Rectangular information lossless linear dispersion codes

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 2, FEBRUARY 2010 517 Rectangular Information Lossless Linear Dispersion Codes Jinsong Wu, Member, IEEE, and Steven D. Blostein, Senior Member, IEEE Abstract—This paper extends square 𝑀 × 𝑀 linear dispersion codes (LDC) proposed by Hassibi and Hochwald to 𝑇 × 𝑀 non-square linear dispersion codes of the same rate 𝑀 , termed uniform LDC, or U-LDC. This paper establishes a unitary property of arbitrary rectangular U-LDC encoding matrices and determines their connection to the traceless minimal nonorthogonality criterion for space-time codes. The U-LDC are then applied to rapid fading channels by constructing traceorthonormal versions, or TON-U-LDC for 2𝐿 and 4𝐿 input symbols, where 𝐿 is a positive integer. Compared to a variety of state-of-the-art codes, the proposed codes are found to perform well in both block and rapid fading channels. In rapid fading, the symbol-wise time diversity order of a 𝑇 × 𝑀 , TON-U-LDC for 2𝐿 input symbols is shown to be min(𝑇 ,2𝑀 ). Index Terms—Space-time coding, MIMO systems, transmit diversity. I. I NTRODUCTION ASSIBI and Hochwald have proposed linear dispersion codes (LDC) as a general framework for arbitrary complex space time codes for block flat-fading channels[1]. In recent years, a number of high-rate block based complex space-time code designs have been have been proposed, including [1]–[4]. However, existing high-rate complex spacetime codes have limited choices of size and have been mainly applied to block fading channels. To better exploit available time diversity, suitable designs for rapid (i.e., fast) fading channels are also needed. In block fading channels, this may be achieved by interleaving input symbols within one spacetime codeword over multiple fading blocks. Even with the use of symbol-interleaving, however, it is not guaranteed that a codeword would necessarily benefit from the available diversity due to rapid fading since wireless channels are highly dynamic. The design of space-time codes that operate well in both block fading and rapid fading channels is therefore critical. In this paper, an approach to better exploit time diversity in rapid fading is investigated, one that employs more flexibly-sized rectangular algebraic code designs. Hassibi and Hochwald proposed a rate-𝑀 linear dispersion codes (LDC) of arbitrary square size 𝑀 × 𝑀 in [1], Eq. (31). This paper extends [1], which we term HH square LDC, to arbitrary rectangular size 𝑇 × 𝑀 with rate-𝑀 , for both 𝑇 ≥ 𝑀 and 𝑇 < 𝑀 . These are termed uniform linear dispersion codes (U-LDC). First, a crucial unitary property of H Manuscript received December 23, 2007; revised July 16, 2009; accepted September 10, 2009. The associate editor coordinating the review of this letter and approving it for publication was A. Nosratinia. The authors are with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, Canada, K7L3N6 (e-mail: [email protected]; [email protected]). This paper was presented in part at Globecom 2005. This research was supported by Natural Sciences and Engineering Research Council of Canada Discovery Grant 41731. Digital Object Identifier 10.1109/TWC.2010.02.090065 arbitrary U-LDC encoding matrices is established. Following this, a connection is made between this unitary property and the traceless minimal non-orthogonality criterion for spacetime codes. Based on U-LDC, this paper proposes traceorthonormal uniform LDC (TON-U-LDC) for 2𝐿 and 4𝐿 input symbols, for integers 𝐿 > 0. Unlike U-LDC, the dispersion matrices for the real and imaginary components of source symbols in TON-U-LDC may differ. While TON-ULDC achieves maximal symbolwise diversity order in block fading channels, in rapid fading channels, the symbol-wise time diversity order of 𝑇 × 𝑀 TON-U-LDC 2𝐿 is shown to be min(𝑇 ,2𝑀 ). Finally, in comparison to a number of other codes, TON-U-LDC are shown to perform effectively in both block and rapid fading channels. The following notation is used: (⋅)𝒯 and (⋅)ℋ for matrix transpose and matrix transpose conjugate, respectively, √ 𝛿 (⋅) for Kronecker delta, Tr (⋅) for matrix trace, 𝑗 for −1, I𝐾 for a 𝐾 × 𝐾 identity matrix, and 0𝐴×𝐵 for an 𝐴 × 𝐵 all-zero matrix. II. LDC E NCODING IN M ATRIX F ORM Assume that uncorrelated input bits have been modulated using complex-valued symbols chosen from an arbitrary constellation. A 𝑇 × 𝑀 LDC matrix codeword, S𝐿𝐷𝐶 , is transmitted from 𝑀 transmit channels and occupies 𝑇 channel uses and encodes 𝑄 source symbols [1]. The matrix codeword S𝐿𝐷𝐶 is expressed as S𝐿𝐷𝐶 = 𝑄 ∑ 𝛼𝑞 A𝑞 + 𝑗𝛽𝑞 B𝑞 , (1) 𝑞=1 where S𝐿𝐷𝐶 ∈ 𝐶 𝑇 ×𝑀 , and A𝑞 ∈ 𝐶 𝑇 ×𝑀 , B𝑞 ∈ 𝐶 𝑇 ×𝑀 , 𝑞 = 1, ..., 𝑄 are called dispersion matrices, the data symbol constellation is 𝑠𝑞 = 𝛼𝑞 + 𝑗𝛽𝑞 , 𝑞 = 1, ..., 𝑄. An alternative dispersion matrix definition [1] is S𝐿𝐷𝐶 = 𝑄 ∑ 𝑠𝑞 C𝑞 + 𝑠∗𝑞 D𝑞 , (2) 𝑞=1 where C𝑞 = 12 (A𝑞 + B𝑞 ) and D𝑞 = 21 (A𝑞 − B𝑞 ), 𝑞 = 1, ..., 𝑄. This paper considers the cases of both A𝑞 = B𝑞 and A𝑞 ∕= B𝑞 , where 𝑞 = 1, ..., 𝑄. For the case of A𝑞 = B𝑞 , 𝑞 = 1, ..., 𝑄, reordering S𝐿𝐷𝐶 by using vec(.) operation, we obtain vec (S𝐿𝐷𝐶 ) = G𝐿𝐷𝐶 s, where G𝐿𝐷𝐶 = [vec(A1 ), ..., vec(A𝑄 )] is LDC encoding matrix, 𝒯 and s = [𝑠1 , ..., 𝑠𝑄 ] . The data symbol coding rate of LDC is defined as [1] c 2010 IEEE 1536-1276/10$25.00 ⃝ 𝑠𝑦𝑚 𝑅𝐿𝐷𝐶 = 𝑄 . 