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B- ng r), R n e e e ) t l h rprMSfrec hne elzto 2][7.This algori [22]–[27]. learning realization machine by channel values. performed like each is their SNR for task and of classification s MCS values to set proper designed SNR been the the the have algorithms of to learning FER, observations FER associated empirical appropriate on an Based mapping of roun of number decoder. or turbo depth a li truncation impairments Viterbi practical like of implementation-d parameters, impact and or the noise SNR to non-Gaussian effective due non-linearities, ideal between not mapping is fixed single MCS in a algorithms Having adaptation effecti link scenarios. of develop use value make to [21] SNR metrics like effective SNR works each Some IEEE MCS. where an tables, in to look-up associated [20] algor of adaptation prediction form to the FER lead in metrics for SNR effective method The default S 802.16e. effective WiM the exponential The as the results. param- metric recommends empirical some example, to with for according values forum, fitted SNR the are the as that of (FER) eters [19] define rate mean are metrics Kolmogorov error SNR a Effective frame study. same under is channel the (AWGN) spatial fading SNR noise experience and effective Gaussian white to carrier The additive an each channel [14]–[18]. for for SNR SNR the (one as effective values defined one SNR to of stream) set altern the An [13]. map enoug adaptation contain to effective not may permit SNR to codewor average information the between error, mapping symbol complicated and error the of that differen is Because experience multiplexi codeword reason values. same The the division multiplexing. in valu symbols spatial frequency different with SNR The MIMO orthogonal single or CSI. a using (OFDM) the with systems of characterized be dimensionality in cannot higher state the channel to due challenging i sta rate channel transmit current information. the ACK/NAK the where with account adaptatio only [12], into link but [11], taking to like without works approach (S modified in different ratio seen A noise be unidimensio to MCS. can signal a MCS each a The to essentially assigning interval [10]. of is consists [9], [7]–[10] that approximations problem in BER problem of selection u use and expressions, the transmi analytical required coded complicated to efficien more approaches in spectral these resulted of the extension maximize The to [8]. 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SNR values [29]. This data-driven formulation was shown to • We derive a closed form approximation of the interuser outperform the effective SNR in [22], and is resiliant to practical interference leakage due to the use of block diagonal- impairments like non-Gaussian noise or amplifier non-linearities ization precoding with zero forcing (ZF) receivers. This [26]. approximation, unlike previous approaches, is not based Prior work in learning-based link adaptation focused in single on a random vector quantization analysis. We exploit the user scenarios [22]–[26]. The extension of these approaches to the particular codebook structure used in IEEE 802.11ac, which more general case of having multiple users served at the same is induced by the use of Givens decompositions, to derive time by the use of space division multiplexing (SDM) is not trivial our approximation. This approximation is shown to be very due to the interaction between user selection, mode selection, accurate for different feedback rates. We restrict our analysis precoding, and MCS selection. For example, the link adaptation to ZF receivers as a worst-case scenario, so our interference algorithm in [22] requires running the classification algorithm estimation will be conservative if the receiver employs more for all possible number of spatial streams (NSS) and selecting the advanced algorithms, such as minimum mean squared error NSS leading to a higher throughput. Applying a similar strategy in (MMSE) or maximum likelihood detectors. the multiuser case would require an additional exhaustive search • We apply previous data-driven approaches [22]–[26], which over the choice of users and the number of spatial streams per are limited to the single user, to a multiuser setting with user. a variable number of spatial streams per user. We show Prior work in learning-based link adaptation also did not that the MCS selection accuracy of the data-driven approach consider the impact of limited feedback precoding [22]–[26]. In outperforms unidimensional metrics such as average SNR multiuser MIMO (MU-MIMO) systems, limited feedback creates or effective SNR. The machine learning classifier is shown quantization error that results in residual interuser interference to be robust to changes in the statistical distribution of the [30]. The resulting interference makes performance a function of channel, i.e., is able to correctly perform MCS selection even the feedback quality. Therefore a smart multiuser link adaptation when the training is done with channel samples taken from algorithm should predict the interuser interference and move from a different statistical distribution. aggressive multiuser transmission to more conservative modes • We use a greedy algorithm that performs mode and user depending on the feedback quality [31]. selection, inspired on previous work like [37]–[39]. This In this paper, we present a data-driven link adaptation method algorithm exploits information on the feedback rate, the SNR that includes user selection, mode selection, precoding, MCS regime and the number of users to perform user and mode selection, and limited feedback. The main objective of the paper is selection. The greedy algorithm sequentially adds spatial to give insight into the effect of limited feedback in MU-MIMO layers from the different users until the maximum number systems with link adaptation. To do so, our solution assumes of spatial streams is reached, or the throughput is no longer perfect CSI at the receivers, and different degrees of CSI at increased. This algorithm, with a complexity that is linear in the transmitter. With this information, and taking into account the maximum number of spatial streams and in the number the degradation due to CSI inaccuracy, the transmitter is able of users, allows solving the user and mode selection problem to perform link adaptation. Our work is different from previous without resorting to an exhaustive search. approaches in several ways. The focus of [32], [33] is to study fairness in a multiuser setting, with limited feedback information The remaining of the paper is organized as follows: Section not taken into account, and MCS selection is performed by II describes the mechanisms in IEEE 802.11ac that enable MU- means of effective SNR. In [34], only a single spatial stream MIMO operation; Section III presents the system model; Section is allowed per user, and the effective SNR approach is followed IV introduces the problem statement. The following sections for MCS selection. In [35], adaptation is performed following a present the main contributions of the paper: Section V describes data-driven approach, and limited feedback is taken into account, the MU-MIMO precoding problem, and presents an approx- but the communication scenario is multicast (i.e., there is only imation for the interference leakage due to limited feedback a common message for all receivers). Our paper studies the precoding; Section VI describes the data-driven MCS selection; broadcast scenario, takes into account limited feedback, includes Section VII presents a greedy algorithm for user and mode the possibility of transmitting more than one spatial stream to selection; Section VIII presents the simulation results; Section each user, and performs MCS selection following a data driven IX concludes the paper. approach. We focus on the multiuser capabilities of IEEE 802.11ac [36] Notation: Matrices are denoted by capital bold letters A, B to develop our link adaptation method. We consider this standard and column vectors by lower case bold letters a, b. A∗ denotes due to its novelty and the challenging constraint of not allowing the Hermitian transpose of matrix A, AT denotes the transpose user allocation among subcarriers, like in LTE or WIMAX. The of matrix A, [A]ij denotes the element in the i-th row and j- adaptation is performed at the access point (AP), thus requiring a th column of matrix A, IK denotes the K K identity matrix, × minimum communication overhead with the user stations (STA) 0K×N denotes the zero matrix of size K N, A B denotes the to obtain CSI. This AP-centric approach is specially suitable for (entrywise) Hadamard product between matrices× ◦A and B, link adaptation in scenarios with cheap STAs, as they might not denotes the number of elements in set , CK denotes the set|A| of implement some of the optional features in IEEE 802.11ac to column vectors with K complex entries,A and CK×N denotes the perform MCS selection. set of K N matrices with complex entries. denotes the The main contributions of the paper are summarized as follows. set difference× operation, i.e., x ( ) A\Bx ,x/ ∈ A\B ⇐⇒ ∈ A ∈B 3
