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Channel estimation based on the PSSMUSIC for millimeterwave MIMO systems equipped with coprime arrays
EURASIP Journal on Wireless Communications and Networking volume 2020, Article number: 17 (2020)
Abstract
The millimeterwave channel estimation problem can be solved by estimating the path directions and the path gains. The previous schemes are nearly all based on the uniform linear arrays (ULAs). However, compared with the ULAs, the coprime array can use fewer array elements to realize larger array aperture, which is beneficial to improve the performance of arrival direction estimation contributing to the estimation performance of angle of arrival (AOA) and angle of departures (AOD). Encouraged by the property of coprime arrays, a novel channel estimation scheme is proposed for twodimensional (2d) coprime arrays. For each path direction, multiple peaks are generated in the spatial spectrum of each subarray and by selecting over any limited sector to search for an arbitrary peak, which can recover the rest peaks. Then by comparing the peaks of the two subarrays, the common peaks are the correct path direction. Compared with the totalspectrum search (TSS) method, this method effectively reduces the complexity. Simulation results show that the proposed algorithm can reduce the computational complexity and maintain accurate channel estimation.
1 Introduction
Massive MIMO (multipleinput and multipleoutput) technologies have been widely used in current 5G wireless communication systems [1,2,3,4,5]. The millimeter wave (mmwave) and massive MIMO are combined to obtain enough signal power and improve the transmission distance. The traditional channel estimation techniques for MIMO systems fail to characterize the spatial sparsity of the mmwave channel; the problem of mmwave channel estimation can be turned into estimating the path directions and gains rather than estimating the MIMO channel matrix [6, 7].
Before discussing the nonuniform arrays, the channel estimation methods based on uniform linear array systems are briefly reviewed. Due to the spatial sparsity in mmwave channel, traditional channel model based on rich scattering is not practical. Taking advantage of the sparse characteristics of mmwave signals which mean that there is only a small amount of important information, the first class of channel estimation method represented by beam training method has been introduced [8, 9]. In addition, in order to reduce the signal pilot training cost, the accuracy of angle estimation is improved by combining compression sensing (CS) with beam training scheme [10, 11]. It is worth mentioning that this method also compares the amplitude of auxiliary beam pair to improve the estimation accuracy. In addition, the orthogonal matching pursuit (OMP) algorithm [12,13,14] is used based on mmwave MIMO channel systems. If there is an assumption that the path direction is distributed discretely in the angle domain where we need to emphasize is that the actual AOA and AOD are continuously distributed in the angle domain [15, 16] can utilize the angular channel sparsity to solve the problem of training overhead when estimating the channel information. Some scholars have proposed an idea based on iterative weight (IR) to deal with the problem of resolution distortion caused by the ongrid angle estimation [17]. Deep learning is also applied to millimeter wave systems improve the performance of communication systems [18, 19].
In addition to the channel estimation scheme already mentioned, classical spatial spectrum estimation, especially MUSIC algorithm, has been used in mmwave channel estimation for a long time. It was originally used to estimate AOA in [20]. Then, this method can be used to estimate both the departure and arrival angles in [21, 22], and among them, the [22] utilized the twodimensional MUSIC. A twodimensional (2d) channel estimation scheme is proposed for the systems with uniform planar arrays (UPAs) in [23]. Channel estimation based on spatial spectrum estimation can theoretically improve the angular estimation accuracy, angular resolution, and other related parameter accuracy of spatial signals in the system processing bandwidth [24]. However, based on this method, matrix decomposition, spectral peak search, and multidimensional parameter optimization require a large number of complex multiplication operations, and the number of operations is proportional to the cube of the number of array elements [25]. Therefore, when the number of array elements is large, the computational complexity of the channel estimation algorithm is extremely high, which is not conducive to engineering implementation. In order to avoid angle ambiguity, the spacing of the equidistant ULA should be no more than half of the wavelength of the radiation, so communication systems based on the ULA can only achieve the expansion of the array aperture by increasing the number of array elements, which is limited to some extent [26]. In order to solve this problem, we can use the new array to reduce the number of array elements without guaranteeing the loss of the array aperture. In recent years, a new geometry of nonuniform linear arrays (ULAs), denoted as coprime arrays attract more attention, which can obtain large array aperture with fewer array elements [27]. These unique advantages of the coprime arrays encourage researchers to conduct a series of related research. Coprime MIMO radar is one of the typical applications in practice; however, there are too few studies on channel estimation in MIMO communication systems equipped with coprime arrays, so next the application of coprime arrays in DOA estimation will be introduced.
