Low Complexity Compressed Sensing Based Channel Estimation in 3D MIMO Systems Ailing Wang, Ying Wang, Jing Xu, Zehua Wei State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications Beijing, P.R. China 100876 Email:
[email protected]
Abstract—By exploiting the spatial correlation in spatial domain, a three-dimensional (3D) pilot aided channel estimation (PACE) has been proposed to improve the mean-square error (MSE) performance in 3D multiple-input multiple-output (MIMO) systems. However, with the development of 3D MIMO technique, there are increasing number antenna ports in a limited space. The pilot overhead in 3D PACE method which increase linearly with the antenna number becomes unacceptable. Since compressed sensing (CS) technique ignoring the theoretic upper limit in the pilot spacing derived by the sampling theorem has been successfully applied to pilot aided 2D sparse channel estimation in orthogonal frequency division multiplexing (OFDM) systems, we introduce the CS technique to 3D pilot aided channel estimation to reduce pilot overhead in 3D MIMO systems. Moreover, a random search method based non-uniform pilot allocation algorithm with low computational complexity is proposed to further improve the CS performance. Simulation results demonstrate that compared to the traditional evenly pilot for 3D PACE, our proposed non-uniform pilot for CS-based channel estimation its average gain can improves about 3.58dB with the same pilot overhead. The result also shows that by employing the CS-based channel estimation, pilot overhead can be sharply reduced without estimation accuracy loss.
I. INTRODCTION To meet the demand for high data rate and high spectrum efficiency in wireless communication systems, it is expected that in future networks, the multiple-input multiple-output (MIMO) technique has become a hot research topic. Through exploiting vertical direction spatially, three-dimensional (3D) MIMO technique can largely decrease inter-cell interference, further improve throughput and spectrum efficiency [1]. On the other hand, orthogonal frequency division multiplexing (OFDM) technique, dividing the broadband channel into overlapping but orthogonal narrowband subchannels, converts the frequency selective channel into several non-frequency-selective channel. So by combining the MIMO and OFDM technique, high data rata can be satisfied and a burning research area is coming up. As an important part of 3D MIMO systems, channel estimation can provide channel state information (CSI) for diversity combination, coherent detection and other transceiver processes. Traditional approach is pilot aided channel estimation (PACE) which can be performed by inserting reference signals (pilots) into transmitted data [2], and then to estimate the channel based on the received data and the known pilot signals. Increasing pilot density with more resources consumption can improve estimation accuracy, but led to the low data effectiveness. Aiming to reduce pilot overhead channel correlation
in frequency and time domain can be exploited. Furthermore, in MIMO systems, due to close antenna spacing and poor scattering environments, channel correlation in spatial domain exists [3]. In macro-cellular deployments, where the base station (BS) antenna array is mounted above rooftop, measurement campaigns suggest that the angles of departing rays are contained within a narrow angular spread [4], the notion of a narrow angular spread gives rise to spatially correlated channels. By utilizing the channel correlation in spatial domain, the accuracy of PACE can be further improved with less pilot overhead. In order to exploit the third dimension degrees of freedom for 3D MIMO systems, in [5]- [6]the elevation impact of channel model has attracted great attention. By utilizing the channel correlation in spatial domain, they extend MIMO-OFDM channel estimation by interpolating in time and frequency to the spatial domain, and establish a theoretical foundation for 3D PACE. By using 3D PACE, the pilot overhead can be reduced without estimation accuracy loss. However, as 3D MIMO develops, there are increasing number antenna ports in a limited space, the sampling theorem give pilot spacing