using a Givens decomposition. The angles resulting from this decomposition are the parameters that are fed back to the AP. The feedback message contains the beamforming matrices plus Fig. 1: Typical message interchange for sounding and feedback. some additional information. This information depends on the selected feedback mode (MU or SU) indicated by the AP in the feedback request message [36, Sec. 8.4.1.48]. II. MU-MIMO IN IEEE 802.11AC • Grouping: Determines if the carriers are grouped in the IEEE 802.11ac is an emerging wireless standard that supports feedback message, or if feedback is provided for every MU-MIMO, single user MIMO, OFDM modulation, and thus carrier. In this paper we assume no grouping. needs sophisticated link adaptation algorithms. In this section • Codebook information: Determines the number of bits for we summarize some of the mechanisms that enable MU-MIMO the codebook entries. This parameter is selected by the link adaptation in IEEE 802.11ac, and explain how they relate STA. In this paper, we treat different feedback rates, but to our system assumptions. Given the functionality required the problem of selecting an appropriate feedback rate is out in this paper, we divide the MU-MIMO operation into three of the scope of this work. different tasks: CSI acquisition, link adaptation and MU-MIMO • Beamforming matrices and SNR information: Includes transmission. preferred beamforming matrix, average SNR for each spatial stream, and SNR for each carrier (or group of carriers) if feedback is in MU mode. A. CSI acquisition Despite being a time division duplexing system, channel reci- B. Link adaptation procity is not natively supported by IEEE 802.11ac to obtain The objective of link adaptation is to perform user selection, CSIT. IEEE 802.11n [4] describes a procedure to obtain CSI mode selection, MCS selection and MIMO precoding design. at the AP by the transmission of a training sequence in the This can be done in a centralized way at the transmitter, taking as uplink [4, Sec. 9.29.2.2] and a calibration procedure to identify input the limited feedback described in the previous section, or the differences between the transmit and receive chains at the in a distributed way by requesting an STA to select the preferred STA. These mechanisms are not present in IEEE 802.11ac, so mode and MCS for a given MIMO configuration. In this paper, acquisition of CSI based on channel reciprocity cannot rely on we follow a centralized approach, since our solution involves joint any information exchange with the receivers. Although there is scheduling and MCS selection. some research in estimating the channel from the normal packet exchange in IEEE 802.11ac [40, Sec. 2.3.3], we only consider CSI acquisition at the AP by the mechanism described in the standard. C. MU-MIMO transmission This mechanism comprises sending a sounding sequence in the Once the users, MCS, mode and precoders are selected, MU- downlink and feeding back the estimated channel to the AP. In MIMO transmission takes place. MU-MIMO in IEEE 802.11ac the following, we describe these two tasks. is non-transparent, meaning that the STAs are aware they will be 1) Sounding: Channel sounding is initiated by an AP by jointly scheduled with other STAs. This allows a given STA to transmitting a very high throughput (VHT - name of the new use the training sequences not only to estimate its own MIMO transmission modes in IEEE 802.11ac) null data packet (NDP) channel, but also the interference [36, Sec. 22.3.11.4], which announcement, which is a control frame that includes the identi- allows performing some advanced tasks such as the modification fiers of the set of users which are potentially going to be polled of the receive precoders, or the correct calculation of the Viterbi for feedback. Together with the identifiers, the frame carries weights for decoding. information on the requested feedback (there are two different types, single user - SU or multiuser - MU) and the number of III. SYSTEM MODEL columns (spatial streams, only for the MU case) in the requested feedback matrix. This frame is described in more detail in [36, Consider the downlink of an N-carrier OFDM wireless net- Sec. 8.3.1.19]. work where an AP equipped with Ntx antennas communicates with U STAs, = 1,...,U where the u-th user has N ,u After the NDP announcement, the AP transmits an NDP, which U { } rx is used by the receivers to estimate the MIMO channel. After receive antennas. At a given time instant, the transmitter conveys information to a subset of the users . In a given time slot estimating the channel, the first user in the list of the NDP T ⊆U announcement sends feedback information, and the remaining all users use all subcarriers. users (if any) transmit their CSI by responding to subsequent We assume that the transmitter employs linear precoding, beamforming (BF) report polls. This operation is depicted in and the receivers use linear equalizers. A single modulation C Figure 1. u = m1 ...m|Mu| is selected for the u-th user, M ⊂ 1 2) Quantization and Feedback: The CSI obtained after the constant over all carriers and spatial streams . For a given carrier Lu training phase is quantized prior to the transmission on the n, su[n] u is the Lu spatial streams modulated signal con- ∈M Ntx×L taining the information for the u-th user, Fu[n] C u is the feedback channel. The most relevant parameters for the MU- ∈ MIMO operation are the preferred beamforming matrices, which 1Although in IEEE 802.11n the use of different modulations in each spatial contain the right eigenvectors associated to the largest singular stream was allowed, it was apparently not implemented in most commercial values of the channel matrix. These matrices are represented devices, and finally discarded for 802.11ac. 4
Nrx ×Ntx transmit precoding matrix for the u-th user, Hu[n] C ,u IV. PROBLEM STATEMENT is the flat-fading MIMO channel from the transmitter∈ to the L ×Nrx In this section, we formulate the link adaptation problem. The u-th receiver, Bu[n] C u ,u is the interference removal ∈ L ×L adaptation problem in the multiuser scenario is different from the matrix, and Gu[n] C u u is the linear equalizer applied at the u-th receiver.∈ We divide the receive processing into two single user scenario. In the single user case, the usual objective different matrices for simplicity in the treatment of the multiuser of link adaptation is to maximize the (unique) link throughput subject to a constraint on the FER. In the MU-MIMO case, each precoding. The objective of Bu[n] is to reject the interuser user has a different rate, so the objective might be to maximize interference, while the equalizer Gu[n] removes the intrauser interference. To limit the transmit power per carrier, we define a function of the rates, subject to a FER constraint p0 > 0 , ∗ (assumed to be equal for all receivers). We consider the sum the power normalization factor P [n] u∈T tr (Fu[n]Fu[n]) . 