𝑇 (3) 518 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 2, FEBRUARY 2010 III. U NIFORM L INEAR D ISPERSION C ODES (U-LDC) A. U-LDC Construction ) ( 2𝜋(𝐾−1) 2𝜋 . Denote Π𝐾 as Denote 𝒟𝐾 = diag 1, 𝑒𝑗 𝐾 , ..., 𝑒𝑗 𝐾 a matrix of size 𝐾 ×𝐾 with zeros except [Π𝐾 [ ]𝑎,𝑎−1 = 1, 𝑎 =] 2, ..., 𝐾 and [Π𝐾 ]𝐾,𝐾 = 1. Denote X = I𝑇 , Z𝑇 ×(𝑀−𝑇 ) where Z𝑇 ×(𝑀−𝑇 ) is a zero matrix. 1) The case of 𝑇 ≤ 𝑀 : Denote Γ = X. The 𝑇 × 𝑀 U-LDC dispersion matrices are: 1 A𝑀(𝑘−1)+𝑙 = B𝑀(𝑘−1)+𝑙 = √ [𝒟𝑇 ]𝑘−1 Γ [Π𝑀 ]𝑙−1 , (4) 𝑇 where 𝑘 = 1, ..., 𝑇 and 𝑙 = 1, ..., 𝑀 . 2) The case of 𝑇 > 𝑀 : Denote Γ = X𝒯 . The 𝑇 × 𝑀 U-LDC dispersion matrices are: 1 𝑘−1 𝑙−1 Γ [𝒟𝑀 ] , (5) A𝑀(𝑘−1)+𝑙 = B𝑀(𝑘−1)+𝑙 = √ [Π𝑇 ] 𝑇 where 𝑘 = 1, ..., 𝑇 and 𝑙 = 1, ..., 𝑀 . B. U-LDC Properties Uniform linear dispersion codes of arbitrary size simultaneously hold the following two properties: 1) Unitary encoding matrix: Property 1: For uniform linear dispersion codes with arbitrary 𝑇 × 𝑀 size dispersion matrices A𝑞 , 𝑞 = 1, ..., 𝑇 𝑀 , the encoding matrix G𝐿𝐷𝐶 = [vec(A1 ), ..., vec(A𝑇 𝑀 )] is unitary. A proof of Property 1 is provided in Appendix A. Property 1 implies other desirable properties mentioned in Appendix A. According to [1], [2], Property 1 ensures capacity and energyoptimality in block-fading space-time channels. For 𝑇 ≥ 𝑀 , U-LDC meets the more restrictive constraint 1 I𝑀 . (6) [A𝑞 ]ℋ A𝑞 = 𝑀 This unitary property ensures arbitrary size 𝑇 × 𝑀 dispersion matrices A𝑞 , 𝑞 = 1, ..., 𝑇 𝑀 satisfy the traceless minimal non-orthogonality criterion for block quasi-static fading channels [5]: [ ] [ ] ℋ ℋ Tr [A𝑞1 ] A𝑞2 = Tr A𝑞1 [A𝑞2 ] = 0, (7) for any 1 ≤ 𝑞1 ∕= 𝑞2 ≤ 𝑇 𝑀 . The traceless minimal non-orthogonality criterion [5] has been related to the error union bound (EUB) [6], [7]. In [7], Tirkkonen and Kokkonen have proven that (7) minimizes the dominant self-interference related to EUB. To the authors’ knowledge, this paper is the first to establish the link between trace-based criteria and unitary criteria for space-time coding designs. 