Rainfall Effect on the Performance of Millimeter-Wave ... - IEEE Xplore

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 9, SEPTEMBER 2015

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Rainfall Effect on the Performance of Millimeter-Wave MIMO Systems Yong-Ping Zhang, Member, IEEE, Peng Wang, Member, IEEE, and Andrea Goldsmith, Fellow, IEEE

Abstract—This paper considers the rainfall effect on the capacities and achievable rates of millimeter-wave (mmW) multipleinput multiple-output (MIMO) systems. We first develop a new channel model for point-to-point mmW MIMO systems to characterize the rainfall effect. This rain propagation model is derived based on stochastic properties of signal propagation in a general random scattering medium. Under this model, we evaluate the channel capacity of the mmW MIMO system at various rain rates and show that rainfall does not always have a negative impact on the system performance, provided that accurate instantaneous channel state information (CSI) is available at both the transmitter and receiver. We also show that a transmit strategy of statistical water-filling (SWF) allows the mmW MIMO system to have near-optimal performance. Index Terms—Millimeter-wave (mmW), rainfall, channel state information (CSI), statistical water-filling (SWF).

I. I NTRODUCTION

T

HE ever-increasing demand on mobile broadband services has led to a global spectrum exhaustion for carriers [1], [2]. To address this problem, the use of large chunks of underutilized spectrum in the millimeter-wave (mmW) bands, which range from 30 GHz to 300 GHz, has recently gained significant interest. The available bandwidth at these higher frequencies can be 200 times more than that used by today’s cellular networks [3]. This additional bandwidth will lead to greatly increased data rates and significantly decreased traffic latency for cellular systems. Moreover, recent advances in subterahertz hardware have enabled low-cost mmW chips [1], [3], which makes the deployment of mmW commercial systems in the near-term more likely. Incorporating mmW systems into commercial wireless networks faces several technical obstacles. The most significant one is the large path loss suffered by mmW signals. Fortunately, the much shorter wavelength of mmW signals allows Manuscript received September 10, 2014; revised February 9, 2015; accepted April 12, 2015. Date of publication April 28, 2015; date of current version September 7, 2015. The work of Y.-P. Zhang was supported by HiSilicon Technologies Co., Ltd. under Huawei Technologies grant. The work of A. Goldsmith was supported by Huawei Technologies and the NSF Center for Science of Information (CSoI) under grant NSF-CCF-0939370. The associate editor coordinating the review of this paper and approving it for publication was C. R. Anderson. Y.-P. Zhang is with the Research Department of Hisilicon, Huawei Technologies Co., Ltd., Beijing 100095, China (e-mail: [email protected]). P. Wang is with the School of Electrical and Information Engineering, University of Sydney, Sydney, N.S.W. 2006, Australia (e-mail: peng.wang@ sydney.edu.au). A. Goldsmith is with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TWC.2015.2427282

for dozens, or even hundreds, of antennas to be packed into a relatively compact space at a cellular base station or WiFi access point. This technique, referred to as massive multipleinput multiple-output (MIMO) [4], allows “pencil beam” antenna patterns to be formed such that sufficient power gain can be achieved to combat severe path attenuation in mmW bands. Consequently, the combination of mmW and massive MIMO can overcome the individual shortcomings of each technique to fully exploit their advantages simultaneously. Recent results from the development of algorithms and prototypes have clearly demonstrated the potential of mmW massive MIMO for next generation (5G) wireless systems [2], [3]. An in-depth understanding of the outdoor mmW MIMO channel is vital for the design of wireless systems operating in this frequency band. Rainfall is one of the most common weather phenomena. The channel impairments associated with rainfall have been studied in [2], [5]. In current wireless system designs, the effect of rainfall is always considered as a fixed propagation attenuation (dB per kilometer) that is typically taken into account during link planning [2], [5]. This is a reasonable approximation in wireless networks operating below 6 GHz, because the typical diameter of rain drops, which ranges from 0.1 mm to 10 mm [6], is much less than the signal wavelength in these systems. Consequently, the signal mainly experiences absorption from the rain, which only causes signal attenuation. However, the situation is completely different in mmW systems. Since the wavelengths of mmW signals are comparable to the rain drop size, these signals undergo scattering when transmitted through rain [7] and consequently suffer both amplitude attenuation and phase fluctuation [8]. Therefore, modeling the rainfall effects as fixed amplitude attenuation [2], [3] is no longer accurate in mmW networks. To the best of our knowledge, characterization of the mmW MIMO channel in rainfall environments remains a challenging and open issue. That is the topic we address in this paper. To investigate the effects of rain on mmW communications, we first develop a channel model based on statistical properties of signal propagation in a random scattering medium [8]. Based on the developed model, we then evaluate the system capacity and numerically show that when accurate instantaneous channel state information (CSI) is available at both the transmitter and receiver, the rainfall does not always reduce the capacity of the mmW MIMO system. In particular, at relatively high signalto-noise ratios (SNRs), the system capacity under the “bestcase rain rate” can be even higher than that of a mmW MIMO channel without rainfall, since by applying the ideal precoder and optimal power adaptation to the channel, the achievable multiplexing gain outweighs the rain attenuation. Finally, we

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Fig. 1. The 2-D geometrical model for a mmW MIMO channel with ULAs at both link ends. The circles in the figure represent rain drops.

