WIRELESS SYSTEMS AND MOBILE COMPUTING
Space-Time Coding for Broadband Wireless Communications
Z. Liu , G. B. Giannakis (contact author), B. Muquet and S. Zhou
Dept. of ECE, Univ. of Minnesota, 200 Union Str. SE, Minneapolis, MN 55455, USA; Tel/Fax: (612) 626-7781/625-4583;
Emails: lzq, georgios,szhou @ece.umn.edu; Work of these authors was supported by ARO grant no. DAAG55-98-10336. Centre de Recherche Motorola Paris, Espace Technologique, Saint-Aubin, 91193 Gif-sur-Yvette, France; Email:
[email protected]. July 30, 2000
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Abstract The development of broadband wireless communication systems must cope with various performance-limiting challenges that include channel fading as well as size and power limitations at the mobile units. As a promising method dealing with these challenges, space-time coding is effective in supporting reliable, high-data-rate transmissions: the major goal in broadband wireless communications. A survey of space-time coding schemes is provided in this paper. Targeting broadband wireless communications, the focus is on space-time coding in the presence of frequency- and time-selective fading and the associated channel estimation and symbol recovery algorithms for both single- and multi-user settings. Keywords diversity, space-time coding, multiple transmit antennas, wireless communications.
I. I NTRODUCTION The next generation of broadband wireless communication systems is expected to provide users with wireless multimedia services such as high speed Internet access, wireless television and mobile computing. The rapidly growing demand for these services is driving the communication technology towards higher data rates, higher mobility, and higher carrier frequencies that are needed to enable reliable transmissions over mobile radio channels. Depending on the Quality-of-Service requirements and different applications per user, many broadband wireless communication systems have been proposed (see [53, 54, 23, 7, 43] and references therein). In these systems, data rates may exceed 100 Mbps while mobile units may move as fast as a high-speed train with user bandwidths that are fixed or dynamically allocated. In order to support wireless multimedia services, research efforts are carried out to develop efficient coding and modulation schemes along with sophisticated signal and information processing algorithms to improve the quality and spectral efficiency of wireless communication links. However, these developments must cope with critical performance-limiting challenges that include mobile radio channel impairments, multiuser interference (MUI) and size/power limitations at the mobile units. Mobile radio channels are subject to time-selective and frequency-selective fading that are induced by carrier phase/frequency drifts, Doppler shifts and multipath propagation, respectively. Channel fading causes performance degradation and renders reliable high-data-rate transmissions a challenging problem. Traditionally, the most effective technique to combat fading has been the exploitation July 30, 2000
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of diversity. According to the domain where the diversity is created, diversity techniques may be divided into three categories, namely, temporal diversity, frequency diversity and spatial diversity [68]. Temporal and frequency diversity normally introduce redundancy in time and/or frequency domain, and therefore induce loss in bandwidth efficiency. Typical examples of spatial diversity are multiple transmit- and/or receive-antenna communications. By employing antennas at the transmitter or the receiver, multiple-antenna communications inherit space diversity to mitigate fading without necessarily sacrificing precious bandwidth resources; thus, they become attractive solutions for broadband wireless applications. Compared to single-antenna transmissions, multiple antenna transmissions increase the channel capacity by an order of magnitude or more [11, 70, 42, 55, 9]. A particular implementation of multiple-antenna communications has been the layered space-time architecture of [10] whose capacity grows linearly with the minimum number of transmit- and receiveantennas under the assumption that the underlying propagation channels are independent, flat and known at the receiver. The effects of fading correlation on the capacity of multiple antenna transmissions have been investigated in [59]. Without requiring channel estimates at either the transmitter or the receiver, the unitary space-time modulation in [22] was shown to be able to achieve substantial fraction of channel capacity. Depending on whether multiple antennas are used for transmission or reception, two types of spatial diversity can be used: receive-antenna diversity and transmit-antenna diversity. In receiveantenna diversity schemes, multiple antennas are deployed at the receiver to acquire separate copies of the transmitted signals which are then properly combined to mitigate channel fading. In fact, receive-antenna diversity has been incorporated in wireless systems such as GSM and IS-136 to improve the up-link (from mobiles to base-stations) transmissions (see [51] and references therein). However, due to size/power limitations at the mobile units, receive-antenna diversity appears less practical for the down-link (from base-stations to mobiles) transmissions. As a result, the down-link becomes the capacity bottleneck in current wireless systems [72, 27], which has motivated rapidly growing research work on transmit-antenna diversity. Transmit-antenna diversity relies on multiple antennas at the transmitter and is suitable for down-link transmissions because having multiple antennas at the base station is certainly feasible. Relative to the frequently deployed receive-antenna
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diversity, there have been only a few industry activities to implement transmit-antenna diversity in real systems [47]. Two major obstacles to implement transmit-antenna diversity are that: i) unlike the receiver, the transmitter does not have instantaneous information about the fading channels and ii) the transmitted signals are mixed spatially before they arrive at the receiver. In order to exploit the embedded diversity from multiple transmissions, transmit-antenna diversity schemes must rely on some additional processing. A number of transmit-antenna diversity schemes have been proposed, and can be divided into two categories: open loop (e.g., [58, 50, 20]) and closed loop (e.g., [76, 24, 28]). The difference between open and closed loop schemes is that the former does not require channel knowledge at the transmitter. On the other hand, the latter relies on some channel information at the transmitter that is acquired through feedback channels. Although feedback channels are present in most wireless systems (for power control purposes), mobility may cause fast channel variations. As a result, the transmitter may not be capable of capturing the channel variations. Thus, usage of open-loop transmit-antenna diversity schemes is well motivated for future broadband wireless systems which are characterized by high mobility. Among various transmit-antenna diversity schemes, particularly popular recently is space-time (ST) coding that relies on multiple antenna transmissions and appropriate signal processing at the receiver to provide diversity and coding gains over uncoded single-antenna transmissions. ST coding has been recently adopted in third generation cellular standards (e.g., [75, 63, 62]) and has been proposed for many wireless applications (e.g., [77, 29]). A detailed survey of space-time coding can be found in [47]. In this paper, we will focus on space-time coding for broadband wireless communications. In particular, we will emphasize space-time coded transmissions through frequency-selective fading channels and time-selective fading channels. II. S PACE -T IME C ODING
FOR FLAT FADING CHANNELS
In this section, a brief summary of ST coding over flat Rayleigh fading channels will be given. First, we will describe a general system model which is applicable to most ST coding schemes. Under this model, some illustrative examples of ST coding will then be presented.
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TX %
RX %
"!
ST Encoder
ST Decoder , -
RX&)
TX &('
* +
#$
Fig. 1. General System Model of ST Coding for Flat Fading Channels
A. System Model Consider a wireless system equipped with .0/ transmit-antennas and .01 receive-antennas, as illustrated in Fig. 1. At the transmitter, information data symbols 2 belonging to the constellation . : TC
YX .;NL>TC
==
.. .
..
