Bit-Interleaved Coded Differential Space-Time Modulation

Transcription

1 Bit-Interleaved Coded Differential Space-Time Modulation Lutz H.-J. Lampe, Student Member, IEEE and Robert Schober, Member, IEEE Abstract Bit-interleaved coded differential space-time modulation for transmission over spatially correlated flat Ricean fading channels is discussed. For improved noncoherent detection without channel state information at the receiver, iterative decoding employing hard-decision feedback and prediction-based metric computation is applied. The performance is assessed based on the associated cutoff rate, analytical expressions for the bit-error rate and the outage probability, respectively, and simulations. It is shown that the proposed scheme offers high power efficiency exploiting both space and time diversity, while computational complexity is kept at a relatively low level. Index terms: Differential space-time modulation, bit-interleaved coded modulation, Ricean fading channels, noncoherent detection, iterative decoding

2 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 2 1 Introduction Space-time modulation has attracted considerable attention because of its ability to provide transmit diversity for communication over flat fading channels, cf. e.g. [1, 2, 3]. More recently, differential space-time modulation (DSTM) has been proposed [4, 5] (cf. also [6]), which does not require knowledge about the coefficients of the flat fading channel at the receiver, i.e., channel state information (CSI) is not necessary. This is particularly interesting for relatively fast time-varying fading prohibiting reliable channel estimation. Efficient designs for DSTM are proposed in [6, 4, 5, 7], whereas improved receiver structures are presented in [8]. These receivers basically exploit an extended observation window of N > 2 received (matrix) symbols in contrast to conventional differential detection with N = 2. In this way, DSTM becomes applicable also in fast fading, and the performance of coherent space-time transmission with perfect CSI is approached [8]. In this paper, DSTM is combined with standard forward error-correction coding to achieve both a diversity and a coding gain. We focus on a low-complexity yet power efficient transmission system. At the transmitter, simple bit-interleaved coded modulation (BICM) [9, 10] and DSTM are concatenated, which makes the introduction of an appropriate labeling of space-time signal points necessary. For the processing at the receiver, we generalize ideas presented by the authors in [8] and [11, 12] for uncoded DSTM and single-antenna noncoherent BICM schemes, respectively. More specifically, iterative decision-feedback differential demodulation (DF-DM) [11] with predictionbased noncoherent metric computation and standard Viterbi decoding are applied. For DF-DM hard decisions of Viterbi decoding are fed back to keep complexity of metric computation low also in the case where an extended observation window (N > 2) is utilized. The predictor structure is favorable as it works without a priori information on channel statistics if standard adaptive algorithms are applied, cf. [13, 14]. Since the combination of BICM and DSTM is most successfully performed for two transmit antennas (see Section 2.1), we concentrate on this important special case for numerical and simulation results, respectively. Both analytical considerations and simulations show in good agreement that DF-DM significantly enhances power efficiency of coded DSTM over general spatially correlated Ricean fading channels. Our investigations further point out the performance improvement offered by transmit diversity compared to single-antenna coded transmission. We note that the use of BICM for coherent ST transmission without space-time

3 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 3 block codes (STBC s) has recently been addressed in [15, 16, 17, 18], where [16, 17, 18] also consider iterative decoding using soft-input soft-output modules. BICM combined with STBC s for coherent transmission is applied in [19, 20]. Although in [20] the concatenation of BICM with DSTM is mentioned, the authors do not elaborate on this issue. The paper is organized as follows. In Section 2, the proposed transmitter and receiver structures for coded DSTM are described, and the multiple-antenna channel model is introduced. Different measures to assess the performance of DSTM with BICM including analytical expressions for the achievable bit-error rate are provided in Section 3. Numerical and simulation results are given and discussed in Section 4. Section 5 concludes this paper. 2 Transmission System We consider a transmission scheme using N T transmit antennas and N R receive antennas. The block diagram of the discrete-time system model is depicted in Figure 1. The channel and all signals are represented in the equivalent low-pass domain, i.e., all quantities are complex valued. 2.1 BICM and Differential Space-Time Modulation A sequence of information bits enters the convolutional encoder. The encoder output symbols are interleaved yielding the sequence of coded bits c[i] (i Z: bit discretetime index). Then, l = log 2 (M) interleaved coded bits are mapped (M( )) to M-ary data-carrying (N T N T matrix) symbols V [k] (k Z: (matrix) symbol discrete-time index). This is the well-known BICM [9, 10]. The transmitted symbols are organized in an N T N T matrix S[k] with elements s µ [N T k + κ] in row κ and column µ, which is obtained by differential encoding from S[k 1] and the information carrying matrix V [k] S[k] = V [k]s[k 1]. (1) At time N T k + κ, 0 κ N T 1, the symbol s µ [N T k + κ] is transmitted by the µth antenna, 0 µ N T 1. We consider unitary matrices V [k] V with elements v µ [N T k + κ] in row κ and column µ, which provide full diversity. The set V = {V 0,..., V M 1 } forms the signal constellation with M = 2 N T R elements, where

4 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 4 R is the data rate in bits/(channel use). Starting from optimum signaling with respect to channel capacity of multiple-antenna block Rayleigh fading channels [21, 22], convenient signal constellations are classified and optimized in [5, 7]. The combination of multilevel signaling with error correction coding generally requires the joint optimization of coding and modulation. This is e.g. true for the design of space-time trellis codes [23]. However, for single-antenna transmission schemes it has been proven [24] that the optimization problem can be separated by applying properly designed multilevel coding (MLC) with multistage decoding (MSD) [25, 24] along with (standard) binary component codes. Furthermore, if the labeling of signal points is chosen such that decoding at higher levels is almost independent from decisions already made at lower levels, parallel decoding of levels can be applied without loss in performance (cf. [24]). Now, replacing different codes for different levels by one average code, simple BICM [9, 10] is obtained. BICM in combination with phase-shift keying (PSK) and quadrature amplitude modulation (QAM) enables high power and bandwidth efficiency, provided that Gray labeling of signal points can be performed, i.e., the most likely error events cause only a single bit error [10, 24]. Adopting the above reasoning for multiple-antenna differential signaling, we consider BICM combined with space-time constellations V for power- and bandwidthefficient transmission. As most relevant scenarios in practice, we are particularly interested in modulation schemes with N T = 1 and N T = 2 transmit antennas offering (uncoded) transmission rates R = bits/(channel use). A) N T = 1 The single-antenna system serves as reference when assessing the performance of multiple-antenna differential signaling. Here, classical M-ary differential PSK (M DPSK) with Gray labeling is applied. B) N T = 2: Differential transmission based on M-ary orthogonal designs [3, 6], more specifically M = 2 2v, v IN, and { [ 1 x V = 2 y y x ] { x, y e j2πm M 0 } } m < M, (2) yields good performance (cf. [7, Tables 1,3,4]) and allows particularly simple detection [3, 6]. As the entries x, y of matrix V [k] are independently detected at the receiver, a labeling suited to BICM is readily found by separately employing Gray labeling of a MPSK constellation for x and y. For subsequent use we denote constellations based on orthogonal designs which support rates up to R = log 2 (M)/2 bits/(channel use) by

