IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY

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1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 4, NO 4, JULY Towards Maximum Achievable Diversity in Space, Time, and Frequency: Performance Analysis and Code Design Weifeng Su, Member, IEEE, Zoltan Safar, Member, IEEE, andkjrayliu,fellow, IEEE Abstract Multiple input multiple output (MIMO) communication systems with orthogonal frequency division multiplexing (OFDM) modulation have a great potential to play an important role in the design of the next-generation broadband wireless communication systems In this paper, we address the problem of performance analysis and code design for MIMO-OFDM systems when coding is applied over both spatial, temporal, and frequency domains First, we provide an analytical framework for the performance analysis of MIMO-OFDM systems assuming arbitrary power delay profiles Our general framework incorporates the space time and space frequency (SF) coding approaches as special cases We also determine the maximum achievable diversity order, which is found to be the product of the number of transmit and receive antennas, the number of delay paths, and the rank of the temporal correlation matrix Then, we propose two code design methods that are guaranteed to achieve the maximum diversity order The first method is a repetition coding approach using full-diversity SF codes, and the second method is a block coding approach that can guarantee both full symbol rate and full diversity Simulation results are also presented to support the theoretical analysis Index Terms Broadband wireless communications, maximum achievable diversity, MIMO-OFDM systems, multiple antennas, space frequency coding, space time frequency coding I INTRODUCTION MULTIPLE INPUT MULTIPLE OUTPUT (MIMO) communication systems have a great potential to play an important role in the design of the next-generation wireless communication systems due to the advantages that such systems can offer By employing multiple transmit and receive antennas, the adverse effects of the wireless propagation environment can be significantly reduced In case of narrowband wireless communications, the fading channel is frequency nonselective, many modulation and coding methods Manuscript received August 18, 2003; revised January 29, 2004; accepted May 9, 2004 The editor coordinating the review of this paper and approving it for publication is Y-C Liang This work was supported in part by US Army Research Laboratory under Cooperative Agreement DAAD W Su is with the Department of Electrical Engineering at the State University of New York (SUNY) at Buffalo, Buffalo, NY USA ( weifeng@engbuffaloedu) Z Safar is with Modem System Design, Technology Platforms, Nokia Danmark A/S, Copenhagen, Denmark ( zoltan2safar@nokiacom) K J Ray Liu is with the Department of Electrical and Computer Engineering and Institute for Systems Research University of Maryland, College Park, MD USA ( kjrliu@engumdedu) Digital Object Identifier /TWC [1] [9], termed as space time (ST) codes, have been proposed to exploit the spatial and temporal diversities available in the multiantenna channel In case of broadband wireless communications, the fading channel is frequency selective, orthogonal frequency division multiplexing (OFDM) modulation can be used to transform the frequency-selective channel into a set of parallel frequency flat channels, providing high spectral efficiency and eliminating the need for high-complexity equalization algorithms To take advantage of both MIMO systems and OFDM modulation, MIMO-OFDM systems have been proposed, resulting in two major channel coding approaches for these systems The first approach is space frequency (SF) coding, coding is applied within a single OFDM block to exploit the spatial and frequency diversities The other approach is space time frequency (STF) coding, the coding is applied across multiple OFDM blocks to exploit the spatial, temporal, and frequency diversities available in frequencyselective MIMO channels Early works on SF coding [10] [15] used ST codes directly as SF codes, ie, previously existing ST codes were used by replacing the time domain with the frequency domain (OFDM tones) The performance criteria for SF-coded MIMO-OFDM systems were derived in [15] and [16], and the maximum achievable diversity was found to be L M r, and M r are the number of transmit and receive antennas, respectively, and L is the number of delay paths in the channel impulse response Bölcskei and Paulraj [16] showed that, in general, systems using ST codes directly as SF codes can achieve only spatial diversity and are not guaranteed to achieve the full spatial and frequency diversity L M r Later, in [17] and [18], systematic SF code design methods that could guarantee to achieve the maximum diversity were proposed To further improve the performance, one may consider STF coding across multiple OFDM blocks to exploit all of the available diversities in the spatial, temporal, and frequency domains The STF coding strategy was first proposed in [19] for two transmit antennas and further developed in [20], [21], and [22] for multiple transmit antennas Both [19] and [22] assumed that the MIMO channel stays constant over multiple OFDM blocks, and we will show later that in this case, STF coding cannot provide any additional diversity compared to the SF coding approach In [21], an intuitive explanation on the equivalence between antennas and OFDones was presented from the viewpoint of channel capacity In [20], the performance /$ IEEE