2) Symbolwise diversity [5], [8]: is a special case of full diversity since protection against single-symbol errors is a necessary condition for full diversity’s protection against multiple-symbol errors. Property 2: Uniform linear dispersion codes of arbitrary size 𝑇 × 𝑀 dispersion matrices A𝑞 , 𝑞 = 1, ..., 𝑇 𝑀 achieve symbolwise diversity order 𝑟 = min {rank (A𝑞 ) , 𝑞 = 1, ..., 𝑄} = min{𝑀, 𝑇 }. Property 2 can be easily proven. To accommodate the requirement of arbitrary size, only full symbolwise diversity is guaranteed, while both [3] and [4] consider the achievement of full diversity. IV. C ONSTRUCTION OF T RACE O RTHONORMAL L INEAR D ISPERSION C ODES A. Introduction Using {C𝑞 , D𝑞 } as dispersion matrices, trace orthonormal linear dispersion codes (TON-LDC) of size 𝐾𝑀 × 𝑀 were proposed in [3]. TON-LDC may achieve the lower bound of the worst-case pairwise error probability of the maximum likelihood detector [3]. We remark that U-LDC is also TONLDC with A𝑞 = B𝑞 , where 𝑞 = 1, ..., 𝑄. However, in this section, we aim to construct TON-LDC with A𝑞 ∕= B𝑞 , where 𝑞 = 1, ..., 𝑄. If the source data symbols are energy-normalized as 𝔼(∥𝑠𝑞 ∥2 ) = 1, then the general conditions of TON-LDC [3] can be re-expressed as follows: 1) For the case of 𝑇 > 𝑀 : ℋ ℋ [C𝑞 ] C𝑞 + [D𝑞 ] D𝑞 = 𝑇 I𝑀 . 𝑄 (8) 2) For the cases of both 𝑇 > 𝑀 and 𝑇 ≤ 𝑀 : ( ) a) Tr C𝑞 [C𝑝 ]ℋ + D𝑝 [D𝑞 ] ℋ = 𝑀𝑇 𝑄 𝛿(𝑝 − 𝑞) ( ) ℋ ℋ b) Tr D𝑞 [C𝑝 ] + D𝑝 [C𝑞 ] =0 (9) (10) where C𝑝 (or C𝑞 ) and D𝑝 (or D𝑞 ) for {𝑝, 𝑞} = 1, ..., 𝑄 are defined in Section II. We remark that in [9], TON-LDC may have the additional condition ℋ ℋ [D𝑞 ] C𝑞 + [C𝑞 ] D𝑞 = 0, (11) where 𝑞 = 1, ..., 𝑄. However, a well-performing TON-LDC may not necessarily satisfy condition (11), e.g., see the 2 × 2 optimal design on page 626 of [9]. In the following, we only assume conditions (8), (9), and (10). B. Construction of trace orthonormal linear dispersion codes for 𝑄 = 2𝐿 1) General procedure: The following provides a general procedure to construct TON-LDC for the case of an even number of data source symbols. In this construction procedure, the new TON-LDC dispersion matrices, denoted by { } (2) (2) (2) (2) A𝑞 , B𝑞 , C𝑞 , D𝑞 , are constructed from an existing { } (1) (1) (1) (1) LDC denoted by matrices A𝑞 , B𝑞 , C𝑞 , D𝑞 , where (1) (1) (1) (1) A𝑞 = B𝑞 . Recall from Eq. (2) that C𝑞 and D𝑞 are (1) (1) functions of A𝑞 and B𝑞 . In the TON-LDC, however, (2) (2) A𝑞 ∕= B𝑞 . Proposition 1: Consider with en] [ a linear dispersion code (1) (1) (1) coding matrix G𝐿𝐷𝐶 = vec(A1 )...vec(A𝑄 ) for 𝑄 data (1) symbols as defined in Section II, where A𝑞 , 𝑞 = 1, ..., 𝑄, where Q is an even number, are the corresponding dispersion matrices of size 𝑇 × 𝑀 . Assume the following holds for 𝑝 = 1, ..., 𝑄 and 𝑞 = 1, ..., 𝑄: (i) For the case of 𝑇 > 𝑀 , [ ]ℋ 𝑇 I𝑀 . (12) A(1) A(1) 𝑞 𝑞 = 𝑄 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 2, FEBRUARY 2010 (ii) For both cases of 𝑇 > 𝑀 and 𝑇 ≤ 𝑀 , ( ]ℋ ) [ [ ]ℋ (1) (1) (1) A vec(A ) = Tr A ) vec(A(1) 𝑝 𝑞 𝑞 𝑝 𝑇𝑀 𝛿(𝑝 − 𝑞). (13) 𝑄 The codes are constructed by defining the following matrices: (i) For 𝑞 = 1, ..., 𝑄/2, = C(2) 𝑞 = D(2) 𝑞 = 1 √ A(1) 𝑞 , 2 1 √ 𝑒𝑗𝜇 A(1) 𝑞+(𝑄/2) . 