2 rate as the performance objective in this paper. If we denote by We use nu[n] 0, σ I to denote the received noise vector ∼ CN P t = [t1 ...tU ] the vector containing the throughput (6) of all at the u-th receiver. Given these definitions, the post-processed U L users, and by t , the sum rate, the LA problem signal at the u-th receiver y [n] C u is ν ( ) u=1 tu u ∈ can be stated as 1 P y [n] = Gu[n]Bu[n]Hu[n] Fi[n]si[n] (1) u maximize ν (t) subject to p p u =1,...,U. i∈T P [n] u 0 X ≤ (7) +Gu[n]Bu[n]nu[n] p We will assume tu =0, pu =0 if u / to be consistent with our E ∗ ∈ T with (su[n]su[n]) = ILu . For the sake of clarity, we de- approach, which involves scheduling a subset of the users. Note 1 fine Hˆ u,i[n] , Gu[n]Bu[n]Hu[n]Fi[n], and wu[n] , that the LA problem can be modified to maximize another utility √P [n] metric besides the sum rate just by defining ν (t) accordingly. Gu[n]Bu[n]nu[n], so For example, we can include proportional fairness in this setting ˆ ˆ U yu[n]= Hu,u[n]su[n]+ Hu,i[n]si[n] + wu[n] . (2) by changing the objective function to ν (t)= u=1 log tu [41]. i∈T , i6=u Trying to solve (7) directly is computationally intractable. X Besides the difficulty of obtaining a mathematicalP model that Interuser interference Noise maps the CSI to the FER pu, the number of design variables is The transmit signal for| each of{z the scheduled} | receivers{z } is quite large and difficult to handle. The design variables include, the result of performing coding, interleaving and constellation among others, the set of active users , the streams per each T mapping operations on a stream of source bits. The MCS for the active user Lu, the precoding matrices Fu[n], the interference re- u-th user cu is selected from a finite set of MCS . The selected C moval matrices Bu[n], equalizers Gu[n] and MCS cu. Moreover, number of spatial streams and MCS for the u-th user has an imperfect CSI at the transmitter creates unknown interference associated rate of η (cu,Lu) bits per second. leakage between the different receivers. We propose a procedure In general, the probability that a frame is not correctly decoded for link adaptation that finds a good solution to the sum rate at the u-th receiver (i.e., the FER), depends on the transmit maximization problem. Our solution has three operational blocks: power, channel matrices, number of scheduled users, selected precoding and equalization with interference estimation, MCS MCS for the u-th user, and selected modulation for the interfering selection and user and mode selection. users. By treating the noise and residual multi-user interference as Gaussian, and assuming a linear receiver, it is reasonable to write the FER of the u-th user pu as a function of the the V. PRECODING AND EQUALIZATION WITH INTERFERENCE STIMATION selected MCS cu and the post-processing SNR values γu = E [γ [1] ,...,γ [N] ,...,γ [1] ,...,γ [N]]T , where u,1 u,1 u,Lu u,Lu In this section, we present the precoding technique for MU-
pu = FER (γu,cu) . (3) MIMO transmission and obtain a closed form approximation for the residual interuser interference caused by limited feedback The post processing SNR of the u-th user in the i-th spatial stream precoding. Given the subset of active users and the number and -th carrier is defined as T n of streams per user Lu, the joint problem of selecting optimum 2 2 precoders Fu, interference rejection matrices Bu, and equalizers [Du,u [n]]ii γu,i[n]= u =1 ...U,i =1 ...Lu (4) G to maximize the sum rate does not have a closed form R u [ u[n]]ii solution. Several approaches have been proposed, e.g. block with diagonalization (BD) [42] or minimizing signal leakage [43]. In ∗ this paper we assume that the precoders and interference rejection Ru[n]= Hˆ u,u[n] Du,u[n] Hˆ u,u[n] Du,u[n] + (5) − − matrices are obtained using BD. We choose this precoding ˆ ˆ ∗ 2 ∗ ∗ algorithm because of its simplicity and its low gap to capacity Hu,j [n]Hu,j [n]+ σ Gu[n]Bu[n]Bu[n]Gu[n] j∈T , j6=u when used in conjunction with user selection algorithms [44]. X The proposed link adaptation framework, however, can be used , the covariance matrix of interference plus noise, and Du,u with other precoding techniques, although we restrict our analysis ˆ Hu,u ILu . Based on the rate η (cu,Lu) and the FER pu, we to BD. define◦ the throughput of user u as 2As the design of precoders is independent for each carrier, we will drop the tu = η (cu,Lu) (1 pu) . (6) n − index [ ] in this section. 5
A. Block diagionalization precoding necessary to design the precoders. Consequently, the precoders ∗ can be found by ensuring that V˜ Fj = 0 u = j. Therefore, BD precoding removes the interference between the different u ∀ 6 users but not the interference between streams associated to the set of precoding matrices can be obtained from the preferred ˜ the same user. It is well suited for IEEE 802.11ac because beamformers Vu. The presence of limited feedback, however, the precoders can be designed based on the feedback infor- creates some unknown interference leakage between the different mation provided. The BD precoder is designed as follows. Let users, which has to be estimated. We present next the quantization ∗ scheme in IEEE 802.11ac, and an analytical approximation to this Hu = UuΣuVu be the SVD decomposition of Hu with the interference leakage. singular values in Σu arranged in decreasing order. Note that N ×N N ×N N ×N Uu C rx,u rx,u , Σu R rx,u tx and Vu C tx tx . The ∈ ∈ ∈ interference rejection matrix Bu is formed by taking the first B. Quantization Lu columns of Uu (corresponding to the left singular vectors The presence of limited feedback creates unknown interference associated to the Lu largest singular values). Let us denote leakage among the different users. The post-processing SNR H˜ u , BuHu, and depends on the interference leakage, so this value has to be esti- T ¯ , mated. This estimation can be easily performed at the receive side, Hu H˜ 1 ... H˜ u−1 H˜ u+1 ... H˜ T . (8) and CSIT can be acquired by the use of the feedback link. This BD requires thath the precoder F satisfies i procedure, however, is not desirable for various reasons. First, there is a circular dependency between user and mode selection H¯ uFu = 0 u. (9) ∀ and interference leakage estimation. User and mode selection The set of precoders achieving (9) can be written as NuPu, requires knowing the post processing SNR and, consequently, where Nu is a basis of the nullspace of H¯ u, and Pu is an the interference leakage, but estimating the interference leakage arbitrary matrix, that can be used to select the directions of is restricted to a certain user/mode configuration. Second, the transmission (in case the nullspace of H¯ u is of dimension higher amount of overhead and training is roughly doubled with respect than Lu) as well as to perform power allocation. We choose Fu to the simple message interchange in Figure 1. Therefore, it is as the matrix containing the Lu singular vectors associated to the desirable to estimate the interference leakage at the transmit side, largest singular values of H˜ uNu, i.e., we perform uniform power without additional feedback from the receivers. allocation along the Lu stronger directions of the equivalent Next we derive an approximation for the interference leakage channel H˜ uNu. caused by the quantization procedure in IEEE 802.11ac. This The post-processed signal at the u-th receiver is approximation can be easily computed at the transmitter by using a statistical characterization of the quantization error. We describe yu = GuBuHuFusu + wu. (10) first the quantization method used in IEEE 802.11ac, and then Equation (10) can be obtained from (2) just by noticing that the derive an approximation for the interference leakage. The objective of the quantization task is to provide the trans- BD condition (9) is equivalent to BuHuFi = 0 u = i and ∀ 6 mitter with a quantized version Vˆ u of matrix V˜ u, which is the therefore, Hˆ u,i[n] = 0, u = i. Assuming limited feedback of ∀ 6 preferred beamforming matrix. The quantization process proceeds CSI, as anticipated in Section II, full knowledge of Hu is not ˜ CNtx,L possible. We now explain how BD can be performed also with as follows. The unitary matrix V is decomposed by using the Givens decomposition [45,∈ Ch. 5] as reduced information. Decompose Hu using the SVD as N ˜ ∗ L tx Σu,0 0 Vu V˜ D G ˜I H = U˜ U˜ . (11) = ℓ (φℓ,1 ...φℓ,Ntx−ℓ+1) n,ℓ (ψℓ,n) (12) u u u,small ˜ ∗ " 0 Σu,1 # " Vu,small # ℓ=1 n=ℓ+1 ! h i Y Y Nrx,u×Lu Nrx,u×(Nrx,u−Lu) where where U˜ u C , U˜ u,small C are the ∈ ∈ matrices containing the left singular vectors associated to the Lu jφℓ,1 jφℓ,Ntx−ℓ+1 CNtx×Ntx Dℓ (φℓ,1 ...φℓ,Ntx−ℓ+1)= diag 1ℓ−1,e ...e . largest singular values and Nrx,u Lu smallest singular values, (13)∈ Lu×Lu − respectively; Σu,0 C is the diagonal matrix containing ˜ ∗ I is a matrix containing the first L columns of an Ntx Ntx ∈ ˜ Lu×Ntx the L largest singular values of H ; and V C , Ntx×Ntx × u u u identity matrix and Gn,ℓ (ψℓ,n) R is a rotation matrix ∗ (N −L )×N ∈ V˜ C rx,u u tx are the matrices containing the Lu ∈ u,small ∈ operating in the ℓ and n coordinates: right singular vectors associated with the Lu largest singular values, and the (Nrx,u Lu) right singular vectors associated to Iℓ − −1 the remaining nonzero singular values. Assuming that the receiver cos ψ sin ψ ˜ ∗ ℓ,n ℓ,n uses Bu = Uu, then the equivalent channel can be written as ˜ ˜ ∗ ˜ Gn,ℓ (ψℓ,n)= In−ℓ−1 . Hu = Σu,0Vu. It can be seen that Hu can be available at sin ψ cos ψ the transmitter (with some quantization error) with the feedback ℓ,n ℓ,n − scheme described in Section II for IEEE 802.11ac, since the ma- IN−n ˜ trix Vu is the beamforming matrix for Lu spatial streams, and the (14) values of Σu,0 can be obtained from the SNR of each subcarrier The feedback consists of quantized versions ψˆℓ,n and φˆℓ,n of and spatial stream. Further, since Σu,0 is invertible, the nullspace the angles ψℓ,n and φℓ,n. The quantization of ψℓ,n and φℓ,n ˜ ˜ ∗ of Hu is the same as that of Vu, so the SNR information is not is performed by the use of bψ and bφ bits, respectively. IEEE 6
