Filter-And-Forward Relay Design for MIMO-OFDM Systems
aa r X i v : . [ c s . I T ] O c t SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 29, 2018 1
Filter-And-Forward Relay Design forMIMO-OFDM Systems
Donggun Kim,
Student Member, IEEE,
Youngchul Sung † , Senior Member, IEEE ,and Jihoon Chung,
Student Member, IEEE,
Abstract
In this paper, the filter-and-forward (FF) relay design for multiple-input multiple-output (MIMO)orthogonal frequency-division multiplexing (OFDM) systems is considered. Due to the consideredMIMO structure, the problem of joint design of the linear MIMO transceiver at the source and thedestination and the FF relay at the relay is considered. As the design criterion, the minimization ofweighted sum mean-square-error (MSE) is considered first, and the joint design in this case is approachedbased on alternating optimization that iterates between optimal design of the FF relay for a given setof MIMO precoder and decoder and optimal design of the MIMO precoder and decoder for a given FFrelay filter. Next, the joint design problem for rate maximization is considered based on the obtainedresult regarding weighted sum MSE and the existing result regarding the relationship between weightedMSE minimization and rate maximization. Numerical results show the effectiveness of the proposed FFrelay design and significant performance improvement by FF relays over widely-considered simple AFrelays for MIMO-ODFM systems.
Index Terms † Corresponding authorThe authors are with the Dept. of Electrical Engineering, KAIST, Daejeon 305-701, South Korea. E-mail: { dg.kim@,ysung@ee., and j.chung@ } kaist.ac.kr. This research was funded by the MSIP(Ministry of Science, ICT & Future Planning),Korea in the ICT R&D Program 2013. October 29, 2018 DRAFT SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 29, 2018
Linear relay, filter-and-forward, weighted mean-square-error, MIMO-OFDM systems, quadraticallyconstrained quadratic program
I. I
NTRODUCTION
Recently, the filter-and-forward (FF) relaying scheme has gained an interest from the researchcommunities as an alternative relaying strategy due to its capability of performance improvementover simple AF relays and still low complexity compared with other relaying strategies such asdecode-and-forward (DF) and compress-and-forward (CF) schemes [1]–[7]. It is shown that theFF scheme can outperform the AF scheme considerably. However, most of the works regardingthe FF relay scheme were conducted for single-carrier systems [1], [3], [5], [6]. Recently, Kim etal. considered the FF relay design for single-input and single-output (SISO) OFDM systems [7],[8], but their result based on worst subcarrier signal-to-noise ratio (SNR) maximization or directrate maximization is not easily extended to the MIMO case since SNR is not clearly defined forMIMO channels and furthermore in the MIMO case the design of the MIMO precoder at thesouce and the MIMO decoder at the destination should be considered jointly with the FF relaydesign. Thus, although there exists vast literature regarding the relay design for MIMO-OFDMsystems in the case that the relay performs OFDM processing ∗ [9]–[14], not many results areavailable for the FF relay design for MIMO-OFDM transmission, which is the current industrystandard for the physical layer of many commercial wireless communication systems.In this paper, we consider the FF relay design for MIMO-OFDM systems. In the MIMO case,the FF relay should not be designed alone without considering the MIMO precoder and decoderat the source and the destination. Thus, we consider the problem of joint design of the linearMIMO transceiver at the source and the destination and the FF relay at the relay. As mentioned,in the MIMO case, it is not easy to use SNR as the design metric as in the SISO case [7]. Thus, ∗ In this case, each subcarrier channel is independent and we only need to consider a single flat MIMO channel.
DRAFT October 29, 2018 we approach the design problem based on the tractable criterion of minimization of weightedsum MSE first and then consider the rate-maximizing design problem based on the equivalencerelationship between rate maximization and weighted MSE minimization with a properly chosenweight matrix [15]–[19]. We tackle the complicated joint design problems by using alternatingoptimization, which enables us to exploit the existing results for the MIMO precoder and decoderdesign when all channel information is given. The proposed alternating optimization is basedon the iteration between optimal design of the FF relay for a given set of MIMO precoder anddecoder and optimal design of the MIMO precoder and decoder for a given FF relay filter. Whilethe linear MIMO transceiver design for a given FF relay filter can be addressed by existing resultse.g. [15], the problem of optimal design of the FF relay for a given MIMO transceiver is newlyformulated based on the block circulant matrix theorem and reparameterization. It is shown thatthe FF relay design problem for a given MIMO transceiver reduces to a quadratically constrainedquadratic program (QCQP) problem and a solution to this QCQP problem is proposed based onconversion to a semi-definite program (SDP). Numerical results show the effectiveness of theproposed FF relay design and significant performance improvement by FF relays over widely-considered simple AF relays, and suggests that it is worth considering the FF relaying schemefor MIMO-OFDM systems over the AF scheme with a certain amount of complexity increase.
A. Notation and Organization
In this paper, we will make use of standard notational conventions. Vectors and matrices arewritten in boldface with matrices in capitals. All vectors are column vectors. For a matrix X , X ∗ , X T , X H , tr ( X ) , and X ( i, j ) indicate the complex conjugate, transpose, conjugate transpose, trace,and ( i, j ) -element of X , respectively. X (cid:23) and X ≻ mean that X is positive semi-definiteand that X is strictly positive definite, respectively. I n stands for the identity matrix of size n (thesubscript is omitted when unnecessary), I m × n denotes the first m × n submatrix of I , and m × n denotes a m × n matrix of all zero elements (the subscript is omitted when unnecessary). The October 29, 2018 DRAFT SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 29, 2018 notation blkToeplitz ( F , N ) indicates a N A × ( N + L f − B block Toeplitz matrix with N rowblocks and [ F , , · · · , ] as its first row block, where F = [ F , F , · · · , F L f − ] is a row blockcomposed of A × B matrices { F k } ; diag ( X , X , · · · , X n ) means a (block) diagonal matrixwith diagonal entries X , X , · · · , X n . The notation x ∼ CN ( µ , Σ ) means that x is complexcircularly-symmetric Gaussian distributed with mean vector µ and covariance matrix Σ . E {·} denotes the expectation. ι = √− .The remainder of this paper is organized as follows. The system model is described inSection II. In Section III, the joint transceiver and FF relay design problems for minimizingthe weighted sum MSE and for maximizing the data rate are formulated and solved by usingconvex optimization theory and existing results. The performance of the proposed design methodsis investigated in Section IV, followed by the conclusion in Section V.II. S YSTEM M ODEL
We consider a point-to-point MIMO-OFDM system with a relay, as shown in Fig. 1, where thesource has N t transmit antennas, the relay has M r receive antennas and M t transmit antennas,and the destination has N r receive antennas. The source and the destination employ MIMO-OFDM modulation and demodulation with N subcarriers, respectively, as in a conventionalMIMO-OFDM system. However, we assume that the relay is a full-duplex † FF relay equippedwith a bank of M t M r finite impulse response (FIR) filters with order L g , i.e., the relay performsFIR filtering on the incoming signals received at the M r receive antennas at the chip rate ‡ ofthe OFDM modulation and transmits the filtered signals instantaneously through the M t transmitantennas to the destination without OFDM processing. Thus, the FF relay can be regarded as anextension of an amplify-and-forward (AF) relay and as an additional frequency-selective fading † In the case of half-duplex, the problem can be formulated similarly. ‡ The FIR filtering is assumed to be performed at the baseband. Thus, up and down converters are necessary for FF operationand one common local oscillator (LO) at the relay is sufficient.
