Power-Balanced Orthogonal Space-Time Block Code

Accepted by IEEE Trans. Vehicular Technology 1 Abstract —In this paper, w e propose two new sy stematic ways to construct amicable orthogonal designs (AOD), with an aim to facilitate the construction of power-balanced orthogonal space- time block codes (O-STBC) with f avorable practical attributes. We also show that an AOD can be constructed from an Amicable Family (AF), and such a construction is crucial for achieving a power-balance d O-STBC. In addition, we develop design guidelines on how to select the “type” p arameter of an AOD so that the resultant O-STBC w ill have better power-distribution and code-coefficient attribut es. Among the new O-STBCs obtained, one is shown to be optimal in terms of power distribution attributes. In addition, one of the proposed construction methods is show n to generalize some oth er construction methods prop osed in the literature. Index Terms —Orthogonal Space-Time Block Code, MIMO Coding, Transmit Diversity, Power Balanced. I. I NTRODUCTION N Orthogonal Space-Time Block Code (O-STBC) can provide full tran smit diversity with simple linear decoding. Due to these advantages, O-STBC has drawn a l ot of research attention. In [1-3], the Alamouti STBC is generalized to square O-STBC for more than two transmit antennas by using different m athemat ical techniques, while i n [4,5], non-square O-STBC are investigated in an attem pt to increase the maximum achievable code rate of O-STBC. Nonetheless, square O-STBC has the advantages of m inimal decoding delay and applicability for differential modulation. Some of the square O-STBCs, such as the ones in [2,3], contain “zero” symbols; while those in [1] have irrational numbers in the code coefficien ts. The regular transm ission of “zeros” implies turning off the transmit antennas at regular intervals. This leads to an undesirable low-frequency interference and some di fficulty in the front -end power amplifie r design [6-8]. Therefore, the problem of designing power-balanced O-STBC with less or no zero sy mbols has been investigated in [6,8-11]. Manuscript received September 25, 2007. C. Yuen is with the Institute fo r Infocomm Research, Singapore (phone: 65-68745679; fax: 65-67744990; e-mail: cy uen@i2r.a-star.edu.sg). Y. L. Guan is with the Department of Electrical and Electronic Engineering, Nanyang Technological University, Singapore (e-m ail: eylguan@ntu.edu.sg). T. T. Tjhung is with the Institute for Infocom m Research, Singapore (e- mail: tjhungtt@i2r.a-star.edu.sg). In [6], a method is proposed such that the transm ission power of existing O-STBC is distributed as equal ly as possible between different antennas; whi le in [8-11], other algebraic techniqu es such as Williamson and Wallis- Whiteman matrices are used to construct power-balanced O- STBC. Besides the zero sym bols problem , irrational numbers in the code coefficien ts require floating-poin t multiplications in both the transmitter and receiver of the O-STBC system. This is inconvenient t o implem ent compared t o having just 1 in the code-coefficients, which requires only simple additions/subtract ions. In this paper, we focus on the construction of O-STBC’s from am icable orthogonal designs (AOD) that avoids zero and irrational coefficient s. We propose two new system atic methods to const ruct higher-order AOD’s or Am icable Family’s (AF) from lower-order AOD’s or AF’s, and use them to construct new square O-STBCs. W e then investigate the relationship between the “type” parameter of the AOD’s and the power-distribution and code-coefficient characteristics of the constructed O-STBC’s. We will evaluate the advantages of the newly constructed O-STBCs over the existing ones. We will also give a comparison of the proposed construction methods to t he methods in [9], and show t hat the proposed construction me thods can be generalized