Quasi-Orthogonal Space-Frequency Block Codes Channels - CiteSeerX
Quasi-Orthogonal Space-Frequency Block Codes for
MIMO OFDM
Channels
Fatemeh Fazel and Hamid Jafarkhani Center for Pervasive Communications and Computing Department of Electrical Engineering and Computer Science University of California, Irvine {fazel, hamidj } @uci.edu Abstract- In this paper, we propose a novel class of SpaceFrequency and Space-Time-Frequency block codes based on Quasi-Orthogonal designs, over a frequency selective Rayleigh fading channel. The proposed codes are able to achieve rate one and full diversity by exploiting the space and multipath diversity gains available in the MIMO-OFDM channel. As the simulation results show, our codes outperform the existing SpaceFrequency block codes in terms of bit error rate performance. Additionally, the proposed Space-Time-Frequency code benefits from a reduced maximum likelihood decoding complexity, which is a huge simplification compared to the existing codes. We also discuss the conditions under which the maximum likelihood decoding for our proposed Quasi-Orthogonal Space-Frequency code is simplified as well. I. INTRODUCTION
A combination of Multiple-Input Multiple-Output (MIMO) and Orthogonal Frequency Division Multiplexing (OFDM) is a promising technique for high data rate broadband wireless systems. A frequency selective channel offers an additional degree of diversity known as multipath or frequency diversity. In a MIMO-OFDM system it is desired to achieve multipath as well as spatial diversity gains. Space-Frequency (SF) and Space-Time-Frequency (STF) codes have been designed to achieve some levels of space and multipath diversity. Basically, SF codes use the two dimensions of space (antenna) and frequency tones (subcarriers) to code over, while STF codes, code across the three dimensions of space, frequency and time. It is proved that a MIMO-OFDM system can achieve a maximum diversity gain equal to the product of the number of its transmit antennas, the number of its receive antennas and the number of multipaths present in the frequency selective channel as long as the channel correlation matrix is full rank. The design criteria to achieve such diversity gains are presented in [1], [2] and [3]. Space-time coded OFDM was first introduced in [4] by using space-time trellis codes over frequency tones. Authors in [5] introduced a space-frequency-time coding method over MIMO-OFDM channels. They used trellis coding to code over space and frequency and space-time block codes to code over OFDM blocks. The authors used the Alamouti block code structure [6] for the case of two transmit antennas and This work was supported in part by an NSF Career Award CCR-0238042.
argued that for larger numbers of transmit antennas one can use Orthogonal Space-Time Block Code (OSTBC) structures introduced in [7]. It is worthwhile to mention that in case of more than two transmit antennas the OSTBC can provide a rate of at most 3/4 and we are not able to have rate-one transmission. In [8], authors point out the analogy between antennas and frequency tones and based on capacity calculation, propose a grouping method that reduces the complexity of code design for MIMO-OFDM systems. The idea of subcarrier grouping is further pursued in [2] and [9] with precoding and in [10] with bit interleaving. Reference [11] proposes a repetition mapping technique to transform the existing spacetime codes, designed for quasi-static flat fading channels, to construct full-diversity codes in frequency selective fading channels. Note that their proposed method provides a tradeoff between diversity and symbol rate. Later on, the authors proposed a rate-one, full-diversity space-frequency block code in [12]. Their proposed scheme can obtain a target diversity gain but the decoding complexity grows exponentially with the desired diversity. We use their design as a reference to compare our proposed structure in terms of performance and
complexity. Quasi-orthogonal code structures were first introduced in [13] and [14]. Original Quasi-orthogonal designs provide fullrate codes and pairwise Maximum Likelihood (ML) decoding but fail to achieve full-diversity. Later on, improved quasiorthogonal codes were proposed through constellation rotation [15]-[19]. A rotated Quasi-Orthogonal Space-Time Block Code (QOSTBC) provides both full diversity and full rate and performs better in low and high SNR, compared to OSTBC. These benefits together with the simple decoding capabilities of rotated quasi-orthogonal codes, motivate us to design Space-Frequency codes based on quasi-orthogonal structures. In this paper, we provide a systematic method of designing rate-one, full-diversity space-frequency codes for two transmit antennas, using quasi-orthogonal space time block codes designed for frequency-flat quasi-static channels. We specifically construct one sample quasi-orthogonal space-frequency code. As the simulation results suggest, the proposed code has a better performance and under certain conditions, provides reduced decoding complexity compared to the existing rate-
5383
U.S. Government work not protected by U.S. copyright This full text paper was peer reviewed at the direction ofIEEE Communications Society subject matter expertsfor publication in the IEEE ICC 2006 proceedings. Authorized licensed use limited to: Univ of Calif Irvine. Downloaded on November 23, 2008 at 14:45 from IEEE Xplore. Restrictions apply.
