BLOCK DATA TRANSMISSION: A COMPARISON OF PERFORMANCE FOR THE MBER PRECODER DESIGNS. Qian Meng, Jian-Kang Zhang and Kon Max Wong

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1 BLOCK DATA TRANSISSION: A COPARISON OF PERFORANCE FOR THE BER PRECODER DESIGNS Qian eng, Jian-Kang Zhang and Kon ax Wong Department of Electrical and Computer Engineering, caster University, Hamilton, Ontario, Canada mengq@grads.ece.mcmaster.ca ABSTRACT Block data transmission is an efficient scheme to mitigate the inter-symbol interference ISI caused by dispersive channels. Recent research work attempt to obtain optimal designs of precoders given various linear and non-linear receivers. In this paper we focus on the systems with optimal precoder designs based on the minimization of bit error rate BER which is the more meaningful criterion in data communications. The linear receivers employed in these systems include zero-forcing and minimum mean square error SE equalizers. Recently, block decision feedback equalization DFE has also been suggested to be used in the design of such block data transmission systems. We also revisit the jointly optimum designs of the precoder coupled with the DFE receivers. First we derive the block-data error rate for all of these different systems with their corresponding optimal precoders. Then we rank their performance accordingly. Simulation results are also obtained to verify our analysis. Later, we evaluate the computational complexity on the various systems and draw some conclusions on their relative merits. Finally, we derive the lower and upper bounds of bit error rate BER considering error propagation for systems employing nonlinear equalizers and give criteria on minimizing the error propagation on the premise of the optimal block error performance. 1. INTRODUCTION In the transmission of digital data over dispersive media, inter-symbol interference ISI is a major performance limiting factor [8]. To mitigate such an effect, it is often helpful to transmit information-bearing data in equal-size blocks [4]. Examples of block based communication systems include important multi-carrier C systems such as orthogonal frequency division multiplexing OFD [7]. Recently, a broad class of linear block-by-block transmission schemes, which includes DT and OFD as special cases, has been studied in detail [9], [10]. The design of block based linear transmitters for block-by-block linear receivers in particular, was studied in [9] and the transmitters which minimize the mean square error of the equalized symbols were derived for both and SE equalizers, under the assumption that the channel state information is known. This work sparked off much further research especially in the design of optimum designs of linear precoders according to the minimum bit error rate BER criterion, which is more meaningful for data communications. These include the BER precoders for and SE equalizers [1], [2]. ore recently, using a different approach, the BER precoder has been extended to a system with a DFE [3] [5], part of which can be realized either using the criterion or the SE criterion. The performance of this optimum precoder fitted with such a DFE appears to be superior to the other optimum precoder equipped with linear equalizers. 2. BER PRECODER DESIGNS WITH LINEAR EQUALIZERS In this paper we employ the generalized block-by-block transceiver model developed in [9]. This model encompasses many modern communication systems, including OFD and DT. To effectively transmit the data block, a data sequence {s 1,...,s } is first passed through a serial-to-parallel converter forming an 1 data vector s = [s 1,...,s ] T for which Ess H = I. This vector is then processed by a N,N = + L L is channel order, linear precoder matrix F 0. This added redundancy is the key to avoiding inter-block interference at the receiver. Two commonly used techniques of such redundancy are the zero padding ZP and the cyclic prefix CP schemes. By the use of ZP transmission, the form of precoder matrix is F 0 = [ F 0 ], where F is an matrix. The vector of transmitted symbols, u [u1,...,u] T for these schemes can be written as u = Fs. The transmission of this symbol vector u through a dispersive channel can then

