SWISS: Spectrum Weighted Identification of Signal Sources for mmwave Systems

Transcription

1 SWISS: Spectrum Weighted Identification of Signal Sources for mmwave Systems Ziming Cheng, Jingyue Huang, Meixia Tao, and Pooi-Yuen Kam Dept. of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China Department of Electrical and Computer Engineering, National University of Singapore, Singapore s: {charlotte311, huangjingyue, Abstract This paper considers the channel estimation problem in millimeter-wave (mmwave) systems where a singleantenna user communicates with a massive multiple-input multiple-output (MIMO) base station (BS) in the uplink. Unlike many existing works which estimate the channel gain under the assumption that the number of channel paths is given a priori, we address first the problem of path-number identification. By taking the weighted discrete Fourier transform (WDFT) of the received noisy signal, we formulate an optimization problem to determine the optimum combination of DFT components in this weighted spectrum that leads to a time-domain reconstructed signal (the channel vector) that is at the minimum Euclidean distance from the received signal. Our algorithm, called SWISS (Spectrum Weighted Identification of Signal Sources), is an accurate and computationally efficient means for identifying the paths in the channel vector, providing the information needed for BS beamforming. Once the paths are identified, their individual directions-of-arrival (DoAs) and complex fading gains can be obtained easily. Simulation results for the case of no power leakage in the DFT are presented to demonstrate the effectiveness of SWISS. I. INTRODUCTION Communication over millimeter wave (mmwave) bands from 30 to 300 GHz is a promising technology for 5G mobile networks to provide multi-gigabit communication services [1]. Beamforming with massive multiple-input multiple-output (MIMO) antennas in mmwave communications can further boost the transmission coverage and reliability. However, to exploit the beamforming gains, the transmitter must know the perfect channel state information (CSI). It is very challenging to obtain the accurate CSI with the deployment of largescale antenna arrays. The conventional training strategy with orthogonal pilots [2] cannot be applied here since it requires that the minimum length of the training sequence must be equal to the number of transmit antennas, which causes severe overhead. Moreover, as the size of channel dimension increases, the matrix operations involved in channel estimation induce prohibitively high complexity in practical systems. Recent contributions to the channel estimation problem in mmwave massive MIMO communication, e.g., [3] [6], mostly exploit the sparse nature of the mmwave channel This work was done while P. Y. Kam was a visiting professor at Shanghai Jiao Tong University, Shanghai, China. This work is supported by the National Natural Science Foundation of China under grants and to reduce the effective number of channel parameters. The authors in [3] consider the joint design of hybrid precoding and channel estimation, and propose a compressed sensingbased channel estimation method for the hybrid architecture. [4] develops an efficient channel estimation algorithm based on an overlapped beam pattern design. In [5], the authors propose a two-dimensional discrete Fourier transform (DFT)- based algorithm to estimate the angular information and the channel gain separately in a downlink 60GHz indoor system. The authors in [6] build a DFT-based spatial basis expansion model (SBEM) to decrease the parameter dimensions of the channel and propose a unified channel estimation strategy for multiuser time-division duplex (TDD)/frequency-division duplex (FDD) massive MIMO systems. Note that all the aforementioned works [3] [6] assume that the number of paths is known a priori. Estimating the number of paths is critically important for reconstructing the actual channel. Given the similarity between directionof-arrival (DoA) estimation and multiple-discrete-frequency signal estimation, the MUSIC algorithm [7] can be applied to estimate the number of paths and the corresponding DoAs. However, the singular value decomposition (SVD) method in the MUSIC algorithm works only if the complex channel gains of the paths are independently fast fading in time. For massive MIMO channel estimation, a block-wise constant fading channel model is more accurate and practical, which does not meet the independence requirement of the SVD method. Moreover, the SVD