Optimal Shape-Gain Quantization for Multiuser MIMO Systems with Linear Precoding
aa r X i v : . [ c s . I T ] N ov Optimal Shape-Gain Quantization for MultiuserMIMO Systems with Linear Precoding
Muhammad N. Islam,
Student Member, IEEE,
Raviraj Adve,
Senior Member,IEEE , Behrouz Khoshnevis,
Member, IEEE
Abstract
This paper studies the optimal bit allocation for shape-gain vector quantization of wireless channelsin multiuser (MU) multiple-input multiple-output (MIMO) downlink systems based on linear precoding.Our design minimizes the mean squared-error between the original and quantized channels throughoptimal bit allocation across shape (direction) and gain (magnitude) for a fixed feedback overhead peruser. This is shown to significantly reduce the quantization error, which in turn, decreases the MUinterference. This paper makes three main contributions: first, we focus on channel gain quantizationand derive the quantization distortion, based on a Euclidean distance measure, corresponding to singularvalues of a MIMO channel. Second, we show that the Euclidean distance-based distortion of a unit normcomplex channel, due to shape quantization, is proportional to − Bs M − , where, B s is the number ofshape quantization bits and M is the number of transmit antennas. Finally, we show that for channelsin complex space and allowing for a large feedback overhead, the number of direction quantization bitsshould be approximately (2 M − times the number of channel magnitude quantization bits. Index Terms
MIMO Broadcast Channels, Limited Feedback of CSI, Optimal Bit Allocation, Sum Mean SquaredError, Shape-Gain Product Quantization.
I. I
NTRODUCTION
The availability of channel state information (CSI) at the transmitter significantly improvesthe performance of multiuser (MU) multiple-input multiple-output (MIMO) systems [1]–[3].
Muhammad N. Islam is with WINLAB, Rutgers - State University of New Jersey, New Jersey, USA. email: [email protected]. Raviraj Adve and Behrouz Khoshnevis are with Electrical & Computer Engineering, University ofToronto, Toronto, Canada. email: [email protected], [email protected]
Specifically, CSI is essential for effective communications in the MU downlink. In frequencydivision duplex systems, in order to provide the base station (BS) with CSI, the receivers needto quantize the CSI and feed the quantized information back to the BS. Clearly this feedback isan overhead to the system and, therefore, must be limited to an acceptable level.This paper focuses on limited-feedback MU MIMO systems, where a single BS communicateswith multiple receivers and each user can potentially receive multiple data streams. Specifically,we restrict our analysis to systems using linear precoding [4]. Our main goal is to present anefficient quantization scheme for these systems. For this purpose, we use sum mean squared erroracross all data streams as the design objective and focus on quantization issues by assumingperfect channel estimation at the receiver (user) side and a noiseless, delay-free, feedback linkto the BS.This work is mainly motivated by the fact that the performance of limited-feedback MU-MIMOsystems is very sensitive to the quality of the CSI available at the BS. Without accurate CSI, thequantization error common performance measures saturate in the high signal-to-noise ratio (SNR)regime because the BS cannot completely pre-cancel the multi-user interference [3], [5], [6]. Itis therefore essential to design a limited feedback system such that the CSI quantization erroris minimized. The optimal design of channel quantization in MU-MIMO systems is, therefore,the core objective of this paper.Most of the works in limited feedback literature focus on either channel magnitude quantiza-tion [7]–[9] or direction quantization [1], [3], [10], [11] but not both. However, in MIMO systems,optimizing the precoder at the transmitter depends on both the channel magnitude (also knownas the channel gain) and the phase of individual channel entries (the channel direction or shape).The authors of [12] specifically showed that one needs both channel gain and quantized directioninformation to achieve multi-user diversity gain. However, the gain information was assumedperfect in [12]. In general, joint vector quantization (VQ) of channel magnitude and directionis a very complex task [13]. To reduce the design complexity, the authors of [14] investigateindependent quantization of the channel gain and shape and develop optimal bit allocation acrossgain and shape of real channel vectors using spherical codes. The authors of [15], [16] also usesuch a product codebook and solve for the optimal bit allocation to minimize the average transmitpower with quality-of-service constraints. Such a structure has several practical advantages andprovides an analytically tractable framework to optimize the limited feedback [14], [15], [17].
