Estimation of Sparse Channels in Millimeter-Wave MU-MIMO Systems

Abstract

This paper considers a channel estimation scheme for millimeter-wave multiuser multiple-input multiple-output systems. According to the proposed method, parts of the beams are selected and the channel parameters are estimated according to the sparsity of channels and the orthogonality of the beams. Since the beams for each channel become distinct and the signal power increases with the increased number of antennas, the proposed approach is able to achieve good estimation performance. As a result, the sum rate can be increased in comparison with traditional approaches, and channels can be estimated with fewer pilot symbols. Numerical results verify that the proposed approach outperforms traditional approaches in cases with large numbers of antennas.

1. Introduction

Millimeter-wave (mm-wave) systems operating from 30 GHz to 300 GHz can provide multi-GHz bandwidth, and are expected to meet the capacity demands of future wireless networks [1], [2]. Meanwhile, the small wavelengths in mm-wave systems enable placing multiple antennas with half-wavelength distance in constrained spaces [3],[4]. These arrangements of large numbers of antennas constitute massive multiple-input multiple-output (MIMO) systems, which are expected to achieve high spectral efficiency in future wireless systems [5],[6].

In mm-wave systems, their performance relies on the employment of efficient channel estimation schemes. Since the wireless channels tend to exhibit sparse-scattering structure as the signal space dimension gets large, the channel sparsity is utilized to improve system performance, as demonstrated in [7-13]. It is well known that mm-wave channels are sparse, thus existing channel estimation approaches for mm-wave systems usually resort to compressed sensing (CS) theory to reduce the estimation overhead [11-15]. In [12], a CS-based channel estimation approach was proposed to estimate the angles and path gains with a limited number of RF chains. Furthermore, a technique for improving the resolution of the angle estimation and reducing the pilot overhead was presented in [13].However, these CS-based approaches are highly complex computationally. For example, even the computational complexity of the least squares solution of the CS-based approaches is proportional to M2, where M is the number of the base station (BS) antennas. Moreover, all these methods were only proposed for point-to-point transmission scenarios, i.e., they are not suitable for mm-wave multiuser MIMO (MU-MIMO) systems.

Recently, in order to exploit spatial channel sparsity in mm-wave systems, the beamspace MIMO concept was proposed [16-19]. A beamspace MIMO transmitter selects a number of orthogonal beams for base-band precoding, and converts the low-dimensional baseband signals to high-dimensional radio-frequency (RF) signals with an analog beamforming front-end. As the complexity of the base-band precoding and the number RF chains are dramatically reduced, beamspace MIMO systems are capable of achieving near-optimal performance with low complexity [16]. Since the beamspace MIMO concept also utilizes channel sparsity, it is feasible to estimate mm-wave channels in beamspace. For example, the approach in [20] utilized partial spatial beam selection for transmission. As the beams are orthogonal and are not reused among the mobile stations (MSs), the interference can be mitigated during the downlink channel estimation process. However, the uplink channel estimation is assumed to be perfect, which requires traditional orthogonal pilots, just as an approach utilized in [21]. Hence, the pilot overhead is extensive in cases where there is a large number of MSs.

In this paper, a channel estimation approach for mm-wave MU-MIMO systems is proposed. By exploiting the beamspace concept, the correlations of the received pilot vector and the beams are used to select the partial beam. Then, the direction-of-arrivals (DOAs) of the MSs are estimated by matching these correlation values with the theoretical values that are known a priori. Finally, path gains are estimated based on the received pilots and the estimated DOAs. The main contributions of this paper are two-fold.

Notations: Lower-case (upper-case) boldface symbols denote vectors (matrices); (⋅)H denotes the conjugate transpose; [⋅]j is the j th element of a vector or the j th column of a matrix; CN(μ,σ2) is a circularly-symmetric complex Gaussian random variable with mean μ and variance denotes the expectation, and i is the imaginary unit.

 

2. System Model

Consider the following system model. A BS equipped with N antennas serves K MSs simultaneously, where each MS has only one antenna1. In addition, it is assumed that the number of RF chains at the BS is the same as the number of antennas at the BS2.

The BS is equipped with a uniform rectangular array (URA) of dimension N = NazNel, where Naz is the number of antennas in the y-axis and Nel is the number of antennas in the z-axis, as shown in Fig. 1.

Fig. 1.The geometry of a cell. The URA is placed in the y-z plane, the MSs are distributed below the BS. The antennas are distributed uniformly in the URA.