∗ 802.11ac uses a uniform quantizer. Since ψℓ,n [0,π/2] and where the last equality is due to the BD constraint Vˆ Fj = 0. ∈ u φℓ,n [0, 2π] [46], the codebook for each of the angles is First, note that the interference covariance matrices Ru,j in ∈ (22) are random variables from the transmitter’s point-of-view, kπ π bψ ψˆℓ,n qψ,k , + , k =0, 1 ... 2 1 (15) since the quantization noise Eu is unknown at the transmitter. We ∈ 2bψ+1 2bψ+2 − define a new covariance matrix by averaging over the realizations of E kπ π bφ u φˆℓ,n qφ,k , + , k =0, 1 ... 2 1 . (16) 1 ∈ 2bφ−1 2bφ − R¯ , E R = G Σ C Σ∗ G∗ (23) u,j u,j P u u,0 u,j u,0 u The quantization of the angles is performed by finding the with minimum distance codeword E ˜ ∗ ∗ ˜ Cu,j = VuFj Fj Vu (24) ˆ , kπ (k + 1)π ψℓ,n = qψ,k if ψl,n ψ,k , (17) V˜ ∈Q 2bψ+1 2bψ+1 and the expectation is taken over the random realization of u given the received feedback Vˆ u. Note that we are implicitly averaging over E , but the particular structure of the quantization kπ (k + 1)π u φˆℓ,n = qφ,k if φl,n φ,k , , . (18) method makes it easier to derive the final result when explicitly ∈Q 2bφ−1 2bφ−1 ˜ ˆ averaging over Vu Vu. | , π , π Let us define φ , (φ ,...,φ ) and φˆ , For the sake of simplicity, we denote δ b and ǫ b +2 , so ℓ ℓ,1 ℓ,Ntx−ℓ+1 ℓ 2 φ 2 ψ ˆ ˆ ˜ that ψ,k = [qψ,k ǫ, qψ,k + ǫ] and φ,k = [qφ,k δ, qφ,k + δ]. φℓ,1,..., φℓ,N −ℓ+1 . With this, we can write Vu following Q − Q − tx Note that this quantization scheme is the LBG quantizer [47], a Givens decomposition, so that the covariance matrix is optimal in the minimum distortion sense, only if the angles ∗ Lu Ntx are independent and the distribution of both angles is uniform, ∗ C = E˜I D (φ ) G (ψ ) F (25) which is not the case even in the well-studied independent and u,j l ℓ n,ℓ ℓ,n k ℓ=1 n=ℓ+1 ! identically distributed Gaussian MIMO channel, see e.g. [48]. Y Y Lu Ntx ∗ F Dl (φ ) Gn,ℓ (ψℓ,n) ˜I k × l C. Interference estimation ℓ=1 n=ℓ+1 ! Y Y Now we characterize the residual interference using only where we parametrized the matrix random variable V˜ u Vˆ u using the quantized CSI. The BD precoders are designed using the ˆ ˆ | ∗ the Givens parameters φℓ,n φℓ,n and ψℓ,n ψℓ,n. If we assume that quantized beamformers Vˆ u to satisfy Vˆ Fj = 0 u = j. | | u ∀ 6 all the angles φ and ψ are independent, then the expected value Because the beamformers are quantized, the interuser interference over all the angles can be decomposed into several expected cannot be completely removed due to the imperfect CSI, so the values, each one over a different angle. This is a reasonable total interference plus noise covariance matrix (5) assuming BD assumption for a MIMO channel with zero-mean Gaussian iid precoding and ZF equalizer at user u is entires [48], for example. To simplify the computation of the 2 ∗ covariance matrix, we work with a vectorized version of Cu,j , Ru = Ru,j + σ GuGu (19) cu,j , vec Cu,j . Using properties of the Kronecker product [49], j∈T \{u} X we have ∗ , ˆ ˆ T where Ru,j Hu,j Hu,j is the covariance matrix of the interfer- Lu Ntx ence from the message intended to user j. Equation (19) follows E ˆT ˆ∗ T T ∗ ck,i = I I Rℓ Wℓ,n vec Fj Fj (26) by assuming perfect CSI at the receiver, so that the ZF equalizer ⊗ ! ℓ=1 n=ℓ+1 removes all the intra-stream interference, i.e., Hˆ = D in Y Y u,u u,u where (5), and by applying the fact that Bu is a unitary matrix. If Gu , E T ∗ Rℓ ˆ Dℓ (φℓ) Dℓ (φℓ) (27) is a ZF equalizer, and the precoders and interference rejection φℓ|φℓ ⊗ matrices are designed following the BD procedure, then and −1 −1 ∗ , E T ∗ ˜ Wℓ,n ˆ Gn,ℓ (ψℓ,n) Gn,ℓ (ψℓ,n) . (28) Gu = (BuHuFu) = Σu,0VuFu . (20) ψℓ,n|ψℓ,n ⊗ We now proceed to approximate R and W . We resort to ˜ ∗ ˆ ∗ ℓ ℓ,n Now, write Vu = Vu + Eu, where Eu is the quantization a well known result in high resolution quantization theory [50] error matrix. If the quantization error is small, (20) can be and approximate the quantization error by a uniform random approximated as variable in the quantization bin. This approximation is exact for −1 −1 ˆ ˆ ˆ ∗ ˆ ∗ the quantization noise φℓ,i φℓ,i (but not for ψℓ,n ψℓ,n) if Gu = Σu,0VuFu + Σu,0EuFu Σu,0VuFu . − − ≈ Hu has independent and identically distributed Gaussian entries, (21) since φˆℓ,i is uniformly distributed in [0, 2π] [46]. Note that for (21) to hold it is only necessary that the entries of We now calculate a closed form expression for (27) and (28) ˆ ∗ Eu are negligible with respect to the entries of Vu. assuming uniform quantization noise. For the sake of clarity, we The multiuser interference can be written as will omit subscripts and matrix angular arguments when their 1 ∗ ∗ ∗ ∗ value is clear. Using the definition in (13), it is possible to take Ru,j = GuΣu, V˜ Fj F V˜ uΣ G (22) P 0 u j u,0 u the Kronecker product and write (27) as 1 ∗ ∗ ∗ ∗ = GuΣu,0E FuF EuΣ G ∗ ∗ jφℓ,1 ∗ jφℓ,Ntx−ℓ+1 ∗ P u u u,0 u diag Dℓ ,..., Dℓ , e Dℓ ,...e Dℓ . (29) 7
∗ Iℓ−1 Gℓ,n ⊗ ∗ ∗ cos ψℓ,nG sin ψℓ,nG ℓ,n − ℓ,n E ∗ Wℓ,n = In−ℓ−1 Gℓ,n (33) ⊗ sin ψ G∗ cos ψ G∗ ℓ,n ℓ,n ℓ,n ℓ,n ∗ INtx−n Gℓ,n ⊗
Iℓ−1 ˆ ˆ sin ǫ cos ψℓ,n sin ǫ sin ψℓ,n ǫ − ǫ E ∗ Gℓ,n = In−ℓ−1 (34) sin ǫ sin ψˆℓ,n sin ǫ cos ψˆℓ,n ǫ ǫ I Ntx−n cos ψˆℓ,n sin ǫ ǫ Iℓ−1 E ∗ cos ψℓ,nGℓ,n = Jℓ,n (35)
cos ψˆℓ,n sin ǫ INtx−n ǫ with ǫ+cos ǫ cos 2ψˆℓ,n sin ǫ cos ǫ cos ψˆℓ,n sin ǫ sin ψˆℓ,n 2ǫ ǫ ˆ − J = cos ψℓ,n sin ǫ (36) ℓ,n ǫ In−ℓ−1 cos ǫ cos ψˆ sin ǫ sin ψˆ ǫ+cos ǫ cos 2ψˆ sin ǫ ℓ,n ℓ,n ℓ,n ǫ 2ǫ and sin ψˆℓ,n sin ǫ ǫ Iℓ−1 E ∗ sin ψℓ,nGℓ,n = Sℓ,n (37)
sin ψˆℓ,n sin ǫ INtx−n ǫ with cos ǫ cos ψˆℓ,n sin ǫ sin ψˆℓ,n ǫ−cos ǫ cos 2ψˆℓ,n sin ǫ ǫ 2ǫ ˆ − S sin ψℓ,n sin ǫ (38) ℓ,n = ǫ In−ℓ−1 . ǫ−cos ǫ cos 2ψˆ sin ǫ cos ǫ cos ψˆ sin ǫ sin ψˆ ℓ,n ℓ,n ℓ,n 2ǫ ǫ
Now we compute the expectations of each term. First observe write Wℓ,n as in (33). that The expectations of the different blocks of the matrix can be ˆ obtained by simple integration, similarly to (31). The results for 1 φℓ,i+δ E ∗ E −jφℓ,i −jφ [Dℓ ]ℓ+i−1,ℓ+i−1 = e = e dφ(30) the different blocks are shown in (34)-(38). 2ǫ ˆ Zφℓ,i−δ Note that the complexity of obtaining the error covariance ˆ e−jφℓ,i sin δ matrix is similar to the complexity of recovering the matrix Vˆ = . δ from the quantized angles. Figure 2 shows the accuracy of the The expectation of the other diagonal terms can be obtained analytical approximation. similarly as ˆ VI. MCSSELECTION ejφℓ,j sin δ E jφℓ,j ∗ e [Dℓ ]i,i = , i =1 ...ℓ 1 (31) The MCS selection block consists of a function µ that takes δ − as input the set of post-processing SNR values of user u and the and number of spatial streams Lu, and computes the higher MCS that 1 if i = j meets the FER constraint for those SNR values, i.e. E jφℓ,j ∗ ˆ ˆ e [Dℓ ]ℓ+i−1,ℓ+i−1 = j(φℓ,j −φℓ,i) 2 (32) e sin (δ) µ (γu,Lu) = argmax η (c,Lu) subject to FER (γu,cu) p0. if i = j c∈C ≤ δ2 6 (39) for . Now, we proceed to obtain a closed The post-processing SNR values are calculated by applying i = 1 ...Ntx ℓ +1 form for (28) with− the uniform error approximation. First, let us the approximations (21) and (26) in (4) to incorporate the 8
L1 = 1,L2 = 5,Nrx,1 = 1,Nrx,2 = 5,N = 6.bφ = bψ + 2 −1 tx spatial streams) of user u. We obtain our feature vector f by 10 selecting a subset of the entries of γ˜u. Other approaches for −2 dimensionality reduction, like principal component analysis [52], 10 User 1 may alternatively be applied, but in our simulations we did not Simulation −3 10 see a significant benefit. Theory