2 1848 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 4, NO 4, JULY 2005 Fig 1 STF-coded MIMO-OFDM system with transmit and M r receive antennas criteria for STF codes were derived, and an upper bound on the maximum achievable diversity order was established However, there was no discussion in [20] whether the upper bound can be achieved or not, and the proposed STF codes were not guaranteed to achieve the full spatial, temporal, and frequency diversities In this paper, we consider the problem of performance analysis and full-diversity STF code design for MIMO-OFDM systems We provide a general framework, taking into account coding over the spatial, temporal, and frequency domains Our model incorporates the ST and SF coding approaches as special cases First, we derive the performance criteria for STF-coded MIMO-OFDM systems, based on the results of [18], [23], and [24], and we show that the maximum achievable diversity order is L M r T, T is the rank of the temporal correlation matrix of the channel Then, we propose two STF code design methods that are guaranteed to achieve the maximum achievable diversity order The first method is a repetition coding approach, which achieves full diversity at the price of symbol rate decrease The advantage of this approach is that any full-diversity SF code (block or trellis) can be used to design full-diversity STF codes The other STF code design method, a block coding approach, provides both data rate (full symbol rate) and performance (full diversity) In this case, the STF codes are constructed using existing results on signal constellation design for single-antenna fading channels The paper is organized as follows In Section II, we describe the MIMO-OFDM system model with an arbitrary power delay profile In Section III, we derive the STF code performance criteria and determine the maximum achievable diversity order In Section IV, two STF code design methods are proposed The simulation results are presented in Section V, and some conclusions are drawn in Section VI II SYSTEM MODEL We consider an STF-coded MIMO-OFDM system with transmit antennas, M r receive antennas, and N subcarriers, as shown in Fig 1 Suppose that the frequency-selective fading channels between each pair of transmit and receive antennas have L independent delay paths and the same power delay profile The MIMO channel is assumed to be constant over each OFDM block period, but it may vary from one OFDM block to another At the kth OFDM block, the channel impulse response from transmit antenna i to receive antenna j at time τ can be modeled as L 1 h k i,j(τ) = αi,j(l)δ(τ k τ l ) (1) l=0 τ l is the delay and αi,j k (l) is the complex amplitude of the lth path between transmit antenna i and receive antenna j The αi,j k (l) s are modeled as zero-mean complex Gaussian random variables with variances E αi,j k (l) 2 = δl 2, E stands for the expectation The powers of the L paths are normalized such that L 1 l=0 δ2 l =1 We assume that the MIMO channel is spatially uncorrelated, so the channel coefficients αi,j k (l) s are independent for different indices (i, j) From (1), the frequency response of the channel is given by L 1 Hi,j( k f) = αi,j(l)e k j2πfτ l (2) l=0 j = 1 We consider STF coding across transmit antennas, N OFDM subcarriers, and K consecutive OFDM blocks Each STF codeword can be expressed as a KN matrix C =[C T 1 C T 2 C T K ]T (3) the channel symbol matrix C k is given by c k 1(0) c k 2(0) c k (0) c k 1(1) c k 2(1) c k (1) C k = (4) c k 1(N 1) c k 2(N 1) c k (N 1) and ci k (n) is the channel symbol transmitted over the nth subcarrier by transmit antenna i in the kth OFDM block The STF code is assumed to satisfy the energy constraint E C 2 F = KN, C F is the Frobenius norm of C During the kth OFDM block period, the transmitter applies an N-point IFFT to each column of the matrix C k After appending a cyclic prefix, the OFDM symbol corresponding to the ith (i =1, 2,, ) column of C k is transmitted by transmit antenna i

3 SU et al: DIVERSITY IN SPACE, TIME, AND FREQUENCY: PERFORMANCE ANALYSIS AND CODE DESIGN 1849 At the receiver, after matched filtering, removing the cyclic prefix, and applying FFT, the received signal at the nth subcarrier at receive antenna j in the kth OFDM block is given by ρ yj k (n) = c k i (n)hi, k j(n)+zj k (n) (5) L 1 Hi,j(n) k = αi,j(l)e k j2πn fτ l (6) l=0 is the channel frequency response at the nth subcarrier between transmit antenna i and receive antenna j, f =1/T is the subcarrier separation in the frequency domain, and T is the OFDM symbol period We assume that the channel state information Hi,j k (n) is known at the receiver but not at the transmitter In (5), zj k (n) denotes the additive white complex Gaussian noise with zero mean and unit variance at the nth subcarrier at receive antenna j in the kth OFDM block The factor ρ/ in (5) ensures that ρ is the average signal-to-noise ratio (SNR) at each receive antenna III PERFORMANCE CRITERIA In this section, we derive the performance criteria for STFcoded MIMO-OFDM systems, based on the results of [18], [23], and [24], and we also determine the maximum achievable diversity order for such systems A Pairwise Error Probability Using the notation c i [(k 1)N + n] = ci k(n), H i,j[(k 1)N + n] = Hi,j k (n), y j[(k 1)N + n] = yj k(n), and z j[(k 1)N + n] = zj k (n) for 1 k K, 0 n N 1, 1 i, and 1 j M r, the received signal in (5) can be expressed as ρ y j (m) = c i (m)h i,j (m)+z j (m) (7) for m =0, 1,,KN 1 We further rewrite the received signal in vector form as ρ Y = DH + Z (8) D is a KNM r KN M r matrix constructed from the STF codeword C in (3) as follows: D = I Mr [ D 1 D 2 D Mt ] (9) denotes the tensor product, I Mr is the identity matrix of size M r M r, and D i = diag {c i (0),c i (1),,c i (KN 1)} (10) for any i =1, 2,, The channel vector H of size KN M r 1 is formatted as in (11) (at the bottom of the page) H i,j =[H i,j (0) H i,j (1) H i,j (KN 1) ] T (12) The received signal vector Y of size KNM r 1 is given by (13) (at the bottom of the page) and the noise vector Z, which has the same form as Y, is given by (14) (at the bottom of the page) Suppose that D and D are two matrices constructed from two different codewords C and C, respectively Then, the pairwise error probability between D and D can be upper bounded as [23] ( ) ( ) 1 P (D D) 2r 1 r ( ) r ρ γ i (15) r r is the rank of (D D)R(D D) H, γ 1,γ 2,,γ r are the nonzero eigenvalues of (D D)R(D D) H, and R = E{HH H } is the correlation matrix of H The superscript H stands for the complex conjugate and transpose of a matrix Based on the upper bound on the pairwise error probability in (15), two general STF code performance criteria can be proposed as follows 1) Diversity (rank) criterion: The minimum rank of (D D)R(D D) H over all pairs of different codewords C and C should be as large as possible H =[H T 1,1 H T,1 H T 1,2 H T,2 H T 1,M r H T,M r ] T (11) Y =[y 1 (0) y 1 (KN 1) y 2 (0) y 2 (KN 1) y Mr (0) y Mr (KN 1) ] T (13) Z =[z 1 (0) z 1 (KN 1) z 2 (0) z 2 (KN 1) z Mr (0) z Mr (KN 1) ] T (14)

4 1850 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 4, NO 4, JULY ) Product criterion: The minimum value of the product r γ i over all pairs of different codewords C and C should be maximized B Performance Criteria and Maximum Achievable Diversity In case of spatially uncorrelated MIMO channels, ie, the channel taps αi,j k (l) are independent for different transmit antenna index i and receive antenna index j, the correlation matrix R of size KN M r KN M r becomes R = diag (R 1,1,,R Mt,1,R 1,2,,R Mt,2,,R 1,Mr,,R Mt,M r ) (16) R i,j = E { H i,j Hi,j H } (17) is the correlation matrix of the channel frequency response from transmit antenna i to receive antenna j Using the notation w =exp( j2π f), from(6),wehave H i,j =(I K W )A i,j (18) w W = τ 0 w τ 1 w τ L 1 w (N 1)τ 0 w (N 1)τ 1 w (N 1)τ L 1 and A i,j is defined at the bottom of the page Substituting (18) into (17), R i,j can be calculated as follows: R i,j = E { (I K W )A i,j A H i,j(i K W ) H} =(I K W )E { A i,j A H i,j} (IK W H ) With the assumptions that the path gains αi,j k (l) are independent for different paths and different pairs of transmit and receive antennas, and that the second-order statistics of the time correlation is the same for all transmit and receive antenna pairs and all paths (ie, the correlation values do not depend on i, j, and l), we can define the time correlation at lag m as r T (m) =E{αi,j k (l)αk+m i,j (l)} Thus, the correlation matrix E{A i,j A H i,j } can be expressed as E { A i,j A H i,j} = RT Λ (19) Λ = diag {δ 2 0,δ 2 1,,δ 2 L 1 }, and R T is the temporal correlation matrix of size K K, whose entry in the pth row and the qth column is given by r T (q p) for 1 p, q K We can also define the frequency correlation matrix, R F,as R F = E{Hi,j k HkH i,j }, H k i,j = [ H k i,j(0),,h k i,j(n 1) ] T Then, R F = W ΛW H As a result, we arrive at yielding R i,j =(I K W )(R T Λ)(I K W H ) = R T (W ΛW H )=R T R F (20) R = I Mt M r (R T R F ) (21) Finally, combining (4), (9), (10), and (21), the expression for (D D)R(D D) H in (15) can be rewritten as (D D)R(D D) H = I Mr = I Mr [ Mt ] (D i D i )(R T R F )(D i D i ) H {[ (C C)(C C) H] } (R T R F ) denotes the Hadamard product 1 Denote (22) =(C C)(C C) H (23) and R = R T R F Then, substituting (22) into (15), the pairwise error probability between C and C can be upper bounded as ( ) ( ) M P (C C) 2νMr 1 ν r ( ) νmr ρ λ i (24) νm r ν is the rank of R, and λ 1,λ 2,,λ ν are the nonzero eigenvalues of R The minimum value of the product ν λ i over all pairs of distinct signals C and C is termed as coding advantage, denoted by ζ STF = min C C ν λ i (25) As a consequence, we can formulate the performance criteria for STF codes as follows 1) Diversity (rank) criterion: The minimum rank of R over all pairs of distinct codewords C and C should be as large as possible 2) Product criterion: The coding advantage or the minimum value of the product ν λ i over all pairs of distinct signals C and C should also be maximized 1 Suppose that A = {a i, j } and B = {b i, j } are two matrices of size m n The Hadamard product of A and B is defined as A B = {a i, j b i, j } 1 i m, 1 j n A i,j =[αi,j 1 (0) α1 i,j (1) α1 i,j (L 1) αk i,j (0) αk i,j (1) αk i,j (L 1) ]T

