Low Complexity Transmitter Architectures for SFBC ... - IEEE Xplore

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 6, JUNE 2012

Low Complexity Transmitter Architectures for SFBC MIMO-OFDM Systems Chih-Peng Li, Senior Member, IEEE, Sen-Hung Wang, Member, IEEE, and Kuei-Cheng Chan

Abstract—Multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems with spacefrequency block coding (SFBC) have a high computational complexity since the number of inverse fast Fourier transforms (IFFTs) required scales in direct proportion to the number of antennas at the transmitter. This paper proposes to generate the SFBC encoded signals of the various antennas in time domain by exploiting the time-domain signal properties and signal correlations among the various transmitter antennas, achieving a significant reduction in computational complexity. In particular, it is demonstrated that the time domain SFBC encoded signals of the various antennas can be obtained from the time domain signal of the first antenna. Therefore, the proposed scheme requires only one IFFT irrespective of the number of transmission antennas. In addition, a low-complexity peak-toaverage power ratio (PAPR) reduction scheme is presented based on the proposed transmitter architectures. Index Terms—Multiple-input multiple-output (MIMO), orthogonal frequency division multiplexing (OFDM), spacefrequency block coding (SFBC), peak-to-average power ratio (PAPR).

I. I NTRODUCTION ULTIPLE-INPUT multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems with space-frequency block coding (SFBC) [1]-[7] have attracted substantial attention due to their robust performance in time selective fading channels. However, the transmitters in SFBC MIMO-OFDM systems have a relatively high computational complexity since they are required to perform one inverse fast Fourier transform (IFFT) operation for each transmission antenna. To reduce the system complexity, time domain encoding approach has been proposed in [6] and [7]. However, the proposed methods are only applicable to a fraction of known SFBC encoding schemes and still have a high computational complexity. The other major drawback of SFBC MIMO-OFDM systems is the high peak-to-average power ratio (PAPR) due to the use of the OFDM technique. Various PAPR reduction methods have been proposed in the literature [8]-[15]. In particular,

M

Paper approved by N. Al-Dhahir, the Editor for Space-Time, OFDM and Equalization of the IEEE Communications Society. Manuscript received October 8, 2010; revised May 1 and November 27, 2011. This paper was presented in part at the IEEE International Conference on Communications, 23-27 May, 2010. C.-P. Li (corresponding author) is with the Institute of Communications Engineering and the Department of Electrical Engineering, National Sun YatSen University, Kaohsiung, Taiwan 804 (e-mail: [email protected]). S.-H. Wang is with the Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan (e-mail: [email protected]). K.-C. Chan is with the Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2012.041212.100613

traditional selected mapping (SLM) scheme is a distortionless method, but requires a bank of IFFTs to generate various candidate signals. This paper presents five time domain signal properties. Then, it is shown that the time domain SFBC encoded signals of the various antennas can be obtained from the time domain transmitted signal of the first antenna. Therefore, the proposed scheme requires only one IFFT irrespective of the number of transmission antennas. In addition, the proposed scheme can be applied to all known SFBC encoding schemes. Finally, a low-complexity peak-to-average power ratio (PAPR) reduction scheme is presented based on the proposed transmitter architecture. The remainder of this paper is organized as follows. Section II describes the SFBC encoding scheme and the SFBC MIMOOFDM system model. Section III presents the five timedomain signal properties used to construct the transmitter architectures and PAPR reduction scheme proposed in this study. Section IV illustrates the approach that we adopted to generate the SFBC encoded signals of various antennas in time domain. Section V describes the proposed low-complexity transmitter architecture and Section VI demonstrates the proposed lowcomplexity PAPR reduction scheme. Section VII evaluates the computational complexity of the proposed transmitter architectures and PAPR reduction schemes. Section VIII presents the simulation results. Finally, Section IX provides some brief concluding remarks. II. E NCODING S CHEMES AND S YSTEM M ODELS FOR SFBC MIMO-OFDM S YSTEMS This paper considers a SFBC MIMO-OFDM system with M transmitter antennas and N sub-carriers. An assumption is made that the N sub-carriers are partitioned into N/U groups, where each group consists of U consecutive sub-carriers. It is further assumed that both N and U are non-negative integer powers of two and N/U is a positive integer. The literature contains various SFBC encoding schemes for MIMO-OFDM systems [16]-[18]. In this study, it is assumed that the MIMOOFDM system uses a general M × U encoding matrix with a code rate of Q/U , i.e. ⎤ ⎡ S1,0 S1,1 · · · S1,U−1 ⎥ ⎢ .. .. .. (1) S = ⎣ ... ⎦, . . . SM,0 SM,1 · · · SM,U−1 Q−1 where Sm,u = q=0 {κm,u,q · X[q] + λm,u,q · X ∗ [q]}, X[q] is the qth data symbol, κm,u,q and λm,u,q are complex coefficients. It is assumed that the elements in the second to

c 2012 IEEE 0090-6778/12$31.00

LI et al.: LOW COMPLEXITY TRANSMITTER ARCHITECTURES FOR SFBC MIMO-OFDM SYSTEMS

M th row of S are linear combinations of the elements in the first row and their complex conjugates, i.e. Sm,u =

