Joint Design of Transceivers for Multiple-Access ... - IEEE Xplore

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 8, OCTOBER 2011

Joint Design of Transceivers for Multiple-Access Channels Using MMSE Decision Feedback Detection Jian-Kang Zhang, Senior Member, IEEE, Kon Max Wong, Fellow, IEEE, Wenwen Jiang, and Alex B. Gershman, Fellow, IEEE

Abstract—We consider the joint design of transceivers for a multiple-access multi-input–multi-output (MIMO) system communicating over intersymbol interference (ISI) channels. The system we consider is equipped with the minimum mean-square error (MMSE) decision-feedback (DF) detector. Here, we explore a novel perspective of designing a transceiver that minimizes the arithmetic mean-square error (MSE) of MMSE-DF detection and satisfies a fixed-sum Gaussian mutual information constraint. For this optimization problem, a direct explicit solution is obtained. We show that the optimal solution is achieved if and only if uniform user mutual information and uniform symbol mutual information are both achieved. Uniform user mutual information is attained by successively solving a series of problems of minimizing individual user transmission power under fixed mutual information, whereas uniform symbol mutual information is maintained by applying the equal-diagonal QRS decomposition. We also show that, in addition to minimizing the arithmetic MSE of MMSE-DF detection, such uniform mutual information distribution has also other desirable properties: 1) The optimal detection order in terms of signal-to-interference-noise ratios (SINRs) for both the users and the symbols is the natural transmission order of the system; and 2) when the sum Gaussian mutual information is large, the performance of the optimum transceiver using the MMSE-DF receiver asymptotically approaches that of the maximum-likelihood (ML) receiver. Index Terms—Equal-diagonal QRS decompositions, minimum mean-square error (MMSE) decision-feedback (DF) detectors, multiple-access multi-input–multi-output (MIMO) channels, sum Gaussian mutual information.

Manuscript received November 3, 2010; revised March 9, 2011 and May 5, 2011; accepted May 30, 2011. Date of publication July 25, 2011; date of current version October 20, 2011. The work of J.-K. Zhang and K. M. Wong was supported in part by the Natural Sciences and Engineering Research Council of Canada and in part by the Alexander von Humboldt Foundation of Germany. The work of A. Gershman was supported by the European Research Council Advanced Investigator Grants Program under Grant 227477-ROSE. The review of this paper was coordinated by Prof. C. P. Oestges. J.-K. Zhang and K. M. Wong are with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON L8S 4L8, Canada (e-mail: [email protected]; [email protected]). W. Jiang is with Hatch Mott MacDonald Ltd., Mississauga, ON L5K 2R7, Canada (e-mail: [email protected]). A. Gershman, deceased, was with the Institut für Nachrichtentechnik, Technische Universität Darmstadt, 64283 Darmstadt, Germany. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2011.2162863

I. I NTRODUCTION

A

MAJOR PROBLEM in data communications arises from the intersymbol interference (ISI) created by a frequencyselective channel [1], which, however, can usually be mitigated by transmitting the data in a block-based fashion [2]. In particular, effective block-by-block detection can be performed if the blocks are designed so that there is no interblock interference at the receiver. Such block-by-block data communication systems include the commonly used discrete multitone (DMT) [3], [4] and orthogonal frequency-division multiplexing [5], [6] schemes. For the single-user block-by-block data communication over an ISI channel with Gaussian noise, it has been shown [7] that to achieve the capacity of the channel, the transmitter must allocate appropriate bit and power to the subcarriers according to the water-filling distribution [8]. To take full advantage of such a design, the maximum-likelihood (ML) detector [1], [9], [11] must be employed at the receiver. In practice, however, the ML detector is too complicated to implement. Zero-forcing (ZF) or minimum mean-square error (MMSE) linear receivers, on the other hand, are simple to implement. However, the corresponding optimum transmitters designed under different criteria [12]–[16], ranging from maximization of the mutual information and SNR to minimization of the mean-square error (MSE) and receiver bit error rate (BER), all show a substantial loss in performance compared with the use of the ML detector at the receiver. An effective alternative is to employ decision-feedback equalization (DFE) at the receiver, which, in general, is a good compromise between implementation complexity and overall performance. The DFE scheme is widely used to combat ISI in linear dispersion channels. To disentangle ISI, each input symbol is first decoded based on the entire received sequence. Its effect on the remainder of the sequence is then subtracted before decoding of the next symbol begins. Canceling of the interference by DFE can be effected using either the ZF or the MMSE criteria, which are designated ZF-DFE and MMSEDFE, respectively. For a single-user system, using various criteria, closed-form jointly optimal transmitters with both ZF decision-feedback (DF) [17], [18] and MMSE-DF [19]–[22] detectors have been obtained using a novel equal-diagonal QRS matrix decomposition [17] (or similar algorithms). It has been shown [19] that with the use of the respective

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ZHANG et al.: DESIGN OF TRANSCEIVERS FOR MULTIPLE-ACCESS CHANNELS USING MMSE-DF DETECTION

optimum transmitters, the MIMO communication system employing MMSE-DFE is superior in performance to that with ZF-DFE. Indeed, the MMSE-DFE receiver has been studied [23]–[26] and is referred to as a canonical receiver, suggesting that by using properly designed codes and under the assumption of having no error propagation, reliable communication at rates approaching the capacity of the block transmission system can be achieved by using independent instances of the same (Gaussian) code in each element of a block. Thus, the good performance of the MMSE-DFE receiver together with its relatively low complexity makes it a very desirable receiver. Our discussion so far has focused on the design of MIMO transceivers based on a single-user scenario. For a multiuser system, achieving capacity requires much more sophisticated resource allocation than water filling [27], [28]. The direct use of nonoverlapping resource allocation schemes in time division (TDMA) or frequency division (FDMA) in an arbitrary fashion will result in multiuser rates far below the capacity [29]. The optimum resource allocation, in general, is difficult to exactly compute [30], and to attain reliable performance at the rates predicted in [27] and [28], joint detection may be needed at the receiver, resulting in an unacceptably high computational load, or certain structures on the transmitted signals may have to be imposed [31]. On the other hand, the aforementioned optimal design solutions for single-user systems employing linear or DFE receivers cannot be directly generalized to a multiuser scenario either. The main technical obstacle here is that the transmitter matrix has a block-diagonal structure such that each subblock is constrained in power individually. Thus far, this difficult problem of designing optimal transceiver pairs for a multiuser case has been only successfully tackled by minimizing the total MSE in the system employing a linear MMSE receiver [32], [33].1 In this paper, we consider the optimal design of the transceiver pair for a synchronous multiple-access MIMO system in which K-user data sequences are separately precoded and transmitted block by block at a full data rate over ISI channels. Due to its many advantages, we employ the MMSE-DF detector at the receiver of this multiple-access MIMO system. Instead of trying to extend the optimum single-user transceiver design to the multiuser system, we explore a novel perspective by forming the dual optimization problem of water filling and developing a direct explicit solution for it, thus minimizing the transmission power for a given value of the sum mutual information. We also build into the transceiver pair design such that the arithmetic MSE for all the users is minimized, arriving at the condition that the share of the fixed-sum mutual information by each user must be directly proportional to the number of subcarriers allocated to the user. This is a situation to which we can directly apply the dual water-filling solution, leading to an optimum design of the transceiver. The optimum transceiver also possesses other attractive properties such as having the received symbols in the optimum detection order as 1 All the aforementioned designs of optimum transceivers necessitate the knowledge of channel state information (CSI) for both the transmitter and the receiver. This is the condition we assume throughout this paper.