to the latter. II. O RTHOGONAL STBC A linear STBC, G, can be represented as: k RI 1 () ii i i i xj x = =+ ∑ GA B (1) where A i and B i are the “dispersion matrices” (both of dimensi on p × n t ), where x i R and x i I represent respectively, the real and imaginary parts of the i th transmitted symbol, p is the code length, n t is the number of transm it antennas, and k is the number of complex symbols being transmitted over p periods of time. Hence the code rate of a STBC is k / p . For an O-STBC, its dispersion matrices, A i and B i , must satisfy the following con straints [2]: ( ) () () HH HH H H HH i , 1 ii , 1 iii 1 tt ii n i i n iq q i i q q i iq q i ik iq k i, q k == ≤ ≤ = −= − ≤ ≠ ≤ =≤ ≤ AA I B B I A A A A BB BB AB B A (2) where the superscript H represents Hermitian (i.e. conjug ate transpose) of a matrix. In order to achieve full diversity, p has Power -Balanced Orthogonal Space-T ime Block Code Chau Yuen, Yong Liang Guan and Tjeng Thiang Tjhung, Member, IEEE A Accepted by IEEE Trans. Vehicular Technology 2 to be greater or equal to n t . Hence a square STBC design with p = n t gives the mini mum possi ble code length. In addition, a square design can be applied to differential unitary space-time coding. So we only consider the design of square O-STBC in this paper. To construct a square O-STBC, one can make use of the AOD. Let us first review the relationship between AOD’s and O-STBC’s [2]: Definition 1 [12]: Let the matrices A = A 1 a 1 + … + A s a s and B = B 1 b 1 + … + B t b t be orthogonal designs of the same order n (i.e. both A and B are n × n ), where A is of “ type ” ( f 1 , …, f s ) on the variables { a 1 , …, a s } and B is of “ type ” ( g 1 , …, g t ) on the variables { b 1 , …, b t }. A and B are said to be Amicable Orthogonal Design (AOD) if HH = AB B A (3) A necessary and sufficient condition for an AOD of “ type ” ( f 1 , …, f s ; g 1 , …, g t ), as defined in Definition 1 , to exist is that there exists a family of matrices { A 1 , …, A s ; B 1 , …, B t } satisfying: () () H H HH H (0) 1 ; 1 i 1 ; 1 ii 1 ; il qm ii i n qq q n il li qm il s qm t fi s gq t il s ∗= ≤ ≠ ≤ ∗= ≤ ≠ ≤ =≤ ≤ =≤ ≤ += ≤ ≠ ≤ AA 0 BB 0 AA I BB I AA AA 0 BB () H HH 1 iii 1 ,1 mq iq q i qm t is q t += ≤ ≠ ≤ −= ≤ ≤ ≤ ≤ BB 0 AB BA 0 (4) where A i and B q consist of only {0, 1, -1}, the sy mbol * represents the Hadamard product , and 0 is a zero matrix. ■ By comparing the constraints in (4) for the AOD matrix with that in (2) for the O-STBC dispersion matrix, it can be seen that the A i and B q matrices defined i n (4) can be used as the dispersion matrices of an O-STBC since they satisfy the design constraints in (2). The “ order ” param eter n of an AOD corresponds to the number of transm it antennas n t of the O- STBC, while the code le ngth p of the constructed O-STBC will be equal to n , since an AOD is a square design. In addition, the “ type ” parameter of an AOD, i.e. ( f 1 , …, f s ; g 1 , …, g t ), is related to the power d istribution of the transmit symbols of an O-STBC. This will be elab orated later in Section IV. Furtherm ore, the total num ber of variables, i.e. s + t , of the AOD represents the number of real symbols, i.e. 2 k , carried by the O-STBC. Hence the code rate of an O-STBC constructed from an AOD will be ( s + t ) / 2 n . It has been shown in [1-3] that the m aximal code rate of a square O- STBC is ¾ for four transmit antennas and ½ for eight transmit antennas by using the follo wing property of an AOD: Lemma 1 [12]: For an AOD of order n , where n =2 a b , a and b are both integers and b is odd, the total num ber of variables in an AOD (i.e. s + t ) is upper bounded by 2 a + 2, and that bound is achieved. ■ Definition 2 [12]: An amicable family (AF) of type ( f 1 , …, f s ; g 1 , …, g t ), of order n is a collection o f matrices { A 1 , .., A s ; B 1 , .., B t } satisfying (4)(i), (ii), and (iii), but no