5 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 5 MOD. Also diagonal signal constellations are well suited to differential space-time modulation [5], i.e., V = { { } diag e j2πu 0 m j2πu NT 1 0 } M,..., e m M m < M, (3) where diag{ } denotes a diagonal matrix and u µ, 0 µ N T 1, are optimized coefficients. The group property of V with respect to matrix multiplication, the existence of improved noncoherent reception strategies [8], and a low-complexity fast detection algorithm [26] make diagonal constellations very attractive for implementation. find a suitable labeling, the distance measure ζ m,n as proposed in [5, Eq. (29)] between pairs V m, V n V, 0 m, n M 1, of signal points is evaluated. Unfortunately, the distance properties of constellations with maximum diversity product (minimum of ζ m,n for all m n) and N T 2 (cf. [5, Table 1]) do not allow for a labeling appropriate for BICM. For this reason, we have tried to optimize diagonal constellations with respect to their diversity product and their suitability to BICM. For rate R = 2 bits/(channel use), out of the four reasonable possibilities (cf. [5]) [u 0, u 1 ] = [1, l], l {1, 3, 5, 7}, our investigations have shown that [u 0, u 1 ] = [1, 3] and the addressing of symbols [m = 0,..., m = M 1 = 15] with binary 4-tuples corresponding to decimal numbers [0, 1, 3, 11, 10, 2, 6, 7, 15, 14, 12, 4, 5, 13, 9, 8] is a favorable choice. This particular di agonal constellation is subsequently referred to as 16DI. C) N T > 2: Real orthogonal designs [3] with entries from ASK constellations and Gray labeling might be applied, but a loss in power efficiency due to ASK modulation results. In case of diagonal constellations and N T > 2, a Gray labeling could be found, but strong dependencies among the address bits (i.e., levels when regarding MLC/MSD) remain. Thus, BICM is not well suited. Nondiagonal group or nongroup constellations tabularized in [7, Tables 3,4] have sizes M which are not an integer power of 2. Hence, they cannot directly be combined with BICM. However, even if certain signal elements are deliberately omitted to reduce M to a power of two, again, a proper labeling could not be found. Consequently, when presenting numerical and simulation results in Section 4, we concentrate on the most interesting cases N T = 1 and N T = 2. Nevertheless, the receiver structure and performance analysis provided subsequently are, in general, applicable to arbitrary N T 1. Finally, it has to be mentioned that iterative noncoherent decoding with hard-decision To

6 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 6 bit feedback as discussed in Section 2.3 has the potential to exploit dependencies between address bits mapping one signal point V [k]. In case of error-free feedback, correctly decided bits support the decision on another address bit the better the more the bit-decision variables depend on each other, i.e., a labeling that maximizes the dependencies, which we refer to as anti-gray labeling, seems to be favorable. On the other hand, the quality of decision variables affected by erroneous feedback due to remaining bit errors after decoding degrades significantly. Balancing these opposed effects and focusing on coherent single-antenna 8PSK/16QAM transmission, in [27, 28] a mixed labeling is successfully introduced. For DSTM and noncoherent iterative DF-DM with N > 2, simulation results indicate that the loss after the first decoding iteration could not be compensated in further iterations when using non-gray labelings, i.e., for bit-error ratios usually of interest sufficient convergence of our iterative noncoherent decoding can only be assured if Gray labeling is performed. This especially applies for large constellation sizes (M > 8) and large observation intervals (N > 2). Furthermore, the gain of iterative DF-DM with N = 2 and non-gray labeling was always lower than that for N > 2 and Gray labeling. Therefore, we apply Gray labeling throughout this paper. 2.2 Correlated Flat Ricean Fading Channel Model Assuming sufficiently slow time variance of the transmit channel and transmit and receive filters with square-root Nyquist characteristics, after T -spaced sampling (T is the modulation interval with respect to s µ [N T k + κ]) the links from the transmit antennas to the receive antennas are modeled as discrete-time flat Ricean fading channels with spatially and temporally correlated fading gains. Accordingly, at time N T k + κ the signal at the νth, 0 ν N R 1, receive antenna follows as r ν [N T k + κ] = N T 1 µ=0 h µν [N T k + κ]s µ [N T k + κ] + n ν [N T k + κ], (4) where h µν [N T k + κ] denotes the complex fading gain between the µth transmit and the νth receive antenna, whereas n ν [N T k + κ] refers to the complex additive white Gaussian noise (AWGN) at receive antenna ν. The AWGN processes at different receive antennas are assumed to be spatially and temporally uncorrelated and to have equal variance σ 2 n = E{ n ν[n T k + κ] 2 }, 0 ν N R 1 (E{ }: expectation). As usual, we assume that the fading processes have the same temporal statistical properties, cf.