We adopt a similar approach where channel gain and shape are independently quantized.The optimal quantizer depends on the transmission scheme and performance measure used.Due to its simplicity and efficiency, we adopt an eigen-based combining (EBC) approach toprecode the data [1], [2]. We study quantization of CSI to minimize the sum mean squarederror (SMSE) over all data streams received, a popular measure in the MU MIMO downlink [4],[18]. As shown in our earlier work [19], there is a one-to-one relationship between the SMSEobjective and the variance of the quantization error. The current work, therefore, focuses onoptimizing the bit allocation across shape and gain given a budget for feedback overhead peruser. Once this bit allocation is optimized, one can use our earlier work in [19] for designing thelimited-feedback system. To the best of our knowledge, the problem of optimal bit allocationin shape-gain vector quantization to minimize the SMSE of a multiuser MIMO system has notbeen investigated before.This paper makes two key contributions:1) We show that the quantization distortion of a uniformly distributed unit-norm vector in C M is upper bounded by: K s × − Bs M − , where M is the total number of transmit antennas, B s is the number of shape quantization bits and K s is a constant that does not depend on B s .2) We also show that, for channels in complex space, the optimal number of channel directionquantization bits should be approximately (2 M − times the optimal number of channelmagnitude quantization bits.Numerical simulations suggest that the proposed bit allocation laws provide a substantialimprovement over full shape quantization or full gain quantization in the SMSE and bit errorrate (BER) performance of a multiuser MIMO system. Notation : Lower case letters denote scalar values while lower case bold face letters representcolumn vectors. Upper case boldface letters denote matrices. The superscripts ( · ) T and ( · ) H denote the transpose and conjugate transpose operators respectively. tr[ · ] denotes the traceoperator. I is reserved for the identity matrix whereas represents the column vector with allone entries. diag ( a , · · · , a n ) denotes the diagonal matrix with diagonal elements a , · · · , a n ;whereas diag ( A , · · · , A n ) represents the block diagonal matrix with the matrices A , · · · , A n on its main diagonal. || · || denotes the L norm of the vector. E ( · ) and S ( · ) denote statisticalexpectation and surface area respectively. The remainder of the paper is organized as follows. Section II outlines the limited feedbackMIMO system model and the corresponding shape-gain product VQ structure. Section III derivesthe distortion measures and provides the optimal bit allocation solution; in general, proofs aredeferred to appendices. This section also presents the linear precoding algorithm that incorporatesthe optimal bit allocation policy. Section IV presents results of numerical simulations illustratingthe theory developed. The paper wraps up with some conclusions in Section V.II. S
YSTEM M ODEL
We begin by developing the system model for linearly-precoded MU-MIMO system followedby the model for CSI feedback and the product shape-gain quantization structure.
A. MU MIMO System Model
Consider a single base station equipped with M transmit antennas communicating with K independent users. User k has N k antennas and receives L k data streams. All data streams areindependent of each other. Let L = P k L k , N = P k N k . To ensure resolvability, we require L ≤ M and L k ≤ N k .Let U ∈ C M × L denote the global precoder, the columns of which are unit-norm. Similarly,let P ∈ R L × L denote the diagonal power matrix whose entries are the powers allocated toindividual streams. Let, P max be the total available power; we require tr[ P ] ≤ P max . The datavector x = [ x , ...., x L ] T = (cid:2) x T , x T , . . . , x TK (cid:3) T , includes all L data streams to the K users. The N k × M block fading channel, H Hk , between the BS and user k is assumed to be flat. The globalchannel matrix is H H , with H = [ H , ..., H k ] . The elements of channel entries are assumed tobe zero mean complex Gaussian random variables with unit variance. User k receives y DLk = H Hk U √ Px + n k , (1)where n k represents the zero mean additive white Gaussian noise at the receiver. User k , inorder to estimate its own transmitted symbols from y DLk , forms ˆ x k = Λ k V Hk y DLk , (2)where V k Λ k is the N k × L k decoder matrix for user k . The columns of V k ∈ C N k × L k are unitnorm while Λ k = diag (cid:16) λ k , λ k , · · · , λ k Lk (cid:17) ∈ R L k × L k contains the gain variables that normalize the received data. Although the gain variables at the receiver side do not affect the signal-to-interference-plus-noise