t the “disjointness statement” (4)(0). ■ It will be shown that an AF plays an important role in the construction of power-balanced O-STBC’s. III. C ONSTRUCTION O F AOD We now propose two new methods to construct higher- order AOD’s from lower-order AOD’s. In the first method we construct an AOD of order 4 n from an AOD of order n , while in the second method we construct an AOD of order 2 n from an AOD of order n . We sh all show that the resultant high er- order AOD achieves the maximum number of variables (and its associated O-STBC achieves the maximum code rate) if the lower-order AOD used to generate it achieves the m aximum number of variables. Construction 1 : If { A i , 1 ≤ i ≤ s ; B q , 1 ≤ q ≤ t } is an AOD / AF of order n with s + t variables of “ type ” ( f 1 , …, f s ; g 1 , …, g t ), then { 11 ⊗ BM , 12 ⊗ BM , 13 ⊗ BM , 4 , 2 i is ⊗≤ ≤ AI ; 11 ⊗ AN , 12 ⊗ AN , 13 ⊗ AN , 4 , 2 q qt ⊗≤ ≤ BI } is an AOD / AF of order 4 n with s + t + 4 variables and “ type ” ( g 1 , g 1 , g 1 , f 2 , …, f s ; f 1 , f 1 , f 1 , g 2 , …, g t ), where ⊗ is the Kronecker product, if the mat rices M i and N i , for 1 ≤ i ≤ 3, satisfy the following con ditions: HH 44 HH H H HH 44 (0) , 1 3 (i) , 1 3 (ii) , 1 3 (iii) , 1 , 3 ( i v ) , , 1 3 (v iq i q ii i i i i iq q i i q q i iq q i ii iq uv i iq iq i ∗= ∗ = ≤ ≠ ≤ == ≤ ≤ + =+ = ≤ ≠ ≤ −= ≤ ≤ ∗= ∗= ≤ ≤ MM 0 NN 0 MM I NN I MM MM 0 N N NN 0 MN N M 0 MI 0 N I 0 HH ) , , 1 3 ii i i i += + = ≤ ≤ MM 0 NN 0 (5) Proof of Construction 1 : The proofs are routine, hence omitted due to space constraint. ■ The following observations can be m ade from Const ruction 1 : - An interesting observation is that since the condit ions (5)(0- iii) are equivalent to the cond itions (4)(0-iii), it implies that { M 1 , M 2 , M 3 ; N 1 , N 2 , N 3 } must themselves be an AOD of order 4 with six variables and t ype ( u 1 , u 2 , u 3 ; v 1 , v 2 , v 3 ). - If the target design is an AF, then conditions (5)(0) and (5)(iv) in Construction 1 can be neglected because they are related to the “disjoin tness statement". On the other hand, if the target design is an AOD, then u i and v i must both be 1 for 1 ≤ i ≤ 3. This is because in an AOD of order 4, if u i or v i is greater than 1, the “disjo intness statement" (5)(0) and (5)(iv) will be violated. - If the lower-order design is an AOD, the resultant higher- order design will also be an AOD. However if the lower- order design is an AF, the resultant higher-order design m ay be an AOD or an AF. For an illustration, an AOD of order 8 constructed from an AF of order 2 will be shown later in Example 1 . Accepted by IEEE Trans. Vehicular Technology 3 Proposition 1 : An AOD of order 4 n constructed by Construction 1 can achie ve the maxi mum number of vari ables if an AOD of order n with the maxim um num ber of variables is used to construct it. Proof of Proposition 1 : From Lemma 1 , the maxi mum num ber of variables for an AOD of order n is bounded by 2 a + 2, where n = 2 a . This implies that the maximum number of variables for an AOD of order 4 n is bounded by 2 a + 6. Therefore the difference between the maximum number of variables of an AOD of order n and an AOD of order 4 n is at most 4. From Construction 1 , the number of vari ables of an AOD of order 4 n is s + t + 4. This is also 4 more than th e number of variables of an AOD of order n used to g enerate it. Hence if an AOD of order n that achieves the maximum number of variables is used, an AOD of order 4 n that achieves the maxim um num ber of variables can be constructed. ■ Examples of { M , N } that satisfy the conditions in (5) are: 123 123 010 0 00 1 0 0 00 1 10 0 0 0 0 0 1 0 0 1 0 ,, 0001 1 0 0 0 0 1 0 0 00 1 0 01 00 1 00 0 01 00 0 01 0 0001 1 0 00 0 00 1 00 1 0 ,, 00 0 1 1 00 0 0100 00 10 0 1 0 0 1 000 ⎡⎤ ⎡⎤ ⎡⎤ ⎢⎥ ⎢⎥ ⎢⎥ −− ⎢⎥ ⎢⎥ ⎢⎥ == = ⎢⎥ ⎢⎥ ⎢⎥ −− ⎢⎥ ⎢⎥ ⎢⎥ −− ⎣⎦ ⎣⎦ ⎣⎦ ⎡⎤ ⎡⎤ ⎡⎤ ⎢⎥ ⎢⎥ ⎢⎥ −− ⎢⎥ ⎢⎥ ⎢⎥ == = ⎢⎥ ⎢⎥ ⎢ −− ⎢⎥ ⎢⎥ ⎢ −− ⎣⎦ ⎣⎦ ⎣⎦ MM M NN N ⎥ ⎥ (6) Example 1 : An AOD of order 8 and “ type ” (2, 2, 2, 2; 2, 2, 2, 2) Consider an AF 1 1 1 1 11 11 ,;, 11 11 1 1 1 1 ⎧− − − − ⎫ ⎡⎤ ⎡ ⎤ ⎡ ⎤ ⎡⎤ ⎪⎪ ⎨⎬ ⎢⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎥ −− − − ⎪⎪ ⎣⎦ ⎣ ⎦ ⎣ ⎦ ⎣⎦ ⎩⎭ of order 2 and “ type ” (2, 2; 2, 2). An AOD of order 8 and “ type ” (2, 2, 2, 2; 2, 2, 2, 2), can be constructed by usi ng Construction 1 and { M , N } in (6): 441 1 2 2 3 3 4 4 1 1 2 233 11 4 4 3 3 22 11 4 4 3 3 22 2 2 3 344 1 1 223 3 4 4 11 33 2 2 1 1 4 4 33 22 1 1 4 4 44 1 1 2 aa a a a a a a aa a a a a a a aa a a a a aa aa a a a a aa aa a a a a a a aaaa a a aa aa a a a a a a aa aa a a a a bb b b b −− − ⎡⎤ ⎢⎥ − ⎢⎥ ⎢⎥ −− − ⎢⎥ −− − −− ⎢⎥ = ⎢⎥ −− − ⎢⎥ −−−− − ⎢⎥ ⎢⎥ −− − ⎢⎥ −− − − − ⎢⎥ ⎣⎦ −− − − = A B 23 3 4 4112233 11 4 4 33 2 2 114 4 33 22 22 3 3 4 4 11 22 33 4 4 11 33 22 1 1 4 4 3322 11 4 4 bb b b b bb bb bb bb b b bb b b bb b b bb bb bb b b b b bb bb bb b b bb bb bb b b b b bbbb bb b b − ⎡⎤ ⎢⎥ −− − − −− − ⎢⎥ ⎢⎥ −− − − − ⎢⎥ −− − ⎢⎥ ⎢⎥ −− − − − ⎢⎥ −− − ⎢⎥ ⎢⎥ −− − − − ⎢⎥ −− − ⎢⎥ ⎣⎦ (7) By letting x i = a i + jb i , where j 2 = -1, and rearranging the symbols x 1 to x 4 , we obtain the O-STBC: ** 1 1 22 33 44 ****** 1 1 223344 *** ** * 22 1 1 4 4 33 ** 22 1 1 44 33 * **** * 33 4 4 1 1 2 2 ** 3344 1 1 22 * ** *** 44 3 3 2 2 1 1 ** 44 33 22 1 1 x xx x x x x x x x xxxxxx x xxx x xx x x xx xx x x x x x x xxxx x x xxx x x x x x xx xx xxx x xx x x xx x ⎡ −−− ⎢ − ⎢ ⎢ −− − ⎢ −− − −− ⎢ = ⎢ −− − ⎢ −−−− − −− − −− −− − ⎣ G8 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎦ (8) G8 has the form: ** 1 ** ** ⎡ ⎤ ⎢ ⎥ −− ⎢ ⎥ = ⎢ ⎥ −− ⎢ ⎥ −− ⎣ ⎦ PQ R S QP S R RS P Q SR Q P Q (9) where ** 11 11 x x x x ⎡ ⎤ = ⎢ ⎥ − ⎣ ⎦ P , 22 ** 22 x x x x ⎡⎤ − = ⎢⎥ ⎣⎦ Q , 33 ** 33 x x x x ⎡ ⎤ − = ⎢ ⎥ ⎣ ⎦ R , 44 ** 44 x x x x ⎡ ⎤ − = ⎢ ⎥ ⎣ ⎦ S . This design is exactly the same as the one proposed in Theorem 1 in [9], hence Theorem 1 in [9] can be treated as a special case of Construct ion 1 . This example also demonstrates that an AOD can be constructed from an AF. By changing the lower-order AOD { A , B } matrices or the { M , N } matrices in Construction 1 , we can obtain new and existing O-STBCs. One such exam ple is shown i n Appendix, which shows that by using anot her set of { A , B } and { M , N } matrices, Theorem 2 in [9] can be obtained. This demonstrates the generality and versatility of the proposed construction method. Example 2 : An AOD of order 8 and “ type ” (2, 2, 2, 2; 2, 2, 2, 2) with complex entries Consider an AF 11 1 1 1 11 1 ,; , jj jj j j j j ⎧ −− − − ⎫ ⎡ ⎤⎡ ⎤⎡ ⎤⎡ ⎤ ⎪ ⎪ ⎨ ⎬ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ −− − − ⎪ ⎪ ⎣ ⎦⎣ ⎦⎣ ⎦⎣ ⎦ ⎩⎭ of order 2 and “ type ” (2, 2; 2, 2). The following O-STBC of order 8 and “ type ” (2, 2, 2, 2; 2, 2, 2, 2) can be const ructed using Construction 1 and { M , N } in (6) (here we allow AF and AOD with complex entries, as it has been shown in [13] that AF and AOD with complex entries have the sam e total number of variables): ** 1 1 22 33 44 ****** 1 1 223344 *** ** * 22 1 1 4 4 3 3 ** 22 1 1 44 3 3 * **** * 33 4 4 1 1 2 2 ** 3344 1 1 22 * ** *** 44 3 3 2 2 1 1 * 4 x x x xx xx x j x j x j xj xj xj xj xj x x xxx x xx x j x j x j xj x j x j xj x j x x x x xxxx x j x j x j x j xj x j xj x j x x xx xx xxx jx jx −−− − −− − −− − −− = −− − −−−− − −− − −− H8 * 43 3 2 21 1 j xj x j xj x j x j x ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −− − ⎣ ⎦ (10) Note that H8 in (10) is the first O-STBC of order 8 that contains no zero entries first reported by us in [13]. The sub- matri x of this code does not follow condit ion 2 of Theorem 1 in [9], hence this code cannot be constructed di rectly using the Accepted by IEEE Trans. Vehicular Technology 4 proposed method i n [9]. Construction 2 : If { A i , 1 ≤ i ≤ s ; B q , 1 ≤ q ≤ t } is an AOD / AF of order n with s + t variables of “ type ” ( f 1 , …, f s ; g 1 , …, g t ), and 1 01 