Antenna Selection Scheme for Wireless Channels Utilizing Differential Space-Time Modulation

Transcription

1 Antenna Selection Scheme for Wireless Channels Utilizing Differential Sace-Time Modulation Le Chung Tran and Tadeusz A. Wysocki School of Electrical, Comuter and Telecommunications Engineering Wollongong University, Northfields Avenue, NSW 2522, Australia Phone: Fax: Abstract In this aer, the authors rove that the differential sace-time modulation techniques roosed in literature rovide a full satial diversity. Based on that, the authors roose an antenna selection scheme called general N-out-of-M antenna selection technique for wireless communications channels utilizing differential sace-time modulation. The simulation shows that differential detection associated with our roosed technique rovides much better bit error erformance over that without antenna selection, and even over coherent detection without antenna selection at certain signal-to-noise ratios.. Introduction Sace-time codes have been examined intensively and various roosals for sace-time codes have been mentioned in literature so far. Secifically, sace-time codes can be decoded coherently when transmission gains of the channels between transmit and receive antennas are assumed to be known at the receiver. This assumtion is suitable for the scenario where channels are assumed to be quasi-static flat fading, i.e. the fade changes so slowly that transmission gains are constant over a frame comrising multile symbol eriods. Therefore, the transmitter is able to transmit the training signals enabling the receiver to estimate the channels. In faster fading channels, however, this assumtion is not reasonable any more. The differential sace-time modulation (DSTM) [, 2, 3, 4] is then a more ractical candidate. In all existing DSTM techniques, transmission gains are not known at either transmitter or receiver. The drawback of differential detection is the worse of bit error erformance. In order to imrove the erformance of sacetime codes in fading channels, several antenna selection techniques (ASTs) have been roosed in [5, 6] and our imroved AST has been mentioned in [7] for coherent detection. The fundamental idea for selecting transmit (and/or receive) antennas in coherent detection is selecting the best channels out of available ones to transmit signals with an assumtion that the transmission gains are known at the receiver. Although, this idea cannot be directly alied in differential detection as the transmission gains are not known at the receiver, it suggests us to search for an AST in case of DSTM. As mentioned in more details in Section 4, we make use of transmitting the initial code block to enable the receiver to redict semiblindly, with the maximum likelihood, the best channels out of all available ones. The receiver then informs the transmitter via a feedback loo enabling the transmitter to select these transmit antennas. It should be emhasized that, although, the transmission gains change faster than those in coherent detection, so that the transmission of training signals is imractical and, consequently, the utilization of DSTM is useful, but they are assumed not to change too fast to transmit a few feedback bits. Otherwise, no closed-loo transmit diversity technique, certainly, is alicable. The rest of this aer is organized as follows. In Section 2, we recall some main oints of DSTM techniques roosed in literature. The diversity of sace-time codes utilizing DSTM is examined in the next section. In Section 4, we roose an antenna selection scheme called (general) N-out-of-M AST for wireless channels utilizing DSTM. The simulation is resented in Section 5 and Section 6 concludes the aer. The detailed mathematical roof of exressions used to rove the full satial diversity of DSTM is given in the Aendix. 2. Differential sace-time modulation In this section, at first, we briefly outline the DSTM scheme roosed by Ganesan et. al. [], which is based on the theory of unitary sace-time block codes. Then, we shortly consider the DSTM schemes roosed by Hughes [2] and by Hochwald et. al. [3]. We consider a system with n transmit and m receive antennas. Let R t, A, N t be the (m n)-sized matrices of received signals at time t, transmission gains between receive and transmit antennas, and noise at the receive antennas, resectively. The κη th element of A, namely a κη, is the gain factor of the ath between the η th transmit antenna and the κ th receive antenna. Transmission gains are assumed to be identically indeendently distributed (i.i.d.) comlex Gaussian ran-