2 be represented in a matrix form as r = Hu + ν, where r = [r 1,...,r N ] T is a N 1 received signal vector, H is a N full rank channel matrix which is known at both the receiver and the transmitter, and ν = [ν 1,...,ν N ] T is a white Gaussian noise vector being statistically independent of the signal and having R νν = Eνν H = σ 2 I. At the receiver, various types of equalizers can be used. A linear equalizer [9] has the advantage of being simple in implementation. If a linear equalizer is used, this can be represented by an N matrix G. This equalizer matrix performs the function of equalizing the response of the channel matrix as well as eliminating the redundancy caused by IBI. By grouping the parts for redundancy in the transmitter and the parts for its elimination in the equalizer together with the channel matrix, we can arrive at the following compact matrix equation for the signal at the output of the receiver: ŝ = GHFs + Gν, 1 This output is passed through a threshold for decision of the transmitted symbol. The optimum designs considered in Eq. 1 aim at choosing an optimal precoder F to minimize the BER of ŝ when the transmission power is constrained to p 0 and the equalizer G is either of The Type [1]: In this case, G = HF, and the optimum precoder is given by F = p 0 trλ 1/2 WΛ1/4 D, where Λ and W are the eigenvalue and eigenvector matrices of H H H 1, respectively, D is a DFT matrix. Then the BER with the equalizer, P ber, is P ber p 0 Based on this optimal design, the optimal block-data error rate,, is = 1 1 P ber 3 Or, The SE Type [2]: In this case, G = F H H H σ 2 I + HFF H H H 1. The correspondence BER precoder is F SE = WΦD, where Φ is an diagonal matrix such that p0 + σ 2 trλ 1/2 φ mm = λmm σ 2 λ trλ 1/2 mm 4 when p 0 σ 2 1 N { 4trΛ 1/2 2 trλ, trλ 1/2 } λ 11 trλ 2 5 Again, the BER with the SE equalizer, P ber SE, is P ber SE p 0 + σ 2 trλ 1 Similarly, the optimal block-data error rate, SE, is given by 6 SE = 1 1 P ber SE 7 3. BER PRECODER DESIGN WITH DFE Decision-feedback equalization DFE [6], [7], [11] is an effective method to eliminate ISI in data communications. Recently [3], the idea has been employed in the form of a block decision feedback equalizer [12] for block data transmission systems. The DFE consists of an N feedforward matrix W operating on the N 1 received data block r. This is followed by a unit element anti-diagonal matrix J and the reversed data vector is then detected using a threshold detector. The output of the detector is then fed back through a strictly upper-triangular matrix B. Here, we seek a joint design of F,W, and B striving to minimize the BER of the scheme. We first establish the relationships between F,W, and B according to whether we desire or SE for the input to the detector. With these relationships, we seek to jointly design F,W, and B minimizing the system BER BER Precoder for -DFE For the case of -DFE, the optimum block bit error rate precoder is F DFE = p 0 VS, where V is the eigenvection matrix of H H R 1 νν H, S is obtained from an iterative algorithm [5] [3]. Here, we also analyze the block-data error rate for the -DFE scheme, arriving at where DFE = 1 1 P DFE 8 P DFE p0 deth H H 1/ σ BER Precoder for SE-DFE On the other hand, for the case of SE, the optimum block bit error rate precoder is given by F SE DFE = V ˆΦU H [3], where U is a unitary matrix, and ˆΦ is a diagonal matrix with ˆφ ii 2 = 1 p 0 + trˆλ 1 9 ˆλ 1 i 10