The system adopts time division duplex (TDD) mode and thus channel reciprocity holds for two-way transmission. Furthermore, it is assumed that the channel undergoes block fading, which means the channel remains constant during the transmission of each coherent block of Tc symbols. The details of the transmission process are as follows.

As can be seen, coherent intervals are used for the three phases, which results into Tc = Nu+Nd+1.

For the uplink transmission, the received pilot vector at the BS is given by

where H∈ℂN×K is the channel matrix from the MSs to the BS, xp is the pilot symbol transmitted by the MSs, and n∈ℂN×1 is the noise vector, which is composed of independent and identically distributed (i.i.d.) CN(0,1) variables. Similarly, at the BS the received data matrix is given by

where Xa∈ℂK×Nu is composed of the data symbols transmitted by the MSs, N∈ℂN×Nu is the noise matrix, which is also composed of i.i.d. CN (0,1) variables. For the downlink transmission, the received data matrix at the MSs is given by

where Ya∈ℂN×Nd is composed of the BS precoded data symbols, and W∈ℂK×Nd is the noise matrix, composed of i.i.d. CN (0,1) variables.

2.1 Channel Model

According to the sparse-scattering property of the mm-wave channels and the cell geometry presented in Fig. 2, the uplink channel of the k-th MS is given by [18]

where βk is the uplink complex path loss and is the array steering vector for the azimuth, and the elevation, components of the DOA, which are also shown in Fig. 2. In addition, NP is the number of multipaths, βk,m is the uplink complex path loss of the m-th multipath of the k-th MS, and are the azimuth and elevation components of the DOA of the m-th multipath of the k-th MS, respectively. Similarly to [18], this paper focuses on pure line-of-sight (LOS) transmission4, and the uplink channel can be re-expressed as

Fig. 2.An example of the relation between different sets of C1 and C2.

In addition, the uplink path loss is written as [23]:

where λ is the transmission wavelength, ϕk is a random phase of the path loss of the k-th MS, dk is the distance between the k-th MS and the BS, and is the array directivity as defined in [24]. According to the geometry of the URA, the array steering vector can be expressed as

where naz = 0,1,⋯Naz-1, nel = 0,1,⋯Nel-1, and n = nelNaz+naz+1.

2.2 Beamspace Processing

According to the principle of beamspace MIMO, the N orthogonal vectors corresponding to the beams form the unitary matrix, U∈ℂN×N, in which the m-th orthogonal vector or beam is

where satisfy m = (mel-1)Naz+maz, maz=1,2,⋯,Naz, and mel = 1,2,⋯,Nel.

Depending on the locations of the MSs, parts of the beams are selected to construct the beamforming matrix Ub∈ℂN×Nb, where is the number of all the beams selected, and is the number of the beams selected for each MS. Then, beamspace precoding and the beamspace detection for data transmission phases can be derived using the minimum mean square error (MMSE). For the uplink transmission, the beamspace detection matrix for Ra in (2) can be expressed as

where is the beamspace channel estimate, is the channel estimate, and ρMS is the transmission power of each MS, i.e., For the downlink transmission, the beamspace precoding for Ta in (3) can be rewritten as

where is matrix of the transmitted data symbols of the BS and ρBS is the total transmission power of the BS, i.e., Comparing (3) and (10), it can be seen that

With the system model presented and the principle of beamspace processing given, the proposed channel estimation approach will be presented in the following section.

 

3. Channel Estimation

The traditional CS-based approach is highly complex computationally in mm-wave systems with large numbers of antennas. In addition, the modified approach in [20] and [21] is based on pilots, so the estimation performance will not benefit from the increased numbers of antennas. In order to avoid these drawbacks, a beamspace channel estimation approach is proposed. In the proposed estimation process, parts of the beams are firstly selected for beamspace transmission. Then, the DOAs of the MSs are and the path losses are estimated.

3.1 Beam Selection

In order to select the beams for transmission, the correlation of the pilot vector and the orthogonal beams is first calculated using

where According to the expressions of H and U in (5) and (8), respectively, we have

where

and the second equation in (12) is derived based on (5). In addition, which are also known as the spatial frequencies.

Let f(x) = sin(πxM) / sin(πx), where |x|<2, and M is an integer not smaller than 2. When f'(x) = 0, f(x) = M, which is a local maximum or minimum. Meanwhile, when x = ±2, f(x) = M. Since |x|<2, it can be seen that |f(x)| achieves its global maximum when f'(x) = 0, and that maximum is equal to M. In addition, it is evident that f'(x) = 0 only when sin (πx/2) = 0, i.e., x = 0. Therefore, the global maximum of |f(x)| is achieved when x = 0. Since based on the above results and (13), there is

and the maximum is achieved when

From the above analysis, we see that, for variable m, |λm,k|, ∀k, reaches the maximum when According to (12), it can also be seen that, ∀k, for variable m reaches its maximum value when We denote the set of the indices of the maximal elements of Then, is close to αk, and is close to γk. Moreover, the element in Ck that corresponds to the maximum in is denoted as ck.

