Space–time coding wireless communications

space time coding theory and practice solution manual and space-time block coding for wireless communications larsson
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4 Space–time coding for wireless communications: principles and applications 4.1 Introduction The essential feature of wireless transmission is the randomness of the communication channel which leads to random fluctuations in the received signal commonly known as fading. This randomness can be exploited to enhance performance through diversity. We broadly define diversity as the method of conveying information through multiple independent instantiations of these random fades. There are several forms of diversity; our focus in this chapter will be on spatial diversity through multiple independent transmit/receive antennas. Information theory has been used to show that multiple antennas have the potential to dramatically increase achievable bit rates 76, thus converting wireless channels from narrow to wide data pipes. The earliest form of spatial transmit diversity is the delay diversity scheme proposed in 81,84 where a signal is transmitted from one antenna, then delayed one time slot, and transmitted from the other antenna. Signal processing is used at the receiver to decode the superposition of the original and time-delayed signals. By viewing multiple- antenna diversity as independent information streams, more sophisticated transmission (coding) schemes can be designed to get closer to theoretical performance limits. Using this approach, we focus on space–time coding (STC) schemes defined by Tarokh et al. 74 and Alamouti 5, which introduce temporal and spatial correlation into the signals transmitted from different antennas without increasing the total transmitted power or the transmission bandwidth. There is, in fact, a diversity gain that results from multiple paths between the base-station and the user terminal, and a coding gain that results from how symbols are correlated across transmit antennas. Significant performance improvements arepossiblewithonlytwoantennasatthebase-stationandoneortwoantennasattheuser terminal, and with simple receiver structures. The second antenna at the user terminal can be used to further increase system capacity through interference suppression. In only a few years, space–time codes have progressed from invention to adoption in the major wireless standards. For wideband code-division multiple access (WCDMA) where short spreading sequences are used, transmit diversity provided by space–time codes represents the difference between data rates of 100 and 384kb/s. Our emphasis is on solutions that include channel estimation, joint decoding and equalization, and where the complexity of signal processing is practical. The new world of multiple transmit and 1404.2 Background 141 receive antennas requires significant modification of techniques developed for single- transmitsingle-receivecommunication.Sincereceivercostandcomplexityisanimportant consideration, our treatment of innovation in signal processing is grounded in systems with one, two or four transmit antennas and one or two receive antennas. For example, the interference cancellation techniques presented in Section 4 enable transmission of 1 Mb/s over a 200 kHz GSM/EDGE channel using four transmit and two receive antennas.Hence,ourlimitationonnumbersofantennasdoesnotsignificantlydampenuser expectations. Initial STC research efforts focused on narrowband flat-fading channels 5,62,74. Successful implementation of STC over multi-user broadband frequency-selective channels requires the development of novel, practical, and high-performance signal processingalgorithmsforchannelestimation,jointequalization/decoding,andinterference suppression. This task is quite challenging due to the long delay spread of broadband channels which increases the number of channel parameters to be estimated and the number of trellis states in joint equalization/decoding, especially with multiple transmit antennas.This,inturn,placessignificantadditionalcomputationalandpowerconsumption loads on user terminals. On the other hand, development and implementation of such advanced algorithms for broadband wireless channels promises even more significant performance gains than those reported for narrowband channels 5,62,74 due to availability of multi-path (in addition to spatial) diversity gains that can be realized. By virtue of their design, space–time-coded signals enjoy rich algebraic structure that can (and should) be exploited to develop near-optimum reduced-complexity modem signal processing algorithms. The organization of this chapter is as follows. We start in Section 4.2 with background material where we set up the broadband wireless channel model assumed, followed by a discussion of transmit diversity and the concept of diversity order. Section 4.3 describes STC design criteria and discusses representative examples with both the trellis and block structure.Wealsogivesomerecentdevelopmentsinspace–timecodes.Section4.4shows through concrete examples from signal processing, coding theory, and networking, how the STC algebraic structure can be exploited to enhance system performance and reduce implementation complexity. The chapter concludes in Section 4.5 with a summary and a discussion of several future challenges. 