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Fig. 1. DF receiver.

well as having asymptotic detection performance approaching that of the ML detector. II. M INIMUM M EAN S QUARE E RROR -D ECISION F EEDBACK E QUALIZER D ETECTOR A PPLIED TO A M ULTIPLE -ACCESS S YSTEM A. MMSE-DFE Receiver for a Single-User MIMO System A general MIMO communication system having N input dimensions and P output dimensions (which includes the single-user block-by-block data communication system over an ISI channel with Gaussian noise) can be represented by the following equation: y = Hx + ξ

(1)

where x = [ x1 x2 · · · xN ]T consists of the block of N symbols to be transmitted, with each symbol xi being independently chosen from finite-size alphabet X and is normalized to have unit energy so that E[xxH ] = I, I being the identity matrix, y = [ y1 y2 · · · yP ]T is the received signal vector consisting of the P symbols at the receiver, H is a P × N channel matrix assumed to be known to both the transmitter and the receiver, and ξ = [ ξ1 ξ2 · · · ξP ]T is the noise vector (not necessarily white in general) with covariance matrix E[ξξ H ] = Σξξ . The task for the receiver is to detect vector x ∈ X N given noisy observation y. Let us apply the MMSE-DFE successive cancelation detector (see Fig. 1) to the MIMO system in (1). The MMSE-DF receiver employs an N × P feedforward filter F to process received data vector y, giving output z = Fy. Estimated symbol vector x is given by subtracting output χ of an N × N feedback filter B from z [23]–[25]. Detection of the nth symbol xn of vector x sequentially proceeds, starting from the last symbol, as follows: 1) Starting from n = N , we set xN = zN . The first detected symbol is then the quantized version of the estimated symbol, i.e., x ˆN = Q[xN ]. 2) This quantized symbol is then fed back through feedback filter B so that xn = zn − χn , for n = ˆ , with bn beN − 1, N − 2, . . . , 1, where χn = N =n+1 bn x ing the coefficients of B. 3) The subsequent detected symbols are then the quantized version of the estimated symbols, i.e., x ˆn = Q[xn ], for n = N − 1, N − 2, . . . , 1. Note that filter B is a strictly upper triangular matrix since detection starts from last symbol x ˆN , and the symbol χn to be canceled is a linear combination of the previously detected symbols. Thus, the output x. Assuming that the detected of DFE is x = FHx + Fξ − Bˆ symbol is the same as the transmitted symbol, the detection ∆ error is e = x − x = (FH − I − B)x + Fξ. Hence, the error ∆ covariance matrix is Σee = E[eeH ], i.e., Σee = (FH − B − I)(FH − B − I)H + FΣξξ FH .

(2)

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Fig. 2. Multiple-access system model.

The MMSE-DFE receiver is designed to minimize mean-square detection error tr(Σee ) according to the orthogonality principle [1], resulting in the relationship between F and B, i.e., −1 H −1 HH Σ−1 F = (B+I)Σxy Σ−1 yy = (B+I) I+H ΣξξH ξξ . (3) Substituting (3) into (2) and simplifying [19], we have −1 Σee = (B + I) I + HH Σ−1 H (B + I)H ξξ = (B + I)G−1 (B + I)H .

(4)

Since the mutual information for the single-user channel [34] is given by I = log det(I + HH Σ−1 ξξ H), here, we designate G in (4) as the mutual information matrix given by G = I + HH Σ−1 ξξ H.

(5)

Applying the QR decomposition [35] to G1/2 such that G1/2 = QR, where Q is an N × N orthonormal matrix and R is an N × N upper triangular matrix with positive diagonal elements, we have H

through the P × M channel matrix Hk . Then, the combined equivalent baseband discrete-time signal vector y received at the base station from all K users can be written as y=

K

Hk Tk xk + ν = Hx + ν

(8)

k=1

and x= where H = [ (H1 T1 ) · · · (HK TK ) ], [ xT1 · · · xTK ]T . Here, each xk is assumed to have zero mean and identity covariance matrix; Hk can be a block Toeplitz tall channel matrix corresponding to zero-padded modulation [12] or, for P = M , an M × M square block-diagonal channel matrix corresponding to vectorized DMT modulation (e.g., [3]); Tk is an M × Nk precoder matrix, which also controls the transmission power of the kth-user data; ν is a P × 1 white Gaussian noise vector with an identity covariance matrix and is independent of input signal vector xk . It is shown that the model in (8) is equivalent to the model in (1). Therefore, in parallel with (5), the sum mutual information matrix for this entire multiuser system can be written as G = I + HH Σ−1 νν H

(9)

H

G = (QR) (QR) = R R 2 2 H 2 = L diag r11 , r22 , . . . , rN N L

and the sum Gaussian mutual information [34] is given by (6)

where we extract diagonal elements {rnn } from matrix R as a diagonal matrix so that L is lower triangular having unity diagonal elements. Comparing (6) and (4), we can conclude that −2 −2 −2 (7) Σee = diag r11 , r22 , . . . , rN N i.e., the errors are uncorrelated (but not white) after the MMSEDFE receiver. Thus, the MSE detection error can be also of the −2 expressed as ε¯2 = (1/N ) N r n=1 nn . We note that the signalto-interference-noise ratio (SINR) for the nth symbol at the 2 − 1. receiver [19] is ρn = rnn B. Multiple-Access MIMO System Equipped With a DF Receiver 1) System Model: Let us now turn our attention to the operation of a synchronous multiple-access MIMO system. The system considered in this paper is shown in Fig. 2. Each of the K-user data sequences is separately precoded by Tk and transmitted block by block at a full data rate over frequencyselective ISI channels. For transmission, the system assigns to each user a signal length, i.e., the kth user transmits signal vector xk consisting of Nk symbols (assigned by the system)

H IG = log det(G) = log det I + HΣ−1 νν H K H H Hk Tk Tk Hk = log det I +

(10)

k=1

where the property Σ−1 νν = I of white Gaussian noise vector ν has been used. 2) MMSE-DFE Detection in Multiuser MIMO: As mentioned in Section II-A, the DF receiver detects the transmitted symbols in reversed order. For a multiuser system, signals are also detected by the DF receiver in the reversed order of the user index, i.e., we first detect the signal from user K, then user K − 1, etc. Thus, for user k, k = K, (K − 1), . . . , 1, we can rewrite the received signal in (8) as yk = y −