5 SU et al: DIVERSITY IN SPACE, TIME, AND FREQUENCY: PERFORMANCE ANALYSIS AND CODE DESIGN 1851 If the minimum rank of R is ν for any pair of distinct STF codewords C and C, we say that the STF code achieves a diversity order of νm r For a fixed number of OFDM blocks K, transmit antennas, and correlation matrices R T and R F,the maximum achievable diversity or full diversity is defined as the maximum diversity order that can be achieved by STF codes of size KN According to the rank inequalities on Hadamard products and tensor products [38], we have rank( R) rank( )rank(r T )rank(r F ) Since the rank of is at most and the rank of R F is at most L, we obtain rank( R) min {L rank(r T ),KN} (26) Thus, the maximum achievable diversity is at most min {L M r rank(r T ),KNM r }, in agreement with the results of [20] However, there is no discussion in [20] whether this upper bound can be achieved or not In the following sections, we show that this upper bound can indeed be achieved We can also observe that if the channel stays constant over multiple OFDM blocks (rank(r T )=1), the maximum achievable diversity is only min{l M r,knm r }Inthis case, STF coding cannot provide additional diversity advantage compared to the SF coding approach Note that the proposed analytical framework includes ST and SF codes as special cases If we consider only one subcarrier (N =1)and one delay path (L =1), the channel becomes a single-carrier time-correlated flat fading MIMO channel The correlation matrix R simplifies to R = R T, and the code design problem reduces to that of ST code design, as described in [24] In the case of coding over a single OFDM block (K =1), the correlation matrix R becomes R = R F, and the code design problem simplifies to that of SF codes, as discussed in [18] IV FULL-DIVERSITY STF CODE DESIGN METHODS We propose two STF code design methods to achieve the maximum achievable diversity order min {L M r rank(r T ), KNM r } in this section Without loss of generality, we assume that the number of subcarriers N is not less than L,sothe maximum achievable diversity order is L M r rank(r T ) A Repetition-Coded STF Code Design In [18], we proposed a systematic approach to design fulldiversity SF codes Suppose that C SF is a full-diversity SF code of size N We now construct a full-diversity STF code C STF by repeating C SF K times (over K OFDM blocks) as follows: C STF = 1 k 1 C SF (27) 1 k 1 is an all one matrix of size k 1 Let STF =(C STF C STF )(C STF C STF ) H and SF =(C SF C SF )(C SF C SF ) H Then, we have [ STF = 1 k 1 (C SF C SF ) ][1 1 k (C SF C SF ) H] Thus, = 1 k k SF STF R =(1 k k SF ) (R T R F ) = R T ( SF R F ) Since the SF code C SF achieves full diversity in each OFDM block, the rank of SF R F is L Therefore, the rank of STF R is L rank(r T ),soc STF in (27) is guaranteed to achieve a diversity order of L M r rank(r T ) We observe that the maximum achievable diversity depends on the rank of the temporal correlation matrix R T If the fading channels are constant during K OFDM blocks, ie, rank(r T )=1, the maximum achievable diversity order for STF codes (coding across several OFDM blocks) is the same as that for SF codes (coding within one OFDM block) Moreover, if the channel changes independently in time, ie, R T = I K, the repetition structure of STF code C STF in (27) is sufficient, but not necessary to achieve the full diversity In this case R = diag ( 1 R F, 2 R F,, K R F ) k =(C k C k )(C k C k ) H for 1 k K Thus, in this case, the necessary and sufficient condition to achieve fulldiversity KL M r is that each matrix k R F be of rank L over all pairs of distinct codewords simultaneously for all 1 k K The proposed repetition-coded STF code design ensures full diversity at the price of symbol rate decrease by a factor of 1/K (over K OFDM blocks) compared to the symbol rate of the underlying SF code The advantage of this approach is that any full-diversity SF code (block or trellis) can be used to design full-diversity STF codes B Full-Rate STF Code Design We can also design a class of STF codes that can achieve a diversity order of Γ M r rank(r T ) for any fixed integer Γ(1 Γ L) by extending the full-rate full-diversity SF code construction method (coding over one OFDM block, ie, the K =1case) proposed in [25] We consider an STF code structure consisting of STF codewords C of size KN by C =[C T 1 C T 2 C T K ]T (28) C k = [ G T k,1 G T k,2 G T k,p 0 T N P Γ ] T (29)

6 1852 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 4, NO 4, JULY 2005 for k =1, 2,,K In (29), P = N/(Γ ), and each matrix G k,p (1 k K, 1 p P ) is of size Γ The zero padding in (29) is used if the number of subcarriers N is not an integer multiple of Γ For each p (1 p P ), we design the code matrices G 1,p,G 2,p,,G K,p jointly, but the design of G k1,p 1 and G k2,p 2, p 1 p 2, is independent of each other For a fixed p (1 p P ),let G k,p = diag (X k,1,x k,2,,x k,mt ), k=1, 2,,K (30) diag (X k,1,x k,2,,x k,mt ) is a block diagonal matrix, X k,i =[x k,(i 1)Γ+1 x k,(i 1)Γ+2 x k,iγ ] T, i = 1, 2,,, and all x k,j, j =1, 2,, Γ, are complex symbols and will be specified later The energy normalization condition is K ΓM t E x k,j 2 = KΓ k=1 j=1 The symbol rate of the proposed scheme is P Γ /N, ignoring the cyclic prefix If N is a multiple of Γ, the symbol rate is 1 If not, the rate is less than 1, but since usually N is much greater than Γ, the symbol rate is very close to 1 We define full rate as one channel symbol per subcarrier