constrains in space, frequency and time domains. With these constrains, the pilot spacing exists a theoretic upper limit. Hence, the pilot overhead in 3D PACE method which increases linearly with the antenna number becomes unacceptable. Recently, emergent attention has been taken on compressed sensing (CS) which has been widely applied in various areas such as signal processing, communication systems and so on. According to CS theory, if a signal has a sparse representation or in transformation domain has a sparse representation, it can use a sample signal which the rate obvious less than the sampling theorem, and the signal can be reconstructed with some optimization algorithm. In fact, the CS theory have already been used in the channel estimation to reduce pilot overhead and enhance the system performance. For instance, in [7] and [8], the CS theory has been employed for sparse channel estimation in OFDM systems, the pilots are randomly allocated in [7] but uniformly placed in [8]. For optimal pilot allocation [9] obtained the near-optimal pilot pattern by the discrete stochastic approximation. For channel recovery method in [10] a modified CS matching pursuit sparse channel estimation method is proposed. However, all these works were only considered the pilot design in frequency or time domain, if directly applied this method to the 3D MIMO systems, with
978-1-4799-8088-8/15/$31.00 ©2015 IEEE
the antenna number increasing the pilot overhead is still a serious problem. By exploiting the spatial correlation in spatial domain, although 3D PACE can reduce pilot overhead, but the pilot interval still be affected by multi-dimensional sampling theorem. Thus, to further reduce the pilot overhead, in this paper a CS-based channel estimation is used in 3D MIMO systems. The key challenge of CS-based channel estimation lies in how to search optimal pilot allocation for channel estimation to ensure minimum mean-square error (MSE). Hence, a random search method based non-uniform pilot allocation algorithm with low computational complexity is proposed to further improve the CS performance. Simulation results show that the proposed non-uniform pilot for CS-based channel estimation greatly improves performance than the evenly pilot in terms of the MSE as well as the less pilot overhead. The remainder of this paper is organized as follows. Section II presents the system and 3D channel model. Section III analyzes channel sparsity and raise a problem of CS channel estimation. Section IV describes the pilot allocation and analyzes the performance. Simulation results are provided in Section V. Finally, conclusions are drawn in Section VI. II. SYSTEM AND 3D CHANNEL MODEL Assume a MIMO-OFDM with the number of subcarriers to be K, L OFDM symbols and Nt transmit antennas. At the transmitter, after serial-to-parallel conversion (S/P), the transmit signal vector of subcarrier k at OFDM symbol l is transmitted by an antenna array with elements. So the incoming data stream forms can denoted by Xk,l = Nt ×1 t , 1 ≤ u ≤ Nt , 1 ≤ k ≤ K, [x1k,l , · · · , xuk,l , · · · , xN k,l ] ∈ C 1 ≤ l ≤ L, where u, k and l are the antenna, subcarrier and symbol indexes, respectively. The sum of the transmit u 2 power is E xk,l =Es . Then, the OFDM modulation performed by K point IFFT, followed by an insertion of cyclic prefix (CP) with length Ncp . The transmit signals propagate through a multipath fading MIMO channel which dimension Nt × Nr , Nr is receive antennas. In order to facilitate the research, we only consider the channel correlation between transmit antennas, so we only take one receive antenna into consideration, that is, Nr = 1. At the receiver, assuming perfect synchronization and frequency offset using the estimation algorithm, so the effect of the frequency offset and timing errors caused can be negligible, after the OFDM demodulation, the received signals can denoted by: yk,l = xk,l T hk,l + zk,l ,
(1)
Nt ×1 t T where hk,l = [h1k,l , · · · , huk,l , · · · , hN stands for k,l ] ∈ C the channel frequency response between transmit antenna u and the receive antenna of subcarrier k at OFDM symbol l; zk,l denotes additive white Gaussian noise (AWGN) with zero mean and N0 variance. For MIMO-OFDM system, in vector notation the signal of OFDM symbol block l can be conveniently expressed as:
yl = Xl hl + zl ∈ CK×1 ,
(2)
] [
0 ,M 0
8/$ [
900 , M
90
subpath
[q ,s
1800
Mq ,s
[
d \
0
[
90 , M 0
0
0
[
Antenna spacing
90 , M 0
90
0
Fig. 1. Subpath s within cluster q at the transmitter with auniform linear antenna array (ULA)
where the two-dimensional transmit signals consists of transmit symbols in spatial and frequency domain, can be denoted by Xl = diag[xT1,l , · · · , xTK,l ] ∈ CK×Nt K . Meanwhile, hl = [hT1,l , · · · , hTK,l ]T ∈ CNt K×1 indicates the channel response of one OFDM symbol. The noise vector can be represented by zl = [z1,l , · · · , zK,l ] ∈ CK×1 . So the received signal of one block of L consecutive OFDM symbols, T T ] ∈ CKL×1 is in the form: y = [y1T , · · · , yL y = Xh + z ∈ CKL×1 ,
(3)
where X = [X1 , · · · , XL ]T ∈ CKL×Nt KL , h = [hT1 , · · · , hTL ]T ∈ CNt KL×1 , z = [zT1 , · · · , zTL ]T ∈ CKL×1 . A. Channel model Compared with 2D MIMO, 3D MIMO take the elevation impact into consideration. And for the multipath fading MIMO channel can be modeled as the sum of multiple clusters with different delay. Each cluster consists of some subpaths [6] illustrates as Fig. 1. The direction of transmitter subpaths can be described by the elevation angular of departure (EAOD) ξq,s and the azimuth angular of departure (AOD) ϕq,s , where q, s symbolize the index of subpath s within cluster q. Assuming that the transmit antenna elements are uniformly spaced with a spacing of d and the structure is a uniform linear antenna array (ULA), the 3D channel frequency response can be molded as [5]: H u (k, l) =
L−1
k hu (k, l)WK ,
(4)
l=0
where WK = in section III.
√1 K
exp(−j2πukl/K). H u (k, l) will be given
B. Channel correlation As [4] described if the BS antenna arrays are mounted above rooftop, the angulars of departure are limited within a narrow
40
20
0 1000
800
Δ
∗ = E {H · HΔ } (u+Δu)∗ u = E Hk,l · Hk+Δk,l+Δl ;
600
400
200
Subcarrier
Δ = [Δu, Δk, Δl]
T
.
(5) It is known that three independent correlation functions in time, and spatial domain comprise the autocorrelation function (5). Hence, the autocorrelation function in (5) can be separated into three parts as follows: (a) The time variations caused by the Doppler effect are described as correlation function in time domain: u u∗ · Hk,l+Δl Rt (Δl) = E Hk,l , (6) = J0 (2πfD max Tsym Δl)
0
1
5
4
3
2
6
7
6
7
OFDM symbol
(b) channel under IFFT
10 Amplitude
R (Δ)
(a) Original channel
Amplitude
angular spread, which can increase the spatially correlation at the transmitter. In this paper, the spatially correlation refers to the correlation between transmit antenna elements associated to neighboring transmit antenna elements. The autocorrelation of the channel response can be defined as [11]:
5
0 1000
800
600
400
200
Subcarrier
0
1
5
4
3
2
OFDM symbol
Fig. 2. Performance of channel sparsity in one OFDM symbol block case : (a) original channel, (b) channel under IFFT (a) Original channel
where τq be the delay associated to cluster q, 1/Tc is the carrier frequency spacing. (c) In case the BS is mounted above rooftop, the spatial correlation between transmit antennas is in the form: (u+Δu)∗ u · Hk,l Rs (Δu) = E Hk,l , (8) d = ej2π λ sin ϕq,s sin ξq,s where d is the transmit antenna spacing, ϕq,s and ξq,s are as illustrated in Fig. 1. By exploits the channel correlation in time, frequency and spatial domain to decrease pilot overhead and improve channel estimation accuracy. III. PROBLEM FORMULATION A. Channel sparsity In order to use CS theory for channel estimation, first we should verify that the channel is sparse. The channel frequency response between transmit antenna and the receive antenna can be formed as [6]:
2π Sq sin ξq,s · sin ηq,s · ej λ du sin ξq,s sin ϕq,s u , H (k, l) = k ·ej2πfD,q,s lTsym · e−2π Tc τq · ejvq,s s=1 (9) where ηq,s is the elevation angular of arrival (EAOA), Sq stands for subpaths, ejvq,s can be understood as the phase of subpath s within cluster q. Adopt an urban macro scenario (Non Line Of Sight (NLOS) clustered delay line model) given in WINNER [4]. As we
40
Amplitude
30 20 10 0
0
100
200
300
400
500 Subcarrier
600
700
800
900
1000
700
800
900
1000