K

Hi Ti xi = Hk Tk xk +

i=k+1

= Hk Tk xk + ζ k

k−1 j=1

Hj Tj xj + ν

ζk

(11)

where ζ k = k−1 j=1 Hj Tj xj + ν is the kth interference-plusnoise vector, which can no longer be considered white, and the


covariance of which is given by k−1

H H Σk = E ζ k ζ H Hj Tj Σ−1 k =I+ j Tj Hj j=1

= Σk−1 I +

k−1

H H +Hk−1 Tk−1 Σ−1 Tk−1 Hk−1 . (12)

=1

The last part of (12) provides us with a recurrent relation. We also note from (10) and (12) that ΣK+1 = G = I + HH Σ−1 νν H

⇒ det ΣK+1 = 2IG . (13)

In (11), the MMSE-DFE receiver is used to detect xk from received signal y by canceling the previously detected signals. Therefore, comparing (11) with (1) and using the results in Section II-A, the error vector for the kth user is given by ek = xk − xk = (Fk Hk Tk − I − Bk )xk + Fk ζ k , with Fk and Bk being the feedforward and feedback filters of the kth user receiver, respectively. Again, the kth DF receiver employs the MMSE strategy, yielding the following relationship between the feedforward and feedback filters: −1 H −1 H −1 Hk Σk Fk = (Bk + I) I + TH k Hk Σk Hk Tk H −1 = (Bk + I)G−1 k Hk Σk

(14)

where, in parallel with (5), we define the mutual information matrix of the kth channel as H −1 Gk = I + T H k Hk Σk Hk Tk

(15)

with the Gaussian mutual information of the kth user given by [34] H −1 Ik = log det(Gk ) = log det I + TH (16) k Hk Σk Hk Tk and Σk = E[ζ k ζ H k ]. The error covariance matrix for user k is then Σeek

−1 H = E ek eH k = (Bk + I)Gk (Bk + I) −2 −2 = diag rk11 , . . . , rkN k Nk

(17)

with Σ1 = I, and rkii being the ith diagonal element of the 1/2 R-factor of the QR decomposition Gk = Qk Rk . Since there are altogether N symbols for transmission, N = K k=1 Nk , we define the average MSE of the successive cancelation detectors of all K users as

ε2 =

K K 1 1 = tr E ek eH tr (Σeek ) . k N N k=1

(18)

k=1

Our goal is as follows: Given the knowledge of each Hk , for such a system using MMSE-DF detection, find an optimum design for all K transceivers that minimizes the average MSE in (18).

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Let us first examine the relation between the different mutual information matrices. 3) Sum and Individual Mutual Information Matrices of the Multiuser System: In the previous sections, we have seen that the channel matrix for the multiuser system is given by H in (8) and that the sum mutual information matrix is given by G in (9). On the other hand, the mutual information matrix for individual user k is Gk , as given by (15). If we now perform 1/2 the QR decompositions, respectively, on G 1/2 and Gk , we can see that the two R-factors have the following relationship. 1/2 Property 1: Suppose Gk = Qk Rk and G 1/2 = Q0 R, where both Qk and Q0 are unitary matrices, and Rk and R are both upper triangular matrices. Then, R is an N × N block upper triangular matrix with the kth diagonal block being of dimension Nk × Nk . The kjth block matrix of R is given by if k = j Rk , Rkj = −1/2 Gk (Hk Tk )H (Σk−1 )−1 (Hj Tj ), if k < j (19) with Σk and Gk given by (12) and (15), respectively. The proof of Property 1 is straightforward but tedious and can be found in [36]. Equation (19) provides us with a deeper insight on the distribution of the sum mutual information of the system: The MIMO model of the multiuser system in (8) has a total of N symbols to be detected by the MMSE-DF receiver. Using the result in [26], we can mutual see that the sum 2 information is given by IG (x; y) = N n=1 log([R]nn ), where [R]nn denotes the nth diagonal element of upper triangular matrix R. Since Rk is the R-factor of the QR decomposition of 1/2 Gk of the individual users and because of the relationship in (19), we can easily see that the mutual information of each user is the sum of the mutual information of the subchannels allocated to the user and that the sum Gaussian mutual information of the whole system is the sum of the mutual information of all the individual users. III. WATER F ILLING AND THE D UAL WATER -F ILLING P ROBLEMS A. Optimum Precoder for Single-User MIMO—Water-Filling Solution Suppose there is only one user, e.g., the kth user, in the multiple-access system. From (11), the received signal model is then reduced to yk = Hk Tk xk + ζ k , where ζ k is Gaussian but not necessarily white. The Gaussian mutual information Ik given by (16) can be maximized by the well-known waterfilling solution [37], [38], which is summarized as follows. Problem Statement: Find a transmitter Tk , subject to a power constraint, that maximizes Ik , i.e., maxTk log det(I + H −1 H TH k Hk Σk Hk Tk ) s.t. tr(Tk Tk ) ≤ Pk . If we let the eigen−1 H Σ value decomposition of HH k k Hk be Uk Λk Uk with eigenvalues λkm for m = 1, . . . , M arranged in a nonincreasing order, then the preceding problem can be solved by first selecting the largest integer k such that   k 1  k . < Pk + λ−1 (20) kj λkk j=1

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˜ k Φk Sk , where Sk is an Optimal Tk is given by Tk = U ˜ arbitrary unitary matrix Uk consists of the first k columns of Uk , and Φk is a diagonal matrix with diagonal entries φkm given by   k 1 Pk +  − λ−1 , λ−1 m = 1, . . . , k . φ2km = kj km k j=1 (21) Finally, the maximum Gaussian mutual information, i.e., the channel capacity, is given by    k k k   k  Pk + λ−1 λkm  . Ck = log − (22) kj k j=1

m=1

Remarks on the Water-Filling Solution: 1) Equation (20) instructs us that only subchannels with noise levels lower than the arithmetic average of the total transmission power and the subchannel noise levels will be used for transmission. 2) Equation (21) yields the optimum transmission power for the mth transmission subchannel, which is equal to the arithmetic average of the sum of the total transmission power and the subchannel noise levels, minus the kth subchannel noise level. ˜ k Φk of optimum precoder Tk ensures that the 3) The part U signal is transmitted through selected k subchannels with corresponding transmission power φ2km . 4) While the extra unitary matrix Sk in Tk does not change the value of the maximum Gaussian mutual information, it adds an extra degree of freedom that can be utilized to minimize the mean square of the detection error (MSE). In [19], the lower bound of the MSE has been shown to be the inverse of the M th root of the quantity in the brackets of (22), and the optimum value of Sk for the precoder was obtained by applying the QRS decomposition algorithm [17], [20] to bring the MSE to this lower bound. Unfortunately, the direct application of the approach in Remark 4 to the case of multiple-access systems is not possible since channel matrix H contains individual transmission precoders, each of which has to be designed and the power of which has to be individually controlled.2 We now introduce a novel approach by turning around the water-filling problem forming the dual problem. A solution for the dual water-filling problem is then proposed. With this solution, we can proceed to solving for the optimum design for the multiuser MIMO system. B. Optimum Precoder for Single-User MIMO—Dual Water-Filling Problem Again, we consider the case of having only one user in the multiple-access system of (8). Here, for user k, we turn the water-filling problem around and formulate the following dual problem. 2 An iterative water-filling algorithm was proposed in [29] to solve the optimum transceiver design problem in a multiuser MIMO system using the criterion of mutual information, which led to a convex optimization problem. The criterion of MSE used here does not provide us with such.