==
Z ; . N
. [ .;NL>TC \==
^_ _
= U . NL>V.;NWADC _ _
X .UNL>V.;NWADC __ .. . `ba
[ .;NL>V.;NIA]C
(1)
where the code symbol dc belongs to the constellation set e . The constellation sets 3
and e may O . N columns of are be identical or different depending on different ST coding schemes. The ;
generated in successive time intervals fhg with each of .0/ coded symbols in a given column sent through one of the .U/ transmit-antennas simultaneously. Because .;N coded symbols from each transmit-antenna correspond to .;: information symbols, the overall transmission rate i is: i
6
. : ; X n3 n L .;Nkj lm
bits/sec/Hz
(2)
a n n where 3 denotes the cardinality of 3 . Let opg and oq: denote the average energy of dc and 2 ,
respectively. In order to constrain the total transmission power to be independent of the ST encoding, we relate oqg to or: by opg 6Ts oq: , where st476 .@:vu .;N.U/ . July 30, 2000
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We assume that the channels’ delay spread is small compared to fhg but their coherence time is larger than .UNYfhg . Under these assumptions, the channels between pairs of transmit- and receiveantennas are flat faded and can be modeled as complex constants within one block. When the channels are flat, no inter-symbol interference (ISI) occurs in the time domain. Indeed, at any time the received signal at each of the .U1 receive-antennas is a noisy superposition of .U/ signals transmitted from the .U/ transmit-antennas. Letting #w be the baseband received signal at the x th receiveantenna, we arrive at our data model #(w L6zy s oq:
{
cG|h~}
cw Yc > w
x 6 C
.U1
(3)
a a([a a w where: ’s are independent samples of a zero-mean complex white Gaussian process with two sided power spectral density .;u
per dimension; and the gain cw models the flat fading channel } from the th transmit-antenna to the x th receive-antenna. , - Given the received signals #w in (3), the ST decoder will decode + using the unique
mapping between
O and + . Different space-time coding schemes distinguish themselves with
different encoding and decoding schemes. We will next discuss the optimal ST encoding and decoding based on the maximum-likelihood (ML) principle. B. Optimal Space-Time Coding Design Detailed analysis of optimal ST codes can be found in [68, 19]. Based on the system model (3), we now summarize the main results of ST coding designs under the assumption that the channels gains cw are i.i.d. complex Gaussian random variables with variance 0.5 per dimension and known } to the receiver. Channel knowledge can be either acquired through training or by employing blind estimation algorithms (see also Section IV). as the event that Dropping the block index , we define the pairwise block error event @ the receiver decodes the block erroneously when the block is actually sent. Let Pr @ be the pairwise block error probability averaged over the fading channels. The goal is to pursue optimal ST coders so that Pr is minimized. Let us denote by
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O
O ( ) the code matrix mapped from ( ) and define the .0/LHM.U/ error matrix
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WIRELESS SYSTEMS AND MOBILE COMPUTING
O
6
as [47]: O
4769 O A
O ( O A
O
(4)
C , it was shown in [68, 25, 47] that Pr is upper-bounded by $ $ s oq: u[.; Pr q @P c (5) cG|hh " a O O 6 6 c C i , are the i nonzero eigenvalues rank is the rank of and where i O a c a are termed diversity advantage and of . In (5), the factors WN 4K6 i .01 and Wg 47a 6¢¡ cG£|h coding advantage, respectively. In order to minimize Pr , it is clear from (5) that ST coding
At high SNR, i.e., or:vu(.;I
should be chosen to maximize both IN and Wg , which leads to the following design criteria: O or equiva1. Rank Criterion: In order to achieve the maximum diversity advantage, the matrix O O O O O¥¤ O lently the matrix A has to be full rank over all possible and ( 6 );
O 2. Determinant Criterion: Suppose is full rank. In order to achieve the maximum coding advan O O O O¥¤ O tage, the minimum determinant of over all possible and ( 6 ) should be maximized.
Some remarks are now in order: Remark 1: The diversity advantage WN and the coding advantage Wg affect the bound in (5) in different ways. At high SNR, it is clear that IN plays a more important role to reduce the upper bound in (5) than Ig . Thus, ST coding designs should first satisfy the rank criterion and then the determinant criterion. If a tradeoff has to be made, the coding advantage should be the candidate; Remark 2: The aforementioned ST coding design criteria are based on ML decoding and provide guidelines to design ST coding schemes. Even with optimal ST coding designs however, the promised diversity and coding advantages may not be necessarily achieved for a specific (non-ML) ST decoding scheme. The basic method to check the achieved diversity and code advantages is to derive the error probability expression as in (5); Remark 3: In order to achieve the maximum diversity advantage, we infer from the dimensionality O that one should select .;N@¦§.0/ . This implies that time-domain processing is indispensable of for ST coding. Recall also that ST coding does not require channel knowledge at the transmitter; Remark 4: Although multiple receive-antennas ( .U1©¨ªC ) are helpful to increase the diversity advantage, they are optional in ST coding designs. All design criteria are meaningful even for a single July 30, 2000
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«
«
«
«
WIRELESS SYSTEMS AND MOBILE COMPUTING
Training Sequence
Data Frame 1
7
Pilot
Data Frame 2
Pilot
Fig. 2. Signaling Structure
receive-antenna provided that the SNR is high. Having obtained the optimal design criteria, we discuss next some design examples. C. ST Coding Examples Based on our system model in Section II-A, we outline three examples of ST coding designs for flat fading channels: ST trellis coding (ST-TC) [57, 69, 68, 48, 66]; ST block coding (ST-BC) [2, 65, 13]; and ST differential coding (ST-DC) [25, 64, 21]. We will explain how performance and complexity are traded off in these alternatives. C.1 Space-Time Trellis Coding Fig. 2 depicts the signaling structure of ST-TC where the information data symbols are transmitted frame by frame with pilot and training sequences inserted periodically [48]. The pilot and training sequences are used for channel estimation, timing and synchronization. We consider the data frames + and fit the ST-TC under our model in Section II-A. ST-TC does not change symbol constellae and .@: 6 the transmission rate i 6
tion, i.e., 3
6
.;N . If we neglect the pilot and training sequences, we can easily see that
XLn 3 n bits/sec/Hz, which implies no bandwidth efficiency loss in STj l
m TC as compared to the uncoded case. ST-TC employs ML decoding and its encoding is optimal in
terms of maximizing both diversity and coding advantages [68]. In order to facilitate low-complexity Viterbi decoding, a trellis is designed to perform encoding in ST-TC. For example, considering 4-PSK 4 states ST-TC with two transmit-antennas ( .0/ 6
). Fig. 3
depicts the corresponding 4-PSK constellation labeling and encoding trellis (see [68] for further details). Using the encoding trellis, the ST encoder maps the data frame to the code matrix O . As an example, when + k6§8 C O ®6
July 30, 2000
C ==vE F , the code matrix is given by Q aL¬a aL¬a~£a ^ = = C C S ` C ¬ ¬ C == ¬ ¬
(6)