2 (14) (ii) For 𝑞 = ((𝑄/2) + 1), ..., 𝑄, (1) C(2) 𝑞 = A𝜎(𝑞−(𝑄/2)) , D(2) 𝑞 = −𝑒𝑗𝜇 A(𝑄/2)+𝜎(𝑞−(𝑄/2)) . (1) (15) where 𝜇 is an arbitrary real number, and 𝜎 (𝑎) is an arbitrary fixed permutation of 𝑎 = 1, ..., 𝑄/2. (2) (2) Then C𝑞 and D𝑞 consist of a set of dispersion matrices, as defined in Section II, of trace orthonormal linear dispersion codes. The proof of Proposition 1 is omitted due to space limitations. Proposition 1 demonstrates that there are an infinite number of TON-LDC constructions which satisfy conditions (8), (9), and (10). Proposition 1 can be used not only for constructing rate-𝑀 TON-LDC, but also for constructing lowrate TON-LDC. Unlike in [9], Proposition 1 provides flexible choices of codeword sizes 𝑇 × 𝑀 and only requires that the number of source data symbols be even. 2) A special subclass of codes - TON-U-LDC-2𝐿: Note that U-LDC are consistent with the assumptions of Proposition 1 in the case of 𝑄 = 𝑇 𝑀 , where 𝑄 is even, i.e., 𝑄{= 2𝐿. } (1) We propose to use dispersion matrices of U-LDC as A𝑞 and apply the procedure in Section IV-B1 to generate traceorthonormal uniform linear dispersion codes, denoted as TONU-LDC-2𝐿. To gain more insight into TON-U-LDC-2𝐿, it is useful to (2) (2) calculate dispersion matrices A𝑞 and B𝑞 , A(2) 𝑞 = = B(2) 𝑞 (2) C(2) 𝑞 + D𝑞 ⎧ (1) (1)  A𝑞 + 𝑒𝑗𝜇 A𝑞+(𝑄/2) ,   ⎨ 𝑞 = 1, ..., 𝑄/2 (1) (1)  A𝜎(𝑞−(𝑄/2)) − 𝑒𝑗𝜇 A(𝑄/2)+𝜎(𝑞−(𝑄/2)) ,   ⎩ 𝑞 = ((𝑄/2) + 1), ..., 𝑄 = C(2) − D(2) ⎧𝑞 (1) 𝑞 (1)  A𝑞 + 𝑒𝑗𝜇 A𝑞+(𝑄/2) ,   ⎨ 𝑞 = 1, ..., 𝑄/2 = (1) (1)  A + 𝑒𝑗𝜇 A(𝑄/2)+𝜎(𝑞−(𝑄/2)) ,   ⎩ 𝜎(𝑞−(𝑄/2)) 𝑞 = ((𝑄/2) + 1), ..., 𝑄 (2) (2) It is easy to verify rank(A𝑞 ) = rank(B𝑞 ) = min {𝑀, 𝑇 }, and thus having real and imaginary components per symbol achieves full component-wise diversity. 519 ] ] [ [ , 1 ≤ 𝑡 ≤ 𝑇. and b𝑡 = 𝑏1𝑡 , ..., 𝑏𝑀 Denote c𝑡 = 𝑐1𝑡 , ..., 𝑐𝑀 𝑡 ]𝒯 [ 𝑡 𝒯 𝒯 The probability of transmitting 𝒞 = [c1 ] , ..., [c𝑇 ] ]𝒯 [ 𝒯 𝒯 and deciding in favor of ℬ = [b1 ] , ..., [b𝑇 ] at the maximum-likelihood decoder is well approximated by [10] )−𝑁 ∏ ( 2 𝑃 (𝒞 → ℬ) ≤ , (16) ∣c𝑡 − b𝑡 ∣ 𝜌/4 𝑡∈𝒱(𝒞,ℬ) where 𝜌 is symbol signal-to-noise-ratio (SNR), 𝒱(𝒞, ℬ) denotes the set of time instances 1 ≤ 𝑡 ≤ 𝑇 such that ∕ 0 and ∣𝒱(𝒞, ℬ)∣ denotes the number of elements ∣c𝑡 − b𝑡 ∣ = of 𝒱(𝒞, ℬ). Thus, the diversity achieved is 𝑁 ∣𝒱(𝒞, ℬ)∣ for space-time rapid fading channels [10]. Note that the code construction