Problem Statement: Subject to a given amount of mutual information Ik > 0, find a transmitter Tk such that the total transmitted power is minimized, i.e., min tr TH k Tk Tk

s.t.

H −1 log det I + TH k Hk Σk Hk Tk = Ik .

(23)

The solution of this optimization problem is provided by the following theorem. Theorem 1: Let the Gaussian mutual information Ik for the received signal model of user k be given by (16). H −1 Let Uk Λk UH k be the eigendecomposition of Hk Σk Hk , where Uk is an M × M (M ≥ 1) unitary matrix, and Λk = diag(λk1 , λk2 , . . . , λkM ) with λk1 ≥ λk2 ≥ · · · ≥ λkM > 0. positive integer not exceeding If we let k be the maximal k λki /2Ik )1/k for m = 1, 2, . . . , k , M such that λkm > ( i=1 then the optimal solution Tkop of (23) is an M × k matrix given by ˜ k Γk VH Tkop = U k

(24)

˜ k is an M × k matrix consisting of the first k where U columns of unitary matrix Uk , Vk is an arbitrary k × k unitary matrix, and Γ = diag(γk1 , γk2 , . . . , γkk ), with each γkm given by 2 γkm

=

2Ik

1/k − λ−1 km ,

k

i=1 λki

m = 1, . . . , k . (25)

The minimum power is Pkmin = k

2Ik

k

i=1 λki

1/k −

k m=1

λ−1 km .

The proof of the preceding theorem is presented in the Appendix. Remarks on the Dual Water-Filling Solution: 1) The solution of the dual problem specifies the transmission subchannels to have a signal-to-noise level higher than the ratio of the geometric mean of the signal-to-noise levels of all the transmission subchannels to the k th root of the mutual information. This is different from the water-filling solution, which uses the arithmetic average of the sum of the transmission power and the subchannel noise levels for the selection of transmission subchannels. 2) Equation (25) indicates the amount of optimum power in the mth transmission subchannel, which is equal to the ratio of the k th root of the mutual information to the geometric mean of the signal-to-noise level of all the transmission subchannels minus the noise level of the subchannel itself. As shown in the Appendix, to ensure having optimum power in all the subchannels, user k must employ a precoder Tkop of the form shown in (24). Furthermore, the total power of user k will be minimized if we choose Nk = k . 3) Theorem 1 provides direct explicit solutions for the optimum transmitter


and the minimum transmission power of the dual water-filling problem.3 IV. J OINT D ESIGN OF T RANSCEIVER PAIRS FOR M ULTIPLE -ACCESS C HANNELS We now consider the joint optimum design of the transceivers for a multiple-access ISI MIMO system equipped with the MMSE-DFE receiver. Trying to obtain an optimum design that minimizes the MSE of the detected symbols for all K transceivers while applying transmission power constraints to each user would encounter difficulties, as stated at the end of Section III-A. However, since the dual water-filling algorithm allows us to have minimum transmission power for all channels given a fixed amount of mutual information, we may be able to achieve our goal of a minimum MSE design using this algorithm. First, we recall that each of the K users is given a transmission signal length Nk so that the total signal length for all the users is N = K k=1 Nk . Our objective is to minimize the overall detection MSE of these N symbols in the DFE detector while minimizing the transmission power of each user. We now 2 in (18) here and examine its rewrite this objective of MSE ε 2 lower bound, i.e., ε = (1/N ) K k=1 tr(Σeek ), k = 1, . . . , K. We apply the trace-determinant inequality [41] for a positive semidefinite (PSD) matrix to Σeek so that (1/Nk )tr(Σeek ) ≥ det(Σeek )1/Nk , with equality holding iff Σeek = αI, where α is a positive scaling factor. In addition, we note from (17) that det(Σeek ) = det(G−1 k ). Then, we can write ε2 ≥ =

K 1 −1/Nk Nk det Gk N

1 N

1 = N

k=1 K k=1 K k=1

N−1 H −1 k Nk det I + TH k Hk Σk Hk Tk Nk

det(Σk )1/Nk det(Σk+1 )1/Nk

(26a)

.

(26b)

The last step in (26b) is due to the fact that det(I + AB) = det(I + BA) for matrices A and B of compatible dimensions (Sylvester’s determinant theorem) [41] so that each determinant in the summation, along with (12), can H −1 be written as det(I + Hk Tk TH k Hk Σk ) = det(Σk + H H −1 Now, Hk Tk Tk Hk ) det(Σk ) = det(Σk+1 )/ det(Σk ). −2 −2 , . . . , rkN ), then equality in from (7), Σeek = diag(rk11 k Nk (26a) holds iff the diagonal elements of the R-factor are all equal, i.e., rk11 = rk22 = · · · = rkNk Nk ,

k = 1, . . . , K.

(27)

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to ensure that the average MSE reaches this lower bound (i.e., for the condition of (27) to be satisfied), unitary matrix Sk must be chosen according to the QRS decomposition algorithm 1/2 [17], which states that: For a PSD matrix Gk , there exists a unitary matrix Sk obtainable efficiently by the quadratic recursive algorithm such that, if a QR decomposition is applied 1/2 to matrix product Gk Sk = Qk Rk , the diagonal elements {rk11 , . . . , rkNk Nk } of Rk are rendered all equal. Employing this algorithm, we can obtain the optimum Sk for the kth user, ensuring that the MSE reaches its lower bound in (26b). Now, we can further develop the lower bound in (26a) by noting that the expression on the right side of (26b) is the arithmetic mean of terms det(Σk )1/Nk / det(Σk+1 )1/Nk , each occurring Nk times. Applying the inequality that µA ≥ µG , where µA and µG are the arithmetic and geometric means of a sequence of numbers, respectively, (26a) can be written as ε2 ≥

K det(Σk )1/Nk 1 Nk N det(Σk+1 )1/Nk k=1

≥ det(ΣK+1 )−1/N = 2−

IG N

(28)

where the very last step of (28) is due to the identity in (13). Equality in the second inequality of (28) holds iff det(Σk ) constitutes a geometrical sequence, i.e.,

det(Σ1 ) det(Σ2 )

1/N1

= ··· =

det(Σk ) det(ΣK+1 )

1/Nk .

(29)

From (26b), we see that each of the terms in (29) is equal to det(Gk )1/Nk . Therefore, to achieve this second lower bound, the sum mutual information must be equally distributed among each subchannel of all users as indicated by (29), and the share of the kth user is given by det(Gk ) = 2(Nk /N )IG . The distribution of the total sum mutual information in this way by each of the K users ensures that the lower bound of the MSE in (26b) reaches its absolute minimum given by the right side of (28). The aforementioned analysis of the lower bound of the MSE reduces our design problem to solving the following optimization problem. Problem 1: For a multiple-access communication system having K users in which channel matrices Hk , k = 1, . . . , K, are known at both the transmitter and the receiver, we are given that the amount of the sum mutual information is IG and that each of the K users transmits a fixed number4 of symbols Nk such that 0 < Nk ≤ Lk , with Lk being the rank of Hk . Divide IG among the K users such that the share for each user is Ik = log det(Gk ) = (Nk /N )IG .