−4 10 User 2 B. Classification The objective of the classification task is to estimate the highest −5 −5.1 10 10 MCS supported by the channel, as characterized by the feature
Interference Leakage −6 vector f. Following a similar approach as in [25], we use a set 10 of classifiers δc,L(f) to discriminate whether the current channel
−7 will support transmission with MCS c and L spatial streams while 10 −5.9 10 meeting the FER constraint. More formally, given a set of M 7 7.5 8 M −8 10 training samples (γi, ptr,i) i=1, with ptr,i the FER of the i-th 0 2 4 6 8 10 12 { } M channel, the input data for the classifier is the set (f i, νi) , b i=1 ψ with { } Fig. 2: Analytical and empirical interference leakage per spatial 1 if ptr,i p0 ν = ≤ (40) 1 i stream tr Cu,j for a two-user scenario. The theoretical ( 1 if ptr,i >p0 Lu − approximation is calculated using (26) and the closed form and f the subset of the ordered SNR of the i-th channel. The expressions for the expected values (30) - (37) . The simulation i classifier is a function of the feature input vector that maps interference leakage was averaged over 100 independent MIMO channels with independent complex Gaussian entries. δc,L : f 1, 1 . (41) → {− } There are several ways to construct a classifier; we choose the interference leakage estimate. An important observation is that popular SVM. An SVM determines the class of a sample by in (39) the rate is being maximized, not the throughput. The the use of linear boundaries (hyperplanes) in high dimensional spaces. Operating in a high dimensional space is enabled by reason is that for small values of p0, the feasible points meet the use of a Kernel function K(x1, x2) that maps x1 and η (c,Lu) (1 FER (γu,cu)) η (c,Lu). This approximation simplifies the− problem, since≈ estimating the actual value of x2 to vectors φ(x1) and φ(x2) lying in a Hilbert space, and performs the inner product in that space φ(x1), φ(x2) . The FER (γu,cu) is not required, rather it is only necessary to h i Kernel function K(x1, x2) has a very simple form for properly discriminate whether it is above the desired threshold p0 or not. We use a machine learning inspired approach to solve (39). chosen φ. In many cases, perfect separation by a hyperplane is Essentially, we classify features derived from the channel into the not possible (or not desirable, since it would lead to non-smooth highest MCS that meets the target FER constraint. The classifier is boundaries) and a penalization term is introduced to take into made up of individual classifiers that distinguish whether a certain account the misclassified training samples. Formally, the classifier is MCS and number of spatial streams are supported by the current M channel. Note that this is slightly different from conventional δc,L(x)= sign αiνiK (x, f i)+ b (42) machine learning in that there is a target average error rate, i=1 ! whereas machine learning usually involves avoiding classification X where αi is obtained as the result of the optimization problem errors altogether. We will follow a supervised learning approach 1 M M M to solve this problem, which includes two separated tasks: feature minimize αiαj K (f i, f j ) αi 2 i=1 j=1 − i=1 extraction and classification. M (43) subject to Pi=1 νiPαi =0 P 0 αi C, i =1,...,M A. Feature extraction P≤ ≤ In machine learning, the curse of dimensionality is well known: and b can be obtained by solving δc,L(f i)νi =1 for any training the larger the dimension of the feature vector, the exponentially sample f i such that 0 < αi < C [28]. The parameter C has to more data is required [51]. To reduce the dimensionality of be adjusted to trade off smoothness and training misclassification rate. High values of C result in very irregular boundaries caused γu, we exploit insights made in [22] about performance in coded bit interleaved MIMO-OFDM systems. In particular, it was by very small training errors, and low values of C result in large recognized that performance was invariant to subcarrier ordering, training errors caused by smooth boundaries. In this paper we use the radial basis function kernel i.e., FER (γu,cu) = FER (Πγu,cu), with Π any permutation matrix. Therefore, the reduced dimension feature vector should x x 2 K(x , x ) = exp k 1 − 2k , (44) be invariant to subcarrier ordering. Similar to [22], we use a 1 2 − ρ2 subset of the ordered SNR values as our feature vector. Define the ! T 2 ordered SNR vector γ˜u = [˜γu,1 ... γ˜u,NLu ] as a vector formed where parameter ρ is used to tradeoff bias and variance: small by ordering the elements in γu in ascending order. For example, values of ρ tend to take into account only the nearby training γ˜u,i denotes the i-th smallest SNR value (among all carriers and points, leading to high variance classifiers, and large values of 9
ρ result in biased results. The parameters ρ and C are selected TABLE I: MCS in IEEE 802.11ac with the corresponding data using a cross-validation approach [28]. rates in 20MHz channels [36]. For a given number of streams Lu, the overall classifier MCS Data Rate (Mb/s) MCS Data Rate (Mb/s) chooses the MCS with a higher rate among those predicted to BPSK 1/2 6.5 64-QAM 2/3 52 meet the FER constraint. The MCS selection function µ (39) is QPSK 1/2 13 64-QAM 3/4 58.5 implemented as QPSK 3/4 19.5 64-QAM 5/6 65 16-QAM 1/2 26 256-QAM 3/4 78 µ (γu,Lu) = argmax η (c,Lu) s.t. δc,L(γu)=1. (45) c∈C { } 16-QAM 3/4 39
VII. USER AND MODE SELECTION Performing optimal user and mode selection requires an ex- VIII. SIMULATION RESULTS haustive search over all possible combinations of users and number of streams per user. To overcome this challenge, we To validate the performance of the proposed link adaptation propose a greedy approach, similar to [37]–[39], where the method, we performed simulations using parameters from the streams are added one by one until the utility function ν (t) does physical layer of IEEE 802.11ac. The studied scenario comprises not increase. In each iteration, one spatial stream is added to the a 4-antenna transmitter communicating with three 2-antenna user whose increment in the number of spatial streams led to a receivers over a 20MHz channel (52 OFDM carriers) with an higher throughput. The algorithm continues until the maximum 800ns guard interval. The frame length was set to 128 bytes. number of spatial streams is reached, or when an increment in Perfect CSI was assumed at the receiver and different levels of the number of spatial streams lead to a lower sum rate. CSI at the transmitter. The FER constraint for the link adaptation An important observation is that the greedy algorithm is not problem was set to p0 = 0.1. The set of MCS for optimization fair, in the sense that a user could be assigned all the spatial with the associated data rate for one spatial stream is shown in streams if it led to a a higher objective function, in our case Table I. the sum rate. The objective function, however, could be modified The training of the classifiers δ was performed as follows. to encourage fairness. For example, metrics that are concave in c,L The training set was generated in a single user setting with perfect the rates would give more utility to assigining spatial streams to CSI by simulating different channels for all MCS and NSS values. unscheduled users. This change in the objective function does not For each of the channels, the complete transmit-receive chain was require any other change in the user selection algorithm, or the simulated, e.g. coding, interleaving, equalization, decoding. The other pieces of our link adaptation procedure. The entire proposed resulting SNR γ˜ values and FER p were stored for each of the link adaptation algorithm is summarized in Algorithm 1. i tr,i channels. The channels were generated in the time domain witha 4-tap MIMO channel with iid Gaussian entries with 30 different Algorithm 1 Link Adaptation Algorithm noise levels, corresponding to SNR values between 5 and 50 dB. Lu =0 u For each training sample, the feature vector f consisted on 4 0 ∀ equispaced SNR values (including the first and last ones) from R ← U while u=1 Lu
TABLE II: Classification errors and accuracy gain of the SVM classifier with respect to the Average SNR classifier (all in %).