2 dom variables with the distribution CN(0, ), which remain constant during every frame comrising several symbol eriods and change from frame to frame. It is imortant to note that, by using the term frame here, the authors do not mean that the considered channels are very slow fading (or quasi-static fading) ones like in coherent detection. We just use this term to make it easier to exlain the roosed AST in a very general case mentioned in Section 4. U to date, in all existing DSTM techniques, the channel gains have been assumed to be constant during, at least, two consecutive code blocks. Therefore, when the Alamouti DSTM, for instance, is used, the size of frames here is four symbol eriods. Again, the fade in the channels considered here is still fast enough, so that the utilization of DSTM is useful. Noises are assumed to be i.i.d. comlex Gaussian random variables with the distribution CN(0, σ 2 ). Let {s j } j= = {sr j +isi j } j= (where i 2 =, s R j and si j are the real and imaginary arts of s j, resectively) be the set of unitary symbols, which are derived from a signal constellation S and transmitted in the t th block. Since the symbols are unitary, each symbol has the unit energy: We define: Z t = j= (X js R j s j 2 = () + iy js I j ), where form a set of matrices of size {X j } j= and {Y j} j= n n, satisfying the following roerties, which are linked to the theory of amicable orthogonal designs [8]: X jx H j = I; Y jy H j = I j (2) X j X H k = X k X H j ; Y j Y H k = Y k Y H j k j (3) X j Y H k = Y k X H j k, j (4) where I is an identity n n matrix and (.) H is the Hermitian transose oeration of the argument matrix. From (), (2), (3) and (4), we have: Z t Z H t = ( s j 2 )I = I j= Hence, Z t is a unitary matrix. In the DSTM scheme roosed in [], at the beginning of every frame, an initial matrix W 0 = I n n is transmitted. Then, the matrix transmitted at time t (t =, 2, 3... ) is given by: W t = W t Z t (5) As Z t is a unitary matrix, the matrix W t is also a unitary one. The model of the channel at the t th transmission time (t = 0,, 2... ) is as follows (the 0 th transmission means the initial transmission): R t = AW t + N t (6) If the transmission gain matrix A is assumed to be constant over two blocks t and t, then the maximum likelihood (ML) detector for the symbols {s j } j= is calculated as follows [], [9]: {ŝ j } j= = Arg max {s j},s j S Re{tr(RH t R t Z t )} (7) where tr(.) is the trace oeration. Hence, the ML detector for the symbol s j is: s j S [Re{tr(RH t R t X j )s R j } + If we denote: +Re{tr(R H t R t iy j )s I j }] (8) D j = Re{tr(R H t R t X j )} + iim{tr(r H t R t Y j )} (9) where Re{.} and Im{.} are the real and the imaginary arts of the argument, resectively, then (8) becomes (see equation (3) in Aendix): s j S Re{D j s j } (0) where Dj is the conjugate of D j. Therefore, at the receiver, we form the statistic D j to decode the symbol s j. Exressions (9) and (0) show that the detection of the symbol s j is carried out without the knowledge of transmission gains. Particularly, the symbol s j can be decoded by using the received signal blocks in the two consecutive transmission times. The enalty of the differential detection mentioned above is that the signal-to-noise ratio (SNR) required for the same bit error rate as in the case of coherent detection is 3 db higher. This is exlicitly roven in [9] (equations (5.5) and (5.30)). Indeendently, Hughes roosed the DSTM scheme based on the grou theory in [2] and Hochwald et. al. devised a DSTM technique based on unitary sace-time codes in [3]. It is interesting that the detector for the symbol s j in these schemes (see equation (6) in [2] and Section V. C in [3]) is similar to that mentioned in (7), although these aroaches are different from each other. This note is imortant in the sense that the consideration mentioned in the next section is true for all these DSTM schemes. 3. The satial diversity of differentially modulated sace-time codes It is roven in Aendix (equation (2)) that the statistic D j in (9) is calculated as follows: D j tr(a H A)s j + Re{tr(W H t A H N t X j)} + Re{tr(N H t AW t X j) + iim{tr(w H t A H N t Y j)} + iim{tr(n H t AW t Y j )} where: X m κ= nx η= a κη 2 s j + η j η j = Re{tr(W H t A H N t X j )} + + Re{tr(N H t AW t X j ) + + iim{tr(w H t A H N t Y j )} + + iim{tr(n H t AW t Y j )}