ratio (SINR), they play an important role in the error performance oftransmissions that include amplitude modulation, e.g., quadrature amplitude modulated systems.Figure 1 illustrates the proposed downlink system.Let VΛ be the N × L block diagonal global decoder matrix, V = diag ( V , ..., V K ) ∈ C N × L and Λ = diag ( Λ , ..., Λ K ) ∈ R L × L . Overall, ˆ x = ΛV H H H U √ Px + V H n = F H U √ Px + V H n , (3)where, n = (cid:2) n T , n T , . . . , n TK (cid:3) T . For the ease of representation, we define the M × L matrix F = HVΛ with F = [ f , . . . , f L ] . The vectors f , . . . , f L are the effective M × vector downlinkchannels of the individual data streams.The MSE of the i th data stream of the k th user is given by , e DLk,i = E h ( b x k,i − x k,i ) ( b x k,i − x k,i ) H i . (4)The min-SMSE optimization problem is: min p , U , V , Λ K X k =1 L k X i =1 e DLk,i ; subject to tr[ P ] ≤ P max , || u ℓ || = || v ℓ || = 1 , (5)To solve this problem, it is computationally efficient to use a virtual dual uplink [4]. In thisuplink the transmit powers are Q = diag [ q , .., q L ] T for the L data streams, while the matrices U and V remain the same as before. B. Feedback Model:
As mentioned earlier, we use an eigen-mode strategy [1], [2]. According to this strategy, the k th user estimates its own channel H k and uses a set of dominant singular values and singularvectors of H k as Λ k and V k respectively.Since the user is aware of H k , V k and Λ k , it can form the product matrix F k = H k V k Λ k ,whose columns act as the effective vector downlink channels for the data streams. Each userquantizes its effective vector downlink channel based on an Euclidean distance measure and feeds Note that we, interchangeably, index streams as being the ℓ th of L streams overall or the i th stream of the k th user. Anyone-to-one mapping between the two notations is acceptable. back the quantized channel to the BS. Details of the CSI quantization policy will be described inthe next section. To model the effect of quantization, we consider the following relation betweenthe original and the quantized variables, f k,i = b f k,i + e f k,i or F = b F + e F . (6)Here, f k,i denotes the effective vector downlink channel of the i th stream of the k th user. F comprises L effective channel vectors with the original channel directions and channel gains. ˆF denotes the L quantized feedback vectors. The matrix ˜F represents the quantization error.The BS assumes that the quantization error matrix ˜F has M × L independent identicallyGaussian distributed (i.i.d.) elements with zero mean and a variance of σ E /M , where σ E is thequantization error variance associated with each quantized vector ˆf k,i and is defined as, σ E = E h || f k,i − b f k,i || i . (7)By using the optimal P and U , the minimum SMSE takes the following form [19]: SM SE = L − M + (cid:18) σ + σ E M P max (cid:19) tr (cid:2) J − (cid:3) , (8)where, J = ˆFQˆF H + (cid:18) σ + σ E M P max (cid:19) I M . (9)where Q is the virtual uplink power allocation matrix.Equations in (8) and (9) show that the SMSE is directly related to the quantization error σ E .The limited feedback system design problem can therefore be formulated as minimization of thequantization error variance subject to a fixed feedback overhead. C. Shape-Gain Product Quantization Model
We intend to find the optimal bit allocation for quantizing the effective vector downlink channel f k,i . From now on, we will use z to represent the effective vector downlink channel to simplifythe notation. According to the eigen-based receiver structure assumed in this work, z representsthe product of a singular value of the channel matrix and its corresponding singular vector.Let ˆz be the quantized effective vector downlink channel and let C = { c , c , · · · , c N tot } denotethe codebook of quantized channels. Here, N tot = 2 B are the total number of quantization levelsusing a total of B bits. This codebook is simplified to a product codebook. Fig. 2 illustrates the product codebook operation based on independent quantization of gain and shape. Let, z = g s where, g = || z || and the unit-norm s = z / || z || denote the gain and shape of the channelrespectively. The BS is provided with the quantized information ˆz = ˆ g ˆ s , where ˆ g and ˆ s denotethe quantized gain and shape respectively.Let B g and B s denote the number of bits allocated to gain and shape quantization anddefine N g = 2 B g and N s = 2 B s . Further, let C g and C s represent the gain and shape codebookrespectively: C g = h c g , c g , · · · , c g Ng i (10) C s = (cid:2) c s , c s , · · · , c s Ns (cid:3) . (11)The product codebook can therefore be represented as, C = C g × C s . (12)The quantized gain and shape variables are computed