U−1

∗ μm,u,v ·S1,v +νm,u,v ·S1,v , m > 1,

(2)

v=0

where μm,u,q and νm,u,q are complex coefficients. Almost all the SFBC encoding schemes presented in the literature satisfy the above described conditions. Let the frequency-domain encoded vector of each antenna be denoted as Xm . Performing IFFT on Xm produces the time-domain signal vectors xm . The nth element of xm is written as N−1 j2πnk 1 xm [n] = √ Xm [k] · exp , N N k=0

m = 1, 2, . . . , M, (3)

where Xm [k] is the kth element of the Xm . Therefore, in the discrete-time case, the PAPR of the signals transmitted from each antenna is defined as max |xm [n]|2 0≤n≤N −1 PAPR(xm ) = , m = 1, 2, . . . , M. (4) E [|xm [n]|2 ] Since the signals of the M antennas are transmitted simultaneously, the overall PAPR of the SFBC MIMO-OFDM system is defined as PAPR(x) = max{PAPR(xm )|m = 1, 2, . . . , M }, where x = [x1 x2 · · · xM ]. The PAPR reduction performance is usually evaluated using the complementary cumulative distribution function (CCDF), defined as the probability of the PAPR exceeding a certain threshold γ, i.e. CCDFPAPR(x) = Pr (PAPR(x) > γ). III. T IME -D OMAIN S IGNAL P ROPERTIES FOR SFBC MIMO-OFDM S YSTEMS Consider an N × 1 encoded data vector Xm of the mth antenna. Then define an interleaved sub-carrier index set TU u ≡ {u + i · U |i = 0, 1, . . . , N − 1}, where u = 0, 1, . . . , U − 1. U U−1 U , where X Xm can then be written as Xm = u=0 XU m,u m,u is an N × 1 vector, and the kth element of XU m,u has a nonU zero value only when k ∈ TU u , i.e. Xm,u consists of data from interleaved sub-carriers. Let the IFFTs of Xm and XU m,u be denoted as xm and xU m,u , respectively. Since IFFT is a linear U−1 operation, it is easily shown from Xm = u=0 XU m,u that U−1

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U U kth element of GU u , Gu [k], equals one if k ∈ Tu . Otherwise, U U Gu [k] = 0. Performing IFFT on both sides of XU m,u = Gu ⊗ Xm yields the desired results. We note that XU m,u contains data from interleaved subcarriers. Property 1 states that the IFFT of XU m,u can be obtained without IFFT operation if the time domain signal U U −1 xm is given, i.e. F −1 {XU m,u } = xm,u = gu · xm , where F denotes the IFFT operation. Property 1 also shows that xm can be partitioned in time domain into U different streams, xU m,u , u = 0, . . . U − 1, by using Eq. (6), where each stream consists of data from different set of interleaved sub-carriers U−1 and xm = u=0 xU m,u . T

U U U U Property 2: xm,u = xU x x · · · x , m,u,0 m,u,1 m,u,2 m,u,U−1

N where xU m,u,0 is an 1 × U vector consisting of the N U U U first U elements of xm,u , xU m,u,q = αu,q · xm,u,0 , and U αu,q = exp(j2πuq/U ), q = 1, 2, . . . , U − 1. Proof: From Eqs. (6) and (7), the nth and N U (n + N U )th elements of xm,u , n = 0, 1, . . . , U − 1, U = can be written respectively as xm,u [n] U−1 −j2πuw/U U e · x [(n + wN/U ) ], and x [n + m N m,u w=0 U−1 −j2πuw/U · xm [(n + wN/U + N/U )N ] = N/U ] = w=0 e U ej2πu/U · xU = exp(j2πu/U ). m,u [n]. Therefore, αu,1 Similarly, it can be shown that αU u,q = exp(j2πuq/U ), q = 1, 2, . . . , U − 1. We note that the signal xU m,u contains the data from interleaved sub-carriers with a sub-carrier spacing of U . Property 2 shows that xU m,u consists of U identical signal fragments multiplied by various complex numbers. The same property can also be found in the OFDMA systems [14]. = Property 3: For U > 4, xU m,u