3 Tx Tx 2 TX... Tx M- Tx M a a 2 a M a M feedback loo RX Fig.. The diagram of systems with the general N- out-of-m antenna selection scheme in the differential detection scenario. The statistic D j has a form of the received signal in the case that the symbol s j is transmitted while the noise of the channel is η j. The coefficient of s j in the statistic is small only when all m n comlex modules of transmission gains are small. In other words, the DSTM schemes roosed in [], [2] and [3] rovide a full satial diversity of m n level. It is shown in [9] (equation (5.30)) that the SNR of the statistic D j is aroximately: m κ= n η= a κη 2 SNR diff tr(a H A) 2σ 2 2σ 2 () Clearly, the larger the SNR diff is, the more recise the detection is. In addition, not only the DSTM techniques roosed in [], [2] and [3], but also the one roosed by Tarokh et. al. [4] is roven to rovide a full satial diversity (equation (26) in [4]). Generally seaking, all existing DSTM techniques have the same roerty that the SN R of the decision metric is linearly roortional to m n κ= η= a κη 2. This note is imortant in the sense that the AST mentioned below is alicable to all existing DSTM schemes. 4. Antenna selection technique for channels utilizing DSTM In this section, we consider a system comrising M transmit antennas and one receive antenna while the transmitted code blocks have a size of N N (N < M). The redundant transmit antennas are used to rovide satial diversity. If the transmission gains are known at the receiver then, from the equation (), one realizes that the otimal AST is selecting the N transmit antennas out of M available ones, from which the comlex modules of the transmission gains of the aths to the receive antenna are maximal. This scheme has been well examined for coherent detection [5]. However, this rincile cannot be directly alied for differential detection as the receiver has no knowledge about transmission gains. Therefore, we roose here an AST for differential detection where the transmitter selects transmit antennas based on the statistical comarison carried out at the receiver between received W L W L 2... W 2 W W 0 (a) W L 2... W 2 W (b) Time Fig. 2. Code blocks transmitted in a frame with (b) and without (a) the antenna selection technique. The delay of transmitting the feedback information is not considered. signals during the initial transmission time of each frame. Let us consider a system comrising M = 4 transmit antennas and only one receive antenna using the DSTM based on the Alamouti code (N = 2) as an examle. The transmission gain matrix is assumed to be constant in a frame comrising 2L symbol eriods. Again, as we emhasize in Section 2, this assumtion does not mean that the channels are very slow fading (or quasi-static fading) ones like in coherent detection. The roosed AST is as follows: At the beginning of every frame, the transmitter sends an initial block W 0 = I M via M transmit antennas, instead of sending an initial block W 0 = I N via N transmit antennas as in every existing DSTM technique. Because the size of the initial matrix W 0 has been changed, the size of the other matrices in (6) have also been changed. We emhasize the change in the size of matrices by using the tilde mark for matrices as below: W 0 W 0 = I 4, Ã = ( a a 2 a 3 a 4 ) Ñ 0 = ( n 0 n 02 n 03 n 04 ) where a η (η=... 4) is the transmission gain of the channel from the η th transmit antenna to the receive antenna, and n 0η is the noise affecting this channel during the initial transmission. The received signal in the initial transmission time is: R 0 = Ã W 0 + Ñ0 = ( a + n 0 a 2 + n 02 a 3 + n 03 a 4 + n 04 ) Let R0 = ( r 0 r 02 r 03 r 04 ). After determining the received matrix R 0, the receiver carries out two tasks. First, the receiver semiblindly estimates the N best channels based on the matrix R 0 by comaring the comlex modules of four received signals, namely r 0, r 02, r 03 and r 04, and then finding out the two received signals corresonding to the first and the second maximum comlex modules. Without