7 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 7 e.g. [29, 30, 31, 32]. The direct component of the fading process is h[n T k + κ] = E{h µν [N T k + κ]} = e j2πf DT (N T K k+κ) e jφ 1 + K q h, (5) where f D, φ, K, and qh 2 = E{ h µν [N T k + κ] 2 } are the Doppler shift of the direct component, a constant uniformly distributed phase, the Ricean factor (i.e., the ratio between the power of the direct component and the diffuse component), and the power of the fading process, respectively. Assuming the fading to be stationary in time, the space-time covariance function reads (k = NT k + κ) ψ hh [λ, µ 1, µ 2, ν 1, ν 2 ] = E{(h µ1 ν 1 [k + λ] h[k + λ])(h µ2 ν 2 [k ] h[k ]) }, (6) 0 µ 1, µ 2 N T 1, 0 ν 1, ν 2 N R 1. For the numerical results presented in Section 4, we use (cf. e.g. [29, 33]) ψ hh [λ, µ 1, µ 2, ν 1, ν 2 ] = ψhh s [µ 1, µ 2, ν 1, ν 2 ] ψhh t [λ] (7) with the simple spatial covariance function [29] ψhh s [µ 1, µ 2, ν 1, ν 2 ] = ρ { 1 if µ1 = µ 2, ν 1 = ν 2 otherwise, (8) and the temporal covariance function ψ t hh[λ] = K q2 h J 0 (2πB f T λ) (9) according to the Clarke model [34] with maximum fading bandwidth B f (B f f D ). Here, ρ, 0 ρ 1, refers to the space-correlation coefficient and J 0 ( ) is the zeroth order Bessel function of the first kind. The model (7) is convenient to separately discuss the effects of temporal and spatial correlation and (8) is realistic for N T 3 and N R = Iterative Decoding Algorithm At the receiver, bit branch metrics λ b [i] are computed using an increased observation interval of N 2 consecutively received (matrix) symbols R[k] (see Section 2.4, Figure 1). After deinterleaving the λ b [i] constitute the soft input for the standard Viterbi 1 For N T = 2 the simple model (8) immediately arises from the symmetry of the antenna arrangement. For N T = 3, such a symmetry might be achieved if the antennas are arranged at the edges of a equilateral triangle.

8 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 8 decoder [35]. To keep complexity low when N > 2, hard decisions ĉ[i] obtained from previous Viterbi decoding are fed back for metric computation. This procedure of iterative decision-feedback differential demodulation (DF-DM) has been introduced and analyzed by the authors [11] for the case of single-antenna transmission. As no previous decoding decisions ĉ[i] are available, in the first demodulation of a received sequence (first decoding iteration) we resort to conventional differential demodulation (C-DM) based on N = 2 consecutively received symbols. Moreover, only the nearest-neighbor signal point is used as trial signal point, which yields very simple metric expressions, cf. [9, 10, 11]. 2.4 Metric Calculation In [8] noncoherent receivers exploiting an increased observation interval N 2 are introduced for uncoded DSTM over Rayleigh fading channels. More specifically, multiplesymbol differential detection (MSDD) and decision-feedback differential detection (DF- DD) well known from single-antenna systems (cf. e.g. [36] and [37], respectively) are adopted. Particularly simple decision rules are obtained for DF-DD and diagonal signal constellations. Then, analogous to single-antenna transmission (cf. e.g. [37, 38, 39]), the optimum noncoherent metric can be interpreted as coherent metric with linear prediction for channel estimation. Adopting again a similar approach as for single-antenna transmission [14, 12], we apply this prediction-based noncoherent metric computation also to (diagonal and nondiagonal) DSTM over general Ricean fading channels as characterized in Section 2.2. In this way, a receiver structure very well suited to adaptive low-complexity implementation results (cf. [14, 12]). On the other hand, the performance loss compared to optimum metric computation, which requires information about various channel parameters (e.g. fading bandwidth, Ricean factor, Doppler shift) have been shown to be negligible in case of M DPSK single-antenna transmission [14, 12]. To obtain simple-to-implement prediction-based bit metrics we assume (i) mutually uncorrelated and (ii) time-invariant fading coefficients over N T modulation intervals. The degradation caused by violation of these assumptions is discussed in Section 4. (ii) implies h µν [N T k + κ] = h µν [N T k], 0 κ N T 1, and thus (4) can be written as R[k] = S[k]H[k] + N[k], (10) where the N T N R matrices R[k] and N[k] have elements r ν [N T k +κ] and n ν [N T k +κ] in row κ and column ν, respectively, and H[k] is an N T N R matrix with h µν [N T k]

9 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 9 in row µ and column ν, 0 µ, κ N T 1, 0 ν N R 1. Regarding (10) and taking into account the unitary property of V [k] and S[k], the derivation of symbol metrics for DSTM is in perfect analogy to the single-antenna MDPSK case, i.e., N T = 1, and prediction-based DF-DD considered in [14], cf. also [12, 8]. Let ˆV [k ξ], 1 ξ N 1, denote decision-feedback symbols remodulated from hard bit decisions after Viterbi decoding. Then, with the coefficients 2 p η, 1 η N 1, of the (N 1)st order linear predictor [40] for prediction of c µν [N T k] = h µν [N T k] + n ν [N T k] from c µν [N T (k η)], 1 η N 1, the metric for trial symbol Ṽ [k] reads { { N 1 η 1 λṽ [k] = tr Re R H [k]ṽ [k] p η η=1 ξ=1 ˆV [k ξ]r[k η] }}. (11) Here, tr{ }, Re{ }, and ( ) H refer to the trace of a matrix, the real part of a complex number, and Hermitian transpose, respectively. Defining the reference matrix ˆR ref [k 1] = N 1 η=1 η 1 p η ξ=1 ˆV [k ξ]r[k η] (12) and denoting its entries in row κ and column ν by ˆr ref,ν [N T (k 1)+κ], λṽ [k] follows as λṽ [k] = N R 1 ν=0 N T 1 µ=0 N T 1 κ=0 Re {r ν [N T k + κ]ṽ µ [N T k + κ]ˆr ref,ν [N T (k 1) + κ]}. (13) Finally, we obtain the bit metric λ b [i] with b {0, 1} as trial bit value by inserting additional l 1 decision-feedback bits ĉ[ ] in (11) [27, 11]: λ b [i] = λṽ [k] (14) with Ṽ [k] = M(ĉ[i γ],..., ĉ[i 1], b, ĉ[i+1],..., ĉ[i+l 1 γ]), i = k l+γ, 0 γ l 1. We observe that computational complexity is almost independent of the observation length N. Furthermore, the coefficients p η can be adaptively adjusted, e.g. via the recursive least-squares (RLS) algorithm [13, 14], i.e., a priori knowledge on channel statistics is not required. If the channel statistics are known, the p η may be obtained from the Wiener-Hopf equation [40]. For C-DM the determination of nearest neighbor signal points is easily accomplished by simply taking the maximum of (13) over all M/2 symbols Ṽ [k] representing the considered binary symbol. 2 Note that the scalar predictor coefficients are a consequence of the assumption of spatially uncorrelated fading. Otherwise, matrix coefficients would be necessary.