10 ⎡⎤ = ⎢⎥ − ⎣⎦ N , 2 01 10 ⎡⎤ = ⎢⎥ ⎣⎦ N , 3 10 01 ⎡⎤ = ⎢⎥ − ⎣⎦ N , then { 11 ⊗ BN , 2 ,1 i is ⊗≤ ≤ AI ; 12 ⊗ BN , 13 ⊗ BN , 2 ,2 q qt ⊗≤ ≤ BI } is an AOD / AF of order 2 n with s + t + 2 variables and “ type ” ( g 1 , f 1 , f 2 , …, f s ; g 1 , g 1 , g 2 , …, g t ). Proof of Construction 2 : Sim ilar to the proof of Constr uction 1 . ■ Proposition 2 : An AOD of order 2 n constructed by Construction 2 can achieve the ma ximum number of variables if an AOD of order n with the maxi mum num ber of variables is used. Proof of Proposition 2 : Similar to the proof of Proposition 1 . ■ Example 3 : An AOD of order 4 and “ type ” (2, 2, 2; 2, 2, 2) Consider an AOD 1 1 11 1 1 11 ,;, 11 1 1 11 1 1 ⎧− − ⎫ ⎡⎤ ⎡⎤ ⎡⎤ ⎡⎤ ⎪⎪ ⎨⎬ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ −− ⎪⎪ ⎣⎦ ⎣⎦ ⎣⎦ ⎣⎦ ⎩⎭ of order 2 and “ type ” (2, 2; 2, 2), a new O-STBC for four transmit antennas, based on an AOD of order 4 and “ type ” (2, 2, 2; 2, 2, 2), denoted herein as G4 , can be constructed: ** 12 12 3 3 ** 12 12 3 3 ** 33 1 2 1 2 * * ** ** 3 3 12 12 xx x x x x xx xx x x x xx x x x x xx x x x ⎡⎤ −+ − ⎢⎥ +− + ⎢⎥ = ⎢⎥ −− + ⎢⎥ −− + − + ⎢⎥ ⎣⎦ G4 (11) Due to the prop erties stated in Proposition 1 and Proposition 2 , all the square O-STBCs constructed so far can achieve the maxim um achievable code rat e of ¾. As the proposed construction met hods are general, they can also be used to construct ma ny existing O-STBCs. IV. C ONSTRUCTION OF O-STBC FROM AOD To quantify the benefit of havi ng less or no zero entries in an O-STBC, we now introduce the power distri bution properties of O-STBC’s, and show that codes designed using Proposition 3, to be presented later in Section IV C, will have more favorable power dist ribution properties. A transmitted signal with good power-distribution characteristics [6] shoul d have: - Low, ideally 1, peak-to-av erage power ratio (peak/ave) - Low, ideally 1, averag e-to-minimum power ratio (ave/min) - Low, ideally 0, p robability P o that an antenna transmits “zero” (i.e. is turned off) In general, space-time codes that have the above “ideal” attributes are “power-balanced ”. We next show the new O- STBCs in Example 1 and Example 3 have desirable attributes for practical implementation. A. O-STBC for Eight Transmit Ante nnas Consider O-STBC designs for eight transmit antennas. Two rate-½ square O-STBCs for eight transm it ante nnas have been proposed in the literature: one by Tirkkonen and Hottinen [3], herein denoted as “ TH ”; the other by Tran, Seberry et al. [8], herein denoted as “ TS ”. The TH code is one of the first O- STBC’s for eight transmit antennas, and it is constructed from the Clifford algebraic technique. The TS code was designed with an aim t o reduce the unused time slots (i.e. num ber of zeros inside the codeword). Both designs have zero coefficients in the codewords. 12 3 4 ** 21 3 4 ** 31 2 4 ** 32 1 4 ** 41 2 3 ** 42 1 3 ** 43 1 2 ** * * 43 2 1 00 0 0 000 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 000 0 xxx x xx x x xx x x x xx x xx x x x xx x x xx x x xx x ⎡ ⎤ ⎢ ⎥ −− − ⎢ ⎥ ⎢ ⎥ −− ⎢ ⎥ − ⎢ ⎥ = ⎢ ⎥ − ⎢ ⎥ −− ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ ⎢ ⎥ −− ⎣ ⎦ TH (12) 13 2 2 3 12 3 3 2 * 32 2 3 1 * 23 32 1 *** * 444 4 ** * * 44 4 4 ** * * 44 4 4 ** * * 44 4 4 4 0 0 0 0 ... 222 2 22 2 2 22 2 2 22 2 2 2 .. RI RI RI RI RI RI RIRI xx j x x j x xx j x x j x xj x xj x x xj x xj x x xxx x xx x x xx x x xx x x xx ⎡ ++ ⎢ −+ − ⎢ ⎢ −+ + ⎢ −+ −− ⎢ = ⎢ −−− − ⎢ −− ⎢ ⎢ −− ⎢ ⎢ −− ⎣ TS 44 4 44 4 4 44 4 4 44 4 4 * 13 2 3 1 21 3 3 1 * 31 1 3 2 31 2 1 3 22 2 22 2 2 22 2 2 22 2 2 0 0 0 0 RI RI R IR I RI RI R IR I xx xx x x xx x x xx x x xj x x xj x x xj x xj x xj x xj x x x jx x x jx ⎤ ⎥ −− ⎥ ⎥ −− ⎥ −− ⎥ ⎥ −− ⎥ −+ − ⎥ ⎥ −− + − ⎥ ⎥ −− − ⎦ (13) In the follo wing, the TH and TS codes will b e compared to a new O-STBC, G8, constructed from Example 1 in (7) . Note that half of the cod eword entries in TH are