Line of Sight 2 x nr MIMO with Random Antenna Orientations
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Line of Sight × n r MIMOwith Random Antenna Orientations
Lakshmi Natarajan, Yi Hong,
Senior Member, IEEE , and Emanuele Viterbo,
Fellow, IEEE
Abstract —Line-of-sight (LoS) multiple-input multiple-output(MIMO) gives full spatial-multiplexing gain when the antennaarray geometry and orientation are designed based on the inter-terminal distance. These known design methodologies, that holdfor antenna arrays with fixed orientation, do not provide fullMIMO gains for arbitrary array orientations. In this paper,we study LoS MIMO channels with random array orientationswhen the number of transmit antennas used for signalling is . We study the impact of common array geometries on errorprobability, and identify the code design parameter that describesthe high signal-to-noise ratio ( SNR ) error performance of anarbitrary coding scheme. For planar receive arrays, the errorrate is shown to decay only as fast as that of a rank channel,and no better than SNR − for a class of coding schemesthat includes spatial multiplexing. We then show that for thetetrahedral receive array, which uses the smallest number ofantennas among non-planar arrays, the error rate decays fasterthan that of rank channels and is exponential in SNR forevery coding scheme. Finally, we design a LoS MIMO systemthat guarantees a good error performance for all transmit/receivearray orientations and over a range of inter-terminal distances.
Index Terms —Antenna array, array geometry, coding scheme,line-of-sight (LoS), multiple-input multiple-output (MIMO),probability of error.
I. I
NTRODUCTION T HE LARGE swathes of raw spectrum available in themillimeter-wave frequency range are expected to providean attractive solution to the high data-rate demands of thefuture 5G cellular networks [1]. The small carrier wavelengthof millimeter-wave frequencies allow for reduced spacingbetween the antenna elements when multiple antennas areused at the transmitter and receiver. This implies that multiple-input multiple-output (MIMO) spatial multiplexing gains canbe obtained even in the presence of a strong line-of-sight (LoS)component when operating in such high frequencies [2].In LoS environments, the MIMO channel matrix H is adeterministic function of the positions of the transmitter andreceiver and the geometry of the antenna arrays used at eitherterminals. If the positions of the communicating terminals arefixed and known apriori, the geometry of the antenna arrays Copyright ©2015 IEEE. Personal use of this material is permitted. However,permission to use this material for any other purposes must be obtained fromthe IEEE by sending a request to [email protected] work was supported by Australian Research Council Discovery Projectunder Grant ARC DP160100528.L. Natarajan is with the Department of Electrical Engineering, IndianInstitute of Technology Hyderabad, Sangareddy 502285, India (email: [email protected]).Y. Hong and E. Viterbo are with the Department of Electrical and Com-puter System Engineering, Monash University, VIC 3800, Australia (e-mail: { yi.hong, emanuele.viterbo } @monash.edu). can be designed to optimize the performance of the commu-nication system. The LoS MIMO channel quality, in terms ofcapacity, multiplexing gain, coverage and channel eigenvalues,have been studied in [2]–[9] as a function of the inter-terminaldistance and the inter-antenna spacing of transmit and receivearrays, when the antennas are to be arranged in a rectangular,circular or a linear array. However, these design techniquesassume that the position and the orientation of the antennaarrays are fixed, and the resulting criteria may be difficult to besatisfied if either of the communicating terminals is mobile orif the positions of the wireless terminals are not known a priori.Systems designed according to these known criteria degradegracefully with variations in the geometric parameters, andmay be adequate in certain scenarios where the changes in theorientation are limited, such as in a sectored communicationcell where the variation of the base station orientation withrespect to the direction of propagation is limited. However,these designs, which utilize two-dimensional antenna arrays,do not provide MIMO spatial multiplexing gains for arbitraryarray orientations.In [10], the mutual information rates of a predominantlyLoS channel with arbitrary antenna array orientations werestudied using simulations and direct measurements in an in-door environment. The results show that the three-dimensionalantenna arrays obtained by placing the antennas on the facesof a tetrahedron or a octahedron provide mutual informationrates that are largely invariant to the rotation of antennaarrays in indoor LoS conditions. Previous studies of three-dimensional antenna arrays for wireless communications havemainly studied the capacity of the resulting MIMO systemin a rich scattering environment. In [11] a compact MIMOantenna was proposed which consists of dipole antennasplaced along the edges of a cube. A -port and a -portantenna were designed in [12] by placing antennas alongthe edges and faces of a cube. In [13] and [14], -port and -port antennas were designed on a cube, respectively, andthe performance of the MIMO system in terms of capacityand channel eigenvalues in a richly scattering environmentwere studied. The objective of [11]–[14] has been to design acompact array by densely packing the antenna elements whileexploiting the degrees of freedom available in an environmentthat provides abundant multipath components.To the best of our knowledge, there has been no priortheoretical study of LoS MIMO channels where the transmitor receive antenna array orientations are arbitrary, as maybe experienced in wireless mobile communications. Further,all previous work have focussed on optimizing the mutualinformation rates of the MIMO channel. In order to achieve TO APPEAR IN IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY the information theoretic limits, we need code design criteriabased on an error performance analysis of the communicationchannel. In this paper, we consider LoS MIMO channels wherethe number of transmit antennas used for signalling is andboth the transmit and receive arrays have random orientations.We study the impact of the geometry of the antenna arrays onthe system error performance and design a LoS MIMO systemthat guarantees a minimum channel quality and good errorperformance for arbitrary transmit and receive orientationsover a range of inter-terminal distances.We model the -transmit antenna n r -receive antenna LoSMIMO channel H using the upper triangular matrix R ob-tained from its QR-decomposition (Section II). This allows usto derive bounds on pairwise error probability and identify thecode parameter that determines the high signal-to-noise ratio( SNR ) error performance of arbitrary coding schemes in LoSMIMO channels.We show that for any planar, i.e., -dimensional, arrange-ment of receive antennas (such as linear, circular and rect-angular arrays), the rate of decay of error probability issimilar to that of a rank LoS MIMO channel wheneverthe receiver undergoes random rotations. Further, for somecoding schemes, including spatial multiplexing [15]–[17], theerror rate with any planar receive array decays no faster than
SNR − even though the channel is purely LoS and experiencesno fading (Section III).We consider the smallest number of receive antennas n r = 4 that can form a three-dimensional, i.e., non-planar, arrange-ment, and derive bounds on error performance when they forma tetrahedral array. In this case, the error probability decaysfaster than that of a rank channel and is always exponentialin SNR irrespective of the coding scheme used (Section IV-A).We then design a LoS MIMO system with a good error per-formance for all transmit and receive array orientations over arange of inter-terminal distances by using a tetrahedral receivearray and adaptively choosing two transmit antennas from atriangular/pentagonal array at the transmitter (Section IV-B).Finally, we present simulation results to support our theoreticalclaims (Section V).
Notation:
Matrices and column vectors are denoted by boldupper-case and lower-case symbols respectively. The symbols A ⊺ , A † and k A k F denote the transpose, the conjugate-transpose and the Frobenius norm of a matrix A . The symbol k · k denotes the -norm of a vector. For a complex number z , arg( z ) and Re( z ) denote its phase and real part, respectively.The expectation operator is denoted by E ( · ) .II. T HE × n r L O S MIMO C
HANNEL
We consider MIMO line-of-sight (LoS) transmission with n t = 2 antennas at the transmitter and n r ≥ antennas atthe receiver. Assuming that the large scale fading effects,such as path loss, are accounted for in the link budget, wetake the magnitude of the complex channel gain between anytransmit-receive antenna pair to be unity. If r m,n is the distancebetween the n th transmit and the m th receive antennas, thenthe ( m, n ) th component of channel matrix H ∈ C n r × is [4] h m,n = exp (cid:18) i πr m,n λ (cid:19) , (1) where λ is the carrier wavelength and i = √− . The re-sulting wireless channel is y Rx = √ SNR Hx + w Rx , where y Rx ∈ C n r is the received vector, x ∈ C is the transmittedvector, w Rx ∈ C n r is the circularly symmetric complex whiteGaussian noise with unit variance per complex dimension,and SNR is the signal-to-noise ratio at each receive antenna.The power constraint at the transmitter is E (cid:0) k x k (cid:1) ≤ . Weassume that the channel matrix H is known at the receiverbut not at the transmitter. Let h , h ∈ C n r denote the twocolumns of H , and H = QR be its QR decomposition where Q ∈ C n r × has orthonormal columns, i.e., Q is a semi-unitarymatrix, and R = k h k h † h k h k r k h k − | h † h | k h k . Let µ denote the correlation between the two columns h and h of H , and θ µ be the phase of h † h , i.e., µ = | h † h |k h k k h k and θ µ = arg (cid:16) h † h (cid:17) . From (1), we have k h k = k h k = √ n r , and hence, R = √ n r (cid:20) iθ µ µ p − µ (cid:21) . (2)Since Q is semi-unitary and w Rx is a white Gaussian noisevector, y = Q † y Rx is a sufficient statistic for x . Hence, in therest of the paper we will consider the following equivalentchannel y = √ SNR Rx + w , (3)where R is given in (2), and w = Q † x is a two-dimensionalcircularly symmetric complex white Gaussian noise with zeromean and unit variance per complex dimension. A. Modelling the R matrix To analyze the error performance of arbitrary codingschemes in LoS MIMO channels, we model the phase θ µ asindependent of µ and uniformly distributed in [0 , π ) . Derivingthe probability distribution of θ µ and µ appears difficult,however, we provide an analytical motivation and numericalexamples to support the validity of our model.We follow the notations from [3], [4] to describe thegeometry of the transmit and receive antenna positions asillustrated in Fig. 1. We denote the inter-antenna distance atthe transmitter by d t , and define the origin O of the three-dimensional reference coordinate system as the mid-pointbetween the two transmit antennas. Define the z -axis of thecoordinate system to be along the line connecting the twotransmit antennas, i.e., the positions of the two transmit an-tennas are (cid:2) , , d t (cid:3) ⊺ and (cid:2) , , − d t (cid:3) ⊺ , respectively. Choosethe x -axis of the coordinate system such that the centroid O ′ of the receive antenna array lies on the x – z plane. Let O ′ beat a distance of R from O and at an angle β to the x -axisi.e., at the point (cid:2) R cos β, , R sin β (cid:3) ⊺ . Consider an auxiliarycoordinate system with O ′ as the origin and the three axes x ′ , y ′ , z ′ defined as follows: the x ′ axis is along the direction ATARAJAN et al. : LINE OF SIGHT × n r MIMO WITH RANDOM ANTENNA ORIENTATIONS 3
Fig. 1. Illustration of the parameters used in the system model. OO ′ , i.e., along the vector (cid:2) cos β, , sin β (cid:3) ⊺ , z ′ axis is on the x – z plane, and y ′ is parallel to y . Let ( d m , θ m , φ m ) be thespherical coordinates of the m th receive antenna with respectto this auxiliary coordinate system, where d m is the radialdistance, θ m is the polar angle and φ m is the azimuthal angle.The distance r m,n between the n th transmit and m th receiveantennas satisfies [5] r m,n ≈ R + d m sin θ m cos φ m + ( − n d t β +( d m sin θ m sin φ m ) + ( d m cos θ m + ( − n d t cos β ) R .
Therefore, the difference r m, − r m, is given by r m, − r m, = d t sin β + ( d m cos θ m + d t cos β ) R − ( d m cos θ m − d t cos β ) R = d t sin β + d t d m cos β cos θ m R . (4)Let F ( β ) = h † h denote the inner product between the twocolumns of H as a function of β . Using (1) and (4), we obtain F ( β ) = h † h = n r X m =1 h † m, h m, = exp (cid:18) i πd t sin βλ (cid:19) n r X m =1 exp (cid:18) i πd t d m cos β cos θ m Rλ (cid:19) (5)Let f ( β ) = exp ( i πd t sin β/λ ) and f ( β ) = n r X m =1 exp (cid:18) i πd t d m cos β cos θ m Rλ (cid:19) . Then F ( β ) = f ( β ) f ( β ) , arg F = arg f + arg f , and since | f | = 1 , we also have | F | = | f | . The angle β is equal to the parameter θ t used in [3], [4]. We now upper bound the magnitude of the derivative of µ with respect to β . The derivative of d f / d β equals n r X m =1 − i πd t d m sin β cos θ m Rλ exp (cid:18) i πd t d m cos β cos θ m Rλ (cid:19) . (6)Note that | d f / d β | ≤ b , where b = 2 πd t P n r m =1 d m Rλ . For aninfinitesimal change ∆ β in the value of β , | f ( β + ∆ β ) | − | f ( β ) | = (cid:12)(cid:12)(cid:12) f ( β ) + d f d β ∆ β (cid:12)(cid:12)(cid:12) − | f ( β ) | . Using the fact that (cid:12)(cid:12) | u + w | − | u | (cid:12)(cid:12) ≤ | w | for any u, w ∈ C ,we have (cid:12)(cid:12)(cid:12) | f ( β + ∆ β ) | − | f ( β ) | (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) d f d β (cid:12)(cid:12)(cid:12)(cid:12) | ∆ β | ≤ b | ∆ β | . It follows immediately that | d | f | / d β | ≤ b . Using the factthat µ = | F ( β ) | /n r = | f ( β ) | /n r , we have (cid:12)(cid:12)(cid:12)(cid:12) d µ d β (cid:12)(cid:12)(cid:12)(cid:12) = 1 n r (cid:12)(cid:12)(cid:12)(cid:12) d | f | d β (cid:12)(cid:12)(cid:12)(cid:12) ≤ bn r . (7)Note that θ µ = arg F = arg f + arg f , and hence, d θ µ / d β = d(arg f ) / d β + d(arg f ) / d β . Now, arg f =2 πd t sin β/λ , and hence, d(arg f ) / d β = 2 πd t cos β/λ . Us-ing (7) and the fact that the range of transmission R is muchlarger than d m , we have d(arg f )d β = 2 πd t cos βλ ≫ πd t λ P n r m =1 d m R n r = bn r ≥ (cid:12)(cid:12)(cid:12)(cid:12) d µ d β (cid:12)(cid:12)(cid:12)(cid:12) . Hence, we expect d θ µ / d β ≫ | d µ/ d β | , i.e., a small change inthe value of β , that causes a negligible change in µ , changesthe phase θ µ by an entire cycle of π rad. This motivates thechannel model where θ µ is independent of µ and uniformlydistributed in the interval [0 , π ) . Example . Consider a × LoS system operating in E-bandat the frequency of GHz over a distance R = 10 m. Let thetwo receive antennas be positioned such that θ = 0 , θ = π , φ = φ = 0 and d = d = d r / . Then, using (5), we have h † h = 2 exp (cid:18) i πd t sin βλ (cid:19) cos (cid:18) πd t d r cos βRλ (cid:19) . It follows that µ = cos (cid:18) πd t d r cos βRλ (cid:19) and θ µ = 2 πd t sin βλ . (8)Suppose the antenna geometry is to be configured so that H is unitary, i.e., µ = 0 , under the assumption that β = 0 . Thiscan be achieved by choosing d t and d r so that d t d r cos βRλ = d t d r Rλ = 12 . This is the criterion for uniform linear arraysgiven in [3]–[5]. With λ = 4 . mm, the choice of d t = d r = p Rλ/ . m yields µ = 0 . With thischoice of d t and d r , through direct computation using (8), weobserve that as β undergoes a small variation in value from rad through . rad ( . ◦ ), the corresponding value of µ changes from to . × − , while θ µ ranges over theentire interval from to π rad. TO APPEAR IN IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY
Fig. 2. The joint probability density function f ( θ µ , µ ) of Example 2. Example . Continuing with the × system of Example 1,now assume that the transmit and receive arrays are affected byindependent random rotations about their respective centroids.The random rotations are uniformly distributed over the spaceof all -dimensional rotations. The channel matrix H , and theparameters θ µ and µ are now random variables. The jointprobability density function f ( θ µ , µ ) obtained using Monte-Carlo methods is shown in Fig. 2. We computed f ( θ µ , µ ) overa rectangular grid of points using randomly generatedinstances of H . For any fixed µ , we observe that f ( θ µ , µ ) isessentially constant across all values of θ µ , implying that θ µ is uniformly distributed in [0 , π ) and is independent of µ . Example . Consider a × LoS MIMO system, with arectangular array at the receiver, carrier frequency of GHz,and inter-terminal distance of R = 10 m. The receive antennasare placed at the vertices of a square whose edges are oflength d r . We choose d t = d r = p Rλ/ , which yieldsthe ideal channel (i.e., µ = 0 ) if the transmit and receivearrays are placed broadside to each other [5]. The jointprobability density function f ( θ µ , µ ) , obtained using Monte-Carlo methods, when the transmit and receive arrays undergouniformly random rotations about their centroids is shown inFig. 3. As in Example 2, the numerical result supports thevalidity of our channel model.In the rest of the paper we model the × n r LoS channelusing the × matrix (cf. (3)) R = √ n r (cid:20) i Θ µ p − µ (cid:21) , (9)where Θ is uniformly distributed in [0 , π ) and µ = 1 n r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n r X m =1 exp (cid:18) i πd t d m cos β cos θ m Rλ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (10) B. Coding schemes
We analyse the error performance of any arbitrary codingscheme for two transmit antennas with a finite transmissionduration. Let T ≥ denote the transmission duration of agiven communication scheme and C ⊂ C × T the finite set Fig. 3. The joint probability density function f ( θ µ , µ ) of Example 3. of all possible transmit codewords. The rows of the code-words X ∈ C correspond to the two transmit antennas andthe columns to the T time slots. All codewords are equallylikely to be transmitted and the optimal decoder, i.e., themaximum-likelihood (ML) decoder, is used at the receiver.We further assume that the communication scheme satisfies theaverage power constraint P X ∈ C k X k F ≤ | C | T . Our analysisholds for arbitrary codes C , including space-time block codes(STBCs) [18].We now briefly recall two specific coding schemes whichwill be used in our simulations (in Section V) to illustrateour analytical results. Spatial multiplexing (SM) [15]–[17],which is also known as
VBLAST in the literature, is a simpleyet powerful scheme where independent information symbolsare transmitted across different antennas and time slots. Thecodebook C ⊂ C × corresponding to SM occupies T = 1 time slot, and is given by C = (cid:26)(cid:20) s s (cid:21) (cid:12)(cid:12)(cid:12) s , s ∈ A (cid:27) , where A is a complex constellation, such as QAM or PSK.The Golden code [19] is an STBC for two transmit antennasoccupying T = 2 time slots, and is given by C = (cid:26)(cid:20) α ( s + τ s ) α ( s + τ s ) i ¯ α ( s + µs ) ¯ α ( s + µs ) (cid:21) (cid:12)(cid:12)(cid:12) s , . . . , s ∈ A (cid:27) , where A is a QAM constellation, τ = (1 + √ / , µ = 1 /τ , α = 1 + iµ and ¯ α = 1 + iτ . Unlike SM, the Golden codespreads the information symbols across time and antennas.Both SM and Golden code have been well studied in thecase of non line-of-sight MIMO fading channels. The SMscheme provides high data rate with low complexity encodingand decoding, while the Golden code provides high data rate,full-diversity as well as a large coding gain at the cost ofhigher decoding complexity in fading channels. C. Error probability analysis for a fixed µ We now analyse the error performance of a given arbitrarycoding scheme for a fixed value of µ . Let C ⊂ C × T be ATARAJAN et al. : LINE OF SIGHT × n r MIMO WITH RANDOM ANTENNA ORIENTATIONS 5 any code and X a , X b ∈ C be two distinct codewords. Let ∆ X = X a − X b be the pairwise codeword difference matrix.The pairwise error probability between X a and X b for a fixed µ and a given realization Θ = θ is [18] PEP ( X a → X b | µ, Θ = θ ) = Q r SNR k R ∆ X k F ! , where Q is the Gaussian tail function. Using the Chernoffbound Q ( x ) ≤ exp (cid:0) − x / (cid:1) , we have the upper bound PEP ≤
12 exp (cid:18) − SNR k R ∆ X k F (cid:19) . (11)Denoting the two rows of the matrix ∆ X as ∆ x ⊺ and ∆ x ⊺ ,we obtain the following expression for the squared Euclideandistance between the codewords at the receiver, k R ∆ X k F = n r (cid:16) k ∆ x k + k ∆ x k + 2 µ Re(e iθ ∆ x † ∆ x ) (cid:17) = n r (cid:16) k ∆ x k + k ∆ x k + 2 µ cos θ ′ | ∆ x † ∆ x | (cid:17) (12)where θ ′ = θ + arg(∆ x † ∆ x ) mod 2 π .
1) Worst-case Error Probability over θ : For a given µ ,the value of θ that minimizes the squared Euclidean distance k R ∆ X k at the receiver is θ ∗ = π + arg(∆ x † ∆ x ) since itleads to cos θ ′ = − in (12). Using the notation d ( µ, ∆ X ) = k ∆ x k + k ∆ x k − µ | ∆ x † ∆ x | , (13)the worst-case squared Euclidean distance is min θ ∈ [0 , π ) k R ∆ X k F = n r d ( µ, ∆ X ) . Thus the worst-case
PEP for a fixed µ satisfies PEP ∗ ( µ ) ≤
12 exp (cid:18) − n r SNR d ( µ, ∆ X )4 (cid:19) . (14)
2) Average Error Probability over Θ : Since Θ is uniformlydistributed in [0 , π ) , so is Θ ′ = Θ + arg(∆ x † ∆ x ) mod 2 π .Using (11) and (12), the error probability averaged over Θ ,for a fixed µ , can be upper bounded as follows E Θ ( PEP ) ≤ E Θ (cid:18)
12 exp (cid:18) − SNR k R ∆ X k F (cid:19)(cid:19) = 12 exp (cid:18) − SNR n r ( k ∆ x k + k ∆ x k )4 (cid:19) × π Z π exp (cid:18) − SNR n r µ cos θ ′ | ∆ x † ∆ x | (cid:19) d θ ′ = 12 exp (cid:18) − SNR n r ( k ∆ x k + k ∆ x k )4 (cid:19) × I (cid:18) SNR n r µ | ∆ x † ∆ x | (cid:19) where I ( x ) = 1 π Z π exp ( x cos θ ′ ) d θ ′ = 12 π Z π exp ( x cos θ ′ ) d θ ′ = 12 π Z π exp ( − x cos θ ′ ) d θ ′ is the modified Bessel function of the first kind and zerothorder. For large x we have [20] I ( x ) = e x √ πx (cid:0) O (cid:0) x − (cid:1)(cid:1) . (15)Using (13) and the first order approximation (15), we get thefollowing approximate upper bound when µ > , E Θ ( PEP ) . q πn r SNR µ | ∆ x † ∆ x |× exp (cid:18) − n r SNR d ( µ, ∆ X ) (cid:19) . (16)Since the exponential function falls more rapidly than SNR − / , the high SNR behaviour is dictated by d ( µ, ∆ X ) .In this section, we derived bounds on PEP for a fixed µ . InSections III and IV we analyze the effects of random rotationsof the terminals on µ and error performance.III. E RROR PERFORMANCE OF PLANAR RECEIVE ARRAYS
Assume that the receive antenna system is affected by a ran-dom three-dimensional rotation U ∈ R × about its centroid O ′ . Let the rotation U be uniformly distributed on the set ofall -dimensional rotations, i.e., the special orthogonal group SO = (cid:8) U ∈ R × | UU ⊺ = I , det( U ) = 1 (cid:9) . In Theorem 1, we provide a lower bound on the averagepairwise error probability over a LoS MIMO channel withplanar receive array. To do so, we derive a lower bound onthe probability that a random rotation U would lead to a ‘bad’channel matrix with µ close to , i.e. µ ≥ − ǫ for some smallpositive ǫ . By analyzing the PEP for this class of bad channels,and letting ǫ decay suitably with SNR , we arrive at a lowerbound for the average
PEP at high
SNR . Theorem 1.