per OFDM block period, so the proposed method can either achieve full symbol rate, or it can perform very close to it Note that this scheme includes the code design method proposed in [25] as a special case when K =1 The following theorem provides a sufficient condition for the STF codes described above to achieve a diversity order of Γ M r rank(r T ) For simplicity, we use the notation X =[x 1, x 1,ΓMt x K,1 x K,ΓMt ] and X =[ x 1,1 x 1,ΓMt x K,1 x K,ΓMt ] Moreover, for any n n nonnegative definite matrix A, we denote its eigenvalues in a nonincreasing order as: eig 1 (A) eig 2 (A) eig n (A) Theorem 1: For any STF code constructed by (28) (30), if K k=1 ΓMt j=1 x k,j x k,j 0 for any pair of distinct symbols X and X, the STF code achieves a diversity order of Γ M r rank(r T ), and the coding advantage is bounded by ( δ min ) Γrank(R T ) Φ ζ STF ( δ max ) Γrank(R T ) Φ (31) δ min = min min x k,j x k,j 2 (32) 1 k K, 1 j Γ δ max = max max x k,j x k,j 2 (33) 1 k K, 1 j Γ rank(r T ) Φ= det (Q 0 ) rank(r T ) [eig i (R T )] Γ (34) and Q 0 = W 0 diag ( δ0,δ 2 1, 2,δL 1) 2 W H 0 (35) w W 0 = τ 0 w τ 1 w τ L 1 (36) w (Γ 1)τ 0 w (Γ 1)τ 1 w (Γ 1)τ L 1 Γ L Furthermore, if the temporal correlation matrix R T is of full rank, ie, rank(r T )=K, the coding advantage is ζ STF = δm KΓ t det (R T ) Γ det (Q 0 ) K (37) δ = min K ΓM t k=1 j=1 x k,j x k,j 2 (38) Proof: Suppose that C and C are two distinct STF codewords that are constructed from G k,p and G k,p (1 k K, 1 p P ), respectively We would like to determine the rank of R, =(C C)(C C) H and R = R T R F For convenience, let G p =[G T 1,p G T 2,p G T K,p ]T for each p =1, 2,,P For two distinct codewords C and C, there exists at least one index p 0 (1 p 0 P ) such that G p0 G p0 We may further assume that G p = G p for any p p 0 since the rank of R does not decrease if G p G p for some p p 0 [38, Corollary 313, p 149] Note that the frequency correlation matrix R F is a Toeplitz matrix With the assumption that G p = G p for any p p 0, we observe that the nonzero eigenvalues of R are the same as those of [(G p0 G p0 )(G p0 G p0 ) H ] (R T Q), Q = {q i,j } 1 i, j ΓMt is also a Toeplitz matrix whose entries are L 1 q i,j = δl 2 w (i j)τ l, 1 i, j Γ (39) l=0 Note that Q is independent of the index p 0, ie, it is independent of the position of G p0 G p0 in C C For any 1 k K, we have G k,p0 G k,p0 = ( diag X k,1 X k,1,x k,2 X k,2,,x k,mt X ) k,mt = diag(x k,1 x k,1,,x k,γmt x k,γmt ) (I Mt 1 Γ 1 )

7 SU et al: DIVERSITY IN SPACE, TIME, AND FREQUENCY: PERFORMANCE ANALYSIS AND CODE DESIGN 1853 so the difference matrix between G p0 and G p0 is defined at the bottom of the page, diag(x X) = diag(x 1,1 x 1,1,,x 1,ΓMt x 1,ΓMt,,x K,1 x K,1,,x K,ΓMt x K,ΓMt ) Thus, we have [ ( G G p0 p0)(g p0 G ) ] H p0 (R T Q) = {diag (X X)[1 K 1 (I Mt 1 Γ 1 )] [1 K 1 (I Mt 1 Γ 1 )] H diag (X X) H} (R T Q) = [diag (X X)(1 K K I Mt 1 Γ Γ ) diag (X X) H] (R T Q) = diag (X X) {R T [(I Mt 1 Γ Γ ) Q]} diag (X X) H = diag (X X)(R T I Mt Q 0 ) diag (X X) H (40) Q 0 = {q i,j } 1 i,j Γ and q i,j is given by (39) In the above derivation, the second equality follows from the identities [1 K 1 (I Mt 1 Γ 1 )] H = 1 1 K I Mt 1 1 Γ and (A 1 B 1 )(A 2 B 2 )(A 3 B 3 )=(A 1 A 2 A 3 ) (B 1 B 2 B 3 ) [38, p 251], and the third equality follows from a property of the Hadamard product [38, p 304] From (40), we observe that if diag (X X) is of full rank, ie, x k,j x k,j 0 for any 1 k K and 1 j Γ, then the rank of [(G p0 G p0 )(G p0 G p0 ) H ] (R T Q) can be determined as rank(r T I Mt Q 0 ), which is equal to rank(r T )rank(q 0 ) Similar to the correlation matrix R F in (20), Q 0 can be expressed as Q 0 = W 0 diag ( δ 2 0,δ 2 1,,δ 2 L 1) W H 0 W 0 is defined in (36) Note that W 0 is a Γ L matrix consisting of Γ rows of a Vandermonde matrix [38], so with τ 0 <τ 1 < <τ L 1, W 0 is nonsingular Thus, Q 0 is of full rank (rank Γ) Therefore, if K ΓMt k=1 j=1 x k,j x k,j 0,the rank of R is Γ rank(r T ) TheassumptionthatG p = G p for any p p 0 is also sufficient to calculate the coding advantage since the nonzero eigenvalues of R do not decrease if G p G p for some p p 0 [38, Corollary 313, p 149] Using the notation ν 0 = Γ rank(r T ), the coding advantage can be calculated as ζ STF = min ν 0 eig i { [( G p0 G p0 )( G p0 G p0 ) H ] (R T Q) ν 0 = min eig i [ diag (X X)(R T I Mt Q 0 ) } diag (X X) H] (41) ν 0 = min θ i eig i (R T I Mt Q 0 ) (42) eig KΓMt [diag (X X)diag (X X) H ] θ i eig 1 [diag(x X)diag (X X) H ] for i =1, 2,,ν 0 In the above derivation, the second equality follows by (40), and the last equality follows by Ostrowski s theorem [37, p 224] Since and ν 0 eig i (R T I Mt Q 0 ) = Γrank(R T ) [eig i (R T Q 0 )] rank(r T ) = det (Q 0 ) rank(r T ) [eig i (R T )] Γ ( eig 1 diag (X X) diag (X X) H) = max max x k,j x k,j 2 1 k K, 1 j Γ ( eig KΓMt diag (X X) diag (X X) H) = min min x k,j x k,j 2 1 k K, 1 j Γ we have the lower and upper bounds in (31) G p0 G p0 = diag (x 1,1 x 1,1,,x 1,ΓMt x 1,ΓMt )(I Mt 1 Γ 1 ) diag (x 2,1 x 2,1,,x 2,ΓMt x 2,ΓMt )(I Mt 1 Γ 1 ) diag (x K,1 x K,1,,x K,ΓMt x K,ΓMt )(I Mt 1 Γ 1 ) = diag (X X)[1 K 1 (I Mt 1 Γ 1 )]