show that transmitter adaptation based on statistical waterfilling (SWF) has near-optimal performance. The main contributions of this paper are summarized as follows. 1) We develop a tractable channel model for mmW MIMO communication that captures the effect of rainfall scattering. 2) Based on the developed channel model, the effect of rain on the performance of mmW MIMO systems is investigated. Two important conclusions are derived from this analysis: rain does not always reduce the capacity of mmW MIMO systems; and SWF can achieve nearoptimal performance in mmW MIMO systems. The remainder of this paper is organized as follows. Section II introduces the system model and the statistical characteristics of wave propagation in a random scattering medium. Section III develops a model for the mmW MIMO channel that captures the rainfall effect. On the basis of this model, we examine the channel capacity of the mmW MIMO system under various rain rates in Section IV. We also investigate in this section the performance of SWF which adapts the transmit strategy based on channel statistics. Some concluding remarks are provided in Section V. II. P RELIMINARIES A. System Model Consider a fixed point-to-point mmW MIMO system with an N-element uniform linear antenna array (ULA) at the transmitter and an M-element ULA at the receiver. As illustrated in the geometrical model in Fig. 1, the transmit ULA lies on the y-axis with its center located at the origin. The receiver ULA, which is parallel to the transmit ULA, is centered on the positive half of the x-axis with distance D from the origin. In this paper, we assume that the above parameters on antenna configuration and communication distance are constant and known at both ends. To simplify the derivation, we further assume that the antenna elements of the transmit and receive antenna arrays have omnidirectional radiation patterns. The incorporation of antenna effects such as pattern or polarization diversities, mutual coupling, co-polarization and cross-polarization is left for future work. We consider the rainfall effect on the above system. We assume that rain drops are in the shape of isotropical and homogeneous spheres and are randomly and independently distributed

in the signal’s propagation path, which is a common assumption in meteorology [10]. Scattering and absorption are the two main consequences of signal waves propagating through the rain. The propagation properties are related to the rain rate, which is measured by calculating the amount of rain that falls to the earth’s surface per unit area per unit time, typically in millimeters per hour (mm/hr). We assume the rain rate is the same along the propagation path from the transmitter to the receiver and known at both ends. In practice, the rain rate can be accurately measured by the rain gauge sensor [11] at the transmitter. Since the wideband channel can be converted into a set of narrowband sub-channels by orthogonal frequency division multiplexing (OFDM), we focus on a narrowband frequencyflat fading channel for simplicity. Denote by H = {hm,n} ∈ CM×N the narrowband channel response matrix of the system shown in Fig. 1. The received signal, y ∈ CM×1 , can be represented as y = Hx + n

(1)

where x ∈ CN×1 is the transmitted vector and n ∈ CM×1 is a vector of independent and identically distributed (i.i.d.) complex additive white Gaussian noise (AWGN) samples with mean zero and variance σ02 . We first assume that H is known at both the transmitter and receiver. In Section IV we will relax this assumption and consider a more practical scenario when only the statistical information of H is available at the transmitter. For a given transmit covariance matrix Q = E(xxH ), where E(·) denotes the expectation operator, the achievable rate of the above channel is calculated by the well-known formula [12, Chapter 10]: 1 H (2) R = log2 det IM + 2 HQH σ0 where IM is an identity matrix of size M. It is obvious that rainfall will affect the achievable rate of (2) due to the variation of H for which the design of the transmit covariance matrix Q should be optimized. As an extreme scenario of the system in Fig. 1, the mmW MIMO channel without rainfall can be modeled as a line-of-sight (LoS) MIMO channel; the in-depth investigation of this scenario has been conducted in [13]. In a LoS MIMO channel, due to the lack of scattering, the highly correlated antennas are mainly used for power gain and the corresponding achievable multiplexing gain is limited by the aperture sizes of the transmit and receive ULAs [13]. Under rainfall, signal attenuation is introduced by rain drops due to their absorption effect, as well as significant signal scattering due to the similar sizes of the mmW signals’ wavelength and the rain drops. This scattering generally reduces the fading correlation across antennas [8]. Thus, from the viewpoint of the achievable multiplexing gain, a mmW MIMO system may experience an increase in capacity due to the reduction in antenna correlation associated with rainfall. In this paper, our task is to explore and characterize the effects of rainfall and to exploit them in adaptive system design. Specifically, we will first develop a simple and tractable channel model for the above mmW MIMO system. Then we will analyze the

ZHANG et al.: RAINFALL EFFECT ON THE PERFORMANCE OF MILLIMETER-WAVE MIMO SYSTEMS

system capacity and propose a robust and practical adaptive transmission scheme based on this model.

In [8], the effects of randomly distributed discrete scatterers on radar signal propagation have been investigated, and some stochastic properties of wave propagation in this random scattering medium have been derived. For the convenience of our subsequent discussions, these stochastic properties are briefly introduced below. Consider a radar system where a point source X and an object are located in an environment with randomly distributed discrete scatterers. The distance between the point source X and the center of the object is D. Denote by h1 and h2 the channel response coefficients from X to two points O1 and O2 on the object, respectively. By assuming the far-field distance between the point source and the object, the expectations of h1 and h2 and their correlation are given by [8]: 2π √ E(hi ) = ρo · αe−j λ dOi ,X , ∀ i = 1, 2 (3) and 2 −j E(h1 hH 2)=α e

2π λ

dO1 ,X −dO2 ,X ⎝

ρo + ρs e

2 −dO 1 ,O2 l2o

antenna as an object point O1 , the expectation of hm,n can be directly obtained from (3) with dO1 ,X replaced by dm,n, i.e., E(hm,n ) =

B. Stochastic Properties of Wave Propagation in a Random Medium

⎛

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(5)