4 TRANSMITTER a. Transmits W 0 =I M b. Calculates W t using the tacit default matrix W 0 =I N 4. Achieves feedback information 5. Transmits via N antennas a b 2 3a 4 3b 5 6 RECEIVER 2. Determines N antennas 3a. Informs transmitter 3b. Forms R 0 6. Decodes symbols Bit Error Probability Differential detection with antenna selection Coherent detection without antenna selection Differential detection without antenna selection Fig. 3. The general N-out-of-M antenna selection technique for the system using DSTM. loss of generality, we assume here that they are the first and the second received signals. Then the receiver informs the transmitter via a feedback loo to select the first and the second channels (antennas) to transmit the rest of data of the considered frame (see Figure ). Again, it is emhasized in Section that, although, the transmission gains change faster than those in coherent detection, so that the transmission of training signals is imractical and, consequently, the utilization of DSTM is useful, but they are assumed not to change too fast to transmit a few feedback bits. Second, the receiver forms the matrix R 0, which is used to decode the code blocks transmitted in the next transmission times, by taking the first and the second elements of the matrix R 0, which are corresonding to the first and the second maximum comlex modules, i.e. R 0 =( a + n 0 a 2 + n 02 ). The transmission of the rest of data in the considered frame after the initial transmission is exactly the same as that in the system using the first and the second transmit antennas only. In other words, at the transmitter, the next transmitted matrices W t (t=, 2, 3,... ) in (5) are calculated by using the tacit default matrix W 0 =I N (in the examle, the default matrix W 0 =I 2 ). The formation of the matrices W t does not necessarily take lace after the transmitter achieves the feedback information. It is worth to note that, in all existing DSTM techniques, the initial matrix W 0 =I N is only used to initialize the transmission (W 0 is utilized to calculate the next transmitted blocks W t (t ), and to form the received matrix R 0, which is indisensable to decode the code blocks transmitted in the next transmission times). Unlike these techniques, in the roosed technique, the initial identity matrix W 0 =I M is transmitted. This matrix has two main roles. It enables the receiver to form the initially received matrix R 0 indirectly (from the received matrix R 0 ), which is used to decode the next code blocks. Simultaneously, in some sense, it also lays a role of training signals, i.e. it rovides the receiver with the statistic of the channels. This is the main difference between the differential modulation with SNR(dB), 2bits/s/Hz (QPSK) Fig. 4. Differential modulation with the roosed antenna selection techniques. our AST and the one without antenna selection. The transmission rocedure of a whole frame including L code blocks is shown in Figure 2. The code block W 0 is transmitted via four transmit antennas in four symbol eriods. The following blocks are transmitted via two transmit antennas in two symbol eriods. We can realize that another difference between the differential modulation with our AST and the one without antenna selection is the number of code blocks transmitted in a frame. If one can transmit L- code blocks W,..., W L carrying 2(L-) symbols in a frame in a differential modulation technique without antenna selection, then the number of those in a frame in the roosed technique is L-2 (or 2(L-2) symbols are transmitted). Additionally, if the channels are required to be constant during, at least, four symbol eriods in all existing DSTM techniques, then they are required to be unchanged during, at