8 1854 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 4, NO 4, JULY 2005 Finally, if R T is of full rank, ν 0 = KΓ From (41), the coding advantage is [ ζ STF = min det diag (X X)(R T I Mt Q 0 ) diag (X X) H] = M KΓ t det (R T I Mt Q 0 ) min = δm KΓ t K ΓM t k=1 j=1 x k,j x k,j 2 det (R T ) Γ det (Q 0 ) K δ is given by (38) Thus, we have proven Theorem 1 completely From Theorem 1, we observe that with the code structure specified in (28) (30), it is not difficult to achieve the diversity order of Γ M r rank(r T ) The remaining problem is to design a set of complex symbol vectors, X = [ x 1,1 x 1,ΓMt x K,1 x K,ΓMt ], such that the coding advantage ζ STF is as large as possible One approach is to maximize δ min and δ max in (31) according to the lower and upper bounds of the coding advantage Another approach is to maximize δ in (38) We follow the latter for two reasons First, the coding advantage ζ STF in (37) is determined by δ in closed form although this closed form only holds with the assumption that the temporal correlation matrix R T is of full rank Second, the problem of designing X to maximize δ is related to the problem of constructing signal constellations for Rayleigh fading channels, which has been well solved [9] [26] In the literature, δ is called the minimum product distance of the set of symbols X [26], [27] We summarize some existing results on designing X in order to maximize the minimum product distance δ as follows For simplicity, denote L = KΓ and assume that Ω is a constellation such as quadratic-amplitude modulation (QAM), pulse-amplitude modulation, and so on The set of complex symbol vectors is obtained by applying a transform over a L-dimensional signal set Ω L [9], [28], [29] Specifically X = S 1 L V (θ 1,θ 2,,θ L ) (43) S =[s 1 s 2 s L ] Ω K is a vector of arbitrary channel symbols to be transmitted, and V (θ 1,θ 2,,θ L ) is a Vandermonde matrix with variables θ 1,θ 2,,θ L [37] V (θ 1,θ 2,,θ L )= θ 1 θ 2 θ L θ1 L 1 θ2 L 1 θl L 1 (44) The optimum θ l s, 1 l L, have been specified for different L and Ω For example, if Ω is a QAM constellation and L = 2 s (s 1), the optimum θ l s were given by [28], [29] θ l =e j 4l 3 2L π, l =1, 2,,L (45) In case of L =2 s 3 t (s 1,t 1), a class of θ l s were given in [29] as θ l =e j 6l 5 3L π, l =1, 2,,L (46) For more details and other cases of Ω and L, we refer the reader to [9], [28], and [29] The STF code design discussed in this subsection achieves full symbol rate, which is much larger than that of the repetition coding approach However, the maximum-likelihood decoding complexity of this approach is high Its complexity increases exponentially with the number of OFDM blocks K while the decoding complexity of the repetition-coded STF codes increases only linearly with K Fortunately, sphere decoding methods [30] [32] can be used to reduce the complexity V S IMULATION RESULTS We simulated the proposed two STF code design methods for different fading channel models and compared their performance The OFDM modulation had N = 128 subcarriers, and the total bandwidth was 1 MHz Thus, the OFDM block duration was 128 µs We set the length of the cyclic prefix to 20 µs for all cases We present average bit error rate (BER) curves as functions of the average SNR A Performance of the Repetition-Coded STF Codes We simulated a block code and a trellis code example The simulated communication system had M r =1receive antenna We considered a two-ray equal power delay profile (L =2), with a delay of 20 µs between the two rays Each ray was modeled as a zero-mean complex Gaussian random variable with variance 05 The full-diversity STF block codes were obtained by repeating a full-diversity SF block code via (27) across K =1, 2, 3, 4 OFDM blocks The used full-diversity SF block code for =2transmit antennas was constructed from the Alamouti scheme [4] with quaternary phase-shift keying (QPSK) modulation via mapping described in [18] The spectral efficiency of the resulting STF codes were 1, 05, 033, and 025 bit/s/hz (omitting the cyclic prefix) for K =1, 2, 3, 4, respectively We simulated the full-diversity STF block code without temporal correlation (R T was an identity matrix) In Fig 2, we can see that by repeating the SF code over multiple OFDM blocks, the achieved diversity order can be increased The simulated full-diversity STF trellis code was obtained from a full-diversity SF trellis code via (27) with K =1, 2, 3, 4, respectively The used full-diversity SF trellis code for =3 transmit antennas was constructed by applying the repetition mapping [18] to the 16-state QPSK ST trellis code proposed in [33] Since the modulation was the same in all four cases, the spectral efficiency of the resulting STF codes were 1, 05, 033, and 025 bit/s/hz (omitting the cyclic prefix) for K =1, 2, 3, 4, respectively Similar to the previous case, we assumed that the channel changes independently from OFDM block to OFDM block The obtained BER curves can be observed in Fig 3 As