one codes. Then, assuming that the channel is quasi-static over two OFDM symbols, we will design a space-time-frequency code which has a reduced decoding complexity under all circumstances. Both SF and STF code structures provide full symbol rate (one symbol per frequency tone per time slot) and achieve any desired multipath (frequency) diversity available in the frequency selective fading channel. The rest of the paper is organized as follows. In Section II, we describe the MIMO-OFDM channel model and the SF code structure in general. In Section III, we introduce a general class of quasi-orthogonal space-time block codes for quasi-static flat fading channels. This class of QOSTBC will then be used as an underlying structure to design rate-one full-diversity spacefrequency codes in Section IV. Then we continue with the design of Quasi-Orthogonal Space-Time-Frequency (QOSTF) block codes in Section V. In Section VI, the decoding of QOSF and QOSTF codes is discussed. Simulation results are presented in Section VII and finally some concluding remarks are provided in Section VIII.
The OFDM transmitter performs an N point IFFT over the frequency tones or equivalently the columns of the codeword matrix C and adds a cyclic prefix to remove the Inter Symbol Interference. At the receiver, the cyclic prefix is removed and an FFT transformation is applied on the frequency tones. The received signal at receive antenna j at the n'th subcarrier is given by lrj (n)
MT
E ci (n)Hij (n) + A(n),
=
where Hij (n) and JVr(n) are the frequency response of the channel and the complex additive white Gaussian noise at the n'th frequency subcarrier. The maximum achievable diversity in the channel model characterized in this section, is LMTMR [1]. To be able to achieve that maximum diversity by using a SF code, the number of subcarriers, N, has to be larger than or equal to the number of independent delay paths, L. III. UNDERLYING CODE STRUCTURE
II. CHANNEL MODEL
In this section, we define the channel model we use throughout the paper. Consider a MIMO-OFDM system with MT transmit and MR receive antennas. We assume that the receiver has perfect channel knowledge while the transmitter does not know the channel. Each channel between transmit antenna and receive antenna j is assumed to have L independent channel taps with a complex amplitude of aij(1) and a delay denoted by Ti associated with the Ith tap. Note that each ovi,j(l) is a zero mean complex Gaussian random variable with a variance of o2. For normalization purposes, we assume that 7j1 5j2 = 1. Also assume that we have N frequency subcarriers. We can represent the channel impulse response for the path between transmit antenna i and receive antenna j at time t as
In this section, we introduce a general class of quasiorthogonal space-time block codes and later on use this class to build space-frequency block codes in Section IV. Let us denote the Alamouti scheme [6] for the two indeterminate variables x1 and X2 by
A(xi,X2)
j++eols2
S
X2
=
Sk+1 +
X3
=
e s2 + Sl-Cjol
(1)
X4
=
S+1
where d is the dirac delta function. A general Space-Frequency (SF) code word is represented by the following N x MT matrix
X2k-1
L-1
E ai,j (1)6(t
-
)
1
Now suppose we have MT = 2k transmit antennas. For a block of 2k symbols, {s1, ...., s2k}, where si's are taken from a constellation A, we define a new set of combined symbols, {Xl, .. X2k}, as follows: =
=
(5)
X2 2
X1
hi,j(t)
(4)
i=l
-
+ *** +C
kio±S+2 * *
1Ok- Sk,
+ * * * + eJOk- S2k,
+ eJOk- 1S1k
ioSk+2 + * * * + eiCk-1S2k,
1=0
Ci (0) C=c) I
C2
(0)
c2(1)
cLi(N-1) C2(N- 1)
...
...