T U U U U U bm,u,0 bm,u,1 bm,u,2 bm,u,3 , where bm,u,0 is an N U 1× N 4 vector consisting of the first 4 elements of xm,u , U U U U bm,u,i = βu,i ·bm,u,0 , and βu,i = exp{j2πiu/4}, i = 1,2,3. Proof: From Property 2, bU m,u,0 can be written as

T U U U U U bU m,u,0 = xm,u,0 αu,1 · xm,u,0 · · · αu,U/4−1 · xm,u,0 . In

addition, the { N4 · i}th to the { N4 · (i + 1) − 1}th elements of xU m,u are given by

U U U U bU m,u,i = αu,iU/4 · xm,u,0 αu,iU/4+1 · xm,u,0 · · · T U αU = ej2πiu/4 · bU m,u,0 . (8) u,(i+1)U/4−1 · xm,u,0

U Therefore, βu,i = exp(j2πiu/4) and the desired result is obtained. u=0 Property 3 shows that, for U > 4, xU m,u consists of Property 1: The IFFT of XU m,u is given by 4 identical signal fragments multiplied by various complex U xU u = 0, 1, . . . , U − 1, (6) numbers. m,u = gu · xm , N Property 4: Let ZU m,u denote an U ×1 vector in which the U U where gu is an N × N circulant matrix. Let gu,p denote the U U kth element is given by Zm,u [k] = Xm [u+k·U ] = Xm,u [u+k·U ], pth row of guU , p = 0, . . . , N − 1 and (·)N denote the modulo N U k = 0, 1, . . . , U − 1. xm,u,0 in Property 2 has the following U N operation. The nth element of gu,p , n = 0, . . . , N − 1, has form: the form 1 j2πu(n−p) U N · RU xU (9) − N m,u,0 = √ u · zm,u , , n ∈ {(w· +p) |w = 0, 1, . . . ,U −1.} e U N U U gu,p [n]= 0, otherwise. N N U U U (7) where zm,u is the IFFT of Zm,u , Ru is an U × U diagonal U U U U U U U Proof: We note that Xm,u = Gu ⊗Xm , where ⊗ denotes matrix, Ru = diag{Ru,0 , Ru,1 , . . . , Ru,N/U−1 }, and Ru,n = N U component-wise multiplication, Gu is an N ×1 vector, and the exp{j2πnu/N }, n = 0, 1, . . . , U − 1.

xm =

xU m,u .

(5)

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of


U Proof: Since zU m,u is the IFFT of Zm,u , the nth element can be written as

zU m,u

U zm,u [n]

= =

N

−1 j2πnkU U U U Zm,u [k] · e N N k=0

N U

−1 j2πnkU U U Xm,u [u + k · U ] · e N . N

(10)

k=0

In addition, the nth element of xU m,u,0 is given by N −1 j2πnk 1 U U xm,u,0 [n] = Xm,u [k] · e N N k=0 N −1 j2πn(u+k ·U ) 1 U U N = Xm,u [u + k · U ] · e N k =0 N −1 j2πn·u j2πn·k ·U 1 U U =e N · Xm,u [u + k · U ] · e N N k =0

j2πn·u 1 U = √ · zm,u [n] · e N . U

(11)

Property 4 shows that the first fragment, xU m,u,0 , of the in Property 1 can also be obtained with signal streams xU m,u -point IFFTs followed by reduced complexity by using U N U proper phase rotations. xU can then be easily obtained from m,u xU m,u,0 by using Property 2. U,s Property 5: Define D ≡ XU,s m,u , where Xm,u denotes the sth U cyclic shift of Xm,u . Performing IFFT on D yields d. The nth j2πsn/N elements of d can be written as d[n] = xU . In m,u [n] · e ∗ ) . Performing IFFT on E gives addition, define E ≡ (XU m,u ∗ e. The nth element of e is given by e[n] = xU , m,u (−n)N where (·)N denotes the modulo N operation. Note that this property is a direct result of IFFT and thus the corresponding proof is omitted. Remark 1: In Property 1, the nonzero elements of guU belong to the set {±1, ±j} when U ≤ 4. Therefore, applying Eq. (6) requires the use only of complex additions. Remark 2: In Property 2, αU u,q belongs to the set {±1, ±j} when U ≤ 4. Therefore, applying Property 2 does not require any complex additions or multiplications. U ∈ {±1, ±j}, i = 1, 2, 3, u = Remark 3: In Property 3, βu,i 0, 1, . . . , U − 1. Therefore, the last 3N/4 elements of xU m,u can be easily obtained from the first N/4 elements of xU m,u without the need for complex multiplications or additions. Remark 4: In Property 4, the diagonal elements of RU u belong to the set {±1, ±j} when 4nu is an integer. N IV. G ENERATING THE SFBC E NCODED S IGNALS OF VARIOUS A NTENNAS IN T IME D OMAIN In the traditional SFBC MIMO-OFDM transmitters, the signal is SFBC encoded in frequency domain and each antenna requires an individual IFFT operation. To overcome this drawback, we propose to generate the SFBC encoded signals for the various antennas in time domain. This section shows that the time-domain SFBC encoded signals of the various antennas can be obtained from the time-domain transmitted