4.2 Background 4.2.1 Broadband wireless channel model A typical outdoor wireless propagation environment is represented in Figure 4.1 where themobilewirelessterminaliscommunicatingwithawirelessaccesspoint(base-station). The signal transmitted from the mobile may reach the access point directly (line-of-sight) orthroughmultiplereflectionsonlocalscatterers(buildings,mountains,etc.).Asaresult,142 Space–time coding for wireless communications Remote Local to base Local to mobile Base station Local to base Remote Figure 4.1. Radio propagation environment. the received signal is affected by multiple random attenuations and delays. Moreover, the mobility of either the nodes or the scattering environment may cause these random fluctuations to vary with time. Furthermore, a shared wireless environment may cause undesirable interference to the transmitted signal. This combination of factors makes wireless a challenging communication environment. For a transmitted signal st, the continuous-time received signal y t can be expressed as c  y t= h tst−d+zt (4.1) c c 1 where h t is the response of the time-varying channel if an impulse is sent at time c t−, and zt is the additive Gaussian noise. To collect discrete-time sufficient statistics we need to sample (4.1) faster than the Nyquist rate. That is, we sample (4.1) at a rate larger than 2W +W , where W is the input bandwidth and W is the bandwidth of the I s I s channel time-variation. In this chapter, we assume that this criterion is met and therefore we focus on the following discrete-time model:   yk=y kT = hklxk−l+zk (4.2) c s l=0 where yk, xk, and zk are the output, input, and noise samples at sampling instant k, respectively, and hkl represents the sampled time-varying channel impulse response 1 Including the effects of transmit/receive filters.4.2 Background 143 (CIR) of finite memory . Any loss in modeling the channel as having a finite-duration impulse response can be made small by appropriately selecting . Three key characteristics of broadband mobile wireless channels are time-selectivity, frequency-selectivity, and space-selectivity. Time-selectivity arises from mobility, frequency-selectivity arises from broadband transmission, and space-selectivity arises from the spatial interference patterns of the radio waves. The corresponding key parameters in the characterization of mobile broadband wireless channels are coherence time, coherence bandwidth, and coherence distance. The coherence time is the time duration over which each CIR tap can be assumed constant. It is approximately equal 2 to the inverse of the Doppler frequency. The channel is said to be time-selective if the symbol period is longer than the channel coherence time. The coherence bandwidth is the frequency duration over which the channel frequency response can be assumed 3 flat. It is approximately equal to the inverse of the channel delay spread. The channel is said to be frequency-selective if the symbol period is smaller than the delay spread of the channel. Likewise, the coherence distance is the maximum spatial separation over which the channel response can be assumed constant. This can be related to the behavior of arrival directions of the reflected radio waves and is characterized by the angular spread of the multiple paths 50,65. The channel is said to be space-selective between two antennas if their separation is larger than the coherence distance. The channel memory causes interference among successive transmitted symbols that results in significant performance degradation unless corrective measures (known as equalization) are implemented. In this chapter, we shall use the terms frequency-selective channel, broadband channel, and intersymbol interference (ISI) channel interchangeably. The introduction of M transmit and M receive antennas leads to the following t r generalization of the basic channel model:   yk= Hklxk−l+zk (4.3) l=0 where the M ×M complex matrix Hkl represents the lth tap of the channel matrix r t M M t r response with x ∈ as the input and y ∈ as the output. The input vector may have independent entries to achieve high throughput (e.g. through spatial multiplexing) or correlated entries through coding or filtering to achieve high reliability (better distance properties,higherdiversity,spectralshaping,ordesirablespatialprofile).Throughoutthis chapter, the input is assumed to be zero mean and to satisfy an average power constraint, 2 M 4 r i.e.Exk ≤P.Thevectorz∈ modelstheeffectsofnoiseandinterference. Itis assumed to be independent of the input and is modeled as a complex additive circularly- symmetric Gaussian vector with z ∼0R , i.e. a complex Gaussian vector with zz 2 The Doppler frequency is a measure of the frequency spread experienced by a pure sinusoid transmitted over the channel. It is equal to the ratio of the mobile speed to the carrier wavelength. 3 The channel delay spread is a measure of the time spread experienced by a pure impulse transmitted over the channel. 4 Including co-channel interference, adjacent channel interference, and multi-user interference.144 Space–time coding for wireless communications mean 0 and covariance R . Finally, we modify the basic channel model to accommodate zz a block or frame of N consecutive symbols. Now, (4.3) can be expressed in matrix notation as follows: y=Hx+z (4.4) N M M N+ N M ×M N+ r t r t where yz∈ , x ∈ , and H ∈ . In each input block, we insert a guard sequence of length equal to the channel memory  to eliminate inter-block interference (IBI). In practice, the most common choices for the guard sequence are the all-zeros sequence (also known as zero stuffing) and the cyclic prefix (CP). When the channel is known at the transmitter, it is possible to increase throughput by optimizing the choice of the guard sequence. The channel model in (4.4) includes several popular special cases. First, the quasi- static channel model follows by assuming the channel to be time-invariant within the transmission block. In this case, using the cyclic prefix makes the channel matrix H block-circulant, hence diagonalizable using the fast Fourier transform (FFT). Second, the flat-fading channel model follows by setting  = 0 which renders the channel matrix H a block diagonal matrix. Third, the channel model for single-antenna transmission, reception, or both follows directly by setting M , M , or both equal to 1, respectively. t r 4.2.2 Transmit diversity Transmit diversity is more challenging to provision and realize than receive diversity because it involves the design of multiple correlated signals from a single information signal without utilizing CSI (typically not available accurately at the transmitter). Furthermore, transmit diversity must be coupled with effective receiver signal processing techniques that can extract the desired information signal from the distorted and