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Int. J. Communications, Network and System Sciences, 2011, 4, 430-435 doi:10.4236/ijcns.2011.47051 Published Online July 2011 (http://www.SciRP.org/journal/ijcns) Copyright © 2011 SciRes. IJCNS A Design Method of Noncoherent Unitary Space-Time Codes Li Peng, Qiuping Peng, Lingling Yang Department of Electronics and Information Engineering, Huazhong University of Science and Technology, Wuhan National Laboratory for Optoelectronics, Wuhan , China E-mail: pengli@mail.hust.edu.cn Received May 13, 2011; revised June 18, 2011; accepted June 24, 2011 Abstract We generalized an constructing method of noncoherent unitary space time codes (N-USTC) over Rayleigh flat fading channels. A family of N-USTCs with T symbol peroids, M transmit and N receive antennas was constructed by the exponential mapping method based on the tangent subspace of the Grassmann manifold. This exponential mapping method can transform the coherent space time codes (C-STC) into the N-USTC on the Grassmann manifold. We infered an universal framework of constructing a C-STC that is designed by using the algebraic number theory and has full rate and full diversity (FRFD) for t symbol periods and same antennas, where M, N, T, t are general positive integer. We discussed the constraint condition that the expo- nential mapping has only one solution, from which we presented an approach of searching the optimum ad- justive factor αopt that can generate an optimum noncoherent codeword. For different code parameters M, N, T, t and the optimum adjustive factor αopt, we gave the simulation results of the several N-USTCs.1 Keywords: Noncoherent Uintary Space-Time Codes (N-USTC), Coherent Space-Time Codes (C-STC), Grassmann Manifold, Degree of Freedom, Exponenatial Map, Full Rate and Full Diversity (FRFD) 1. Introduction The noncoherent unitary space-time code (N-USTC) in [1-4] provided a potential solution for the multiple an- tennas communication in fading channel that neither transmitter nor receiver knows the channel state imfor- mation (CSI). This paper generalized an constructing method for a family of the N-USTCs based on the Grassmann manifold. The system models on noncoherent and cohenent channel are comparatively built. Starting from the basic theory of the Grassmann manifold [4], a basic thought of designing the Grassmannian unitary space-time matrix was described. That is the exponent mapping method [5,6] from the M t C-STC to the TM N-USTC for the MIMO system with M trans- mit and N receive antennas, where t and T are co- herent and noncoherent symbol periods, respectively, and ,,, M NTt are general positive integer, TtM and TMt. In order to map the M t C-STC into the TM N-USTC, firstly, one must consider how to construct the M t C-STC. Many literatures [7-11] discussed multi- farious methods of constructing the C-STCs. Enlightened by [7-11] and other literatures (omitted in reference as the limitation of length), we discussed a method of con- structing the M t universal C-STCs with FDFR based on the algebraic number theory. Therefore, we created four kinds of matrices: uncoded symbol matrix S , lin- ear combinatorial matrix L , rotated matrix R and lin- ear combinatorial symbol matrix Z that is =ZLSR formed by the linear combinatorial technique of the symbols of constellations, such as q-PSK or q-QAM, and then we get the the coded matrix of a C-STC by trans- forming matrix Z. In the mapping process from the M t C-STC into the TM N-USTC, we discussed the constraint condi- tion of the only one solution of the the exponent map, from which we discover that the optimum codeword of the Grassmannian N-USTC can be obtained by searching the optimum adjustive factor opt . Simulation tests show that for BPSK constellation symbols, when T is un- changed and antenna number = M N increases, the 1This work is supported by National Natural Science Foundation o f China under Grant No. 61071069. L. PENG ET AL. Copyright © 2011 SciRes. IJCNS 431 spectral efficiency increases and the performance of the bit error rate (BER) also advances, or when = M N is unchanged and T increases, so do the spectral effi- ciency and the BER performance; for QPSK constella- tion symbols, when =2MN and 5T, the spectral efficiency achieves 2.4 bits/Hz/s but at the cost of sacri- ficing the BER performance. 