We note that, by attaching a unitary operator Sk to the precoder Tk , the lower bound of (26a) does not change. This can be k )−1 = k = Tk Sk , and then, we have det(G shown by letting T H H H −1 −1 −1 det(I+Sk Tk Hk Σk Hk Tk Sk ) = det(Gk ) . However,

a) Find a sequence of matrices {Tk }K k=1 such that the total power for the kth user is minimized subject to the constraint that the mutual information for user k is Ik . b) Find unitary matrices Sk to form Tkop = Tk Sk so that (27) is satisfied.

3 Approaches similar to the dual water filling have been pursued in [39] and [40] to derive delay-limited capacity of fading channels and to characterize power and capacity regions under rate and power constraints, respectively. In both cases, only numerical methods were given with no explicit solution.

4 The assignment of N to each user is an important constraint in this optimal k design problem because this prevents undesirable outcomes such that users having inferior channels end up with transmitting no signal.

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In view of the preceding discussion, the procedure of solving Problem 1 can now be outlined here. 1) First, divide the sum mutual information IG among the K users according to the shares stated in Problem 1 so that a fixed amount of mutual information allocated for each user and an achievable lower bound 2−(IG /N ) of the MSE given by (28) are established. 2) Apply the dual water-filling algorithm described in Section III for each of the K users to achieve Step a) of Problem 1. This involves the following procedure: i) Perform the eigendecomposition of the M × M matrix −1 H HH k Σk Hk = Uk Λk Uk for k = 1, 2, . . . , K, where Λk = diag(λk1 , . . . , λkM ) with λk1 ≥ · · · ≥ λkM . ii) For user k, let k be the largest positive integers k λki /2Ik )1/k , m = 1, . . . , k . such that λkm > ( i=1 Γk whose diagonal Form an Nk × Nk diagonal matrix k 1/Nk λ ) − (λkn )−1 for entries are γkn = (2Ik / N i=1 ki n = 1, . . . , Nk if Nk ≤ k . If Nk > k , then the diagonal entries of Γk are assigned as  1/k  2I k − λ−1 k kn , n = 1, . . . , k γkn = λki i=1  0, n = k + 1, . . . , Nk . ˜ k Γ1/2 , k = 1, . . . , K, iii) Construct precoder Tk = U k ˜ k contains the first Nk columns of Uk for where U minimum transmission power. ˜ k Γ1/2 Sk , k = 1, . . . , K, the optimum 3) Form Tkop = U k precoder matrices, where Sk is an Nk × Nk unitary matrix obtained by applying the QRS decomposition [17] 1/2 to the mutual information matrix such that Gk Sk = Qk Rk , with Rk having equal-diagonal elements. This precoder Tkop not only maintains minimum transmission power for each user for the subchannels to carry the allocated amount of mutual information, but by incorporating the unitary matrix Sk , it also satisfies the condition of (27), allowing the lower bound in (26a) to be reached. 4) The corresponding optimum feedback and feedforward filters Bk and Fk of the receiver for each user can be obtained, respectively, using (17) and (14) in Section II-A. Remarks on the Optimum Design Algorithm: a) The objective of our design is given as follows: For a given sum mutual information and a given length of transmission symbols for each user, design a transmitter–receiver system that yields the minimum detection MSE at the receiver. Embedded in this objective is the use of an MMSE-DFE receiver that has a fixed order of detection (an extra constraint) such that the symbols of the Kth user are detected first, then the (K − 1)th user, etc. In addition, within each user, the order of symbol detection follows a fixed order such that the Nk th symbol is detected first, then the (Nk − 1)th symbol, etc. Following this order of detection, our algorithm achieves the minimum detection MSE by evenly distributing the sum mutual information among all the K users according to their given lengths of transmission symbols. We then

utilize the extra degree of freedom left over and design the transmitters so that minimum individual transmission power is used by each user. b) Often, for each user, it is desired to have an optimum detection order such that the symbol with the highest SINR is detected first (similar to the detection order in VBLAST [42]). In such a case, for our multiuser system design, the optimal detection order for each user is the natural order, i.e., xN → xN −1 → · · · → x1 . This is because the SINRn of the nth symbol is SINRn = [R]2n − 1 (see Section II-A). Since in our optimum design, all [R]n are equal, implying that the SINR of any symbol is the same, the natural order of detection here, while yielding the minimum MSE, is also the best in detection order. c) The computation complexity of the QRS decomposition on G1/2 necessary for the implementation of Tkop is linear [17]. Thus, the overall complexity of our optimal design is dominated by that of the eigendecomposition of the channel matrices, which, in general, is O(M 3 K). (Note that, in the particular case of a DMT system, the complexity of this process is only O(M 2 K).) V. S UM M UTUAL I NFORMATION D ISTRIBUTION AND A SYMPTOTIC P ERFORMANCE OF THE N EW D ESIGN As discussed in Section II-B3, in the multiuser MIMO system, the mutual information of each user is the sum of the mutual information of the subchannels allocated to the user, which is represented by the log of the square of the corresponding diagonal element of the R-factor in the QR decomposition 1/2 of Gk . We have also seen that the sum Gaussian mutual information of the entire system is the sum of all the mutual information of the individual users. Let us now examine the distribution of the mutual information and the performance of the multiuser system equipped with the optimally designed transceiver described in Section IV. For a given transmitter channel matrix H, its singular values are fixed under any unitary transformation, and hence, its eigensubchannel mutual information remains unchanged. However, the R-factor diagonal values of the mutual information matrix change with the unitary transformation so that the capacity of each subchannel corresponding to each R-factor diagonal element will also change. In other words, different unitary transmitters lead to different R-factors and, hence, different subchannel capacities and different detection error performances for the MMSE-DF detector. By applying the QRS decomposition to the root mutual information matrix and by utilizing the S-factor in the transmission precoder, we have arrived at an optimum design in Section IV, which allows the mutual information to be uniformly distributed (equal-diagonal elements of the R-factor), thereby yielding the minimum MSE of detection. In addition to minimizing the MSE of MMSE-DF detection, as described in Section IV, the uniform decomposition of the sum Gaussian mutual information also has the following effect on the performance of the multiuser MIMO system.