noisy received signal. Transmit diversity is more practical than receive diversity for enhancing the downlink (which is the bottleneck in broadband asymmetric applications such as Internetbrowsinganddownloading)topreservethesmallsizeandlowpowerconsumption features of the user terminal. A common attribute of transmit and receive diversity is that both experience “diminishing returns” (i.e. diminishing SNR gains at a given error probability) as the number of antennas increases 50. This makes them effective, from a performance–complexitytrade-offpointofview,forsmallnumbersofantennas(typically lessthanfour).Thisisincontrastwithspatialmultiplexinggainswheretheratecontinues to increase linearly with the number of antennas (assumed equal at both ends). There are two main classes of multiple-antenna transmitter techniques: closed-loop and open-loop. The former uses a feedback channel to send CSI acquired at the receiver back to the transmitter to be used in signal design while the latter does not require CSI. Assuming availability of ideal (i.e. error-free and instantaneous) CSI at the transmitter, closed-loop techniques have an SNR advantage of 10log M  dB over open-loop 10 t techniques due to the “array gain” factor 5. However, several practical factors degrade the performance of closed-loop techniques including channel estimation errors at the receiver, errors in the feedback link (due to noise, interference, and quantization effects),4.2 Background 145 and feedback delay which causes a mismatch between the available and the actual CSI. All of these factors combined with the extra bandwidth and system complexity resources needed for the feedback link make open-loop techniques more attractive as a robust means for improving downlink performance for high-mobility applications while closed- loop techniques (such as beamforming) become attractive under low-mobility conditions. Our focus in this chapter will be exclusively on open-loop spatial transmit diversity 5 techniques due to their applicability to both scenarios . Beamforming techniques are discussed extensively in several tutorial papers such as 38,39. The simplest example of open-loop spatial transmit diversity techniques is delay diversity 81,84, where the signal transmitted at sampling instant k from the ith antenna is xk = xk−l for 2 ≤ i ≤ M and x k = xk, where l denotes the time delay i i t 1 i (in symbol periods) on the ith transmit antenna. Assuming a single receive antenna, the 6 D-transform of the received signal is given by   M t  l i yD=xD h D+ D hD +zD (4.5) 1 i i=2 It is clear from (4.5) that delay diversity transforms spatial diversity into multi-path diversity that can be realized through equalization 67. For flat-fading channels, we can set l =i−1 and achieve full (i.e. order-M ) spatial diversity using a maximum- i t b M −1 b t likelihood (ML) equalizer with 2  states, where 2 is the input alphabet size. However, for frequency-selective channels, a delay of at least l = i−1+1 is i needed to ensure that coefficients from the various spatial FIR channels do not interfere with each other causing a diversity loss. This, in turn, increases equalizer complexity to b M −1+1 t 2  states, which is prohibitive even for moderate b, M , and . In Section 4.3, t we describe another family of open-loop spatial transmit diversity techniques known as space–time block codes that achieve full spatial diversity with practical complexity even for frequency-selective channels with a long delay spread. 4.2.3 Diversity order Error probability is particularly important as a performance criterion when we are coding over a small number of blocks (low-delay) where the Shannon capacity is zero 63 and, therefore, we need to design for low error probability. By characterizing the error probability, we can also formulate design criteria for space–time codes in Section 4.3. Sinceweareallowedtotransmitacodedsequence,weareinterestedintheprobability that an erroneous codeword e is mistaken for the transmitted codeword x. This is called the pairwise error probability (PEP) and is then used to bound the error probability. This is analyzed under the condition that the receiver has perfect channel state information. However, a similar analysis can be performed when the receiver does not know the channel state information but has statistical knowledge of the channel. 5 It is also possible to combine closed-loop and open-loop techniques as shown recently in Soni et al. (2002).  N−1 6 N−1 k The D-transform of a discrete-time sequence xk is defined as xD= xkD . It is derived from k=0 k=0 −1 the Z-transform by replacing the unit delay Z by D.146 Space–time coding for wireless communications For simplicity, we shall first present the result for a flat-fading Rayleigh channel (where  =0). In the case when the receiver has perfect channel state information, we can bound the PEP between x and e (denoted by Px→e) as follows 41,74:  M r 1 Px→e≤  (4.6)  M E t s 1+  n n=1 4N 0 ∗ where are the eigenvalues of the matrix Axe=B xeBxe and n ⎛ ⎞ x 1−e 1  x 0−e 0 1 1 M M t t ⎜ ⎟ Bxe= (4.7) ⎝ ⎠ x N −1−e N −1 x N −1−e N −1 1 1 M M t t If q denotes the rank of Axe (i.e. the number of non-zero eigenvalues) then we can rewrite (4.6) as   −M   r −qM q r  E s Px→e≤ (4.8) n 4N 0 n=1 Thus, we define the notion of diversity order as follows. ¯ Definition4.1 A scheme which has an average error probability P SNR as a function e of SNR that behaves as ¯ logP SNR e lim =−d (4.9) SNR→ logSNR is said to have a diversity order of d. Inwords,aschemewithdiversityorderdhasanerrorprobabilityathighSNRbehaving −d ¯ as P SNR≈SNR . Given this definition, we can see that the diversity order in (4.8) e  q 1/q is at most qM . Moreover, in (4.8) we obtain an additional coding gain of   . r n n=1 Note that in order to obtain the average error probability, one can calculate a naive union bound using the pairwise error probability given in (4.8). However, this bound may not be tight and a more careful upper bound for the error probability can be derived 68,89. However, if we ensure that every pair of codewords satisfies the diversity order in (4.8), then clearly the average error probability satisfies it as well. This is true when the transmission rate is held constant with respect to SNR. Therefore, code design for the diversity order through the pairwise error probability is a sufficient condition, although more detailed criteria can be derived based on a more accurate expression for the average error probability.4.2 Background 147 The error probability analysis can easily be extended to the case where we have quasi-static ISI channels with channel taps modeled as i.i.d. zero mean complex Gaussian random variables (see, for example 90 and references therein). In this case, the PEP can be written as  M r 1 Px→e≤  (4.10)  M  E t s ˜ 1+  n=1 n 4N 0 ∗ ˜ ˜ ˜ ˜ where the eigenvalues are those of Axe=B xeBxe, n ⎛ ⎞ T T x˜ 0−e˜ 0 ⎜ ⎟ ˜ Bxe=  (4.11) ⎝ ⎠ T T x˜ N −1−e˜ N −1 and T T T x˜k=x kx k− (4.12) ˜ SinceAxeisasquarematrixofsizeM ,clearlythemaximaldiversityorderachievable t for quasi-static ISI channels is M M . r t Finally, if we have a time-varying ISI channel, we can generalize (4.10) to  1 Px→e≤  (4.13) E s I + FR ⊗I  M NM  h M M  4N r t r t 0 where ⊗ denotes a Kronecker product, R is the N ×N correlation matrix of the channel h tap process, and F=diag C0CN −1 with   T T Ck= x˜ k−e˜ k ⊗I (4.14) M r Again,itisclearthatthemaximaldiversityattainableisM M N,butforagivenchannel r t tap process, N is replaced by the number of dominant eigenvalues N of the fading dom correlation matrix. This parameter is related to the Doppler spread of the channel and the block duration. 4.2.4 Rate–diversity trade-off A natural question that arises is how many codewords can we have which allow us to attainacertaindiversityorder.ForaflatRayleighfadingchannel,thishasbeenexamined 7 in 58,74 and the following result was obtained. 7 A constellation size refers to the alphabet size of each transmitted symbol. For example, a quadrature phase-shift keying (QPSK) modulated transmission has a constellation size of 4.148 Space–time coding for wireless communications b Theorem 4.2 If we use a constellation of size 2 and the diversity order of the system is qM , then the rate R that can be achieved is bounded as r R≤M −q+1log  (4.15) t 2 in bits per transmission. One consequence of this result is that for maximum (M M ) diversity order we can t r transmit at most b bits/s/Hz. An alternate viewpoint in terms of the rate–diversity trade-off has been explored in 89 from a Shannon-theoretic point of view. Here the authors are interested in the multiplexing rate of a transmission scheme. Definition 4.3 A coding scheme which has a transmission rate of RSNR as a function of SNR is said to have a multiplexing rate r if RSNR lim =r (4.16) SNR→ logSNR Therefore, the system has a rate of rlogSNR at high SNR. One way to contrast it with the statement in Theorem 4.2, is that the constellation size is also allowed to become larger with SNR. However, note that the naive union bound of the pairwise error probability (4.6) has to be used with care if the constellation size is also increasing with SNR. There is a trade-off between the achievable diversity and the multiplexing opt rate, and d r is defined as the supremum of the diversity gain achievable by any scheme with multiplexing rate r. The main result in 89 is given as the following theorem. Theorem 4.4 ForNM +M −1, and K =minMM , the optimal trade-off curve t r t r opt opt d r is given by the piecewise linear function connecting points in kd kk = 0K, where opt d k=M −kM −k (4.17) r t The interesting interpretation of this result is that one can get large rates which grow with SNR if we reduce the diversity order from the maximum achievable. This diversity–multiplexing trade-off implies that a high multiplexing rate comes at the price of decreased diversity gain and is a manifestation of a corresponding trade-off between the error probability and the rate. Adifferentquestionwasproposedin22,23,whereitwasaskedwhetherthereexistsa strategy that combines high-rate communications with high reliability (diversity). Clearly the overall code will still be governed by the rate–diversity trade-off, but the idea is to ensurethereliability(diversity)ofatleastpartofthetotalinformation.Thisallowsaform of communication where the high-rate code opportunistically takes advantage of good channel realizations whereas the embedded high-diversity code ensures that at least part of the information is received reliably. In this case, the interest was not in a single pair4.3 Space–time coding principles 149 of multiplexing rate and diversity order rd, but in a tuple r d r d  where rate a a b b r and diversity order d were ensured for part of the information with the rate–diversity a a pair r d  guaranteed for the other part. A class of space–time codes with such desired b b characteristics will be discussed in Section 4.3.5. From an information-theoretic point of view Diggavi and Tse 26,27 focused on the case when there is one degree of freedom (i.e. minMM  = 1). In that case if t r we consider d ≥ d without loss of generality, the following result was established a b in 26,27. Theorem 4.5 When minMM =1, then the diversity–multiplexing trade-off curve is t r successively refinable, i.e. for any multiplexing rates r and r such that r +r ≤1, the a b a b diversity orders d ≥d , a b opt opt d =d r  d =d r +r  (4.18) a a b a b opt are achievable, where d r is the optimal diversity order given in Theorem 4.4. Since the overall code has to still be governed by the rate–diversity trade-off given in opt Theorem 4.4, it is clear that the trivial outer bound to the problem is that d ≤d r  a a opt and d ≤d r +r . Hence Theorem 4.5 shows that the best possible performance can b a b be achieved. This means that for minMM  = 1, we can design ideal opportunistic t r codes. This new direction of enquiry is being currently explored (see 25,27). 