(b) channel under IFFT 8 6 Amplitude
where fD max indicates the maximum Doppler frequency, Tsym is the symbol duration and J0 () stands for the zero order Bessel function of the first kind. (b) The correlation function in frequency domain denotes channel distortion due to multi-path propagation, which can be shown as: u u∗ · Hk+Δk,l Rf (Δk) = E Hk,l , (7) Δk = e−j2π Tc τq
4 2 0
0
100
200
300
400
500 Subcarrier
600
Fig. 3. Performance of channel sparsity in one OFDM symbol case : (a) original channel, (b) channel under IFFT
know if a signal has a sparse representation or in transformation domain has a sparse representation, it can use CS theory to reconstruct original signal. So in this paper, we first use IFFT transform the original channel into transform channel. Fig. 2 shows the performance of channel sparsity in one OFDM symbol block case, compared with the original channel after IFFT to transform, the channel only a few peak amplitude is large, the rest of the magnitude is very small which can approximate as 0. In order to indicate the channel sparsity more clearly, Fig. 3 (a) and Fig. 3 (b) denote the performances of original channel and the channel which transformed by IFFT, respectively. B. Problem formulation CS is a technique to reconstruct sparse signals accurately from a limited number of measurement. So in order to facilitate understanding the problem, CS theory is discussed first. Given a known sparsity signal vector s ∈ CN ×1 , and s is K-sparse (it has at most K nonzero entries), the compressed sensing equation can be expressed as:
y = Θs + η,
(10)
where Θ ∈ CM ×N represents a known measurement matrix; y stands for the observed vector; M is usually smaller than N , and η ∈ CM ×1 denotes a noise vector. The objective of CS theory is to reconstruct s from the information of y and Θ. It has been proved that s can be reconstructed correctly when Θ satisfies the restricted isometry property (RIP). In our study, assume selected Np = (Nu , Nk , Nl ) point to send pilot symbol from the block data (Nt , K, L) , where 1 ≤ Nu ≤ Nt , 1 ≤ Nk ≤ K, 1 ≤ Nl ≤ L and Nu , Nk , Nl represents the selected antenna, subcarrier and symbol to send pilot symbol, respectively. According to Section II, the problem is formulated as: Yp = Xp Wp h + Zp ,
(11)
where Yp denotes the received pilot symbols matrix; Xp represents the transmit pilot symbols matrix; h is M-sparse. Let N = Nt × K × L, W p ∈ CNp ×N is a partial FFT matrix which selection Np rows from a standard N × N dimensional FFT matrix, and its (u, k, l)s component is given by W p = √1K exp(−j2πukl/K), Np denotes selected pilot allocation. Here, Yp , Xp and W p are all known to the receiver. So we can use some methods to estimate h from (11). Then, the channel frequency response H can be achieved by H = Wh. Consequently it is a typical problem of sparse signal reconstruction, which can be solved by a series of algorithms based on CS theory. When Np N , comparing to conventional channel estimation algorithm, CS-based channel estimation algorithm would reduce the pilot overhead and also obtain a better MSE performance. Reformulated (11) as:
where ai represents the i-th column of A. According to [12] we can know that the smaller μ(A) is, the better approximation of h can be achieved. Hence, we can optimize pilot allocations according to μ(A). Clearly, (14) can be reached by exhaustive search in N N combinations, since A is a large and needs search Np matrix, so it is not an easy problem to solve. Therefore, in this paper, we propose a random search method based non-uniform pilot allocation algorithm with low computational complexity. Following a random search method the problem of (14) can be solved with a suboptimization. Assume Ψ represents all
N possible pilot allocation with the size of , Λ denotes the Np selected optimal pilot allocation index. The detailed random search algorithm can be described as Algorithm I : Algorithm I Pilot allocation selection method Step 1: Initialization •
t =
•
for t = 1 : Q calculating µt (A)
Δ
=
max
|aT i aj |
1≤i,j≤N,andi=j ai ·aj
, t
=
1, 2, . . . , Q; if µm (A) ≥ µt (A) µm (A) = µt (A); Λm = Λ t ; else µm (A) = µm (A); Λm = Λ m ; end end
Step 3: Finalization Λm is achieved, output.