as: ˆ g = arg min c g ∈C g ( g − c g ) (13) ˆs = arg min c s ∈C s || s − c s || . (14)The Lloyd-Max algorithm is the optimal solution to find the codebook for the gain of thechannel vector with the MSE objective [20]. We use the K -means approach, as described in [21],for numerical implementation of the Lloyd-Max algorithm. The optimal codebook of unit normvectors with a Euclidean measure is not yet known. Therefore, we adopt random VQ to find theshape codebook. With this approach, the unit norm quantized shape vectors are randomly andindependently distributed on the complex unit hyper-sphere in C M .The remaining question is, given B , what is the optimal choice of B s and B g ?III. D ISTORTION ANALYSIS AND OPTIMAL BIT ALLOCATION SOLUTION
A. Design Objective
Our main problem is to optimize the shape-gain bit allocation as formulated below, [ B ∗ s , B ∗ g ] = arg min Bs,Bg E (cid:2) || z − ˆ g ˆs || (cid:3) (15) subject to : B s + B g = B , B s ≥ , B g ≥ , ˆ g ∈ C g , ˆs ∈ C s . Hamkins et al. [14] have shown that, for high resolution quantization (large B s and B g ), thedistortion measure takes the following form [14]: E (cid:2) || z − ˆ g ˆs || (cid:3) ≈ E (cid:2) ( g − ˆ g ) (cid:3) + E (cid:2) g (cid:3) E (cid:2) || s − ˆs || (cid:3) (16) ≈ D g + E (cid:2) g (cid:3) D s , (17)where, E [ g ] denotes the variance of the gain and D g = E [( g − ˆ g ) ] is the gain quantizationdistortion. On the other hand, D s = E [ || s − ˆs || ] represents the distortion due to unit-norm shapequantization. Since D g and D s are independent of each other in (17), the optimal bit allocationproblem can be solved using the following three steps:1) Find D g , gain distortion, for a given B g .2) Find D s , shape distortion, for a given B s .3) Provide optimal bit allocation to minimize the overall distortion, i.e., E [ g ] D s + D g . B. Distortion due to Gain Quantization
The distortion due to quantizing the gain is given by D g = E (cid:2) ( g − ˆ g ) (cid:3) = Z ∞ ( r − ˆ g ( r )) f g ( r ) dr. (18)Here, ˆ g ( r ) is the quantized value of r and f g ( r ) is the probability density function (pdf) of thegain. Using Bennett’s integral ( [17], page-186), the distortion in (18) takes the form, D g = 112 N g || f g ( r ) || , (19)where, N g = 2 B g and || f g ( r ) || = (cid:18)Z ∞ | f g ( r ) | dr (cid:19) . (20) Lemma 1:
For Rayleigh fading and based on the pdf of the dominant eigenvalues of Wishartmatrix and Jacobian transformation in [22] and [23], we have, || f g ( r ) || = 3 × L ( e ) β L ( e ) − (cid:18) L ( e ) + 13 (cid:19) , (21)where, L ( e ) = ( M − e )( N k − e ) , M represents the total number of transmit antennas at the BS, N k denotes the number of receiver antennas of the k th user. e denotes the index of the orderedeigenvalues where represents the most dominant one, denotes the 2nd most dominant oneand so on. Finally, β = ˜ λ e /L ( e ) where ˜ λ e is the mean of the e th eigenvalue. Proof:
See Appendix A. (cid:4)
Using (19) and (21), the gain distortion at high resolution can be expressed as, D g = 112 N g || f g ( r ) || (22) = 116 N g L ( e ) β ( L ( e ) − (cid:18) L ( e ) + 13 (cid:19) (23) = K g − B g , (24)where, K g =
116 3 L ( e ) β ( L ( e ) − Γ (cid:16) L ( e )+13 (cid:17) is a constant with respect to B g . Equation (24) suggests thatthe gain distortion due to quantization is proportional to − B g .Figure 3 shows the distortion due to gain quantization of the dominant singular value of a × MIMO channel. As the figure verifies, the analytical expression converges to the simulationresult as B g increases. C. Shape Quantization Distortion
This section focuses on the shape quantization distortion of a unit-norm vector in C M , interms of the Euclidean distance. The Euclidean distance of two points in a C M plane has aone-to-one relation with the distance of two points in a R M plane. Therefore, we can focus onquantization of unit-norm vectors in R M instead of C M .Figure 4 shows a two dimensional view of the problem where OB = s , OA = ˆs . Here, || s || = || ˆs || = 1 . The Euclidean distance between s and ˆs is defined by, d = || s − ˆs || . Define U M as the unit hypersphere in R M . The surface area of U M is given by [15] S ( U M ) = 2 M C M , (25)where, C M = π M Γ( M + 1) . (26)Define the spherical cap D , i.e., the region ABC around s in Fig. 4, as: D = ( ˆs ∈ U M ||| s − ˆs || ≤ d ) , (27)and let ∠ AOB = θ be the angular distance between s and ˆs . Since || OA || = || ˆs || = 1 , we have AD = sin( θ ) and OD = cos( θ ) . Also, since || OB || = || s || = 1 , we have BD = 1 − cos( θ ) .Therefore, AB = AD + BD = sin ( θ ) + (1 − cos( θ )) = 2 − θ ) . (28) Here, if we define b = d , we will have: θ = cos − (1 − . b ) . (29)The surface area of D is given by [15], S ( D ) = (2 M − C M − Z θ sin M − φ dφ. (30)Now, if we assume a small spherical cap of radius d centered on s , the quantized vector can lieanywhere on this cap. Hence, P r [ || s − ˆs || ≤ b ] = S ( D ) S ( U M ) . (31)Using (25), (26), (29) and (30) in (31) we get P r [ || s − ˆs || ≤ b ] = (2 M − C M − R cos − (1 − . b )0 sin M − φ dφ M C M . (32)Since all the quantized vectors are randomly chosen, the probabilities that the square of theEuclidean distance between any vector in the codebook and the corresponding channel is higherthan b , are independent of each other. Therefore, P r [ min i ∈ [1 ,N s ] || s − ˆs i || ≥ b ] = − (2 M − C M − R cos − (1 − . b )0 sin M − φdφ M C M ! N . (33)Hence, expected value of the distortion error due to shape quantization can be calculated asfollows: E ( b ) = Z P r [min i ∈ N || s − ˆs i || ≥ b ] db. (34)The limits of integration in (34) follow from the fact that the square of the Euclidean distancebetween two points on a unit radius sphere has a range of to . Lemma 2 : E ( b ) < K s − Bs M − , (35)where, K s = π M − Γ( M )2 π M Γ (cid:0) M − + 1 (cid:1) ! − M − . (36)is a constant.Proof: See Appendix B. (cid:4) Figure 5 shows that the upper bound of the shape distortion in (35) has a fixed gap withrespect to the simulation result. Therefore, we can safely approximate the shape distortion withthe analytical expression in (37). Thus, D s = E (cid:0) || s − ˆs || (cid:1) ≈ π M − Γ( M )2 π M Γ (cid:0) M − + 1 (cid:1) ! − M − − Bs M − = K s − Bs M − . (37) D. Optimal Bit Allocation
Having analyzed the individual terms in (17), we are now able to answer the core question ofthis paper: the allocation of bits across gain and shape. In (17), the overall distortion measurewas shown to take the following form, D = E (cid:2) || z − ˆ g ˆs || (cid:3) ≈ D g + E (cid:2) g (cid:3) D s . (38)Using the gain and shape distortion measures of (24) and (37), D can be approximated as, D ≈ E (cid:2) g (cid:3) K s − Bs M − + K g − B g (39) ≈ ¯ K s − Bs M − + K g − B − B s ) , (40)where ¯ K s = K s E [ g ] . With these relations in hand, the optimal shape-gain bit allocation canbe formulated as follows, B ∗ s = arg min B s ¯ K s − Bs M − + K g − B − B s ) (41) Theorem 1:
The optimal bit allocation problem has the following solution: B s = 2 M − M B + 2 M − M log (cid:18) ¯ K s K g (2 M − (cid:19) (42) B g = B − B s = 12 M B − M − M log (cid:18) ¯ K s K g (2 M − (cid:19) . (43)Here, ¯ K s and K g are the terms defined in the previous subsections. Proof:
See Appendix C. (cid:4)
Note that, ¯ K s and K g in (42) and (43) depend on M but not B . Therefore, as B goes toinfinity, B s ≈ M − M B (44) B g ≈ M B. (45) The analytical expressions of (44) and (45) can be intuitively explained as follows: The normof a C M vector varies across a one dimensional line. However, the shape of a C M vector isuniformly distributed in the surface of a (2 M − dimensional hypersphere. Therefore, given M number of bits to quantize a C M vector, one should expend approximately and (2 M − bits to quantize the gain and shape of the vector respectively. It is worth noting that, froma different point of view and using a very different analysis, the work in [15], [16] leads to asimilar expression and explanation. However, this similarity is only for a high available feedbackrate.Finally, the overall quantization error for a fixed feedback overhead takes the following form: D = ¯ K s − Bs M − + K g − B g (46) = ¯ K s − M − (cid:16) M − M B + M − M log (cid:16) ¯ KsKg (2 M − (cid:17)(cid:17) + K g − (cid:16) M B − M − M log (cid:16) ¯ KsKg (2 M − (cid:17)(cid:17) =2 − BM log (cid:18) ¯ K s K g (2 M − (cid:19) (cid:16) ¯ K s − M − K g − M − M (cid:17) (47) = D c − BM , (48)where, D c = log (cid:16) ¯ K s K g (2 M − (cid:17) (cid:16) ¯ K s − M − K g − M − M (cid:17) is a constant. E. Overall Linear Precoding Algorithm
In the previous section we derived the optimal allocation of available bits across gain andshape. Here we use this information to develop the overall linear precoding algorithm. Thematerial exploits previous work in [19], [24]. The algorithm steps are:1) A gain codebook of B g bits is generated based on the dominant singular values of a randomGaussian matrix and using the K -means algorithm [21]. A shape codebook of B s randomunit-norm vectors, uniformly and independently distributed in C M , is also generated. Boththese codebooks are generated off-line and the codebooks are shared between the BS andthe users.2) The BS sends common pilot symbols so that the receivers can estimate H k .3) The receivers calculate the dominant singular values Λ k and the corresponding singularvectors V k and form F k = H k V k Λ k .4) The receivers use the codebooks to quantize the gain and shape of the column vectors in F k and feedback the quantization indices to the BS.