Due to the high computational complexity and low adaptability, several commonly used DOA estimation methods [28,29,30] can result in huge computational complexity if applied directly to the coprime array. Therefore, some scholars have specifically proposed estimation algorithms for systems with coprime arrays. The TSSMUISC scheme using the entire spectral search first appeared in [31]. In this article, DOA is obtained by selecting common peaks in two separated arrays. It is not difficult to find that the search scope is so large leading to a sharp rise in the computational dimension. Considering the maximum peaks generated by each signal source are distributed uniformly in the period range, it has been proposed to change the original entire search to partial search which greatly reduces the number of searches. That is PSSMUSIC [32] where for each DOA, multiple peaks are generated on the spatial spectrum of each subarray and by selecting any limited sector to search for an arbitrary peak to recover the other ones. This is followed by [33] which demonstrates a lowcomplexity DOA estimation method based on ESPRIT. In order to replace the MUSIC algorithm, it uses the ESPRIT algorithm to perform angle estimation. In this way, spectrum search is completely avoided to further reduce computational complexity. An improved RootMUSICbased DOA was proposed in [34], and the estimation accuracy has been improved. In this article, we will mainly improve the method in [31, 32] to estimate the path direction.
In this paper, we propose a computationally efficient channel estimation scheme for 2d coprime arrays where the directions can be uniquely estimated by finding the common peaks of the two decomposed subarrays. It can substantially reduce the complexity while maintaining high resolution, as compared with the TSSMUSIC method.
The main contributions of this paper are as follows:
 a)
From the perspective of the array, the channel estimation accuracy is improved by employ a new array. The influence of the two arrays on the channel parameters is compared under the premise of the same algorithm, which proves the superiority of the coprime array.
 b)
Encouraged by the advantages of the coprime arrays, a novel channel estimation scheme is proposed for 2d coprime arrays. We have proved the superiority of the new algorithm theoretically by comparing the complexity analysis with the traditional method.
 c)
Change the signaltonoise ratio and the number of snapshots separately, we can find the relationship between the mean square error and these two variables. The simulation results show that the proposed channel estimation scheme can reduce the computational complexity. When the time of spectral search are similar, the novel algorithm has better estimation performance.
To be more specific, the rest of the paper is organized as follows: Section 2 introduces the methodology. Section 3 describes the signal model of communication system with coprime planar array. The next section describes the problem and clarifies the reason. Section 5 derives the proposed method. Section 6 discusses the performance of the proposed method and provides the simulation results. And Section 7 concludes the work.
The specific meaning of the symbols in this article is shown below: A is a matrix, a is a vector and diag(Α) is a vector formed by the diagonal elements of A. A^{T}, Α^{∗}, and A^{H} are the transpose, the conjugate, and the conjugate transpose, respectively. The inverse and pseudoinverse is represented by A^{−1} and A^{†}. ‖A‖_{F} is its Frobenius norm.
2 Methodology
In this article, we first introduce existing channel estimation methods and finds that they are nearly all based on the communication systems equipped with ULAs. The research background and related methods are presented in Section 1. There are many factors that affect the performance of channel estimation. This article creatively considers improving the performance of the system from the perspective of array shape. Compared with the ULAs, the coprime array can use fewer array elements to realize larger array aperture which can reduce the cost of the system and decrease the subsequent computational complexity. Moreover, because of the property of the coprime array, we do not need to consider the problem of angular ambiguity caused by too large array element spacing. Thus, changing the structure of the array reasonably is beneficial to improve the performance of arrival direction estimation contributing to the estimation performance of angle of arrivals (AOA) and angle of departures (AOD). During the study of the system model, Fig. 2 shows the relationship between the coprime array and the uniform array in detail. In the simulation comparison, in order to better verify the effectiveness of the algorithm, we use a complex twodimensional array model, that is, an Lshaped array consist of coprime arrays. This array has two outstanding advantages: one is that it does not ignore the azimuth compared to the linear array, and the other is to use a coprime array to form the two parts of the Lshaped array.
Encouraged by the advantage of the coprime arrays, a computationally efficient channel estimation scheme for 2d coprime arrays is presented where the AOA and AOD can be uniquely estimated by finding the common peaks of the two decomposed subarrays. Compared with the totalspectrum search (TSS) method, the channel estimation scheme based on the PSS algorithm can search over a limited area which can effectively reduce the complexity and does not change the resolution level.