operation in MUSIC is of high computational complexity and is not practical for implementation in large scale systems. Thus, in this paper, we develop a simple and effective algorithm to detect the number of paths accurately for a massive-mimo, mmwave channel based on a block-wise constant fading model. This frequency-domain algorithm enables us to also estimate the DoA of each path and its complex fading gain. In this paper, we assume that the channel signal arriving at the BS from a single-antenna user consists of an unknown number of discrete paths, each with unknown complex amplitude and DoA. By taking the DFT of the received signal, we can, in principle, determine its structure in the angular (frequency) domain, if not for the received additive white Gaussian noise (AWGN). Nevertheless, despite the presence of the AWGN, we can determine the optimum combination of

2 angles (frequencies) that best matches the received signal in the least square sense. To this end, we propose a novel algorithm named Spectrum Weighted Identification of Signal Sources (SWISS) for path number detection, and for estimation of the DoA and complex fading gain of each path. The idea is to apply a weight vector w to the spectrum of the received signal, one component for each DFT point, and then take the inverse DFT of this weighted spectrum to reconstruct the channel signal. By minimizing the Euclidean distance between the received noisy signal and the reconstructed channel signal, subject to a constraint on the norm of w being less than or equal to unity, we can obtain the optimum w, denoted by w. This convex optimization problem can be solved by the method of Lagrange multipliers, and we can derive a closedform expression for w, which involves the corresponding dual variable. We then apply the bisection method to find the optimal dual variable efficiently, and recover w numerically. Finally, we obtain the number of paths by identifying the number of significant components in w. We should emphasize the conceptual and computational simplicity of our SWISS, which will be clear after its presentation. The recovered w is the key result for SWISS. For simplicity and lack of space, this paper will focus only on the simulated performance of SWISS for the case of no power leakage, i.e., the DoAs coincide with the DFT points. The simulation results first show that the components of w are always nearly zero for those DFT points that contain noise only and no signal in the corresponding DoAs. The components of w corresponding to the presence of signals are usually much larger than those that correspond to noise alone, and therefore making the paths much easier to identify than by using the original DFT. Thus, SWISS can efficiently reduce the impact of the noise on path identification. To detect the number of paths, we determine the minimum number of components of w whose sum of squares exceeds a certain threshold η that is close to one (e.g. 99.5%) and take this minimum number as the number of paths. Simulations show that SWISS can correctly identify the number of paths with high probability if the signal-to-noise ratios (SNR) of the paths are not too low. In the no-signal case, all the components of w are near to zero, and SWISS can thus serve as a signal detector. Once the paths are identified, we can determine their DoAs (i.e., the frequency values at the corresponding DFT points), and the maximum likelihood (ML) estimates of their corresponding complex amplitudes (magnitudes and phases) are simply the complex amplitudes of the DFT values at these points [8]. In summary, SWISS can determine not only the number of paths but also the DoA and complex amplitude of each path in the channel vector of the user, as an extension of the work in [8]. For lack of space here, we will focus only on the performance of determining the number of paths. We will present the full theoretical analysis of the performance of SWISS in a future report. There, we will show that while the setting of the abovementioned threshold η is done here in an ad hoc manner, a related threshold can be set using the Neymann-Pearson criterion [9]. For instance, holding a constant false alarm rate (the probability of declaring a DFT point to contain a signal when there is actually no path there), we can set the threshold to maximize the detection probability of a path at any DFT point when it is actually there. Notations: Boldface uppercase letters denote matrices and boldface lowercase letters denote vectors. R and C denote the sets of real numbers and complex