DRAFT October 29, 2018 channel between the source and the destination. We assume that there is no direct link betweenthe source and the destination and that the source-to-relay (SR) and relay-to-destination (RD)channels are multi-tap filters with finite impulse responses and their state information is knownto the system. PSfrag replacements s s s n V V V n x x x n A A A n ˆ s ˆ s ˆ s n IDFTIDFTIDFT DFTDFTDFTP/SP/SP/S &&&&&&CPCPCP S/PS/PS/PCPRCPRCPRFF relayFIR filter ... ......... ..................... ... .................. ......... ............ ...... N t M r M t N r Fig. 1: System modelThe considered baseband system model is described in detail as follows. At the source,a block of N input data vectors of size Γ × , denoted as { s n = [ s n [1] , s n [2] , · · · , s n [Γ]] T , n = 0 , , · · · , N − } , is processed for one OFDM symbol time. Here, s n is the input datavector for the effective parallel flat MIMO channel at the n -th subcarrier provided by MIMO-OFDM processing and Γ ≤ min( N t , M r , M t , N r ) is the number of data streams for the effectiveflat MIMO channel at each subcarrier. We assume that each data symbol is a zero-mean in-dependent complex Gaussian random variable with unit variance, i.e., s n [ k ] ∼ CN (0 , for k = 1 , , · · · , Γ and n = 0 , , · · · , N − . Let the concatenated data vector be denoted by s = [ s TN − , s TN − , · · · , s T ] T . Although MIMO precoding can be applied to the concatenated vector s , such processing is complexity-wise inefficient and thus we assume that MIMO precoding isapplied to the effective flat MIMO channel of each subcarrier separately, as in most practicalMIMO-OFDM systems, with a precoding matrix V n for the n -th subcarrier MIMO channel. The October 29, 2018 DRAFT SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 29, 2018
MIMO precoded N symbols for each transmit antenna are collected and processed by inversediscrete Fourier transform (IDFT). By concatenating all IDFT symbols for all transmit antennas,we have the overall time-domain signal vector x , given by x = ( W N ⊗ I N t ) Vs (1)where V = diag ( V N − , V N − , · · · , V ) (2) W N ( k + 1 , l + 1) = 1 √ N e ι πklN , k, l = 0 , , · · · , N − , (3)and x is cyclic-prefix attached and transmitted. The cyclic prefix attached signal vector x cp canbe expressed as x cp = I N I N cp ⊗ I N t | {z } ∆ = T cp x , (4)where N cp is the cyclic prefix length, and in (4) is an N cp × ( N − N cp ) all-zero matrix. Weassume that the length of the overall FIR channel between the source and the destination is notlarger than that of the OFDM cyclic prefix, i.e., N cp ≥ L f + L r + L g − , where L f , L r , and L g denote the SR channel length, the FIR filter order at the relay, and the RD channel length,respectively.The transmitted signal x cp passes through the SR channel, the relay FIR filter, and theRD channel; is corrupted by white Gaussian noise; and is received at the destination. Then,the transmitted signal vector at the relay and the received signal vector at the destination arerespectively given by y t = RFx cp + Rn r and y d = GRFx cp + GRn r + n d , (5) DRAFT October 29, 2018 where y d = (cid:2) y Td,N − , y Td,N − , · · · , y Td, (cid:3) T , (6) y t = h y Tt,N − , y Tt,N − , · · · , y Tt, , y Tt, − , · · · , y Tt, − L g +1 i T , (7) x cp = h x TN − , x TN − , · · · , x T , x T − , · · · , x T − L g − L r − L f +3 i T , (8) n r = h n Tr,N − , n Tr,N − , · · · , n Tr, , n Tr, − , · · · , n Tr, − L g − L r +2 i T , (9) n d = (cid:2) n Td,N − , n Td,N − , · · · , n Td, (cid:3) T , (10) G = blkToeplitz ( G , N ) , R = blkToeplitz ( R , N + L g − , F = blkToeplitz ( F , N + L g + L r − , (11) G = [ G , G , · · · , G L g − ] , R = [ R , R , · · · , R L r − ] , F = [ F , F , · · · , F L f − ] . (12) Here, y d,k and n d,k are N r × vectors; y t,k is a M t × vector; x k is a N t × vector; n r,k is a M r × vector; G k is a N r × M t matrix; R k is a M t × M r matrix; and F k is a M r × N t matrix.The entries of the noise vectors, n r,k and n d,k , are independently and identically distributed (i.i.d)Gaussian with n r,k [ i ] i.i.d. ∼ CN (0 , σ r ) and n d,k [ i ] i.i.d. ∼ CN (0 , σ d ) . Then, the (cyclic-prefix portionremoved) N -point vector DFT of the received vector at the destination is given by y = ( W HN ⊗ I N r ) GRFx cp + ( W HN ⊗ I N r ) GRn r + ( W HN ⊗ I N r ) n d , = ( W HN ⊗ I N r ) GRFT cp ( W N ⊗ I N t ) Vs + ( W HN ⊗ I N r ) GRn r + ( W HN ⊗ I N r ) n d , = ( W HN ⊗ I N r ) H c ( W N ⊗ I N t ) Vs + ( W HN ⊗ I N r ) GRn r + ( W HN ⊗ I N r ) n d , (13) = DVs + ( W HN ⊗ I N r ) GRn r + ( W HN ⊗ I N r ) n d , (14)where y = [ y TN − , y TN − , · · · , y T ] T , y n is a N r × received signal vector at the n -th subcarrier, W HN is the normalized DFT matrix of size N , H c is a N N r × N N t block circulant matrixgenerated from the block Toeplitz overall channel matrix GRF from the source to the destination,and D = ( W HN ⊗ I N r ) H c ( W N ⊗ I N t ) is a block diagonal matrix generated by the block circulantmatrix theorem described in the next section. The n -th subcarrier output of the N -point vectorDFT is processed by a linear receiver filter U n of size Γ × N r to yield an estimate of s n . The October 29, 2018 DRAFT SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 29, 2018 overall receiver processing for all the subcarrier channels can be expressed as ˆ s = UDVs + U ( W HN ⊗ I N r ) GRn r + U ( W HN ⊗ I N r ) n d , (15)where U = diag ( U N − , U N − , · · · , U ) . A. Derivation of the subcarrier channel and mean square error