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If x and x are two different signals chosen from a finite signal constellation X , we define the minimum distance as dmin (X ) = min x,x ∈X [(x − x )H (x − x )]1/2 and the free x=x

distance [9] of an M × N channel matrix H as dFr (H) = [min x,x ∈X (x − x )H HH H(x − x )]1/2 . x=x

The following property shows the asymptotic behavior of the free distance for a channel whose square root mutual information matrix has an equal-diagonal R-factor. Property 2: If the square root mutual information matrix G 1/2 of H has an equal-diagonal R-factor, then d2Fr (H) = d2min (X ), where I¯G = IG /N I¯G →∞ (2IG − 1) lim

is the average mutual information per active subchannel. The proof of this property is given in the Appendix. Property 2 has profound implications on the performance of the optimum equal-diagonal R-factor transceiver: For the ML receiver, the symbol error rate at a high SNR is dominated # by dFr (H) [10] and is given by PeML ≈ C · Q( d2Fr ), √ $ ∞ −ζ 2 /2 where C is a constant and Q(z) = (1/ 2π) z e dζ. Now, for the MMSE-DFE receiver, the SINR of the nth symbol is SINRn = [R]2n − 1. Therefore, for N transmitted symbols, the average symbol # error rate is given N = (1/N ) α · Q( ([R]2n − 1)d2min ) = by P n=1 # eDFE α · Q( ([R]2n − 1)d2min ) since [R]n are all equal, implying that the SINR of any symbol is the same. PeDFE = Since % I¯G = (1/N ) log det G = log[R]2n , # ¯ α · Q( (2IG − 1) · d2min ) = α · Q( d2Fr ). This means that the performance of the optimum transceiver asymptotically approaches that of the system using the ML detector. VI. N UMERICAL E XPERIMENTS Here, we examine the performance of the optimal multiuser transceiver designed by the method outlined in Section IV and compare it with multiuser systems using other transceivers. We present two examples in which each element of the transmitted signal vectors is a symbol independently selected with an equal probability from the 4-quadrature-amplitude modulation (QAM) constellation. Example 1: Here, we consider the scenarios of a) a twouser scheme and b) a three-user scheme in a DMT system. The environment of communication for both cases is such that the users communicate with a base station independently and employ DMT modulation having in total 32 available subcarriers. The channel is modeled as a finite-impulse response filter with ten taps. Each user is equipped with the optimum transceiver designed according to the procedure in Section IV. The performance of the system under test is averaged over 1000 channel realizations. For each realization, the channel tap coefficients, as well as the additive Gaussian noise, are generated independently from a zero-mean circular complex Gaussian distribution, and the noise is normalized to unit energy. a) Fig. 3(a) shows the averaged BER performance of the two users. Three cases are examined.

Fig. 3. BER versus sum Gaussian information. (a) Two-user scenario. (b) Three-user scenario.

• The number of subcarriers assigned to users 1 and 2 is 16 each. In addition, the numbers of symbols assigned to each user are N1 = 16 and N2 = 16, i.e., the number of symbols assigned to both users here is equal to the number subcarriers in the system. Thus, it is likely that there are no shared subcarriers between the two users if all the subcarriers are used. • N1 = 16, and N2 = 17, i.e., there is at least one shared subchannel. • N1 = 17, and N2 = 17, i.e., there are at least two subchannels shared by these two users. Since we have established the problem such that, for a given amount of sum mutual information, we design the respective optimum precoders, therefore, in evaluating the performance, it is natural to show the BER (averaged over the two users) under different given amounts of sum Gaussian mutual information of the system averaged over 1000 channel realizations. b) A three-user system is tested in a similar environment. Fig. 3 shows the comparison of BER against sum mutual information for the different cases having different numbers of subcarriers (see the legend in Fig. 3) assigned

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to the different users. Similar observation in the BER performance as in the two-user case can be noted, i.e., the more subchannels that are shared by the users, the higher the probability of the detection error will be. Example 2: To assess the effectiveness of our optimum transceiver in a system employing the MMSE-DFE receiver, we compare its system performance with those of other designs of “optimum” transceivers, which also necessitate the knowledge of CSI for both the transmitter and the receiver. The designs chosen for comparison in this example include the following: I) the optimum transceiver in a system employing the MMSE-DFE receiver proposed in this paper; II) the optimum linear transceiver proposed in [33] minimizing the receiver MSE; III) the optimum transmitter (employing a ML receiver) developed in [29] maximizing the Gaussian sum mutual information; IV) using the optimum transmitter developed in this paper but employing the ML detector in the receiver. Again, we study a two-user scenario. Due to the complexity of the ML detection involved, the number of available subcarriers in each of the approaches is limited to four in this example. The rest of the simulation environment is similar to that in Example 1. To ensure a fair comparison of all the systems, for system II) using the linear receiver, we generate 200 random channel matrices for each value of the power constraint, and for each of the matrix channel realizations, the optimum sum mutual information and the numbers of subcarriers assigned are calculated from the corresponding algorithm in [33]. This amount of mutual information and the same subcarrier assignment are then applied to obtain the designs of the optimum precoders and receivers, as outlined in Section IV for system I) using the MMSE-DFE receiver, and then, its BER is evaluated. The same precoders (and therefore the same amount of mutual information) are now employed in system IV) equipped with the ML receiver, and its BER is also evaluated. Furthermore, the error performance of system III) is also tested. The overall performance of each of the four different transceiver systems is evaluated by averaging the BER test results over the 200 channel matrix realizations. a) Fig. 4(a) shows the average BER of all the four transceiver systems under different SNRs for the 4-QAM constellation. It can be observed that both systems I) and IV), which employ our optimal transmitter design but are equipped with the MMSE-DFE and ML receivers, respectively, outperform systems II) and III) by a wide margin. This is because the linear receiver, while simple to implement, is well known to be poor in performance. On the other hand, for system III) while it is equipped with the ML detector, its design objective is not targeted at the minimization of receiver errors.It can be observed that the performances of systems I) and IV) are very close. However, it must be pointed out that, while the precoder used in system I) is an optimum design minimizing the average MSE for the system using the MMSE-DFE receiver, it may not be an optimum design in terms of error performance for system IV), which employs the ML

Fig. 4. Comparison of performance of different system detection. (a) 4-QAM. (b) 16-QAM.

receiver. (The design of an optimum multiuser precoder minimizing the error performance for the ML receiver is yet an unsolved problem.) Nevertheless, the comparison of the two systems in Fig. 4(a) validates that the performance of system I) using MMSE-DFE asymptotically approaches that of the system using the ML detector, as indicated by Property 2 in Section V. This property signifies the importance of an optimum precoder design, which may bring the performance of a system using an MMSE-DFE receiver close to that using an ML receiver. b) To examine the effect of the error propagation of the DFE receiver on the performance of our optimal design [see system I)], we repeat the preceding simulations using a larger signal constellation of 16-QAM having 32 subcarriers. (We omit the test of system IV) here due to the complexity of the computation involved in ML detection.) Fig. 4(b) shows the BER performance of our optimum system with the performance of systems II) and III). It can be seen that, while the performances of all the three systems have substantially deteriorated compared with the case of 4-QAM, our optimum design of system I) still maintains a significant gain over the other two systems for the same BER under a high SNR.