signal of the first antenna. Therefore, our proposed transmitter only requires a single IFFT operation (for generating the signal of the first antenna). In SFBC MIMO-OFDM systems, the time-domain trans, m = 1, 2, . . . , M , mitted signals of the various antennas, xm U−1 can be obtained from Eq. (5), i.e. xm = u=0 xU m,u , where U U xm,u ≡ gu · xm as shown in Property 1. Since the frequencydomain signal elements of the second to the M th antennas are linear combinations of the frequency-domain signal elements of the first-antenna and their complex conjugates (see Eq. (2)), U the FFT of xU m,u , i.e. Xm,u , can be written as XU m,u =

U−1

U ∗ μm,u,v ·XU , m > 1, (12) +ν · X1,v m,u,v 1,v

v=0

XU 1,v

where is the FFT of xU 1,v . Therefore, the nth time-domain , n = 0, 1, . . . , N − 1, can be obtained signal element of xU m,u from Property 5 as U−1 j2π(u−v)n/N xU [n] = μm,u,v · xU m,u 1,v [n] · e v=0

U ∗ j2π(u−v)n/N + νm,u,v · x1,v [(−n)N ] · e . (13) As shown in Eq. (5), the time-domain signals of the various antennas xm can be obtained from the summation of xU m,u , which in turn can be generated by linearly combining xU 1,u and their complex conjugates as shown in Eq. (13). Because xU 1,u is obtained from the time-domain transmitted signal of U the first antenna by using Property 1, i.e. xU 1,u = gu · x1 , we reach an important conclusion that the time-domain SFBC encoded signals of the various antennas can be obtained by using the time-domain transmitted signal of the first antenna x1 . Therefore, our proposed architecture only requires a single IFFT operation to generate x1 . V. T HE P ROPOSED L OW C OMPLEXITY T RANSMITTER A RCHITECTURE Figure 1 illustrates the basic blocks of our proposed transmitter. In the stage 1 (Signal Partition, SP), the data vector X is SFBC encoded to obtained the frequency domain signal of the first antenna, X1 , and then the IFFT operation is applied to obtain the time domain signal x1 . By using Property 1, x1 is partitioned into U streams, xU 1,u , u = 0, 1, . . . , U − 1, where each stream consists of signals from interleaved subcarriers. In the stage 2 (Antenna Signal Generator, ASG), the SFBC encoded signals of various antennas can be generated by linearly combining these U signal streams and their complex conjugates, where Property 5 and Eqs. (13) and (5) are utilized. A. Proposed Architecture Figure 2 presents a block diagram of the proposed transmitter architecture (designated hereafter as Proposed Architecture and annotated as PA), where two additional complexity reductions are made to Fig. 1. First of all, the various signal streams xU 1,u in Fig. 1 are obtained by using an N -point IFFT followed by U time-domain signal partition operations. As shown in


Stage 1

X1 N-point x1 IFFT

gU0 g1U g

U U 1

xU2,0

U x1,0

x

U 1,0

U 1,1

x

U 1,1

*

n N

(Property 5)

xU2,1

*

n N

(Property 5)

x U 1,U 1

x

Signal Partition (SP) (Property 1) Fig. 1.

x1

Stage 2

Eq. (13)

X

SFBC Encoder for Antenna 1

1715

x1,UU 1 n N

(Property 5)

*

ș x2 Eq. (5)

U 2,U 1

x

xUM ,0 xUM ,1

ș

xUM ,U 1

xM

Eq. (5)

Signal Generator (SG) Antenna Signal Generator (ASG)

The basic function blocks of the proposed transmitter architecture.