5) The BS calculates the optimum virtual uplink power allocation matrix as, Q opt = min Q (cid:18) σ + σ E P max M (cid:19) tr( J − ) , such that tr( Q ) ≤ P max . Here, σ E = D as in (48) and J is calculated according to (9).6) The precoding matrix of the k th user is calculated as, U mmsek = J − ˆ F k √ q k . Here, q k =[ q k , · · · , q k Lk ] contains the virtual uplink power variables of the L k streams of the k th user.7) Using a recent result [25], the downlink transmit power variables are determined as, p = q .Here, q = [ q , · · · , q L ] is the virtual uplink transmit powers of the L streams.IV. N UMERICAL S IMULATIONS
This section provides the results of simulations to study the effect of shape-gain quantization onthe performance of the MU MIMO linear precoding scheme described. We assume the followingscenario in the simulation setup: the base station has two transmit antennas and serves tworeceivers in the downlink. Each receiver has 2 receive antennas and receives 1 data stream. Thefeedback overhead per user, B , is 16 bits. We show performance curves for different shapequantization bits, B s , in Figs. 6, 7 and 8. The corresponding number of gain quantization bitsis given by, B g = B − B s .Figure 6 illustrates the effect of bit allocation on the quantization error and suggests that B s = 13 and B g = 3 are the optimal bit allocations for this scenario. The analytical results in(42) and (43) lead to, B g = 2 . , B s = 13 . which matches the numerical result.Fig. 7 plots the SMSE of the same system with the transmitter using 16-QAM. The figureshows that B s = 12 leads to the minimum SMSE. The SMSE performance of B s = 13 , i.e.,the optimal solution obtained from analytical results, is very close to that of B s = 12 bits. Theminor difference between the simulation and analytical result stems from the fact that the derivedgain distortion holds only for large number of gain quantization bits. Note that, B s = 16 , i.e.,quantizing the shape exclusively, leads to much higher SMSE. Therefore, optimal bit allocationacross gain and shape feedback provides better performance in terms of SMSE.Fig. 8 shows that the bit allocation B s = 12 or B s = 13 also lead to better performance interms of BER. If one uses all the bits for direction quantization, i.e., B s = 16 , the effect ofmultiuser interference on the norm of the received signal cannot be rectified. This leads to theinferior performance of B s = 16 . We did not plot the performance of all possible bit allocations, e.g., B s = 0 to B s = 8 ,so that the figures look clearer. However, the performance trend of B s = 11 to B s = 9 bitclearly suggests that full gain quantization ( B s = 0 , B g = 16 ) will also perform much inferiorto the optimal bit allocation in shape-gain quantization. Thus, optimal shape-gain quantizationcan improve over full gain or full shape quantization and lead to a lower BER in multiuserMIMO systems. In wireless ethernet [26] systems, where a small number of bit errors may leadto the whole packet drop [27], optimal bit allocation in shape-gain quantization can significantlyreduce the packet loss rate and save packet re-transmission time.V. C ONCLUSION
This paper studies the optimal bit allocation across gain and shape quantization in a MUMIMO downlink system by minimizing the SMSE of the system for a fixed feedback overheadper user. We show that the distortion due to gain and shape quantization are proportional to − B g and − Bs M − respectively, suggesting that, in the asymptotic region of high feedback overhead, thenumber of shape quantization bits should be approximately (2 M − times than the number ofgain quantization bits. The analysis and importance of bit allocation is borne out by the simulationresults that show significant worse performance for the usual approach (in MU MIMO downlinksystems) of only quantizing the gain or shape but not both.Our work with respect to the gain distortion calculation is quite general, since the gainquantization distortion of other distributions like Rician, Nakagami and Weibull fading canalso be calculated using Bennett’s integral. However, the optimal bit allocation results mightbe different from the Rayleigh fading case considered in this paper.A PPENDIX AP ROOF OF L EMMA f ( λ e ) = 1( L ( e ) − λ L ( e ) − e β L ( e ) exp (cid:18) − λ e β (cid:19) . (49)Here, λ e denotes the e th eigenvalue of the Wishart matrix (i.e., H H H or HH H ). e denotes theindex of the ordered eigenvalues. L ( e ) = ( M − e )( N k − e ) . β is a constant whose value is given through the following equation, β = ˜ λ e L ( e ) . (50)Here ˜ λ e is the mean of the eigenvalue. (49) provides the probability distribution function of theeigenvalue of the Wishart matrix, λ e . In our proposed algorithm, we are trying to quantize g ,the singular values of the Gaussian matrix H . Now, λ e = g .Using Jacobian transformation [23], the probability distribution of the singular values of theGaussian matrix can be found as follows, f g ( r ) = 1( L ( e ) − r ) L ( e ) − β L ( e ) exp (cid:18) − r β (cid:19) r. (51)Therefore, || f g ( r ) || = 2( L ( e ) − β L ( e ) (cid:18)Z ∞ r L ( e ) − exp (cid:18) − r β (cid:19) dr (cid:19) . (52)Using standard mathematical tables of [28] ( P - 380, eqn - 662), we find Z ∞ x n exp ( − ax p ) dx = Γ (cid:16) n +1 p (cid:17) pa ( n +1 p ) . (53)Comparing (53) with (52), we find, n = L ( e ) − , a = β , p = 2 . Therefore, (cid:18)Z ∞ r L ( e ) − exp (cid:18) − r β (cid:19) dr (cid:19) = Γ (cid:18) L ( e ) − +12 (cid:19) (cid:16) β (cid:17) L ( e ) −
13 +12 (54) (cid:18)Z ∞ r L ( e ) − exp (cid:18) − r β (cid:19) dr (cid:19) = 12 (3 β ) L ( e )+13 Γ (cid:18) L ( e ) + 13 (cid:19) (55) (cid:18)Z ∞ r L ( e ) − exp (cid:18) − r β (cid:19) dr (cid:19) = 18 (3 β ) L ( e )+1 Γ (cid:18) L ( e ) + 13 (cid:19) . (56)Using (56) in (52), we get, || f g ( r ) || = 2( L ( e ) − β L ( e )
18 3 L ( e )+1 β L ( e )+1 Γ (cid:18) L ( e ) + 13 (cid:19) (57) = 3 × L ( e ) β L ( e ) − (cid:18) L ( e ) + 13 (cid:19) . (58) A PPENDIX BP ROOF OF L EMMA
P r [min i ∈ N || s − ˆs i || ≥ b ] = − (2 M − C M − R cos − (1 − . b )0 sin M − φ dφ M C M ! N (59) = − K Z cos − (1 − . b )0 sin M − φ dφ ! N (60) ≈ − K Z cos − (1 − . b )0 φ M − dφ ! N (61) = (cid:16) − K (cid:0) cos − (1 − . b ) (cid:1) M − (cid:17) N . (62)In (60), we assumed K = (2 M − C M − MC M . (61) follows from the fact that, given a large numberof quantization vectors, i.e., at high bit rate, the complementary cumulative distribution function(CCDF) is significant only for smaller values of φ . For these smaller angles, we can assume sin φ ≈ φ . Equation (62) follows from assuming K = K M − .Figure 9 compares the simulated shape distortion with the original and approximate analyticalshape distortion of a × C M vector. Here, the CCDF of the original and approximate analyticalexpressions are superimposed with the simulated CCDF. Hence, (60) and (61) accurately modelthe actual distortion. This justifies the transition from (60) to (61).Now, using (34), we find, E ( b ) = Z P r [min i ∈ N || s − ˆs i || ≥ b ] db (63) = Z a (cid:16) − K (cid:0) cos − (1 − . b ) (cid:1) M − (cid:17) N db (64) = 2 Z ψ (cid:0) − K θ M − (cid:1) N sin( θ ) dθ (65) ≈ Z ψ (cid:0) − K θ M − (cid:1) N θdθ (66) ≈ Z (cid:0) − K θ M − (cid:1) N θdθ (67) = 2 Z N X i =0 (cid:18) Ni (cid:19) ( − i K i θ i (2 M − ! dθ (68) = 2 N X i =0 (cid:0) Ni (cid:1) ( − i K i i (2 M −
1) + 2 . (69)The transition from (63) to (64) can be explained as follows: the similarity between (60) and(61) holds only for smaller values of b since sin φ = φ for larger φ . Therefore, although thesquare of the Euclidean distance between two random unit norm vectors can vary from 0 to4, (62) holds only for a smaller range of b . At the presence of a large number of codewords,the squared distance between the original and the quantized channel takes large values with anegligibly small probability. Therefore, we can truncate the range of b as long as the CCDF ofthe original function is negligible outside the range, i.e., the limited range of b does not have anysignificant affect on the calculation of the expected value of the distortion. Using this analysis,in (64), we use a as the truncated range, i.e., we assume that b can vary from to a .In (65) we assumed, θ = (cos − (1 − . b )) . Therefore, ψ = (cos − (1 − . a )) . Since onlysmaller angles of θ contribute to E ( b ) , we assumed sin θ ≈ θ in (66). In (68), we assumed ψ = 1 to simplify the other calculations.Fig. 10 justifies the approximations that we used in the derivations of shape distortion cal-culation. Here, approx1 and approx2 denote sin( θ ) ≈ θ (ref: eq. 66) and ψ ≈ (ref: eq. 67)respectively. As Fig. 10 shows, the three curves are superimposed with each other. Therefore,our justifications are valid for high bit rate quantization.Applying (cid:0) Ni (cid:1) = ( − i ( − N ) i i ! , where ( − N ) i = Γ( − N + i )Γ( − N ) [29], (69) takes the following form, N X i =0 ( − i ( − N ) i ( − i K i i !( i (2 M −
