An Enhanced Channel Estimation Method for Millimeter ... - IEEE Xplore

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IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 9, SEPTEMBER 2015

An Enhanced Channel Estimation Method for Millimeter Wave Systems With Massive Antenna Arrays Yuexing Peng, Member, IEEE, Yonghui Li, Senior Member, IEEE, and Peng Wang, Member, IEEE Abstract—In this letter, we focus on the efficient channel estimation problem for millimeter wave (MMW) systems with massive antenna arrays and RF constraints, aiming at achieving a fast and high resolution angle-of-arrival/angle-of-departure (AoA/AoD) estimation. We first propose a presentation of antenna array with virtual elements (AAVE) by appending additional virtual antenna elements into the original antenna array. On the basis of the AAVE structure, we explore the channel sparsity in the angular domain and develop an efficient angle estimation algorithm by using compressive sensing theories. We then proposed a training design and prove that the sensing matrix in the proposed training can guarantee the accurate detection of angles with a high probability. Both the analytical and simulation results show that, without changing the physical antenna arrays, the proposed approach can achieve not only a lower overhead, but also a significantly higher resolution in angles estimation, compared to the existing algorithms. Index Terms—Millimeter wave (MMW) communication, angle estimation, channel estimation, compressive sensing, massive MIMO.

I. I NTRODUCTION

T

HE rapidly growing demand for new wireless services require ever-increasing data throughputs. The availability of large swathes of spectrum at millimeter wave (MMW) frequencies provides great opportunities for achieving Gigabit data rates to meet such a demand [1]. However, at such high MMW frequencies the signal experiences severe propagation loss. On the other hand, the wavelength of MMW is very small and this will enable the packing of a large number of antenna elements into the transceiver to compensate for the severe propagation loss. Therefore, existing MMW systems can exploit large scale antenna arrays at both the transmitter and receiver. In such a system, channel state information (CSI), especially the information about the angle-of-arrival (AoA) and angleof-departure (AoD) of each path, is essential for performing beamforming/precoding and receiver detection. Due to the high cost of RF units with very high operating frequencies, MMW systems usually can only afford a limited number of RF chains. Therefore, one key challenge will be the design of an efficient angle estimation algorithm with limited RF chains. Recently Manuscript received March 26, 2015; revised May 28, 2015; accepted July 1, 2015. Date of publication July 16, 2015; date of current version September 4, 2015. The work of Y. Peng was supported by the NSFC under grant 61171106, the National Key Technology R&D Program of China with grant 2015ZX03002009-004 and the 863 Program with grant 2014AA01A705; the work of Y. Li and P. Wang was supported by the ARC under grant DP150104019, DP120100190, and FT120100487. The associate editor coordinating the review of this paper and approving it for publication was H. Mehrpouyan. Y. Peng is with the Key Laboratory of Universal Wireless Communication, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing 100876, China. Y. Li and P. Wang are with the Centre of Excellence in Telecommunications, School of Electrical and Information Engineering, University of Sydney, Sydney, N.S.W. 2006, Australia. Digital Object Identifier 10.1109/LCOMM.2015.2456872

many angle estimation methods have been proposed for MMW communications, such as [2], but the RF chain constraint is seldomly considered except [3], [4]. In [3], an adaptive compressive sensing (ACS) algorithm has been developed for a MMW channel with the RF chain constraint. This adaptive algorithm can reliably estimate AoA/AoD even in the low signal-to-noise ratio (SNR) region. However, for an MMW system with N antennas and M RF chains at both the base station (BS) and mobile station (MS), it requires kL2 kL/M logk (N/L) time slots and needs to design kL precoding and measurement vectors to find the angles of the L paths, where k is a design parameter with constraint of kL ≤ N because at most N beams can be formed for an antenna array with N elements. Even in this extreme case of kL = N, ACS method still requires NL time slots. This leads to extensive overhead and delay. For fast varying MMW channels, this method might not be able to efficiently track the CSI. On the basis of the work in [3], an overlapped beam pattern design was proposed in [4] to yield a k2 /log22 (k + 1) reduction in measurement time slots with penalty of some degradation of angle detection performance. These two algorithms can only achieve a resolution of O(1/N) in an MMW system with N antennas. This bounded angular resolution will in turn limit the system performance. To overcome this problem, in this letter we propose a concept of antenna array with virtual elements (AAVE), based on which we develop an efficient angle estimation method using compressive sensing (CS) to achieve an enhanced resolution. By following the virtual channel model in [3], [5], a sparse representation model of the MMW channel with AAVE is developed. A transmission design is proposed by jointly optimizing the constraint mapping from the physical array to the AAVE, training sequence, and transmit beamforming/receive combining. The constraint mapping maps the original array of N antennas to the AAVE of K antennas, and at the same time ensures that the signals are not transmitted via the K − N virtual elements such that no physical change is introduced to the antenna arrays. The beamforming/combining design ensures that the resulting sensing matrix can guarantee the reliable recovery of AoAs/AoDs. In contrast to [3], [5], which can only achieve the angular estimation resolution of O(1/N) for a system with N transmit/receive antennas, the proposed method with AAVE can achieve an enhanced angle estimation resolution O(1/K). The resolution will be improved as K increases with a marginal SNR loss. Both the analytical and simulation results show that the proposed design achieves a significantly higher resolution in angles estimation with a much lower overhead, compared to the existing algorithms. Our contributions in this letter are that i) we propose the concept of AAVE, which can improve the angle detection resolution without altering the physical antenna array; ii) based on the AAVE, we develop a compressive angle estimation method with much less overhead and delay than ACS method; iii) we prove that the constructed sensing matrix in our compressive angle estimation method can ensure the reliable angle

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PENG et al.: ENHANCED CHANNEL ESTIMATION METHOD FOR MMW SYSTEMS WITH MASSIVE ANTENNA ARRAYS

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estimation. The major differences between ours and [3] include i) The ACS method proposed in [3] can only reach the resolution of O(1/N) with N being the number of the physical antenna array, while our method can achieve the resolution of O(1/K) by constructing an AAVE with K(K ≥ N) elements from the physical antenna array; ii) Although the signal model for compressive channel sensing in [3] is similar to ours, they have different physical meaning and processing methods. [3] is based on the angle quantization and thus its achievable resolution depends on the training pattern which needs to be designed to ensure that the constructed sensing matrix satisfies the exact recovery property. In contrast, our signal model is based on the AAVE concept, and its achievable resolution is ensured by the designed constraint mapping and training pattern.

II. S PARSE P RESENTATION OF MMW C HANNEL AND A NTENNA A RRAY W ITH V IRTUAL E LEMENTS Consider an MMW communication system with a base station (BS) and a mobile station (MS). The BS is equipped with NB antennas and MB RF chains, while the MS is equipped with NM antennas and MM RF chains. Usually the number of RF chains is less than that of antennas, namely, NB > MB and NM > MM . We assume that both the BS antennas and MS antennas form a uniform linear antenna array (ULA) with a half-wavelength antenna spacing. We consider a block-fading channel, where the channel coefficients remain constant within one block but change independently from one to another. Let’s consider the uplink transmission. Let x ∈ CMM ×1 be the signal to be transmitted at the MM RF chains of MS at a certain PM time slot, satisfying E[xxH ] = M I with PM being the average M total transmit power. Let W B ∈ CNB ×MB and W M ∈ CNM ×MM be, respectively, the combing and beamforming matrices at the BS and MS. Denote by z ∈ CMM ×1 the white Gaussian noise vector with E[zzH ] = σn2 I. The signals after combining process at the BS can be written as y = W TB HW M x + z,

(1)

where H ∈ CNB ×NM denotes the channel matrix with its (m, n)-th element given by [5] Hmn =

L √  α β ej2π(m−1) sin(θ )/2ej2π(n−1) sin(φ )/2 ,

(2)

=1

√ where L is the number of channel paths, α denotes the common large scale fading of all paths including path loss and shadow, β models the small scale fading of the -th path, θ and φ are, respectively, the corresponding AoA and AoD of √ the l-th path. We assume that α is known to both the BS and MS. To improve the angular estimation resolution, we propose a concept of AAVE. The AAVE is constructed by extending the real antenna array to a new one via appending some virtual antennas to the original physical arrays at both BS and MS. Then (1) can be expressed as ˜ M WM x + z y = W TB ETB HE ˜ M x + z, = TB H