9 SU et al: DIVERSITY IN SPACE, TIME, AND FREQUENCY: PERFORMANCE ANALYSIS AND CODE DESIGN 1855 Fig 2 Performance of the repetition block codes Fig 4 Six-ray power delay profile of the TU channel model Fig 3 Performance of the repetition trellis codes Fig 5 Performance of the full-rate STF codes, ε =0 apparent in the figure, the STF codes (K >1) achieved higher diversity order than the SF code (K =1) B Performance of the Full-Rate STF Codes In this part, we used a more realistic six-ray typical urban (TU) power delay profile [36], which is shown in Fig 4, and simulated the fading channel with different temporal correlations We assumed that the fading channel is constant within each OFDM block period but varies from one OFDM block period to another according to a first-order Markovian model [34], [35] α k i,j(l) =εα k 1 i,j (l)+ηk i,j(l), 0 l L 1 (47) the constant ε (0 ε 1) determines the amount of the temporal correlation and ηi,j k (l) is a zero-mean complex Gaussian random variable with variance δ l (1 ε 2 ) 1/2 Ifε = 0, there is no temporal correlation (independent fading), while if ε =1, the channel stays constant over multiple OFDM blocks We considered three temporal correlation scenarios: ε =0; ε = 08; and ε =095 The simulated full-rate STF codes were constructed by (28) (30) for =2transmit antennas with Γ=2Theset of complex symbol vectors X was obtained via (43) by applying Vandermonde transforms over a signal set Ω 4K for K = 1, 2, 3, 4 The Vandermonde transforms were determined for different K values according to (45) and (46) The constellation Ω was chosen to be binary phase-shift keying Thus, the spectral efficiency of the resulting STF codes was 1 bit/s/hz (omitting the cyclic prefix), which is independent of the number of jointly encoded OFDM blocks K The performance of the full-rate STF codes are depicted in Figs 5 7 for the three different temporal correlation scenarios From the figures, we observe that the diversity order of the STF

10 1856 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 4, NO 4, JULY 2005 Fig 6 Performance of the full-rate STF codes, ε =08 fading channels, while in the SF-coded broadband MIMO- OFDM systems, the maximum achievable diversity order is L M r The factor L comes from the additional frequency diversity due to the spread fading In this paper, we explored different channel coding approaches for MIMO-OFDM systems by taking into account all opportunities for performance improvement in both the spatial, the temporal, and the frequency domains in terms of the achievable diversity order First, we developed a general framework for the performance analysis of STF-coded MIMO-OFDM systems, incorporating the ST and SF coding approaches as special cases Then, we derived the code performance criteria and showed that the maximum achievable diversity order is L M r T, T is the rank of the temporal correlation matrix of the channel Moreover, we proposed two STF code design methods that are guaranteed to achieve the maximum achievable diversity order The simulation results showed that by coding across multiple OFDM blocks, the diversity order of the code can be increased significantly, and the achieved improvement depends on the temporal correlation Fig 7 Performance of the full-rate STF codes, ε =095 codes increases with the number of jointly encoded OFDM blocks K However, the improvement of the diversity order depends on the temporal correlation The performance gain obtained by coding across multiple OFDM blocks decreases as the correlation factor ε increases For example, without temporal correlation (ε =0), the STF code with K =4achieves an average BER of about at an SNR of 16 db In case of the correlated channel model and ε =08, the STF code with K =4has an average BER of only at an SNR of 16 db Finally, in case of the correlated channel model and ε =095, the STF code with K =4has an average BER of around 10 4 at an SNR of 16 db VI CONCLUSION In narrowband MIMO wireless communications, the maximum achievable diversity order is M r for quasi-static REFERENCES [1] J-C Guey, M P Fitz, M R Bell, and W-Y Kuo, Signal design for transmitter diversity wireless communication systems over Rayleigh fading channels, in Proc IEEE Vehicular Technology Conf (VTC) 96, Atlanta, GA, pp [2], Signal design for transmitter diversity wireless communication systems over Rayleigh fading channels, IEEE Trans Commun, vol 47, no 4, pp , Apr 1999 [3] V Tarokh, N Seshadri, and A R Calderbank, Space time codes for high data rate wireless communication: Performance criterion and code construction, IEEE Trans Inf Theory, vol 44, no 2, pp , Mar 1998 [4] S Alamouti, A simple transmit diversity technique for wireless communications, IEEE J Sel Areas Commun, vol 16, no 8, pp , Oct 1998 [5] V Tarokh, H Jafarkhani, and A R Calderbank, Space time block codes from orthogonal designs, IEEE Trans Inf Theory, vol 45, no 5, pp , Jul 1999 [6] B M Hochwald and T L Marzetta, Unitary space time modulation for multiple-antenna communication in Rayleigh flat fading, IEEE Trans Inf Theory, vol 46, no 2, pp , Mar 2000 [7] B M Hochwald, T L Marzetta, T J Richardson, W Swelden, and R Urbanke, Systematic design of unitary space time constellations, IEEE Trans Inf Theory, vol 46, no 6, pp , Sep 2000 [8] M O Damen, K Abed-Meraim, and J C Belfiore, Diagonal algebraic space time block codes, IEEE Trans Inf Theory, vol 48,no 3,pp , Mar 2002 [9] Y Xin, Z Wang, and G B Giannakis, Space time diversity systems based on linear constellation precoding, IEEE Trans Wireless Commun, vol 2, no 2, pp , Mar 2003 [10] D Agrawal, V Tarokh, A Naguib, and N Seshadri, Space time coded OFDM for high data-rate wireless communication over wideband channels, in Proc IEEE Vehicular Technology Conf (VTC) 98, Ottawa, Canada, May 1998, pp [11] K Lee and D Williams, A space frequency transmitter diversity technique for OFDM systems, in Proc IEEE Global Telecommunications Conf (GLOBECOM), San Francisco, CA, 2000, vol 3, pp [12] R Blum, Y Li, J Winters, and Q Yan, Improved space time coding for MIMO-OFDM wireless communications, IEEE Trans Commun, vol 49, no 11, pp , Nov 2001 [13] Y Gong and K B Letaief, An efficient space frequency coded wideband OFDM system for wireless communications, in Proc IEEE Int Conf Communications (ICC) 02, New York, vol 1, pp [14] Z Hong and B Hughes, Robust space time codes for broadband OFDM systems, in Proc IEEE Wireless Communications and Networking Conf (WCNC), Orlando, FL, Mar 2002, vol 1, pp