1) + 2) = 22 M − N X i =0 ( − N ) i K i i !( i + M − ) (70) = 22 M − N ! M − (cid:0) M − (cid:1) N K − M − (71) = N !Γ (cid:0) M − (cid:1) Γ (cid:0) N + 1 + M − (cid:1) K (72) = N Γ( N )Γ (cid:0) M +12 M − (cid:1) Γ (cid:0) N + M +12 M − (cid:1) K (73) = N β (cid:18) N, M + 12 M − (cid:19) K . (74)(71) was found using ( [30], 6.6.8). In (72), we assumed K = K − M − . (74) was obtained usingthe relation between the gamma and beta function, β ( a, b ) = Γ( a )Γ( b )Γ( a + b ) [31]. Following a similar work in [3], we find, N β (cid:18) N, M + 12 M − (cid:19) = 2 B Γ(2 B )Γ(1 + M − )Γ(2 B + 1 + M − ) (75) ≤ B Γ(2 B )Γ(2 B + 1 + M − ) (76) = Γ(2 B + 1)Γ(2 B + 1 + M − ) . (77)The preceding inequality in (76) is justified by the following reasoning: due to the convexityof the gamma function [3] and the fact that Γ(1) = Γ(2) = 1 , Γ( x ) ≤ for ≤ x ≤ . Let, y = 2 B + M − , t = 1 − M − , so that, y + t = 2 B + 1 , y + 1 = 2 B + 1 + M − . By applyingKershaw’s inequality for the gamma function [32], Γ( y + t )Γ( y + 1) < (cid:18) y + t (cid:19) t − ∀ y > , < t < . (78)Using (78), Γ(2 B + 1)Γ(2 B + 1 + M − ) < (cid:18) B + 22 M − . − M − (cid:19) − M − (79) = (cid:18) B + 12 M − . (cid:19) − M − (80) < − B M − . (81)Using (81) and the value of K we find, N X i =0 ( − i ( − N ) i ( − i K i i !( i (2 M −
1) + 2) < (cid:18) C M − M C M (cid:19) − M − − B M − . (82)Using the values of C M − and C M one can obtain, E ( b ) < K s − Bs M − , (83)where, K s = (cid:18) π M − Γ( M )2 π M Γ ( M − +1 ) (cid:19) − M − is a constant with respect to B s .A PPENDIX CP ROOF OF T HEOREM dDdB s = ¯ K s (ln 2)2 − Bs M − (cid:18) − M − (cid:19) + K g (ln 2) (cid:0) − B − B s ) (cid:1) (84) d Dd B s = ¯ K s (ln 2) − Bs M − (cid:18) − M − (cid:19) + K g (2 ln 2) (cid:0) − B − B s ) (cid:1) . (85) From (85), d Dd B s ≥ . Therefore, the optimal bit allocation problem is convex [33]. Now, equatingthe 1st derivative to be zero, ¯ K s M − − Bs M − = K g − B − B s ) (86) − B +2 B s + Bs M − = ¯ K s K g (2 M − (87) M B s M − B + 12 log (cid:18) ¯ K s K g (2 M − (cid:19) (88) B s = 2 M − M B + 2 M − M log (cid:18) ¯ K s K g (2 M − (cid:19) . (89)Therefore, at the optimal point, B s = 2 M − M B + 2 M − M log (cid:18) ¯ K s K g (2 M − (cid:19) (90) B g = 12 M B − M − M log (cid:18) ¯ K s K g (2 M − (cid:19) . (91)R EFERENCES [1] F. Boccardi, H. Huang, and M. Trivellato, “Mutliuser eigenmode transmission for MIMO broadcast channels with limitedfeedback,” in
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Convex Optimization , Cambridge University, Cambridge, UK, 2004. Fig. 1. MU MIMO system model in the downlinkFig. 2. Gain-shape product quantization −4 −3 −2 −1 Bg (Gain quantization bits) G a i n qu a n ti za ti on d i s t o r ti on Theoretical DistortionSimulated Distortion
Fig. 3. Quantization distortion of the dominant singular value of 2x2 MIMO channelFig. 4. Shape quantization block diagram −2 −1 Bs (quantization bits) S h a p e D i s t o r ti on Simulated DistortionAnalytical Expression of the upper bound
Fig. 5. Comparison of the simulated distortion with the theoretical upper bound (2x1 complex vector) −2 −1 Shape Quantization Bits (Bs) Q u a n ti za ti on e rr o r v a r i a n ce Fig. 6. Effect of bit allocation in the quantization of the product of dominant eigenvalue & the corresponding eigenvector ofa 2 x 2 MIMO channel Shape Quantization Bits (Bs) S M S E Full CSIBs = 16 bitBs = 15 bitBs = 14 bitBs = 13 bitBs = 12 bitBs = 11 bitBs = 10 bitBs = 9 bit
Fig. 7. Effect of bit allocation in the SMSE of 16-QAM system, M = 2 , N = [2 2] , L = [1 1] , B = 16
10 15 20 2510 −2 −1 SNR (in dB) A v e r a g e B E R M = 4, N = [2 2], L = [1 1], 16 bit per user, 16−QAM
Bs = 10bitBs = 11bitBs = 12bitBs = 13bitBs = 14bitBs = 15bitBs = 16bitFull CSI
Fig. 8. Effect of bit allocation in the BER of 16-QAM systems, M = 2 , N = [2 2] , L = [1 1] , B = 16 Square of the distance CC D F Original Analytical DistortionApproximated Analytical DistortionSimulated Distortion