Property 4, the same xU 1,u signal streams can be obtained with reduced complexity by using U N U -point IFFTs followed by √1 · RU · zU , where proper phase rotations, i.e. xU 1,u,0 = u 1,u U N N U U zU is the -point IFFT of Z and Z is an 1,u 1,u 1,u U U × 1 vector U [k] = X1 [k · U + u]. with the kth element given by Z1,u Secondly, the complexity is further reduced by utilizing Property 2 (see the Signal Fragment Repetition and Multiplication I (SFRM I) block) since each signal stream xU 1,u in , m = 2, 3, . . . , M , in the ASG I block the SP I block and xU m,u consists of U identical signal fragments multiplied by various N complex numbers αU u,q . As a result, only the first U elements need be obtained. Consider the case of U ≤ 4. Remark 2 indicates that applying Property 2 involves no complex operations. Therefore, the Proposed Architecture is particularly suitable for the case of U ≤ 4.

(CSPRs) are performed on the set of signal streams xU 1,u (p) in order to obtain a set of modified signal streams x1,u . Specifically, in obtaining the pth candidate signal set of the various antennas, the CSPR block applies u,p cyclic shifts and a phase rotation of ru,p to each of the U outputs of SP I or MSP I, where u,p ∈ {0, 1, . . . , LN − 1} and (p) ru,p ∈ {±1, ±j}. The nth element of x1,u has the form (p) x1,u [n] = ru,p · xU 1,u [(n − u,p )LN ]. The pth candidate signal (p) set, xm , m = 2, 3, . . . , M , is constructed by the ASG I or U−1 (p) (p) MASG I using Eqs. (5) and (13) where xm = u=0 xm,u . (p) The nth element of xm,u is written as U−1 j2π(u−v)n (p) x(p) μm,u,v · x1,v [n] · e LN [n] = m,u

B. Modified Proposed Architecture (for U > 4)

After generating P candidate signal sets, the signal set with the lowest PAPR is selected for transmission and a total of log2 P side information bits are required for decoding. It should be noted here that the proposed PAPR reduction scheme is basically a modified SLM scheme.

For the case of U > 4, applying Property 2 requires the use of complex multiplications. Consequently, as illustrated in Fig. 3, a modified architecture (designated hereafter as Modified Proposed Architecture and annotated as MPA) is proposed. U U As shown, once the first N U elements of x1,u , i.e. x1,u,0 , have been obtained, Property 2 is applied to obtain the first N4 U elements of xU 1,u , i.e. b1,u,0 . Property 3, i.e. the SFRM II block, is then used to obtain the remaining 3N 4 elements of xU . As indicated in Remark 3, applying Property 3 requires 1,u no complex operations. VI. T HE P ROPOSED L OW C OMPLEXITY PAPR R EDUCTION S CHEME Figure 4 presents the proposed low-complexity PAPR reduction scheme, where various cyclic shifts and phase rotations

v=0

+ νm,u,v ·

(p) {x1,v [(−n)LN ]}∗

·e

j2π(u−v)n LN

.

(14)

VII. A NALYSIS OF C OMPUTATIONAL C OMPLEXITY In performing the analysis, the complexity of the SFBC encoding process is deliberately excluded since the encoding complexity is the same in every case. In addition, the analysis consider the case in which only one of νm,u,v and μm,u,v in Eq. (13) has a non-zero value, i.e. U xU m,u [n] = μm,u,v · x1,v [n] · e

or

j2π(u−v)n N

U ∗ j2π(u−v)n N ·e . xU m,u [n] = νm,u,v · x1,v [(−n)N ]

(15)

(16)


2U−1

N−U N−U+1 N−1

X1

Z1,UU

1

N/U

x

z

R1U

N/U

0

1 1

(Property 4)

0

U 2,1,0

x

U 1,U 1

U x1,1

(U-1)N/U N-1

0 : N U 1 U U 1,1

N/U

U U 1,U 1

U 1,U 1

x

(U-1)N/U N-1

Signal Fragment Repetition and Multiplication I (SFRM I) (Property 2)

Signal Partition I (SP I) Fig. 2.

x

(U-1)N/U N-1

U 1,1

1

x

1

U 2,0,0

0 : N U 1

U 1,U 1,0

U 1,U 1

RUU

1

0 U 1,1,0

U 1,1

z

1

Eq. (5)

U x1,0

xU2,U 1,0

U 2,0

x

SFRM I

RU0

Σ x1

xU2,1 xU2,U 1

xUM ,0,0

xUM ,0

xUM ,1,0

xUM ,1

xUM ,U 1,0

SFRM I

U Z1,1

U x1,0,0 0 : N U 1

U z1,0

SG (See Fig. 1)

U−1 U U+1

N/U-point IFFT

U Z1,0

N/U-point IFFT

X


0 1

N/U-point IFFT

1716

xUM ,U 1

Σ x2 Eq. (5)

Σ xM Eq. (5)

Antenna Signal Generator I (ASG I)

Block diagram of the proposed low-complexity transmitter architecture (Proposed Architecture, PA).