zero, hence four of the eight transmit antennas will have to be turned off at any one time for this cod e, and it is constructed from an AOD of “ type ” (1, 1, 1, 1; 1, 1, 1, 1). Simila rly, the code TS which is constructed from an AOD of “ type ” (1, 1, 1, 4; 1, 1, 1, 4) also requires one of the transm it antennas to be turned off at any one time. In contrast, the G8 code, which is constructed from an AOD of “ type ” (2, 2, 2, 2; 2, 2, 2, 2), has no zero coefficients in the codeword and hence does not require any transmit antenna to be turn ed off at any one time. The fact that the G8 code comes from an AOD that is constructed from an AF (refer to Example 1 ), shows that our construction method of an AOD from an AF plays an important role in the generation of O-STBC’s without zeros in its code matrix. The power distribution characteristics of our new ½-rate O- Accepted by IEEE Trans. Vehicular Technology 5 STBC for eight transmit ante nnas are compared against existing O-STBCs, TH and TS . From Table 1, we can see that the new G8 code has much better power-distri bution characteristics than the TH and TS codes. In fact, G8 is optimally power-balanced as it has the ideal power- distribution attribut es, i.e. peak / ave = 1, ave / min = 1, and P o = 0. Table 1 Power distribution charact eristics of eight-antenna O- STBC with QPSK m odulation Peak Ave Ave Min P o Σ “ type ” Σ “ type ” ≥ 16 ? TH [2] 2 ∞ 50% 8 No TS [8] 2 ∞ 12.5% 10 No G8 1 1 0 16 Yes B. O-STBC for Four Transmit Antennas Here we denote the rate-¾ O-STBCs proposed in [1] by Tarokh, Jafarkhani and Calderbank as “ TJC ”, and the rate-¾ O-STBCs proposed in [2] by Ganesan and Stoi ca as “ GS ”: () () () () 12 3 3 ** 21 3 3 ** ** ** 11 22 22 11 33 ** ** ** 22 11 1 1 22 33 22 22 22 22 22 22 xx x x xx x x xxxx xxxx xx xxx x xx x x xx ⎡⎤ ⎢⎥ −− ⎢⎥ ⎢⎥ −− + − −− + − ⎢⎥ = ⎢⎥ ⎢⎥ ⎢⎥ ++− ++− ⎢⎥ −− ⎢⎥ ⎣⎦ TJC (14) 12 3 ** 13 2 ** 23 1 ** 32 1 0 0 0 0 x xx x xx xx x x xx ⎡⎤ − ⎢⎥ ⎢⎥ = ⎢⎥ −− ⎢⎥ − ⎢⎥ ⎣⎦ GS (15) It can be seen that the TJC code contains the irrational number 1/ √ 2 in som e of the codeword entries. This is because the TJ C code was formed from an AOD of “ type ” (1, 1, 2; 1, 1, 2), which means that t he symbols wit h type “2” have twice the power as the symbols with type “1”. In order to norm alize the power per symbol t o be the same for all symbols, the scaling factor 1/ √ 2 is needed in the TJC . Such multiplication operation is inconvenient to i mplem ent as compared to the addition/subtracti on operation associated with just having ± 1 in the code coefficients of G4 . On the other hand, for the GS code in (15) whi ch is constructed from an AOD of “ type ” (1, 1, 1; 1, 1, 1), each transmit antennas has to be turned off once in every four code symbol durat ions. This shortcom ing does not exis t for the G4 code if its x 1 and x 2 symbols are t aken from different rot ated constellations, e.g. x 1 from QPSK and x 2 from rotated-QPSK (note: constellation rotation doe s not affect the orthogonality of G4 ). The power distribution characteristics of our new ¾-rate O- STBC for four transmit antennas is compared against existing O-STBCs in Table 2. From Tabl e 2, our newly constructed G4 code achieves better peak/ave ratio and ave/min ratio than the GS codes in [2], as well as the “power-balanced” version of GS codes in [6]. Although the TJC code has better power- distribution characteristics than G4 , it contains as mentioned earlier, irrational-number coefficients inside its cod eword. Due to the above reasons, G4 code is more advantageous than the TJC and GS cod es in terms of practical implementation. Table 2 Power distribution charact eristics of four-antenna O- STBC with QPSK m odulation Peak Ave Ave Min P o Σ “ type ” Σ “ type ” ≥ 8 ? TJC [1] 1.33 1.5 0 8 Yes GS [2] 1.33 ∞ 25% 6 No Power-balanced GS #1 [6] 3 3 0 NA NA Power-balanced GS #2 [6] 2.6 17.5 0 NA NA G4 2.28 