Let the receive antenna array be any planararrangement of n r antennas, n r ≥ , undergoing a uniformlydistributed random rotation U about its centroid. At high SNR ,for any transmit orientation β , we have E ( PEP ) ≥ exp (cid:16) − n r c | ∆ x † ∆ x | (cid:17) n r SNR q π | ∆ x † ∆ x | (cid:16) k ∆ X k F + √ n r SNR (cid:17) × exp (cid:18) − n r SNR d (1 , ∆ X ) (cid:19) , (17) where c = max n r m =1 πd t d m /Rλ .Proof: Let { e x , e y , e z } be the standard basis in R . Whenthe receive system undergoes no rotation, i.e., when U = I , letthe position of the m th receive antenna relative to the centroid O ′ of the receive antenna system be d m r m , where r m ∈ R isa unit vector. Since the receive array is planar and the randomrotation U is uniformly distributed, without loss of generality,we assume that the vectors r , . . . , r n r are in the linear spanof e x and e z . From Fig. 1 we see that θ m in (5) is the anglebetween the orientation Ur m of the m th receiver and the unitvector ˜v = (cid:2) − sin β, , cos β (cid:3) ⊺ along z ′ -axis, i.e., cos θ m = r ⊺ m U ⊺ ˜v . Note that U ⊺ has the same distribution as U , and TO APPEAR IN IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY v = U ⊺ ˜v is uniformly distributed on the unit sphere in R .The resulting random variable | e ⊺ y v | is known to be uniformlydistributed in the interval [0 , .For a small positive number δ > , consider the event E : | e ⊺ y v | ≥ − δ . The probability of E is P ( E ) = P (cid:16) | e ⊺ y v | ≥ p − δ (cid:17) = 1 − p − δ ≈ δ , for small values of δ . We will now derive an upper boundfor the PEP for the case when E is true. Using the fol-lowing inequalities, we first show that | cos θ m | ≤ δ , for all m = 1 , . . . , n r , | cos θ m | = | r ⊺ m v | ≤ | e ⊺ x v | + | e ⊺ z v | (since r m ∈ span( e x , e z ))= k v k − | e ⊺ y v | ≤ − (1 − δ ) = δ . Let c m = 2 πd t d m cos β/Rλ and c max = max { c , . . . , c n r } .From (10), we have µ = 1 n r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n r X m =1 exp ( ic m cos θ m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We will now show that the value of µ is close to when E istrue. If ǫ m = 1 − exp( ic m cos β ) , then | ǫ m | = (1 − cos( c m cos θ m )) + sin ( c m cos θ m )= 2 − c m cos θ m ) ≈ − (cid:18) − c m cos ( θ m )2 (cid:19) = c m cos ( θ m ) ≤ δ c , where the approximation follows from the Taylor’s se-ries expansion of the cos( · ) function and the fact that | c m cos θ m | ≤ c m δ is small. Now, µ = 1 n r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n r X (1 − ǫ m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 n r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n r − n r X ǫ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ − n r n r X | ǫ m | ≥ − δc max . Thus µ ≥ − δc max whenever E is true.The pairwise error probability for fixed µ and Θ = θ is Q (cid:16)p SNR k R ∆ X k F / (cid:17) . Since we need a lower bound onthe probability of error, we use the following lower bound forthe Gaussian tail function [21] Q ( x ) ≥ √ π (cid:0) x + √ x + 4 (cid:1) exp (cid:18) − x (cid:19) , for x ≥ . Using x + 4 ≤ ( x + 2) for x ≥ , we obtain a more relaxedbound Q ( x ) ≥ √ π ( x + 1) exp (cid:18) − x (cid:19) . In our case x = p SNR k R ∆ X k F / , and we use the exactvalue of x from (12) for the exponent, and the following upperbound for the denominator x = r SNR k R ∆ X k F ≤ r SNR k R k F k ∆ X k F = p n r SNR k ∆ X k F . Thus, we have the following lower bound for a fixed µ and Θ = θ , PEP ≥ exp (cid:0) − SNR k R ∆ X k F (cid:1) √ π (cid:0) √ n r SNR k ∆ X k F + 1 (cid:1) . (18)Since the denominator is independent of the phase Θ , we canuse the same method as in Section II-C2 to obtain the averageof the above lower bound over the uniformly distributedrandom variable Θ . Averaging (18) over Θ and using theapproximation to the Bessel function (15), we obtain E Θ ( PEP ) & exp (cid:0) − n r SNR d ( µ, ∆ X ) (cid:1) n r SNR q π µ | ∆ x † ∆ x | (cid:16) k ∆ X k F + √ n r SNR (cid:17)
Using the trivial upper bound µ ≤ in the denominator, E Θ ( PEP ) & exp (cid:0) − n r SNR d ( µ, ∆ X ) (cid:1) n r SNR q π | ∆ x † ∆ x | (cid:16) k ∆ X k F + √ n r SNR (cid:17) . (19)Since d ( µ, ∆ X ) is a decreasing function of µ , if E is true,the numerator in the RHS of (19) can be lower boundedby exp (cid:0) − n r SNR d (1 − δc max , ∆ X ) (cid:1) . The expression (19) isa lower bound on the average PEP for a given µ . We nowderive a lower bound for the PEP when averaged over both µ and Θ as follows E ( PEP ) = P ( E ) P ( X a → X b |E ) + P ( E c ) P ( X a → X b |E c ) ≥ P ( E ) P ( X a → X b |E ) ≥ δ exp (cid:0) − n r SNR d (1 − δc max , ∆ X ) (cid:1) n r SNR q π | ∆ x † ∆ x | (cid:16) k ∆ X k F + √ n r SNR (cid:17) . (20)From the definition (13) of d ( µ, ∆ X ) , we have d (1 − δc max , ∆ X ) = d (1 , ∆ X ) + 2 δc max | ∆ x † ∆ x | , Using the above relation and choosing δ = SNR − , which issmall for high SNR , we obtain E ( PEP ) ≥ exp (cid:16) − n r c max | ∆ x † ∆ x | (cid:17) exp (cid:0) − n r SNR d (1 , ∆ X ) (cid:1) n r SNR q π | ∆ x † ∆ x | (cid:16) k ∆ X k F + √ n r SNR (cid:17) . Using cos β ≤ in c m = 2 πd t d m cos β/Rλ we obtain c max ≥ max m πd t d m /R min λ . This completes the proof.We compare the lower bound (17) on PEP for planarreceive arrays undergoing random rotations, with the upperbound (16) for a channel with fixed µ = 1 . The dominantterm dictating the rate of decay of error probability for boththese channels is exp (cid:0) − n r SNR min ∆ X d (1 , ∆ X ) (cid:1) , where the ATARAJAN et al. : LINE OF SIGHT × n r MIMO WITH RANDOM ANTENNA ORIENTATIONS 7 minimization is over all non-zero codewords difference ma-trices ∆ X = X a − X b of the code C . Note that µ = 1 minimizes the performance metric d ( µ, ∆ X ) , and correspondsto the worst-case scenario where both H and R have rank .While planar receive arrays, such as the well-studied linear,rectangular and circular arrays, provide an array gain (an n r -fold increase in received SNR ), their asymptotic coding gain min ∆ X d (1 , ∆ X ) provides no improvement over that of anyrank channel.Theorem 1 further implies that when min ∆ X d (1 , ∆ X ) = 0 ,the error probability is no more exponential in SNR , butdecays at the most as fast as