It is noted that most existing encoding matrices meet this constraint. Because of page limitation, the computation complexity is presented in TABLE I without detail explanations. Numerical results reveal that the computational complexity of our proposed transmitter architecture is lower than that of the traditional scheme. For Jafarkhani’s code with N = 1024, M = 4, and U = 4, the proposed scheme reduces the number of complex multiplications and complex additions by 65% and 50%, respectively, compared to the traditional transmitter. Figure 5 presents the computational complexity of the various PAPR reduction schemes with M = 4, N = 1024, and U = 4. For P = 64 candidate signal sets and S = 64 cyclic shifts, the number of complex multiplications and additions required in PA is just 15.31% and 30.31%, respectively, of that required in the traditional SLM scheme. Similarly, for P = 64 and S = 1, the number of complex multiplications required in PA is just 0.55%. VIII. S IMULATION R ESULTS Simulation experiments are conducted to evaluated the PAPR reduction performance of the proposed scheme and the traditional SLM scheme, where the SFBC MIMO-OFDM system consists of four transmitter antennas and uses the Jafarkhani encoding matrix. In addition, it is assumed that the data are 16-QAM modulated and are transmitted using N = 1024 sub-carriers. Simulations (results omitted here for reasons of page limitation) have shown that the performance of the proposed PAPR reduction scheme is marginally poorer than that of the traditional SLM scheme. For P = S = 64 and Pr(PAPR > γ) = 10−3 , the maximum performance loss is no more than 0.38 dB. However, the proposed PAPR reduction scheme reduces the number of complex multiplications and additions to just 15.31% and 30.31%, respectively. For P = 64

and S = 16, the proposed scheme results in a maximum performance loss of no more than 0.46 dB, but reduces the number of complex multiplications to just 4.06%. We note that although the phase-rotation operation does not require additional complex operations to realize Eq. (14), simulation results show that the cyclic-shift operation results in a better PAPR reduction performance. Our study shows that S = 16 represents an ideal compromise. IX. C ONCLUSION This study has presented a low-complexity transmitter architecture for SFBC MIMO-OFDM systems. The numerical results have shown that the proposed architecture yields a significant reduction in the computational complexity of the SFBC MIMO-OFDM transmitter. In particular, for Jafarkhani’s code with N = 1024, M = 4, and U = 4, the proposed scheme reduces the number of complex multiplications and complex additions by 65% and 50%, respectively, compared to the traditional SFBC MIMO-OFDM. Based on the proposed transmitter architecture, a PAPR reduction scheme for SFBC MIMO-OFDM systems has also been proposed. The simulation results have shown that the performance of the proposed PAPR reduction scheme is only marginally poorer than that of the traditional SLM scheme. For example, for the case of P = 64, S = 16, and Pr(PAPR > γ) = 10−3 , the proposed scheme results in a maximum performance loss of no more than 0.46 dB compared to that of the traditional SLM scheme, but reduces the number of complex multiplications to just 4.06% of that required in the traditional SLM scheme. ACKNOWLEDGMENT The authors would like to thank the National Science Council and the National Science & Technology Program

U x1,0,0 0 : N U 1

R

1

U 0

U Z1,1

2U−1

N/U-point IFFT

0 U 1,1,0

x

U z1,1

R

0

N−U N−U+1

Z1,UU

1

N−1

z1,UU

N/U-1

N/4-1

0 : N U 1

R

0

U 1,1,0

b

U U 1 4

N/4-1

N/U-1

N/2

0 : N

U 0,3

1 N/4

0

b

U

U 1, 1 U 1 4

N/4-1

N/U-1

1

U 1,2

N/2

U 1,3

U x1,1

3N/4 N-1

4 1

U U 1,2

U U 1,3

x1,UU 1

N/4 N/2 3N/4 N-1 Signal Fragment Repetition and Multiplication II (SFRM II) (Property 3)

bU2,U 1,0

xU2,0 xU2,1

ș x2

xU2,U 1

Eq. (5)

bUM ,0,0

xUM ,0

bUM ,1,0

xUM ,1

bUM ,U 1,0

xUM ,U 1

ș xM Eq. (5)

0

Modified Antenna Signal Generator I (MASG I)

Modified Signal Partition I (MSP I) Fig. 3.

bU2,1,0

3N/4 N-1

0 : N U U 1,1

U x1,0

4 1

U 1,1

U 1,U 1,0

(Property 2)

