On the Capacity of MIMO Rician Broadcast Channels

Transcription

1 On the Capacity of IO Rician Broadcast Channels Alireza Bayesteh Kamyar oshksar Amir K. Khani Abstract In this paper, a downlink communication system, in which a Base Station BS equipped with antennas communicates with N N single-antenna users, in a Rician fading environment is considered. The asymptotic in terms of the number of users sum-rate capacity of the system, as well as the capacity-achieving strategies, are derived. The main results of the paper are as follows: i in the region of K = o, where K denotes the Rician factor, the sum-rate capacity scales as log + P η, where P denotes the SNR η, +K which is achieved by Zero-Forcing Beam-Forming ZFBF along with a low-complexity user selection algorithm that considers only the scattered component of the users channels, ii in the region K = ω, in the case of co-located transmit antennas, the capacity scales as log + P, which is achieved by Time Division ultiple Access TDA, iii in the region K = ω, in the case of isotropically-distributed specular components, the sum-rate capacity behaves as log + P, which is achieved by ZFBF, along with a user selection algorithm that considers only the specular component of the users channels. I. INTRODUCTION ultiple-input multiple-output IO systems have proved their ability to achieve high bit rates on a scattering wireless network [], [2]. In a IO Broadcast Channel IO-BC, the base station equipped with multiple antennas communicates with several users. Recently, there has been a lot of interest in characterizing the capacity region of this channel [4] [7]. In these works, it has been demonstrated that the sumrate capacity of IO broadcast channels can be achieved by applying Dirty-Paper Coding DPC [8] at the transmitter. Despite the fact that the sum-rate capacity of Gaussian IO-BC is known, it is still interesting to study the behavior of sum-rate capacity in various scenarios. [9] compares the achievable sum-rate of IO-BC for DPC to that achieved by using linear precoding schemes, characterizes the gap between the achievable sum-rates in the high SNR regime. [0] compares the achievable sum-rate of DPC to that of TDA for a Gaussian IO-BC. [] considers a IO- BC with a large number of users shows that i the sum-rate capacity of the system scales as log, when N is the number of users in the network, ii a simple scheme of Rom Beam-Forming asymptotically achieves the sumrate capacity as N. References [2] [4] consider the same network set-up prove that one can achieve the sumrate capacity of the system by utilizing Zero-Forcing Beam- Forming at the transmitter, provided that the user selection is performed wisely. In [5] the scaling laws of the sum-rate for fading IO Gaussian broadcast channels using time-sharing to the strongest user, DPC beamforming, is derived for the asymptotic case of N. In all the mentioned papers [9]- [5], the channel model is assumed to be Rayleigh fading. Therefore, it is of interest to investigate the sum-rate capacity of IO-BC, assuming more general channel models. One of the most widely-used models for the wireless channels is Rician fading. This model is suitable for wireless links when there is a line of sight LOS link between the transmitter receiver. Several papers in the literature consider Rician fading in the context of point-to-point IO communications. In [6], the authors derive the exact capacity of IO Rician channel, when perfect Channel State Information CSI is available at the receiver, but the transmitter has neither instantaneous nor statistical CSI. Reference [7] studies the capacity of IO Rician channel in the high low SNR regimes, for both coherent non-coherenet communications. it is shown in [7] that in the low SNR regime, the specular component of the channel completely determines the form of the optimum signal whereas in the high SNR regime it has no effect on the optimum signal structure. In [8], the authors consider the min-capacity of a IO Rician channel with an unknown deterministic specular component. [9] analyzes the capacity of a IO Rician channel with isotropically rom rank-one specular component, when the channel is unknown at both the transmitter receiver sides. In this paper, we consider a Rician IO-BC, in which a transmitter equipped with antennas communicates with N N single-antenna users. The channels are assumed to be perfectly known at both the transmitter receiver sides. The asymptotic in terms of the number of users sum-rate capacity of the system, as well as the capacity-achieving strategies, are derived. The main results of the paper are as follows: i in the region of K = o, where K denotes the Rician factor, the sum-rate capacity scales as log+ P η, where P denotes the SNR η +K, which is achieved by ZFBF along with a low-complexity user selection algorithm that considers only the scattered component of the users channels, ii in the region K = ω, in the case of co-located