CMT (O)
CMT(1)
CMT((N2-1)
-A(Xi, 2X2)
where ci (n) is the symbol transmitted by the i'th transmit antenna at the n'th frequency subcarrier. A Space-TimeFrequency (STF) codeword has an additional dimension of time added to the above SF codeword. In general we can express a STF codeword as
C2 ... CM], where the superscript in C' denotes the time index. C= [ C1
sl+ejols2+--- eJOk- sk, eJOk- 1S2k . (6) X2k Sk+1 + Cio Sk+2 + * We then present a general class of QOSTBCs for the MT transmit antennas, over a quasi-static flat-fading channel as follows
(3)
L
o
A o(X3 X4) 0
o
°k . ..
1..
*
(7)
A(X2k- 1 , X2k)
Let us denote the set of transmitted symbols by {sl, s2,. ... ,S2k} and the set of decoded symbols by {u1,U2, ..., U2k}, where sc, vi C A, Vi C {1, 2,. . ., 2k}. We construct the sets of combined symbols, {X1,X2,. ... ,X2k} and {Y1, Y2, . Y2 ,
52384 This full text paper was peer reviewed at the direction ofIEEE Communications Society subject matter expertsfor publication in the IEEE ICC 2006 proceedings. Authorized licensed use limited to: Univ of Calif Irvine. Downloaded on November 23, 2008 at 14:45 from IEEE Xplore. Restrictions apply.
corresponding to si's and ui's respectively, by using the set of equations given in (6). Now let us define the set of differences {D1, D2, .. ., D2k}, where Di = Xi -Yi, Vi C {1,2,...,2k}. In what follows, we present the design criteria for our proposed QOSTBC scheme given in (7). The rotation angles {O1, 02,... ., O-1} for the class of QOSTBC given by (7), are chosen such that for all distinct sets of {S1-... s2kI} and {u1,... , u2}, the following two conditions are satisfied: 1) To guarantee full-diversity, the following condition should be satisfied:
IDjl where di
=
Idj+ejold2 + ...+ eJOk- ldk
#0,
si- ui, Vsi, ui A. Note that if we switch si and ui for any i =
{2,.. . k, we can get Dj :t 0, Vj C {1,3,...,2k -1} as well. 2) To maximize the coding gain, the following max-min optimization problem should be solved: min
max
01. *Ok -1 Dl,D3 ....D2k- 1
D1,D3 ... D2k-1
X3
S1 +32, X2 S1 - 2 X4
IV. QUASI-ORTHOGONAL SPACE-FREQUENCY CODE STRUCTURE It has been shown in [1] that by applying the existing orthogonal space-time block codes to space-frequency block codes, we are not guaranteed to achieve the multipath diversity gains of a frequency selective fading channel. In this section we provide a guideline for constructing space-frequency block codes based on quasi-orthogonal designs, that is guaranteed to exploit any desired level of multipath diversity. Consider a MIMO-OFDM system with L independent delay paths and MT = 2 transmit antennas. Suppose we want to have full spatial diversity and a multipath diversity of A = k < L where as discussed in Section II, L < N. We use the QOSTBC designed for MT = 2k transmit antennas in quasi-static channel model given by Equation (7). A general SF codeword, based on the aforementioned quasi-orthogonal design, is expressed as
C
GTm
O
=
=3 -4.
GmT G
1T
...
.
. ..
I.
A(xTn_,, XTn)] T,
si1 + S2 1 -S1* -41* -3
S3 + 4 1* + 31* 1 2 1 1
§)) (s~32(3* 3-4* 2
s1* 1
_
1
+
2 2*
~ 2*2
1
S3-S4
...