3 when 1 ˆλ ii 1 p 0 + j=1 ˆλ 1 j i = 1,2,, 11 where ˆΛ is the eigenvalue matrix of H H R 1 νν H. We derive the corresponding block-data error rate for the SE-DFE scheme, SE DFE, and show it to be SE DFE = 1 1 P SE DFE 12 P SE DFE p 0 + trˆλ 1 detˆλ 1/ COPARISON OF BLOCK ERROR RATES OF THE DIFFERENT SCHEES We now analyze and compare the derived block error rate formulae in Eqs. 3, 7, 8, and 12 and show the ranking of the performance of the various schemes accordingly. In the following subsections, we assume that the condition 5 and 11 hold and SE From Eqs. 2 and 6, according to Cauchy Schwarz inequality [14], then trλ trλ 1/2 2, we have P ber p 0 p 0 + σ 2 trλ trλ 1/2 2 Hence, = P ber SE and -DFE SE 15 From Eqs. 2 and 9, since trλ 1/2 detλ 1/2 1/, we have P ber p 0 p 0 σ 2 detλ 1/ p 0 deth H H 1/ σ 2 = P DFE 16 As a result, 4.3. SE and SE-DFE Comparing Eqs. 6 with 13, we have DFE 17 P ber SE p 0 + σ 2 trλ 1 p 0 + σ 2 trλ σ 2 2 detλ 1 1/ p 0 + trˆλ 1 detˆλ 1/ 1 = P SE DFE 18 notice that trˆλ 1 = σ 2 trλ and detˆλ 1 = σ 2 detλ for white noise. It means that, SE SE DFE DFE and SE-DFE Comparing Eqs. 9 with 13, again using trλ 1 detλ 1 1/, we arrive at, P SE DFE p 0 + trˆλ 1 detˆλ 1/ 1 Q p 0 detˆλ 1/ p0 deth H 1/ σ 2 This implies that 4.5. SE and -DFE = P DFE 20 SE DFE DFE 21 We cannot established a direct inequality between the performance of these two schemes for all SNR. However, from Eq. 14, we see that for high transmitted SNR i.e., σ 2 0, P ber SE = P ber. Thus, we can conclude from Eq. 16 that under high SNR, SE DFE 22

4 5. COPUTER SIULATION RESULTS AND COPUTATIONAL COPLEXITY COPARISON Figure 1 shows the computer simulation results comparing the performance of the various schemes. We also apply the imum likelihood L detection algorithm to a block data transmission system without a precoder F = I for comparison. The computer simulation results in Fig. 1 Block data error rate SE DFE SE DFE L SNRdB Fig. 1. Block error rate comparison among L,, SE, -DFE and SE-DFE equalizers verify that our analysis of the different schemes are indeed correct. We also evaluate the computational complexities [13] for each of the schemes and these are presented in Table 1 where K is the size of the transmitted signal constellation and L is the length of the channel impulse response. 6. UPPER BOUND OF BER FOR NONLINEAR EQUALIZER SYSTE For the nonlinear equalizer system, it is hard to drive an exact BER because of error propagation. In this section, we will derive the lower and upper bounds of BER for -DFE and SE-DFE systems under the assumption of optimal block error performance DFE System Let s consider a general multiple-input and multiple-output IO channel model r = Ĥs + ν 23 where Ĥ = HF, H is the full rank channel matrix, F is the precoder matrix, ν is a white Gaussian noise vector with L or -DFE or SE SE-DFE computation complexity OK L Olog, L O 4 Table 1. computational complexity comparison Eν H ν = σ 2 I, and r is the received vector. According to [5], using -DFE equalizer at receiver is same as using a QR-decomposition-based successive cancellation. Then, Eq. 23 can be written as r = Rs + ν where r H r, ν H ν. R is an upper triangular matrix and Q is an orthogonal matrix for QR-decomposition of Ĥ. To obtain the optimal block error performance, the diagonal elements of R must be equal, that is, R 11 = R 22 = = R = p 0 dethh H 1/2. Assuming the transmitted signal is BPSK, we can follow the strategy in [8] to prove P LB P ber DFE P LB1 + E 24 where P ber DFE is the arithmetic average bit error rate for the system with -DFE, and P LB is a lower bound of the average BER, P LB = 1 Rii i=1 Q σ. In addition, the error E is defined by E = 2 Q R11 2 σ 2 1 2R 2 P LB R 11 Here R is determined by R = 1 i 1 j=i+1 R ij 25 Therefore, to further optimize the system, we can find a precoder F among all optimal block-data error rate precoders F DFE such that the quantity R is minimized, since there are infinite such F DFE when > 2 [5] SE-DFE System We define A as the diagonal matrix and Y as the upper triangular matrix in the Cholesky decomposition of I + snrh H H, I + snrh H H = Y H AY, snr is the transmitted signal to noise ratio. B is the feedback matrix, B = Y I. To obtain the optimal block error performance, the diagonal elements of A must be equal [5] [3], that is, p A 11 = A 22 = = A = 0+trˆΛ 1 detˆλ 1/. Similarly, for SE-DFE system, we can derive P LB P ber SE DFE P LB1 + E 26