of TON-U-LDC-2𝐿 is quite flexible, and a general analysis of 𝑁 ∣𝒱(𝒞, ℬ)∣ is difficult. However, we introduce the concepts of symbol and (real and imaginary) component-wise diversity ∣𝒱(𝒞, ℬ)∣ for a single input source symbol or component error as ⎧  𝒞 ⎫  𝑠𝑞 = 𝑠ℬ  𝑞 , 𝑞 = 1, ..., 𝑄,  ⎨  ⎬  ∣𝒱(𝒞, ℬ)∣𝑠 = min ∣𝒱(𝒞, ℬ)∣  except ∃𝑘, 1 ⩽ 𝑘 ⩽ 𝑄,    ⎩ ⎭  𝑠𝒞 ∕= 𝑠ℬ 𝑘 𝑘 and  𝒞 ⎫  𝑠𝑞 = 𝑠ℬ  𝑞 , 𝑞 = 1, ..., 𝑄,     except ∃𝑘, 1 ⩽ 𝑘 ⩽ 𝑄 ⎬  ∣𝒱(𝒞, ℬ)∣𝑐 = min ∣𝒱(𝒞, ℬ)∣  , 𝒞 ℬ    either 𝛼𝒞𝑘 = 𝛼ℬ  𝑘 , 𝛽𝑘 ∕= 𝛽𝑘     ⎩ ⎭  or 𝛼𝒞 ∕= 𝛼ℬ , 𝛽 𝒞 = 𝛽 ℬ ⎧    ⎨ 𝑘 𝑘 𝑘 𝑘 } { } { ℬ ℬ respectively, where 𝑠𝒞𝑞 = 𝛼𝒞𝑞 + 𝑗𝛽𝑞𝒞 and 𝑠ℬ 𝑞 = 𝛼𝑞 + 𝑗𝛽𝑞 are source symbol sequences for ℬ and 𝒞, respectively. For 𝑇 ≥ 2𝑀 , it can be verified that TON-U-LDC-2𝐿 always achieves ∣𝒱(𝒞, ℬ)∣𝑠 = ∣𝒱(𝒞, ℬ)∣𝑐 = 2𝑀 , while the TON-LDC proposed in [9] only achieves ∣𝒱(𝒞, ℬ)∣𝑠 = ∣𝒱(𝒞, ℬ)∣𝑐 = 𝑀 . Further, for 𝑇 ≤ 2𝑀 , ∣𝒱(𝒞, ℬ)∣𝑠 = ∣𝒱(𝒞, ℬ)∣𝑐 = 𝑇 is guaranteed to hold for TON-U-LDC-2𝐿. C. Construction of trace orthonormal linear dispersion codes for 𝑄 = 4𝐿 } { (1) Using dispersion matrices of U-LDC as A𝑞 , another set of dispersion matrices of trace orthonormal uniform linear dispersion codes for 𝑄 = 4𝐿, denoted as TON-U-LDC 4𝐿, may be constructed as follows: 1) For 𝑞 = 1, ..., 𝑄/4, 1 (1) 1 (1) (2) C(2) 𝑞 = √ A𝑞 , D𝑞 = √ A𝑞+(𝑄/2) . 2 2 (17) 2) For 𝑞 = 1 + (𝑄/4), ..., 𝑄/2, C(2) 𝑞 = D(2) 𝑞 = 1 √ 𝑒𝑗𝜇 A(1) 𝜏 (𝑞−(𝑄/4))+(𝑄/4) , 2 1 √ 𝑒𝑗𝜇 A(1) 𝜏 (𝑞−(𝑄/4))+(3𝑄/4) . 2 (18) 3) For 𝑞 = ((𝑄/2) + 1), ..., 3𝑄/4, 1 (1) (−1) (1) (2) C(2) 𝑞 = √ A𝑞−(𝑄/2) , D𝑞 = √ A𝑞 . 2 2 (19) 520 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 2, FEBRUARY 2010 4) For 𝑞 = 1 + (3𝑄/4), ..., 𝑄, = D(2) 𝑞 = −1 10 (20) In the above, 𝜇 is an arbitrary real-valued constant and 𝜏 (𝑎) is an arbitrary fixed permutation of 𝑎 = 1, ..., 𝑄/4. −2 10 BER C(2) 𝑞 1 √ 𝑒𝑗𝜇 A(1) 𝜏 (𝑞−(3𝑄/4))+(𝑄/4) , 2 1 √ 𝑒𝑗𝜇 A(1) 𝜏 (𝑞−(3𝑄/4))+(3𝑄/4) . 2 TON−U−LDC 2L TON−U−LDC 4L U−LDC HH TAST GD Golden Codes TON 2x4 (Zhang) −3 10 D. Remarks Note that a specific permutation determines a specific TONU-LDC construction, and thus both TON-U-LDC 2L and 4L are a set of codes due to allowance of different permutations. According to our experiences, TON-U-LDC under different permutations approach similar performance for 2L and 4L cases, respectively. Note that a specific permutation determines a specific TONU-LDC construction. The TON-U-LDC 2L and 4L above each refer to a set of codes, allowing for different permutations. The authors have observed, however, that the performances of TON-U-LDC 2L and 4L are not affected significantly by these different permutations. −4 10 