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where we observe that this example is exactly the delay diversity scheme in [58]. ST-TC achieves the maximum diversity advantage ( WN 6 .U/£.U1 ) and the maximum coding advantage. However, for a fixed number of transmit antennas, its decoding complexity increases exponentially with the transmission rate [68]. As discussed before, when a tradeoff has to be made between coding advantage and diversity advantage, it is appropriate to sacrifice the coding advantage. We show next how ST-BC trades decoding simplicity with possibly reduced coding advantage. C.2 Space-Time Block Coding ST-BC was first proposed for two transmit-antennas [2] and was later generalized to any number of transmit-antennas [65]. Transmit-antenna linear beamforming have also been reported in [13, 14] (see also [56] for optimal linear ST precoding). One property of ST-BC is its decoding simplicity which makes it attractive especially when receiver complexity is at a premium. Compared to ST-TC, it turns out that ST-BC achieves maximum diversity advantage ( WN 6 .U/¯.U1 ), but it does not gain as much coding advantage as ST-TC and loses bandwidth efficiency when more than two transmitantennas are employed [65]. To compensate for this loss, one method is to increase the constellation size at the expense of coding advantage. The signaling structure of ST-BC is the same as that of ST-TC (see Fig. 2). Unlike ST-TC, the block length .@: in ST-BC is equal to .U/ and can not be arbitrary. Accordingly, ST-BC divides each K data frame into blocks + of size .@:¯H°C which are then encoded by ST encoder (see Fig. 1). As a simple but practically important example for ST-BC, we discuss next ST-BC with two-transmitantennas ( .0/ 6 ) and one receive-antenna (.U1 6 C ). Note that ST-BC with .U/ 6 is the only ST-BC with C
bandwidth efficiency. ±
² 00 01 02 03
1
2
0
10 11 12 13
20 21 22 23
3
³ 30 31 32 33
Fig. 3. 4-PSK Constellation Label and Encoding Trellis
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Letting + P47698 2 2 >TC -EGF , the ST encoder maps Q A2=´ >TC O k6 S 2 2 >µC 2 ´
into: ^ `
(7)
Since complex conjugation is involved in (7), the constellation set 3 may not be preserved in ST-BC. Substituting (7) into (3), it follows that two consecutive received samples are given by # L6 # >¶C L6
y oq: u 8 2 } y oq: u 8 A }
> }
X 2 >TC - E >
2 ´ >µC > }
X 2 ´ -E > >µC
(8a) (8b)
a and 2 · > C , where we dropped index x because there is only one receive-antenna. To decode 2 the ST decoder is designed by forming the two consecutive output samples, ¸ and ¸ >¹C , as: Q ^ Q ^ Q ^ X # ¸ ´ S ` 6 S} ` } ` S (9) # X ¸ >TC ´ A ´ >TC º } »(¼ } ½ ¾ |"¿
Substituting (8a) and (8b) into (9), we obtain: Q ^ Q ^ Q ^ X X ¸ S 2 S ` 6 y oq: u n n > n X n À ` >ÂÁ X S ` (10) } } ¸ >TC 2 µ > C ´ µ > C a and decode 2 from ¸ . Exploiting the fact that Á X defined in (9) is unitary, it can be readily deduced that the decoding in (9) is ML and maximum diversity advantage WN 6
is achieved [2].
Note that it is impossible to transmit four symbols in two time slots through two transmit-antennas and one receive-antenna and still be able to recover 2 from ¸ , simply because there are fewer equations ( ) than unknowns (Ã ). C.3 Space-Time Differential Coding The (coherent) decoding in both ST-TC and ST-BC requires channel estimates at the receiver that are acquired either through training [32], or, via blind estimation algorithms [38, 39, 41]. Since multiple .U/ .U1 channels have to be estimated, channel estimation implicitly assumes that the underlying channels remain invariant for a long time, which may not be true for a number of applications. Motivated by the conventional (single-antenna) differential coding where the need of channel July 30, 2000
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estimation is circumvented, several ST differential coding (ST-DC) schemes have been proposed recently to achieve maximum diversity advantage without requiring knowledge of channel estimates [25, 64, 21]. The basic idea behind these schemes is to introduce proper encoding between two consecutive code matrices so that the decoding at the receiver is independent of the underlying channels. As expected, the price paid for is code advantage in addition to 3 dB loss in SNR compared to coherent decoding. We illustrate next ST-DC by outlining the so-termed differential unitary ST modulation (DSTM) of [25]. With the notational conventions of Section II-A, DSTM has parameters: .@: 6 C , .U/ 6 .;N X n 3 n . The differential encoding in DSTM relies on a finite group and therefore i 6 C u(.0/ Ä j l m Å Å of .0/pHD.0/ unitary matrices where ÆÇ È , ÇÉÇ 6 Ç Ç 6ËÊ . To ensure unique correÅ 476 n Å n 6 n 3 n . Let us define spondence between and 3 , their cardinalities are equated, i.e., Ì Å 476 ÇÍ Ç·Î and 3 476 Ï2 2=Î , and establish, without loss of generality, the
a((a ([a following one-to-one (ordered) mappingabetween elements of 3 2 c] and ReTr denotes the real
a part of the trace. We see from (13) that the decoding does not require knowledge of the channels.
Besides the ST coding schemes we summarized in this section, a number of variants and extensions have appeared. For a detailed discussion, the reader is referred to [47] and references therein. As mentioned in the introduction though, channels in broadband wireless communications are characterized by frequency-selectivity and/or time-selectivity. This motivates our devoting the rest of this paper to ST coding designs that are robust to such fading effects. July 30, 2000
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III. S PACE -T IME C ODING F OR F REQUENCY-S ELECTIVE
11
CHANNELS
So far, most work on ST coding assumes frequency-flat fading channels: an assumption that is not well justified in future broadband wireless communication systems. In broadband wireless communications, the symbol duration becomes smaller than the channel delay spread and consequently channel frequency-selectivity arises. Targeting broadband wireless communications, it is important to investigate how to design ST codes in the presence of frequency-selectivity. We will first link ST coding designs in frequency-selective fading channels to those in flat fading channels and then proceed to summarize some existing work. Our focus will be on the ST coding scheme proposed in [34, 35] which guarantees diversity gains and symbol recovery regardless of frequency-selective channels. A. Effects of Frequency-Selectivity on ST Coding Designs The effects of frequency-selectivity on ST coding designs have been investigated in [66, 17], where the pairwise block error probability is re-derived based on a simple two-ray channel model and assuming ML decoding. It was shown that ST codes designed to provide a certain diversity advantage in flat fading channels still provide at least the same diversity advantage even in frequencyselective channels (see [66, Theorem 5.1]). With regards to coding advantage, the theoretical analysis of [17] revealed that the coding advantage might decrease considerably in the presence of temporal ISI induced by frequency-selective channels unless some additional processing is employed. Results in [66] suggest that optimal ST codes in frequency-selective fading channels may achieve higher diversity advantage than those in flat fading channels. However, these results are obtained based on ML decoding that is computationally heavy. Practically, feasible ST decoding designs in frequency-selective channels are on the other hand challenging because the transmitted signals are mixed both temporally and spatially. For example, Alamouti’s simple linear ML decoding for ST-BC is not directly applicable in the presence of frequency-selectivity. In order to maintain the decoding simplicity and take advantage of existing ST coding designs for flat fading channels, most work on ST coding designs pursues (suboptimal) two-step approaches. The basic idea behind these approaches is to: step 1) mitigate the ISI and convert frequency-selective fading channels into flat fading ones; July 30, 2000