least, six symbol eriods in our roosed AST. In the above consideration, the delay of transmitting the feedback information from the receiver to the transmitter is not considered. We call the scheme mentioned above the general 2-out-of-4 AST. The scheme can be generalized to aly for other sace-time block codes of a larger dimension N as well as for any number of transmit antennas M (N < M) without any difficulty. This scheme is called the general N-out-of-M AST (or just N-out-of-M AST whenever there is no ambiguity) and resented in Figure Simulation Results In this Section, the erformance of differential modulation with the roosed AST is resented. The Alamouti code and the QPSK signal constellation are considered. In this simulation, we consider the 2-out-of-4 AST, i.e. N=2 and M=4, and the receiver uses 3 feedback bits to inform the transmitter. The SN R is defined to be the ratio between the

5 total ower that the receiver receives during each symbol eriod and the ower of noise. It can be seen from Figure 4 that differential detection without antenna selection has a 3 db worse erformance comared to coherent detection (without antenna selection). The erformance of differential detection with the general 2-out-of-4 AST is worse than that of the coherent detection at low SNRs. However, at SN Rs > 8 db, the antenna selection remarkably imroves the erformance of differential detection. For instance, at BER = 0 4, the erformance of differential modulation is 5 db better than that of coherent detection and, therefore, 8 db better than that of differential detection without antenna selection. This is interreted as follows. The received signal r 0η (η=... M) can be exressed: r 0η = a η + n 0η, where a η and n 0η are defined in Section 4. Because both a η and n 0η are the i.i.d. Gaussian random variables with the distribution CN(0, ) and CN(0, σ 2 ) resectively, r 0η is also an i.i.d. Gaussian random variable with zero exectation and its variance is given as follows: V ar(r 0η )=V ar(a η )+V ar(n 0η )=+σ 2 where V ar(.) is the variance oeration. It is imortant to notice that, essentially, the N-out-of-M AST roosed in Section 4 is based on selecting the N best antennas, which corresond to the N received signals of the highest ower (or highest variance). Therefore, if the SN R is large enough so that the variance of transmission gains is large enough, comared to the variance of noise σ 2, i.e. σ 2, then the variance of the received signal can be roughly aroximated to that of the transmission gain, i.e. V ar(r 0η ) V ar(a η ). As a result, at high SNRs, the roblem of selecting the N maximum comlex modules among r 0η (η=... M) may be considered as the roblem of selecting the N maximum values among a 0η. Consequently, at low SN Rs, selecting the maximum norms of the received signals does not always lead to selecting the otimal antennas which are corresonding to the transmission gains of the biggest norms, because the variance of noises σ 2, which is inversely roortional to the SNR, is not small enough comared to the variance of the transmission gains. In other words, the roosed AST does not always select the best antennas. However, at high SNRs, secifically at SNRs > 8dB, the contrast scenario usually haens. It means that, at larger SN Rs, the roosed antenna selection scheme is more recise. 6. Conclusion In this aer, we rove that all existing DSTM techniques rovide a full satial diversity of the m n level. Then we roose an antenna selection scheme called (general) N-out-of-M AST for wireless channels utilizing differential detection. Instead of transmitting the initial matrix W 0 = I N (N is the required number of transmit antennas), we transmit the initial matrix W0 =I M (M is the number of available transmit antennas). In all existing DSTM techniques, the signals received when the initial matrix W 0 is transmitted are used to initialize the transmission only. However, in our roosal, these signals are also utilized to enable the receiver to make the maximum likelihood decision about the best channels. The receiver then informs the transmitter via a feedback loo to select those channels. The channels are much faster fading ones