[A(Xn, XTn) A(XTn, XTn)
S3 +34,
S- )2
[ G1T G2T
(9) (10)
and the superscript m C {1...., L ]} denotes the block number. Note that if N, the number of subcarriers, is not a multiple of 2A, we need to pad the space-frequency codeword with zeros. For simplicity, let us assume from now on that N = 2Ap, for some integer p. Thus the SF codeword, C, given by Equation (9), consists of p blocks {G1, ... , GP}. Next, we provide an example for the class of QOSF codes defined by (9). Consider a channel with MT = 2 and suppose we would like to have a multipath diversity of two (A = 2). We construct the QOSF code structure as follows
Note that s cio s is the rotated version of the symbol s. Now substituting the above parameters into the code structure given by (7), we get the following quasi-orthogonal codeword [20]: 0 0 S3 + S4 Sl + S2 §* C--S3*-S4 Sl* + 0
O
=
where,
Note that the decoding of the quasi-orthogonal STBC structure in (7) is done for k symbols at a time. Thus the decoding complexity grows exponentially with k. The above class of QOSTBCs provides rate-one, full-diversity block codes for any number of transmit antennas at the expense of higher decoding complexity compared to the orthogonal space-time block codes. For the case of MT= 2 transmit antennas or equivalently k = 1, the code in (7) reduces to the well-known Alamouti code. For the case of MT= 4 transmit antennas and consequently k 2, we have the following set of parameters: X1
'/2M (for M odd) and for QAM is 7w/4. [18] [21] [22].
C
1
+ §22 -S2* -42* -3 3i1
2
2
3(3-4* )
§1* 2
332+ 3442 2 3 2*
32* 2
(1 1)
(8)
By using an underlying quasi-orthogonal structure corresponding to k > 2, we can come up with similar designs to exploit As mentioned before, the optimum rotation angle, 01, is additional diversity gains but with higher decoding complexity. determined such that the code is full-diversity and the coding In general, there is a tradeoff between the amount of multipath gain is maximized. The code in Equation (8) has a behavior diversity one can achieve and the corresponding decoding similar to the rotated quasi-orthogonal codes for four transmit complexity. antennas discussed in [15]-[17]. It is full-rate and achieves full-diversity and has a pairwise maximum likelihood decod- A. Diversity Analysis In this section, we provide the analysis to show that the ing. The minimum coding gain structure of the code in (8) is also similar to the minimum coding gain of the existing quasi- SF code given by Equations (9) and (10) provides a diversity orthogonal codes. Therefore the optimum rotation angels for of 2k over any two-antenna frequency selective channel with this code, for MPSK constellation is 7/M (for M even) and L > k. -(S3
-S4*)
S*
-
§
5385 This full text paper was peer reviewed at the direction ofIEEE Communications Society subject matter expertsfor publication in the IEEE ICC 2006 proceedings. Authorized licensed use limited to: Univ of Calif Irvine. Downloaded on November 23, 2008 at 14:45 from IEEE Xplore. Restrictions apply.
Assuming that N > 2L, the diversity order of a spacefrequency code is determined by the minimum rank of the N x 2L matrix F(C, E) given as [1]
F(C,E)=[(C-E) 'I(C-E) where I = diag{w k}01 and
...
AFL-1(C -E)] = e-i '. Note that
w C denotes the transmitted codeword and E denotes the decoded codeword. Let us denote the difference between the transmitted symbol s,, and the decoded symbol u', to be d' for a block index m C {1,... , p}. The minimum distance error event happens when only one of the sets {d7,... , dn } or {dm±, ... , dn1 } is non-zero, for some block index m. Without loss of generality, let us assume .d.... ., d) } is the non-zero set. Thus only the first 2k rows of F(C, E) have non-zero elements. In that case, let us denote the non-zero part of F(C, E) by the 2k x 2k matrix F(C, E), where
minant is non-zero. It is easy to show that
det(Fodd (C,E)) DiD13D5 ... D2k-, det(W).