10 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 10 Instead of (14), which includes l 1 decision-feedback bits ĉ[ ], an alternative bit metric λ b [i] could be obtained by averaging the corresponding M/2 symbol metrics exp(2λṽ [k]/σe), 2 where σe 2 is the variance of the linear-prediction error (cf. e.g. [40, 14]). In this way, symbol-decision feedback is exclusively employed as in the case of uncoded transmission. However, computational complexity is significantly increased, because now M instead of l + 1 = log 2 (M) + 1 symbol metrics have to be calculated and because of the computationally complex exponential function. Thus, here λ b [i] is used for implementation. Nevertheless, in Section 4.1, we evaluate cutoff rates for both λ b [i] and λ b [i] assuming genie-aided feedback. A comparison of the corresponding cutoff rates allows to indicate the dependencies between the address bits labeling one ST symbol and hence, to assess the suitability of the adopted labelings for BICM. 3 Performance Analysis In this section, measures to assess the performance of the proposed system are provided. As appropriate information theoretical parameter when employing convolutional codes, the associated channel cutoff rate for DSTM and iterative DF-DM is introduced. Regarding particular codes, analytical expressions for the achievable bit-error ratio (BER) are given. For both cutoff rate and BER ideal (infinite) bit interleaving is assumed, i.e., full time diversity is available. To indicate the advantage of multiple transmit antennas if transmission delay and therefore interleaving depth are strongly limited, we also consider the probability that the instantaneous BER exceeds a desired value (outage probability). Since it is difficult to take into account the effect of erroneous feedback, genie-aided DF-DM, i.e., all binary feedback symbols are assumed to be correct, is used for our analysis as it is customary in literature (cf. e.g. [41, 42, 39, 11, 8]). According to the convergence analysis for single-antenna DF-DM over Rayleigh [11] and Ricean fading channels [12], for BICM with Gray labeling the performance of genie-aided DF-DM is readily approached by realizable, implementable DF-DM for BER s usually of interest. 3.1 Cutoff Rate According to [10], the cutoff rate in bits per symbol assuming BICM with ideal interleaving is given by R 0 = l (1 log 2 (B + 1)), (15)

11 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 11 with the so-called average Bhattacharyya factor { B = 1 l 1 ( ) } λ b[γ] λ b [γ] E exp l σe 2/2, (16) γ=0 where λ b [γ] is taken from (14) with i replaced by γ, i.e., here and subsequently k = 0 is adopted without loss of generality, and b denotes the complement of b. To quantify the effect on the achievable performance due to address-bit dependencies, λ b [γ] in (16) is simply replaced by λ b [γ] using symbol-decision feedback only. 3.2 BER for Ideal Bit Interleaving Starting from the union bound, we now derive analytical expressions which, in general, provide tight approximations for the BER of genie-aided DF-DM for DSTM 3. Since the subsequent analysis is based on (10), the effect of fast fading during N T modulation intervals is not accounted for. For DSTM with diagonal constellations, whose performance is not affected by time-varying fading coefficients during the transmission of one symbol S[k], true upper bounds on BER are obtained. Using the union bound for binary convolutional codes of rate R c bit-error rate is upper bounded by BER 1 k c d=1 = k c /n c, the W (d)f(d), (17) where W (d) is the total input weight of error events of Hamming distance d, and f(d) denotes the pairwise error probability for two code words with Hamming distance d. For (17) to be valid, we assume ideal bit interleaving and symmetric BICM channels (cf. [10]). In [10, Eq. (47)] an expurgated bound f ex (d) for the pairwise error probability f(d) is derived, which is a true upper bound, if for each V m V γ b there is only one relevant error event {V m V n } with V n V γ b. Here, V γ b represents the subset of all M/2 symbols V V whose label has the value b {0, 1} in position γ. For the metric computation (14) with genie-aided bit-decision feedback the trial symbol V m V γ b with V m = M(c[0],..., c[γ 1], b, c[γ + 1],..., c[l 1]) is considered, i.e., there is only one alternative symbol V n V γ b with b replaced by b. Consequently, f ex (d) can be applied to the situation at hand. In accordance with the results in [10] the obtained upper bound/approximation is very tight for BER s which are usually of interest. 3 While this paper was under review, we learned of the independent work by Chindapol and Ritcey [28], which provide a similar BER analysis for coherent transmission with N T = N R = 1.

12 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 12 A convenient form of f ex (d) is [10, Eqs. (48),(49)] f ex (d) = 1 2πj α+j α j 1 lm l 1 1 γ=0 b=0 V m V γ b Φ (m,n) (s) d ds s, (18) where Φ (m,n) (s) is the Laplace transform of the probability density function (pdf) of the difference (m, n) = λ b [γ] λ b[γ] (19) between the metric for the γth bit of the true data matrix symbol V m and the alternative trial symbol V n, and α > 0 lies in the region of convergence of Φ (m,n) (s). A closed-form solution for the integral in (18) may be obtained by the residue method, cf. [43]. Since for the Ricean fading case essential singularities occur, we apply a more efficient approach and compute (18) numerically based on a change of variable and Gauss-Chebyshev quadratures as shown in [44]. To find an expression for Φ (m,n) (s) we start from (11). Assuming genie-aided DF-DM, (19) can be expressed as { ( ) H ( ) (m, n) = tr Re{ H[k] + N [k] I NT (V m S[k 1]) H V n S[k 1] with N 1 η=1 }} p η (H[k η] + N [k η]) (20) = g H F g, (21) g = [x T y T ] T (22) [ ] F = 0NT N R C (23) C H C = I NR 0 NT N R ( ) I NT (V m S[k 1]) H V n S[k 1] (24) x = [x 00 x x NT 10 x x NT 1N R 1] T (25) y = [y 00 y y NT 10 y y NT 1N R 1] T (26) x µν = hµν [N T k] + n ν [N T k + µ] (27) N 1 y µν = η=1 p η (h µν [N T (k η)] + n ν [N T (k η) + µ]) (28) and N [k] = S H [k]n[k] (29)