2. System Model and Background Knowledge 2.1. System Model We focus on the block fading channel model on which the fading coefficients are assumed to be constant during T periods of one codeword and to change independ- ently from one codeword to the next. Under the assump- tion of no inter-symbol interference, the noncoherent channel model with codeword periods TM is TNTM MNTN TYXHW (1) For the convenience of comparison and application later on, we simultaneously give the coherent channel model with codeword periods tM: N tNMMtNt YHBW (2) where Y is TN received signal matrix for nonco- herent model or Nt matrix for coherent model, H is M N or NM fading coefficients matrix and W is TN or Nt additive noise matrix. Elements of H and W are assumed to be the independent and identically distributed complex Gaussian random vari- ables respectively from distribution 0,1CN and 2 0,CN . TM X and M t B are noncoherent and co- herent transmit signal matrices, respectively. 2.2. Grassmann Manifolds and Its Tangential Space Manifold is a topologic space which is locally homeo- morphic to the Euclidian space. More formally, Every point on n-dimensional manifold has a neighborhood homeomorphic to n-dimensional Euclidian space n R. We consider a set of all M -dimension linear sub- spaces in T-dimension complex space. This set has the structure of manifold, called Grassmann manifold and denoted by , C TM G, and its definition [12] is: † , C TM M GΦΦΦ I (3) where “†” denotes transpose for real number or conju- gate transpose for complex number; Φ denotes the subspace spanned by M column vectors in an TM unitary matrix Φ. , C TM G can also be represented by the quotient space of the unitary group nU[12], i.e. , C TM TMTMGU UU (4) As the real dimension of the unitary group nU is 2 dimRnn U, one can obtain the real dimension of , C TM G: 2 22 , dim 2 C RTM TM TMMTM G [13] according to (4). So the complex dimension of , C TM G is , dim C CTM M TMG which means that the N-USTCs on , C TM G have M TM degrees of freedom, and the maximal symbol rate is 1 M MT [3]. Literature [5,6] introduces that the tangential space of any a point on , C TM G forms a set of matrices as follows: † 0 0 TM Δ B QB (5) where M TM BC and the point Q can be chosen arbitrarily, i.e., for simplified calculation, one can choose TMM MTM M † - ,BII 0as a reference subspace on , C TM G. The dimension of the tangential space defined by (5) is also M TM. According to the theory of Lie- group, i.e., the point of the tangential space on , C TM G can be projected into the point of , C TM G by exponent map, the point X of , C TM G can be denoted by the ex- ponent form of the tangential space: † 0 exp 0 TM TM B X I B (6) (6) shows a complicated computing task, but it can be simplified by the technique of the singular value decom- pose (SVD) of matrix. B is disposed by the SVD as follows: † MM M TMTM TM ΛBUV (7) where U and V are unitary matrices, and the form of Λ is: 1000 0 00 000 0 M Λ (8) where 1,, M are the singular values of matrix B. Putting (7) into (6), one can obtain the simplified TM X : † † TM TM UCU XVSU (9) where 1 cos 0 0cos MM M C, L. PENG ET AL. Copyright © 2011 SciRes. IJCNS 432 † 1 sin 0 0 0sin TM M M S. 3. Coherent Space-Time Codes Most methods of constructing the C-STCs with FRFD are to efficaciously combine all information symbols i s (1, 2,,itM ) to form the coding matrix M t B , where all i s belong to one of constellations, such as q-PSK or q-QAM, etc. If we adopt the technique of linear combi- nation to design M t B , then the rank and determinant properties of M ti s B is equivalent to them of M tiMt j s sBB , where ,1,ijMt and ij . Let r be the minimal rank available of any codeword matrix M ti s B. According to the design criterion of determi- nate in [10], under the linear combination of all i s of forming M t B , we can obtain a FRFD matrix and its rank rM, so the maximum coding gain 11 11 rM rM jj jj can be guaranteed. Being enligh- tened by using the algebraic number theory to construct the C-STCs in [8,10,11], we investigate how to design the universal coherent matrix M t B with FRFD for ,, M Nt being general positive integer. The applied de- sign step is shown as follows. a) We first create three kinds of matrices: uncoded symbol matrix