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ßKà
á WIRELESS SYSTEMS AND MOBILE COMPUTING
PSfrag replacementsÞYßKà
á
12
ÞdßàÏá
â %
+ ST Block Encoder
ãBä C , to The space-time encoder takes as input two consecutive precoded blocks, and +
output the following JHM code matrix: Q S >µC
^
A ´ µ > C ` ´
(14)
where each block columns is transmitted in successive time intervals with its two blocks sent through the two transmit-antennas, respectively. Note that without blocking ( 6 C ) the code matrix in (14)
, is the same as the ST block code matrix in (7). For convenience, we denote by c 6 C , the a a that ?HVC block transmitted through the th transmit-antenna at the time , and observe from (14) , c is related to + via
, ®6 é
+
, >TC L6
, X ®6
T > C
, X é >µC ®6 é a 8 E
6 ë Let beé the H FFT matrix with its entry a by the matrix transmission, the inverse FFT (IFFT) described a A ´ >TC
´ a
(15)
Ö¯ AWx u . Before , is applied to c to obtain
, c . Let c , 6 be the chip-rate sample discrete-time baseband equivalent } £atransmit-antenna [a th order FIR channel from the th to the receive-antenna. We emphasize that we do not require knowledge of c at the transmitter, except for an upper bound on its maximum } channel order. In order to eliminate the inter-block-interference (IBI) caused by the FIR channel, we rely on a cyclic prefix (CP) of length inserted at the beginning of c which are discarded c 54K6
ê the receiver. As detailed in [74, 79] and shown in Fig. 5, the CP insertion can be described by at gGë 476 8Ê FgGë Ê F E F , where Ê gGë is formed by the last rows of the H identity matrix Ê . The oper í Ê E ation of discarding the first received symbols can be described by the matrix gGë 476¥8 . H Ü> Toeplitz matrix Á c with Accordingly, the FIR channel c is described by the Í> } the th entry c A . } a ê IBI-free transmission enables one to focus on each user’s received symbol block separately. Let í Á c 476 gGë=Á c g ë denote the equivalent channel matrix after eliminating the IBI. The bHMC IBI-free
received symbol block ! (see Fig. 5) is given by: í ! ®6#"k > gGë%$ July 30, 2000
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é
é
WIRELESS SYSTEMS AND MOBILE COMPUTING
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, , where: "k 74 6 Á > Á X X is the received symbol block from the two transmit é> H C additive white Gaussian noise vector. Given ! , the retrieval antennas; $ is the · of the information blocks + at the receiver proceeds in three steps as follows. First, the FFT , , described by the matrix is performed on ! to obtain ! ; two consecutive blocks, ! , K , and ! > C , are then processed by the space-time decoder to produce the block & þ with transmit-diversity gains. Finally, the linear decoding matrix is employed to recover +
regardless of the FIR channels. Consequently, our system will achieve: i) transmit-diversity gains; and ii) symbol recovery guarantees, regardless of the underlying FIR channels. We will show next ê how the aforementioned features become possible, starting with the following two facts: é é í 6 c c Fact 1: The equivalent channel matrix Á g ëÁ gGë is a JH' circulant matrix [79, 73].
Fact 2: The circulant Á é c cané be diagonalized by pre- and post-multiplication with
and , i.e.,
w wX-,/. 0 1 6 ( c 6 diag 8* c + * c + E Á c ) 5 a(a a 5 * 2 ; 7 4 4 6 3
6 2 c c c | where is the frequency response of the channel at the point 2 [16, p. } } 202 ]. é vector ! , can be written as: é Exploiting Facts 1 and 2, the , ! 54K6
í
, , gGë7! L6)( > ( X X >
í
gGë7$
(17)
( ( X Assuming knowledge of and (or equivalently knowledge of the channels) at the receiver, we , , consider two consecutive blocks, ! and ! >tC and first decode the precoded block + with é we express ! , and ! , >µC as: transmit-diversity gains. Plugging the mapping (15) into (17), í é g ë7$ í A ( ´ µ > C > ( X ´ > gG%ë $ T > C
, ! L6#( > ( X + T > C > , ! >µC L6
(18a) (18b)
Eqs. (18a) and (18b) show that the blocks are transmitted twice in two consecutive time in-
tervals through two different channels. In order to exploit the embedded diversity through repeated , K transmissions, we design the space-time decoder (see Fig. 5) by forming its two consecutive
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> C , as: output blocks, & and & T Q ^ & S ` 6 & >TC S
Q
( ´
( X
( X´
A (
Q `
^
^ ! S ` ! ´ >TC
(19)
Substituting (18a) and (18b) into (19), we arrive at:
& ®6Õ8( ´ ( > ( X´ ( X + >:9
(20)
a -EGF is given by: ^ ` (21) >µC X 7 > = 0 , it follows that the covariIf $ is a white Gaussian noise vector with covariance matrix ; < Ê . ance matrix of 9 is given by: F 9 À F é >TC where the additive noise vector 9 5476Õ8 9 À Q ^ Qé í ( ´ ( X g ë7$
® 6 S ` S 9 í ( X´ A ( ´ gGë%$ ´
Q
( ´ ( > ( X´ ( X L 6 S Cov 9 ; < X
( ´ ( > ( X´ ( X
^ `
(22)
( 6 c Because C , are diagonal matrices, we infer from (22) that the elements of 9 in (20) a a are independent. Thus, the processing of the blocks & can be separated without sacrificing performance. Let ( 476)( ´ ( > ( X´ ( X and re-write (20) as ý + & L6)( >:9 (23) X X w X 3 cG|h n * c + wvX-,. 0?1 n X E , we infer that Eq. (23) Observing that ( 4K6 diag 8 3 cG|h n * c + n a been achieved. implies that diversity advantage of order twoahas ý We also deduce from (23) that the recovery of + from & requires ( to be full column
rank. As shown in [74, 34], the full column rank is guaranteed by selecting: > i) 6 ; ý ii) any rows of to be linearly independent. ý To satisfy conditions i) and ii), a special choice of is given by [74] QR .BA 0 ^__ R 2 = = @ 2 R C _ ý 6 .. .. . . S .BA 0 ` ==@2 a C 2 July 30, 2000
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L=2, K=6
L=2, K=6
0
D
10
0
D
10 with precoding
with precoding
w/o precoding
w/o precoding -1
10 -1
10
-2
average BER
BER
10
-2
10
-3
10
-4
10 -3
10
-5
10
C
-4
10
0
2
4
6
-6
8 Eb/N0 (dB)
10
12
14
16
(a) Specific Channels
10
0
2
4
6
8
10 Eb/N0 (dB)
12
14
16
18
20
(b) Random Channels
Fig. 6. Improvements with Redundant Precoding
where 2FE;6
Ö¯8 x£G H A]C u E .
So far, we have designed our system to achieve transmit-diversity gains and guarantee symbol recovery. The direct advantages of our designs will be improved BER performance and robustness against channel fading. To appreciate the importance of redundant precoding, we compare the ST 6 6JI OFDM with and without precoding. Suppose and . Two discrete-time equivalent K L ¸ and * X 6§ CÀ>bx¸ ( CÀ> ¸ . Clearly, channels are chosen to be * 69 CÀ>Éx ¸ [ CL> BERs with and without redundant precoding are two channels share a common zero at ¸ 6 AWx . The shown in Fig. 6(a), where we see that the system with precoding improves performance considerably at least for this particular pair of channels. With the same setup, Fig. 6(b) confirms the superior performance of PST-OFDM over ST-OFDM in
independent channel realizations.