Fig. 1. Transceiver structure of the BS with AAVE.

where EB = [eK (1) eK (2) · · · eK (NB )] ∈ CK×NB and EM = [eP (1) eP (2) · · · eP (NM )] ∈ CP×NM with eK (i) ∈ CK×1 being a unit vector with 1 at its i-th entry and 0 elsewhere, termed constraint mapping matrices, respectively map the original physical antenna arrays to the AAVEs and at the same time prevent the signal from being transmitted and received from virtual elements, M  EM W M and B  EB W B define the beamforming/combining direction of AAVEs. Taking the BS as an example, in Fig. 1 we depict the structure of AAVE where K − NB virtual antennas are appended to the physical array of NB antennas, and accordingly MB (K − NB ) virtual links are created to connect these virtual antennas to MB RF ˜ ∈ CK×P is the virtual channel matrix which is linked chains. H ˜ M , and its (m, n)-th element has the same to H by H = ETB HE representation as (2) except the values of m and n are extended from NB and NM up to, respectively, K and P. Let us rewrite the channel model in (2) as  1  1 2 2 Hmn = h(θ, φ)ej2π(m−1)θ ej2π(n−1)φ dθ dφ, (4) − 12

    √  where h(θ, φ)  α L=1 β δ θ − sin2θ δ φ − sin2φ is the channel impulse response (CIR) in the angular domain. In this letter, similar to [3], we assume that the AoAs and AoDs are taken from a uniform grid of K and P points, respectively. Then a discrete sparse representation model of the MMW channel with AAVE can be obtained from (4) as follows [5]. ˜ mn (k, p) = H

K/2−1  P/2−1 

h(k/K, p/P)

k=−K/2 p=−P/2

× ej2π(m−1)(k+K)/K ej2π(n−1)(p+P)/P =

(3)

− 12

K−1  P−1  k=0 p=0

g(k, p)ej2π(m−1)k/K ej2π(n−1)p/P,

(5)

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IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 9, SEPTEMBER 2015

Fig. 2. (a) AAEE versus K for MMW systems with N = 128, M = 8, L = 1; (b) APEE versus r for MMW systems with NB = 64, NM = 32, MB = 10, MM = 6, L = 1, r = K/NB = P/NM .

where g(k, p)  h([[k + K]]K /K, [[p + P]]P /P) with [[·]]K representing modulo K operation, is referred to as the virtual CIR. The angular estimation resolution can be improved when K > NB and P > NM are selected. By using (5), the channel matrix can be expressed as ˜ =  TB G M , H

(6)

where  B ∈ CK×K is a discrete Fourier transform (DFT) matrix whose (k, m)-th entry is ej2πmk/K , 0 ≤ m, k ≤ K − 1 and  M ∈ CP×P is a DFT matrix whose (p, n)-th entry is ej2πpn/P, 0 ≤ p, n ≤ P − 1. The matrix G ∈ CK×P is referred to as the virtual channel matrix in the angular domain. Then the discrete representation of (3) is given by y = TB  TB G M M x + z.

(7)

RM  {1, 2, . . . , NM } where PM is randomly chosen from RM . Similarly the n-th column of W B is chosen as W B (:, n) = eNB (in ), n = 1, 2, . . . , MB and PB  {i1 , i2 , . . . , iMB } ⊂ RB  {1, 2, . . . , NB } with PB being randomly selected from RB . For the proposed design, the resulting sensing matrix guarantees the reliable recovery of AoA/AoDs as shown in the following Lemma. Lemma 1: With the precoding and combining matrices W M and W M designed in the above mentioned approach, the resulting sensing matrix    can ensure the recovery of the angles with a high probability. Proof: Please refer to Appendix A.  Thus, classic compressive recovery algorithms can be used to estimate the angles reliably with resolution π/K for AoA and π/P for AoD. IV. S IMULATION R ESULTS

III. CS BASED C HANNEL E STIMATION In our design, the training signals span continuous MS (MS ≥ MM ) time slots and are transmitted from MM transmit RF chains. The corresponding received signals at BS can be denoted as Y = [y1 y2 · · · yMS ] = TB  TB G M M X + Z.