(Property 4)

N/4

U 0,2

bU2,0,0

1, 1

1

U U 1

U 0,1

1 0

x1,UU 1,0 0 : N U 1

1

X1

U b1,0,0 0 : N 4 1

1

1

U 1

ș x1 Eq. (5)

SFRM II

U z1,0

SFRM II

U−1 U U+1

1717

SG (See Fig. 1)

U Z1,0

N/U-point IFFT

X


0 1

N/U-point IFFT


Block diagram of the Modified Proposed Architecture (MPA) for the case of U > 4. TABLE I C OMPUTATIONAL C OMPLEXITY Transmitter Architectures

Traditional

SP I

PA U ≤4

Complex

Complex

Complex

Multiplications

Additions

Multiplications

Additions

· log2 (N )

N · log2 ( N ) 2 U +( N − 1) · (U − U

1)

N·

log2 ( N U

N · log2 ( N ) U

MASG I

N U · (M − 1)/4

N · (U − 1) · (M − 1)

U 1,1

SP I x or MSP I

x1,UU 1

N · (U − 1) · (M − 1)

( p) x1,0

x1( p )

( p) 1,1

( p) 2

x

x1,( pU)1

ASG I or MASG I

x

x(Mp )

x1 x2 xM

Fig. 4. Architecture of the proposed PAPR reduction scheme for SFBC MIMO-OFDM systems.

for Telecommunications of Taiwan, Republic of China, for financially supporting this research under Contract No. NSC

2)

N S · (M − 1) N 2

Compare PAPR and Select Transmit Signals

Number of

· log2 (N )

N · log2 ( N ) 2 U N +( U − 1) · (U − +( N − 1) U

)

MSP I

Cyclic Shifts and Phase Rotations (CSPR) Eq. (14)

M NP 2

M N · log2 (N )

N · log2 ( N ) 2 U N +( U − 1) · (U − 1) log U −3 +N · i=02 2−i−2 U −1 2

U x1,0

X1

Complex

N · (M − 1)

U >4

X

Number of

ASG I

MPA


Number of

MN 2

Scheme

PAPR Reduction Schemes

Number of

M N P · log2 (N )

N · log2 ( N ) U N · (U − 1) ·(M − 1) · P

· log2 ( N ) + (N − 1) U U

·(U − 2) + ( N − 1) U N log2 U −3 + 2 · i=0 −i−2 U −1 2 N U S · (M − 1)/4

N · log2 ( N ) U

N · (U − 1) ·(M − 1) · P

97-2221-E-110-005-MY2, NSC 99-2219-E-007-006, and NSC 100-2219-E-007-010. R EFERENCES [1] M. Torabi, S. Aïssa, and M. R. Soleymani, “On the BER performance of space-frequency block coded OFDM systems in fading MIMO channels,” IEEE Trans. Wireless Commun., pp. 1366–1373, 2007. [2] W. Su, Z. Safar, and K. J. R. Liu, “Full-rate full-diversity spacefrequency codes with optimum coding advantage,” IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 229–249, Jan. 2005. [3] U.-K. Kwon, D. Kim, and G.-H. Im, “Amplitude clipping and iterative reconstruction of MIMO-OFDM signals with optimum equalization,” IEEE Trans. Wireless Commun., vol. 8, no. 1, pp. 268–277, 2009.

1718


10

Number of Complex Operations

for SFBC MIMO-OFDM systems,” IEEE Signal Process. Lett., vol. 16, no. 11, pp. 941–944, Nov. 2009. [16] S. M. Alamouti, “A simple transmit diversity technique for wireless commuications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451– 1458, Oct. 1998. [17] O. Tirkkonen, A. Boariu, and A. Hottinen, “Minimal nonorthogonality rate 1 space-time block code for 3+ tx antennas,” in Proc. 2000 IEEE ISSSTA, pp. 429–432. [18] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 1456–1467, July 1999.

Multiplications Additions Traditional SLM PA (S=P) PA (S=P/4) PA (S=1)

7

6

10

5

10

4

10

0

10

20 30 40 50 Number of Candidate Signal Sets P

60

70

Fig. 5. Number of complex operations of various PAPR reduction schemes (M = 4, N = 1024, U = 4).