10 12 14 16 18 SNR (dB) Fig. 1. BER performance comparison in space time block fading channels, 𝑀 = 𝑁 = 2, 𝐶𝐶𝐼 = 4 V. P ERFORMANCE −2 10 BER In this section, under the same spectral efficiency (bits per time channel use), we compare the U-LDC and TONU-LDC codes with other well-known full-diversity rate-𝑀 designs: HH [1], TAST [11], FDFR [4], TON-LDC [3], [9], Golden codes [12], Heath-Paulraj (HP) codes [2], and GoharyDavidson (GD) codes [13]. In the comparisons, MIMO flat fading channels are assumed. The matrix channel is assumed to be constant over a block of an integer number of symbol time slots, which we denote as the channel change interval (𝐶𝐶𝐼), and changing independently and identically distributed between blocks. Data symbols use 4-QAM modulation in all simulations. The numbers of transmit and receive antennas are 𝑀 and 𝑁 , respectively, where 𝑀 = 𝑁 . Each LDC codeword is of size 𝑇 × 𝑀 . Maximum likelihood decoding is performed at the receiver. Average symbol SNR per receive antenna is reported in all figures. As shown in Figs. 1 and 2, in 𝑀 = 𝑁 = 2 MIMO block fading channels (𝐶𝐶𝐼 = 4), TON-U-LDC 2𝐿 of size 4 under 𝜇 = 41 𝜋 and permutation 𝜎 (𝑎) = (𝑄/2) - a+1 performs second-best, while, in rapid fading channels (𝐶𝐶𝐼 = 1), TON-U-LDC 2𝐿 and TON-U-LDC 4𝐿 (under 𝜇 = 14 𝜋 and permutation 𝜏 (𝑎) = 𝑎) outperform the others. As shown in Figs. 3 and 4, in 𝑀 = 𝑁 = 3 MIMO block fading channels (𝐶𝐶𝐼 = 4), U-LDC of size 5 × 3 performs the best, TON-U-LDC 2𝐿 of size 4 × 3 under 𝜇 = 32 𝜋 and permutation 𝜎 (𝑎) = 𝑎 performs second best, while, in rapid fading channels (𝐶𝐶𝐼 = 1), TON-U-LDC 2𝐿 again performs the best. Fig. 5 illustrates the effects of 𝜇 on the performance of TON-U-LDC 2𝐿 under permutation 𝜎 (𝑎) = (𝑄/2) − 𝑎 + 1 in block fading channels at 15dB SNR. However, in the case of rapid fading channels (𝐶𝐶𝐼 = 1), it has been observed that the performance of TON-U-LDC 2L is insensitive to the choice of 𝜇. 8 −3 10 TON−U−LDC 2L TON−U−LDC 4L U−LDC HH TAST GD Golden Codes TON 2x4 (Zhang) −4 10 −5 10 8 10 12 14 16 18 SNR (dB) Fig. 2. BER performance comparison in space-time rapid fading channels, 𝑀 = 𝑁 = 2, 𝐶𝐶𝐼 = 1 In summary, it can be observed that even though U-LDC and TON-U-LDC are not claimed to possess full diversity, they are able to outperform several well-known full diversity codes in the literature in certain cases. Although U-LDC, TON-ULDC 2𝐿, and TON-U-LDC 4𝐿 have been compared with different space-time dimensions, we do not claim that they have superior performance over all other codes of arbitrary size. Rather, we conclude