(8)

We design the X in such a way that XX H = I, then we have y˜ = Vect(YXH )     = TM ⊗ TB ·  TM ⊗  TB · g˜ + z˜ =  g˜ + z˜,

(9)

where   (TM ⊗ TB ) is referred to as the sampling matrix in CS theory,   ( TM ⊗  TB ) is the transform matrix, g˜  Vect(G) ∈ CKP×1 is the virtual channel vector in the angular domain whose sparsity (the number of non-zero elements) is L, and z˜  Vect(ZXH ) ∈ CMM MB ×1 is the noise vector. To guarantee the reliable recovery of g˜ from (9), we design the matrices W M and W B in the following way. The m-th column of W M is chosen as W M (:, m) = eNM (im ), m = 1, 2, . . . , MM . For convenience, we define PM  {i1 , i2 , . . . , iMM } ⊂

In this section we evaluate the proposed method and compare it with the reference ACS algorithm [3]. In simulations the transmit power is normalized to 1 so that SNR = σn−2 . At the receiver, the OMP algorithm [6] is employed in both the proposed algorithm and that in [3]. For simplicity we assume NB = NM  N, MB = MM = MS  M and K = P. Let us first consider a single-path Gaussian channel. Fig. 2 illustrates the average an 

1 gle estimation error (AAEE), defined as E L L=1 |θ − θˆ | with θˆ being the estimate of angle θ , and probability

average 1 L of detection error (APEE), defined as E L =1 I(θ = θˆ ) with I(x) being the indicator function. From Fig. 2(a), it can be observed that the proposed method achieves a lower AAEE than the ACS whose resolution lower bound is the same as the proposed method when K = N. Most importantly the proposed method can overcome the AAEE limit confined by the antenna number N and approach the resolution of π/K by introducing the AAVE of K antennas. For example, when K = 4N, our method achieves an AAEE of 0.0088, which is significantly lower than the AAEE of ACS. In Fig. 2(b) we compare the performance for different r  K/NB = P/NM values, and the simulation setting is the same as that in [3] and the simulation results of the ACS algorithm are borrowed directly from [3]. It can be seen that the proposed method achieves a lower APEE with the same angle estimation resolution when r = 1. As r

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change of the antenna array with a marginal SNR loss. Moreover, the proposed algorithm is much more efficient in channel estimation process than the existing algorithms. A PPENDIX A P ROOF OF L EMMA 1

Fig. 3. AAEE comparison among three channel models for MMW systems with K = 256, N = 128, M = 16.

increases to 4, the proposed AAVE achieves a much lower AAEE compared to the scheme in [3] but this gain is achieved at the cost of 1.5dB SNR loss at the APEE of 10−2. Next we evaluate the AAEE performance of our method for various numbers of paths and different channel models. Three kinds of channel models are simulated: i) two-path Rician fading channel with Rician K-factor 5 and whose AoAs/AoDs are modeled by truncated Laplacian distribution over [−π/2, π/2) with the angle spread of 10◦, ii) Rayleigh fading channel featuring two eqally-powered paths and uniformly-distributed angles, iii) Single-path Gaussian channel. As shown in Fig. 3, both the AAEE of three channel models and that of each path of the Rican fading channel are given. We can observe that i) the AAEE performance degrades as the path number increases due to the decreased power of each path and the increased sparsity measure L/K under the same undersampling ratio M/K, but they both approach the lower bound at high SNR; ii) the AAEE of the two-path Rician fading channel mainly depends on the weak non-line-of-sight path; iii) the SNR gap between the first path of the Rician fading channel and the one-path Guassian channel is about 1.3 dB, which corresponds to the difference in the average power of the first path in the Rician fading and that in the Gaussian fading and the SNR loss due to the increased sparsity measure. These results show that the angle detection performance of our method is quite independent of the channel distribution and instead mainly depends on the SNR, sparsity measure and undersampling ratio. At last we compare the number of time slots consumed by these two schemes. In the first simulation with N = 128, M = 8, L = 1, from (8), the proposed scheme requires 2M = 16 time slots for the BS and MS to estimate the angles, while the ACS method requires at least NL = 128 time slots. On the other hand for the proposed algorithm, when the antenna number of AAVE increases r = K/NB times, the computational complexity of the OMP algorithm increases r times accordingly. V. C ONCLUSION In this letter we have proposed an AAVE concept for MMW communications with massive antenna arrays and RF constraints, based on which a simple compressive channel sensing method was developed. Compared to the existing methods, the proposed algorithm overcomes the limit on angular estimation resolution restricted by the physical antenna number, and thus can achieve an enhanced resolution without any physical