To facilitate the optimization problem formulation in the next section, we need to derive anexplicit expression for the received signal vector y n , n = 0 , , · · · , N − , at the n -th subcarrier. Lemma 1: If H c is a block circulant matrix with K = [ H , H , · · · , H N − ] as its first rowblock, then it is block-diagonalizable as Λ b = ( W HN ⊗ I N r ) H c ( W N ⊗ I N t ) where Λ b is a block diagonal matrix defined as Λ b = K ( √ N w HN − ⊗ I N t ) T . . . K ( √ N w H ⊗ I N t ) T with √ N w Hk denoting the − ( k − N ) -th row of the DFT matrix √ N W HN , and K ( √ N w Hk ⊗ I N t ) T = N − X n =0 H n e − ι π n ( N − k − N . Proof : In [20], it is shown that a circulant matrix can be diagonalized by a DFT matrix. Thiscan easily be extended to the block circulant case. (cid:3)
By lemma 1, to derive the diagonal blocks of D in (14), we only need to know the firstrow block of H c in (13). Let the first row block of the RD channel matrix G be denoted by a N r × M t ( N + L g − matrix e G = [ G , G , · · · , G L g − , , · · · , ] . Then, the first row block ofthe effective channel filtering matrix GRF is given by e GRF . Note that the cyclic prefix addingand removing operations make
GRF into the block circulant matrix H c by truncating out the DRAFT October 29, 2018 blocks of
GRF outside the first N × N blocks and by moving the lower ( L g + L r + L f − × ( L g + L r + L f − blocks of the truncated part to the lower left of the untruncated N × N block matrix, where each block is a N r × N t matrix. Therefore, the first row block e H c of H c issimply the first N blocks of e GRF , given by e H c = e GRFT and T = I NN t ( L f + L r + L g − N t × NN t (16)where T is a truncation matrix for truncating out the remaining blocks of e GRF except the first N column blocks. By using the first row block e H c and Lemma 1, we obtain the diagonal blocksof D as D = diag ( e H c ( √ N w HN − ⊗ I N t ) T , e H c ( √ N w HN − ⊗ I N t ) T , · · · , e H c ( √ N w H ⊗ I N t ) T ) . (17)Based on (16) and (17), the received signal vector on the n -th subcarrier at the destination isexpressed as y n = √ N e GRFT W Tt,n V n s n + W r,n GRn r,n + W r,n n d,n , (18) = ˆ y n + z n , (19)where W t,n = w Hn ⊗ I N t , W r,n = w Hn ⊗ I N r , ˆ y n = √ N e GRFT W Tt,n V n s n , and z n = W r,n GRn r,n + W r,n n d,n . This received signal vector y n is filtered by the receive filter U n and its output is givenby ˆ s n = √ N U n e GRFT W Tt,n V n s n + U n W r,n GRn r,n + U n W r,n n d,n . (20) October 29, 2018 DRAFT0 SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 29, 2018
Finally, the weighted MSE between s n and ˆ s n is given bytr ( Θ n M n ) = tr (cid:0) Θ n E (cid:8) (ˆ s n − s n )(ˆ s n − s n ) H (cid:9)(cid:1) , = tr (cid:0) Θ n E (cid:8) ( U n y n − s n )( U n y n − s n ) H (cid:9)(cid:1) , = tr (cid:0) Θ n (cid:0) E { U n y n y Hn U Hn } − E { s n y Hn U Hn } − E { U n y n s Hn } + E { s n s Hn } (cid:1)(cid:1) , = tr (cid:0) Θ n E { U n ˆ y n ˆ y Hn U Hn } (cid:1) + tr (cid:0) Θ n E { U n z n z Hn U Hn } (cid:1) − tr (cid:0) Θ n E { s n ˆ y H U Hn } (cid:1) − tr (cid:0) Θ n E { U n ˆ y n s Hn } (cid:1) + tr (cid:0) Θ n E { s n s Hn } (cid:1) , (21)where M n ∆ = E (cid:8) (ˆ s n − s n )(ˆ s n − s n ) H (cid:9) is the MSE matrix at the n -th subcarrier and Θ n is a Γ × Γ diagonal positive definite weight matrix.III. P ROBLEM F ORMULATION AND P ROPOSED D ESIGN M ETHOD
In this section, we consider optimal design of the FIR MIMO relay filter { R , R , · · · , R L r − } and the linear precoders and decoders { V n , U n , n = 0 , , · · · , N − } . Among several optimalitycriteria, we first consider the minimization of the weighted sum mean-square-error (MSE) forgiven weight matrices, and then consider the rate maximization via the weighted sum MSEminimization based on the fact that the rate maximization for MIMO channels is equivalent tothe weighted MSE minimization with properly chosen weight matrices { Θ n } [15]. (Here, thesummation is across the subcarrier channels.) The first problem is formally stated as follows. Problem 1:
For given weight matrices { Θ n } , SR channel F , RD channel G , FF relay filterorder L r , maximum source transmit power P s,max , and maximum relay transmit power P r,max ,optimize the transmit filter V = diag ( V , · · · , V N − ) , the relay filter R , and the receive filter U = diag ( U , · · · , U N − ) in order to minimze the weighted sum MSE: min V , R , U N − X n =0 tr ( Θ n M n ) s.t. tr ( VV H ) ≤ P s, max and tr ( y t y Ht ) ≤ P r, max . (22)Note that Problem 1 is a complicated non-convex optimization problem, which does not yieldan easy solution. To circumvent the difficulty in joint optimization, we approach the problem DRAFT October 29, 20181 based on alternating optimization. That is, we first optimize the relay filter for given transmitand receive filters under the power constraints. Then, with the obtained relay filter we optimizethe transmit and receive filters. Problem 1 is solved in this alternating fashion until the iterationconverges. A solution to each step is provided in the following subsections.