Recall the geometrical model in Fig. 1. The coordinates of the m-th receive antenna and the n-th transmit antenna can be rep(m) (n) (m) resented by (D, yr ) and (0, yt ), respectively, where yr = (n) (2m − 1 − M) d2r and yt = (2n − 1 − N) d2t with dr and dt being, respectively, the receive and transmit antenna spacings. Following (3) and (4), we assume a far-field distance between (n) the transmitter and receiver [13], i.e., D y(m) r and yt , ∀ m, n. (m) (n) 2 Under this assumption, we have (yr − yt ) /D2 1 and therefore the distance between the m-th receive antenna and the n-th transmit antenna, dm,n , can be approximately calculated as

(n) 2

2 y(m) r − yt (n) dm,n = D2 + y(m) ≈D+ r − yt 2D 1 2 2 (2m − 1 − M) dr + (2n − 1 − N)2 dt2 =D+ 8D 1 − (2m − 1 − M)(2n − 1 − N)dr dt . (6) 4D From (5) and (6), the expectation of H can be written as

⎞ ⎠

2π √ ρo · αe−j λ dm,n .

E(H) =

(4)

√ λ η 4πD

is the large-scale propagation factor in free where α = space, λ is the signal wavelength, η is the product of the transmit and receive antenna gains, (·)H denotes the Hermitian transpose operator, dX,Y , denotes the distance between two points of X and Y, ρo = e−τo , ρs = e−τa − e−τo , and lo = 3λ2 αp (1−e−τs ) . π 2 τs

In the above expressions, τa and τs are the absorption and scattering depth, which express the power loss in the beam by absorption and scattering, respectively, during its path through the random scattering medium, τo is the optical depth, which expresses the total power loss by absorption and scattering, and αp is the anisotropy factor, which represents the average cosine of the scattering angle with respect to power. Detailed explanations of these parameters can be found in [8]. Note that when the rain rate is zero, the above four parameters are all equal to zero. The expressions in (3) and (4) serve as the basis of our channel model to be developed. We will elaborate on this in more detail in the next section. III. C HANNEL M ODELING FOR mmW MIMO IN R AINFALL S CATTERING A. Stochastic Characteristics of the mmW MIMO Channel in Rainfall We first derive explicit expressions for the expectation and correlation matrices of H, i.e, E(H), E(HHH ) and E(HH H), based on (3) and (4), starting with E(H). By regarding the n-th transmit antenna as the point source X and the m-th receive

√ ˆ ρo · α H

(7)

ˆ = {hˆ m,n} ∈ CM×N is a matrix with where H 2π D −j hˆ m,n = e−j λ · e

(m) (n) φr +φt

· ej

S(2m−1−M)(2n−1−N) 2

(8)

π(2n−1−N)2d 2

π(2m−1−M)2d 2

t r where φr(m) = , φt(n) = and S = 4λD 4λD √ πdr dt λD . It is worth noting that, apart from a constant scalar ρo , (7) is same as the expression of a LoS MIMO channel response matrix [13]. Next we derive the receive correlation matrix E(HHH ). We again regard the n-th transmit antenna as the point source X and regard the m-th and m -th receive antennas as two object points O1 and O2 respectively. Then the correlation between the channel coefficients hm,n and hm ,n can be obtained from (4) with dO1 ,X , dO2 ,X and dO1 ,O2 replaced by dm,n , dm ,n , and |m − m |dr , respectively, i.e.,

2 −j 2π λ (dm,n −dm ,n ) E hm,n hH m ,n = ρo α e

2 −j 2π λ (dm,n −dm ,n )

+ ρs α e

e

2 −(m−m ) dr2 l2 o

.

(9)

From (9), each entry of E(HHH ) can be written as N N

H hm,n hm ,n = E hm,n hH E m ,n n=1

= α 2 ρo

n=1 N n=1

2 hˆ m,nhˆ H m ,n + α ρs e

−(m−m )2 dr2 l2 o

N

hˆ m,n hˆ H m ,n . (10)

n=1

ˆH ˆ H (reFor convenience, we define a constant matrix Gr = H ˆ where each entry of Gr , denoted call (8) for the definition of H),

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) by g(m,m (m, m ∈ {1, 2, · · · , M}), is written as r

) g(m,m =e r

=e

N

(m) (m ) −j φr −φr jπS(m−m)(2n−1−N)

e

(m) (m ) −j φr −φr

n=1

sin SN(m − m ) . sin (S(m − m ))

(11)

Then, from (10) and the definition of Gr , E(HHH ) can be written as E(HHH ) = α 2 ρo Gr + α 2 ρs r Gr (m,m )

(12) (m,m )

where r = {ψr } ∈ CM×M is a full matrix with ψr = )2 d 2 /l2 −(m−m r o e and denotes the Hadamard product. Finally, we derive the transmit correlation matrix E(HH H). To this end, we regard the m-th receive antenna as the point source X and regard the n-th and n -th transmit antennas as two object points O1 and O2 , respectively. According to the duality between forward and reverse channel coefficients of a communication link [14], the channel coefficients from X to H two points O1 and O2 can be represented as hH m,n and hm,n respectively. Then, similar to (10), we have N H E hm,n hm,n n=1

= α ρo 2

N

2 ˆ hˆ H m,n hm,n +α ρs e

−(n−n )2 dt2 l2o

n=1

N ˆ hˆ H m,n hm,n

(13)

n=1

and the transmit correlation matrix E(HH H) can be expressed as E(HH H) = α 2 ρo Gt + α 2 ρs t Gt

(14)

ˆ HH ˆ whose entries, {gt(n,n ) }, are given by where Gt = H

M (n) (n ) sin SM(n − n ) j φt −φt (n,n ) H ˆ ˆ gt (15) hm,n hm,n = e = sin (S(n − n )) m=1