VII. C ONCLUSION AND D ISCUSSION In this paper, we have examined the optimum joint design of the precoder and the feedforward and feedback filters of an ISI multiple-access MIMO communication system employing a block-by-block transmission scheme and an MMSE-DF receiver. The employment of the MMSE-DF receiver imposes a given order on the detection of the user signals and also within the symbols each user transmits. The design minimizes the average MSE under fixed-sum Gaussian mutual information and is based on the assumption that CSI is fully known at both the transmitters and the receivers. To facilitate the design for a particular user, the “dual water-filling” solution for a single user has been introduced to obtain a precoder that minimizes the transmission power given a fixed amount of Gaussian mutual information. Analysis of the optimality condition of the MMSE reveals that the sum Gaussian mutual information has to be evenly distributed among the subchannels allocated to all the users. Therefore, given the distribution of the sum Gaussian mutual information among the users, the optimum precoder is obtained by 1) applying the dual water-filling algorithm to find a minimum transmission power precoder for each user; 2) applying the equal-diagonal QRS decomposition arriving at a properly chosen unitary matrix so that, while the minimum transmission power is maintained, a uniform distribution of Gaussian mutual information in each active user subchannel is also achieved. In addition to minimizing the MSE of MMSE-DF detection, the optimal design possesses two further detection properties. • The optimal user- and symbol-detection orders in terms of the SINR follow the natural order of the detection by the MMSE-DF receiver. • The performance of the optimum transceiver asymptotically approaches that of the ML receiver. Although our attention here is focused on a specific design of minimizing the MSE of MMSE-DF detection for a multiuser system, the methodology developed in this paper can be extended to problems Nk having an objective function of the form minTk K n=1 f (rnn,k ) while keeping the constraints k=1 unchanged, provided f (2θ ) is convex with respect to θ. This class of optimization problems has a direct explicit solution obtainable by the technique given in this paper. Thus, the solution strategy depends only on the features of the MMSEDF receiver but does not depend on the specific structure of the objective function f (·). Furthermore, in this paper, the minimization of transmission power for each user is attained by the application of the dual water-filling algorithm subject to the uniform distribution of Gaussian mutual information among the subchannels allocated to all the users. This implies that the overall total transmission power of the system may not be minimized. Therefore, it is conceivable that, in the formulation of the optimum design, we try to seek for a three-way tradeoff between detection MSE, sum mutual information, and transmission power while sacrificing the uniform mutual information distribution among users. The formulation of this approach may be quite complicated. Alternatively, we also note that in complying with the operation

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of the DFE receiver, the detection for the users and their symbols follows a fixed reverse order. It is also conceivable that if this fixed order is relaxed such that the users may be detected in any order, this may result in a lower total transmission power. However, this second approach may also lead to an intractable problem because the power consumption of a user is intricately related to the power consumption of all the other users. Thus, how this total power may be incorporated into the problem and be optimized together with the other objectives remains yet unresolved. A PPENDIX A P ROOF OF T HEOREM 1 The Lagrangian function for (23) is L(Tk ) = tr(TH k Tk ) − H −1 H Σ H T ), where µ is the Lagrange µ log det(I + TH k k k k k multiplier. Requiring the gradient of L(Tk ) to vanish for maximum, we have −1 −1 H H −1 = 0. Tk − µ HH k Σk Hk Tk I + Tk Hk Σk Hk Tk (30) yields Left multiplying both sides of (30) by TH k −1 H H −1 = µI. (31) TH k Tk + µ I + Tk Hk Σk Hk Tk Perform singular value decomposition Tk = Wk Γk VkH , where Wk is an M × k columnwise orthonormal matrix, Vk is an k × k unitary matrix, and Γk is a diagonal matrix Γk = diag(γk1 , γk2 , . . . , γkk ) with γk1 ≥ γk2 ≥ · · · ≥ γkk > 0. Substituting this decomposition into (31) results in H H −1 −1

H = µI. ΓH k Γk + µ I + Γk Wk Hk Σk Hk Wk Γk (32) −1 Σ H W must be diagonal with nonFrom (32), WkH HH k k k k −1 negative diagonal elements. Let WkH HH k Σk Hk Wk = Θk = diag(θk1 , θk2 , . . . , θkk ), θkn ≥ 0. Then, (32) can be rewritten H −1 = µI. Since both Γk and Θk as ΓH k Γk + µ(I + Γk Θk Γk ) are diagonal matrices, we can easily equate the diagonal elements, resulting in % −1 γkn = µ − θkn . (33) The Gaussian mutual information constraint can now be expressed in terms of γkn and θkn so that det(I + ΓH k Θk Γk ) = k 2 Ik (1 + γ θ ) = 2 . Combining this with (33) yields kn n=1 kn k Ik 1/k µ = (2 / n=1 θkn ) , resulting in the power in the mth subchannel given by 1/k 2Ik 2 −1 γkm = k − θkm . (34) n=1 θkn We observe that θk1 , θk2 , . . . , θkk must satisfy two constraints. 1) The power of each subchannel in (34) is positive; therefore, we have 1/Nk Nk n=1 θkn for m = 1, . . . , Nk . (35) θkm > 2Ik

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2) Since θk1 , . . . , θkk are the eigenvalues of a channel matrix obtained by having only part of the subchannels −1 of original channel matrix HH k Σk Hk whose eigenvalue matrix is Λ = diag(λk1 , . . . , λkM ), then [41] λk1 ≥ θk1 ≥ λk2 ≥ θk2 ≥ · · · ≥ λkk ≥ θkk .

(36)

Thus, the total power of transmission Pk (θk1 , . . . , θkk ) in all the Nk subchannels can be written as 1/Nk Nk 2Ik −1 − θkm . Pk (θk1 , . . . , θkk ) = Nk Nk θ n=1 kn m=1 (37) The proof of Theorem 1 is thus reduced to finding the optimal {θk , k = 1, . . . , k } that minimizes Pk (θk1 , . . . , θkk ) subject to the aforementioned two constraints of (35) and (36). To do this, we present two important properties of the total power function of (37), the proofs of which are straightforward and can be found in [36]. A) Pk (θk1 , . . . , θk ) ≥ Pk (λk1 , . . . , λk ), for < M . Equality holds when λkm = θkm , m = 1, 2, . . . , . B) Total power function Pk is a decreasing function with respect to , and Pk is minimized when Nk = k . From Property A), we can see that, for θkm = λkm , the transmission power is minimum for all k subchannels. To render θkm = λkm , we can choose the left singular-vector matrix ˜ k , where U ˜ k is an M × k matrix of Tk such that Wk = U consisting of the first k columns of Uk , the unitary eigenvector −1 H H −1 matrix HH k Σk Hk , resulting in Wk Hk Σk Hk Wk = Θk = diag(λk1 , . . . , λkk ). For this optimum choice of θkm , (34), representing the transmission power of subchannel m, is the same as (25), which is the minimum transmission power for subchannel m, as stated in Theorem 1. Now, applying Property B, if we choose the number of symbols Nk transmitted to be the same as the number of subchannels k used by user k, the transmitted power Pk for user k is minimized. This completes the proof of Theorem 1. A PPENDIX B P ROOF OF P ROPERTY 2 Noise ν is white for the multiuser MIMO system. From the expression of G in Section II-B, we have HH H = (G − I). Now, consider two different signal vectors x = [x1 , . . . , xN ]T and x = [x1 , . . . , xN ]T . If xk = xk for k = 2, . . . , N , but x1 = x1 , then 2 (x−x )H HH H(x−x ) = (x−x )H G(x−x )−|x1 −x1 | 2 = [R]21 −1 |x1 − x1 | . (38) Hence, by taking the minima of both sides of (38), we obtain d2Fr (H) ≤ min x,x ∈X ([R]21 − 1)|x1 − x1 |2 = x=x

¯

(2IG − 1)d2min (X ). The last equality is because I¯G = (1/N ) log[R]2n = log[R]2n , since all [R]n are equal. This leads to ¯

d2Fr (H) ≤ (2IG − 1)d2min (X ).