A. Relay Filter Optimization
Whereas the linear precoder V n and decoder U n are applied to each subcarrier channelseparately, the relay filter affects all the subcarrier channels simultaneously since the FF relaydoes not perform OFDM processing. Here we consider the relay filter optimization for giventransmit and receive filters, and the problem is formulated as follows. Problem 1-1:
For given weight matrices { Θ n } , SR channel F , RD channel G , FF relay filterorder L r , transmit filter V , receive filter U , and maximum relay transmit power P r,max , optimizethe relay filter R in order to minimize the weighted sum MSE: min R N − X n =0 tr ( Θ n M n ) s.t. tr ( y t y Ht ) ≤ P r, max . (23)To solve Problem 1-1, we first need to express each term in (23) as a function of the designvariable R . Note that the relay block-Toeplitz filtering matrix R is redundant since the truedesign variable R is embedded in the block Toeplitz structure of R . (See (11).) Hence, taking R as the design variable directly is inefficient and we need reparameterization of the weightedMSE in terms of R . This is possible through successive manipulation of the terms constructingthe weight MSE shown in (21). First, using similar techniques to those used in [7], we can October 29, 2018 DRAFT2 SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 29, 2018 express the first term of (21) in terms of R as follows:tr ( Θ n E { U n ˆ y n ˆ y Hn U Hn } )= N tr ( Θ n E { U n e GRFT W Tt,n V n s n s Hn V Hn W ∗ t,n T H F H R H e G H U Hn } ) , ( a ) = N tr ( V Hn W ∗ t,n T H F H R H e G H U Hn Θ n U n e GRFT W Tt,n V n E (cid:8) s n s Hn (cid:9) ) , = N tr ( V Hn W ∗ t,n T H F H R H e G H U Hn Θ n U n e GRFT W Tt,n V n ) , = N tr (cid:0) Θ / n U n e GR FT W Tt,n V n V Hn W ∗ t,n T H F H | {z } =: K n R H e G H U Hn Θ / n (cid:1) , ( b ) = N h vec ( R T e G T U Tn Θ / n ) i T K n h vec ( R T e G T U Tn Θ / n ) i ∗ , ( c ) = N (cid:2) vec ( R T ) (cid:3) T ( Θ / n U n e G ⊗ I Q ) T K n ( Θ / n U n e G ⊗ I Q ) ∗ (cid:2) vec ( R T ) (cid:3) ∗ , ( d ) = N r T E ( Θ / n U n e G ⊗ I Q ) T K n ( Θ / n U n e G ⊗ I Q ) ∗ E H r ∗ , = r H Q ,n r , (24)where K n = I Γ ⊗ K n ; I Q = I ( N + L r + L g − M r ; r = vec ( R T ); Q ,n = N E ∗ ( Θ / n U n e G ⊗ I Q ) H K ∗ n ( Θ / n U n e G ⊗ I Q ) E T ; and E is defined in Appendix A. Here, (a) holds due to tr ( UBC ) = tr ( CUB ) ; (b) holdsdue to tr ( XK n X H ) = vec ( X T ) T K n vec ( X T ) ∗ ; (c) holds due to the kronecker product identity,vec ( IBC ) = ( C T ⊗ I ) vec ( B ) ; and (d) is obtained because R = blkToeplitz ( R , N + L g − andvec ( R T ) = E T r . In a similar way, the remaining terms of (21) and the relay power constraintcan also be represented as functions of the design variable r . That is, the second term of (21) DRAFT October 29, 20183 can be rewritten astr ( Θ n E (cid:8) U n z n z Hn U Hn (cid:9) )= tr (cid:0) Θ n E (cid:8) U n W r,n GRn r,n n Hr,n R H G H W Hr,n U Hn (cid:9) + Θ n E (cid:8) U n W r,n n d,n n Hd,n W Hr,n U Hn (cid:9)(cid:1) , = tr ( R H G H W Hr,n U Hn Θ n U n W r,n GR E (cid:8) n r,n n Hr,n (cid:9) ) + tr ( Θ n U n W r,n E (cid:8) n Hd,n n d,n (cid:9) W Hr,n U Hn ) , = σ r tr ( R H G H W Hr,n U Hn Θ n U n W r,n G | {z } =: M n R ) + σ d tr ( Θ n U n W r,n W Hr,n U Hn ) , = σ r tr ( R H M n R ) + σ d tr ( Θ n U n ( w Hn ⊗ I N r )( w n ⊗ I N r ) U Hn ) , ( a ) = σ r vec ( R ) H M n vec ( R ) + σ d tr ( Θ n U n ( w Hn w n ⊗ I N r ) U Hn ) , ( b ) = σ r r H E M n E H r + σ d tr ( Θ n U n U Hn ) , = r H Q ,n r + c n , (25)where M n = I ( N + L g + L r − M r ⊗ M n , Q ,n = σ r E M n E H , c n = σ d tr ( Θ n U n U Hn ) , and E is defined in Appendix A. Here, (a) follows from the kronecker product identity ( UB ⊗ CD ) = ( U ⊗ C )( B ⊗ D ) , and (b) is obtained due to vec ( R ) H = r H E . The third term of (21)can be rewritten astr (cid:0) Θ n E { s n ˆ y Hn U Hn } (cid:1) = √ N tr (cid:16) Θ n E (cid:8) s n s Hn (cid:9) V Hn W ∗ t,n T H F H R H e G H U Hn (cid:17) , = √ N tr (cid:16) Θ n V Hn W ∗ t,n T H F H R H e G H U Hn (cid:17) , = √ N tr (cid:16) R H e G H U Hn Θ n V Hn W ∗ t,n T H F H (cid:17) , = √ N vec ( R ) H vec ( e G H U Hn Θ n V Hn W ∗ t,n T H F H ) , = √ N r H E vec ( e G H U Hn Θ n V Hn W ∗ t,n T H F H ) , = r H q n , (26) October 29, 2018 DRAFT4 SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 29, 2018 where q n = √ N E vec ( e G H U Hn Θ n V Hn W ∗ t,n T H F H ) . Finally, the relay transmit power can berewritten as E { tr ( y t y Ht ) } = tr (cid:0) RFT cp ( W N ⊗ I N t ) V E { ss H } V H ( W HN ⊗ I N t ) T cpH F H R H (cid:1) + tr (cid:0) R E { n r n Hr } R H (cid:1) , = tr (cid:0) RFT cp ( W N ⊗ I N t ) VV H ( W HN ⊗ I N t ) T cpH F H R H (cid:1) + σ r tr (cid:0) RR H (cid:1) , = tr R ( FT cp ( W N ⊗ I N t ) VV H ( W HN ⊗ I N t ) T cpH F H + σ r I ) | {z } Π R H , = vec ( R T ) T Π vec ( R T ) ∗ , = r H e Πr , (27)where Π = I ( N + L g − M t ⊗ Π and e Π = E ∗ Π ∗ E T .Based on (24), (25), (26), and (27), the weighted MSE for the n -th subcarrier channel isexpressed as tr ( Θ n M n ) = r H Q n r − r H q n − q Hn r + z n (28)where Q n = Q ,n + Q ,n and z n = c n + tr ( Θ n ) , and Problem 1-1 is reformulated as min r r H Qr − r H q − q H r + z s.t. r H e Πr ≤ P r, max . (29)where Q = P Nn =1 Q n , q = P Nn =1 q n , and z = P Nn =1 z n .The key point of the derivation of (29) is that Problem 1-1 reduces to a quadraticallyconstrained quadratic programming (QCQP) problem with a constraint. It is known that QCQPis NP-hard in general. However, QCQP has been well studied in the case that the number ofconstraints is small. Using the results of [21] and [22], we obtain an optimal solution to Problem1-1 as follows. Let r /t = r , where t ∈ C , and e r = [ r T , t ] T ∈ C ( M t L r M r +1) × . Then, we rewrite DRAFT October 29, 20185 (29) equivalently as min e r e r H B e r s.t. e r H B e r ≤ (30)where B = Q − q − q H z and B = e Π 00 − P r,max . By defining R := e r e r H and removing the rank-one constraint rank ( R ) = 1 , we obtain thefollowing convex optimization problem: min R tr ( B R ) s.t. tr ( B R ) ≤ (31)which is a semi-definite program (SDP) and can be solved efficiently by using the standardinterior point method for convex optimization [23]–[26]. With an additional constraint rank ( R ) =1 , the problem (31) is equivalent to Problem 1-1. That is, if the optimal solution of (31) has rankone, then it is also the optimal solution of Problem 1-1. However, there is no guarantee that analgorithm for solving the problem (31) yields a rank-one solution. In such a case, a rank-onesolution from R can always be obtained by using the rank-one decomposition procedure [22]. B. Transmit and receive filter optimization