In the ideal condition that the sets, Ck and Ck', are disjoint for all k≠k′, there are K local maxima of the function

wherea m,maz,mel are defined in the same manner as in the case of (8). Let us assume that g(maz(k),mel(k)) is the local maximum of the function g(maz,mel) that corresponds to the k-th MS. For maz,mel that are close to maz(k),mel(k)), the corresponding g(maz,mel) is also close to g(maz(k),mel(k)). Hence, for such that maz,mel that are close to maz(k),mel(k), the corresponding values of g(maz,mel) are also close to this local maximum. As can be seen, the indices of this local maximum, and the corresponding values are exactly the elements of Ck. Thus, the indices of the K local maxima of g(maz,mel) and the corresponding values are the elements in Ck,k = 1,2,⋯,K. The element in Ck that corresponds to one of the local maxima in g(maz,mel) is ck.

As an example, Fig. 2 illustrates the relation between different sets, Ck and Ck', ∀k≠k'. The axes represent the indices, maz,mel, the area rounded by the red line is the range of the beam indices for positions in the cell, and the blue circles represent the sets of selected indices for two MSs. For the indices in set C1,c1 is an index that corresponds to a local maximum of g(maz,mel). The other elements in C1 correspond to the other indices that are close to c1, and correspondingly result in close values of g(maz,mel).

According to the definition of in (11), it can be seen that

can be used as an estimate of g(maz,mel) in (15). Therefore, these beams can be selected in the same way with the aid of Then, the beamforming matrix, Ub in (9) and (10), can be constructed in the following manner. The K columns of U that have the elements of Ck as indices are selected to form the th to the th columns of Ub.

When the ideal condition that ∀k≠k', Ck and Ck' are disjoint is not satisfied, the same beam may be selected by different MSs, as shown in Fig. 2. In [20], various beam selection methods were discussed in to circumvent this problem. Here, a simple approach is employed. In this approach, the beams are selected by the MSs sequentially, i.e., the beams that have been already selected will be excluded in the subsequent selection. For the example in Fig. 2, once the elements in C1 have all been selected, those that are common with C2, i.e. those in the intersection of C1,C2, will be excluded in the selection of elements in C2. In addition, there are two principles in the selection process:

3.2 DOA Estimation

According to (12), it can be seen that

From the beam selection part, it is known that ∀m∈Ck, the value of αk is close to and, correspondingly, the value of γk is close to and that, additionally, can be used instead of in the ideal case that ∀k≠k', Ck and Ck' are disjoint. Based on this property, the spatial frequencies αk and γk can be estimated as

where m is such that and and

In addition, and are defined below (8) and (14), and ck is defined above (15). This constraint on m means that only the element in Ck that is close to ck is used for estimation and that the search ranges for αk and γk, i.e., and can be obtained, respectively. Once these spatial frequencies are estimated, the DOAs can be estimated as

When the ideal condition that ∀k≠k', Ck and Ck' are disjoint is not satisfied, the constraint on m mentioned above is not applicable. This means that there is no element in Ck that is close to ck. In this case, the above estimation is not suitable, so the meanings of which are explained in the previous paragraph, are used as estimates of αk and γk, respectively.

3.3 Channel Parameter Estimation

With the estimated DOAs, it can be seen that only the uplink path loss βk remains an unknown in (5). According to (11) and (12), βk can be estimated as

where are estimated by replacing αk and γk, and m with and ck in (13) and (14), respectively. Note that although the above estimation is derived under the ideal condition that Ck and Ck' are disjoint, it is also applicable when this condition does not hold.

Now the channel parameters have been estimated. In a system with perfect synchronization and power control, the path loss of each MS can be coarsely estimated prior to channel estimation. Thus, the known path losses of the MSs can be utilized to resolve the ambiguity of the MSs, i.e., the corresponding estimated channel vector of each MS can be separated from other estimated channel vectors. A similar assumption and a corresponding method for distinguishing MSs can be seen in [25]. In the case that orthogonal pilots are employed, the pilots can also be used to distinguish the MSs.

With the proposed channel estimation approach presented, the performance of the system will be investigated in the following section.