(n,n )

(n,n )

and t = {ψt } ∈ CN×N is a matrix with ψt = 2 2 2 −(n−n ) d /l t o e . In the next section, the stochastic characteristics of the channel model for a mmW MIMO system in rainfall will be developed based on (7), (12), and (14). B. The Proposed Statistical Channel Model Next we develop a statistical channel model for the mmW MIMO system to characterize the randomization caused by rain drops. We first consider the channel response of a single antenna link (SISO). When a plane radio wave impinges on a rain drop, the rain drop absorbs and scatters some power from the wave while allowing the rest pass through it [10]. The scattered wave does not disappear and a part of it may be further scattered by other rain drops and arrive at the receive antenna via indirect paths. Thus it is straightforward to model the received waveform by the sum of two terms, i.e., the directlyattenuated wave and the indirectly-scattered waves. Under the far-field assumption, the communication distance between both ends is much larger than the mmW signal wavelength as well as

the rain drop size, and so there are a sufficiently large number of statistically independent rain drops between the transmitter and receiver. According to the Central Limit Theorem, the second term can be modeled as a complex Gaussian random variable with zero mean and a certain variance [12, Chapter 3]. We now consider the MIMO channel. Again under the farfield assumption, the distances between the different transmitreceive antenna links are approximately same. Hence it is reasonable to model the indirectly-scattered terms of all antenna links to have the same variance represented by a common factor. Moreover, due to the close placement of antennas, the indirectly-scattered terms of adjacent antennas links are spatially correlated with each other. Due to sufficient independent rain drops in the channel, the indirect-path component can be characterized by the Kronecker model [15]. Hence we model the mmW channel in rainfall as ¯ + βA 21 WB 12 , H=H

(16)

¯ ∈ CM×N is a constant matrix representing the directlywhere H attenuated component, β is a constant scalar representing the variance of all the indirectly-scattered terms, A = {am,m } ∈ CM×M and B = {bn,n } ∈ CN×N are both constant Hermitian matrices with normalizations tr(A) = M and tr(B) = N, where tr(·) denotes the trace of a matrix, W = {wm,n } ∈ CM×N is a random matrix with i.i.d. entries following a distribution of CN(0, 1). The closed-form expressions of the constant terms ¯ β, A, and B are given in Proposition 1 below. The correH, sponding detailed derivations can be found in Appendix A. Proposition 1: Based on (7), (12), and (14), the parameters ¯ β, A, and B in the proposed channel model (16) can be of H, written as ˆ ¯ = √ρo · α H, H √ β = ρs · α, 1 A = r Gr , N 1 B = t Gt . M

(17) (18) (19) (20)

From the definitions of ρo , ρs , r , and t , we see that these ¯ β, A, and B) are all functions of the propagation parameters (H, characteristics, i.e., τa , τs , τo and αp , which depend on the density and the size of the rain drops. In meteorology, the droplet size distribution (DSD) of the rain drops is a function of the rain rate and has an exponential form according to the Marshall-Palmer law [16]. Thus, the channel model shown in (16) is related to the rain rate via the propagation characteristics and DSD. It is worth noting that when the rain rate is zero, τa , τs , τo and αp are equal to zero. Then we have ρo = 1 and ρs = 0. By ¯ = substituting ρo = 1 and ρs = 0 into (17) and (18), we have H ˆ and β = 0. From (16), we further have H = α H, ˆ which is αH same as the expression of the LoS channel matrix derived in [13]. This analysis indicates that the developed model, i.e., (16), matches well with the LoS mmW MIMO channel model of [13] when the rain rate is zero.


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C. Channel Model Validation To verify its validity, we compare our proposed channel model against measured mmW propagation data through rain. Unfortunately, despite an extensive literature search for measurements of the mmW MIMO channel in rainfall, the only measurements we found suitable for evaluating the validity of our model is the measured data in [5] for a SISO system. Specifically, in [5, Fig. 3], the authors provided the measured values of the rain attenuation ε for mmW SISO signals under various rain rates. To make a comparison with this data, we reduce our proposed MIMO model to a SISO one by setting ¯ A and B reduce to scalars and (16) can be M = N = 1. Then H, rewritten as 2π D √ √ (21) H = ρo · αe−j λ + ρs · αw Consequently, the average power gain of the channel can be calculated from (21) as 2π D √ E(HH H ) = ρo α 2 + ρo ρs · α 2 e−j λ E(wH )

2π D √ + ρo ρs · α 2 ej λ E(w) + ρs α 2 E |w|2

(a)

= (ρo + ρs )α 2 = e−τa α 2 ,

paper is in good agreement with the measured data of [5] in terms of rain attenuation. IV. S YSTEM P ERFORMANCE A NALYSIS

(22)

where (a) follows from the fact that w is a standard complex Gaussian random variable with zero mean and unit variance. It is worth noting that the average power gain calculated in (22) is the combined effect of both the free space propagation loss and rain attenuation, where the former can be calculated from (22) by setting the rain rate to zero (or equivalently by letting τa = 0), and equals α 2 . Therefore, the rain attenuation ε in our proposed channel model can be given by ε = e−τa .

Fig. 2. Comparison of the rain attenuation with the measured data in [5].