Now consider the joint design of the transmit and receive filters { ( V n , U n ) , n = 0 , , · · · , N − } for a given relay FIR filter. Note that when the transmit power P n,max ( ≥ tr ( V n V Hn )) foreach n and the relay filter are given, the problem simply reduces to N independent problemsof designing the transmit filter V n and the receive filter U n for the n -th subcarrier MIMOchannel for n = 0 , · · · , N − , as in typical MIMO-OFDM systems. This is because we get anindependent MIMO channel per subcarrier owing to MIMO-OFDM processing. However, we October 29, 2018 DRAFT6 SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 29, 2018 have an additional freedom to distribute the total source transmit power P s, max to N subcarrierssuch that P s, max = P N − n =0 P n, max , and should take this overall power allocation into consideration.So, we solve this problem by separating the power allocation problem out and applying theexisting result [15] to this problem. First, consider the transmit and receive filter design problemwhen the transmit power P n,max for each n and the relay filter are given: Problem 1-2:
For given weight matrices { Θ n } , maximum per-subcarrier transmit power P n,max for n = 0 , , · · · , N − , SR channel F , RD channel G , relay filtering matrix R , jointly optimize ( V n , U n ) in order to minimize the weighted MSE at the n -th subcarrier MIMO channel: min V n , U n tr ( Θ n M n ) s.t. tr ( V n V Hn ) ≤ P n, max , for n = 0 , , · · · , N − . (32)Problem 1-2 has already been solved and the optimal transceiver structure for Problem 1-2is available in [15] and [27]. It is shown in [15] that the optimal transmit filter and receivefilter diagonalize the MIMO channel into eigen-subchannels for any weight matrix. Lemma 1and Theorem 1 of [15] provide the optimal transmit filter V n and receive filter U n , and thesolution can be expressed as V n = e V n e P n , where e V Hn e V n = I Γ and e P n is a diagonal matrixwith nonnegative entries s.t. tr ( e P n ) = P n, max determining the transmit power of each of Γ datastreams of the n -th subcarrier MIMO channel. (Please refer to [15].)Note that the solution to Problem 1-2 only optimizes the power allocation within multiple datastreams for each subcarrier when the transmit power is allocated to each subcarrier. Now, considerthe problem of total source power allocation P s, max to subcarrier channels. Here, we exploit the diagonalizing property [15] of the solution to Problem 1-2, take the direction information onlyfor the transmit filter from the solution to Problem 1-2, and apply alternating optimization. Thatis, when the relay filtering matrix R from Problem 1-1 and the normalized transmit filters { e V n } and the receive filters { U n } from Problem 1-2 are given, each subcarrier MIMO channel isdiagonalized into eigen-subchannels. Thus, the effective parallel MIMO channel (20) for the DRAFT October 29, 20187 n -th subcarrier is rewritten as ˆ s n = √ N U n e GRFT W Tt,n V n s n + U n W r,n GRn r,n + U n W r,n n d,n = √ N U n e GRFT W Tt,n e V n e P n s n + U n W r,n GRn r,n + U n W r,n n d,n , (33) = D n e P n s n + U n W r,n GRn r,n + U n W r,n n d,n (34)where D n = diag ( d n [1] , d n [2] , · · · , d n [Γ] ) is obtained from the optimal transceiver ( ˜ V n , U n ) ofProblem 1-2 with each d n [ k ] being a non-negative value [15], and e P n = diag ( p n [1] , p n [2] , · · · , p n [Γ] ).Therefore, we obtain N Γ parallel eigen-subchannels for the overall MIMO-OFDM system as ˆ s n [ k ] = d n [ k ] p n [ k ] s n [ k ] + n n [ k ] , for n = 0 , , · · · , N − and k = 1 , , · · · , Γ , (35)where n n [ k ] = U Hn,k W r,n ( GRn r,n + n d,n ) and U Hn,k is the k -th row of U n . The total power P s, max should now be optimally allocated to these N Γ parallel channels to minimize the weighted sumMSE, where the weighted sum MSE of N Γ parallel eigen-subchannels is derived as N − X n =0 B X k =1 θ nk E {| ˆ s n [ k ] − s n [ k ] | } = N − X n =0 Γ X k =1 θ nk ( d n [ k ] p n [ k ] − d n [ k ] p n [ k ] + c n [ k ]) (36)where c n [ k ] = σ r U Hn,k W r,n GRR H G H W Hr,n U n,k + σ d U Hn,k U n,k + 1 , and θ nk is properly derivedfrom Θ n . Thus, the problem of overall source power allocation to minimize the weight sumMSE subject to the source power constraint is stated as follows. Problem 1-3:
For given any weight matrices { Θ n } , SR channel F , RD channel G , relayfiltering matrix R , maximum source power P s,max = P N − n =0 P n,max , normalized transmit filters { e V n } , and receive filters { U n } , min p n [ k ] N − X n =0 Γ X k =1 θ nk ( d n [ k ] p n [ k ] − d n [ k ] p n [ k ] + c n [ k ]) s.t. N − X n =0 Γ X k =1 p n [ k ] = P s,max . (37)Note that Problem 1-3 is a convex optimization problem with respect to p n [ k ] . The optimalsolution to Problem 1-3 is given in the following proposition: October 29, 2018 DRAFT8 SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 29, 2018
Proposition 1:
The optimal solution to Problem 1-3 is given by p n [ k ] = (cid:18) θ nk d n [ k ] θ nk d n [ k ] + µ (cid:19) + s.t. N − X n =0 Γ X k =1 (cid:18) θ nk d n [ k ] θ nk d n [ k ] + µ (cid:19) = P s,max . (38) Proof : See Appendix BThe solution in Proposition 1 allocates power inverse-proportionally to the power of theeffective channel d n [ k ] in most cases similarly to the method in [27].Now summarizing the results, we propose our method to design the linear transceiver at thesource and the destination and the FF relay filter jointly to minimize the weighted sum MSE,based on alternating optimization solving Problem 1-1, Problem 1-2, and Problem 1-3 iteratively. Algorithm 1:
Given parameters: { Θ n } , F , G , L r , P s,max , and P r,max Step 1: Initialize { e P n } , { e V n } , and { U n } for n = 0 , , · · · , N − . For example, p n [ k ] = P s,max N Γ , e V n = I N t × Γ , and U n = I Γ × N r .Step 2: Solve Problem 1-1 and obtain R .Step 3: Solve Problem 1-2 and obtain { e V n , U n } .Step 4: Solve Problem 1-3 and obtain { e P n } .Step 5: Go to Step 2 and repeat until the change in the weighted sum MSE falls within a giventolerance.The weighted sum MSE is a function of R and { e V n , U n , e P n } denoted by M ( R , e V n , U n , e P n ) .Let X ( i ) denotes the solution at the ( i ) -th step. Then, it is easy to see that M ( R (0) , e V (0) n , U (0) n , e P (0) n ) ≥ M ( R (1) , e V (0) n , U (0) n , e P (0) n ) ≥ M ( R (1) , e V (2) n , U (2) n , e P (0) n ) ≥ M ( R (1) , e V (2) n , U (2) n , e P (3) n ) ≥ · · · ≥ because the optimal solution is obtained at each step and the possible solution set of the currentstep includes the solution of the previous step. In this way, the proposed algorithm converges bythe monotone convergence theorem although it yields a suboptimal solution and the initializationof the algorithm affects its performance. DRAFT October 29, 20189