5 where P ber SE DFE is the arithmetic average bit error rate, and P LB is a lower bound of the average BER, P LB = 1 i=1 Q E = 2 Q A ii 2 j=1 Aij 2 +σ 2, the error E is defined as A 2 11 σ 2 + A isi 1 2B 2 P LB A 11 where A isi = j=1 A ij 2. Here B is determined by B = 1 i 1 j=i+1 B ij 27 So we give a criterion that minimizes the B. Using this criterion, we can construct a specified optimal precoder F which not only minimizes the block bit error rate, but also minimizes the error propagation among the transmitted signals from infinite selections [5]. 7. CONCLUSION In this paper, the block error performance and computation complixity comparisons among the systems employing, SE, -DFE and SE-DFE equalizers were given. From the performance analysis and comparison, we can see that the block data transmission system fitted with optimum precoder and DFE equalizers are superior in performance to those fitted with linear either or SE equalizers. However, the nonlinear DFE are computationally more complex. Furthermore, the SE version in either the linear or the nonlinear equalizer class has superior performance to the version. We also gave one criterion for -DFE equalizer system and the other criterion for SE-DFE equalizer system. Following these criteria, the specified precoders can be found to reduce error propagation on the premise of the optimal block error performance. 8. REFERENCES [4] G. D. Forney and. V. Eyuboǧlu, Combined Eualization and Coding Using Precoding, IEEE Communication agzine, 29-12, pp , Dec [5] J.-K. Zhang, A. Kavčić, X. a and K.. Wong, Design of Unitary Precoders for ISI Channels, ICASSP02 Orlando USA, ay [6] J.. Cioffi, G. P. Dudevoir,. V. Eyuboǧlu and G. D. Forney SE Decision-Feedback Equalizers and Coding Part I: Equalization Results and Part II: Coding Results, IEEE Trans. Commun., 43-10, pp , Oct [7] J. A. C. Bingham, ulticarrier odulation for Data Transmission: An Idea Whose Time Has Come, IEEE Communications agazine, 28-5, pp. 5-14, ay [8] J. G. Proakis, Digital Communications, Third Edition New York: cgraw-hill. [9] A. Stamoulis, G. B. Giannakis and A. Scaglione, Redundant Filterbank Precoders and Equalizers, Parts I and II, IEEE Trans. SP, 47-7, pp , July [10] A. Scaglione, G. B. Giannakis, Block FIR Decision- Feedback Equalizers for Filterbank Precoded Transmissions with Blind Channel Estimation Capabilities, IEEE Trans. Commun., 49-1, pp , Jan [11]. K. Varanasi, Decision Feedback ultiuser Detection: A Systematic Approach, IEEE Trans. Inform. Theory, 45-1 pp , Jan, [12] D. Williamson, R. A. Kennedy and G. W. Pulford, Block Decision Feedback Equalization, IEEE Trans. Commun., 40-2, pp , Feb [13] G. H. Golub and C. F. V. Loan, atrix Computations, Third Edition, The John Hopkins University Press, [14] R. A. Horn and C. R. Johnson, atrix Analysis, Combridge, A: Combridge University Press, [1] Y. Ding, T. N. Davidson, Z.-Q. Luo and K.. Wong, inimum BER Block Precoders for Zero- Forcing Equalization, IEEE Trans. SP, 51-9, pp , Sept [2] S. S. Chan, T. N. Davidson and K.. Wong, Asymptotically inimum Bit Error Rate Block Precoders for inimum ean Square Error Equalization, to appear in IEE Proc.-Comm.. [3] F. Xu, Design of Block-by-Blcok Transceivers with Decision Feedback Equalization,. Eng. Thesis, caster Univ., Hamilton, ON., Oct