(23)

Recall that τa in (23) is the absorption depth, which expresses the power loss caused by absorption. It is a deterministic function of the rain rate and communication distance, and can be calculated as follows. • First, the contribution of a single rain drop with an arbitrary diameter L (i.e., the droplet size) to τa , which is denoted by τ¯a (L), is computed via Mie theory (see [7] for more details); • Second, the DSD of all the rain drops in the communication area, denoted by D(·), is calculated according to the enhanced Marshall-Palmer law [9]; • Finally, the absorption depth τa can be obtained by the following numerical integration: ∞ πL2 τ¯a (L)D(L)dL. τa = D (24) 4 0 Next, we calculate the rain attenuation ε by (23) under the same communication distance of 1 kilometer as in [5] and various rain rates. The comparison with the data in [5] is shown in Fig. 2, where we can see that the difference between the rain attenuation in our model and that in [5] is marginal, e.g., less than 0.8 dB at 38 GHz and less than 1.4 dB at 75 GHz. We conclude that the statistical channel model proposed in this

We now provide some numerical examples of the mmW MIMO system performance in rain based on the channel model developed in Section III. We assume that the exact channel H is always known at the receiver in the following scenarios. Consider a static mmW link between the transmitter and receiver, which are equipped with ULAs of two different sizes (0.1 m and 0.2 m). The antenna spacing is assumed to be a halfwavelength. Two typical mmW frequencies, i.e., 38 GHz and 75 GHz, are considered. For simplicity, we assume the largescale propagation factors of all antenna links are normalized, i.e., α = 1. The transmit SNR γ is varied in the simulations. In the channel generation, the propagation parameters τa , τs , τo and αp are obtained via the similar calculation of τa in Section III-C. Tables I and II list some examples for these parameters at 38 GHz and 75 GHz. A. Capacity With Full CSI at the Transmitter Denote by Ropt the ergodic capacity of the mmW MIMO system in rain, which is achieved by assuming the channel H is known at the transmitter which adapts to this CSI to maximize capacity. From (2), Ropt is given by 1 H (25) Ropt = E log2 det IM + 2 HQoptH {H} σ0 where Qopt = Vopt PoptVH opt , Vopt is the principal right singular vector of the instantaneous channel matrix H, and Popt ∈ CN×N is diagonal matrix, whose diagonal entries are generated by waterfilling the total transmit power P over each sub-channel based on the eigenvalues of instantaneous HH H. In Fig. 3, the channel capacity Ropt at a transmit SNR of γ = 0 dB is plotted relative to the rain rate. Several interesting observations can be made. Observation 1: The impact of the rainfall on the capacity is not always negative. The curves exhibit a unimodal property. They first increase steeply with the rain rate. After achieving

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TABLE I P ROPAGATION PARAMETERS IN R AIN AT 38 GHz

TABLE II P ROPAGATION PARAMETERS IN R AIN AT 75 GHz

their maximum point at the “best-case rain rate” (e.g., 6 mm/hr in a 75 GHz system with M = N = 101), the curves decrease as the rain rate increases. For a clear explanation for the above phenomenon, in Fig. 4, we plot the trace and several eigenvalues of the instantaneous value of HHH (averaged over the distribution of W in (16)) at 38 GHz. A MIMO channel can be transformed into 1 to min(M, N) parallel independent sub-channels. The quality associated with each sub-channel depends on the eigenvalues of HHH . In a LoS scenario, due to the close placement and the lack of scatterers, the transmitted signals from the different antennas will travel through the same path and experience the same fading before reaching the receiver. As a result, only 1 subchannel is with high quality. Hence, no diversity or multiplexing can be exploited. From Fig. 4 we see that when the rain rate is zero, except for the maximum one, the other eigenvalues of HHH all have extremely small values close to zero. In other words, due to rainfall, a large number of rain drops, whose sizes are comparable to the wavelengths of mmW signals, cause rich scattering in the mmW MIMO channel. A rich scattering environment yields that the channel is full rank and can support more simultaneous spatial streams under typical SNRs. Thus, in Fig. 4, we see that although the curves of the trace and the maximum eigenvalue of HHH decrease as the rain rate increases, the values of the other eigenvalues increase with the rain rate when the latter is low. This means that the qualities of the sub-channels of a mmW MIMO channel become close to each other as the rain rate increases. At a high SNR, the equal channel quality leads to significant multiplexing gain. Combining the above analysis on the eigenvalues of a mmW MIMO channel in rainfall, the phenomenon in Fig. 3 can be

Fig. 3. Ergodic capacity as a function of rain rate at a transmit SNR of γ = 0 dB. (a) 38 GHz, (b) 75 GHz.

Fig. 4. The trace and average eigenvalues of HHH at 38 GHz as a function of rain rate.

explained by the combined mechanisms of power loss caused by rain absorption and multiplexing gain caused by rain scattering. On the one hand, absorption always degrades system performance by reducing the power gain of the wireless channel. On the other hand, a large number of rain drops in free space cause significant scattering in the mmW MIMO channel, which