13 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 13 with entries n ν[n T k+µ] having the same statistical properties as the noise n ν [N T k+µ]. I K, 0 K, and denote the K K identity matrix, the K K all-zero matrix, and the Kronecker product, respectively. Generally, (m, n) depends on the symbol S[k 1] through (24). Conditioned on S[k 1], (m, n) (21) constitutes a Hermitian quadratic form of complex Gaussian distributed random variables with mean vector ḡ = E{g} (30) and covariance matrix Ψ gg = E{(g ḡ)(g ḡ) H }, (31) respectively. ḡ and Ψ gg depend exclusively on h[k], ψ hh [λ, µ 1, µ 2, ν 1, ν 2 ], and σ 2 n defined in Section 2.2. Thus, the Laplace transform Φ (m,n) (s) reads [45] Φ (m,n) (s) = 1 S S[k 1] S exp( sḡ H (F 1 + sψ gg ) 1 ḡ) det{i 2NT N R + sψ gg F } where S denotes the set of S[k] with size S. Generally, V is a subset of S., (32) In some important special cases, averaging over S[k 1] in (32) can be avoided. A) Rayleigh Fading with Independent Diversity Branches Using, I NT (V m S[k 1]) H V n S[k 1] = U H [k 1](I NT Λ m,n )U[k 1] (33) with unitary matrix U[k 1] and diagonal matrix Λ m,n whose main-diagonal elements are the eigenvalues of V H m V n, for spatially uncorrelated Rayleigh fading the channel and noise matrices in (20) can be multiplied by U[k 1] without changing the statistical properties of fading and noise, respectively. Hence, (m, n) does not depend on S[k 1] and Φ (m,n) (s) is solely determined by the eigenvalues of V H m V n (cf. also [5]). B) N T = 1 or Diagonal Constellations with Arbitrary N T Here, U[k 1] = I NT is always true. Hence, S[k 1] does not influence (m, n). C) Orthogonal Designs and Independent Diversity Branches and f D = 0 In case of differential transmission based on orthogonal designs as described in Section 2.1, it can be shown that for spatially uncorrelated Ricean fading with zero Doppler shift f D the eigendecomposition (33) can be applied and that the terms of the sum in (32) are independent of U[k 1] and thus of S[k 1], of course. Finally, we note that for C-DM, (17) with f ex (d) can be used as an approximation for BER. Such an approach yields good results for unitary DSTM (cf. [10] for coherent single-antenna PSK).

14 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation Outage Probability Analysis For transmission over slow fading channels with strict delay constraints and consequently insufficient interleaving to eliminate statistical dependencies between decision variables, the outage probability, i.e., the probability that the instantaneous bit-error rate BER inst exceeds a specified value BER max, might be a more appropriate performance measure than the average BER. To enable analysis we assume transmission over a block fading channel, i.e., all channel weights remain constant during transmission of one data block. Using the joint pdf p(snr inst ) of the instantaneous signal-to-noise ratios SNR inst = [ h00 [N T k] 2 /σn 2,..., h N T 1N R 1[N T k] 2 /σn 2 ]T (34) of all diversity paths, the outage probability follows as P out = Pr{BER inst > BER max } = p(snr inst ) dsnr inst. (35) BER(SNR inst )>BER max To calculate BER(SNR inst ) we apply the upper bound (17) as introduced in the previous section 4. We note that because of the slow fading assumption the bit-error rate is independent of the N 1 predictor coefficients p η as for each data block they all take identical real values, respectively. The computation of (35) reduces to numerical integration of a single one-dimensional integral for the interesting special case of transmit diversity with N T = 2 (N R = 1) and uncorrelated block Rayleigh fading. 4 Results and Discussion Now, the expressions derived in Section 3 are evaluated to quantify and discuss the performance of DSTM with DF-DM. In particular, we focus on the effects of transmit diversity, i.e., the case of one receive antenna (N R = 1) is exclusively considered 5. As performance limits, we show the results for coherent space-time transmission assuming perfect CSI. Additionally, for uncorrelated diversity branches the BER for DSTM with DF-DM and N is included, which is bounded as described in Section 3.2 by using the minimum prediction error variance for N to obtain ḡ and Ψ gg, cf. [14, 46]. For both cutoff rate and simulated BER, fading coefficients h µν [N T k +κ] are generated 4 For the static channel a relatively short interleaver depth is sufficient to justify the use of (17). 5 At least for uncorrelated diversity branches the influence of multiple receive antennas is quite obvious.

15 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 15 as specified in Section 2.2. In particular, although for analysis and metric computation h µν [N T k+κ] is assumed to be constant for 0 κ N T 1, for the presented simulation results it varies with κ if B f > 0 and f D > Cutoff Rate The BICM cutoff rate for spatially uncorrelated Rayleigh fading with B f T = 0.01 over Ēs/N 0 (Ēs: average energy per received symbol, N 0 : single-sided noise power spectral density) is shown in Figure 2. DSTM for N T = 2 with, respectively, an orthogonal design (16OD) and the diagonal constellation (16DI) specified in Section 2.1, is compared to 4DPSK (N T = 1). Observation intervals N = 3 and N = 10 are exemplarily used for DF-DM. As performance bounds, the respective curves for coherent transmission are given. Regarding 16OD (left), considerable gains in power efficiency are achievable by transmit diversity with N T = 2 over N T = 1. We observe that the application of two transmit antennas becomes more advantageous for higher transmission rates, i.e., higher code rates. Clearly, by employing a lower rate code and operating closely at the cutoff rate measure, time diversity is more effectively utilized for detection and additional space diversity is less beneficial. The application of DF-DM is rewarding for both N T = 1 and N T = 2. DF-DM with N = 3 and N = 10 improves power efficiency by 2-3 db over C-DM. The performance of idealized coherent transmission is approached 6. Since the effective fading bandwidth, i.e., the bandwidth B f N T which corresponds to the fading process regarding successive matrix symbols S[k], is higher for N T = 2 than for N T = 1, the remaining gap to the coherent curve is somewhat larger (cf. [8] for a detailed discussion). For N T = 1 and for N T = 2 with 16OD, 4PSK symbols (see signal set (2)) and standard Gray labeling are used. Hence, the two (N T = 1) and four (N T = 2) binary symbols mapped to one symbol V [k], respectively, are transmitted independently of each other, i.e., sole symbol-decision feedback and additional bit-decision feedback yield identical performance. In terms of complexity, bit-feedback is favorable and therefore applied. For 16DI (right), the results for sole symbol-feedback (λ b [i]) and for additional bit-feedback (λ b [i]) are presented, respectively. First, comparing N T = 1 and N T = 2 using symbol-feedback only, we recognize that space diversity is only advantageous for 6 Note that even with N the performance of coherent transmission with perfect CSI cannot be achieved if B f T > 0 (cf. e.g. [14, 8]).