SVM Av. SNR Gain Av. SNR Eff. SNR Gain Eff. SNR L=1 L=2 L=3 L=4 L=1 L=2 L=3 L=4 L=1 L=2 L=3 L=4 L=1 L=2 L=3 L=4 L=1 L=2 L=3 L=4 BPSK 1/2 0.45 1.33 1.05 0.88 9.4 2.9 1.78 2.82 95.2 54.0 41.1 68.6 0.866 1.2 1.08 1.46 48.1 -11.1 3.08 39.8 QPSK 1/2 0.433 1.12 1 0.683 3.92 7.87 2.33 3.32 95.2 54.0 41.1 68.6 1.1 1.35 1.18 0.9 60.6 17.3 15.5 24.0 QPSK 3/4 1.43 3.57 4.78 2.08 9.22 6.38 6.7 3.78 84.4 44.1 28.6 44.9 2.75 1.4 3.35 2.83 47.9 -154 -42.8 26.5 16-QAM 1/2 0.417 0.6 1.22 0.8 9.73 4.11 1.97 2.45 95.7 85.4 38.1 67.3 1.51 1.58 1.68 1.8 72.5 62.0 27.7 55.6 16-QAM 3/4 1.58 3.87 3.63 1.9 7.63 7.15 4.92 3.28 79.3 45.9 26.1 42.1 3.07 3.63 3.1 2.7 48.4 -6.42 -17.2 29.6 64 QAM 2/3 0.433 1.5 1.66 1.15 4.5 5.11 3.31 2.36 90.3 70.6 49.7 51.4 1.72 2.52 2.33 2.87 74.8 40.4 28.6 59.9 64 QAM 3/4 1.03 2.68 3.05 2.05 6.53 6.1 5.47 3.12 84.2 56.0 44.2 34.2 2.57 3.28 3.08 4.13 59.7 18.3 1.08 50.4 64-QAM 5/6 0.783 1.85 1.57 2.33 7.87 6.38 2.82 5.67 90.0 71.0 44.3 58.8 2.57 3.03 1.88 3.01 69.5 39.0 16.8 22.6 Average 1.65 5.03 67.2 2.24 26.3 training set error. Formally, the average SNR classifier is in IEEE 802.11ac, but the results are not shown as they are indistinguishable from the perfect CSI ones. In all cases a MU Lu N Av. SNR 1 feedback was assumed (i.e., including SNR information of all δc,L (γ)= sign γl [n] γth . (46) NLu − carriers), even for , which is only available for l=1 n=1 ! (bψ =4,bφ = 6) X X SU mode in IEEE 802.11ac. The misclassified samples, i.e., The exponential effective SNR consists on a generalized mean of the cases where the selected MCS led to a FER value over the SNR values the predefined threshold, were penalized and computed as zero L N 1 1 u throughput. This penalization is set to remove the advantage γeff = log exp( βγl [n]) (47) of selecting an MCS with a higher throughput, but without −β NLu − l n=1 ! X=1 X meeting the outage constraint, and is a common procedure when that is compared with a threshold γeff, th, designed to minimize evaluating link adaptation algorithms [22], [55]. The channels the training set error. The parameter β depends on the MCS, and were generated independently from the training samples, but is also selected to minimize the training set error. We can write with the same statistical distribution. The channels for the three the effective SNR classifier as receivers were generated independently and with the same distri-
Eff. SNR bution, so fairness is automatically induced in this scenario. In δ (γ)= sign (γeff γeff,th) . (48) c,L − other scenarios with users with different average received power, In Table II we show the accuracy results for the tested fairness can be introduced by changing the objective function, as classifiers. We can see that the SVM classifier outperforms the explained in Section IV. average and effective SNR classifiers, and in many cases the Figure 3a illustrates the sum throughput as a function of classification error is below 1%. For example, for 16-QAM 1/2, the noise variance. As expected, in the perfect CSI case, the L = 1, the SVM classifier misclassifies around 1 sample out throughput increases with the SNR, as the interuser interference is of 200 (0.417%), whereas the average SNR classifier makes completely avoided in this setting, and lower noise levels enable approximately 20 times more errors (9.73%). In that case, the the use of higher rate MCS. For the limited feedback CSI we error rate of the effective SNR classifier is 1.51%, approximately study two different cases. The first case, represented by solid 4 times more than the SVM classifier. The classification gain lines, is the result of following the complete link adaptation with respect to the Av. / Eff. SNR classifier is calculated algorithm, including the interference estimation procedure. The as Error Error /Error , second case, in dashed lines, does not include the interference Av. / Eff. SNR − SVM Av. / Eff. SNR and in the 16-QAM 1/2 case for the Av. SNR classifier is estimation block (i.e., assumes C = 0), thus is overestimating u,j approximately 20−1 0.95. This classification gain reflects the actual SNR. For the interference estimation case, we see that 20 ≈ the percentage of errors made by the average or effective SNR (bψ =5,bφ = 7) follows the trend of the perfect CSI curve with classifier that would be corrected by the SVM. For example, a slightly lower throughput until an SNR value of around 35dB, a classification gain of 100% means that the SVM classifier is and then flattens with a maximum throughput around 220Mb/s. perfect, and a classification gain of 0% means that the SVM Performance loss is more significant with the lower rate feedback has the same accuracy as the average SNR classifier. We can (bψ =4,bφ = 6) for moderate SNR values, reaching an error see that in some MCS the Eff. SNR classifier outperforms the floor around 190Mb/s in the interference limited regime. We can SVM classifier, but in average the SVM performs 22.34% better see that the perfect CSI curve flattens at 312Mb/s, which is the than the Eff. SNR classifier. The average gain with respect to throughput of 4 streams using 256-QAM 3/4, the maximum rate the Av. SNR classifier is much higher, above 67%. This result MCS in Table I. If the interference estimation is not performed confirms that SVM-based link adaptation algorithms outperform (the curves tagged as No Est) then the results are dramatically state-of-the-art effective SNR classifiers, currently used widely different. At low SNR values, the evolution of these curves is in LTE, among other communication systems [54]. similar to the more sophisticated algorithm, as in this region the Complete system simulations were run for three different CSI system is noise limited rather than interference limited. As the levels at the transmitter: perfect CSI and limited feedback CSI SNR increases, there is a huge degradation with respect to the with (bψ =5,bφ = 7) and (bψ =4,bφ = 6). Simulations were perfect CSI case. Performance is penalized at high SNR where also run for the (bψ =6,bφ = 8) case, the higher rate available throughput decreases dramatically. This unexpected behavior is 11
Sum throughput with penalization 0 Frame Error Rate 350 10 Perfect CSI Perfect CSI
300 bψ = 5, bφ = 7 bψ = 5, bφ = 7
bψ = 4, bφ = 6 bψ = 4, bφ = 6
250 bψ = 5, bφ = 7 No Est bψ = 5, bφ = 7 No Est
bψ = 4, bφ = 6 No Est bψ = 4, bφ = 6 No Est 200 −1 10 FER 150 Throughput (Mb/s) 100
50 −2 10
0 5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50 2 2 10 log ¡1/σ ¢ 10 log ¡1/σ ¢
(a) Sum throughput. (b) Frame error rate. Fig. 3: Simulation results for different feedback rates, with and without interference estimation. 4-tap Gaussian channel model.
Perfect CSI bψ = 4, bφ = 6 1 No Transm. 1
BPSK 1/2 0.9 0.9 QPSK 1/2 0.8 0.8 QPSK 3/4
0.7 16-QAM 1/2 0.7
16-QAM 3/4 0.6 0.6 64 QAM 2/3
0.5 64 QAM 3/4 0.5
Frequency 0.4 64-QAM 5/6 Frequency 0.4 256-QAM 3/4 0.3 0.3
0.2 0.2
0.1 0.1
0 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 2 2 10 log ¡1/σ ¢ 10 log ¡1/σ ¢
Fig. 4: Evolution of the frequency of the different MCS with the average SNR for the perfect CSI case and limited feedback with interference estimation. caused by the overestimation of the SNR due to the interuser performing the interference estimation procedure. interference that is not taken into account, which causes the clas- In Figure 4, we plot the fraction of the selected MCS for sifier to choose an MCS with a higher rate than the channel can different SNR values, for both perfect CSI and limited feedback. support. This leads to misclassified samples, leading to high FER In the perfect CSI case, the number of scheduled users is usually values that drive the throughput towards zero. From a learning low for low SNR values (the No Transmission frequency, repre- perspective, the feature set does not contain enough information senting the case where a user is not scheduled for transmission, to perform the adaptation. This information is implicitly included is quite high) and the MCS are robust, while for higher SNR in the SNR values when performing interference estimation. the No Transmission frequency decreases and higher rate MCS In Figure 3b, we plot the evolution of the FER with the average are employed for transmission. In the limited feedback case the SNR. We can see that in the perfect CSI case as well as in the behavior is similar to perfect CSI for low SNR values, but cases where the interference is estimated, the FER constraint for higher SNR the interuser interference forces to use more robust MCS and approximately one third of the time the No of 0.1 is always met. This result shows the robustness of the proposed approach even with limited feedback. If the interference Transmission MCS is selected. is not estimated, then the FER grows up to 1 due to the mismatch Now we study the frequency of a given number of users being between the selected MCS and the actual SNR values, which selected in Figure 5. In the perfect CSI case the number of users causes the throughput to decrease, as previously explained. The grows up to three (i.e., all users are scheduled most of the time), classifier is able to correctly perform MCS selection in a multiuser while in the limited feedback case transmits to only two users setting despite being trained in a single user scenario, thus most of the time. The reason is that in the limited feedback case showing also the robustness of the classifiers against changes the interuser interference increases with the scheduled number of on the channel distribution. This plot shows the importance of users, which is the limiting factor in the high SNR regime. The 12
Perfect CSI bψ = 4, bφ = 6 1 1 1 User 0.9 2 Users 0.9 3 Users 0.8 0.8
0.7 0.7
0.6 0.6
0.5 0.5 Frequency Frequency 0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 2 2 10 log ¡1/σ ¢ 10 log ¡1/σ ¢
Fig. 5: Evolution of the frequency of the number of users scheduled for MU transmission with the average SNR for the perfect CSI case and limited feedback with interference estimation.
