCs,Ỹ 1 is the received signals quantized by the low-resolution ADCs, and Y 1 isỸ 1 before quantization. The input signals are subject to noise during the transmission due to interference, where Gaussian noise demonstrates good mathematical ease of handling, so we adopt the additive quantization noise model (AQNM) to describe the n th received signal through the low-resolution ADCs: where α ∈ (0, 1) is a linear gain that is positively correlated with the number of quantization bits of the ADC, q is the additive Gaussian noise vector.

B. CHANNEL MODEL
Considering the geometric channel model of a mmWave channel with multiple paths, the channel matrix H can be modeled as where P is the effective number of paths in the transmission process, α p denotes the complex gain of the p th path, φ p and θ p are the arrival and departure angle of the p th path, respectively, and a R (a T ) denotes the array response vector at the receiver (transmitter). We assume that the antenna arrangement conforms to the relatively simple uniform linear array (ULA) form, then the array response vector at both sides can be expressed as where λ is the signal wavelength, d is the antenna spacing and satisfies d = λ 2 . Formula (5) can be expressed in a more concise form as where is a diagonal matrix of size P × P, G r ∈ C N r ×P and G t ∈ C N t ×P are matrices containing a R (φ p ) and a T (θ p ) in the columns, respectively.

C. LS (LEAST SQUARE)
The LS is a non-blind channel estimation algorithm, which can be operated without the known pilot sequence in the wireless system. Therefore, the LS algorithm requires fewer factors to be considered and is relatively simple to implement. However, the LS algorithm ignores the impact of noise, which leads to a degree of inaccuracy from the estimated channel information to the actual channel, especially in low SNR scenarios. To show the derivation process of the LS algorithm more clearly, the general received signal model can be defined as where A is the channel matrix whose elements are distributed independently, x is the transmit signal and v is the noise vector. The principle of LS estimation is to estimate A which makes the "distance" between Ax and y minimum, even if the following cost functions are minimum: Let the above cost function have zero partial derivatives with respect to A, The channel estimation of LS algorithm can be given by The estimated channel can be obtained using the LS algorithm. Since the channel element of mmWave has strong sparse characteristics, further reduction of the estimation error can be attempted by the OMP algorithm.

III. CHANNEL ESTIMATION ALGORITHMS
This section presents the compressed sensing technology that applies the traditional algorithms to estimate the effective channel state information by taking advantage of mmWave sparsity.

A. PROBLEM DESCRIPTION
Compressed sensing is a signal sampling technique that enables reconstruction of the original signal from the receiver at low sampling rates and largely reduces the complexity of the acquisition side. The sensing process can be divided into three main steps: The sparse representation of the signal to achieve compression of the signal; the design of the observation matrix to obtain the observed values; and the reconstruction of the signal to obtain the recovered signal.
Assuming that a high-dimensional signalx ∈ R N×1 can be mapped to a low-dimensional space by an orthogonal basis ∈ R N×N , then the sparse representation of the signalx can be written asx where the sparsity of the signalx is K,ŝ ∈ R N×1 is called the sparse coefficient and is also a sparse column vector, y ∈ R M×1 is a one-dimensional known measurement of length M, which can be expressed aŝ where denotes the measurement matrix, satisfying that each element of the matrix is independently and identically distributed over a particular distribution, = is the sensing matrix. Applying the OMP to calculate, the received signal before quantization in the m−th frame in the equation (1) can be simplified to as where N m = W H m n m . By vectorizing the channel matrix in formula (8), the vectorized concatenated channel matrix H can be obtained where the AoA's and AoD's are unknown in above formula.
To solve the compressed sensing problem, we can utilize the virtual channel method to decomposition (16) as where U T ∈ C N t ×G t and U R ∈ C N r ×G r are discrete Fourier transform (DFT) matrices, and h denotes the transformed sparse vector possessing the diagonal elements of . By bringing (17) into (15), we can rewrite the vectored received frame signal as Then the received signal within the overall M frames range can be superimposed as where = (U T ⊗ U R ) is the sparse basis matrix, is the measurement matrix, which can be expanded as follows (20) and N = [N 0 , N 1 , . . . , N M−1 ] . The perception matrix and the received signal in the above formula are known, so the CS estimate h can be obtained.

B. CHANNEL ESTIMATION BASED ON OMP
The OMP is a classical greedy algorithm in the compressive sensing field, which finally achieves signal reconstruction by continuously selecting the atom that is most relevant to the current residual. Due to the orthogonalization of the selected atoms during the execution of the algorithm, the number of measurements and iterations are effectively reduced. The specific steps are as follows: Finding the iteration index based on the maximum inner product value.
where r 0 = y is defined as the initial residual and y is the observation vector, θ i represents the i th column of the sensing matrix ∈ C M×N . Add the most relevant column index to the index set k , and simultaneously updating the data corresponding to the index in the sensing matrix to the reconstructed atomic set θ k .
The LS method is used to find the approximate solution.
Update the residual, we have By repeating the above process until the number of iterations k = K, eventually obtaining the reconfiguration vector s K .