Since the sensing matrix is the Kronecker product of two random partial Fourier matrices which is no more a partial Fourier matrix, to the best of our knowledge, there is no existing work to derive its restricted isometry property (RIP) condition. It is a non-trivial task and non-closed form expressions can be derived to obtain the analytical expression of measurement number for this kind of sensing matrix, required to achieve the RIP condition. Instead, we resort to an alternative condition to RIP by proving that the measurement matrix  is incoherent with the sparsifying basis  in (9) in the sense that the row vectors (m, :) in matrix  cannot sparsely represent the column vectors (:, n) in  and vice versa [7]. According to the design of W B and W M , B and M in (8) can be expressed as M = [eP (i1 ) eP (i2 ) · · · eP (iMM )] with {i1 , i2 , . . . , iMM } ⊂ RM and B = [eK (j1 ) eK (j2 ) · · · eK (jMB )] with {j1 , j2 , . . . , jMB } ⊂ RB . Then the ((s − 1)MB + t)-th row vector of measurement matrix  in (9) can be rewritten as  ((s − 1)MB + t, :) = TM (:, s) ⊗ TB (:, t) = eTP (is ) ⊗ eTK (jk ) = eTKP (K(is − 1) + jt ) .

(A-1)

Similar to W B and W M ,  consists of unit vectors and thus can be regarded as a random sampling matrix. As for the transform matrix  in (9), its ((p − 1)K + k)-th column vector can be rewritten as  ((p − 1)K + k, :) =  TM (p, :) ⊗  TB (k, :)  (p−1) (p−1)(P−1) T P = 1 ej2π P · · · ej2π  (k−1) (k−1)(K−1) T K ⊗ 1 ej2π K · · · ej2π . (A-2) Clearly each element in  is non-zero. From (A-1) and (A-2), it is easy to know that the row vectors in  cannot sparsely represent every column vector in  and vice versa. R EFERENCES [1] Z. Pi and F. Khan, “An introduction to millimeter-wave mobile broadband systems,” IEEE Commun. Mag., vol. 49, no. 6, pp. 101–107, Jun. 2011. [2] W. Hou and C. Lim, “Structured compressive channel estimation for large-scale MISO-OFDM systems,” IEEE Commun. Lett., vol. 18, no. 5, pp.765–768, May 2014. [3] A. Alkhateeb, O. Ayach, G. Leus, and R. Heath, “Channel estimation and hybrid precoding for millimeter wave cellular systems,” IEEE J. Sel. Topics Signal Process., vol. 8, no. 5, pp. 831–846, Oct. 2014. [4] M. Kokshoorn, P. Wang, Y. Li, and B. Vucetic, “Fast channel estimation for millimeter wave wireless systems using overlapped beam patterns,” in Proc. ICC, London, U.K., Jun. 8–12 2015, pp. 2916–2921. [5] W. U. Bajwa, A. Sayeed, and R. Nowak, “Compressed sensing of wireless channels in time, frequency, and space,” in Proc. ACSSC, Pacific Grove, CA, USA, Oct. 26–29, 2008, pp. 2048-2052. [6] J. Tropp and A. Gilbert, “Signal recovery from random measurements via orthogoanl matching pursuit,” IEEE Trans. Inf. Theory, vol. 53, no. 12, pp. 4655–4666, Dec. 2007. [7] R. G. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag., vol. 24, no. 4, pp. 118–120, 124, Apr. 2007.