It is of interest to investigate the price paid for
these improvements choosing transmission rate as > our figure of merit. Including the cyclic prefix of length , a block of > ( 6 ) symbols
is transmitted to transmit
Notice that if
symbols per block. Thus, the overall transmission rate of our design is: M 6 6 (25) Ü> >D MON C . Hence, our PST-OFDM designs do not induce much loss in we have
transmission rate. July 30, 2000
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WIRELESS SYSTEMS AND MOBILE COMPUTING
18
So far, ST-OFDM designs require estimates of multiple channels at the receiver. Clearly, the success of ST-OFDM depends on the channel estimation accuracy. We will devote the next section to summarize relevant work on multi-channel estimation. We will focus on semi-blind channel estimators based on subspace methods. IV. M ULTI -C HANNEL E STIMATION
WITH
S PACE -T IME C ODING
Channel knowledge is indispensable at the receiver to decode space-time coded transmissions. For some transmitter diversity schemes, e.g., delay diversity for OFDM systems [30], the overall transmitter can be viewed as if only one antenna is effectively transmitting. In such cases, most identification algorithms already developed for SISO channels can be employed to estimate the channels. However, for most ST coding schemes, specific multi-channel estimation algorithms are needed. Traditionally, known training symbols are transmitted periodically and thus the receiver can deduce the multiple channel responses. This is proposed for instance in [67] for the MIMO frequency-flat channel model and in [32] for the ST-OFDM scheme described in Section III-A. However, training sequences consume bandwidth and thus incur spectral efficiency losses especially in rapidly varying environments. For this reason, blind channel estimation methods receive growing attention and a plethora of blind estimation algorithms have been proposed for various applications and contexts (see [71] and the references therein). More specifically, many blind algorithms have been developed for estimating the MIMO channels corresponding to multiple transmit and receive antennas [18]. However, only few works have been reported so far on MIMO/MISO channel estimation by capitalizing on the specific properties of ST codes. Relying on nonredundant (nonconstant modulus) precoding, [3] proposed a blind channel identification and equalization algorithm for OFDM-based multiple-antenna systems using cyclostationary statistics. For ST-OFDM, an deterministic constant modulus (CM) blind channel estimator was proposed in [38, 41], which can identify the channels under certain conditions. Here, we propose a blind channel identification algorithm based on a subspace approach. This algorithm possesses three attractive properties: i) it can be applied to arbitrary signal constellations; ii) by proper system design, it guarantees channel identifiability regardless of channel zero locations; and iii) it can identify multiple channels up to a scalar ambiguity only. July 30, 2000
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19
In the following, we consider the same system model as described in Section III-B except that we ý vary the redundant precoder from block to block. Specifically, we design: PQ R ý + if is even, + ®6 QS (26) ý X + if is odd Using (26), the data blocks ! Q , ! S , ! ´ >µC º »(¼ ¾ |UV T .XW70
, can be re-written from (18a) and (18b) as (c.f. Fig. 5): ^ Q ^ Q ^ Q ^ ý ( ( X ` 6 S ` S ` S ` > noise ý X ( X´ A ( ´ >µC a ½ º »(¼ ½ º »(¼ ½ | \T .XW70 |¾ Y Z ¾[
] ] Y + ! k6 > noise 4K6 "k > noise ] ] ] a Y where ! ( HÂC ), ( ?H¹ ] ), and + ( H C ) are defined in (27), and "I U476 Â ] denotes the noise-free version of ! . and equivalently,
(27)
]
Y
(28) ] +
We will start with channel identifiability issues from the noiseless vectors "k , since we are concerned with basic feasibility questions first. Henceforth, we also adopt the following design conditions on the block lengths and the linear precoders: a1) J¨ > ; ý a2) c ®È 8 C E is designed so that any of its rows are linearly independent. a a1) and a a a2) guarantee symbol recovery regardless of channel zero locations as discussed Note that
in Section III-B. c -EGF , 6 C , denote the channel impulse response between the th Let ^ c 4K6 8 c } } X Y a((receive a transmit- and the single ] antenna. a To estimate the channels ] ^ c cG|h (or] equivalently ), the "k .\A§C -E and forms receiver collects . blocks of "I to a] ]HÂ. ] matrix _ 6 8 "k
Ya` ` Y ` + . -E . At the receiver-end, a([a we also select: _ _ 6 , where 47698 ` ` a3) . ¦µ to be large enough so that a [a is of full rank .
Assumption a3) is known as the “persistence of excitation” assumption [78] and is usually satisfied by most signal constellations and for values of . comparable to .
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20
Y Under a1) and a2), it has been shown in [45] that has always full column rank . Therefore, Y together with a3), we have rank _ _ 6 and the range space b _ _ 6 b . Thus, the nullity of _ _ is c _ _ L6 ·AV . Further, the eigendecomposition Q
_ _ 6Õ8ed
d E S
XgA XgA
f .KX XgAh0 XgA
XgA .7X XgAh0 .7X XgAh0 .K X XgAh0
^
Q `
S
d d
^ `
(29)
d matrix whose columns span the null space i _ _ . Because the yields the H V A Y Y 6aF XgA latter is orthogonal to b _ _ L6 b , it follows that B , È 8C j
@A° E , where
denotes the th column of d . a ] ] * F * Let us now split the vector to its upper and lower parts as: U6 8 F EGF , where and
w wvX-,. 0?1 -EGF 7 4 9 6 e 8 * + * + c c c are °HbC vectors. Let ^ and k mlÀ standa for the diagonal matrix a(of [a the vector l . Therefore, Y can be expanded as with diagonal entries from the elements Q ^ Q ^ ] ý ( ( X 8 * E S ` S ` 6) (30) ý X ( X´ A ( ´ a a ]
Q
^ Q ^ * ý
Ank 8 ^ F ^ X E S k ] ´ 6 ` S ` a (31) * ( ý X
´ ´ k k a With o being the °H u , one >ÂC Vandermonde matrix with  > C >VC st entry Ö¯ AIx£G a can write ^ c 6 op^ c and express (31) as:
and rewritten as
Q 8 ^ F ^ X E S a
o F
o
º
Stacking (32) for each , 6
»(¼ ¾ |q C
^
Q ` ½ º
* k ] ´ S k ´
ÍAV
^
Ark = ` * k
»(¼ 0 ¾ | ( . t s
a[a 8 ^ F ^ X E q 8 (Ý yx a 8 from which one can solve for ^ F ^ X GE F .
]
Q S ½ º
ý
ý X ´
»(¼ ¾ |vu
^ ` 6w
(32)
½
, we obtain:
(Ý X gX A yxEh6a aa
(33)
a The natural question here is whether the solution of (33) is unique. As a special case, if we ý ý X choose 6 , we can find that both ^ X 6 8 ^ F ^ X E F and ^ X 6 8 ^ FX Ar^ E , and their linear a July 30, 2000
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WIRELESS SYSTEMS AND MOBILE COMPUTING
21
combinations, satisfy (33) and thus the channel estimator does not yield the desired output (see also [45]). This fact is due to the intrinsic symmetry between antenna pairs and can be easily understood by rewriting (28) as:
] " k6 k S
Q
(
( X
( X´
A ( ´
Q `
^
^ ý + S ` 6 ý >TC S
Q
( X A ( ´
^
A ( ` A ( X´
Q
^ ý + µ > C S ` ý + A
For symmetric constellations, it is impossible to determine whether the transmitter has sent + and + >C on channels ^ and ^ X or + >VC and A + on channels ^ X and An^ respectively [38]. To break the symmetry and identify the channel uniquely, [41] has proposed a pre-weighting approach, which transmits different power over different antennas. However, the price paid is reduced Bit Error Rate (BER) performance [38]. Here, we break this symmetry by introducing distinct ý ý linear precoder matrices and X , which can be designed to have balanced power. The channel identifiability can be guaranteed as summarized in the following theorem [45]: , Theorem 1: Suppose a1), a2) and a3) hold true; and let k denote a diagonal matrix with unit ý ý ý ý amplitude diagonal entries, and z , z X be formed from any ÉA rows of , X respectively. If , ý ý ý ¤ ý z and z X satisfy: k z È{b z X , the solution of (33) is unique up to a scalar constant and thus channel identifiability (within a scalar) is guaranteed. ý ý To select the appropriate and X , we can, for instance construct them as in (24) but from , ý ¤ ý distinct generators and then check whether kz È4b z X by simulation. Therefore, from the received data and one pilot tone, which is used to resolve the scalar ambiguity (inherent to all blind methods), multiple channels can be estimated simultaneously with linearly precoded ST-OFDM transmissions. In the presence of white noise, we replace _ _ in (29) by the received data covariance matrix ] ] í T T 4K6 V V E ! ! and we sort the eigenvalues in (29) in decreasing order. In practice, ] ] the ensemble . t 0 í correlation matrix is replaced by the sample average: V T V T 6 C u(. ] v3 W=| ! ! , which k "
converges in the mean square sense to the true correlation matrix since has finite moments.