comared to those in coherent detection, and consequently, the utilization of DSTM is useful (the transmission of training signals is imractical), but they are assumed not to change too fast to transmit some feedback information. The simulation shows that differential detection with the roosed technique rovides a much better bit error rate over that without antenna selection. Moreover, in comarison with the erformance of coherent detection, the roosed scheme also rovides a better bit error rate at any SNR >8 db. Therefore, the roosed AST remarkably imroves the erformance of wireless channels utilizing DSTM. 7. Aendix In this section, the authors mention the exression of the statistic D j mentioned in (9). Then, we rove that the detector for the symbol s j is given by: s j S Re{D j s j } Before roceeding further, it is imortant to note that:. {tr(ψa H A)} is real if Ψ is a Hermitian matrix, i.e. Ψ=Ψ H. Consequently, Im{tr(ΨA H A)}=0. 2. tr(ωυ)=tr(υω) if Ω and Υ are square matrices. 3. Z H t W H t =W H t 4. Z t = k= (X ks R k + iy ks I k ) 5. {X k } and {Y k } satisfy the following roerties: One has: a. X k Xk H = I; Y kyk H = I k b. X k Xl H = X l Xk H; Y kyl H = Y l Yk H c. X k Yl H = Y l Xk H; k, l k l R H t R t = (AW t Z t + N t ) H (AW t Z t + N t ) = Z H t W H t A H AW t + + Z H t W H t A H N t + + N H t AW t + N H t N t If the noise variance is small enough, the term Nt H N t is negligible. We have the following trans-

6 forms: Dj R = Re{tr(Rt H R t X j )} Re{tr(X H j Wt A H H AW t X j )s R j } + + Re{tr[ Re{tr( (Xk H s R k iyk H s I k) k= k= + Re{tr(Nt H AW t X j )} tr(a H A)s R j + W H t A H N t X j ]} iyk H Wt A H H AW t X j s I k)} + Re{tr(Zt H Wt A H H N t X j )} Im{tr( X j Y H k A H As I k)} + k= + Re{tr(Nt H AW t X j )} tr(a H A)s R j + Re{tr(Wt H A H N t X j )} Im{tr( k= + Re{tr(N H t AW t X j )} X j Y H k s I ka H A)} + Let Ξ= k= X jyk HsI k. Clearly, Ξ =ΞH, i.e. Ξ is a Hermitian matrix. Therefore: Im{tr( k= X jy HsI k AH A)}=0 and: D R j tr(a H A)s R j + Similarly, we have: k + Re{tr(W H t A H N t X j )} + + Re{tr(N H t AW t X j )} Dj I = Re{tr(Rt H R t iy j )} tr(a H A)s R j + Re{tr(Wt H A H N t iy j )} + Re{tr(N H t AW t iy j )} The statistic for decoding the symbol s j is given below: D j = Dj R + idj I = tr(a H A)s j + + Re{tr(W H t A H N t X j )} + + Re{tr(N H t AW t X j ) + + iim{tr(w H t A H N t Y j )} + + iim{tr(n H t AW t Y j )} (2) Decoding the symbol s j is equivalent to minimizing the following exression (note that s j 2 =): D j tr(a H A)s j 2 = D j D j + (tr(ah A)) 2 2 tr(a H A)Re{D j s j} Therefore, the detector of the symbol s j is: s j S [Re{tr(RH t R t X j )s R j } + + Re{tr(R H t R t iy j )s I j }] = Arg max s j S Re{D j s j } (3) Equations (2) and (3) are the aim of the roof. References [] G. Ganesan and P. Stoica, Differential modulation using sace-time block codes, IEEE Sign. Process. Lett., vol. 9, no. 2, , Feb [2] B. L. Hughes, Differential sace-time modulation, IEEE Trans. Inform. Theory, vol. 46, no. 7, , Nov [3] B. M. Hochwald and W. Sweldens, Differential unitary sace-time modulation, IEEE Trans. Commun., vol. 48, no. 2, , Dec [4] V. Tarokh and H. Jafarkhani A differential detection scheme for transmit diversity, IEEE J. Select. Areas Commun., vol. 8, no. 7, , July [5] D. Gore and A. Paulraj, Sace-time block coding with otimal antenna selection, Proc. IEEE International Conference on Acoustics, Seech, and Signal Processing, vol 4, , 200. [6] M. Katz, E. Tiirola and J. Ylitalo, Combining sace-time block coding with diversity antenna selection for imroved downlink erformance, IEEE Veh. Technol. Conf. VTC 200, vol., , 200. [7] L. C. Tran, T. A. Wysocki and A. Mertins, Imroved antenna selection technique to enhance the downlink erformance of mobile communications systems, Proc. 7th International Symosium on Signal Processing and Its Alications ISSPA 03, Paris, France, -4, July [8] A. V. Geramita and J. Seberry, Orthogonal designs: quadratic forms and Hadamard matrices, Lecture notes in ure and alied mathematics, vol. 43, Marcel Dekker, New York and Basel, 979. [9] G. Ganesan, Designing sace-time codes using orthogonal designs, Doctoral dissertation, Usala University, Sweden, 2002.