that U-LDC, TON-U-LDC 2𝐿, and TON-U-LDC 4𝐿 are all of flexible size, mathematically tractable, and possess desirable properties as discussed in Sections III and IV. We remark that the performance of ULDC of larger dimensions for the case of MIMO-OFDM channels appears in [14]. In this paper, channels are assumed to be uncorrelated. In the future, code design for correlated channels could be studied. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 2, FEBRUARY 2010 521 −3 −1 10 x 10 3.2 3 −2 10 2.8 BER BER 2.6 −3 10 U−LDC TON−U−LDC 2L TAST HP FDFR HH −4 10 2.4 2.2 2 −5 10 8 10 12 14 16 18 1.8 0 50 100 Fig. 3. BER performance comparison in space-time block fading channels, 𝑀 = 𝑁 = 3, 𝐶𝐶𝐼 = 60 −1 10 200 250 300 350 Fig. 5. Effects of 𝜇 on the performance of TON-U-LDC 2L in space-time block fading channels, 𝑀 = 𝑁 = 2, 𝑇 = 4„ 𝐶𝐶𝐼 = 60 where C −2 10 BER 150 μ(o) SNR (dB) [ ]ℋ 𝑘 −1 𝑘 −1 = [Γ]𝒯 𝒟𝑇𝑝 𝒟𝑇𝑞 Γ ( ) 𝑘 −𝑘 𝒟𝑇𝑞 𝑝 0𝑇 ×(𝑀−𝑇 ) = . 0(𝑀−𝑇 )×𝑇 0(𝑀−𝑇 )×(𝑀−𝑇 ) −3 10 Note that the above proof uses the following facts U−LDC TON−U−LDC 2L TAST HP FDFR HH −4 10 ℋ a) ΠM and Γ is real matrix, thus [ΠM ] = [ΠM ] ℋ 𝒯 and , ]∗ 𝑗𝜃= [Γ] −𝑗𝜃 [ 𝑗𝜃 [Γ] 𝑞 𝑃 𝑒 = 𝑒 𝑃 𝑒𝑗𝜃𝑞 = 𝑒𝑗(𝜃𝑞 −𝜃𝑃 ) . b) 𝑒 −5 10 8 9 10 11 12 13 SNR (dB) 14 15 16 17 Fig. 4. BER performance comparison in space-time rapid fading channels, 𝑀 = 𝑁 = 3, 𝐶𝐶𝐼 = 1 Also, note that Π𝒯M C rotates the rows of C upward, and CΠM rotates the columns of C leftward. So, Π𝒯M CΠM just rotates the diagonal elements of C from top left corner to bottom ) as C is a diagonal matrix ( right corner and hence, Tr Π𝒯M CΠM = Tr (C). Also, Tr A PPENDIX A A U NIFIED P ROOF OF THE U NITARY P ROPERTY Proof: Consider 𝑝 = 𝑀 (𝑘𝑝 − 1) + 𝑙𝑝 and 𝑞 = 𝑀 (𝑘𝑞 − 1) + 𝑙𝑞 , where 1 ≤ {𝑘𝑝 , 𝑘𝑞 } ≤ 𝑇 and 1 ≤ {𝑙𝑝 , 𝑙𝑞 } ≤ 𝑀 . Denote ) ( ℋ ℋ = [vec(A𝑝 )] vec(A𝑞 ) Δ𝑝,𝑞 = Tr vec(A𝑞 ) [vec(A𝑝 )] ) ( = Tr [A𝑝 ]ℋ A𝑞 . 1) The case of 𝑇 ≤ 𝑀 : The following always holds for U-LDC, Δ𝑝,𝑞 = = ([ ]ℋ [ ]ℋ 1 𝑙 −1 𝑘 −1 ℋ Tr ΠM𝑝 [Γ] 𝒟𝑇𝑝 𝑇 ) 𝑘 −1 𝒟𝑇𝑞 ΓΠM 𝑙𝑞 −1 (( ) )𝑙𝑝 −1 1 𝒯 𝑙𝑞 −1 [ΠM ] Tr CΠM , 𝑇 𝒯 (( ) { )𝑙𝑝 −1 Tr(C), 𝒯 [ΠM ] CΠM 𝑙𝑞 −1 = 0, 𝑙𝑝 = 𝑙𝑞 𝑙𝑝 ∕= 𝑙𝑞 Thus, Δ𝑝,𝑞 = { Tr(C), 0, 𝑙𝑝 = 𝑙𝑞 and 𝑘𝑝 = 𝑘𝑞 , 𝑖.𝑒., 𝑝 = 𝑞 𝑙𝑝 ∕= 𝑙𝑞 2) The case of 𝑇 > 𝑀 : Using Tr (XY) = Tr (YX), ([ ]ℋ [ ]ℋ 1 𝑙 −1 𝑘 −1 Tr 𝒟M𝑝 Γ Π𝑇𝑝 Δ𝑝,𝑞 = 𝑀 ) 𝑘 −1 Π𝑇𝑞 Γ𝒟M 𝑙𝑞 −1 ( [ ]ℋ 1 𝑘 −1 𝑙 −1 𝒯 Tr Π𝑇𝑞 Γ𝒟M 𝑙𝑞 −1 𝒟M𝑝 = [Γ] 𝑀 [ ]𝒯 ) 𝑘𝑝 −1 Π𝑇 . The rest of the proof for the case of 𝑇 > 𝑀 is similar to that of the case of 𝑇 ≤ 𝑀 . 522 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 2, FEBRUARY 2010 R EFERENCES [1] B. Hassibi and B. M. 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