In order to verify the effectiveness of the algorithm, we have conducted a variety of experiments to obtain comparison results. First, we first explain why we choose a coprime array. Experiments show that the same algorithm performs better than a uniform array when using a coprime array. Then, the complexity is calculated and the superiority of the new algorithm is proved mathematically. Finally, the new channel estimation algorithm is compared with the TSSMUSIC algorithm. The specific analysis can be found in Section 6.
3 System model
In this paper, we focus on a mmwave MIMO communication system equipped with the hybrid precoding [22] which has been shown in Fig. 1.
N_{t} and N_{r} represent the number of the antenna array at the transmitter and receiver. One end of the analog beamformer F_{RF} is connected the transmitter, and the other end is connected to M_{t} RF chains. The receiver has the same hybrid structure, the ends of W_{RF} are connected to receiver and M_{r} RF chains. In the practice systems, the number of antennas is greater than that of RF chains, such that N_{t} > M_{t}, while the F_{BB} and W_{BB} denote the digital precoders and combiners.
The signal obtained at the receiver can be expressed by the following formula
where F = F_{RF}F_{BB} and W = W_{RF}W_{BB} are the hybrid precoder and hybrid combiner, respectively. s is the vector of transmitted baseband signal, H is the channel matrix, and the noise n follows the distribution \( \mathcal{CN}\left(0,{\sigma}_n^2{I}_N\right) \).
Supposing there are two uniform linear arrays which are consisting of M and N array elements where M and N are coprime integers. As illustrated in Fig. 2a, the number of array elements in subarray above is M where Nd is the interelement spacing. Similarly, the number of array elements in subarray below is N where Md is the interelement spacing. Figure 2b shows a coprime array model setting M = 4 and N = 3 where the simple values are set for a more intuitive introduction. The d is usually set as half wavelength λ/2. The number of array elements in the entire array is M + N − 1 because the first element positions of the two subarrays are coincident.
Then, introducing a x − y plane Lshaped array which consists of two coprime subarrays shown like Fig. 3. The Lshaped array has M + N − 1element arranged along the Xaxis and an M + N − 1element arranged along the Yaxis. The spacing between adjacent elements of two subarrays is Md or Nd, respectively. The first element of the two subarrays serves as a common reference point.
Defining the elevation and azimuth by θ and ϕ, then the channel model can be expressed as Eq. (2).
where the number of paths is L, Λ_{G}(q) = diag {g_{1}(q), ⋯g_{L}(q)}, A_{R} = [a_{r}(θ_{r1}, ϕ_{r1}), ⋯, a_{r}(θ_{rL}, ϕ_{rL})], andA_{T} = [a_{t}(θ_{t1}, ϕ_{t1}), ⋯, a_{t}(θ_{tL}, ϕ_{tL})].
Defining \( {\alpha}_{rx}={e}^{j2\pi {d}_x\cos {\phi}_{rx}\sin {\theta}_{rx}/\lambda } \), \( {\alpha}_{ry}={e}^{j2\pi {d}_y\sin {\phi}_{ry}\sin {\theta}_{ry}/\lambda } \), \( {\beta}_{rx}={e}^{j2\pi {d}_x\cos {\phi}_{tx}\sin {\theta}_{tx}/\lambda } \), and \( {\beta}_{ry}={e}^{j2\pi {d}_y\sin {\phi}_{ty}\sin {\theta}_{ty}/\lambda } \), we have
The channel direction and path have different fading scale where the former belongs to large scale fading and the latter belongs to small scale fading. The path directions will not change during a frame when the receiver can obtain N_{b} timefading block information which can be utilized to estimate the path directions. The gain of the path can be obtained according to the path directions. Based on the above theory, we can see the importance of estimating AOA and AOD.
4 Problem formation
4.1 The impact of the arrays
In the above channel estimation algorithm, the estimation of the angle parameter is a very important part of it, so we will analyze the effect of the antenna array structure on the direction estimation. It is worth noting that in the existing DOA estimation algorithm, uniform linear array is the most commonly used array structure where interelement spacing d is usually set as half wavelength λ/2 to eliminate angle ambiguity. In this case, the array aperture is limited by the number of array elements. In summary, traditional algorithms need more expensive physical array elements to obtain larger array apertures and better direction of arrival estimation. In order to solve this problem, we proposed a novel scheme based on the coprime arrays to reduce costs and improve performance.