4.3 Space–time coding principles Space–time coding has received considerable attention in academic and industrial circles 3,4 due to its many advantages. First, it improves the downlink performance without the need for multiple receive antennas at the terminals. For example, for wideband code- division multiple access (WCDMA), STC techniques were shown in 64 to result in substantial capacity gains due to the resulting “smoother” fading which, in turn, makes power control more effective and reduces the transmitted power. Second, it can be elegantly combined with channel coding, as shown in 74, realizing a coding gain in addition to the spatial diversity gain. Third, it does not require channel state information (CSI) at the transmitter, i.e. it operates in open-loop mode, thus eliminating the need for an expensive and, in the case of rapid channel fading, unreliable reverse link. Finally, they have been shown to be robust against non-ideal operating conditions such as antenna correlation, channel estimation errors, and Doppler effects 62,73. There has been extensive work on the design of space–time codes since their introduction in 74. The combination of the turbo principle 8,9 with space–time codes has been explored (see, for example, 7 and 55 among several other references). In addition, the150 Space–time coding for wireless communications application of linear low-density parity check (LDPC) codes 36 to space–time coding hasbeenexplored(see,forexample57andreferencestherein).Wefocusourdiscussion on the basic principles of space–time codes and next describe the two main flavors: trellis and block codes. 4.3.1 Space–time code design criteria In order to design practical codes that achieve a performance target we need to glean insights from the analysis to arrive at design criteria. For example, in the flat-fading case of (4.8) we can state the following rank and determinant design criteria 41,74. • Rank criterion. In order to achieve maximum diversity M M , the matrix Bxe t r from (4.7) has to be full rank for any codewordsxe. If the minimum rank ofBxe over all pairs of distinct codewords is q, then a diversity order of qM is achieved. r  q 1/q • Determinant criterion.Foragivendiversityordertargetof q,maximize   n n=1 over all pairs of distinct codewords. Asimilarsetofdesigncriteriacanbestatedforthequasi-staticISIfadingchannelusing the PEP given in (4.10) and the corresponding error matrix given in (4.11). Therefore, if we need to construct codes satisfying these design criteria, we can guarantee performance in terms of diversity order. The main problem in practice is to construct such codes that do not have a large decoding complexity. This sets up a familiar tension on the design in terms of satisfying the performance requirements and having low-complexity decoding. If coherent detection is difficult or too costly, one can employ non-coherent detection forthemultiple-antennachannel46,88.Thoughitisdemonstratedin88thatatraining- based technique achieves the same capacity–SNR slope as the optimal, there might be a situation where inexpensive receivers are needed because channel estimation cannot be accommodated. In such a case, differential techniques which satisfy the diversity order might be desirable. There has been significant work on differential transmission with non-coherent detection (see, for example 45,47 and references therein) and this is a topic we discuss briefly in Section 4.4.1. The rank and determinant design criteria given above are suitable for transmission when we have a fixed input alphabet. As mentioned in Section 4.2.4, the rate–diversity trade-off can also be explored in the context of the multiplexing rate (see Definition 4.3). Therefore, it is natural to ask for the code-design criteria in this context. For the diversity–multiplexing guarantees, it is not clear that the rank and determinant criterion is the correct one to use. In fact, in 29, it is shown that when designing codes with the multiplexing rate in mind, the determinant criterion is not necessary for specific fading distributions. However, it has been shown that the determinant criterion again arises as a sufficient condition when designing codes for the diversity– multiplexing rate trade-off for specific constructions (see 32,86 and references therein). For these constructions, it is shown that the determinant of the codeword difference matrix plays a crucial role in the diversity–multiplexing optimality of the codes.4.3 Space–time coding principles 151 Another multiplexing rate context in which the codeword difference matrix plays a crucial role in the space–time code design is in the design of approximately universal codes 75. Traditionally space–time codes are designed for a particular distribution of the channel. Universal codes are designed to give an error probability which decays exponentially in SNR for all channels that are not in outage. Therefore, this provides a robust design rule which gives performance guarantees over the worst-case channel, rather than averaging over the channel statistics. For the multiple-transmit single-receive (MISO) channel, the code design is related to maximizing the smallest singular value of thecodeworddifferencematrix.Thiscorrespondstoaworst-casechannelthatalignsitself with the weakest direction of the codeword difference matrix. This is in contrast to the average case, where we are interested in maximizing the product of the singular values (i.e. the determinant). In fact for MIMO channels, in certain cases, the maximizing the determinant of the codeword difference matrix again arises as the universal code design criterion 75. 