Sum throughput with penalization 0 Frame Error Rate 250 10 Perfect CSI
Perfect CSI bψ = 5, bφ = 7 200 bψ = 5, bφ = 7 bψ = 4, bφ = 6
bψ = 4, bφ = 6 bψ = 5, bφ = 7 No Est
bψ = 5, bφ = 7 No Est b = 4, b = 6 No Est 150 ψ φ −1 bψ = 4, bφ = 6 No Est 10 FER
100 Throughput (Mb/s)
50
−2 10
0 5 10 15 20 25 30 35 5 10 15 20 25 30 35 2 2 10 log ¡1/σ ¢ 10 log ¡1/σ ¢
(a) Sum throughput (b) Frame Error Rate Fig. 6: Simulation results for different feedback rates, with and without interference estimation. TGac channel model B. Classifier trained with Gaussian channel model. user selection algorithm is able to identify the suboptimality of feature vector suffices to characterize the performance of the an aggressive multiuser transmission by estimating the residual system even in the presence of a change in the environment. The interuser interference. The effect of scheduling only two users throughput and frame error rate results are shown in Figure 6. We out of three was also observed in Figure 4, where for high SNR can see that the link adaptation algorithm is able to keep the FER values the No transmission MCS is selected 33% of the time. under the required threshold. Once again, interference estimation This partial scheduling of the users can lead to unfair scenarios is shown to be necessary to correctly select the number of users where the maximization of the sum rate causes one user not to and MCS. be scheduled. This behavior can be corrected by changing the objective function of the LA problem, as explained in Section IV. We compared the performance of the proposed adaptation pro- cedure with the same algorithm, but using an effective SNR and We also run simulations using TGac channel model B [56] to average SNR classifier instead of SVM. We run the simulations show the robustness of the proposed approach against changes in under TGac channel model B, and assuming perfect CSI at the the environment or channel model. The classifier is the same as transmitter. The results are shown in Figure 7. The average SNR in the previous section, i.e., is trained with 4-tap MIMO channels classifier offers a bad performance, as predicted by the results in with iid Gaussian entries. The 802.11 channel model B presents Table II. The effective SNR has a slightly worse performance realistic characteristics not present in the 4-tap channel, such as than SVM, especially at low SNR values. At moderate SNR delay taps with different power, and correlation among antennas. values (around 20dB), SVM offers approximately a 20% gain The objective of this experiment is to show if the ordered SNR in throughput. 13
Sum throughput with penalization SNR vector would lead to different FER values depending on 250 the complexity of the detection algorithm). A similar limitation is the assumption of perfect CSI at the receivers. In practice, Perfect CSI SVM the receivers will estimate the channel based on the incoming 200 Perfect CSI Eff. SNR Perfect CSI Av. SNR 18% signal, thus creating some CSI inaccuracy. This error is likely to affect the design of both the precoders and receive equalizers. 150 Also, different receivers may use different channel estimation 50% algorithms (e.g., a basic receiver is expected to use the pilot sym- bols to estimate the channel, whereas more advanced receivers 100 may be able to improve channel estimation by exploiting the Throughput (Mb/s) data symbols), and this behavior should be also captured by the learning algorithm. 50 We also assumed prior knowledge of a training set. In practice, the training samples have to be acquired online, thus suggesting an exploration versus exploitation tradeoff. Finally, prototyping 0 5 10 15 20 25 30 35 the system would require taking into account complexity and 2 10 log ¡1/σ ¢ storage requirements. Fig. 7: Comparison between SVM, Eff. SNR and Av. SNR classifiers REFERENCES [1] A. Rico-Alvari˜no and R. W. Heath Jr., “Learning-based link adaptation in multiuser MIMO-OFDM,” in Proc. EUSIPCO, (Marrakech, Morocco), Sept. 2013. IX. CONCLUSION AND FUTURE WORK [2] “LTE; evolved universal terrestrial radio access (E-UTRA); physical layer procedures,” ETSI TS 136 213 V8.8.0 (2009-10). In this paper, we presented a framework for link adapta- [3] “IEEE standard for air interface for broadband wireless access systems,” tion in multiuser MIMO-OFDM networks with limited feedback IEEE Std 802.16-2012 (Revision of IEEE Std 802.16-2009), pp. 1 –2542, information. The link adaptation problem is formulated as a 17 2012. [4] “IEEE standard for information technology–telecommunications and infor- maximization of the sum rate subject to a FER constraint. mation exchange between systems local and metropolitan area networks– Focusing on the multiuser capabilities in IEEE 802.11ac, we specific requirements part 11: Wireless lan medium access control (MAC) developed a data-driven algorithm that performs user selection, and physical layer (PHY) specifications,” IEEE Std 802.11-2012 (Revision of IEEE Std 802.11-2007), pp. 1–2793, 29 2012. mode selection, MCS selection and takes into account limited [5] “Digital video broadcasting (DVB); second generation framing structure, feedback information. We showed that estimating the interference channel coding and modulation systems for broadcasting, interactive ser- due to imperfect CSI is crucial to achieve a good performance vices, news gathering and other broadband satellite applications (DVB-S2),” ETSI EN 302 307 V1.2.1 (2009-08). in the multiuser MIMO setting. Depending on the feedback rate, [6] “Digital video broadcasting (DVB); second generation DVB interactive and by estimating the residual interference, the transmitter is able satellite system (DVB-RCS2); part 1: Overview and system level speci- to identify the error floor caused by multiuser transmission in the fication,” ETSI TS 101 545-1 V1.1.1 (2012-05). [7] A. Goldsmith and S.-G. Chua, “Variable-rate variable-power MQAM for high SNR regime. We conclude that machine learning classifiers fading channels,” IEEE Trans. Commun., vol. 45, pp. 1218 –1230, Oct. can be used in a multiuser setting, even in limited feedback 1997. scenarios. Performing interference estimation, however, is crucial [8] S. T. Chung and A. Goldsmith, “Degrees of freedom in adaptive modulation: a unified view,” IEEE Trans. Commun., vol. 49, pp. 1561 –1571, Sept. 2001. for the algorithm performance. The machine learning classifier is [9] A. Goldsmith and S.-G. Chua, “Adaptive coded modulation for fading also shown to be robust to changes in the statistical distribution channels,” IEEE Trans. Commun., vol. 46, pp. 595 –602, May 1998. of the channel. For example, the information acquired in an open [10] Q. Liu, S. Zhou, and G. Giannakis, “Cross-layer combining of adaptive modulation and coding with truncated arq over wireless links,” IEEE Trans. environment can be used to effectively perform adaptation in an Wireless Commun., vol. 3, pp. 1746 – 1755, Sept. 2004. office setting, where more frequency selectivity is expected. [11] C. Wang, “Dynamic ARF for throughput improvement in 802.11 WLAN Future work includes incorporating more realistic system as- via a machine-learning approach,” Journal of Network and Computer Applications, vol. 36, no. 2, pp. 667 – 676, 2013. pects. For example, we assumed a constant codeword length, [12] A. Kamerman and L. Monteban, “Wavelan R -II: a high-performance wire- whereas in IEEE 802.11ac the codeword length can vary from 1 less LAN for the unlicensed band,” Bell Labs Technical Journal, vol. 2, to 65535 bytes, thus severely affecting the FER performance. It is no. 3, pp. 118–133, 1997. [13] M. Lampe, H. Rohling, and W. Zirwas, “Misunderstandings about link not feasible to have a different classifier for every possible length, adaptation for frequency selective fading channels,” in Proc. IEEE PIMRC, so a different approach has to be studied. With the proposed vol. 2, (Lisboa, Portugal), pp. 710 – 714 vol.2, Sept. 2002. framework, a possible solution could be to include the packet [14] K. Brueninghaus, D. Astely, T. Salzer, S. Visuri, A. Alexiou, S. Karger, and G.-A. Seraji, “Link performance models for system level simulations length as an extra feature in the MCS classifier [57]. of broadband radio access systems,” in Proc. IEEE PIMRC, vol. 4, (Berlin, Another limitation of the proposed framework is the assump- Germany), pp. 2306–2311 Vol. 4, Sept. 2005. tion of ZF receivers. In a practical scenario, different receivers can [15] Y. Blankenship, P. Sartori, B. Classon, V. Desai, and K. Baum, “Link error prediction methods for multicarrier systems,” in Proc. IEEE VTC Fall, vol. 6, implement different detection algorithms, and the link adaptation (Los Angeles, CA), pp. 4175–4179 Vol. 6, Sept. 2004. algorithm should be able to adapt to that as well. Although it is [16] I. Dagres, A. Polydoros, and A. Zalonis, “DR.3.3 final report: AMC design unlikely that the transmitter is able to know the exact algorithm towards next generation wireless systems,” NEWCOM++, 2010. [17] M. Lamarca and F. Rey, “Indicators for PER prediction in wireless systems: the receiver is using, having a different MCS classifier for each A comparative study,” in Proc. IEEE VTC Spring, vol. 2, (Stockholm, receiver should be able to capture its effect (for example, the same Sweden), pp. 792–796 Vol. 2, May 2005. 14