] ] We then summarize the algorithm ] in the following steps: K . U 0 í s1. Collect the received data blocks ! and compute V T V T 6§ C u(. 3 W=| ! ! ; s2. Determine the eigenvectors , V6 C corresponding to the smallest JA¶ JAµ aa July 30, 2000
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WIRELESS SYSTEMS AND MOBILE COMPUTING
eigenvalues of matrix
22
í . T tT 0 V V ;
s3. From these eigenvectors, estimate 8 ^ F ^ X E F as the nontrivial solution of (33) by determining the q a left eigenvector corresponding to the smallest eigenvalue of matrix 8|(]8l }x ==~(Ýml X XgA yxE . An inherent problem to most blind methods is their rather slow convergence rate [33]. Note that our proposed algorithm needs to collect at least blocks as per a3). To facilitate convergence
and also enable tracking of slow channel variations, a semi-blind implementation of the subspace based method can be devised by capitalizing on training sequences, which are usually provided for synchronization or quick channel acquisition in practical systems. Following the lines of [44], this semi-blind implementation is summarized below: * * 1. obtain initial channel estimates ^ and ^ X through training (using e.g., [32]); and estimate the í autocorrelation matrix V T V T as: í .T T 0 6 V V
Q
* k ^ S * ´ k ^ X
^ * X k ^ ` * ´ Ank ^ S
Q
ý ý
ý X ý X
Q `
^
* k ^ S * ´ k ^ X
^ * X k ^ ` * ´ Ank ^
(34)
]
2. refine iteratively the autocorrelation matrix each time a new symbol block ! . becomes available using:
where
È 8
í .KT UT 0 6 V V
] ] K . 0 í T T ! . ! . V V > CpA
C E is the forgetting factor.
(35) a
í . T tT 0 a 3. perform the subspace algorithm based on V V .
To test the blind and semi-blind algorithms we resort to simulations. The figure of merit here is * X v 3 c h | t the averaged Normalized Mean Square Error (NMSE) of the channels defined as: C u ^ c A X X 6 6I ^ c u ^ c . We set 6 C I , , 6 > Ã ; and generate the channels according ¬ to the channel model A specified by ETSI in [6]. Fig. 7 shows the NMSE for the blind, semi-blind
algorithms and the training based approach proposed in [32] (with one training block from each antenna) at a typical SNR of o-u(.U 6 C dB and a terminal speed 6 m/s. We can infer from estimators outperform the training ¬ Fig. 7 that the blind and semi-blind channel based approach and
are able to track slow channel variations.
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23
HiperLAN/2 channel model A −Terminal speed =3m/s − SNR=10 dB
0
10
Training approach Blind algorithm Semi blind algorithm
−1
NMSE
10
−2
10
0
50
100
150
200 250 # of symbol blocks
300
350
400
Fig. 7. Comparisons between different channel estimators
V. S PACE -T IME C ODING F OR T IME -S ELECTIVE C HANNELS One desirable feature of future wireless communications is to support high-mobility up to those encountered when communicating with e.g., high-speed trains [12]. A direct effect of high mobility is the appearance of time-selectivity in mobile radio channels. The goal of this section is to study ST coding for transmissions through time-selective rapidly fading channels. For simplicity, we only consider time-selective but frequency-flat fading channels. The first issue addressed here is accounting for time-selectivity in ST coding. After summarizing some existing work, we will discuss a novel ST double differential coding scheme which is capable of achieving maximum diversity advantage in time-selective channels with low receiver complexity. A. Effects of Time-Selectivity on ST Coding Designs The system model of ST coding in time-selective channels is exactly the same as that in Section IIA except that cw in (3) is replaced by cw . The time-index in cw implies that time-selective } } } c w channels are allowed to vary from symbol to symbol. Theoretical analysis of ST coding } designs in the presence of time-selectivity was performed in [68], where the so-termed distance and product criteria were obtained following steps similar to those used to derive (5) (c.f. Section IIB). Combining these design criteria with those for flat fading channels, the so-called smart-greedy ST codes were constructed to guarantee the desired diversity gains in both flat and time-selective
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WIRELESS SYSTEMS AND MOBILE COMPUTING
24
fading channels. Simulations there showed that smart-greedy codes achieve higher diversity gains over time-selective fading channels than over flat ones. This suggests that time-selectivity can be exploited to provide additional (time) diversity. On the other hand, the results of [66, Theorem 6.1] assert that ST codes providing a certain diversity advantage in flat fading channels guarantee at least the same diversity advantage in time-selective fading channels. In theory, availability of time-selectivity is an advantage because it provides additional time diversity. A practical question though is how to take advantage of time-selectivity. The analysis in [68, 66] adopts ML decoding and assumes that channel gains c w ’s are i.i.d. random variables for } different , x , ; and are known to the receiver. This assumption facilitates proof of the results in .01 , there [68, 66], but it is ill-posed because for every .U1 incoming received data #(w x 6 C a ais(impossible. (a appear .U1d.0/ more independent unknowns ( c w ’s), and hence channel estimation } As a result, ST coding designs based on the distance and product criteria may not yield the desirable
performance improvement in real time-selective channels. We underscore here that our discussion excludes cases where time-selectivity is artificially generated by some methods, e.g., via interleaving and deinterleaving. To the best of our knowledge, taking advantage of time-selectivity in ST coding designs is still an open topic. Most works on ST coded transmissions through time-selective channels try to achieve the same diversity advantage as that in flat fading channels [37, 36], or, simply use the existing codes for flat fading channels with additional processing to compensate for the effects of time-selectivity at the receiver [40, 8]. For example, the iterative ST receiver in [8] treats channel gains cw ’s as } unknown random variables and decodes information symbols using the EM algorithm. Unlike [8], the iterative ST receiver of [40] models channel time-evolution as an AR process and uses Kalman Filtering to track the channel variations. While the ST codes designed for flat fading channels are used in [8, 40], the ST coding designs in [37, 36] take the channel variations explicitly into account and will be described next. B. ST Double Differential Coding In mobile radio channels, time-selectivity is mainly caused by frequency offsets. Two main sources of carrier frequency offset exist: one comes from Doppler shifts caused by the relative July 30, 2000
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WIRELESS SYSTEMS AND MOBILE COMPUTING
25
K motion between the transmitter and the receiver. For example, when a train is moving as fast as LFL km/hr, the induced Doppler shift may be up to Hz if the carrier frequency is GHz; the other source is the carrier frequency mismatch between transmitand receive-oscillators that arises due to