4.3.2 Space–time trellis codes (STTC) Thespace–timetrellisencodermapstheinformationbitstreamintoM streamsofsymbols t b 8 (each belonging in a size-2 signal constellation) that are transmitted simultaneously. STTC design criteria are based on minimizing the PEP bound in Section 4.2.3. As an example, we consider the eight-state 8-PSK STTC for two transmit antennas introduced in 74; the trellis description is given in Figure 4.2, where the edge label c c 1 2 means that symbol c is transmitted from the first antenna and symbol c from the second 1 2 2 3 1 4 0 7 5 6 00 01 02 03 04 05 06 07 0 1 50 51 52 53 54 55 56 57 20 21 22 23 24 25 26 27 2 3 70 71 72 73 74 75 76 77 40 41 42 43 44 45 46 47 4 5 10 11 12 13 14 15 16 17 60 61 62 63 64 65 66 67 6 7 30 31 32 33 34 35 36 37 Figure 4.2. Eight-state 8-PSK space–time trellis code with two transmit antennas and a spectral efficiency of 3 bits/sec/Hz. 8 The total transmitted power is divided equally among the M transmit antennas. t152 Space–time coding for wireless communications Constellation mapper To T × 1 Information x(k) source One symbol To T × 2 delay x(k – 1) Pk p = 1 if x(k – 1) even k p = –1 if x(k – 1) odd k Figure 4.3. Equivalent encoder model for an eight-state 8-PSK space–time trellis code with two transmit antennas. antenna. The different symbol pairs in a given row label the transitions out of a given state, in order, from top to bottom. An equivalent and convenient (for reasons to become clearshortly)implementationoftheeight-state8-PSKSTTCencoderisdepictedinFigure 4.3. This equivalent implementation clearly shows that the eight-state 8-PSK STTC is identical to classical delay diversity transmission 67 except that the delayed symbol from the second antenna is multiplied by −1 if it is an odd symbol, i.e. ∈ 1357 . This slight modification results in an additional coding gain over a flat-fading channel. We emphasize that this STTC does not achieve the maximum possible diversity gains (spatial and multi-path) on frequency-selective channels; however, its performance is 9 near optimum for practical ranges of SNR on wireless links 34. Furthermore, when implementing the eight-state 8-PSK STTC described above on a frequency-selective channel, its structure can be exploited to reduce the complexity of joint equalization and decoding. This is achieved by embedding the space–time encoder in Figure 4.3 in the two channels h D and h D, resulting in an equivalent single-input single-output (SISO) 1 2 data-dependent CIR with memory +1 whose D-transform is given by STTC h kD=h D+p Dh D (4.19) 1 k 2 eqv where p =±1 is data-dependent. Therefore, trellis-based joint space–time equalization k +1 and decoding with 8 states can be performed on this equivalent channel. Without 2 exploiting the STTC structure, trellis equalization requires 8 states and STTC decoding requires eight states. ThediscussioninthissectionjustillustratesoneSTTCexample.Severalotherfull-rate full-diversity STTCs for different signal constellations and different numbers of antennas were presented in 74. 9 For examples of STTC designs for frequency-selective channels see, for example 54.4.3 Space–time coding principles 153 4.3.3 Space–time block codes (STBC) The decoding complexity of STTC (measured by the number of trellis states at the decoder) increases exponentially as a function of the diversity level and the transmission rate 74. In addressing the issue of decoding complexity, Alamouti 5 discovered an ingeniousspace–timeblockcodingschemefortransmissionwithtwoantennas.According tothisscheme(seealsoAppendix4.1),inputsymbolsaregroupedinpairswheresymbols x and x are transmitted at time k from the first and second antennas, respectively. k k+1 ∗ Then, at time k+1, symbol −x is transmitted from the first antenna and symbol k+1 ∗ x is transmitted from the second antenna, where ∗ denotes the complex conjugate k transpose (cf. Figure 4.4). This imposes an orthogonal spatio-temporal structure on the transmitted symbols. Alamouti’s STBC has been adopted in several wireless standards such as WCDMA 77 and CDMA2000 78 due to the following attractive features. First, it achieves full-diversity at full transmission rate for any (real or complex) signal constellation. Second, it does not require CSI at the transmitter (i.e. open-loop). Third, maximum-likelihood decoding involves only linear processing at the receiver (due to the orthogonal code structure). TheAlamoutiSTBChasbeenextendedtothecaseofmorethantwotransmitantennas 72 using the theory of orthogonal designs. There it was shown that, in general, full- rate orthogonal designs exist for all real constellations for two, four, and eight transmit antennas only, while for all complex constellations they exist only for two transmit antennas (the Alamouti scheme). However, for particular constellations, it might be possible to construct full-rate orthogonal designs for larger numbers of transmit antennas. Moreover, if a rate loss is acceptable, then orthogonal designs exist for an arbitrary number of transmit antennas 72. The advantage of orthogonal design is the simplicity of the decoder. However, using a sphere decoder, space–time codes that are not orthogonal, but are linear over the complex field can also be decoded efficiently. A class of space–time codes known as linear dispersion codes (LDC) was introduced in 43 where the orthogonality constraint is relaxed to achieve a higher rate while still enjoying (expected) polynomial decoding complexity for a wide SNR range by using the sphere decoder. This comes at the expense Signal 2 Constellation ST block code mapper Information c –c 1 2 c c source 1 2 c c 2 1 Signal 1 Figure 4.4. Spatial transmit diversity with Alamouti’s space–time block code.154 Space–time coding for wireless communications of signal constellation expansion and not guaranteeing maximum diversity gains (as in orthogonal designs). With M transmit antennas and a channel coherence time of T, the t T ×M transmitted signal space–time matrix X in LDC schemes has the form t Q  X=  A +j B  (4.20) q q q q q=1 where the real scalars  and  are related to the Q information symbols x (that belong q q q b to a size-2 complex signal constellation) by x =  +j for q = 12Q. This q q q LDC has a rate of Q/Tlog M. Several LDC designs were presented in 43 based on a 2 judicious choice of the parameters T,Q and the so-called dispersion matrices A and B q q to maximize the mutual information between the transmitted and received signals. Analternatewaytoattaindiversityistobuildthediversityintothemodulationthrough constellation rotations. This basic idea was proposed by Boulle and Belfiore 10 and Kerpez 51, and developed for higher-dimensional lattices by Boutros and Viterbo (11 and references therein). Therefore, one can construct modulation schemes with built-in diversity,withthecaveatthattheconstellationsizeisactuallyincreasing.Thepointtonote