SNR − . Hence, although thechannel is purely LoS and experiences no fading, the errorperformance with a planar arrangement of antennas can decayslowly, similar to a fading channel.The parameter d (1 , ∆ X ) satisfies the following tight in-equality d (1 , ∆ X ) = k ∆ x k + k ∆ x k − | ∆ x † ∆ x |≥ k ∆ x k + k ∆ x k − k ∆ x k k ∆ x k = ( k ∆ x k − k ∆ x k ) . (21)The second line follows from the Cauchy-Schwarz inequalitywhich is tight if and only if ∆ x and ∆ x are linearlydependent. Thus, d (1 , ∆ X ) = 0 if and only if ∆ x and ∆ x are linearly dependent and k ∆ x k = k ∆ x k , i.e., if and only if ∆ x = α ∆ x for some complex number α of unit magnitude.We use this observation in Example 4 below to show that thewidely used spatial multiplexing coding scheme suffers fromsuch a slowly decaying error probability with planar receivearrays. Example . Performance of Spatial Multiplexing with PlanarReceive Array. The codeword difference matrices of the SMscheme are of the form ∆ X = (cid:20) ∆ s ∆ s (cid:21) , where ∆ s , ∆ s ∈ ∆ A and ∆ A = { x − y | x, y ∈ A} is theset of pairwise differences of the complex constellation A .When ∆ s = ∆ s the two rows of the codeword differencematrix ∆ X are equal resulting in d (1 , ∆ X ) = 0 . Hence, forthe SM scheme, min ∆ X d (1 , ∆ X ) = 0 , and from Theorem 1,the rate of decay of the average error probability will be nofaster than SNR − . Note that this result is valid for any numberof antennas n r used in any planar arrangement of the receivearray. This theoretical result is validated by our simulations(see Fig. 10 and Fig. 13) in Section V.IV. E RROR P ERFORMANCE OF T ETRAHEDRAL R ECEIVEARRAY
The smallest number of antennas that can form a non-planararrangement is . In this section we consider the case where n r = 4 receive antennas are placed at the vertices of a regulartetrahedron, see Fig. 4. The inter-antenna distance d r is thesame for any pair of receive antennas, and this is related tothe distance d m of each antenna from the centroid O ′ of the Fig. 4. The receive antennas are placed at the vertices , . . . , of thetetrahedron. Also shown in the figure are the centroid O ′ , the distances d and d of the antennas and from O ′ , and the inter-antenna distance d r . receive array as d m = p / d r , m = 1 , . . . , . Let us definethe deviation factor η as in [3], [4] as follows η = R λ d t d r cos β . (22)In the case of a tetrahedral receiver, using (10) and (22), µ = 14 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m =1 exp i πη r
38 cos θ m !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The parameter η captures both the distance R and the transmitorientation β , while the variables θ , . . . , θ jointly determinethe receive orientation U . In order to upper bound the errorprobability using (14), we need the maximum value of µ overall possible η and U . Let µ ∗ ( η ) = max U ∈ SO (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m =1 exp i πη r
38 cos θ m !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (23)be the maximum channel correlation over all receive orien-tations as a function of η . If one is aware of the range ofvalues that R and β may assume, then one can upper boundthe worst-case PEP using (14) as
PEP ∗ ≤
12 exp (cid:18) − n r SNR d (max η µ ∗ ( η ) , ∆ X ) (cid:19) = 12 exp (cid:18) − SNR d (max η µ ∗ ( η ) , ∆ X ) (cid:19) . (24) A. An upper bound on µ ∗ ( η ) In this sub-section we derive an upper bound on µ ∗ ( η ) forall η ≥ . This result will allow us to show that the high SNR error performance of the tetrahedral array is better than anyplanar receive array when η ≥ and the receiver undergoesa uniformly random rotation. To derive this upper bound, wefirst show that when η ≥ , irrespective of the receive arrayorientation, the × channel matrix H contains at least one × submatrix H sub such that the correlation µ sub betweenthe two columns of H sub is at the most cos (cid:0) π/ √ η (cid:1) . Thislatter problem is equivalent to finding the maximum distortion TO APPEAR IN IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY
Fig. 5. The tetrahedron arrangement illustrating the vertices , . . . , , thereference O ′ at the centroid of the tetrahedron, and the directions of a fewof the unit vectors r m and g m,ℓ . when a unit vector in R is quantized using a codebook G consisting of unit vectors that correspond to the edges ofthe tetrahedron along with the polarities ± . The computationof this maximum distortion is then simplified by showing that G is a group code [22].We first introduce some notation to capture the geometricalproperties of the tetrahedral array. Consider the tetrahedronshown in Fig. 5 with the centroid O ′ . Let r m ∈ R bethe unit vector in the direction of the m th receive antennawith respect to the reference O ′ . Hence, the position vectorof the m th receive antenna is d m r m . If one applies a -dimensional rotation U ∈ R × on the receive system about O ′ , the position of the m th receive antenna is d m Ur m . Itis straightforward to show that the polar angle θ m of the m th rotated receive antenna (cf. Fig. 1) satisfies cos θ m = r ⊺ m U ⊺ ˜v ,where the unit vector ˜v = (cid:2) − sin β, , cos β (cid:3) ⊺ . Since U is anarbitrary rotation matrix, the set of all possible values assumedby the vector v = U ⊺ ˜v is the sphere S consisting of all unitvectors in R . From (10), the correlation µ for a tetrahedralreceiver is µ = 14 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m =1 exp (cid:18) i πd t d m cos β cos θ m Rλ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where cos θ m = r ⊺ m U ⊺ ˜ v = r ⊺ m v , and v ∈ S captures theeffect of the rotation undergone by the receive array. For any m = ℓ , the unit vectors r m and r ℓ satisfy k r m − r ℓ k = p / .Let g m,ℓ = r m − r ℓ k r m − r ℓ k = r
38 ( r m − r ℓ ) be the unit vector along r m − r ℓ , i.e., along the edge of thetetrahedron between the vertices m and ℓ (see Fig. 5).Let H sub be the × submatrix of H formed using the m th and ℓ th rows. Note that H sub is the channel responseseen through the receive antennas m and ℓ . Using the fact that d m = d ℓ = p / d r , the correlation between the columns of H sub can be written as µ sub = 12 (cid:12)(cid:12)(cid:12) exp (cid:18) i πd t d m cos β r ⊺ m v Rλ (cid:19) +exp (cid:18) i πd t d ℓ cos β r ⊺ ℓ v Rλ (cid:19) (cid:12)(cid:12)(cid:12) = 12 (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) i πd t d m cos β ( r m − r ℓ ) ⊺ v Rλ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i πd t d m p / β g ⊺ m,ℓ v Rλ !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 12 (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) i πη g ⊺ m,ℓ v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) cos (cid:18) π η g ⊺ m,ℓ v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (25)where the fourth equality follows from (22) and the lastequality uses straightforward algebraic manipulations. Givenan ‘orientation’ v , we intend to find the submatrix H sub withthe least correlation µ sub . If η ≥ , we have (cid:12)(cid:12)(cid:12)(cid:12) π η g ⊺ m,ℓ v (cid:12)(cid:12)(cid:12)(cid:12) ≤ π . Since cos is decreasing function in the interval [0 , π/ ,from (25), the problem of finding µ sub translates to findingthe edge g m,ℓ of the tetrahedron that has the largest innerproduct with v .We will now show that for any v ∈ S there exists a g m,ℓ such that p / ≤ g ⊺ m,ℓ v ≤ . Since k v − g m,ℓ k = k v k + k g m,ℓ k − g ⊺ m,ℓ v = 2 (cid:16) − g ⊺ m,ℓ v (cid:17) this is equivalent to finding the maximum squared Euclideanerror when the set of vectors G = { g m,ℓ | m = ℓ } is used asa codebook for quantizing an arbitrary unit vector v in R .The set G contains vectors, corresponding to the edgesof the tetrahedron together with the polarity ± . Proposition 1.