 

4. Performance Analysis

In this section, the proposed approach is compared with the modified approach in [20] and [21] and the performance of the proposed approach in systems with large numbers of antennas is analyzed.

4.1 Comparison with Traditional Approaches

Although mm-wave MU-MIMO systems can achieve extremely high data rates, the channel estimation approach employed in these systems is not specific, as evident the traditional pilot-based approach in [20] and [21]. When the traditional pilot-based approach is modified for channel estimation in the mm-wave MU-MIMO system presented here, the number of uplink pilot symbols is K, and the received pilot matrix is HXp+N, where Xp∈ℂK×K is the matrix of pilots, and N∈ℂN×K is the noise matrix. Then, the uplink channel matrix is estimated by multiplying with the received pilot matrix. The beams can be then selected based on the uplink channel estimate.

According to the previous analysis, the number of pilot symbols is K for the traditional approach, which is the dimension of Xp in the previous paragraph. On the contrary, only one pilot symbol is necessary in the proposed approach, which is the dimension of xp in (1). Thus, the advantage of the proposed approach is that it can transmit more data symbols in one coherence interval and is thus expected to achieve a higher sum rate.

On the other hand, with the traditional approach in [20] and [21], the channels can be directly estimated with the pilots and the estimation error is only caused by noise. However, with the proposed approach, the channels are estimated through searching the DOAs and the estimation error is not only caused by the noise but also by interference. Therefore, the disadvantage of the proposed approach is that the channel estimation error may be larger than that of the traditional approach. Fortunately, this disadvantage can be effectively mitigated by increasing the number of antennas, as will explained in the next subsection.

4.2 Asymptotic Analysis

In this subsection, the performance of the proposed approach in the asymptotic limit is investigated, where the asymptotic limit means Naz,Nel→∞. The asymptotic property of λm,k in (13) is presented first.

Theorem 1:1) When do not tend to zero and one of them tends to infinity, λm,k/N = 0 ; 2) When are finite, λm,k/N ≠ 0.

Proof: For the first part of the theorem, there is

where and are finite.

For the second part, there is

where are finite.

For any αk and γk, the second property in the theorem above is satisfied with by a value of m∈{1,2,⋯,N}. Thus, for the set Ck, the element m∈Ck corresponds to the one of the maximal values of λm,k. Since the distances between the MSs are non-zero, the differences between the DOAs of two MSs as well as the differences between αk and γk of two MSs are also non-zero, which means the condition of the first part of the theorem above is satisfied for m∈Ck',∀k≠k'. Then, for the second part of the theorem, it can be seen that ∀k≠k', Ck and Ck', are disjoint. In other words, the ideal condition mentioned in the previous derivation is satisfied. Additionally, for m∈Ck, is satisfied in the asymptotic limit. Thus, the DOAs as well as the path losses can be estimated with no error.

In reality, the number of antennas is finite, i.e., Naz,Nel are finite. In this case, the estimation in (18) can be expressed as

which is based on (11). It can be seen that the estimation error is inevitable because of the interference and the noise Therefore, the estimation performance of the proposed channel estimation approach is not as good as its corresponding asymptotic performance.

According to this asymptotic analysis, it can be seen that the proposed approach benefits from an increased number of antennas. However, for the modified approach in [20] and [21], the channels are directly estimated using the pilots, as shown in the previous subsection. Hence, the channel estimation performance of the traditional approach is invariant with the number of the antennas. Thus, the proposed approach is also expected to achieve a higher sum rate than the modified approach in [20] and [21] even when the same number of pilot symbols is used in the two approaches.

 

5. Simulation Results

For the simulation, the system was set up as follows. The radius of the cells was in the range [10m, 100m]. The range of the azimuth angle of the cell was [-60°,60°]. The height of the BS was 10 meters. The frequency of the two-way channels was set to80 GHz and K = 10, Given the symmetry of two-way transmissions, in the channel coherence interval only uplink channel estimation and uplink data detection were carried out for the simulation. In addition, the transmission SNR in the simulation ranged from 60 dB to 100dB. As explained in [18], the actual transmission SNR ranges from 52dB to 112dB, which roughly corresponds to a received SNR in the range of -40 dB to 20 dB, due to high path loss. In addition, the corresponding transmission power is in the range of -20 dBm to 40 dBm. Thus, the range of the transmission SNR in this simulation was close to the actual range and is reasonable for implementation. In the following simulation, the “traditional approach” mentioned is the modified approach presented in [20] and [21] that was presented in Section 4.1.