numbers, respectively. ( ) T, ( ) H are the transpose, and Hermitian transpose, respectively. CN(μ, σ 2 ) represents a complex Gaussian distribution with mean μ and variance σ 2. y N(0, Q) means y is a Gaussian random vector with mean 0 and covariance matrix Q. II. SYSTEM MODEL Consider a uplink transmission of a mmwave system form a single-antenna user to a BS equipped with M antennas. Assume that the channel between the BS and the user is blockwise constant over the duration of T pilot-symbols, denoted as h C M 1. Let r(t) C M 1 be the received signal at the t-th time point given by r(t) =hs(t)+n(t),t=0, 1,..., T 1, (1) where s(t) is the pilot symbol and n(t) CN(0,σ 2 I M ) is the AWGN. By applying the mmwave channel model in [10], the corresponding channel vector can be written as M P h = α p a(φ p ). (2) P p=1 Here, P is the number of paths, α p is the complex fading gain of the p-th path, φ p represents the DoA of the p-th path, and a(φ p ) denotes the antenna array response vector at DoA φ p. We assume an uniform linear array (ULA) whose array response vector can be expressed as a(φ) = 1 [1,e j 2π λ d sin(φ),,e j(m 1) 2π λ d sin(φ)] T, (3) M where λ is the wavelength and d is the antenna spacing. While the general situation in practice would involve more than one user, we focus on the one-user case here to clarify the structure of our novel SWISS algorithm. III. PROPOSED ALGORITHM Without loss of generality, we set the pilot symbols to be s(t) =1, t {0, 1,..., T 1}. Then an initial estimation of the channel vector can be obtained via the least square (LS) method as ĥ LS = r, (4) where r C M 1 is the average received noisy signal given by r = 1 T 1 r(t) =h + 1 T 1 n(t). (5) T T t=0 t=0 However, the LS estimate does not provide the detailed information on the channel vector h, i.e., the number of paths and their DoAs. Such detailed information is necessary for designing beamforming at the BS. In what follows, we propose our approach for obtaining this information by the use of the DFT and the idea of optimally weighting the spectrum.

3 A. The DFT of the Channel Vector It is proved in [6] that when the number of antennas at the BS, M, the normalized DFT of the channel vector h expressed in (2) only has finite number of nonzero points, and the number of nonzero points equals the number of paths. More specifically, if there is only one path in h, i.e., h = αa(φ), then the DFT of h will have only one nonzero point as M. Denote the index of the nonzero point as q 0, for q 0 {0, 1,..., M 1}, and the DoA of the signal can be calculated as φ = arcsin(q 0 λ/md). For a multipath channel, each path will be mapped to a distinct DFT point as M. Defining the indexes of these nonzero points as q 1,q 2,..., q P, the DoA of each path can be calculated as φ p = arcsin(q p λ/md),p= 1, 2,..., P. Based on the above finding in [6], we define the normalized DFT of the estimated channel vector ĥls as g LS = FĥLS, where F is the M M DFT matrix whose (m, n)-th element is [F] mn = 1 M e j 2π M mn, 0 m, n M 1. From the normalized DFT of the estimated channel vector, one can in principle identify all the paths from the number of nonzero points of the spectrum. However, noise can easily cause errors in identifying the paths. Thus, in the next subsection, we introduce a weighted DFT approach to amplify the desired signal components and suppress the components purely due to noise. B. Spectrum Weighting for Channel Vector Reconstruction Define a weight vector w = [w 1,w 2,..., w M ] T, with w Then the weighted Discrete Fourier Transform (WDFT) of the estimated channel vector can be expressed as ĝ LS = Wg LS = WFĥLS, (6) where W = diag(w). Since the M M DFT matrix F is unitary, the inverse DFT matrix is F H, and we reconstruct the channel vector, denoted as ĥ, by taking the inverse of the WDFT as ĥ = F H ĝ LS. (7) The weighting vector w is chosen to minimize the Euclidean distance between the average received signal r (or equivalently the initial channel estimate ĥls) and the reconstructed channel vector ĥ. We thus formulate an optimization problem as follows: P : min r w ĥ 2 (8a) s.t. w (8b) This problem is a convex quadratically constrained quadratic programming (QCQP) problem, which can be solved by a general-purpose solver through interior-point methods in general. In what follows, we exploit the specific structure of problem P, and obtain some insights from the expression of the optimal solution. Define y = T F r, then problem P can be reformulated as P 1 : min Q (9a) w s.t. w (9b) where Q = w H Aw 2Re{b T