(15)
Using the well-known ideas in the theory of determinants [23], one can write the determinant of Fodd (C, E) as follows 2k-1
det (Fodd (C, E))
( I| Di ) det (W) i=l i:odd 2k-1
' H i=1 i:odd
k-2
Di)(fH
m=0
k-1
H (w 2n n=m+1
W2m). (16)
The first term in Equation (16), which is the product of Di's for odd values of i, is non-zero because of the full-diversity condition of the underlying QOSTBC. The second term is also F(C, E) non-zero because we have assumed that N > 2L > 2k, so 0 0 D1 D1 e-i NKt< 1, Vl C {1,* .* 2(k -1)}, therefore wt f wJ, Vi # O O D1 ... wk-lD* j. Thus Fodd(C,E) is full-rank and similarly Feven(C E) 0 ... w2(k- 1) D3 0 D3 is full-rank as well. Consequently F(C, E) has a minimum rank of 2k because there are 2k linearly independent columns. Thus we have proved that the code in Equation (9) achieves 0 D2k-1 0 ... w 2(k-1)(k -)D2k-1 a diversity of 2k, where two levels of diversity are due to 0 Lo 92k-1 ... w(k1(2k
MIMO OFDM
Channels
Fatemeh Fazel and Hamid Jafarkhani Center for Pervasive Communications and Computing Department of Electrical Engineering and Computer Science University of California, Irvine {fazel, hamidj } @uci.edu Abstract- In this paper, we propose a novel class of SpaceFrequency and Space-Time-Frequency block codes based on Quasi-Orthogonal designs, over a frequency selective Rayleigh fading channel. The proposed codes are able to achieve rate one and full diversity by exploiting the space and multipath diversity gains available in the MIMO-OFDM channel. As the simulation results show, our codes outperform the existing SpaceFrequency block codes in terms of bit error rate performance. Additionally, the proposed Space-Time-Frequency code benefits from a reduced maximum likelihood decoding complexity, which is a huge simplification compared to the existing codes. We also discuss the conditions under which the maximum likelihood decoding for our proposed Quasi-Orthogonal Space-Frequency code is simplified as well. I. INTRODUCTION
A combination of Multiple-Input Multiple-Output (MIMO) and Orthogonal Frequency Division Multiplexing (OFDM) is a promising technique for high data rate broadband wireless systems. A frequency selective channel offers an additional degree of diversity known as multipath or frequency diversity. In a MIMO-OFDM system it is desired to achieve multipath as well as spatial diversity gains. Space-Frequency (SF) and Space-Time-Frequency (STF) codes have been designed to achieve some levels of space and multipath diversity. Basically, SF codes use the two dimensions of space (antenna) and frequency tones (subcarriers) to code over, while STF codes, code across the three dimensions of space, frequency and time. It is proved that a MIMO-OFDM system can achieve a maximum diversity gain equal to the product of the number of its transmit antennas, the number of its receive antennas and the number of multipaths present in the frequency selective channel as long as the channel correlation matrix is full rank. The design criteria to achieve such diversity gains are presented in [1], [2] and [3]. Space-time coded OFDM was first introduced in [4] by using space-time trellis codes over frequency tones. Authors in [5] introduced a space-frequency-time coding method over MIMO-OFDM channels. They used trellis coding to code over space and frequency and space-time block codes to code over OFDM blocks. The authors used the Alamouti block code structure [6] for the case of two transmit antennas and This work was supported in part by an NSF Career Award CCR-0238042.
argued that for larger numbers of transmit antennas one can use Orthogonal Space-Time Block Code (OSTBC) structures introduced in [7]. It is worthwhile to mention that in case of more than two transmit antennas the OSTBC can provide a rate of at most 3/4 and we are not able to have rate-one transmission. In [8], authors point out the analogy between antennas and frequency tones and based on capacity calculation, propose a grouping method that reduces the complexity of code design for MIMO-OFDM systems. The idea of subcarrier grouping is further pursued in [2] and [9] with precoding and in [10] with bit interleaving. Reference [11] proposes a repetition mapping technique to transform the existing spacetime codes, designed for quasi-static flat fading channels, to construct full-diversity codes in frequency selective fading channels. Note that their proposed method provides a tradeoff between diversity and symbol rate. Later on, the authors proposed a rate-one, full-diversity space-frequency block code in [12]. Their proposed scheme can obtain a target diversity gain but the decoding complexity grows exponentially with the desired diversity. We use their design as a reference to compare our proposed structure in terms of performance and
complexity. Quasi-orthogonal code structures were first introduced in [13] and [14]. Original Quasi-orthogonal designs provide fullrate codes and pairwise Maximum Likelihood (ML) decoding but fail to achieve full-diversity. Later on, improved quasiorthogonal codes were proposed through constellation rotation [15]-[19]. A rotated Quasi-Orthogonal Space-Time Block Code (QOSTBC) provides both full diversity and full rate and performs better in low and high SNR, compared to OSTBC. These benefits together with the simple decoding capabilities of rotated quasi-orthogonal codes, motivate us to design Space-Frequency codes based on quasi-orthogonal structures. In this paper, we provide a systematic method of designing rate-one, full-diversity space-frequency codes for two transmit antennas, using quasi-orthogonal space time block codes designed for frequency-flat quasi-static channels. We specifically construct one sample quasi-orthogonal space-frequency code. As the simulation results suggest, the proposed code has a better performance and under certain conditions, provides reduced decoding complexity compared to the existing rate-
5383
U.S. Government work not protected by U.S. copyright This full text paper was peer reviewed at the direction ofIEEE Communications Society subject matter expertsfor publication in the IEEE ICC 2006 proceedings. Authorized licensed use limited to: Univ of Calif Irvine. Downloaded on November 23, 2008 at 14:45 from IEEE Xplore. Restrictions apply.