2 transmit antennas, the capacity scales as log+p, which is achieved by TDA, iii in the region K = ω, in the case of isotropically-distributed specular components, the sum-rate capacity behaves as log+p, which is achieved by ZFBF, along with a user selection algorithm that considers only the specular component of the users channels. This paper is organized as follows. In section II, we introduce the system model. Section III is devoted to analyzing the sum-rate capacity of the underlying system, section IV concludes the paper. Throughout this paper, the norm of the vectors are denoted by., the Hermitian operation is denoted by. H, the determinant the trace operations are denoted by.., respectively. E. represents the expectation, notation log is used for the natural logarithm, the rates are expressed in nats. For any given functions gn, = OgN is equivalent to lim N gn <, = ogn is equivalent to lim N = 0, gn = ΩgN is equivalent to lim N gn > 0, = ωgn is equivalent to lim N gn =, = ΘgN is equivalent to lim N gn = c, where 0 < c <, gn is equivalent to lim N gn =. II. SYSTE ODEL In this work, a IO-BC in which a base station equipped with antennas communicates with N users, each equipped with a single antenna, is considered. The received signal by user k can be written as y k = H k x + n k, where x C is the transmitted signal, H k C is the channel vector from the transmitter to the kth user, which is assumed to be perfectly known at the receiver side provided to the BS via a noiseless feedback channel, n k CN0, is the AWGN at this receiver. Under Rician channel model, H k can be written as H k = r k G k + r k b k, 2 where G k is a circularly symmetric zero mean unit variance Gaussian vector, reflecting the scattered component b k is a unit-norm vector representing the specular component of the channel, r k is a constant related to the Rician factor K k 2 via r k = K k K k +. We consider two scenarios for b k: i The entries of H k are i.i.d Gaussian with mean b k variance b k 2, where b k is a complex number satisfying b k 2 = r k. In this case, it is easy to observe that b k = ejθ k, where is the vector of all ones. This model corresponds to the case that the transmit antennas are co-located, consequently, In fact, the BS does not need to have the perfect CSI about all the users channels. However, the partial CSI that the BS receives through feedback is based on the perfect CSI that the receivers have. 2 Rician factor is defined as the ratio of the power of the specular component to the power of the scattered component. the specular components from all transmit antennas to each of the users are equal 3. ii The vector b k is isotropically distibuted in the unit sphere. This model has been used in [9]. It is assumed that r k is fixed for all the users during the whole transmission period is equal to a constant r, i.e., r = r 2 = = r N = r. We assume that the transmitter has an average power constraint P, i.e. E xx P. The power constraint is assumed to be per frame. In other words, the power constraint is independent of the channel realization. The channels are assumed to be quasi-static block fading, in which each channel H k is drawn romly at the start of each transmission frame remains constant for the whole transmission frame, changes independently to another realization in the start of the next frame. The frame itself is assumed to be long enough to allow communication at rates close to the capacity. Defining the sum-rate capacity of the system in the channel realization H H k N as R OptH, the average sum-rate capacity, denoted as R Opt, is defined as the average over time of R Opt H, which is by the ergodicity of the channel, equal to E H R Opt H. R Opt is shown in [4] to be equal to R Opt = E H max P k P Pk =P N log I + H k P kh k, 3 where P k is the transmit power allocated to the kth user. III. ASYPTOTIC ANALYSIS; CAPACITY COPUTATION In this section, we compute the capacity of IO-BC under Rician fading, in the asymptotic scenario of N. To this end, we consider two cases; i K = o ii K = ω. For each case, we provide a lower-bound upper-bound for the capacity prove that as N, these bounds converge to each other. A. K = o Theorem The capacity of the underlying IO-BC in the case of K = o equals R Opt = log + P + o, 4 + K which is asymptotically achievable by ZFBF. Proof - The proof is based on the upper-bound lower-bound given as follows: 3 Note that however, the specular components from each transmit antenna to different users are not necessarily equal.

3 Upper-bound: Using 2, the upper-bound for the sumrate capacity can be derived as 4 R Opt E log + P H 2 max log + P + K E G 2 max + 2 K G max E P + K + + K E P + K +. 5 In [20], it is shown that 2 K G max E A + G 2 = o, 6 max K E P + K + = o. 7 Substituting in 5, the upper-bound on the sum-rate capacity can be written as R Opt log + P + K E G 2 max + o = log + P + o, 8 + K where the second line follows from the fact that E G 2 max = + Olog []. 2 Achievability: Scheduling based on the scattered component: Consider the follwoing algorithm: Algorithm Set the threshold t = + 3log Among the users in the following set: S k G k 2 > t, 9 select one user at rom. Call this user s, define S S s. For m = 2 to, repeat the following: Denote the set of selected users up to the m th step as A m s,, s m. Define S m S A m. Define P m as the sub-space spanned by the scattered channel components of the users selected in the previous steps, i.e., v sj m j=, where v k G k G k, k =,, N. Let Φ j m j= be m orthonormal bases for P m. Then, s m = arg min k S m m j= v k Φ H j 4 For the details of the proofs, the reader is referred to [20].. 0 In the above algorithm, the user selection is solely performed based on the scattered component of the channel. First, the users with scattered channel gains above the threshold t are cidated. After that, the algorithm tries to find a set of semiorthogonal channel vectors out of the cidate users. To this end, at each step of the algorithm, the user whose channel vector is the most orthogonal to the sub-space spanned by the previously selected users is chosen. After selecting the users, the BS performs ZFBF on the whole channel vectors of the selected users. Defining H [ H T s H T s ] T u = [u,, u ] T as the information vector for the selected users, we have x = H u. Therefore, the achievable sum-rate of this scheme can be written as R = E H [H H H]. 2 Defining B as the event that L S >, C as δg H > +2 2, in which G [ ] G T s G T T s δa denotes the orthogonality defect [2] of A, D as the event that G 2 max t +, where t + + log G 2 max max k G k 2, we have R E H [H H H] B, C, D PrB, C, D. 3 In [20], it has been shown that PrB, C, D = + o conditioned on B, C, D, [H H H ] K + [ + O µ ], 4 where µ 4 R log. Substituting in 3 yields P + K + O µ. 5 = olog K, it follows that log P +K = + o. Noting this fact comparing the Since K log + P +K above lower-bound with the upper-bound derived in 8 the proof of Theorem follows. B. K = ω Co-located transmit antennas: In this scenario, the specular components from all transmit antennas to each receiver are equal. In other words, b k = eiθ k, where is the allone vector with size. However, the scattered component of all users channels follow the circularly symmetric complex Gaussian distribution. The following theorem gives the capacity of IO-BC in this scenario:

4 Theorem 2 The capacity of IO-BC in the case of K = ω co-located transmit antennas scales as R Opt = log + P + o, 6 which is achievable by TDA. Proof - Similar to the proof of Theorem, we first give an upper-bound on the sum-rate capacity then, by giving an achievable rate which is asymptotically equal to the upperbound the theorem is proved. Upper-bound: Writing the sum-rate capacity of IO-BC from 3, we have R Opt = E = E max P P k Pk =P max P k P Pk =P log I + QP log det I + log I + r log + P + E N H H k P k H k N b H k P k b k + max log I + QP P P k Pk =P 7 where Q r r N [ G H k P k b k + b H k P ] kg k + r N bh k P kb k, P. I + rp H Denoting the second term in the right h side of the above equation by R 2, in [20], it has been shown that Consequently, R 2 = o. 8 R Opt log + P + o. 9 Achievability - In order to show that the sum-rate given in 6 is achievable, we propose a TDA scheme, in which the transmitter romly selects one user at a time communicates with that user. Therefore, the maximum achievable rate is equal to the capacity of a ISO Rician channel, expressed as bellow: R = E log + P H k 2 E log + P r r G k 2.20 Let us define E as the event that G k 2 <. R can be lower-bounded as R E log + P r r G k 2 E Pr E a = log + P + o. 2 In the above equation, a follows from i the assumption of K = ω, which implies that conditioned on E, r Gk = G k K+ = o, ii as G k 2 has Chi- Square distribution with 2 degrees of freedom, Pr E log N!N K+ = o iii as r = K K = ω, we have r = + o. This completes the proof of achievability hence, the proof of Theorem 2. 2 Isotropic specular components: In this case, it is assumed that the specular component of all users channels, i.e., b k, k =,, N, are isotropically distributed in the unit sphere. The difference between this case the previous case is that in the case of co-located transmit anennas, there is only one available coordinate in the system the coordinate of for transmission, as a result, we do not have the -fold capacity increase. However, in this case, by wisely selecting the users one can achieve the -fold capacity increase. The following theorem gives the capacity in this case: Theorem 3 The capacity of Rician IO-BC in the case of K = ω isotropic specular components is equal to R Opt = log + P + o. 22 Proof - Upper-bound: R Opt E log + P H 2 max E log + P r + 2 r G max a log + P E 2 r + r G max b = log + P r + o 2 = log + rp + o c = log + P + o. 23 In the above equation, a follows from the concavity of log function along with the Jensen s inequality, b results from the fact that G max = O since r = +K = o, we have r G max = o, c results from r = + o. Achievability; Scheduling based on specular component Consider the following algorithm: Algorithm 2 select one user at rom. Call this user s, define S S s. For m = 2 to, repeat the following: Denote the set of selected users up to the m th step as A m s,, s m. Define S m S A m. Define P m as the sub-space spanned by the specular channel components of the users selected in the previous steps, i.e., b sj m j=.

5 Let Φ j m j= be m orthonormal bases for P m. Then, s m = arg min k S m m j= b k Φ H j. 24 After selecting the users, the BS performs zero-forcing beam-forming on the whole channel vectors of the selected users. Defining H [ ] H T s H T T s u = [u,, u ] T as the information vector for the selected users, we have x = H u. 25 Defining the event F δb H < + ǫ Q G H G <, where B = [ ] b T T s b s, G = [ ] G T T s G s, ǫ 2N 2, similar to 3, we have R E H [H H H] F, Q PrF, Q. 26 In [20], it has been shown that PrF, Q = + o N conditioned on F, Q, [H H H ] +o. Substituting in 26, we have R log + P + o, 27 which completes the proof of Theorem 3. Remark - Comparing the sum-rate capacity of the system in the two cases of co-located transmit antennas isotropic specular components when K = ω, it follows that in the first case, the capacity grows logarithmically with, while in the second case it scales linearly with. oreover, since + x > + x, x,, it follows that R isotropic Opt R co located Opt. 28 IV. CONCLUSION In this paper, we have derived the asymptotic sum-rate capacity of IO-BC with large number of users in a Rician fading environment. It is observed that in the region K = o, the capacity achieving strategy is exactly the same as the Rayleigh fading case. In the region K = ω, the sum-rate capacity depends on the distribution of the specular component; in the case of co-located transmit antennas, it is demonstrated that TDA achieves the sum-rate capacity the capacity grows logarithmically with the number of transmit antennas. 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