hereisthatTheorem4.2referstoarateversusdiversitytrade-offforagivenconstellation size.Therefore,inordertoconsidertheefficiencyofcodingschemesbasedontherotated constellations, one needs to take into account the expansion in the constellation size. As an alternative to alphabet constraints, other constellation constraints have been studied in order to produce codes with maximal rate as well as maximal diversity order (see 30 andreferencestherein).Therefore,constellationrotationswithoutalphabetconstraintscan yield the maximal performance of both rate (in terms of information constellation size) and diversity order. Note that in these cases there is a difference between the information constellation size and the constellation size of the transmitted codeword, which could be much larger. Therefore, in this sense, the rotated codes are actually more suitable in the context of the diversity–multiplexing trade-off discussed in Section 4.2.4, where there are no transmit alphabet constraints. In fact, using such rotation-based codes, several diversity– multiplexing rate optimal codes have been constructed (see 32,75,86 and references therein). Recently, STBCs have been extended to the frequency-selective channel case by implementing the Alamouti orthogonal signaling scheme at the block level instead of the symbol level. Depending on whether the implementation is done in the time or the frequency domain, three STBC structures for frequency-selective channels have been proposed: time-reversal (TR)-STBC 53, OFDM-STBC 56, and frequency-domain- equalized (FDE)-STBC 1. As an illustration, next we present the space–time encoding scheme for FDE-STBC. Denote the nth symbol of the kth transmitted block from antenna k k k i by x . At times k=024 pairs of length-N blocks x n and x n (for 0 ≤ i 1 24.3 Space–time coding principles 155 n≤N −1) are generated by the mobile user. Inspired by Alamouti’s STBC, we encode the information symbols as follows 1: k+1 ∗k ∗k k+1 x n=−x −n  and x n=x −n  1 2 N x 1 N (4.21) for n=01 N −1 and k=024 where · denotes the modulo-N operation. In addition, a cyclic prefix of length  N (the maximum order of the FIR wireless channel) is added to each transmitted block to eliminate IBI and make all channel matrices circulant. We refer the reader to 2 for a detailed description and comparison of these schemes. The main point we would like to stress here is that these three STBC schemes can realize both spatial and multi-path diversity gains at practical complexity levels. For channels with a long delay spread, the frequency-domain implementation using a fast Fourier transform (FFT) either in a single-carrier or multi-carrier fashion becomes more advantageous from a complexity point of view. 4.3.4 A new non-linear maximum-diversity quaternionic code In this section, we show how the STC algebraic structure inspires new code designs with desirable rate–diversity characteristics and low decoding complexity. The only full-rate complex orthogonal design is the 2×2 Alamouti code 5, and as the number of transmit antennas increases, the available rate becomes unattractive. 1 For example, for four transmit antennas, orthogonal STBC designs with rates of and 2 3 were presented in 72. This rate limitation of orthogonal designs caused a recent 4 shift of research focus to non-orthogonal code design. These include a quasi-orthogonal design 48 for four transmit antennas that has rate 1 but achieves only second-order diversity. Full-diversity can be achieved by including signal rotations which expand the constellation. Another approach is the design of non-orthogonal but linear codes 18 for which decoding is efficient albeit not linear in complexity. In this project, we revisit the problem of designing orthogonal STBC for 4 TX. Another reason for our interest in orthogonal designs is that they limit the SNR loss incurred by differential decoding to its minimum of 3 dB from coherent decoding. The proposed code in this proposal is constructed by means of 2×2 arrays over the quaternions, thus resulting in a 4×4 array over the complex field. The proposed code is rate-1, full-diversity (for any M- PSK constellation), orthogonal over the complex field, but is not linear. For QPSK, the code does not suffer constellation expansion and enjoys a simple maximum-likelihood decoding algorithm. Consider the 4×4 space–time block code  pq ∗ ∗ X= (4.22) q p q ∗ −q 2 q 156 Space–time coding for wireless communications where each block entry is a quaternion. There is an isomorphism between quaternions q =q +iq +jq +kq and 2×2 complex matrices as follows: 0 1 2 3   c c q 0q 1 q ↔ =Q (4.23) ∗c ∗c −q 1q 0 c c where q 0=q +iq , q 1=q +iq . Therefore, we may replace the quaternions p 0 1 2 3 and q by the corresponding 2×2 complex matrices to obtain a 4×4 STBC with complex 3 entries. There is a classical correspondence between unit quaternions and rotations in R ∗ given by q −→T p −→ q pq (details of the transformation T are given in 14). q q For QPSK, maximum-likelihood decoding requires a size-256 search. Through linear combining operations and appropriate application of the transformation T , we showed q how to exploit the quaternionic structure of this code to reduce the complexity of ML decoding to a size-16 search without loss of optimality. In Figure 4.5, the significant performance gains achieved by the code in (4.22) in the IEEE 802.16 WiMAX environment 83 as compared to single-antenna transmission/reception translate to a 1.5- and 2.6-fold increase in the cell coverage area −3 at 10 bit error rate when used with one and two receive antenna(s), respectively. We also compare in Figure 4.5 the effective throughput of our proposed quaternionic code 3 with the rate- full-diversity Octonion code given in (A4.1) assuming QPSK modulation 4 and an outer RS(1511) code for both. We observe that our proposed code achieves a throughput level of 1 46 bits per channel use, whereas the achievable throughput for the Octonion code is 1 1 bits per channel use (33% increase). 