16 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 16 rates R 1.5 bits/(channel use). If time diversity can be effectively exploited through coding, the beneficial effect of space diversity, which improves the quality of bit metrics in general, is limited. The loss of 16DI compared to single-antenna 4DPSK for R 1.5 bits/(channel use) is owing to the specific two-antenna signal constellation. Comparing 16DI with and without additional bit feedback, there is a gain of about 1 db due to bit feedback. Although the labeling has been optimized for BICM (see Section 2.1), dependencies between bits addressing one signal point remain. Hence, when further assessing 16DI the influence of bit feedback on performance has to be kept in mind. Of course, with genie-aided feedback, which is assumed here, anti-gray labeling is optimum. But with realistic feedback convergence of iterative decoding suffers. Nevertheless, the proposed labeling for 16DI ensures sufficient convergence (see results in the next section). Finally, we note that 16OD is superior to 16DI with gains of up to 2 db depending on the target rate. 4.2 BER for Ideal/Long Bit Interleaving Simulated BER s together with the respective upper bounds/approximations (17) for genie-aided DF-DM for N T = 1 and N T = 2 are now presented. If not otherwise stated (punctured) convolutional codes are taken from [47]. For simulated BER s, bitinterleaving with randomly generated interleavers for each transmitted block of 4000 channel uses is applied to provide time diversity. For each simulation point about 1000 bit errors have been measured BER for Spatially Uncorrelated Fading In Figure 3, BER over Ēb/N 0 (Ēb: average receive energy per information bit) for 4DPSK (left), 16OD (middle), and 16DI (right) and rate 3/4 16-states convolutional coding (R = 1.5 bits/(channel use)) is depicted. Transmission over spatially uncorrelated Ricean fading with 10 log 10 (K) = 3 db, B f T = 0.02, f D = 0.01 is assumed. The simulated curves of DF-DM represent BER s obtained after only two iterations and BER s with genie-aided feedback, respectively. As performance limits, the respective bounds for DF-DM with N and coherent space-time transmission with perfect CSI are also included. Regarding simulated curves, convergence of DF-DM with two iterations, i.e., decisionfeedback is performed only once, to genie-aided DF-DM is achieved for BER s usually of interest in all cases. Next, we note that the simulated BER s are very well approx-

17 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 17 imated by the evaluation of the analytical result of Section 3.2. An analytical result for 16DI with C-DM is not given, because the true BER is somewhat underestimated due to the dependencies between address bits (see discussion on cutoff rate). The occasional intersection between bounds 7 and simulated genie-aided curves for 4DPSK and 16DI might be attributed to insufficient interleaving and inaccuracy of simulation. From the comparison of simulation and analytical results for 16OD we conclude that suboptimum metrics neglecting the variation between N T = 2 successive fading coefficients during transmission of V [k] do not lead to a noticeable performance degradation even for relatively large fading bandwidths. The improvement due to DF-DM with enlarged observation window N over C-DM is significant for both N T = 1 and N T = 2. Since the effective fading bandwidth (B f N T ) is increased for N T = 2, the gap between coherent space-time transmission with perfect CSI and DSTM with N is larger than for N T = 1. Nevertheless, DSTM based on orthogonal design yields gains of about 2 db for BER 10 4 over standard 4DPSK. Although 16DI is superior to 4DPSK for low BER s and N > 2, 16OD is clearly the favorable choice. As the slope of the BER curves for N T = 2 is steeper than for N T = 1, gains increase with decreasing target BER (the slopes of 16OD and 16DI become the same for higher signal-to-noise ratios). In Figure 4, the BER s for 8DPSK (left) and 64OD (right), respectively, with rate 2/3 16-states convolutional coding (R = 2 bits/(channel use)) transmitted over spatially uncorrelated Ricean fading with 10 log 10 (K) = 3 db, B f T = 0.01, and zero Doppler shift are compared. Again, DF-DM with only two iterations is considered. Similar statements as above apply. As can be seen, DF-DM bridges the gap between idealized coherent and conventional differential reception. Furthermore, at BER 10 4 gains of about 1.5 db are achieved with two transmit antennas compared to single-antenna transmission. Since the convergence of realistic DF-DM to genie-aided DF-DM is sufficiently fast and the analytical approximations are sufficiently tight, we subsequently use exclusively approximations to highlight further properties of DSTM with DF-DM BER for Spatially Correlated Fading For illustration of the effect of spatial correlation between N T = 2 transmit antennas, Figure 5 provides approximated BER s for OD with 16 (left) and 64 (right) signal points, respectively. In each case, 16-states rate 1/2 convolutional coding and Rayleigh 7 We note that for diagonal constellations true upper bounds are obtained.