C. Rate maximization
Now we consider the problem of rate maximization. In general, the rate maximization problemis not equivalent to the MSE minimization problem. However, they are closely related to eachother. The relationship has been studied in [15]–[17]. By using the relationship, the rate max-imization problem for MIMO broadcast channels and MIMO interference-broadcast channelshas recently been considered in [18] and [19]. In the case of the joint design of the FF relayat the relay and the linear transceiver at the source and the destination, the result regardingthe weighted sum MSE minimization in the previous subsection can be modified and used tomaximize the sum rate based on the existing relationship between the weighed MSE and therate. It was shown in [15] that the rate maximization for the n -th subcarrier MIMO channel (33)is equivalent to the weighted MSE minimization when the weight matrix Θ n is set as a diagonalmatrix composed of the eigenvalues of H H Σ − n H , where H = √ N e GRFT W Tt,n is the effectiveMIMO channel matrix and Σ n is the effective noise covariance matrix of the n -th subcarrierMIMO channel (33). (See Lemma 3 of [15].) Exploiting this result, we propose our algorithmto design the linear transceiver and the relay filter to maximize the sum rate below. Algorithm 2:
Given parameters: F , G , L r , P s,max , and P r,max Step 1: Initialize { Θ n } , { e P n } , { e V n } , and { U n } for n = 0 , , · · · , N − . For example, Θ n = I , p n [ k ] = P s,max N Γ , e V n = I N t × Γ , and U n = I Γ × N r .Step 2: Solve Problem 1-1 and obtain R .Step 3: § Solve Problem 1-2 and obtain { e V n , U n , Θ n } .Step 4: Compute { e P n } for the N Γ parallel scalar channels obtained from Step 3 by water-filling.Step 5: Go to Step 2 and repeat until the change in the weighted sum MSE falls within a giventolerance. § When R is given, all the parallel subcarrier MIMO channels are determined and a solution { e V n , U n , Θ n } is given byLemma 1 and Theorem 1 of [15]. October 29, 2018 DRAFT0 SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 29, 2018
Note that the weight matrices { Θ n } in Algorithm 2 are updated in each iteration so that theweighted MSE minimization is equivalent to the rate maximization for an updated relay filter,whereas the weight matrices are fixed over iterations in Algorithm 1.Now consider the complexity of the proposed algorithms. Note that solving Problem 1-2involves N separate small MIMO systems of size N r × N t , and the solution to Problem 1-3(Algorithm 1) and the water-filling power allocation solution (Algorithm 2) are explicitly given.Thus, the main complexity of the proposed algorithms lies in solving Problem 1-1 that requiressolving an SDP problem of size M t M r L g . Due to the existence of fast approximate algorithmsfor solving SDP problems [28], [29], the proposed algorithm is implementable if the number ofiterations for convergence is not so large, which will be seen in Fig. 5. For other practical issuessuch as channel estimation and self-interference caused by full-duplex operation, please see [7].IV. N UMERICAL RESULTS
In this section, we provide some numerical results to evaluate the performance of the proposedFF relay design in Section III. Throughout the simulation, we fixed the number of OFDMsubcarriers as N = 16 with a minimal cyclic prefix covering the overall FIR channel length ineach simulation case. In all cases, each channel tap coefficient of the SR and RD channel matrices, F k and G k , was generated i.i.d according to a Rayleigh distribution, i.e., F k ( i, j ) i.i.d. ∼ CN (0 , σ f ) and G k ( i, j ) i.i.d. ∼ CN (0 , σ g ) , where σ f = σ g = 1 . The SR channel length and the RD channellength were set as L f = L g = 3 , and N t = M r = M t = N r = 2 . The relay and the destinationhad the same noise power σ r = σ d = 1 , and the source transmit power was 20 dB higher than thenoise power, i.e., P s,max = 100 . (From here on, all dB power values are relative to σ r = σ d = 1 .)We first evaluated the MSE performance of the proposed FF relay design method, Algorithm1, to minimize the sum MSE subject to a source power constraint and a relay power constraint.Figs. 2 and 3 show the resulting sum MSE over all subcarriers. For the curves in the figures, DRAFT October 29, 20181 S u m M SE L r = 1 (AF) L r = 2 L r = 3 L r = 5 L r = 9 L r = 11 Fig. 2: Sum MSE versus FF relay transmit power.200 channels were randomly realized with L f = L g = 3 and each plotted value is the averageover the 200 channel realizations. As expected, it is seen in Figs. 2 and 3 that the performanceof the FF relay improves as the FF relay filter length increases, and the FF relay significantlyoutperforms the simple AF relay ( L r = 1 ). It is also seen that most of the gain is achieved byonly a few filter taps for the FF relay.Next, we investigated the BER performance corresponding to Fig. 2. Here, we assumeduncoded QPSK modulation for each subcarrier channel. From the result of Fig. 2, we obtainedthe SNR of each subcarrier channel of the total N = 16 subcarrier channels for the designed FFrelay filter, transmit filer, receive filter and source power allocation. Based on this, we computedthe subcarrier BER based on the SNR of each subcarrier and averaged all the subcarrier channelBERs to obtain the overall BER, and the result is shown in Fig. 4. It is seen in Fig. 4 that theFF relay significantly improves the BER performance over the AF relay. Next, we tested theconvergence property of the proposed algorithm, and Fig. 5 shows the result. It is seen that theproposed algorithm converges with a few iterations. October 29, 2018 DRAFT2 SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 29, 2018 S u m M SE P r,max = 8 dB P r,max = 11 dB P r,max = 14 dB P r,max = 17 dB Fig. 3: Sum MSE versus relay filter length. −4 −3 −2 −1 Relay transmit power (dB) O v e r a ll BE R L r = 1 (AF) L r = 2 L r = 3 L r = 5 L r = 9 L r = 11 Fig. 4: Overall BER versus FF relay transmit power.