can be exploited to obtain multiplexing gain. When the rainfall is light, although the power of the channel matrix is degraded by absorption, the available multiplexing gain dominates the rainfall effect, and hence the channel capacity increases from light rainfall. We define the “best-case” rain rate as the rate of rain for which the channel capacity is maximized. If the rain rate increases further above this rate, since the power loss caused by absorption will dominate the rainfall effect, the capacity will fall off. Observation 2: The channel capacity at 75 GHz has a much larger dynamic range compared with that of the channel capacity at 38 GHz. This observation can be explained based on the shorter signal wavelength at 75 GHz and the associated larger number of antennas (101×101, 51×101, and 51×51 arrays for 75 GHz vs. 51×51, 26×51, and 26×26 for 38 GHz). The consequence of this larger antenna array is that more power and multiplexing gains can be obtained. On the other hand, the shorter wavelength makes the mmW signals at 75 GHz more sensitive to the rainfall (see the propagation parameters in Tables I and II). Consequently, under extremely heavy rainfall, compared with the signal at 38 GHz, the signal at 75 GHz suffers more severe absorption and associated power loss, which leads to a lower channel capacity. Fig. 5 shows the channel capacity at different transmit SNRs at 38 GHz and at 75 GHz for different numbers of antennas. We see that at an SNR of −20 dB, the capacity of systems operating at 38 GHz and 75 GHz falls off monotonically as the rain rate increases. This is because the power loss caused by the absorption dominates the effect of rainfall at low SNRs; the achieved multiplexing gain is not high enough to compensate for the performance loss caused by the attenuation, and therefore the channel capacity deceases monotonically as the rain rate increases. The same is true at an SNR of −10 dB for 38 GHz and less than 51 antennas (Fig. 5(a)), however the decrease is not smooth under light rainfall (≤ 5 mm/hr) as the multiplexing gain somewhat compensates for the increased absorption. The behavior at 75 GHz with a larger number of antennas (Fig. 5(b)) is markedly different. In this case, capacity increases with rain rate up to a given maximum, achieved at the “best-case” rain rate. It is at this rain rate when the difference between the multiplexing gain and absorbtion is maximized. For higher rates, capacity decreases monotonically. We consider scenarios with more antennas under 75 GHz (Fig. 5(b)) versus 38 GHz (Fig. 5(a)) since typically at higher frequencies (lower wavelengths) a larger array can be used for the same area with closer antenna spacing. B. Statistical Water-Filling In the current design of mmW MIMO systems, the performance loss caused by rainfall attenuation is compensated by scaling up the transmit power [2], [5]. Hence the power consumed at the transmitter increases when it rains. In Section IV-A, we showed that when perfect CSI from the receiver is available via feedback at the transmitter, the mmW MIMO systems can benefit from rainfall by exploiting the multiplexing gain. However, the assumption of ideal instantaneous

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Fig. 5. Ergodic capacity as a function of rain rate when the transmit SNR γ varies. (a) 38 GHz, (b) 75 GHz.

CSI at the transmitter is generally infeasible in a mmW MIMO system for two reasons. First, the large number of transmit antennas implies an extremely heavy feedback overhead. Second, a mmW MIMO channel with rainfall changes very quickly as the rain drops are continuously falling to the ground at an unpredictable speed depending on the wind, and therefore the small coherence time yields a high overhead for real-time CSI feedback. In this section, to exploit the abovementioned capacity gain in a practical mmW MIMO system, we design a transmission strategy with much lower overhead in the CSI feedback than instantaneous CSI. Recall from the assumption in Section II that besides the system configuration parameters (e.g., S, M, N), the propagation parameters in rain (e.g., τo ,τs , τa , and αp ) are all determined by the rain rate that is assumed to be known at both ends. Thus, given the system configuration parameters and the rain rate, we can obtain the parameters of α, ρo , ρs , Gt , and t and then the correlation matrix at the transmitter E(HH H). The benefit of a transmission strategy based on E(HH H) has been demonstrated in the classic fading channel in microwave bands [12, Chapter 10]. In this section, we will investigate the performance of this transmission strategy based on E(HH H), which is referred

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Fig. 6. The performance comparison in mmW MIMO systems with symmetric configuration. (a) 38 GHz, (b) 75 GHz.

Fig. 7. The performance comparison in mmW MIMO systems with asymmetric configuration. (a) 38 GHz, (b) 75 GHz.

to as SWF, in the mmW MIMO system. In particular, from eigenvalue decompositions, we have E(HH H) = Vt t VH t

(26)

where Vt is a unitary matrix whose columns comprise the eigenvectors of E(HH H) and t is a diagonal matrix with its diagonal entries being the eigenvalues of E(HH H) denoted by {μ(n) t |n = 1, 2, · · · N}. In the proposed SWF, Vt is chosen as the precoding matrix. The power allocation matrix Pswf is obtained by the water-filling formula over the eigenvalues of E(HH H). Then the corresponding transmit covariance matrix Qswf is given by Qswf = Vt Pswf VH t .

(27)

Combining (2) and (27), we can obtain the achievable rate of SWF, denoted by Rswf , as 1 . (28) Rswf = E log2 det IM + 2 HQswfHH {H} σ0 Next we compare the performance achieved by the SWF strategy with the capacity assuming full CSI at the transmitter, as a function of the transmit SNR γ . Figs. 6 and 7 show the

performance of SWF under different MIMO configurations. Several observations are worth noting. First, we see that at low SNRs (below −10 dB), SWF can achieve almost the same rate as the capacity with perfect instantaneous CSI. In fact, at low SNRs, all antennas are used for power gain and no multiplexing gain can be achieved, so the system is less sensitive to the need for CSI. Second, as SNR increases from moderate to high, the achievable rate of SWF converges to the capacity. We also see that in the asymmetric configuration of MIMO, under a given min(M, N), SWF has better performance in the scenario of M > N compared to that in the scenario of M < N. This is as expected since the system with more transmit antennas relies on accurate CSI at the transmitter more heavily than the system with less transmit antennas. In a practical implementation, since the system configurations can be stored in advance and the rain rate can be measured in real time by the rain gauge sensor [11] at the transmitter, no additional CSI feedback is required by the SWF strategy. The low overhead of CSI feedback coupled with the minimal rate loss relative to capacity with ideal CSI at moderate to large SNRs makes SWF a compelling transmission strategy in practical mmW MIMO systems.