C. COMPLEXITY ANALYSIS
In this section, we analyze the computational complexity associated with the LS algorithm and OMP algorithm in channel estimation, respectively. The least squares method requires the calculation of the pseudo-inverse of the transmitted signal x, which is more complicated. Assuming that x is an N-dimensional vector, the corresponding computational complexity is O (MN 3 ). Setting the channel sparsity to K, the computational complexity of the OMP algorithm to find the maximum relevant atoms in each iteration is O (MN).
In addition, the computational complexity of the iterative Algorithm 1 The OMP Algorithm for Recovering Sparse Signals Require: The observation vector y, the sensing matrix , and the sparsity K. Ensure: r 0 = y, k = 1, 0 = ∅.
1: while k ≤ K do 2: k = k + 1 8: end while 9: return The reconfiguration vector s K . process is O(K 3 ) after ignoring some low-complexity operations. Therefore, the total computational complexity of the OMP algorithm is O(K 3 ) + O(KMN).

IV. NUMERICAL RESULTS
In this section, we evaluate the performance of the OMP and LS algorithms for different ADC quantization by numerical simulations of the following parameters. We consider a mmWave system with N t = 32 transmitter antennas and N r = 16 receiver antennas, which is equipped with 4 RF chains for signal transmission and satisfies N RF = N s . The remaining parameters are as follows: = 0.2, G t = 64, G r = 32, N c = 4, P = 2 and M = 80. We will explain if there is any change of parameters in the simulation.
We utilize the normalized mean square error (NMSE) as the performance measure of the algorithm in this simulation: whereĤ denotes the channel recovered by the algorithm. Fig. 2 shows the effect of different bits of ADC on the performance of the OMP and LS algorithms in a certain SNR range. It can be observed from the figure that under the one-bit ADC quantization condition, it is difficult for both algorithms to reduce the estimation error when the SNR is greater than 10 dB, while the performance of the algorithms under the four-bit ADC quantization condition is much better. In addition, the OMP algorithm performed better than LS at every resolution in the figure, and we found that the LS algorithm with four-bit resolution performed worse than the OMP algorithm with two-bit resolution. Fig. 3 depicts the performance comparison of the LS and OMP algorithms in mixed-ADC architecture. Compared with the LS algorithm and OMP algorithm in one-bit ADC, the OMP algorithm in mixed-ADC has better performance. However, we found that there was not a big performance gap between the two. Because in the mixed-ADC architecture we originally designed, the proportion of high-resolution ADC  is very small, and most of them used the pure one-bit resolution ADC architecture. In subsequent experiments, we test the mixed-ADC architecture with combinations of different resolutions.
The performance of the OMP algorithm under one-bit ADC and four-bit ADC quantization is plotted in Fig. 4 and Fig. 5 Fig. 4 is that the NMSE gradually decreases in the process of SNR from −20 to 0 dB, while the SNR curve increases in a reverse direction from 0 to 15 dB, and the same variation occurs in Fig. 3, which is a phenomenon called stochastic resonance [35]. Noise is often considered a disadvantage during signal analysis. However, the presence of noise can enhance the detection of weak signals in some specific nonlinear systems, such as the pure one-bit resolution ADC architecture.    5 shows that the channel estimation performance using 100 frames is better than that using 10 frames, and the estimation error can also be effectively reduced by increasing the number of RF chains. Therefore, for devices that can only be equipped with a single RF chain, it is possible to try to increase the number of frames to optimize the algorithm performance, and the results are even better than the case with four RF chains.
We define the number of bits of the low-resolution ADC in the mixed-ADC architecture as ADC low . Fig. 6 shows the effect of different ADC low on the performance of the algorithms. We can see that the estimation error of both algorithms gradually reduces as ADC low increases, but adding the number of bits of ADC low after ADC low > 5 does not have much impact on the overall performance, so we judge that when the low-resolution ADC reaches 5 bit, the mixed-ADC enables maximum balance of performance and hardware cost. In addition, under the comparison of 20 frames and 80 frames, the performance gap of LS under different frames is much larger than that of the OMP algorithm, and the  estimation error of the LS algorithm at 80 frames is even higher than that of the OMP at 20 frames. Therefore, we can determine by comparison that the OMP algorithm still has better performance even at low frames.
We derive the ADC low value required to achieve the most balanced mixed-ADC from the above simulation results. Fig. 7 compares the estimation error of the high-resolution ADC with the mixed-ADC at ADC low = 5 for the OMP algorithm, and considers the case of different SNRs. From the figure we can clearly see that the difference between the mixed-ADC and the high-resolution ADC is already small, which reflects the reasonableness of the mixed ADC. Generally speaking, the OMP algorithm can achieve lower estimation error by using more frames, but when the SNR becomes large enough, increasing the number of frames does not necessarily guarantee better estimation performance. For example, under the condition of SNR = 15 dB, the estimation error is not dramatically changed when the number of frames is raised from 50 to 100, and the error even increases at the 100 frames, so we can consider other factors to further reduce the error.

V. CONCLUSION
In this paper, we proposed the sparse channel estimation algorithms for mmWave massive MIMO systems with mixed-ADC architecture, which was based on the sparsity of the channel. Experiments were carried out to verify the performance of the proposed method. The numerical results showed that the performance of the OMP algorithm had better than that of the LS algorithm in a certain SNR range, such as the performance of the OMP algorithm using onebit quantization is better than that of the LS algorithm using the four-bit resolution of ADC device. For the mixed-ADC architecture, simulation results also verified that a variety of combinations of ADCs with different resolutions can obtain a near optimal performance when ADC low = 5. In the future work, we shall study the channel estimation problem for IRS-assisted mmWave massive MIMO systems, and explore the impact of different scenarios on the cascaded channel generation.