For any v ∈ S , there exist m, ℓ ∈ { , , , } , m = ℓ , such that g ⊺ m,ℓ v ≥ p / .Proof: With some abuse of notation we will denote theelements of G as g , . . . , g . For each i = 1 , . . . , , let D i = (cid:8) v ∈ S | g ⊺ i v ≥ g ⊺ j v , for all j = i (cid:9) (26)be the set of unit vectors that are closer to g i than any other g j ∈ G . Since ∪ i D i = S , it is enough to show that min i min v ∈D i g ⊺ i v = r . As we now show, the regions D , . . . , D are congruent toeach other. Let H be the symmetry group of the tetrahedron,i.e., the set of all orthogonal transformations on R that mapthe tetrahedron onto itself. It is known that the group H is isomorphic to the symmetric group S of degree , andevery element of H is uniquely identified by its action onthe set of vertices, which is isomorphic to the action of thecorresponding element in S on the set { , , , } ; see [23].Since for any two given pairs ( m , ℓ ) and ( m , ℓ ) , with ATARAJAN et al. : LINE OF SIGHT × n r MIMO WITH RANDOM ANTENNA ORIENTATIONS 9
Fig. 6. An illustration of the cones S and R used in the proof ofProposition 1. The cone S is circular with axis g (dashed line). The cone R is bounded by hyperplanes, and its edges are along the vectors q , . . . , q .The edge q is the farthest from the axis g and lies on the surface of S . m = ℓ and m = ℓ , there exists a permutation on { , , , } that maps m to m and ℓ to ℓ , we see thatthere exists an orthogonal transformation M ∈ H such that r m = Mr m and r ℓ = Mr ℓ . This can be extended to a group action on G as Mg m ,ℓ = M (cid:18) r m − r ℓ k r m − r ℓ k (cid:19) = r M ( r m − r ℓ )= r
38 ( r m − r ℓ ) = g m ,ℓ . Thus we see that the group H acts transitively on G , i.e., G = { Mg i | M ∈ H} for every i = 1 , . . . , . This makes G a group code, and consequently, the regions D , . . . , D are congruent to each other [22], i.e., for every ≤ i < j ≤ , there exists an orthogonal transformation M ∈ H such that D j = M D i = { Mv | v ∈ D i } . Since orthogonal transformations conserve inner products andsince g i ∈ D i for all i , we have min v ∈D i g ⊺ i v = min v ∈D j g ⊺ j v for any i = j. Thus, to complete the proof it is enough to show that min v ∈D g ⊺ v = r . We now restrict ourselves to one particular region D andfind the smallest value of g ⊺ v . Note that when v ∈ S , theinner product of v with g i decreases with increasing distance k v − g i k . Thus, from (26), D is the intersection of S withthe set of all points in R that are closer to g than any other g i ∈ G . The region D is called a fundamental region ofthe group code G and is bounded by two-dimensional planespassing through the origin [22]. The half-spaces P i that definethis fundamental region are P i = (cid:8) x ∈ R | k x − g k ≤ k x − g i k (cid:9) = (cid:8) x ∈ R | ( g − g i ) ⊺ x ≥ (cid:9) , and are related to D as D = S ∩ R , where R = ∩ i =2 P i . The group code G and the half-spaces P i can be explicitlycalculated starting from the geometry of the tetrahedron, and itcan be verified that R , and hence D , is bounded by exactly planes arising from of the eleven half-spaces P i . The region R is a convex cone [22] generated from the edges runningalong the vectors q , . . . , q that are the intersections betweenthe hyperplanes, i.e., R is the infinite cone generatedfrom the convex hull of the set { q , . . . , q } . Fig. 6 showsan illustration of the geometry considered in this proof (thedepiction of q , . . . , q is not exact). Since min v ∈D g ⊺ v = min x ∈R g ⊺ x k x k , (27)and since g ⊺ x / k x k is the cosine of the angle between x and g , our problem is to find a vector in R which makes thelargest angle with g . The set of points that make a constantangle with g form the surface of an infinite circular conewith g as its axis. Thus (27) is equivalent to finding thesmallest circular cone S , with g as the axis, that contains theconical region R . Since R is generated by q , . . . , q , S isthe smallest circular cone that contains the vectors q , . . . , q ,and has g as the axis. It follows that S contains on its surfacethe vector q i , from among q , . . . , q , that makes the largestangle with g . Thus, min v ∈D g ⊺ v = min x ∈R g ⊺ x k x k = min x ∈ S g ⊺ x k x k The numerical value min i ∈{ ,..., } g ⊺ q i / k q i k = 1 / √ is ob-tained by a direct computation of the half-spaces P , . . . , P ,and the resulting vectors q , . . . , q arising from the tetrahe-dral geometry. Proposition 2.
If a tetrahedral array is used at the receiverand η ≥ , then for every receive orientation U , there existsa × submatrix H sub of the channel matrix H such that ≤ µ sub ≤ cos (cid:18) π √ η (cid:19) , where µ sub is the correlation between the two columns of H sub .Proof: From Proposition 1, there exist m = ℓ such that g ⊺ m,ℓ v ≥ p / . Let H sub be the submatrix of H formedby the m th and ℓ th rows. From (25) and the hypothesis that η ≥ , we have µ sub = (cid:12)(cid:12)(cid:12) cos (cid:16) π η g ⊺ m,ℓ v (cid:17)(cid:12)(cid:12)(cid:12) ≤ cos (cid:16) π η q (cid:17) .The following upper bound on µ ∗ ( η ) follows immediatelyfrom Proposition 2. Theorem 2.