V. C ONCLUSION

Substituting (33) into (31) yields

We have proposed a new channel model for mmW MIMO systems in rainfall scattering. Based on this model, we observe an interesting phenomenon that at practical SNRs and low rain rates, channel capacity first increases with the rain rate due to the reduction on the antenna correlation from the scattering of raindrops, then decreases as the rain rate further increases. Our results indicate that the mmW MIMO system with adaptive transmission based on perfect CSI can benefit from rainfall by exploiting the multiplexing gain. To avoid the significant overhead required for the perfect CSI feedback, we propose an SWF strategy. We show that by taking advantage of the system configurations and rainfall characteristics, SWF results in a small capacity loss in the mmW MIMO system relative to capacity under perfect instantaneous CSI. A PPENDIX A P ROOF OF P ROPOSITION 1 We first take the expectation at the both sides of (16):

(a) ¯ + βA 21 E(W)B 12 = H ¯ ¯ + βA 12 WB 12 = H E(H) = E H (29) where (a) follows from E(W) = 0. Since the expectation of the developed channel model is equal to (7), we then have ˆ ¯ = √ρo · α H, H

(30)

which completes the proof of (17). Then from (16), the correlation of the developed model at the receiver, i.e., E(HHH ) can be written as H ¯ + βA 21 WB 12 H ¯ + βA 12 WB 12 E(HHH ) = E H

H 1 H (a) ¯H ¯ H+β 2A 12 E WB 12 B 12 =H WH A 2 (b)

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1

1

E(HHH ) = α 2 ρo Gr + β 2 NA 2 IM A 2 = α 2 ρo Gr + β 2 NA. (34) Since the correlation of the developed model at the receiver is equal to (12), we have α 2 ρo Gr + β 2 NA = α 2 ρo Gr + α 2 ρs r Gr Solving for A yields the following equation: A=

ρs 2 α r Gr . β 2N

2

2

H

= ρo α Gr + β A E(WBW )A

1 2

(31)

where (a) follows from E(W) = 0, and (b) follows from the assumption that A and B are Hermitian matrices. Assume C = {cm,m } = WBWH ∈ CM×M . The expectation of each entry of C can be rewritten as N N H wm,k bk,l wm ,l E(cm,m ) = E l=1 k=1

=

N N

E wm,k wH m ,l bk,l .

(32)

l=1 k=1

m

Since the elements in W are i.i.d., when m = or l = k, we have E(wm,k wH m ,l ) = 0. Furthermore, since we assume the diagonal entries of B are normalized, E(C) can be rewritten as E(C) = E(WBWH ) = NIM .

(33)

(36)

From the definitions r and r , the diagonal entries of A of G ρs 2 can be written as β 2 α (m = 1, 2, · · · , M), which are with an equal value. Based on the assumption of the normalized diagonal entries of A, we obtain β=

√

ρs · α 2 ,

(37)

which completes the proof of (18). Substituting (37) into (36) yields A=

1 r Gr , N

(38)

which completes the proof of (19). Next we derive the expression of B. Based on the similar derivations for (34), we can write E(HH H) as H ¯ + βA 12 WB 12 ¯ + βA 12 WB 12 E(HH H) = E H H ˆ HH ˆ + β 2 MB. = α 2 ρo H

(39)

Since the correlation of the developed model at the transmitter is equal to (14), we have α 2 ρo Gt + α 2 ρs MB = α 2 ρo Gt + α 2 ρs t Gt .

1 2

(35)

(40)

Solving for B yields the following equation: B=

1 t Gt . M

(41)

From (41) and the definitions of Gt and t , we can see that the diagonal entries of B are uniformly normalized, which coincides with our assumption on B in (16). Thus we complete the proof of (20), and thus the proof of Proposition 1. ACKNOWLEDGMENT The first author gratefully acknowledges Dr. Feng Yang (Google) for an enlightening discussion on the intelligent control for illumination, Dr. Leiming Zhang (Huawei) and Dr. Peter Almers (Huawei) for their invaluable suggestions on antenna polarization, and Dr. Philipp Zhang (Huawei) and Shulan Feng (Huawei) for their ongoing support of this work.

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R EFERENCES [1] T. S. Rappaport, J. N. Murdock, and F. Gutierrez, “State of the art in 60 GHz integrated circuits and systems for wireless communications,” Proc. IEEE, vol. 99, no. 8, pp. 1390–1436, Aug. 2011. [2] 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. [3] S. Rangan, T. S. Rappaport, and E. Erkip, “Millimeter-wave cellular wireless networks: Potentials and challenges,” Proc. IEEE, vol. 102, no. 3, pp. 366–385, Mar. 2014. [4] E. G. Larsson, F. Tufvesson, O. Edfors, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” IEEE Commun. Mag., vol. 52, no. 2, pp. 186–195, Feb. 2014. [5] “E-band technology: Overview of the 71–76 & 81–86 GHz frequency bands, E-Band Communications Corp.” [Online]. Available: http://www.e-band.com/index.php?id=226 [6] R. de Charette et al., “Fast reactive control for illumination through rain and snow,” in Proc. IEEE Int. Conf. Comput. Photography, Seattle, WA, USA, Apr. 2012, pp. 1–10. [7] A. Ishimaru, Wave Propagation and Scattering in Random Media. Piscataway, NJ, USA: IEEE Press, 1978. [8] A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Multipel scattering effects on the radar cross section (RCS) of objects in a random medium including backscattering enhancement and shower curtain effects,” Waves Random Media, vol. 14, pp. 499–511, 2004. [9] R. Uijlenhoet, “Raindrop size distributions and radar reflectivity-rain rate relationships for radar hydrology,” Hydrol. Earth Syst. Sci., vol. 5, no. 4, pp. 615–627, 2001. [10] S. Okamura and T. Oguchi, “Electromagnetic wave propagation in rain and polarization effects,” Proc. Jpn. Acad. Ser. B Phys. Biol. Sci., vol. 86, no. 6, pp. 539–562, Jun. 11, 2010. [11] H. Messer, A. Zinevich, and P. Alpert, “Environmental monitoring by microwave communication networks,” Science, vol. 312, no. 5774, p. 713, May 2006. [12] A. Goldsmith, Wireless Communications. New York, NY, USA: Cambridge Univ. Press, 2005. [13] P. Wang, Y. Li, X. Yuan, L. Song, and B. Vucetic, “Millimeter wave wireless transmissions at E-band channels with uniform linear antenna arrays: Beyond the Rayleigh distance,” in Proc. IEEE Int. Conf. Commun., Sydney, NSW, Australia, Jun. 2014, pp. 5455–5460. [14] P. Wang and P. Li, “On maximum eigenmode beamforming and multi-user gain,” IEEE Trans. Inf. Theory, vol. 57, no. 7, pp. 4170–4186, Jul. 2011. [15] J. Kermoal, L. Schumacher, K. I. Pedersen, P. Mogensen, and F. Frederiksen, “A stochastic MIMO radio channel model with experimental validation,” IEEE J. Sel. Areas Commun., vol 20, no. 6, pp. 1211–1226, Aug. 2002. [16] J. S. Marshall and W. M. K. Palmer, “The distribution of raindrops with size,” J. Meteorol., vol. 5, no. 4, pp. 165–166, Aug. 1948.