Figure 4 shows a uniform linear array formed by 9 elements and a coprime array formed by 6 elements. The coprime array in the above picture reduces the overhead of three array elements while ensuring the same array aperture and the number of reduced array elements can be expressed as MNd − Nd − M − N + 1.
4.2 Angle ambiguity
Coprime array is composed of uniform linear arrays. When the spacing between elements is a multiple of half the wavelength, the angle ambiguity will occur. However, due to the property of coprime arrays, this problem can be easily solved. The sine of both AOA and AOD can be expressed as u_{i}. Every angle corresponds to a map and the map of theithpath can be defined by\( {\varTheta}_i^L \). L is the number of array element which is M or N. Because of the periodicity of the negative exponential function, the guidance vectors of all elements in this subarray are not different. This is the reason for the ambiguity of angle.
For a certain u_{i}, \( {\varTheta}_i^N \) and \( {\varTheta}_i^M \) can be obtained respectively as the following.
The property of \( {\varTheta}_i^L \) and the corresponding proof are given below.
4.2.1 Property
Given that u_{i} and u_{i + 1} have the same steering vectors, \( {\varTheta}_1^L \) is equal to \( {\varTheta}_2^L \) for a subarray when the interelement spacing is L d.
4.2.2 Proof
The steering vectors of u_{i} can be defined as a_{i}
Assuming u_{1} and u_{2} has the same steering vectors, then we get a_{1} = a_{2} and the Eq. (7)
Substitute Eq. (7) into Eq. (5), we get
If there is \( w\in {\varTheta}_i^M \) and \( w\in {\varTheta}_i^N \), according to Eq. (4) and Eq. (5).
There are two situations at this time shown by Eq. (10).
M and N are coprime integers, so w ≥ 1 or w < 1; however, this is contradictory to the assumption, so
In order to explain the above theory more vividly, the simulation results is given here. When M = 5 and N = 3, the MUSIC spectrums are depicted in Fig. 5 where we found a unique peak of coincidence and we get the actual sineu = − 0.5. When the subarray has five elements, three peaks can be found in the spatial spectrum and when the subarray has three elements, five peaks can be found in the spatial spectrum. In fact, only the two peaks of the eight ones are meaningful, and the rest of the peak information is invalid. This phenomenon is called angle ambiguity.
5 Proposed channel estimation scheme for coprime arrays
The novel channel estimation is based on partial spectral search method which can tackle the angle ambiguity.
Before introducing the proposed channel estimation scheme for coprime arrays, we first review the channel model for mmwave system in paper [8].
5.1 Estimation of path direction
Supposing there are M_{t} pilot signals at the transmitter where M_{t} is the number of RF chains, the total transmission power is defined as E = M_{t}P and the transmit power allocated equally for each pilot signal. The received signal y(q) is obtained during one fading block as
Substituting (2) into (12), we get
In the following, we can split a coprime Lshaped array into two uniform linear arrays as Fig. 6 so that it can be processed separately.
The received signal model can be divided into y_{1} and y_{2}.
First, we process the signal y_{1} in the first array. Equation (14) can be further formulated as (16), where\( {\mathbf{Z}}_{GT1}={\boldsymbol{\Lambda}}_G{\mathbf{A}}_{T1}^{\mathrm{T}}\left({\theta}_t,{\phi}_t\right)\mathbf{F} \) and B_{R1}(θ_{r}, ϕ_{r}) = PW^{H}A_{R1}(θ_{r}, ϕ_{r}), in other words, taking PW^{H}A_{R1}(θ_{r}, ϕ_{r}) as a whole with the information of AOA and taking \( {\boldsymbol{\Lambda}}_G{\mathbf{A}}_{T1}^{\mathrm{T}}\left({\theta}_t,{\phi}_t\right)\mathbf{F} \) as whole with the information of path gain.
In order to estimate the path direction, we collect the measurement vectors from the N_{b} blocks to compose the covariance matrices as follows.
Substituting (16) into (17), the eigenvalue decomposition of R_{1} is expressed as Eq. (18) where U_{s} and U_{n} are signal subspace and noise subspace, respectively.