2.56 0 12 Yes NA: Not applicable. C. Guidelines for Designing Good Practic al O-STBC From the above exampl es and other earlier observations, we can draw the guidelines in Proposition 3 below, for the design of a practical O-STBC: Proposition 3 : To design an O-STBC wi thout irrational number coefficients and zero entries, t he following guidelines can be applied: - To avoid irrational number coeffi cients inside an O-STBC codeword, an AOD having a constant “ type ” value for all variables, i.e. f i = g i = constant ∀ i , is desired. - To avoid zero entries inside an O-STBC codeword, an AOD with the sum of “ type ” for all its variab les equal to or greater than 2 n , i.e. Σ ( f i + g i ) ≥ 2 n , is desired. Proof of Proposition 3 : From the above exam ples and from 2(i) and 4(i), we know that the “ type ” of a variable in an AOD is related to the power of the corresponding transmitted symbols. If all the variables have equal “ type ”, all the transmitted symbols will have equal transmission power, th is will eliminate the need for power normalization for indiv idual symbols and hence irrati onal number coefficients i n the O- STBC codeword. Next, the “ type ” of a variable in an AOD is also related to the num ber of o ccurrence of that variable in a row of the AOD matrix. To e lim inate zeros inside the codeword of an O-STBC, it is desired to have all the positions inside the codeword matrix at least filled by a symbol. So the sum of occurrence of every com plex symbol s must be equal to or greater than n , or the sum of “ type ” of all variabl es must be equal to or greater than 2 n (since a complex symbol consists of two real symbols, and i s represented by two variables i n an AOD). However, this is only a necessary, but not sufficient, condition to eliminate the zeros insid e an O-STBC codeword. ■ Both Table 1 and Table 2 also show that codes that sat isfy the second design guideline of Proposition 3 , i.e. sum of “ type ” for all variables greater than or equal to 2 n , will achieve P o = 0. It can be easily shown that all the new O- STBCs proposed in this paper have exactly the same code rate, coding gain and diversity gains as the existing O-STBC s Accepted by IEEE Trans. Vehicular Technology 6 [1,2,6], hence their practical advantages (better power distribution propert ies) do not come wi th rate or perform ance penalty. V. C ONCLUSION In this paper, we have proposed two new systemat ic ways to construct higher-order AOD’s or AF’s from lower-order AOD’s or AF’s. We found that the “ type ” parameter in an AOD plays an important role in shaping the power distribution and code coeffici ent characteristics of the O- STBC. From these insights, we propose two guideli nes on how to select the “ type ” parameter of AOD for constructing new O-STBC with favorable implementation attributes. New square O-STBCs for four and eight transm it antennas are constructed using the proposed AOD construction methods and design guidelines. They are shown to possess better power distribution or rat ional c ode coefficient characteristics than the existing O-STBC . In particular, one of the newly constructed O-STBCs for eight transmit ant ennas is found t o be the first to achieve the optimal power distribution properties. Interestingly, new AOD’s that lead to good practical O-STBC’s can be c onstructed from lower-ordered AF’s. The proposed construc tion m ethod is general and inclusive of some of t he construction met hods proposed in the literature. A PPENDIX : E QUIVALENCE OF T HEOREM 1 AND 2 OF [9] We will show that Th eorem 1 and 2 of [9] can be unified by our proposed construction m ethod. Example 4 : An AOD of order 8 and “ type ” (2, 2, 2, 2; 2, 2, 2, 2) with new set of { M , N } Use the set of { M , N } as shown below: 123 123 01 00 0 01 0 0001 1 0 00 0 00 1 00 1 0 ,, 00 0 1 1 00 0 0100 00 10 0 1 0 0 1 000 0100 00 10 0 00 1 1 000 00 0 