DRAFT October 29, 20183
Finally, we examined the rate performance of the proposed rate-targeting design method,Algorithm 2. (Rate maximization may be the ultimate goal of design in many cases.) Fig. 6shows the result. Again, for the figure 200 channels were randomly realized with L f = L g = 3 and each plotted value is the average over the 200 channel realizations, and the sum rate isthe sum over the total subcarrier channels. It is shown in Fig. 6 that the FF relay improves therate performance as the FF relay filter length increases, and the improvement gap shows thatit is worth considering FF relays over simple AF relays even though FF relays require moreprocessing than AF relays. S u m M SE L r = 1 (AF) L r = 2 L r = 9 Fig. 5: Sum MSE versus the number of iteration.V. C
ONCLUSION
In this paper, we have considered the joint design of the linear transceiver and the FF relay forMIMO-OFDM systems for weighted sum MSE minimization and sum rate maximization, andhave proposed algorithms for this purpose based on alternating optimization that iterates betweenoptimal design of the FF relay for a MIMO transceiver at the source and the destination and
October 29, 2018 DRAFT4 SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 29, 2018 S u m r a t e L r = 1 (AF) L r = 2 L r = 3 L r = 9 L r = 11 Fig. 6: Sum rate versus FF relay transmit power.optimal design of the MIMO transceiver for a given FF relay filter. We have shown that theFF relay design problem for a given MIMO transceiver reduces to a quadratically constrainedquadratic program (QCQP) and have proposed a solution to this QCQP problem based onconversion to a semi-definite program (SDP). We have provided some numerical results toevaluate the performance gain of the FF relaying scheme over the simple AF scheme for MIMO-OFDM systems. Numerical results show the effectiveness of the proposed FF relay design andsuggest that it is worth considering the FF relaying scheme over the widely-considered simpleAF scheme for MIMO-ODFM systems.
DRAFT October 29, 20185 A PPENDIX A E AND E MATRICES E and E are M t L r M r × M t ( N + L r + L g − N + L g − M r matrices and defined asfollows: E = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I0 ...... ......... |{z} N + L g − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ... ......... |{z} N + L g − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ......
0I 0 ......... |{z} N + L g − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)| {z } M t ( N + L r + L g − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ......... |{z} I0 ...... ......... |{z} N + L g − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ......... |{z} ... ......... |{z} N + L g − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ......... |{z} ......
0I 0 ......... |{z} N + L g − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)| {z } M t ( N + L r + L g − · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ......... |{z} N + L g − I0 ...... (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ......... |{z} N + L g − ... (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ......... |{z} N + L g − ...... (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)| {z } M t ( N + L r + L g − ⊗ I M r (39) where I = I L r . October 29, 2018 DRAFT6 SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 29, 2018 E = E ...... ... E M t ...... E M t +1 E ... ... E M t + M t E M t ... · · · E ( L r − M t +1 ......... E ... E ( L r − M t + M t ......... E M t | {z } L r -th block E ( L r ) M t +1 ......... E M t +1 ... E ( L r ) M t + M t ......... E M t + M t · · · E ( N + L g − M t +1 ......... E ( N + L g − L r − M t +1 ... E ( N + L g − M t + M t ......... E ( N + L g − L r − M t + M t | {z } ( N + L g − -th block E ( N + L g − M t +1 ...... E ( N + L g − L r ) M t +1 ... E ( N + L g − M t + M t ...... E ( N + L g − L r ) M t + M t | {z } ( N + L g ) -th block · · · ...... E ( N + L g − M t +1 ... ...... E ( N + L g − M t + M t | {z } ( N + L g + L r − -th block and E k = e Tk e T ( N + L g − M t + k e T N + L g − M t + k ... e T ( M r − N + L g − M t + k (40) where e Ti is the i -th row of I ( N + L g − M t M r . DRAFT October 29, 20187 A PPENDIX B Proof of Proposition 1
The Lagrangian of (37) is given by L ( p n [ k ] , µ ) = N − X n =0 B X k =1 θ nk ( d n [ k ] p n [ k ] − d n [ k ] p n [ k ] + c n [ k ]) + µ ( N − X n =0 B X k =1 p n [ k ] − P s,max ) − N − X n =0 B X k =1 λ n,k p n [ k ] (41)where µ ∈ R and λ n,k ≥ are dual variables associated with the source power constraint andthe positiveness of power, respectively.Then, the following KKT conditions are necessary and sufficient for optimality because theproblem (37) is a convex optimization problem: p n [ k ] ≥ , N − X n =0 B X k =1 p n [ k ] − P s,max = 0 , (42) µ ∈ R , λ n,k ≥ , (43) λ n,k p n [ k ] = 0 (44) ∇ p n [ k ] L = 2 θ nk d n [ k ] p n [ k ] − θ nk d n [ k ] + 2 µp n [ k ] − λ n,k = 0 (45)for n = 0 , , · · · , N − and k = 1 , · · · , B .The gradient (45) can be rewritten as λ n,k = 2( θ nk d n [ k ] + µ ) p n [ k ] − θ nk d n [ k ] . Plugging thisinto (43) and (44), we get µp n [ k ] ≥ θ nk d n [ k ] − θ nk d n [ k ] p n [ k ] (46) (( θ nk d n [ k ] + µ ) p n [ k ] − θ nk d n [ k ]) p n [ k ] = 0 (47)Let us consider the case that p n [ k ] = 0 . Then, (46) is satisfied only if d n [ k ] = 0 because d n [ k ] ≥ . If p n [ k ] > , p n [ k ] = (cid:16) θ nk d n [ k ] θ nk d n [ k ] + µ (cid:17) by the complementary slackness (47). This alsosatisfies (46). Therefore, we get the desired result satisfying the primal constraints (42). (cid:3) October 29, 2018 DRAFT8 SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS, OCTOBER 29, 2018 R EFERENCES [1] A. El Gamal, M. Mohseni, and S. Zahedi, “Bounds on capacity and minimum energy-per-bit for AWGN relay channels,”