For a tetrahedral receive array and η ≥ , µ ∗ ( η ) ≤ (cid:18) (cid:18) π √ η (cid:19)(cid:19) . Proof:
Let H = [ h m,n ] be the × channel matrix.From Proposition 2, assume without loss of generality that the × submatrix formed from the first two rows has correlation µ sub ≤ cos (cid:0) π/ √ η (cid:1) . Then, µ = 14 (cid:12)(cid:12)(cid:12) h † , h , + h † , h , + h † , h , + h † , h , (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) h † , h , + h † , h , (cid:12)(cid:12)(cid:12) + 14 (cid:12)(cid:12)(cid:12) h † , h , + h † , h , (cid:12)(cid:12)(cid:12) = 12 µ sub + 14 (cid:12)(cid:12)(cid:12) h † , h , + h † , h , (cid:12)(cid:12)(cid:12) ≤
12 cos (cid:18) π √ η (cid:19) + 24 , where the last inequality follows from Proposition 2 and thefact that all h m,n have unit magnitude.The upper bound (cid:0) (cid:0) π/ √ (cid:1)(cid:1) / on µ ∗ ( η ) is lessthan for η ≥ . Since d ( µ, ∆ X ) is a decreasing function of µ , we have d ( µ ∗ ( η ) , ∆ X ) > d (1 , ∆ X ) . Hence, the geometryof the tetrahedral arrangement allows the error probability todecay faster than that of rank LoS MIMO channels, andprovides performance improvement over any planar arrange-ment n r = 4 of antennas, irrespective of the code used at thetransmitter. Note that this gain of the tetrahedral arrangementover planar arrays is not due to larger inter-antenna distances d t and d r .From (21), we have d (1 , ∆ X ) ≥ ( k ∆ x k − k ∆ x k ) . Us-ing µ ∗ < , we obtain d ( µ ∗ , ∆ X ) > d (1 , ∆ X ) ≥ ( k ∆ x k − k ∆ x k ) ≥ . Hence, unlike the planar case, the error probability of atetrahedral receiver is exponential in
SNR for any code C . Example . Performance of Spatial Multiplexing with Tetra-hedral Receive Array . Consider the SM scheme signalledover n t = 2 antennas using -QAM symbols. Let the trans-mit orientation β = 0 be fixed, the inter-terminal distance R = 10 m, λ = 4 . mm, and d t = d r = 0 . m.Then, η = Rλ/ (2 d t d r cos β ) = 1 , and from Theorem 2, µ ∗ ( η ) ≤ . . An exhaustive numerical computation overall pairs of codewords yields min ∆ X d (0 . , ∆ X ) = 0 . .Using (24), the pairwise error probability of SM for fixedtransmit orientation and random receive orientation can beupper bounded as E ( PEP ) ≤ PEP ∗ ≤
12 exp ( − SNR µ ∗ (1)) ≤
12 exp ( − SNR × . . On the other hand, as shown in Example 4, for any planarreceiver array, the error rate is not better than
SNR − . B. System design for arbitrary array orientations
In Section IV-A, we assumed that η was fixed, i.e., the trans-mit orientation β and inter-terminal distance R were fixed, andwe studied the effect of an arbitrary rotation U of the receivearray on µ and error probability. We now design a system thatallows arbitrary transmit and receive array orientations and arange of values R min ≤ R ≤ R max . It is desirable that the Fig. 7. Triangular arrangement of transmit antennas.
LoS MIMO system guarantees a minimum channel quality i.e., µ ≤ µ max , for some µ max < . Using (24), for such a system, E ( PEP ) ≤ PEP ∗ ≤
12 exp (cid:18) − n r SNR d ( µ max , ∆ X ) (cid:19) . Using union bound, the average codeword error rate and biterror rate of the system can be upper bounded by | C | (cid:18) − n r SNR ∆ X d ( µ max , ∆ X ) (cid:19) . Hence, the coding gain of an arbitrary coding scheme C overthis LoS MIMO system is min ∆ X d ( µ max , ∆ X ) .When the number of transmit antennas n t = 2 , by choosing β = π/ , we observe from (10) that the worst case correlation µ max = 1 irrespective of the array geometry used at thereceiver. Hence, in order to have µ max < , we need morethan antennas at the transmitter.Suppose the transmitter uses an array of n t ≥ antennas.Based on the transmit array orientation, one can choose ofthe n t antennas for signal transmission so that the angle β corresponding to the chosen pair of antennas is minimum. Forexample, let n t = 3 antennas be placed at the vertices of anequilateral triangle with inter-antenna distance d t , as shown inFig. 7. Let t m,n be the unit vector in R in the direction ofthe position of transmit antenna m with respect to the positionof transmit antenna n . Note that the vectors t m,n vary withchanges in the transmit array orientation. If antennas m and n are used for transmission and if u ∈ R is the unit vectoralong the direction OO ′ of transmission, then sin β = u ⊺ t m,n (cf. Fig. 1, where tx and tx correspond to tx m and tx n ,respectively). The six vectors in the set T = { t m,n | m, n = 1 , , , m = n } are arranged symmetrically in a two-dimensional plane atregular angular intervals of π/ . Let u k and u ⊥ be thecomponents of u parallel and perpendicular to the plane of T , respectively. Since the vectors in T divide the plane intoregular conical regions of angular width π/ , there exists atleast one vector t m,n ∈ T such that the angle between t m,n and u k lies in the interval [ − π/ , + π/ , i.e., | u ⊺ k t m,n |k u k k ≤ sin (cid:16) π (cid:17) = 12 . We can thus upper bound | u ⊺ t m,n | as follows | u ⊺ t m,n | = | u ⊺ ⊥ t m,n | + | u ⊺ k t m,n | ≤ k u k k ≤ . ATARAJAN et al. : LINE OF SIGHT × n r MIMO WITH RANDOM ANTENNA ORIENTATIONS 11
Fig. 8. The functions µ ∗ , µ ∗ pent , upper bound on µ ∗ and the line µ max = 2 / . Thus there exists a t m,n such that | sin( β ) | = | u ⊺ t m,n | ≤ / , i.e., β ∈ [ − π/ , π/ . Hence, if the transmit array is anequilateral triangle, by appropriately choosing out of the available antennas for signalling, one can ensure | β | ≤ π/ .The upper bound on µ ∗ ( η ) of Theorem 2 is not tight and isavailable only for η ≥ . Since this bound can not be used as agood estimate of µ ∗ ( η ) and the analytical computation of theexact expression (23) of µ ∗ ( η ) appears to be difficult, we usenumerically computed values of µ ∗ ( η ) for system design. Thefunction µ ∗ ( η ) and the upper bound of Theorem 2 are shownin Fig. 8. Using the exact function µ ∗ ( η ) , the requirement onchannel quality µ ≤ µ max can be translated into a criterion η ∈ [ η min , η max ] . From (22), for fixed d t , d r , λ , and | β | ≤ β max , we have η min = R min λ d t d r and η max = R max λ d t d r cos β max . (28)The range [ R min , R max ] can thus be obtained from (28). Example . Suppose we require µ max = 2 / with λ =4 . mm. Using a triangular transmit array we have β max = π/ . From Fig. 8, the criterion µ ∗ ( η ) ≤ / is equivalent to η min = η = 0 . and η max = η = 1 . . If each side ofthe triangular transmit array has length d t = 6 cm, and thetetrahedral receive array has d r = 25 cm, then from (28) wehave R min = 4 . m and R max = 7 . m.The narrow range of [ R min , R max ] in Example 6 can beattributed to the small value of η − η in Fig. 8. This can beimproved by using a pentagonal transmit array as follows. Asshown in Fig. 9, with a regular pentagon, the choice of thetransmit antenna pair can be divided into the following twocases: (i) the two antennas are the neighbouring vertices ofthe pentagon with inter-antenna distance equal to the length d t of the edge of the regular pentagon, or (ii) the antennas arenon-neighbouring with inter-antenna distance (cid:0) √ (cid:1) d t / .Irrespective of the class from which the antenna pair ischosen, it is straightforward to show that | β | ≤ π/ canbe always guaranteed. While the value of η for the first caseis given by (22), in the second case it reduces by a factor of Fig. 9. Left: any pair of neighbouring antennas in a pentagonal array has aninter-antenna distance of d t . Right: Any pair of non-neighbouring antennashas distance (1 + √ d t / . (cid:0) √ (cid:1) / because of the larger inter-antenna distance. Thusthe maximum correlation with pentagonal transmit array is µ ∗ pent ( η ) = min (cid:26) µ ∗ ( η ) , µ ∗ (cid:18) η √ (cid:19)(cid:27) , where µ ∗ ( η ) is given in (23). From Fig. 8, the value of η max improves from η to η , thereby widening [ R min , R max ] . Example . As in Example 6, let µ max = 2 / , λ = 4 . mm, d t = 6 cm and d r = 25 cm. With a pentagonal transmitarray, β max = π/ , and using the function µ ∗ pent , wehave η min = η = 0 . and η max = η = 2 . Using (28), R min = 4 . m and R max = 12 . m.V. S IMULATION R ESULTS