Because of the orthogonality of signal subspace and noise subspace, the spatial spectrum search for the subarray 1 can be described as
Then, we process the signal y_{2} in the second array and we get Eq. (20) in the same way
Next, we should look for the common peaks in P_{R1} and P_{R2}; however, the actual element spacing is not equal to half wavelength but a multiple of λ/2. Even if there is only one true angle value, multiple peaks will be found when performing a spatial spectrum search. Therefore, he actual twodimensional DOA and the remaining DOA are distributed regularly over the period, based on this fact, the ambiguous 2D DOAs can be represented by θ_{a}, ϕ_{a} and the actual ones can be represented as θ_{c}, ϕ_{c}. According to Eq. (7), we have
By performing a transformation as μ = sin(⋅) and ν = cos(⋅), the relationship in Eq. (21) can be converted as
From the above formula, we need to search for an arbitrary peak and then calculate the coordinates of other peaks based on the position of the first peak. This allows us to start searching within a small range instead of a full search, so that our new algorithm can greatly reduce the calculation range. Before the algorithm is executed, the search interval of the two arrays needs to be evenly divided and a random interval is selected to obtain the position of the peak value. Then, we can recover all the others using the relationship Eq. (22).
It just follows from the above property in Section 3.2 that each of the two subarrays will have multiple peaks. In theory, the true angle estimation is obtained by finding the positions of common peaks in the spectrum. Because the noise cannot be eliminated, there is virtually no peak that will completely coincide. So we need to compare the distance between all the peaks to find the closest ones to estimate AOA in Eq. (23) where \( \left({\overset{\frown }{\theta}}_{rM},{\overset{\frown }{\phi}}_{rM}\right) \) and \( \left({\overset{\frown }{\theta}}_{rN},{\overset{\frown }{\phi}}_{rN}\right) \) denote 2d MUSIC peak for the subarray 1 and subarray 2 in Fig. 6, respectively.
Then, we use the same method to estimate the AOD, getting the conjugate transpose \( {\mathrm{y}}_1^H(q) \) and \( {\mathrm{y}}_2^H(q) \)
Next, we should look for the common peaks by repeat the Eq. (16) to Eq. (23) formula and we get \( {\overset{\frown }{\theta}}_t \) and \( {\overset{\frown }{\phi}}_t \).
5.2 Estimation of channel
The PSSMUSIC method has been applied to obtain the path directions. In this section, the estimated angles can be used to obtain the path gain. We firstly calculate \( {\tilde{\mathbf{B}}}_{R1} \) in Eq. (16) and then uses LS method to estimate \( {\tilde{\mathbf{Z}}}_{GT1} \). Then the gain \( {\tilde{\boldsymbol{\Lambda}}}_G \) can be computed. In particular, the gain here depends on \( {\tilde{\mathbf{B}}}_{R1} \), because the angle change belongs to largescale fading and the gain change belongs to smallscale fading.
Therefore, the gain is estimated according to the obtained path direction during each time block. In the end, \( \tilde{\mathbf{H}} \) can be estimated as shown in Eq. (27).
6 Results and discussion
In this section, numerical simulation results are provided to assess the performance of the proposed 2d channel estimation scheme for coprime arrays. Firstly, by comparing the performance of the same algorithm in different array antenna systems, the advantages of the coprime array compared to the uniform array are further proved in Fig. 7. Considering 2 mmwave MIMO communication systems with hybrid precoding are equipped with different Lshaped antenna arrays where one is a coprime array and the other is a uniform array. The number of uniform Lshaped array elements in the Xaxis and Yaxis directions is N_{X − uniform} and N_{Y − uniform}. Correspondingly, the number of coprime arrays are N_{X − coprime} and N_{Y − coprime}. It should be noted that in order to make the two arrays have the same aperture, the number of elements in the uniform array is more than that in the coprime array. In order to compare the estimation performance of the proposed algorithm under two different array conditions, the remaining parameters are set to be the same. There are M_{t} RF chains at the transmitter and M_{r} RF chains at the receiver for the two kinds of array models. The detail simulation parameters are defined as follow where the N_{b} represents the number of block’s pilots. The experimental parameter settings in Fig. 7 are as shown in Table 1.
Assuming that there is only one channel path, the AOA is (17.55°, 32.85°) and AOD is (19.45°, 36.10°). The results of the following simulation effectively show the value of the new array structure. From Fig. 7, we can see that channel estimation using a coprime array is less complex than using a conventional uniform antenna array and the estimation accuracy is higher in the same time. The normalized mean squared error (NMSE) of the channel is defined as follows.
Through the above picture, we have proved the superiority of coprime array. Next, we study the effect of the number of array elements on the channel estimation performance in the coprime arrays. Set parameters M = 13, N = 11 in one array and then set M = 31, N = 29 in another array. Figure 8 shows the effect of the number of array elements on the channel estimation performance in a coprime, we can see that the more the number of arrays, the worse the estimated performance.