S , linear combinatorial matrix L (also named left-multiplied matrix) and rotated matrix R (also named right-multiplied matrix), they have next ge- neral forms: 11 11 22 12 2 21 221 1 12 21 1 , 1 1, 1 10 0 00 00 MtM MtM MM Mt M MM MMM t ss s ss s ss s jj j jj j S L R where 12 ,, , tM s ss take from the constelltions; let t ; choosing j makes the determinate of matrix L be unequal to zero. Let i e which is an alge- braic number [10], here 1i and is a parameter of needing the optimization design so that is searched in 0, π2 to maximize the coding gain. b) For S left-multiplied by L and right-multiplied by R, one can get the linear combinatorial symbol matrix M t Z like (10) as follows: c) In the linear combinatorial symbol matrix M t Z, circulant-right-shifting the second row one time, the third row two times,…, the final row (i.e., the M th row) 1 M times, respectively, one can get the coded matrix like (11) as follows: 21 11 1(1) 221 1 22 2(1) 1 12 21 2 11 121 22 100 1 00 1 00 1 ... M MMt MM MMt Mt t MMM MM Mt MM MMMM ss s ss s jj j ss s jj j ssss ss ZLSR 1 11 21 1 1 11 21 11 11 121 22 11 11 11 21 1211 21 121 22 ... ... t Mt Mt M Mt t Mt Mt MM tt MMM M MM Mt tM Mt Mt MMMMMM MM MM ss s sjs sjsjssjsjs js sjs sj sjssj sjs j 21MM Mt s (10) L. PENG ET AL. Copyright © 2011 SciRes. IJCNS 433 1 11 21 11 121 221 2 1 12 22 11)2111 12 11 11 1 11 11 21 t Mt Mt MM MMM MM Mt t t Mt Mt Mt MtMM M MM MM Mt Mt Mt tM Mt Mt M ss ssssss s sjs sjs sjsj s js js sjs j B 1 12 121 12 21 21 2 M MM MMM MM M M Mt sjs sj sjs js s (11) Several examples of the C-STCs are presented as fol- lows. Let 2Mt, according to (10), we have: 13 22 24 12 34 1234 110 10 ss ss j ss ss sjs sjs ZLSR If 1j, then 22 0 Z which means the linear combinatorial form of information symbols is lost. If 1j , the linear combinatorial operation is retained. For π4 2ei , we can get the 22 C-STC matrix which is same as those in [5,6] 12 34 22 34 12 s sss ss ss B Again let 3M , 4t , we have 2 14710 22 342 5 8 112 22 36912 3 10 00 1000 100 0 100 0 ssss j jssss jjssss ZLSR when π 6 4ei , 2π 3 ei j, 34 Z is full rank. Thus we can get the 34 C-STC matrix like (12) as follows: 222232 1234567891011 12 322 2222222 341011121 234567 89 222 3222222 789101112123 456 ssssssssss ss s jsj ss jsj ssjsj ssjsj s s jsjssjsjssjs jss jsjs B (12) Similarly, we can get 23 B , 24 B , 25 B and 33 B which and all above will be applied to simulation testing later on. 4. Noncoherent Space-Time Codes Literatures [5,6] introduce the design criterion of the Grassmann N-USTC. Let i and j denote the sub- spaces spanned by the column vectors of i X and j X , respectively. Let 12 ,,, M denote the principal angles between i and j , then the chordal product distance between two points i X and j X on , C TM G is: 2 , 1 sin M ij m m (13) The design criterion of the Grassmann N-USTC C is to make the minimal chordal product distance achieve the maximum, i.e. , max min ij ij XXC C . It is known from the expression (6) that the product ,0i Θ of the chordal dis- tances between the subspace i and the reference sub- space 0 is equal to the product of all singular values in matrix i B whose M column vectors span the sub- space i . The design criterion of a C-STC is to maxi- mize its coding gain, which is equal to maximizing the minimum product of singular values of codeword matrix. Therefore we can use the matrix M t B of (11) to design the matrix B in (6). The exponential map from M t B to TM X must be the monotone and reversible, which requires that the expo- nential map of (6) is the reversible map, i.e., (9) exists the reversible matrix. So cos m and sin m in (9) should be the monotone function, then the constraint L. PENG ET AL. Copyright © 2011 SciRes. IJCNS 434 condition of m is: max π/2,0,1, ,1 mMt mBmM (14) where mMt B is the mth singular value of any codeword M t B which is equal to the mth principal angle between any and 0 . Therefore, a conceiv- able skill is that taking a scale , called the adjustive factor which multiplies the codeword matrix M t B, can guarantee the map to be monotone and reversible. Thus (6) can be rewritten as follows: † 0 exp 0 TM TM B X I B (15) Obviously, only affects the singular value of the matrix M t B. Let max 0, denote a range of values, the method of optimum searching is described as follows. Let , ,C ij TM