We use the system parameters λ , d t , d r , R max and R min from Example 7. We assume that the transmit and receivearrays undergo independent uniformly random -dimensionalrotations about their centroids, and the distance R between theterminals is uniformly distributed in [ R min , R max ] . In all thesimulations the channel matrix H was synthesized using (1)and the exact distances { r m,n } between the transmit andthe receive antennas. We consider the following three codingschemes with the transmission rate of bits per channel use:( i ) the Golden code [19] using -QAM alphabet, ( ii ) spatialmultiplexing (SM) [15]–[17] with -QAM, and ( iii ) uncoded -QAM transmitted using only one transmit antenna (single-input multiple-output SIMO). Gray mapping is used at thetransmitter to map information bits to constellation points, andunless otherwise stated, maximum-likelihood (ML) decodingis performed at the receiver. While we used pairwise errorprobability for performance analysis in Sections II, III and IV,we simulate the bit error rate to compare the average errorperformance. A. Error performance with n r = 4 Fig. 10 shows the performance of the three schemes withtwo different antenna geometries: ( i ) uniform linear array(ULA) at the transmitter with n t = 2 , and uniform rectan-gular array (URA) at receiver with n r = 4 , ( ii ) selecting The performance of uniform linear array at receiver is worse than that ofURA, and hence has been omitted. ✷ ✹ ✻ ✽ ✶(cid:0) ✶✷ ✶✹✶(cid:0)✲✁✶(cid:0)✲✂✶(cid:0)✲✄✶(cid:0)✲☎✶(cid:0)✲✆✶(cid:0)✲✝ ❙✞✟ ✠✡☛ ☞✌✍❇✎✏✑✒✒✓✒✒✔✏✑ ■☞✕✖✗●✘✗☞✕☛ ✙✘☞✕❙✚✖✛✡✖✗ ✜✢✣❙✡☛✤✗✕ ✖☛✛✕☛☛✖ ❯✥✦ ✣ ❯✟✦P✕☛✛ ✣ ✧✕✛★
Fig. 10. Comparison of Pent × Tetr with ULA × URA.Fig. 11. Coding gain for bit rate of bits per channel use. antennas from a pentagonal array at the transmitter, andusing a tetrahedral array at the receiver. The values of d t , d r are ideal for the ULA × URA configuration [5] at thedistance R = 2 d t d r /λ = 7 . m, which is near the mid-pointof the interval [ R min , R max ] . The performance of the single-antenna transmission scheme is independent of the receiveantenna geometry since, from (1), all the channel gainsof the SIMO channel have unit magnitude. Also, Fig. 10shows the performance of the ideal channel with µ = 0 , i.e., R = √ n r I , which is a pair of parallel AWGN channels eachcarrying a -QAM symbol. From Fig. 10, we see that, withULA × URA, the performance of both SM and the Goldencode are worse than SIMO at high
SNR . Further, since min ∆ X d (1 , ∆ X ) = 0 for SM, the error probability decaysslowly with SNR , confirming our theoretical results. With theproposed pentagon × tetrahedron geometry both codes showimproved performance, close to that of the ideal channel.The above error performance is succinctly captured by thecoding gain min ∆ X d ( µ, ∆ X ) shown in Fig. 11 as a functionof µ . From Example 7, µ ≤ / for the new antenna geometry.From Fig. 11 we see that the coding gains of SM and theGolden code are both equal to for all µ ≤ / and arelarger than the SIMO coding gain for µ ≤ / , which explains ✷ ✹ ✻ ✽ ✶(cid:0) ✶✷ ✶✹✶(cid:0)✲✁✶(cid:0)✲✂✶(cid:0)✲✄✶(cid:0)✲☎✶(cid:0)✲✆✶(cid:0)✲✝ ❙✞✟ ✠✡☛ ☞✌✍✎❇✏✑✒✓✓✔✓✓✕✑✒ ■☞✖✗✘●✙✘☞✖☛ ✚✙☞✖❙✛✗✜✡✗✘ ✢✣✤ ❯✥✦❚✧✡✗☛★✘✖P✖☛✜✗★✙☛ Fig. 12. Performance of different tx arrays with tetrahedral rx array. ✲(cid:0) ✲✁ ✲✂ ✲✄ ✵ ✄ ✂ ✁ (cid:0) ✶✵ ✶✄✶✵☎✆✶✵☎✝✶✵☎✞✶✵☎✟✶✵☎✠ ❙✡☛ ☞✌✍ ✎✏✑✒❇✓✔✕✖✖✗✖✖✘✔✕ ❚✙✚✛✛ ✜✢✣✛✤✥✢✦✤✧★ ✧✚✚✧✩❘✛✪✫✧✤✬✭★✧✚ ✧✚✚✧✩✻✮ ✧✤✫✛✤✤✧✥ ✯✻ ✧✤✫✛✤✤✧✥
Fig. 13. Error probability of spatial multiplexing with triangular transmitarray when the receive array is (i) three-dimensional, and (ii) rectangular.Results are shown for n r = 16 and n r = 64 antennas. their superiority to SIMO. On the other hand, the codinggain for linear and rectangular arrays is min ∆ X d (1 , ∆ X ) . For µ = 1 , from Fig. 11 we observe that SIMO has the largestcoding gain followed by the Golden code and then SM. Theerror performances in Fig. 10 show this same trend for therectangular array at high SNR .Fig. 12 compares the performance of different transmit arraygeometries when a tetrahedral array is used at the receiver.The n t = 2 case (ULA) performs poorly since µ max = 1 .While the triangular array with the Golden code achievesmost of the available gain, the pentagonal array has near idealperformance. B. Error Performance with large number of receive antennas
The LoS MIMO system analysed in Section IV employs thetetrahedral receive array – a three-dimensional antenna arrayfor n r = 4 antennas – to enable smaller error rates than planararrays. The geometry of the receive array is relevant evenif the number of receiving antennas n r is large. Theorem 1and Example 4 show that the probability of error of the SMscheme is lower bounded up to a constant factor by SNR − ATARAJAN et al. : LINE OF SIGHT × n r MIMO WITH RANDOM ANTENNA ORIENTATIONS 13 for any value of n r , if a planar receive array is used. Onthe contrary, from Example 5, the SM scheme can achieveexponential rate of decay of error probability if n r = 4 antennas are placed at the vertices of a regular tetrahedron.It follows that for any n r ≥ , a careful three-dimensionalarrangement of n r antennas can ensure that the error rate isexponential in SNR . For instance, if the three-dimensionalarrangement includes a subset of antennas that form atetrahedron, it immediately follows from Example 5 that asub-optimal decoder that bases its decision only on the signalsreceived by these antennas achieves exponential error rate.Hence, the optimal ML decoder that utilizes all the n r receiveantennas achieves an exponential error probability as well.Fig. 13 compares the error performance of SM schemeunder planar and three-dimensional receive antenna arrayswhen n r = 16 , . A triangular array is used at the transmitter, -QAM is chosen as the modulation scheme and ML decodingis performed at the receiver. For both values of n r , we considera URA (rectangular arrangement of receive antennas) for theplanar arrangement of antennas. The three-dimensional arrayis chosen as a set of n r points on the surface of a sphere sothat the minimum distance between the points is large. A tableof such arrangements of points, which are known as sphericalcodes , is available online [24]. For fairness, the diameter ofthe sphere is set equal to the width of the rectangular array.The coordinates of the n r points on the sphere were obtainedfrom [24]. As with previous simulations, we set the valuesof d t , λ, R max and R min as in Example 7. The inter-antennadistance d r of the URA is chosen to be . cm when n r = 16 and to be . cm when n r = 64 . This is the optimal inter-antenna distance for the URA when the transmit and receivearrays are oriented broadside to each other and the inter-terminal distance R = 7 . m [5].It is evident from Fig. 13 that array geometry is an importantdesign parameter even when n r is large. The error rates ofrectangular arrays shown in Fig. 13 decay as SNR − at high SNR . The gain due to the three-dimensional array is about dB at an error rate of − for both n r = 16 and .VI. C ONCLUSION
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