Yong-Ping Zhang (M’11) received the B.Eng. degree in telecommunication engineering and the M.Eng. degree (with highest honors) in pattern recognition and intelligent systems from Xidian University, Xi’an, China, in 2001 and 2004, respectively. In 2004, he was as a Program Engineer with China Telecom, Shenzhen Branch. From 2005 to 2006, he was with Ricoh Software Research Center, Beijing, China, where he led a project on print-and-scan watermarking. He is currently a Senior Research Engineer with the Research Department of HiSilicon, Huawei Technologies Company, Ltd., Beijing. From 2013 to 2014, he was a Visiting Scholar with the Department of Electrical Engineering, Stanford University, Stanford, CA, USA. His current research interests include MIMO techniques and millimeter-wave communications.

Peng Wang (S’05–M’10) received the B.Eng. degree in telecommunication engineering and the M.Eng. degree in information engineering from Xidian University, Xi’an, China, in 2001 and 2004, respectively, and the Ph.D. degree in electronic engineering from the City University of Hong Kong, Hong Kong SAR, in 2010. He was a Research Fellow with the City University of Hong Kong and a visiting Post-Doctor Research Fellow with the Chinese University of Hong Kong, Hong Kong SAR, both from 2010 to 2012. Since 2012, he has been with the Centre of Excellence in Telecommunications, School of Electrical and Information Engineering, the University of Sydney, Australia, where he is currently a Research Fellow. His research interests include channel and network coding, information theory, iterative multi-user detection, MIMO techniques and millimetre-wave communications. He received the best paper award in IEEE International Conference on Communications (ICC) in 2014. He has published over 40 peerreviewed research papers in the leading international journals and conferences, and has served on a number of technical programs for international conferences such as ICC and WCNC.

Andrea Goldsmith (S’90–M’93–SM’99–F’05) received the B.S., M.S., and Ph.D. degrees in electrical engineering from U.C. Berkeley. She is the Stephen Harris Professor in the School of Engineering and a Professor of Electrical Engineering at Stanford University. She was previously on the faculty of Electrical Engineering at Caltech. She co-founded and served as CTO for two wireless companies: Accelera, Inc., which develops softwaredefined wireless network technology for cloud-based management of WiFi access points, and Quantenna Communications, Inc., which develops high-performance WiFi chipsets. She has previously held industry positions at Maxim Technologies, Memorylink Corporation, and AT&T Bell Laboratories. She is a Fellow of the IEEE and of Stanford, and has received several awards for her work, including the IEEE ComSoc Edwin H. Armstrong Achievement Award as well as Technical Achievement Awards in Communications Theory and in Wireless Communications, the National Academy of Engineering Gilbreth Lecture Award, the IEEE ComSoc and Information Theory Society Joint Paper Award, the IEEE ComSoc Best Tutorial Paper Award, the Alfred P. Sloan Fellowship, and the Silicon Valley/San Jose Business Journal’s Women of Influence Award. She is author of the book Wireless Communications and co-author of the books MIMO Wireless Communications and Principles of Cognitive Radio, all published by Cambridge University Press, as well as an inventor on 28 patents. Dr. Goldsmith has served as editor for the IEEE T RANSACTIONS ON I NFORMATION T HEORY, the Journal on Foundations and Trends in Communications and Information Theory and in Networks, the IEEE T RANSACTIONS ON C OMMUNICATIONS, and the IEEE Wireless Communications Magazine as well as on the Steering Committee for the IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS . She participates actively in committees and conference organization for the IEEE Information Theory and Communications Societies and has served on the Board of Governors for both societies. She has also been a Distinguished Lecturer for both societies, served as President of the IEEE Information Theory Society in 2009, founded and chaired the Student Committee of the IEEE Information Theory Society, and chaired the Emerging Technology Committee of the IEEE Communications Society. At Stanford she received the inaugural University Postdoc Mentoring Award, served as Chair of Stanford’s Faculty Senate in 2009, and currently serves on its Faculty Senate, Budget Group, and Task Force on Women and Leadership.