XXG be two distinct N-USTC code- words, the SVD of the matrix † j i X X is †† ji Σ X XUV, where Σ is a diagonal matrix formed by the singular values 1,, M . The M principal angles between the subspaces i and j spanned respectively by i X and j X are 1 cos mm , 0,1,,1mM, substi- tuting 1 cos mm for the expression (13), we can get: 2 , 1 1 M ij m m (16) Under the condition of making ,ij maximize, by searching in max 0, , we can get the optimum opt . Now we design TM X by mapping exponentially M t B to , C TM G. The design step is: (a) For M transmit antennas and t coherent peri- ods, according to (10) and (11), design the C-STC matrix M t B from the information symbol 12 ,, , tM s ss. (b) Substitute M t B for B in (15), search the opti- mum adjustive factor , and construct the exponential mapping matrix † 0 0 M t Mt B EB. (c) According to (7) and the above E matrix, apply- ing the SVD to opt Mt B, we get the noncoherent codeword TM X like (9), and TMt. □ 5. Examples and Numerical Simulation Results According to the above presented method, this section gives several examples of the N-USTCs whose numeri- cal simulation curves are shown as Figure 1. Let DMTM denote the degree of freedom. Suppose modulation is q-PSK with symbol number p. So the spectral efficiency of the N-USTC is 2 log TM pDT bits/Hz/s. Example 1: Compare two curves of solid line with black dot and dash line with circle in Figure 1. For sys- tem 2MN and QPSK modulation with 4p , let 2t , we construct the C-STC 22 B ,where π4 2ei , 1j . When 3t, we get 23 B , where π4 3ei , 1j . As TMt, having 4T for 2t and 5T for 3t, corresponding to 4D and 6D , we compute 42 2 bits/Hz/s and 52 2.4 bits/Hz/s, respectively. We map the C-STC 22 B into the N-USTC 42 X on 4,2 C G and 23 B into 52 X on 5,2 C G which correspond to the op- timum adjustive factor 42 ,0.29 opt Q and 52 ,0.25 opt Q , respectively. At 5 10 bit error rate (BER), 52 X out- perform 42 X about 3 dB. Obviously, under all para- meter being same except T increasing, the N-USTC BER performance and the spectral efficiency are im- proved. Example 2: Compare four curves of solid line with black square, dash line with white square, solid line with black diamond and dash line with white diamond in Fig- ure 1. For system 2MN and BPSK modulation with 2p , let 2, 3, 4, 5t , get 4,5, 6, 7T and 4, 6,8,10D , so 421 , 52 1.2 , 62 1.33 and 72 1.43 bits/Hz/s, respectively. We map 22 B , 23 B , 24 B and 26 B into 42 X , 52 X , 62 X and 72 X whose factors are 42 ,0.41 opt B , 52 ,0.36 opt B , 62 ,0.32 opt B and 72 ,0.29 opt B . At 5 10 BER, the performance of the N-USTC improve about 0.5 - 1.0 dB and the spectral efficiency increases along with T in- creasing. Example 3: Compare two curves with solid line with black triangle and dash line with white triangle in Figure 1. For system 3MN and BPSK modulation with 2p , let 3,4t , get 6, 7T and 9,1 2D , so 63 1.5 and 73 1.71 bits/Hz/s, respectively. We map 33 B and 34 B into 63 X and 73 X whose fac- Figure 1. Performance comparison of several N-USTCs. L. PENG ET AL. Copyright © 2011 SciRes. IJCNS 435 tors are 63 ,0.24 opt B and 73 ,0.22 opt B . At 5 10 BER, the performance of the N-USTC improve about 0.8 dB from 6T to 7T, and the spectral efficiency also increases 0.2 bits/Hz/s. 6. Conclusions A specific step that maps the coherent space-time matrix into the noncoherent space-time matrix by means of the exponent form of the tangential space of Grassmann manifold was summed up for designing the N-USTCs. Especially, our work makes the structural parameters ,,, M NTt with regard to both the N-USTC based on the Grassmann manifold and the C-STC based on the alge- braic number theory be able to be designed more flexibly. We also discovered that in the discussed family of Grassmannian N-USTC, the optimum codeword can be obtained by searching the optimum adjustive factor opt . It is noticed that the design of the parameter j in left- multiplied matrix L is open problem, we will track this problem in the future. 7. References [1] T. L. Marzetta and B. M. Hochwald, “Capacity of a Mo- bile Multiple-Antenna Communication Link in Rayleigh flat Fading,” IEEE Transactions on Information Theory, Vol. 45, No. 1, January 1999, pp. 139-157. doi:10.1109/18.746779 [2] B. M. 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