[18] T. Jensen, S. Kant, J. Wehinger, and B. Fleury, “Mutual information metrics [37] G. Dimic and N. Sidiropoulos, “On downlink beamforming with greedy user for fast link adaptation in IEEE 802.11n,” in Proc. IEEE ICC, (Beijing, selection: performance analysis and a simple new algorithm,” IEEE Trans. China), pp. 4910 –4915, May 2008. Signal Process., vol. 53, pp. 3857–3868, Oct. 2005. [19] A. Kolmogorov, Sur la notion de la moyenne. Atti della R. Accad. dei [38] M. Kobayashi and G. Caire, “Joint beamforming and scheduling for a Lincei, 1930. MIMO downlink with random arrivals,” in Proc. IEEE ISIT, (Seattle, WA), [20] R. Jain, C. So-In, and A.-k. Al Tamimi, “System-level modeling of IEEE pp. 1442–1446, July 2006. 802.16E mobile wimax networks: Key issues,” IEEE Wireless Commun. [39] R. Chen, Z. Shen, J. Andrews, and R. W. Heath Jr., “Multimode transmission Mag., vol. 15, pp. 73–79, Oct. 2008. for multiuser MIMO systems with block diagonalization,” IEEE Trans. [21] G. Martorell, F. Riera-Palou, and G. Femenias, “Cross-layer fast link adapta- Signal Process., vol. 56, pp. 3294–3302, July 2008. tion for MIMO-OFDM based WLANs,” Wireless Personal Communications, [40] Cisco Systems, “White paper: 802.11ac: The fifth generation of wi-fi,” 2012. vol. 56, no. 3, pp. 599–609, 2011. [41] H. Kim, K. Kim, Y. Han, and S. Yun, “A proportional fair scheduling for [22] R. Daniels, C. Caramanis, and R. W. Heath Jr., “Adaptation in convolution- multicarrier transmission systems,” in Proc. IEEE VTC-Fall, (Los Angeles, ally coded MIMO-OFDM wireless systems through supervised learning and CA), pp. 409–413 Vol. 1, Sept. 2004. SNR ordering,” IEEE Trans. Veh. Technol., vol. 59, pp. 114–126, Jan. 2010. [42] Q. Spencer, A. Swindlehurst, and M. Haardt, “Zero-forcing methods for [23] Z. Puljiz, M. Park, and R. W. Heath Jr., “A machine learning approach to downlink spatial multiplexing in multiuser mimo channels,” IEEE Trans. link adaptation for SC-FDE system,” in Proc. IEEE GLOBECOM, (Houston, Signal Process., vol. 52, pp. 461–471, Feb. 2004. TX), pp. 1–5, Dec. 2011. [43] M. Sadek, A. Tarighat, and A. Sayed, “A leakage-based precoding scheme [24] R. Daniels and R. W. Heath Jr., “Link adaptation in MIMO-OFDM with for downlink multi-user MIMO channels,” IEEE Trans. Wireless Commun., non-uniform constellation selection over spatial streams through supervised vol. 6, pp. 1711 –1721, may 2007. learning,” in Proc. IEEE ICASSP, (Dallas, TX), pp. 3314–3317, Mar. 2010. [44] Z. Shen, R. Chen, J. G. Andrews, R. W. Heath Jr., and B. L. Evans, “Low [25] R. Daniels and R. W. Heath Jr., “Online adaptive modulation and coding complexity user selection algorithms for multiuser MIMO systems with with support vector machines,” in Proc. European Wireless Conference, block diagonalization,” IEEE Trans. Signal Process., vol. 54, pp. 3658– (Lucca, Italy), pp. 718–724, Apr. 2010. 3663, Sept. 2006. [26] R. Daniels and R. W. Heath Jr., “An online learning framework for [45] G. Golub and C. Van Loan, Matrix computations. Johns Hopkins University link adaptation in wireless networks,” in Proc. Information Theory and Press, 1996. Applications Workshop, (San Diego, CA), pp. 138–140, Feb. 2009. [46] J. Roh and B. Rao, “Efficient feedback methods for MIMO channels based [27] S. Wahls and H. V. Poor, “Link adaptation for BICM-OFDM through on parameterization,” IEEE Trans. Wireless Commun., vol. 6, pp. 282 –292, adaptive kernel regression,” in Proc. IEEE ICASSP 2013, (Vancouver, Jan. 2007. Canada), pp. 5136–5140, 2013. [47] Y. Linde, A. Buzo, and R. Gray, “An algorithm for vector quantizer design,” [28] T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical IEEE Trans. Commun., vol. 28, pp. 84 – 95, Jan. 1980. Learning: Data Mining, Inference and Prediction. Springer, 2 ed., Feb. [48] M. A. Sadrabadi, A. K. Khandani, and F. Lahouti, “Channel feedback 2009. quantization for high data rate MIMO systems,” IEEE Trans. Wireless [29] R. Daniels and R. W. Heath Jr., “Modeling ordered subcarrier SNR in Commun., vol. 5, pp. 3335 –3338, Dec. 2006. MIMO-OFDM wireless links,” Physical Communication, vol. 4, no. 4, [49] H. V. Henderson and S. Searle, “The vec-permutation matrix, the vec pp. 275–285, 2011. operator and Kronecker products: A review,” Linear and multilinear algebra, [30] N. Jindal, “MIMO broadcast channels with finite-rate feedback,” IEEE vol. 9, no. 4, pp. 271–288, 1981. Trans. Inf. Theory, vol. 52, pp. 5045–5060, Nov. 2006. [50] R. Zamir and M. Feder, “On lattice quantization noise,” in Proc. DCC, [31] J. Zhang, R. W. Heath Jr., M. Kountouris, and J. G. Andrews, “Mode (Snowbird, UT), pp. 380 –389, Mar. 1994. switching for the multi-antenna broadcast channel based on delay and [51] G. Hughes, “On the mean accuracy of statistical pattern recognizers,” IEEE channel quantization,” EURASIP J. Adv. Signal Process, vol. 2009, pp. 1:1– Trans. Inf. Theory, vol. 14, pp. 55 – 63, Jan. 1968. 1:15, Feb. 2009. [52] S. Wold, K. Esbensen, and P. Geladi, “Principal component analysis,” [32] M. Esslaoui, F. Riera-Palou, and G. Femenias, “Fast link adaptation for Chemometrics and Intelligent Laboratory Systems, vol. 2, no. 1-3, pp. 37 opportunistic multiuser MIMO-OFDM wireless networks,” in Proc. ISWCS, – 52, 1987. Proceedings of the Multivariate Statistical Workshop for (Aachen, Germany), pp. 372 –376, Nov. 2011. Geologists and Geochemists. [33] M. Esslaoui, F. Riera-Palou, and G. Femenias, “A fair MU-MIMO scheme [53] C.-C. Chang and C.-J. Lin, “LIBSVM: A library for support vec- for IEEE 802.11ac,” in Proc. ISWCS, (Paris, France), pp. 1049 –1053, Aug. tor machines,” ACM Transactions on Intelligent Systems and Tech- 2012. nology, vol. 2, pp. 27:1–27:27, 2011. Software available at [34] Z. Chen, W. Wang, M. Peng, and F. Cao, “Limited feedback scheme based http://www.csie.ntu.edu.tw/∼cjlin/libsvm. on zero-forcing precoding for multiuser MIMO-OFDM downlink systems,” [54] T. Tao and A. Czylwik, “Performance analysis of link adaptation in LTE in Proc. ICST WICON, (Singapore), pp. 1 –5, Mar. 2010. systems,” in International ITG Workshop on Smart Antennas, pp. 1–5, 2011. [35] S. Yun, C. Caramanis, and R. W. Heath Jr., “Distributed link adaptation [55] Y.-S. Choi and S. Alamouti, “A pragmatic PHY abstraction technique for for multicast traffic in MIMO-OFDM systems,” Physical Communication, link adaptation and MIMO switching,” IEEE J. Sel. Areas Commun., vol. 26, vol. 4, no. 4, pp. 286–295, 2011. no. 6, pp. 960–971, 2008. [36] “IEEE draft standard for IT - telecommunications and information exchange [56] TGac Channel Models Special Committee, “TGac channel model adden- between systems - LAN/MAN - specific requirements - part 11: Wireless dum,” IEEE 802.11-09/0308r0, May 2009. LAN medium access control and physical layer specifications - amd 4: [57] A. Rico-Alvari˜no and R. W. Heath Jr., “Link adaptation in MIMO-OFDM Enhancements for very high throughput for operation in bands below 6ghz,” with practical impairments,” in Proc. Asilomar SSC, (Pacific Grove, CA), IEEE P802.11ac/D3.0, June 2012, pp. 1–385, jun 2012. Nov. 2013.