one codes. Then, assuming that the channel is quasi-static over two OFDM symbols, we will design a space-time-frequency code which has a reduced decoding complexity under all circumstances. Both SF and STF code structures provide full symbol rate (one symbol per frequency tone per time slot) and achieve any desired multipath (frequency) diversity available in the frequency selective fading channel. The rest of the paper is organized as follows. In Section II, we describe the MIMO-OFDM channel model and the SF code structure in general. In Section III, we introduce a general class of quasi-orthogonal space-time block codes for quasi-static flat fading channels. This class of QOSTBC will then be used as an underlying structure to design rate-one full-diversity spacefrequency codes in Section IV. Then we continue with the design of Quasi-Orthogonal Space-Time-Frequency (QOSTF) block codes in Section V. In Section VI, the decoding of QOSF and QOSTF codes is discussed. Simulation results are presented in Section VII and finally some concluding remarks are provided in Section VIII.
The OFDM transmitter performs an N point IFFT over the frequency tones or equivalently the columns of the codeword matrix C and adds a cyclic prefix to remove the Inter Symbol Interference. At the receiver, the cyclic prefix is removed and an FFT transformation is applied on the frequency tones. The received signal at receive antenna j at the n'th subcarrier is given by lrj (n)
MT
E ci (n)Hij (n) + A(n),
=
where Hij (n) and JVr(n) are the frequency response of the channel and the complex additive white Gaussian noise at the n'th frequency subcarrier. The maximum achievable diversity in the channel model characterized in this section, is LMTMR [1]. To be able to achieve that maximum diversity by using a SF code, the number of subcarriers, N, has to be larger than or equal to the number of independent delay paths, L. III. UNDERLYING CODE STRUCTURE
II. CHANNEL MODEL
In this section, we define the channel model we use throughout the paper. Consider a MIMO-OFDM system with MT transmit and MR receive antennas. We assume that the receiver has perfect channel knowledge while the transmitter does not know the channel. Each channel between transmit antenna and receive antenna j is assumed to have L independent channel taps with a complex amplitude of aij(1) and a delay denoted by Ti associated with the Ith tap. Note that each ovi,j(l) is a zero mean complex Gaussian random variable with a variance of o2. For normalization purposes, we assume that 7j1 5j2 = 1. Also assume that we have N frequency subcarriers. We can represent the channel impulse response for the path between transmit antenna i and receive antenna j at time t as
In this section, we introduce a general class of quasiorthogonal space-time block codes and later on use this class to build space-frequency block codes in Section IV. Let us denote the Alamouti scheme [6] for the two indeterminate variables x1 and X2 by
A(xi,X2)
j++eols2
S
X2
=
Sk+1 +
X3
=
e s2 + Sl-Cjol
(1)
X4
=
S+1
where d is the dirac delta function. A general Space-Frequency (SF) code word is represented by the following N x MT matrix
X2k-1
L-1
E ai,j (1)6(t
-
)
1
Now suppose we have MT = 2k transmit antennas. For a block of 2k symbols, {s1, ...., s2k}, where si's are taken from a constellation A, we define a new set of combined symbols, {Xl, .. X2k}, as follows: =
=
(5)
X2 2
X1
hi,j(t)
(4)
i=l
-
+ *** +C
kio±S+2 * *
1Ok- Sk,
+ * * * + eJOk- S2k,
+ eJOk- 1S1k
ioSk+2 + * * * + eiCk-1S2k,
1=0
Ci (0) C=c) I
C2
(0)
c2(1)
cLi(N-1) C2(N- 1)
...
...