(12)
where A = Xp Wp . An estimator of h based on convex optimization was proposed as: = arg min h , s.t. y − Ah ≤ ε, h 1 2
⊂ Ψ,
Step 2: Calculation
•
y = Ah + z ,
Initialize Randomly generating Q index, Λt 1, 2, . . . , Q; µm (A) = µ1 (A); Λm = Λ1 ;
(13)
h
where ε > 0 indicates permissible error, and A can be regarded as measurement matrix. IV. PILOT ALLOCATION As mentioned above, seeking optimal pilot allocation for channel estimation to ensure minimum MSE is extremely important. Designing the measurement matrix A with a certain property would increase channel estimation success probability. Measurement matrix is determined by the pilot symbols and pilot allocations. In order to simplify the computational complexity, in this paper we fix pilot power and only optimize the pilot allocations. The mutual coherence of the measurement matrix A is defined as:
T
a aj Δ i , (14) max μ (A) = 1≤i,j≤N,andi=j ai · aj
This algorithm can reduce the computation times
suboptimal N from to Q, and if Q large enough, Λm will be closest Np to the result of optimization of (14). V. SIMULATION RESULTS The channel estimation error MSE can be obtained by:
2
ˆ MSE = E H − H , (15) ˆ stands for where H denotes practical channel matrix, H estimate channel matrix. Although the channel estimation MSE is a commonly properties to demonstrated the performance, but not assess the performance of a MIMO-OFDM systems. So the following analysis the SNR degradation due to channel estimation errors. Es , Assume that the total transmit power is Es , SNR is γ = N 0 takes into account channel estimation errors the effective SNR γ˜ can be described by: γ˜ =
γ Es = . N0 + Es · MSE 1 + γ · MSE
(16)
Eq. (16) shows that, as MSE reduces γ˜ will increases, to this end, in order to get higher γ˜ , we should set MSE small enough.
0
10
2D PACE with Np= 880 3D PACE with Np= 880 CS with Np= 880 CS with Np= 700 −1
10
Consider an MIMO-OFDM systems with Nt = 5, the number of subcarriers to be K, L OFDM symbols is 1024 and 7, respectively. The transmit antennas spacing is d = λ/2. The carrier frequency, symbol duration and carrier spacing are 2GHz, 28.8us and 1/25.6MHz respectively. Moreover, the other urban macro scenario parameters are same as reference [6]. In 2D PACE and 3D PACE, for the pilot spacing in space, frequency and time domain is Ds = 1, Df = 14, Dt = 4 and Ds = 2, Df = 7, Dt = 4. In these cases, both 2D PACE and 3D PACE in one resource block (RB) exhibit the same pilot overhead Np = DsNDttBDf = 880, and the channel estimation method employing least square (LS) algorithm. By using CS-based channel estimation algorithm, adopt University of British Columbia (UBC) released Matlab SGPL1 [13] toolbox spg bpdn () function BP algorithm simulation and reconstruction error ε is set 1 × 10 - 3 , in this case the pilot overhead were selected with Np = 880 and Np = 700. B. Results Analysis Fig. 4 shows the MSE versus SNR plots of the proposed method to optimal pilot allocation in comparison with evenly pilot patterns. With the same pilot overhead, estimation accuracy can be improved by utilizing the channel correlation in time, frequency and space domains at the same time from the comparison of 2D PACE and 3D PACE. Compared with blue dotted line and the two solid lines in Fig. 4, the results show that the CS-based channel estimation algorithm can greatly improves the estimation accuracy than evenly pilots in 2D PACE and 3D PACE, and the MSE performance can be its average improved about 6.71dB and 3.58dB, respectively. Furthermore, in Fig. 4 the black dotted line shows that by employing the CS-based channel