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no 5, pp , 2002 [32] Z Safar, W Su, and K J R Liu, A fast sphere decoding framework for space frequency block codes, in Proc IEEE Int Conf Communications (ICC), Paris, France, 2004, pp [33] Q Yan and R Blum, Optimum space time convolutional codes, in Wireless Communications and Networking Conf, Chicago, IL, 2000, vol 3, pp [34] H S Wang and N Moayeri, Finite-state Markov channel A useful model for radio communication channels, IEEE Trans Veh Technol, vol 44, no 1, pp , Feb 1995 [35] H S Wang and P-C Chang, On verifying the first-order Markovian assumption for a Rayleigh fading channel model, IEEE Trans Veh Technol, vol 45, no 2, pp , May 1996 [36] G Stuber, Principles of Mobile Communication Boston, MA: Kluwer, 2001 [37] R A Horn and C R Johnson, Matrix Analysis Cambridge, UK: Cambridge Univ Press, 1985 [38], Topics in Matrix Analysis Cambridge, UK: Cambridge Univ Press, 1991 Weifeng Su (M 03) received the BS and PhD degrees in mathematics from Nankai University, Tianjin, China, in 1994 and 1999, respectively He also received the PhD degree in electrical engineering from the University of Delaware, Newark, in 2002 From June 2002 to March 2005, he was a Postdoctoral Research Associate at the Department of Electrical and Computer Engineering and the Institute for Systems Research (ISR), University of Maryland, College Park Currently, he is an Assistant Professor at the Department of Electrical Engineering, State University of New York (SUNY) at Buffalo His research interests span a broad range of areas from signal processing to wireless communications and networking, including space time coding and modulation for multiple-input multiple-output (MIMO) wireless communications, cooperative communications for wireless networks, and ultrawideband (UWB) communications He has published more than 16 journal papers and 24 conference papers in the related areas Dr Su received the Signal Processing and Communications Faculty Award from the University of Delaware in 2002 as an outstanding graduate student in the field of signal processing and communications He also received the Competitive University Fellowship Award from the University of Delaware in 2001 Zoltan Safar (M 04) received the University Diploma in electrical engineering from the Technical University of Budapest, Budapest, Hungary, in 1996, and the MS and PhD degrees in electrical and computer engineering from the University of Maryland, College Park, MD, USA, in 2001 and 2003, respectively From September 2003 to March 2005, he was an Assistant Professor at the Department of Innovation, IT University of Copenhagen, Copenhagen, Denmark Currently, he is a Senior Design Engineer at Modem System Design, Nokia, in Copenhagen, Denmark His research interests include wireless communications and multimedia signal processing, with particular focus on multiple-input multiple-output (MIMO) wireless communication systems and space time and space frequency coding Dr Safar received the Outstanding Systems Engineering Graduate Student Award from the Institute for Systems Research, University of Maryland, in 2003 K J Ray Liu (F 03) received the BS degree from the National Taiwan University, Taipei, Taiwan, in 1983, and the PhD degree from University of California, Los Angeles (UCLA), in 1990, both in electrical engineering He is Professor and Director of Communications and Signal Processing Laboratories of Electrical and Computer Engineering Department and Institute for Systems Research, University of Maryland, College Park His research contributions encompass broad aspects of wireless communications and networking; information forensics and security; multimedia communications and signal processing; signal processing algorithms and architectures, and bioinformatics, in which he has published over 350 refereed papers Dr Liu is the recipient of numerous honors and awards including the IEEE Signal Processing Society 2004 Distinguished Lecturer, the 1994 National Science Foundation Young Investigator Award, the IEEE Signal Processing Society s 1993 Senior Award (Best Paper Award), IEEE 50th Vehicular Technology Conference Best Paper Award, Amsterdam, 1999, and EURASIP 2004 Meritorious Service Award He also received the George Corcoran Award in 1994 for outstanding contributions to electrical engineering education and the Outstanding Systems Engineering Faculty Award in 1996 in recognition of outstanding contributions in interdisciplinary research, both from the University of Maryland Dr Liu is the Editor-in-Chief of IEEE Signal Processing Magazine, the prime proposer and architect of the new IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, and the founding Editor-in-Chief of EURASIP Journal on Applied Signal Processing