IEEE Trans. Inform. Theory , vol. 52, pp. 1545 – 1561, Apr. 2006.[2] A. del Coso and C. Ibars, “Linear relaying for the Gaussian multiple-access and broadcast channels,”
IEEE Trans. WirelessCommun. , vol. 8, pp. 2024 – 2035, Apr. 2009.[3] H. Chen, A. Gershman and S. Shahbazpanahi, “Filter-and-forward distributed beamforming in relay networks with frequencyselective fading,”
IEEE Trans. Signal Process. , vol. 58, pp. 1251 – 1262, Mar. 2010.[4] Y. Liang, A. Ikhlef, W. Gerstacker and R. Schober, “Cooperative filter-and-forward beamforming for frequency-selectivechannels with equalization,”
IEEE Trans. Wireless Commun. , vol. 10, pp. 228 – 239, Jan. 2011.[5] Y. Sung and C. Kim, “The capacity for the linear time-invariant Gaussian relay channel,”
ArXiv.http://arxiv.org/abs/1109.5426 , Sep. 2011.[6] C. Kim, Y. Sung and Y. H. Lee, “A joint time-invariant filtering approach to the linear Gaussian relay problem,”
IEEETrans. Signal Process. , vol. 60, pp. 4360 – 4375, Aug. 2012.[7] D. Kim and J. Seo and Y. Sung, “Filter-and-Forward Transparent Relay Design for OFDM systems,”
IEEE Trans. Vech.Technol. , vol. 62, pp. 1 – 16, Nov. 2013.[8] D. Kim and J. Seo and Y. Sung, “Filter-and-forward relay design for OFDM systems for quality-of-service enhancement,”in
Proc. of APSIPA ASC , Hollywood, CA, Dec. 2012.[9] I. Hammerstr¨om and A. Wittneben, “On the optimal power allocation for nongenerative OFDM relay links,” in
Proc. ofICC , vol. 10, pp. 4463–4468, June. 2006.[10] T. Ng and W. Yu, “Joint optimization of relay strategies and resource allocations in cooperative cellular networks,”
IEEEJ. Sel. Areas Communi. , vol. 25, pp. 328 – 339, Feb. 2007.[11] M. Dong and S. Shahbazpanahi, “Optimal spectrum sharing and power allocation for OFDM-based two-way relaying,” in
Proc. of ICASSP , pp. 3310–3313, Mar. 2010.[12] W. Dang, M. Tao, H. Mu and J. Huang, “Subcarrier-pair based resource allocation for cooperative multi-relay OFDMsystems,”
IEEE Trans. Wireless Commun. , pp. 1640 – 1649, May. 2010.[13] Z. Fang and Y. Hua and J. C. Koshy, “Joint source and relay optimization for a non-regenerative MIMO relay,” in Proc.of 2006 IEEE SAM , 2006.[14] S. Simoens and O. Munoz-Medina and J. Vidal and A. del Coso, “On the Gaussian MIMO relay channel with full channelstate information,”
IEEE Trans. Signal Process. , vol. 57, Sep. 2009.[15] H. Sampath and P. Stoica and A. Paulraj, “Generalized linear precoder and decoder design for MIMO channels using theweighted MMSE criterion,”
IEEE Trans. Commun. , vol. 49, pp. 2198 – 2206, Dec. 2001.[16] D. Guo and S. Shitz and S. Verdu, “Mutual information and minimum mean-square error in Gaussian channels,”
IEEETrans. Inform. Theory , vol. 51, pp. 1261 – 1282, Apr. 2005.
DRAFT October 29, 20189 [17] D. Palomar and S. Verdu, “Gradient of mutual information in linear vector gaussian channels,”
IEEE Trans. Inform. Theory ,vol. 52, pp. 141 – 154, Jan. 2006.[18] S. S. Christensen and R. Argawal and E. de Carvalho and J. M. Cioffi, “Weighted sum-rate maximization using weightedMMSE for MIMO-BC beamforming design,”
IEEE Trans. Wireless Commun. , vol. 7, pp. 1 – 7, Dec. 2008.[19] Q. Shi and M. Razaviyayn and Z. -Q. Luo and C. He, “An iteratively weighted MMSE approach to distributed sum-utilitymaximization for a MIMO interfering broadcast channel,”
IEEE Trans. Signal Process. , vol. 59, pp. 4331 – 4340, Sep. 2011.[20] R. M. Gray,
Toeplitz and Circulant Matrices : A Review . Now publishers, 2006[21] Y. Huang and S. Zhang, “Complex matrix decomposition and quadratic programming,”
Math. Oper. Res. , vol. 32, pp. 758– 768, Aug. 2007.[22] W. Ai and Y. Huang and S. Zhang, “New results on Hermitian matrix rank-one decomposition,”
Math. Programm. , vol.128, pp. 253 – 283, Aug. 2009.[23] S. Boyd and L. Vandenberghe,
Convex Optimization . New York: Cambridge University Press, 2004.[24] C. Helmberg, “Semidefinite programming,”
European Journal of Operation Research , vol. 137, pp. 461 – 482, 2002.[25] J. F. Sturn, “Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones,”
Optimization Methods ansSoftware, Special Issue on Interior Point Methods , vol. 11/12, pp. 625 – 563, 1999.[26] M. Grant and S. Boyd, { CVX } : Matlab Software for Disciplined Convex Programming. ver. Version 1.21, 2011.[27] H. Sampath and A. Paulraj, “Joint transmit and receiver optimization for high data rate wireless communications usingmultiple antennas,” in Proc. of Asilomar Conf. Signals, Systems and Computers , vol. 1, 1999.[28] S. Arora and E. Hazan and S. Kale, “Fast algorithms for approximate semidefinite programming using the multiplicativeweights update method,” in Proc. of 46th FOCS , pp. 339–348, 2005.[29] S. Arora and E. Hazan and S. Kale, “The Multiplicative Weights Update Method: a Meta-Algorithm and Applications,” Theory of Computing (TOC) , vol. 8, pp. 121 – 164, 2012., vol. 8, pp. 121 – 164, 2012.