4.3.5 Diversity-embedded space–time codes The trade-off between rate and diversity was explored within the framework of fixed alphabets by Tarokh, Seshadri, and Calderbank 74 and by Zheng and Tse 89 within an information-theoretic framework. Common to both is the observation that to achieve a high transmission rate, one must sacrifice diversity and vice versa. Consequently a large body of literature has mainly emphasized the design of codes that achieve a certain level of diversity (typically maximal diversity order), and a corresponding rate associated with it, i.e. a particular point on this rate–diversity trade-off (see 31,59 and references therein). As explained at the end of Section 4.2.4, a different point of view was proposed by Diggavi et al. 22,23, where the code was designed to achieve a high rate but has embedded within it a higher-diversity (lower-rate) code (see Figure 4.6). Moreover, in this work it was argued that diversity can be viewed as a systems resource that can be allocated judiciously to achieve a desirable rate–diversity trade-off in wireless communications. In particular, it was argued that if one designs the overall system for a fixed rate–diversity operating point, we might be over-provisioning a resource which could be flexibly allocated to different applications. For example, real-time applications need lower delay and therefore higher reliability (diversity) compared to non-real-time applications. By giving flexibility in the diversity allocation, one can simultaneously accommodate multiple applications with disparate rate–diversity requirements 23.4.3 Space–time coding principles 157 0 10 4Tx−2Rx Transmission 4Tx−1Rx Transmission –1 10 1Tx−1Rx Transmission –2 10 –3 10 –4 10 –5 10 –6 10 2 4 6 8 10 12 14 16 18 SNR in dB (a) 2 Rate 3/4 STBC with RS (15,11) 1.8 Proposed STBC with RS (15,11) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 SNR in dB (b) Figure 4.5. (a) Compares the quaternionic code with the single-antenna case for WiMAX. (b) Compares the effective throughput of the quaternionic code with an octonion in a quasi-static channel. Stream 1 a(0), a(1), . . . Transmitted CODEBOOK x(0), x(1), . . . sequence b(0), b(1), . . . Stream 2 Figure 4.6. Embedded codebook. Bit error rate Effective throughput158 Space–time coding for wireless communications Let  denote the message set from the first information stream and  denote that from the second information stream. The rates for the two message sets are, respectively, R and R. The decoder jointly decodes the two message sets with average error ¯ ¯ probabilities, P  and P , respectively. We design the code Xab, such that a e e certain tuple R D R D  of rates and diversities are achievable, where R =R= a a b b a log/T, R =R=log/T and analogously to 89 we define b ¯ ¯ logP  logP  e e D = lim D = lim (4.24) a b SNR→ SNR→ logSNR logSNR For fixed-rate codes it has been shown in 23 that to guarantee the diversity orders D D we need to design codes such that a b min min rankBx x ≥D /M (4.25) a b a b a r 1 1 2 2 a =a ∈ 1 2 b b ∈ 1 2 min min rankBx x ≥D /M  (4.26) b r a b a b 1 1 2 2 a a ∈ b =b ∈ 1 2 1 2 where B is the codeword difference matrix as defined in (4.7). Basically, this implies that if we transmit a particular message a ∈, regardless of which message is chosen in message set, we are ensured a diversity level of D for this message set. A similar a argument holds for message set. Using this criterion several diversity-embedded codes have been constructed and will be discussed in the following. This design rule is a generalization of the design rule for traditional space–time codes given in Section 4.3.1. Linear diversity-embedded codes In 23, linear constructions of diversity-embedded codes were given. These code designs are linear over the complex field in order to be able to decode them efficiently using the sphere decoder algorithm 19, which has an average complexity that is only polynomial (not exponential) in the rate, making it an attractive choice for decoding high-rate codes. Another constraint that we impose in our code designs is to not expand the transmitted signal constellation, in contrast with other designs based on constellation rotations. For illustration we focus on one code example given in 23. Code example Let  come from the message set a0a1 ∈  and  come from 1 b0b1b2b3 ∈. Hence, R = log and R =log, leading to a total rate a b 2 3 of R +R = log. a b 2 ⎡ ⎤ a a b b 1 2 3 4 ∗ ∗ ∗ ∗ ⎢ ⎥ −a a b −b 2 1 4 3 ⎢ ⎥ X=X +X =  (4.27) a b ∗ ⎣ ⎦ b b a −a 1 2 1 2 ∗ ∗ ∗ −b b a a 2 1 2 1 where X is a function of variables a , a and X is a function of variables b , b , b , a 1 2 b 1 2 3 b . This code is linear over the complex field so that it can be decoded using the sphere 44.3 Space–time coding principles 159 0 10 –1 10 –2 10 –3 10 –4 10 Diversity-3 layer with estimated CSI –5 10 Diversity-2 layer with estimated CSI Diversity-3 layer with perfect CSI Diversity-2 layer with perfect CSI –6 10 5 10 15 20 25 SNR in dB Figure4.7. Performanceofadiversity-embeddingspace–timeblockcodewithperfectandestimated CSI. decoder 20, where the average complexity is polynomial rather than exponential in the rate. The proof that this code achieves diversity 3 for variables a , a and diversity 2 for 1 2 variables b , b , b , b makes essential use of quaternion arithmetic. The code does not 1 2 3 4 require channel knowledge at the transmitter and it outperforms time-sharing schemes 23. The performance of this code with perfect and estimated CSI over a quasi-static flat-fading Rayleigh channel is depicted in Figure 4.7. Non-linear diversity-embedded codes Constructions of a class of non-linear diversity-embedded codes were given in 12,25, and we explain the principles behind these constructions here. The basic idea of this class of non-linear codes is to use rank properties of binary matrices to construct codes in the complex domain with the desired diversity-embedding property. Given two message sets , they are mapped to the space–time codeword X as shown below: ⎡ ⎤ K11  K1T f 1 ⎢ ⎥  −→K= ⎣ ⎦ KM 1 KMT t t ⎡ ⎤ x11  x1T f 2 ⎢ ⎥ −→X=  ⎣ ⎦ xM 1 xMT t t Bit error rate

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