(39)

On the other hand, we note that (x − x )H HH H(x − x ) = (x − x )H G(x − x ) − x − x 2 &2 &   & N & & &N & rij (xj − xj )&& − x − x 2  . = & & & i=1 j=i

(40)

For x = x , let k be an integer such that xi = xi , for i > k, but xk = xk . Then, from (40), and that R is an upper triangular matrix, we have (x − x )H HH H(x − x ) & &2   & k & & k & & = rij xj − xj && − x − x 2  & & & i=1 j=i ≥ [R]21 − 1 |xk − xk |2 − x − x 2 ¯ ≥ (2IG − 1)d2min (X ) − x − x 2 . Taking the minima of both sides of (41) yields ¯ d2Fr (H) ≥ (2IG − 1)dmin (X ) − x − x 2max .

(41)

(42)

Since constellation X is finite, x 2max is bounded, and as a result of (42), we obtain d2Fr (H) ≥ d2min (X ). ¯ I¯G →∞ 2IG − 1 lim

Comparing (39) and (43) directly leads to the result.

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Jian-Kang Zhang (SM’09) received the B.S. degree in information science (mathematics) from Shaanxi Normal University, Xi’an, China, in 1983, the M.S. degree in information and computational science (mathematics) from Northwest University, Xi’an, in 1988, and the Ph.D. degree in electrical engineering from Xidian University, Xi’an, in 1999. He is currently an Assistant Professor with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada. He has held research positions with McMaster University and Harvard University, Cambridge, MA. His research interests include multirate filterbanks, wavelet and multiwavelet transforms and their applications, and number theory transform and their applications in signal processing. His current research focuses on random matrices, channel capacity, and coherent and noncoherent multi-input–multi-output communication systems. Dr. Zhang currently serves as an Associate Editor of the IEEE SIGNAL PROCESSING LETTERS and the Journal of Electrical and Computer Engineering. He is the coauthor of the paper that received the IEEE Signal Processing Society Best Young Author Award in 2008.

Kon Max Wong (F’02) received the B.Sc.(Eng.), D.I.C., Ph.D., and D.Sc.(Eng.) degrees from the University of London, London, U.K., in 1969, 1972, 1974, and 1995, respectively, all in electrical engineering. In 1969, he joined the Transmission Division, Plessey Telecommunications Research Ltd., Poole, U.K. In October 1970, he was on leave from Plessey pursuing postgraduate studies and research at Imperial College of Science and Technology, London. In 1972, he rejoined Plessey as a Research Engineer and worked on digital signal processing and signal transmission. In 1976, he joined the Department of Electrical Engineering, Technical University of Nova Scotia, Halifax, NS, Canada, and in 1981, he moved to McMaster University, Hamilton, ON, Canada, where he has been a Professor since 1985 and served as the Chairman of the Department of Electrical and Computer Engineering during 1986–1987, 1988–1994, and 2003–2008. He was on leave as a Visiting Professor with the Department of Electronic Engineering, The Chinese University of Hong Kong, Kowloon, Hong Kong, from 1997 to 1999. He currently holds the Canada Research Chair in Signal Processing at McMaster University. He has published more than 240 papers. His research interest is in signal processing and communication theory. Prof. Wong is a Fellow of the Institution of Electrical Engineers, the Royal Statistical Society, the Institute of Physics, the Canadian Academy of Engineering, and the Royal Society of Canada. He was an Associate Editor of the IEEE TRANSACTION ON SIGNAL PROCESSING during 1996–1998 and served as the Chair of the Sensor Array and Multichannel Signal Processing Technical Committee of the IEEE Signal Processing Society (SPS) during 2002–2004. He was the recipient of the IEE Overseas Premium for the Best Paper in 1989 and is also the coauthor of the papers that received the IEEE SPS Best Young Author Award in 2006 and 2008. He was the recipient of the Alexander von Humboldt Research Award in 2010 and the McMaster Faculty of Engineering Research Achievement Award in 2011.

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Wenwen Jiang received the B.Eng. degree in electronic information engineering from the University of Science and Technology Beijing, Beijing, China, and the M.A.Sc. degree in electrical and computer engineering from McMaster University, Hamilton, ON, Canada. She was a Research Assistant while completing the master’s degree at McMaster. She is currently a Railway Signal Designer with Hatch Mott MacDonald, Mississauga, ON. Her research areas are in multiuser transceiver designs, channel capacity, and multi-input–multi-output communication systems.

Alex B. Gershman (M’97–SM’98–F’06) received the Diploma and Ph.D. degrees in radiophysics from Nizhny Novgorod State University, Nizhny Novgorod, Russia, in 1984 and 1990, respectively. From 1984 to 1999, he held several full-time and visiting research appointments in Russia, Switzerland, and Germany. In 1999, he joined the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada, where he became a Professor in 2002. Since April 2005, he has been with the Technische Universität Darmstadt, Darmstadt, Germany, as a Professor of communication systems. His research interests are in the area of signal processing and communications, with primary emphasis on array processing, statistical signal processing; beamforming; multiantenna, multiuser, and cooperative communications; and estimation and detection theory. Prof. Gershman was a General Cochair of the First IEEE Workshop on Computational Advances in Multi-Sensor Adaptive Processing, Puerto Vallarta, Mexico, December 2005; a Technical Cochair of the Fourth IEEE Sensor Array and Multichannel Signal Processing Workshop, Waltham, MA, June 2006; a Tutorial Chair of the European Signal Processing Conference, Florence, Italy, September 2006; the Chair of the Sensor Array and Multichannel Technical Committee of the IEEE Signal Processing Society (SPS) during 2007–2008; and a General Cochair of the Fifth IEEE Sensor Array and Multichannel Signal Processing Workshop, Darmstadt, July 2008. He was an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING during 1999–2005 and the Editor-in-Chief of the IEEE SIGNAL PROCESSING LETTERS during 2006–2008. He has received several awards, including the 2000 Premier’s Research Excellence Award, Ontario, Canada; the 2001 Wolfgang Paul Award from the Alexander von Humboldt Foundation, Germany; the 2002 Young Explorers Prize from the Canadian Institute for Advanced Research; the IEEE Aerospace and Electronic Systems Society Barry Carlton Award for the Best Paper published in 2004; and the 2004 IEEE SPS Best Paper Award. He is also the coauthor of the paper that received the 2005 IEEE SPS Young Author Best Paper Award.