w} A = diag[ y 1, y 2,..., y M ] T b =[ y 1, y 2,..., y M ] T. The Lagrangian of problem P 1 is given by (10a) (10b) (10c) L μ = Q + μ( w 2 2 1), (11) where μ 0 is the dual variable associated with the inequality constraint w By setting the derivative of L μ to zero with respect to w, the optimum w can be expressed as w =(A + μi M ) 1 b. (12) To find the optimum dual variable μ, we substitute the result (12) back into the inequality w 2 2 1, and obtain f(μ) M i=1 y i 4 ( y i 1 0. (13) + μ) 2 Taking a closer look at the function f(μ) in (13), it is clear that f(μ) is monotonically decreasing in the region μ 0. According to the complementary slackness condition [11], we have μf(μ) =0, for some value of μ. Since f(0) = M 1 > 0 because M >> 1 in massive MIMO, f(μ) must have a zero-crossing in the region μ > 0, and the optimum μ can be found efficiently using a root finding method, such as bisection search method. The optimum w can thus be computed numerically from (12) after finding the optimum μ. C. Path Identification After obtaining the optimum w, we first sort { wi 2 } M i=1 in descending order, and then select the minimum number of the points in a set, which is defined as S, whose sum of the squares of the weights exceeds a pre-defined threshold, η < 1, i.e., i S w i 2 η. The cardinality of set S is the number of paths P. The subscripts i in the set S will then give us the locations of the DFT points of the paths, denoted as {q 1,q 2,...q P }. Under the assumption of no power leakage, i.e., the DoA of each path falls right on a DFT point, we can calculate the DoA of these paths as φ p = arcsin(q p λ/md),p=1, 2,..., P. IV. NUMERICAL RESULTS AND DISCUSSIONS In this section we illustrate the performance of the proposed SWISS using numerical results. The total number of antennas is set to be M = 128 and the number of pilot symbols is set to be T =10. In practical systems, the number of transmit antennas can be large but not infinite, and thus the power leakage may happen. To avoid the power leakage, we assume in the simulations that the angles coincide with DFT points, i.e., φ p = arcsin(q p λ/md),p=1, 2,..., P.

4 almost 100% with 9 DFT points Fig. 1. The squares of weights in noiseless case (P =1) Fig. 3. The squares of weights in noiseless case (P =9) 10-2 over 99.5% with 9 DFT points Fig. 2. The squares of weights in noisy case (P =1, SNR = 10dB) Fig. 4. The squares of weights in noisy case (P =9, σ 2 =1) A. The Single-Path Case First we consider the special case where there is only one path, i.e., h = α 0 a(φ 0 ). We set the actual DoA to be φ 0 = arcsin(q 0 λ/md),d= λ 2 and choose q 0 =60. The gain of the channel is set to be α 0 = 10dB. Assume that there is no noise first. By using the proposed method, the square of the optimal weights is shown in Fig. 1. From the figure, we can see that the weight becomes almost 1 at the DFT point q 0 =60, while the weights at all other points are nearly 0. It is thus clear that the number of path is 1 in the noiseless scenario. Then we consider the noisy case. The variance of the noise is normalized to σ 2 =1so that the SNR of the path is 10dB. Fig. 2 plots the squares of the weights at different DFT points using the proposed method. It can be seen that at q 0 =60the square of the weight is still much larger than those at other points. In this case we can also find that the number of path is 1. Note that the SNR of the path is 10dB, which shows that SWISS can locate the angle even though the amplitude is very weak and the square of weight of the path is almost 1. B. The Multipath Case Next we consider the multi-path case. Suppose there are P =9paths and the DoAs of the paths are chosen as φ i = arcsin(q i λ/md),d= λ,q i =20+5(i 1),i=1,..., P. The gain of each path α p is chosen from a uniform distribution over the interval of [ 5dB, 5dB]. Fig. 3 plots the squares of the weights when there is no noise. We can see that the sum of the squares of all these nine weights is almost 1 and the weights of noise points are nearly zero. Thus we can easily identify that the path number is 9. Fig. 4 plots the squares of the weights when the variance of the noise is σ 2 =1. It can be seen that the weights of the nine paths are greatly larger than others. If we select the points whose sum of squares of weights exceeds η =99.5%, i.e., i S 99.5%, we will get exactly nine paths. From Figs. 1 4 with both the single-path case and the multipath case, we can see that the weights of the noise points are almost zero in both noiseless and noisy cases. This indicates that the