The effects of different arrays on channel estimation are discussed above. The following shows how the computational complexity is reduced. Angle estimation method based on MUSIC and an improved version of PSSMUSIC are analyzed respectively. The results are shown in Table 2. The number of array elements of two uniform arrays constituting a coprime array is M and N, respectively. Besides covariance matrix estimation and eigenvalue decomposition, the complexity of DOA algorithm is also affected by spectral search. The times of spectral search can be expressed as N_{search}, and the number of snapshots can be represented as K. The corresponding complexities of MUSIC are O((M + N − 1)^{2}K), O((M + N − 1)^{3}), and O(N_{search}(M + N − 1)^{2}), respectively. The complexity of PSSMUSIC in the proposed scheme is O((M^{2} + N^{2})K), O(M^{3} + N^{3}), and O(2N_{search}(M^{2}/N + N^{2}/M)).
The abscissa of Fig. 9 is the number of complex elements, and the ordinate is the complex multiplication number. The parameters of the system are set as follows: K = 200 and N_{search} = 3600. Under the condition of M = 31 and N = 29, the complex multiplications of the MUSIC algorithm reach 4.7 × 10^{6} and the complex multiplications of the PSSMUSIC are 4.0 × 10^{6}. We can get that the computational complexity of the latter is 62% of the former, which is the basis for our proposed algorithm to reduce computational complexity. It is expected that when the number of array elements is large, the PSSMUSIC based channel estimation will have lower computational complexity. Therefore, the proposed estimation scheme can realize faster channel estimation.
Before demonstrating the new method channel estimation performance, we first verify the accuracy of the angular accuracy estimation accuracy of the new algorithm. The MSE of the estimated path angles are defined as
Figure 10 depicts the MSE versus SNR via 10000 Monte Carlo simulations and shows that the PSSbased angle estimation performs better than the traditional TSSMUSIC method
We set M = 7 and N = 5 in the above simulation system where one channel path is assumed, and the AOA is (7.95°, 10.85°) and AOD is (11.45°, 16.10°). The search time depends not only on the size of the search grid, but also on the range of search angles. It is known from the previous analysis that instead of total spectral search, the channel estimation scheme based on the PSS algorithm can search over a limited area. In order to narrow the difference in the times of searching between the two methods, the searching grid for MUSIC is set to 0.05° and the one for PSSMUSIC is set to 0.01°.
Figure 11 shows that the proposed channel estimation performs better than the method based on TSSMUSIC. One channel path is assumed as above Fig. 10 where M = 7 and N = 5. For the same reason as above, the searching grid for MUSIC is set to 0.05° and the one for PSSMUSIC is set to 0.01°.
Figure 12 plots the reverses number of block’s pilots via 10000 Monte Carlo simulations with SNR = 10 dB. From the figure, we can see that the performance of the two channel estimation methods gets better with the increase of the number of block’s pilots, but overall, the performance of the new algorithm is always better than the old algorithm. It illustrates the superiority of the proposed scheme from another aspect.
7 Conclusions
In this paper, we proposed a channel estimation scheme for coprime arrays and the proposed scheme is based on the PSSMUSIC method. The simulation results show that the novel method can effectively estimate the channel state information in coprime arrays while greatly reducing the computational complexity. However, we have to admit that our research also has a limitation, that is, as the number of channel paths increases, spectrum searching between different paths may be confused. For future work, it would be interesting to work on how to solve this problem which may help to further improve the accuracy of channel estimation coprime.
Availability of data and materials
Not applicable
Abbreviations
 2d:

Twodimensional
 AOA:

Angle of arrival
 AOD:

Angle of departure
 DOA:

Directionofarrival
 mmwave:

Millimeter wave
 PSSMUSIC:

Partial spectral search MUSIC method
 TSS:

Total spectral search
 ULAs:

Uniform linear arrays
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The authors acknowledged the three anonymous reviewers and editors for their efforts in constructive and generous feedback.
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This work was supported by the National Nature Science Funding of China (NSFC): 61401407 and the Fundamental Research Funds for the Central Universities.
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SL contributed to the conception and design of the study. GC is the main author of the current paper. LJ and HW commented on the work. All authors read and approved the final manuscript.
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Li, S., Cao, G., Jin, L. et al. Channel estimation based on the PSSMUSIC for millimeterwave MIMO systems equipped with coprime arrays. J Wireless Com Network 2020, 17 (2020). https://doi.org/10.1186/s1363802016374
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DOI: https://doi.org/10.1186/s1363802016374