The sum rate was averaged over the channel coherence interval, i.e., the effect of the pilot is taken into account in calculating the sum rate. The channel coherence interval is Tc = 500, which was regarded as a reasonable value in [26] and the reference therein. The antennas’ configurations were as follows. For N = 252, the number of antennas in each of the two dimensions was Naz = 7, Nel = 36 : for the second case, N = 2716 with Naz = 16, Nel = 76 ; and N = 4832, with Naz = 32, Nel = 151.

The sum rates versus the SNR are shown in Fig. 3. It can be seen that the sum rates of all approaches increase with the increase of the SNR, while, the increase of the number of antennas also results into an increase of the sum rate. This is because the received signal power is increased with the increase of the number of the transmitting or receiving antennas. In other words, the increase of the sum rate with the number of the antennas reflects the diversity gain, and the diversity gain depends on the number of the antennas rather than the employed approaches. Additionally, the proposed approach tends to achieve a higher sum rate than the existing approach when the number of the antennas increases. This is because the accuracy of the channel estimate of the traditional approach does not vary, while the accuracy of the DOA estimate of the proposed approach improves as the number of the antennas increases. In other words, the increase of the sum rate with the increase of the number of the antennas demonstrates that the estimation performance of the proposed approach improves at the same time. However, the increase of the sum rate with the number of the antennas of the traditional approach only demonstrates the diversity gain. This result verifies the asymptotic analysis of the proposed approach in Theorem 1.

Fig. 3.Illustration of the sum rates versus the SNR.

The sum rates versus the channel coherence interval are shown in Fig. 4 for a transmission SNR of 100 dB. It can be seen that the sum rates of both approaches increase with the increase of the channel coherence interval. In addition, the proposed approach is applicable in the case that the channel coherence interval is less than 10, while the traditional approach is not applicable in this case. It can also be seen that the proposed approach has a higher sum rate for the whole range of channel coherence interval values when the number of antennas is N = 4832. Note that in the case of long channel coherence intervals, i.e., when Tc = 500 in Fig. 4, the proposed approach also performs better than the traditional approach. The ratio of the number of the pilots versus the channel coherence interval is 1/500 for the proposed approach and 10/500 for the traditional approach. Thus, the reduced number of pilot symbols for the proposed approach can save little time for data transmission. In other words, the better performance of the proposed approach in the case of Tc = 500 demonstrates that the proposed approach can estimate the channels more accurately. These results verify that the proposed approach can estimate channels with higher precision and is feasible for a wider range of the channel coherence interval.

Fig. 4.Illustration of the sum rates versus the channel coherence interval.

However, when the number of antennas is 252 or 1216, the proposed approach tends to perform worse than the traditional approach as the channel coherence interval increases. The reason is as follows. The channel estimation accuracy of the proposed approach relies on the number of antennas, while the traditional approach does not. When the number of antennas is not large enough, e.g., 252 or 1216, the estimation error of the proposed approach is larger than that of the traditional approach. When the channel coherence interval is short, the proposed approach benefits from the reduced number of pilot symbols and thus performs better than the traditional approach in these cases. When the channel coherence interval is long, the proposed approach does not benefit from the reduced number of pilot symbols, so it is outperformed by the traditional approach.

Finally, the sum rates versus the number of the pilot symbols are shown in Fig. 5. The transmission SNR is 100 dB, and the channel coherence interval is Tc = 500. When the number of the pilot symbols increases, the proposed approach can also be modified to utilize the additional pilots. With the increased number of the pilot symbols, the MSs can be partitioned into different groups, where the MSs in the same group share the same pilot sequence while MSs in different groups employ an orthogonal pilot sequence. Then, the proposed approach can estimate the DOAs and path losses of the MSs group by group. In Fig. 5, the number of the pilot symbols for the traditional approach is fixed, and the number of the pilot symbols is K = 10. It can be seen that the sum rate of the proposed approach increases with the increase of the number of the pilot symbols. In addition, the proposed approach tends to perform comparably to the traditional approach for N = 252 and N = 1216, but outperforms it for N = 4832. Since the difference between the DOAs of two MSs increases with this partition of the MSs, the proposed approach can distinguish the MSs more easily. Thus, the proposed approach also benefits from an increase of the number of pilot symbols.

Fig. 5.Illustration of the sum rates versus the number of pilot symbols.

 

6. Conclusion

In this paper, a channel estimation approach is proposed for mm-wave MU-MIMO systems. This method uses fewer pilot symbols than existing approaches and can estimate the channels without interference in the large system limit. Thus, the proposed approach is feasible for mm-wave MU-MIMO systems.