drifts from their nominal frequencies. In the presence of frequency offset, one approach to designing ST coding schemes is to first estimate the offset and then compensate its effects so that ST codes designed for flat fading channels can be applied. Clearly, the success of this approach depends on estimation accuracy, which is ultimately limited by the rate at which the frequency offset changes, the complexity and the cost which the receiver can afford for processing. In order to maintain receiver simplicity, an alternative approach to overcome frequency offsets is to design ST codes that render decoding at the receiver insensitive to frequency offset. The idea behind this approach is similar to the conventional (single-antenna) double differential coding [60, 52, 49, 15] and thus we term it ST double differential coding (ST-DDC). The system model of ST-DDC is the same as that in Section II-A with .@: 6 C and .0/ 6 .;N . We .B
0 .N 0 consider frequency-offset induced time-selectivity and denote by and the carrier frequency offset and Doppler shift at the th receive-antenna, respectively. The channel gain from the th transmit-antenna to the th receive-antenna is modeled as: }Z
®6
7+ } Z
wX-, s W
6 C
.U/
6 C
.U1
(36)
a a[a a a[a a .N 0
captures multipath fad>a fhg is the normalized frequency offset; and }Z ing effects. It is assumed that remains invariant during at least three consecutive information }> .N 0 symbols. To obtain (36), we also assume that the Doppler shift (being independent of ) is
where: 74 6Ë
.X
0
common to all transmit antennas. This assumption is valid when the receiver (e.g., a high-speed train) does not have any local scatterers so that multipath components originate far away and arrive at the receiver with a common angle. Under the channel model (36), the received signal # at the th receive-antenna is given by: wX-, s W { ©6 C # ®6 y s oq: + > + .01 (37) |h }Z a a((a Defining the received data block ! + q476 8 #
.;N Ý.;NIATC -EGF , we can cast (37) into July 30, 2000
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WIRELESS SYSTEMS AND MOBILE COMPUTING
26
a matrix/vector form as:
! + ®6
y sÜ or: +
wvX-,7 s W=>
O k ^ : >
6 C
.U1
(38)
a a[a a wX-, s wX-, s .> 0 °
7 4 6 + = = + +
K 4 6 8
< = =
diag C , where k .UN .;N5>µ.UNA C E F and a a ^ U476Õ8 L== E F .a } } The basic idea behind ST-DDC is to process three consecutive received data blocks, namely, ! A , ! + A§C and ! + from each receive antenna, to recover the data symbol 2 by
exploiting a judiciously designed encoding relationship among
O O O AM , A¹C and . Note
that this process does not require knowledge of the channels and frequency offsets as long as our channel model (36) holds true. O O O Similar to ST-DC in Section II-C.3, our encoding among AV , A]C and is based Å on a unitary matrix group . However, we will find it necessary to restrict our unitary matrices to Å be diagonal. With a little abuse of notation, let be the group of .0/kH¹.U/ ( .U/ 6 .;N ) unitary and Å diagonal matrices ( Ç°Ç 6 Ç Ç 6 Ê , and Ç is diagonal, ÆBÇ È ). The encoding is designed to
satisfy the following recursion: O ®6)Ü O A]C
¦ C
(39)
a a Ü
where the so-termed generating matrix obeys the following second recursion:
Ü L6
Ç g AÝC
¦µ
Ü C ~ 6µÊ
(40)
a Å
È In the second recursion, the matrix Ç conveys the information symbols and is chosen to correspond one-to-one with 2 according to the mapping rule defined in (11). Hence, knowing or decoding Ç at the receiver, determines uniquely 2 .
Exploiting recursions (39) and (40), it is shown in [36, 37] that 2 , or equivalently Ç , can be decoded from ! , ! A]C and ! AV by choosing: $ * { Ñ×Ö 8 8 ADC ! -E diag 8 ! AV ! AÝC -E ´ Ç ®6µÑ
Ó ØkÙ(Ú |h ReTr diag ! m§Ô (41) H diag 8 ! ! >Â! A]C ! A]C >!
A ! AV E Ç a July 30, 2000
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WIRELESS SYSTEMS AND MOBILE COMPUTING
27
where diag · be the diagonal matrix obtained by nulling the off-diagonal elements of the square matrix . Based on the ST decoding scheme described in (41), we resort to analyzing its pairwise block error probability in order to obtain the diversity advantages of ST-DDC as [36]: WN 6 i 476 where
rank Ç and Ç
4K6
Ç
i .U1
AÝÇ ( Ç
(42) a
AÝÇ . When Ç
is full rank, i.e., i 6
.U/ ,
the corresponding coding advantage is given by: C Wg 6@ Ã
j lm
X®n 3 n
Interestingly, except for the scaling before det
.U/
det Ç
(43)
Ç that incurs a loss in SNR, the diversity and
coding advantages take the same form as those for ST-BC. Therefore, the so-called rank and deÅ terminant criteria can be also applied to design the group in ST-DDC. Our main difference here is that we have to restrict our group code matrices to be diagonal and unitary. The designs of the unitary diagonal group for ST-DDC can be summarized in the following lemma [26, 36]: ë Lemma 1: For Ì 4K6 n 3 n 6 , every full-rank group of .0/IH¹.U/ diagonal unitary matrices with nÅ n 6
is equivalent to an Ì Ì
==y Ì
cyclic group, for some odd numbers a([a
. In particular, letting
QR
R R Î R ý
R
Î
R
476
S
the diagonal unitary group can be represented as: ý Ê
July 30, 2000
5
Î Ô
_ _
a
_
..
_
==
. Î
_
(44)
` a
ý Î
a a .0/ .U1 and Xg1 c X Ì j lm cG|h> Ì .U/
Using Lemma 1, it is easy to verify that IN 6 Ig 6
^_
==
Å 6
(45)
(46) DRAFT
WIRELESS SYSTEMS AND MOBILE COMPUTING
28
i
X
Wg
2
0.5
1
1
0.5
4
1
1
1, 3
0.25
8
1.5
1
3, 5
0.179
16
2
1
7,9
0.0732
Ì
TABLE I O PTIMAL
'¡ ¢7£}¤-¢¦¥§
DIAGONAL UNITARY GROUP
( ¨ª©v«¬ )
Thus, designing our diagonal unitary group codes is equivalent to choosing ®°¯²±7±7±%¯yj³ such that ´¶µ is maximized (see Table I for the optimal codes when ·¹¸»º½¼ ). We investigate the performance of ST-DDC by presenting two complementary examples: Example 1: To motivate the design of ST-DDC over ST-DC, we investigate the performance of DSTM in Section II-C.3 when the frequency error ¾G¿ does exist, but is not compensated for. In this example, we consider a system with ·À¸¶ºÁ¼ transmit antennas and a single receive antenna at SNR ºÃÂ7Ä dB. The simulation results are shown in Fig. 8(a) where we observe that the DSTM is very sensitive to frequency offset. When ¾¿Åº4Æ ± Æ Ç , the BER increases from ÂÆjÈÉ to Â7ÆÈÊ . When
¾¿¹ËÌÆ ±  , the system is useless because BER ËÍÆ ± Ç . For voice transmissions, the data rate is about ÂÆ kbps (κÏÂÆ ÈjÐ ). Thus, ¾G¿ÑºÌÆ ± Æ Ç is equivalent to ¾Ó¿ ÒXÔÕ Ö ¾Ó¿ ÒX×~Õ ºØÇ Æ Æ Hz which may happen in practice especially when the carrier frequency is up to GHz. Example 2: Recall that we assumed ¾ ¿ ÒX×6Õ to be common to all transmit antennas. To test robustness against cases where ¾ ¿ ÒX×~Õ ’s are different for different transmit antennas, we simulate our system with ·¹¸ÙºÍ¼ transmit antennas and a single receive antenna. The optimal Ú Û>Ü% ¯ ÂGÝ group codes are used. Let ÞžG¿ denote the difference between ¾¿ ’s for the two transmit antennas. We emphasize that our system is designed for a common ¾G¿ ; thus, only the Þ ¾G¿ should be taken into account. We check performance when ÞžG¿ºßÆ and when Þž¿ºßÆ ±  . Fig. 8(b) shows that Þž¿àºßÆ ±  increases the BER from ÂÆÈÊ to Çâá)ÂÆÈÊ at ÂÄ dB. Noting that ¾G¿ãºäÆ ±  led DSTM in Example 1 to an unacceptable performance loss, we appreciate the importance of ST-DDC.