estimation, pilot overhead can be sharply reduced without estimation accuracy loss. VI. CONCLUSION In this paper, we introduce the CS technique to estimate channel in 3D MIMO systems and propose a random search method based non-uniform pilot allocation algorithm with low computational complexity. Simulation results show that with the proposed optimization algorithm on pilot allocation, MSE performance improves significantly compared with evenly pilot allocation. Moreover, by utilizing the CS-based channel estimation the pilot overhead can be largely reduced without sacrificing the estimation accuracy. In the next research work, great efforts will be made to optimization of channel estimation algorithm and more effective search schemes should be developed for optimal pilot allocation. ACKNOWLEDGMENT This work is supported by National Major Science and Technology Project (2013ZX03001025-002), National 863
MSE
A. Numerical Example
6.71dB
−2
10
3.58dB
−3
10
−4
10
0
5
10
15 SNR[dB]
20
25
30
Fig. 4. Comparison of the channel estimation performance among the 2D PACE, 3D PACE and the proposed CS method
Project (2014AA01A705), National Nature Science Foundation of China (61421061). R EFERENCES [1] E. Telatar, “Capacity of multi-antenna gaussian channels,” European transactions on telecommunications, vol. 10, no. 6, pp. 585–595, 1999. [2] S. Coleri, M. Ergen, A. Puri, and A. Bahai, “Channel estimation techniques based on pilot arrangement in ofdm systems,” Broadcasting, IEEE Transactions on, vol. 48, no. 3, pp. 223–229, 2002. [3] M. K. Ozdemir, H. Arslan, and E. Arvas, “Mimo–ofdm channel estimation for correlated fading channels,” in Proc. IEEE Wireless and Microwave Technol. Conf, vol. 1, 2004, pp. 1–5. [4] I. Winner, “D1. 1.2,winner ii channel models,,” URL-http://www. istwinner. org/deliverables. html, 2007. [5] G. Auer, “Bandwidth efficient 3d pilot design for mimo-ofdm,” in Wireless Conference (EW), 2010 European. IEEE, 2010, pp. 701–705. [6] X. He, J. Zhang, and W. Bao, “Pilot aided 3d channel estimation for mimo-ofdm systems,” in Vehicular Technology Conference (VTC Spring), 2013 IEEE 77th. IEEE, 2013, pp. 1–5. [7] G. Taubock, F. Hlawatsch, D. Eiwen, and H. Rauhut, “Compressive estimation of doubly selective channels in multicarrier systems: Leakage effects and sparsity-enhancing processing,” Selected Topics in Signal Processing, IEEE Journal of, vol. 4, no. 2, pp. 255–271, 2010. [8] C. R. Berger, S. Zhou, W. Chen, and P. Willett, “Sparse channel estimation for ofdm: Over-complete dictionaries and super-resolution,” in Signal Processing Advances in Wireless Communications, 2009. SPAWC’09. IEEE 10th Workshop on. IEEE, 2009, pp. 196–200. [9] C. Qi and L. Wu, “A study of deterministic pilot allocation for sparse channel estimation in ofdm systems,” Communications Letters, IEEE, vol. 16, no. 5, pp. 742–744, 2012. [10] N. Wang, Z. Zhang, G. Gui, and P. Zhang, “Improved sparse channel estimation for multicarrier systems with compressive sensing,” in Wireless Personal Multimedia Communications (WPMC), 2011 14th International Symposium on. IEEE, 2011, pp. 1–5. [11] G. Auer, “3d mimo-ofdm channel estimation,” Communications, IEEE Transactions on, vol. 60, no. 4, pp. 972–985, 2012. [12] X. He and R. Song, “Pilot pattern optimization for compressed sensing based sparse channel estimation in ofdm systems,” in Wireless Communications and Signal Processing (WCSP), 2010 International Conference on. IEEE, 2010, pp. 1–5. [13] M. Friedlander and E. van den Berg, “Toolbox spgl1,” Univ. British Columbia. Vancouver, BC, Canada [Online]. Available: http://www. cs. ubc. ca/labs/scl/spgl1, 2011.