CMT (O)
CMT(1)
CMT((N2-1)
-A(Xi, 2X2)
where ci (n) is the symbol transmitted by the i'th transmit antenna at the n'th frequency subcarrier. A Space-TimeFrequency (STF) codeword has an additional dimension of time added to the above SF codeword. In general we can express a STF codeword as
C2 ... CM], where the superscript in C' denotes the time index. C= [ C1
sl+ejols2+--- eJOk- sk, eJOk- 1S2k . (6) X2k Sk+1 + Cio Sk+2 + * We then present a general class of QOSTBCs for the MT transmit antennas, over a quasi-static flat-fading channel as follows
(3)
L
o
A o(X3 X4) 0
o
°k . ..
1..
*
(7)
A(X2k- 1 , X2k)
Let us denote the set of transmitted symbols by {sl, s2,. ... ,S2k} and the set of decoded symbols by {u1,U2, ..., U2k}, where sc, vi C A, Vi C {1, 2,. . ., 2k}. We construct the sets of combined symbols, {X1,X2,. ... ,X2k} and {Y1, Y2, . Y2 ,
52384 This full text paper was peer reviewed at the direction ofIEEE Communications Society subject matter expertsfor publication in the IEEE ICC 2006 proceedings. Authorized licensed use limited to: Univ of Calif Irvine. Downloaded on November 23, 2008 at 14:45 from IEEE Xplore. Restrictions apply.
corresponding to si's and ui's respectively, by using the set of equations given in (6). Now let us define the set of differences {D1, D2, .. ., D2k}, where Di = Xi -Yi, Vi C {1,2,...,2k}. In what follows, we present the design criteria for our proposed QOSTBC scheme given in (7). The rotation angles {O1, 02,... ., O-1} for the class of QOSTBC given by (7), are chosen such that for all distinct sets of {S1-... s2kI} and {u1,... , u2}, the following two conditions are satisfied: 1) To guarantee full-diversity, the following condition should be satisfied:
IDjl where di
=
Idj+ejold2 + ...+ eJOk- ldk
#0,
si- ui, Vsi, ui A. Note that if we switch si and ui for any i =
{2,.. . k, we can get Dj :t 0, Vj C {1,3,...,2k -1} as well. 2) To maximize the coding gain, the following max-min optimization problem should be solved: min
max
01. *Ok -1 Dl,D3 ....D2k- 1
D1,D3 ... D2k-1
X3
S1 +32, X2 S1 - 2 X4
IV. QUASI-ORTHOGONAL SPACE-FREQUENCY CODE STRUCTURE It has been shown in [1] that by applying the existing orthogonal space-time block codes to space-frequency block codes, we are not guaranteed to achieve the multipath diversity gains of a frequency selective fading channel. In this section we provide a guideline for constructing space-frequency block codes based on quasi-orthogonal designs, that is guaranteed to exploit any desired level of multipath diversity. Consider a MIMO-OFDM system with L independent delay paths and MT = 2 transmit antennas. Suppose we want to have full spatial diversity and a multipath diversity of A = k < L where as discussed in Section II, L < N. We use the QOSTBC designed for MT = 2k transmit antennas in quasi-static channel model given by Equation (7). A general SF codeword, based on the aforementioned quasi-orthogonal design, is expressed as
C
GTm
O
=
=3 -4.
GmT G
1T
...
.
. ..
I.
A(xTn_,, XTn)] T,
si1 + S2 1 -S1* -41* -3
S3 + 4 1* + 31* 1 2 1 1
§)) (s~32(3* 3-4* 2
s1* 1
_
1
+
2 2*
~ 2*2
1
S3-S4
...