[4] S. D. Muruganathan and A. B. Sesay, “A low-complexity decisiondirected channel-estimation scheme for OFDM systems with spacefrequency diversity in doubly selective fading channels,” IEEE Trans. Veh. Technol., vol. 58, no. 8, pp. 4277–4291, Oct. 2009. [5] S. Lu, B. Narasimhan, and N. Al-Dhahir, “A novel SFBC-OFDM scheme for doubly-selective channels,” IEEE Trans. Veh. Technol., vol. 58, no. 5, pp. 2573–2578, June 2009. [6] A. Lodhi, F. Said, M. Dohler, and A. H. Aghvami, “Time domain implementation of space-time/frequency block codes for OFDM systems,” IEE Proc.-Commun., vol. 153, pp. 633–638, 2006. [7] A. Ahmed and F. Said, “Unified model for time-domain implementation of space block coded OFDM,” Electron. Lett., vol. 45, no. 25, pp. 1336– 1337, Dec. 2009. [8] C.-L. Wang and Y. Ouyang, “Low-complexity selected mapping schemes for peak-to-average power ratio reduction in OFDM systems,” IEEE Trans. Signal Process., vol. 53, no. 12, pp. 4652–4660, 2005. [9] T. Jiang, W. Xiang, P. C. Richardson, D. Qu, and G. Zhu, “On the nonlinear companding transform for reduction in PAPR of MCM signals,” IEEE Trans. Wireless Commun., vol. 6, pp. 2017–2021, 2007. [10] Y. Wang, W. Chen, and C. Tellambura, “PAPR reduction method based on parametric minimum cross entropy for OFDM signals,” IEEE Commun. Lett., vol. 14, no. 6, pp. 563–565, June 2010. [11] H. Lee and M. P. Fitz, “Unitary peak power reduction in multiple transmit antennas,” IEEE Trans. Commun., vol. 56, no. 2, pp. 234–244, Feb. 2008. [12] Z. Latinovi´c and Y. Bar-Ness, “SFBC MIMO-OFDM peak-to-average power ratio reduction by polyphase interleaving and inversion,” IEEE Commun. Lett., vol. 10, no. 4, pp. 266–268, Apr. 2006. [13] C.-P. Li, S.-H. Wang, and C.-L. Wang, “Novel low-complexity SLM schemes for PAPR reduction in OFDM systems,” IEEE Trans. Signal Process., vol. 58, no. 5, pp. 2916–2921, May 2010. [14] S.-H. Wang, J.-C. Sie, and C.-P. Li, “A low-complexity PAPR reduction scheme for OFDMA uplink systems,” IEEE Trans. Wireless Commun., vol. 10, no. 4, pp. 1242–1251, Apr. 2011. [15] S.-H. Wang and C.-P. Li, “A low-complexity PAPR reduction scheme

Chih-Peng Li received his B.S. degree in Physics from National Tsing-Hua University, Hsin-Chu, Taiwan, in June 1989 and the Ph.D. degree in Electrical Engineering from Cornell University, NY, USA, in January 1998. From January 1998 to October 2000, he was a Member of Technical Staff with the Lucent Technologies. From October 2000 to January 2002, he was a Design Engineering and Manager for the Acer Mobile Networks. In 2002, he joined the faculty of the Institute of Communications Engineering, National Sun Yat-sen University, as an assistant professor. He has been promoted to Professor in 2010. His current research interests include baseband signal processing and circuit design, OFDM, and data networks. From August 2003 to July 2005, he has served as the member of the Communications Promotion Committee, Academia-Industry Consortium for Southern Taiwan Science Park. From August 2005 to October 2008, he has served as the Chief of the Division of Industry Collaboration, Office of Research Affairs, NSYSU. From October 2008 to July 2009, he has acted as the Division Chief of the Engineering Technology Research and Promotion Center, NSYSU. From August 2008 to September 2010, he has also served as the General Secretary of the Taiwan Institute of Electrical and Electronic Engineering. He is currently the Associate Editor of the IEEE T RANSACTIONS ON B ROADCASTING, the Chapter Chair of the IEEE Tainan Section Broadcasting Technology Society, and the Member of Editorial Board of the Far East Journal of Electronics and Communications. Sen-Hung Wang received the B.S. degree in electrical engineering from National Dong Hwa University, Hualien, Taiwan, in 2004, the M.S. degree in communications engineering and the Ph.D. degree in electrical engineering from National Sun Yat-Sen University, Kaohsiung, Taiwan, in 2006 and 2010, respectively. His research interests include wireless communication, orthogonal frequency division multiplexing, and baseband signal processing and circuit design.

Kuei-Cheng Chan received the B. S. and M. S. degrees from National Sun Yat-Sen University, Kaohsiung, Taiwan, in 2006 and 2008 respectively. He has been pursuing the Ph.D. degree in the Graduate Institute of Communication Engineering at National Taiwan University since 2008. His current research interests include wireless communication, OFDM systems, statistical signal processing, and digital signal processing.