1 0 01 0 ,, 0001 1 0 00 0 1 0 0 00 1 0 01 00 1 00 0 ⎡⎤ ⎡⎤ ⎡⎤ ⎢⎥ ⎢⎥ ⎢⎥ −− ⎢⎥ ⎢⎥ ⎢⎥ == = ⎢⎥ ⎢⎥ ⎢⎥ −− ⎢⎥ ⎢⎥ ⎢⎥ −− ⎣⎦ ⎣⎦ ⎣⎦ ⎡⎤ ⎡⎤ ⎡⎤ ⎢⎥ ⎢⎥ ⎢⎥ −− ⎢⎥ ⎢⎥ ⎢⎥ == = ⎢⎥ ⎢⎥ ⎢ −− ⎢⎥ ⎢⎥ ⎢ −− ⎣⎦ ⎣⎦ ⎣⎦ MM M NN N ⎥ ⎥ (16) And the AF 11 1 1 1 1 1 1 ,;, 11 11 1 1 1 1 ⎧− − − − ⎫ ⎡⎤ ⎡ ⎤ ⎡ ⎤ ⎡⎤ ⎪⎪ ⎨⎬ ⎢⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎥ −− − − ⎪⎪ ⎣⎦ ⎣ ⎦ ⎣ ⎦ ⎣⎦ ⎩⎭ of order 2 and “ type ” (2, 2; 2, 2) on Construction 1 , we obtain the followings O-STBC F8, which has the fo rm in Q 2 : ** 1 1 22 33 44 ****** 1 1 223344 ** * * * * 22 1 1 44 3 3 ** 22 1 1 44 33 ** * ** * 33 4 4 1 1 2 2 ** 33 44 1 1 22 ** * ** * 44 33 2 2 1 1 ** 44 33 22 1 1 x xx x x x x x x x xxxxxx x xxx x xx x x x x xxx xx x xx x xx x x x x xxx xxx x xx x xx x x x xxx xx x x ⎡ −−− ⎢ − ⎢ ⎢ −− − ⎢ −− −−− ⎢ = ⎢ −−− ⎢ −− −−− −− − −−−− − ⎣ F8 ** 2 ** ** ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎦ ⎡⎤ ⎢⎥ −− ⎢⎥ ⇒= ⎢⎥ −− ⎢⎥ −− ⎣⎦ PQ R S QP S R RS P Q SR Q P Q (17) Although it may not be obvious, it can be shown that by interchanging the first two and the last two co lumns and negating the last two rows of Q 2 , it becomes exactly the Theorem 2 proposed in [9]. ** 2 ** ** 10 0 0 0010 01 0 0 0001 00 1 0 1 000 00 0 1 0 100 = Theoream 2 of [9] ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥ −− ⎢ ⎥⎢ ⎥ ⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎢ ⎥ −− − ⎢ ⎥⎢ ⎥ ⎢ ⎥ −− − ⎣ ⎦⎣ ⎦ ⎣ ⎦ RS P Q SR Q P PQ R S QP S R Q We have also shown earlier in Example 1 that Q 1 and Theorem 1 of [9] are of the same design. Hence the proposed Construction 1 uni fies both Theorem 1 and 2 of [9] by usi ng different set of { M , N }. ■ R EFERENCES [1] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs”, IEEE Trans. on Info. Theory , vol. 45, pp. 1456-1467, Jul. 1999. [2] G. Ganesan and P. Stoica, “Space- time block codes: a maxim um SNR approach”, IEEE Trans. on Info. Theo ry , vol. 47, pp.1650- 1656, May 2001. [3] O. Tirkkonen and A. Hottinen, “S quare-matrix embeddable space-tim e block codes for complex signal constellations”, IEEE Trans. on Info . Theory , vol. 48, pp.384-395, Feb. 2002. [4] X. Liang, “Orthogonal designs with maxim al rates”, IEEE Trans. on Info. Theory , vol. 49, pp. 2468 –2503, Oct. 2003. [5] K. Lu, S. Fu, and X. Xia, “Close form s designs of complex orthogonal space-time block codes of rates (k +1)/(2k) for 2k-1 or 2k transmit antennas”, IEEE ISIT 2004 , pp. 307. [6] O. Tirkkonen and A. Hottinen, “C om plex space-time block codes for four transmit antennas”, IEEE Globecom 2000 , vol. 2, pp. 1005-1009. [7] Y. Xin, Z. Wang, and G. B. Gia nnakis, “Space-time diversity system s based on linear constellation precoding”, IEEE Trans. on Wireless Communications , vol. 2, pp. 291-309, Mar. 2003. [8] L. C. Tran, J. Seberry, Y. Wang, and et. al ., “Two new complex orthogonal space-time block code s for 8 transm it antennas”, IEE Electronics Letters , vol. 40, pp. 55-56, 2004. [9] L. C. Tran, T. A. Wysock i, J. Sebe rry, A. Mertins, and S. A. Spence, “Generalized Williamson and Wallis -Whitem an constructions for improved square or der-8 CO STBCs”, IEEE PIMRC 2005 . [10] J. Seberry, S. A. Spence, and T. A. Wysocki, “A construction technique for generalized com plex orthogonal desi gns and applications to wireless comm unications”, Linear Alg. and Applic. , vol. 405, pp. 163-176, 2005. [11] J . S e be r r y , L . C . T r a n , Y . W a n g , B . J . Wysocki, T. A. Wysocki, T. Xia, and Y. Zhao, “New complex orthogona l space-tim e block codes of order Accepted by IEEE Trans. Vehicular Technology 7 eight”, Signal Processing for Telecommunications and Multimedia , Springer, New York, pp. 173- 182. [12] A. V. Geramita and J. Seberry, Orthogonal designs, quadratic forms and Hadamard matrices , Marcel Dekker, 1979. [13] C. Yuen, Y. L. Guan, and T. T. Tjhung, “ Orthogonal space-time block code from am icable co m plex orthogonal design”, IEEE ICASSP 2004 , pp. 469-472.