[A(Xn, XTn) A(XTn, XTn)
S3 +34,
S- )2
[ G1T G2T
(9) (10)
and the superscript m C {1...., L ]} denotes the block number. Note that if N, the number of subcarriers, is not a multiple of 2A, we need to pad the space-frequency codeword with zeros. For simplicity, let us assume from now on that N = 2Ap, for some integer p. Thus the SF codeword, C, given by Equation (9), consists of p blocks {G1, ... , GP}. Next, we provide an example for the class of QOSF codes defined by (9). Consider a channel with MT = 2 and suppose we would like to have a multipath diversity of two (A = 2). We construct the QOSF code structure as follows
Note that s cio s is the rotated version of the symbol s. Now substituting the above parameters into the code structure given by (7), we get the following quasi-orthogonal codeword [20]: 0 0 S3 + S4 Sl + S2 §* C--S3*-S4 Sl* + 0
O
=
where,
Note that the decoding of the quasi-orthogonal STBC structure in (7) is done for k symbols at a time. Thus the decoding complexity grows exponentially with k. The above class of QOSTBCs provides rate-one, full-diversity block codes for any number of transmit antennas at the expense of higher decoding complexity compared to the orthogonal space-time block codes. For the case of MT= 2 transmit antennas or equivalently k = 1, the code in (7) reduces to the well-known Alamouti code. For the case of MT= 4 transmit antennas and consequently k 2, we have the following set of parameters: X1
'/2M (for M odd) and for QAM is 7w/4. [18] [21] [22].
C
1
+ §22 -S2* -42* -3 3i1
2
2
3(3-4* )
§1* 2
332+ 3442 2 3 2*
32* 2
(1 1)
(8)
By using an underlying quasi-orthogonal structure corresponding to k > 2, we can come up with similar designs to exploit As mentioned before, the optimum rotation angle, 01, is additional diversity gains but with higher decoding complexity. determined such that the code is full-diversity and the coding In general, there is a tradeoff between the amount of multipath gain is maximized. The code in Equation (8) has a behavior diversity one can achieve and the corresponding decoding similar to the rotated quasi-orthogonal codes for four transmit complexity. antennas discussed in [15]-[17]. It is full-rate and achieves full-diversity and has a pairwise maximum likelihood decod- A. Diversity Analysis In this section, we provide the analysis to show that the ing. The minimum coding gain structure of the code in (8) is also similar to the minimum coding gain of the existing quasi- SF code given by Equations (9) and (10) provides a diversity orthogonal codes. Therefore the optimum rotation angels for of 2k over any two-antenna frequency selective channel with this code, for MPSK constellation is 7/M (for M even) and L > k. -(S3
-S4*)
S*
-
§
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Assuming that N > 2L, the diversity order of a spacefrequency code is determined by the minimum rank of the N x 2L matrix F(C, E) given as [1]
F(C,E)=[(C-E) 'I(C-E) where I = diag{w k}01 and
...
AFL-1(C -E)] = e-i '. Note that
w C denotes the transmitted codeword and E denotes the decoded codeword. Let us denote the difference between the transmitted symbol s,, and the decoded symbol u', to be d' for a block index m C {1,... , p}. The minimum distance error event happens when only one of the sets {d7,... , dn } or {dm±, ... , dn1 } is non-zero, for some block index m. Without loss of generality, let us assume .d.... ., d) } is the non-zero set. Thus only the first 2k rows of F(C, E) have non-zero elements. In that case, let us denote the non-zero part of F(C, E) by the 2k x 2k matrix F(C, E), where
minant is non-zero. It is easy to show that
det(Fodd (C,E)) DiD13D5 ... D2k-, det(W).
(15)
Using the well-known ideas in the theory of determinants [23], one can write the determinant of Fodd (C, E) as follows 2k-1
det (Fodd (C, E))
( I| Di ) det (W) i=l i:odd 2k-1
' H i=1 i:odd
k-2
Di)(fH
m=0
k-1
H (w 2n n=m+1
W2m). (16)
The first term in Equation (16), which is the product of Di's for odd values of i, is non-zero because of the full-diversity condition of the underlying QOSTBC. The second term is also F(C, E) non-zero because we have assumed that N > 2L > 2k, so 0 0 D1 D1 e-i NKt< 1, Vl C {1,* .* 2(k -1)}, therefore wt f wJ, Vi # O O D1 ... wk-lD* j. Thus Fodd(C,E) is full-rank and similarly Feven(C E) 0 ... w2(k- 1) D3 0 D3 is full-rank as well. Consequently F(C, E) has a minimum rank of 2k because there are 2k linearly independent columns. Thus we have proved that the code in Equation (9) achieves 0 D2k-1 0 ... w 2(k-1)(k -)D2k-1 a diversity of 2k, where two levels of diversity are due to 0 Lo 92k-1 ... w(k1(2k