18 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 18 fading with B f T = 0.01 is regarded. Exemplarily, the BER s for C-DM, DF-DM with N = 10, and coherent transmission are shown. The correlation coefficient is chosen as ρ = 0, ρ = 0.5, and ρ = 0.9, respectively. We particularly note that the correlation ρ is not known to the receiver and the metric calculation according to Section 2.4 is performed. As references, BER s for N T = 1, i.e., 4DPSK and 8DPSK, respectively, are also plotted. It turns out that even a relatively large spatial correlation with ρ = 0.5 offers high spatial diversity. The difference in power efficiency between ρ = 0 and ρ = 0.5 for N T = 2 is rather small compared to the gain over N T = 1. Hence, DSTM with the proposed receiver structure is fairly robust against spatial correlations between diversity paths. Clearly, with N T = 2 and ρ = 0.9 the single-antenna situation is more closely approached BER for Different Convolutional Codes It is insightful to discuss the influence of code rate R c and code memory on the performance of DSTM over spatially uncorrelated fully interleaved fading channels. For N T = 1 fading diversity can only be utilized through coding, whereas for N T = 2 spatially diversity is effective even for uncoded transmission. In Figure 6, the required Ē b /N 0 to achieve BER = 10 4 for 4DPSK and 16OD, respectively, for Rayleigh fading with B f T = 0.01 is depicted as a function of the code rate (top, 16-states codes, punctured codes taken from [48]) and code memory (bottom, rate 1/2 code taken from [49]), respectively. C-DM, DF-DM with N = 10 and N, and coherent reception are considered. As expected the performance improvement due to spatial diversity becomes more pronounced for less powerful coding, i.e., for higher code rate and lower code memory, respectively. The chosen code rate has a particularly strong influence on the performance gap and the results for the specific 16-states convolutional codes are in good agreement with the cutoff-rate results depicted in Figure 2 regarding a transmission rate R = 2 R c bits/(channel use) close to R Outage Probability Analysis Finally, transmission over spatially uncorrelated block Rayleigh fading is considered, which models the scenario of relatively slow fading with strict delay constraints. In such situations, spatial diversity is expected to be highly beneficial. The outage probabilities corresponding to BER max = 10 4 are calculated as described in Section 3.3 and plotted in Figure 7 for 16-states convolutional codes with rate 1/2 (left) and rate 3/4 (right). Again, 4DPSK, 16OD, and 16DI are compared. Results for C-DM, DF-DM with

19 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 19 N = 10, and coherent reception are shown. As anticipated, transmit diversity yields significant gains in required Ēb/N 0 to ensure a certain value of P out. We particularly observe that the slope of the curves is steeper for N T = 2 than for N T = 1. This is true for both 16OD and 16DI since both constellations offer the same diversity order. The advantage of 16OD over 16DI is obvious from a constant gap between the respective curves. 5 Conclusions The combination of BICM with DSTM for transmission over spatially correlated flat Ricean fading channels and improved noncoherent detection without CSI at the receiver have been introduced. The focus of this paper was on low-complexity processing and simple implementation. Nevertheless, the proposed schemes offer high power efficiency. The evaluation of the associated cutoff rate and analytical bit-error rate expressions for the two transmit-antenna case show in good agreement with simulation results the achievable gains both due to transmit diversity and due to decision-feedback noncoherent detection. Two decoding iterations turn out to be sufficient for realistic DF-DM. Although the receiver has been designed for spatially uncorrelated and over N T channel uses constant fading, it operates fairly robust in spatially correlated and fast fading environments. In summary, coded DSTM with iterative DF-DM is a promising solution to exploit both space and time diversity with very moderate complexity.

20 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 20 References [1] S.M. Alamouti. A Simple Transmitter Diversity Scheme for Wireless Communications. IEEE J. Select. Areas Commun., 16(7): , October [2] J.-C. Guey, M.P. Fitz, M.R. Bell, and W.-Y. Kuo. Signal Design for Transmitter Diversity Wireless Communication Systems Over Rayleigh Fading Channels. IEEE Trans. Com., 47(4): , April [3] V. Tarokh, H. Jafarkhani, and A.R. Calderbank. Space-Time Block Codes from Orthogonal Designs. IEEE Trans. Inf. Theory, 45(5): , July [4] B.L. Hughes. Differential Space-Time Modulation. IEEE Trans. Inf. Theory, 46(7): , November [5] B.M. Hochwald and W. Sweldens. Differential Unitary Space-Time Modulation. IEEE Trans. Com., 48(12): , December [6] V. Tarokh and H. Jafarkhani. A Differential Detection Scheme for Transmit Diversity. IEEE J. Select. Areas Commun., 18: , July [7] A. Shokrollahi, B. Hassibi, B.M. Hochwald, and W. Sweldens. Representation Theory for High-Rate Multiple-Antenna Code Design. IEEE Trans. Inf. Theory, 47(6): , September [8] R. Schober and L.H.-J. Lampe. Noncoherent Receivers for Differential Space-Time Modulation. accepted for publication in IEEE Trans. Com., [9] E. Zehavi. 8-PSK Trellis Codes for a Rayleigh Channel. IEEE Trans. Com., 40(5): , May [10] G. Caire, G. Taricco, and E. Biglieri. Bit-Interleaved Coded Modulation. IEEE Trans. Inf. Theory, 44(3): , May [11] L.H.-J. Lampe and R. Schober. Iterative Decision-Feedback Differential Demodulation of Bit-Interleaved Coded MDPSK for Flat Rayleigh Fading Channels. IEEE Trans. Com., 49(7): , July [12] L.H.-J. Lampe and R. Schober. Low-Complexity Iterative Demodulation for Noncoherent Coded Transmission over Ricean Fading Channels. IEEE Trans. Veh. Tech., 50(6), November [13] R.J. Young and J.H. Lodge. Detection of CPM Signals in Fast Rayleigh Flat-Fading Using Adaptive Channel Estimation. IEEE Trans. Veh. Tech., pages , May [14] R. Schober and W.H. Gerstacker. Decision-Feedback Differential Detection Based on Linear Prediction for MDPSK Signals Transmitted over Ricean Fading Channels. IEEE J. Select. Areas Commun., 18: , March [15] E. Biglieri, G. Taricco, and E. Viterbo. Bit-Interleaved Time-Space Codes for Fading Channels. In Conf. on Inf. Sciences and Systems (CISS), pages WA , Princeton University, March [16] J.J. Boutros, F. Boixadera, and C. Lamy. Bit-Interleaved Coded Modulations for Multiple-Input Multiple-Output Channels. In Proc. IEEE 6th Int. Symp. on Spread- Spectrum Techn. & Appl. (ISSSTA 2000), pages , New Jersey, September 2000.