proposed SWISS can efficiently suppress the noise components and therefore identify the correct number of paths under the assumption of no power leakage. We further evaluate the performance of SWISS by the probability of detecting the correct path number. The benchmark is selected as the original DFT method without any weighting. We perform the DFT of the initial estimated channel vector and identify the number of significant components in the DFT vector directly using a similar threshold. First we consider a two-path case and fix the gain of one path, e.g., α 1 =3dB in Fig. 5. The noise variance is set to be σ 2 =1. Note that these parameters are only chosen for illustration; similar results are observed for other choices of parameters. We set the number of pilot symbols as T =5and 10, respectively. η is set to be 95% and 90%. The number of simulation runs is Fig. 5 plots the probability of identifying exactly two paths versus different gains for the second path. When T =5and η = 95%, the original DFT method is unable to identify exactly two paths, while SWISS

5 probability SWISS,T=5, =95% SWISS,T=10, =95% SWISS,T=5, =90% SWISS,T=10, =90% DFT method,t=5, =95% DFT method,t=10, =95% DFT method,t=5, =90% DFT method,t=10, =90% the gain of the second path / db Fig. 5. The probability of identifying exactly two paths in the twopath case Fig. 7. The performance of SWISS in the no-signal case (σ 2 =1) SWISS, =95% SWISS, =99.5% DFT method, =95% DFT method, =99.5% 10-4 probability the identified number of paths Fig. 6. The probability of path identification (P =9, σ 2 =1) can always identify exactly two paths as the gain of the second path becomes larger than 6dB. Moreover, when we increase the number of pilot symbols to 10, the probability of correctly determining the number of paths becomes almost 1 even when the gain of the second path is somewhat smaller than 10dB, while the original DFT method performs very poorly. When we set η = 90%, SWISS can find two paths with probability 1 in both cases where T = 5 and 10, while the original DFT method still have poor performance when T =5. The comparison indicates that the weighting method is necessary and extremely useful. Next we consider the nine-path case. The DoAs of the paths are the same as in Fig. 4. The gain of each path α p is chosen from a uniform distribution over the interval of [0, 5dB]. The variance of the noise is σ 2 =1and the number of simulation runs is set at η is set to be 90%, 95% and 99.5%, respectively. Fig. 6 plots the probability of correct path number identification. It can be seen that SWISS can identify exactly nine paths with probability of nearly 1, while the estimated number of paths by the original DFT method is much larger than nine. From Figs. 5 and 6, we can see that SWISS can efficiently reduce the influence of the noise by choosing optimum weights and has high accuracy in identifying the number of paths Fig. 8. The performance of SWISS in the no-signal case (σ 2 =4) C. The No-Signal Case In this part, we evaluate the capability of SWISS for deciding whether there is a signal or no signal. Figs. 7 and 8 show the squares of weights when there is no signal but with noise variances of σ 2 =1and 4, respectively. It is seen that all the weights are very small (< 10 1 ) and almost uniformly distributed over all the DFT points. The values of the weights depend on the number of the antennas. We can show that if we have an infinite number of antennas, all the weights will be nearly zero, which means that SWISS can serve as a signal detector. V. CONCLUSION SWISS is a novel algorithm for detecting a signal, i.e., for deciding whether a signal is coming from a certain DoA. If a signal is detected, it enables a rapid identification of the number of paths/rays present, and a determination of their individual complex path gain and DoA. A rigorous theoretical development of the algorithm and its performance analysis will be presented in future work. REFERENCES [1] Z. Pi and F. Khan, An introduction to millimeter-wave mobile broadband systems, IEEE Communications Magazine, vol. 49, no. 6, pp , June [2] B. Hassibi and B. M. Hochwald, How much training is needed in multiple-antenna wireless links? IEEE Transactions on Information Theory, vol. 49, no. 4, pp , April [3] A. Alkhateeb, O. E. Ayach, G. Leus, and R. W. Heath, Channel estimation and hybrid precoding for millimeter wave cellular systems, IEEE Journal of Selected Topics in Signal Processing, vol. 8, no. 5, pp , Oct 2014.

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