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WIRELESS SYSTEMS AND MOBILE COMPUTING
29
SNR=16 dB, R=1.0 bit/sec/Hz
0
R=1
0
10
10
-1
10 -1
BER
BER
10
-2
10
-2
10
∆ fk = 0
-3
10
∆ fk=0.1
-4
-3
10
0
0.05
0.1 0.15 Frequency Error (fk )
0.2
0.25
10
0
2
4
6
8
å
10 E /N (dB) b
(a) DSTM
12
14
16
18
20
0
(b) ST-DDC
Fig. 8. Performance in the presence of Time-Selectivity
VI. C ONCLUSIONS In this paper, we summarized ST coding in the presence of various fading channel impairments that are recognized to constitute major limitations in performance and capacity of envisioned broadband wireless communication systems. In particular, we explained the effects of channel frequencyand time-selectivity on the design of ST codes, and we presented several representative examples together with simulations illustrating their performance. It was shown that ST coding is capable of improving system performance considerably without necessarily sacrificing precious bandwidth resources. In addition to superior performance, ST coding possesses low complexity at the receiverend, making it an attractive solution especially for down-link applications.
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30
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[2]
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[3]
H. B¨olcskei, R. W. Heath Jr., and A. J. Paulraj, “Blind channel identification and equalization in OFDM-based multi-antenna systems”, IEEE Trans. Signal Processing, 2000 (submitted).
[4]
W. Choi and J. M. Cioffi, “Multiple input/multiple output (MIMO) equalization for space-time block coding,” in Proc. of IEEE Pacific Rim Conference on Communications, Computers and Signal Processing, Victoria, Canada, Aug. 22-24, 1999, pp. 341 –344.
[5]
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[6]
ETSI Normalization Commitee, “Broadband radio access networks (BRAN); HIgh PErformance Radio Local Area Networks (HIPERLAN) Type 2; System Overview,” Norme ETSI, document ETR0230002, European Telecommunications Standards Institute, Sophia-Antipolis, Valbonne, France, 1999.
[7]
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[79] S. Zhou and G. B. Giannakis, “Finite-alphabet based channel estimation for OFDM and related multi-carrier systems,” IEEE Transactions on Communications, Feb. 2000 (submitted).
Zhiqiang Liu received the B. S. degree from the Department of Radio and Electronics, Peking University, China, in 1991 and the M. E. degree from the Institute of Electronics, Chinese Academy of Science, China, in 1994. From 1995 to 1997, he worked as a research scholar at the Department of Electrical Engineering, National University of Singapore. From 1997 to 1998, he was a research assistant with the University of Virginia. He is currently working toward the Ph. D. degree at the Department of Electrical and Computer Engineering, University of Minnesota, Twin Cities. His research interests include space-time coding, multiuser detection, multi-carrier, and blind channel estimation algorithms.
G. B. Giannakis received his Diploma in Electrical Engineering from the National Technical University of Athens, Greece, 1981. From September 1982 to July 1986 he was with the University of Southern California (USC), where he received his MSc. in Electrical Engineering, 1983, MSc. in Mathematics, 1986, and Ph.D. in Electrical Engineering, 1986. After lecturing for one year at USC, he joined the University of Virginia (UVA) in 1987, where he became a professor of Electrical Engineering in 1997, Graduate Committee Chair, and Director of the Communications, Controls, and Signal Processing Laboratory in 1998. He was awarded the School of Engineering and Applied Science Junior Faculty Research Award (UVA) in 1988, and the UVAEE Outstanding Faculty Teaching Award in 1992. Since January 1999 he has been with the University of Minnesota as a professor of Electrical and Computer Engineering. His general interests span the areas of communications and signal processing, estimation and detection theory, time-series analysis, and system identification – subjects on which he has published more than 100 journal papers. Specific areas of expertise have included (poly)spectral analysis, wavelets, cyclostationary, and non-Gaussian signal processing with applications to SAR, array and image processing. Current research topics focus on transmitter and receiver diversity techniques for equalization of single- and multi-user communication channels, mitigation of rapidly fading wireless channels, compensation of nonlinear amplifier effects, redundant precoding for block transmissions, multicarrier, and wide-band communication systems. G. B. Giannakis received the IEEE Signal Processing Society’s 1992 Paper Award in the Statistical Signal and Array Processing (SSAP) area, and co-authored the 1999 Best Paper Award by Young Author (M. K. Tsatsanis). He co-organized the 1993 IEEE Signal Processing Workshop on Higher-Order Statistics, the 1996 IEEE Workshop on Statistical Signal and Array Processing, and the first IEEE Signal Processing Workshop on Wireless Communications in 1997. He guest (co-)edited two special issues on high-order statistics (International Journal of Adaptive Control and Signal Processing, and the EURASIP journal Signal Processing), and the January 1997 special issue on signal processing for advanced communications (IEEE Transactions on Signal Processing). He has served as an Associate Editor for the IEEE Transactions on Signal Processing and the IEEE Signal Processing Letters, a secretary of the Signal Processing Conference Board, a member of the SP Publications board and a member and vice-chair of the SSAP Technical Committee. He now chairs the Signal Processing for Communications Technical Committee and serves as the Editor in Chief for the IEEE Signal Processing Letters. He is a Fellow of the IEEE, a member of the IEEE Fellows Election Committee and the European Association for Signal Processing.
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Bertrand Muquet was born in Lisieux, France, in 1973. He received the Ing. degree in electrical engineering ´ ´ from the Ecole Sup´erieure d’Electricit´ e (Sup´elec), Gif-sur-Yvette, France in 1996 and the teaching degree ´ ”Agr´egation” in Applied Physics from the Ecole Normale Sup´erieure de Cachan, Cachan, France, in 1997. He ´ is currently pursuing a PhD at the Motorola Research Center in Paris in cooperation with the Ecole Nationale Sup´erieure des T´el´ecommunications (ENST), Paris, France. His general research interests lie in the area of signal processing and digital communications with emphasis on multicarrier systems, blind channel estimation and equalization and iterative and turbo algorithms.
S. Zhou (student member) was born in Anhui, China, in 1974. He received his B.S. degree in 1995 and the M.Sc. degree in 1998, from the University of Science and Technology of China (USTC), all in electrical engineering and information science. He is now working towards the Ph.D. degree in the Department of Electrical and Computer Engineering at the University of Minnesota. His broad interests lie in the areas of statistical signal processing and communications, including transceiver optimization, blind channel estimation and equalization algorithms, multi-carrier, space-time signal processing, spread-spectrum systems.
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