21 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 21 [17] A.M Tonello. Space-Time Bit-Interleaved Coded Modulation with an Iterative Decoding Strategy. In Proc. of IEEE Vehicular Technology Conference (VTC), pages , Boston, September [18] A. Stefanov and T.M. Duman. Turbo-Coded Modulation for Systems with Transmit and Receive Antenna Diversity over Block Fading Channels: System Model, Decoding Approaches, and Practical Considerations. IEEE J. Select. Areas Commun., 19(5): , May [19] G. Bauch. Concatenation of Space-Time Block Codes and Turbo-TCM. In Proc. IEEE Int. Conf. Commun. (ICC), pages , Vancouver, June [20] Z. Hong and B.L. Hughes. Robust Space-Time Trellis Codes Based on Bit-Interleaved Coded Modulation. In Conf. on Inf. Sciences and Systems (CISS), pages , The Johns Hopkins University, March [21] T.L. Marzetta and B.M. Hochwald. Capacity of a Mobile Multiple-Antenna Communication Link in Rayleigh Flat Fading. IEEE Trans. Inf. Theory, 45: , January [22] B.M. Hochwald and T.L. Marzetta. Unitary Space-Time Modulation for Multiple- Antenna Communications in Rayleigh Flat Fading. IEEE Trans. Inf. Theory, 46: , March [23] V. Tarokh, N. Seshadri, and A.R. Calderbank. Space-Time Codes for High Data Rate Wireless Communication: Performance Criterion and Code Construction. IEEE Trans. Inf. Theory, 44(2): , March [24] U. Wachsmann, R.F.H. Fischer, and J.B. Huber. Multilevel Codes: Theoretical Concepts and Practical Design Rules. IEEE Trans. Inf. Theory, 44(3): , May [25] H. Imai and S. Hirakawa. A New Multilevel Coding Method Using Error Correcting Codes. IEEE Trans. Inf. Theory, 23: , [26] K.L. Clarkson, W. Sweldens, and A. Zheng. Fast Multiple Antenna Differential Decoding. IEEE Trans. Com., 49(2): , February [27] X. Li and J.A. Ritcey. Trellis-Coded Modulation with Bit Interleaving and Iterative Decoding. IEEE J. Select. Areas Commun., 17(4): , April [28] A. Chindapol and J.A. Ritcey. Design, Analysis, and Performance Evaluation for BICM- ID with Square QAM Constellations in Rayleigh Fading Channels. IEEE J. Select. Areas Commun., 19(5): , May [29] J. Ventura-Traveset, G. Caire, E. Biglieri, and G. Taricco. Impact of Diversity Reception on Fading Channels with Coded Modulation-Part I: Coherent Detection. IEEE Trans. Com., 45: , May [30] T.-A. Chen, M.P. Fitz, W.-Y. Kuo, M.D. Zoltowski, and J.H. Grimm. A Space-Time Model for Frequency Nonselective Rayleigh Fading Channels with Application to Space- Time Modems. IEEE Journal on Sel. Areas in Commun. (JSAC): Wireless Communications Series, 18: , July [31] J.D. Parsons and A.M.D. Turkmani. Characterisation of mobile radio signals: model description. IEE Proceedings-I, 138: , December 1991.

22 Lampe, Schober: Bit-Interleaved Coded Differential Space-Time Modulation 22 [32] J.D. Parsons and A.M.D. Turkmani. Characterisation of mobile radio signals: base station crosscorrelation. IEE Proceedings-I, 138: , December [33] M. Stojanovic and Z. Zvonar. Differential Coherent Diversity Combining Techniques for DPSK Over Fast Rayleigh Fading Channels. IEEE Trans. Veh. Tech., 49: , September [34] W.C. Jakes, Jr. Microwave Moble Communications. John Wiley & Sons, Inc., New York, [35] A.J. Viterbi. Error Bounds for Convolutional Codes and an Asymptotically Optimum Decoding Algorithm. IEEE Trans. Com., 13(2): , April [36] D. Divsalar and M.K. Simon. Multiple-Symbol Differential Detection of MPSK. IEEE Trans. Com., 38(3): , March [37] P.Y. Kam and C.H. Teh. Reception of PSK Signals Over Fading Channels Via Quadrature Amplitude Estimation. IEEE Trans. Com., 31: , August [38] J. H. Lodge and M. L. Moher. Maximum Likelihood Sequence Estimation of CPM Signals Transmitted Over Rayleigh Flat-Fading Channel. IEEE Trans. Com., 38: , June [39] R. Schober, W.H. Gerstacker, and J.B. Huber. Decision-Feedback Differential Detection of MDPSK for Flat Rayleigh Fading Channels. IEEE Trans. Com., 47(7): , July [40] S. Haykin. Adaptive Filter Theory. Prentice-Hall, Englewood Cliffs, NJ, third edition, [41] F. Edbauer. Bit Error Rate of Binary and Quarternary DPSK signals with Multiple Differential Feedback Detection. IEEE Trans. Com., 40: , March [42] F. Adachi and M. Sawahashi. Decision Feedback Multiple Symbol Differential Detection of M ary DPSK. Electronics Letters, 29(15): , July [43] J.K. Cavers and P. Ho. Analysis of the Error Performance of Trellis-Coded Modulations in Rayleigh-Fading Channels. IEEE Trans. Com., 40:74 83, January [44] E. Biglieri, G. Caire, G. Taricco, and J. Ventura-Traveset. Computing Error Probabilities over Fading Channels: A Unified Approach. Europ. Trans. Telecom. (ETT), 9:15 25, Jan/Feb [45] M. Schwartz, W. Bennett, and S. Stein. Communication Systems and Techniques. McGraw-Hill, New York, [46] R. Schober and L.H.-J. Lampe. Differential Modulation Diversity. accepted for publication in IEEE Trans. Veh. Tech., [47] J.B. Huber. Trelliscodierung. Springer-Verlag, Berlin, [48] D. Haccoun and G. Bégin. High-Rate Punctures Convolutional Codes for Viterbi and Sequential Decoding. IEEE Trans. Com., 37(11): , November [49] M. Cedervall and R. Johannesson. A Fast Algorithm for Computing Distance Spectrum of Convolutional Codes. IEEE Trans. Inf. Theory, 35(6): , November 1989.