approach for increasing data rate over wireless channels is to employ coding techniques appropriate for multiple transmit antennas, namely space-time coding.
Enhanced Underwater Acoustic Communication Performance Using Space-Time Coding and Processing Subhadeep Roy
Tolga Duman
Department of Electrical Engineering Arizona State University Tempe, AZ 85287-5706
Leo Ghazikhanian
Abstract— Recent advances in information theory and terrestrial wireless communication show that significant performance gains are achievable with increased signaling diversity through the use of multiple transmit and receive arrays. An effective approach for increasing data rate over wireless channels is to employ coding techniques appropriate for multiple transmit antennas, namely space-time coding. This paper investigates the feasibility and effectiveness of space-time trellis and layered space-time codes for the shallow-water, acoustic, frequencyselective channel. Using data collected during a recent experiment in the Mediterranean, we show that systems using multiple transmit and receive transducers outperform more conventional single-transmit single-receiver configurations.
I. I NTRODUCTION Recent information theoretic studies [1], [2] have shown that significant capacity improvement is possible with the use of multiple-input, multiple-output (MIMO) systems, as opposed to the more conventional single-input, single-output (SISO) systems. As an example, for a system employing transmit and receive antennas over a flat Rayleigh fading channel, the information theoretic capacity grows linearly with the minimum of and . This tremendous increase in capacity, which directly translates into a corresponding increase in the achievable data rate, motivates us to investigate the performance of MIMO systems, for real underwater acoustic (UWA), frequency selective fading channels. Typically, UWA channels are characterized by fast temporal variations and long multipath spreads that cause intersymbol interference (ISI). The optimal methods for detecting signals impaired by ISI are the maximum likelihood sequence estimation (MLSE) technique and the maximum a-posteriori probability (MAP) technique. The computational complexity of MLSE and MAP, however, grows exponentially with the channel memory. Moreover, for systems with multiple transmit antennas, this complexity is also exponential in the number of transmit antennas. The high complexity of systems with a large number of transmitters and a long ISI span renders the optimal detection practically infeasible, thereby requiring sub-optimal low-complexity techniques.
Vincent McDonald
Space and Naval Warfare Systems Center San Diego
Dr. John Proakis is an adjunct professor at the University of California, San Diego.
John Proakis
James Zeidler
Department of Electrical and Computer Engineering University of California, San Diego
In this paper, we consider the MIMO communication scenario for frequency selective, shallow-water, UWA channels by using space-time coding at the transmitter, and MIMO decision feedback equalization at the receiver. In particular, we consider the use the space-time trellis codes (STTC) [3] and layered space-time codes (LSTC) [4]. The receiver consists of a MIMO decision feedback equalizer (DFE) [5], which is an extension of the structure first proposed in [6] for SISO systems. The structure consists of an explicit phasetracking and timing-recovery loop for each link of the system (from each transmitter to each receiver), whose operation is jointly optimized with the equalizer coefficients. For the case of STTC, we propose a modification of the above structure, which facilitates joint equalization and decoding of STTC. The proposed structure strives to utilize the powerful trellis structure of the STTC in the equalization process, and reduce error propagation effects inherent in the DFE. For the case of layered space-time codes, we extend the structure proposed in [7] for MIMO systems and perform iterative (turbo) equalization. By successfully decoding several MIMO data sets obtained in the Mediterranean sea, we demonstrate: ( ) the feasibility of MIMO systems and space-time coding for shallow-water, UWA channels over a range of km to km and ( ) show that MIMO systems can achieve considerable performance improvement over SISO systems, both in terms of added signaling diversity and increased data rate. The rest of the paper is organized as follows: We describe the system model and the jointly optimized MIMO DFE structure in Section II. In Section III we describe the space-time trellis codes briefly and outline the proposed receiver structure with the embedded STTC decoder using the modified Viterbi algorithm. Section IV talks about MIMO communication using LSTC and also describes the extension of the iterative DFE structure [7] for MIMO systems. We present experimental results in Section V and conclusions in Section VI. II. MIMO-DFE A. System Model
WITH J OINTLY
O PTIMIZED PLL
Consider a MIMO system with transmit and receive antennas. The signal transmitted from the transmit antenna
$
%&
'
$#
!
Feed-forward filter bank for Transmitter 1
From DFE 2
From DFE
Feedback filter bank for Transmitter
$
%(
Feed-forward filter bank for Transmitter
')
$ #
" !
From DFE 1 From DFE 2
Feedback filter bank for Transmitter
Fig. 1.
MIMO DFE Structure.
banks, where the bank aims to equalize the data stream. Each bank consists of finite impulse response *,+-)./1032 4657+(- /8,-.:9 (1) (FIR) filters, one for each of the incoming received signal = Y? ? Y? streams. The B filter of the bank ( = ? where 5 + - / is the space-time coded A -ary symbol transmit- B ) is followed by a phase compensator (de-rotator), ted from the transmit antenna at time instant , 8-tc +WD7-
/
T
O+-
/ /3! % =
(15)
WITH
S PACE -T IME
A. Space-Time Trellis Codes for Flat Fading Channels
/
i +ID
/10
III. J OINTLY O PTIMIZED MIMO-DFE T RELLIS C ODES
where is the known transmitted symbol during the training mode and is replaced by 5J +(- / during the decision directed mode. The parameters for equalization of the stream are obtained [ by minimizing the mean squared error, MSE+ 0 ^ O + - / b . The parameters to be jointly optimized are the equalizer ] coefficient i vectors + and + , and the synchronization pa ? ? ? ? rameters, X +ID and iL +WD ( = B ). As proposed in [6], due to its fast convergence property, the RLS algorithm is used to estimate the equalizer tap weights. The RLS algorithm [8] is applied to the composite data vector + - / 0 ^ + - = diL / = - / b , which recursively estimates ] + b . the composite weight vector + 0_^ + = 9 The channel phase and the symbol timing parameters are estimated using a second-order gradient based algorithm. The i gradient of the MSE, w.r.t. X +ID and iL +WD are given by 5J+(-
and - / G and - / [ are the proportional and the integral gain constants respectively. A similar equation for the timing parameter can be obtained by forming the instantaneous gradient of the MSE w.r.t. iL +WD .
/, Maximum likelihood detection is then performed on 5 + - / , is made. and a hard decision on the transmitted symbol, The estimation error is defined as O+-
T
0 +WD7-
(9) /
where 0 +WDJi +WD , i.e., X
stream is then given by
/K9 +-
i +WD -
Space-time trellis codes are introduced by Tarokh et al. in 1998 [3]. These codes are described by a trellis structure; an example is shown in Fig. 2. The incoming symbol stream is first encoded using the trellis structure and the encoded stream is then distributed among the transmit antennas. The trellis is designed to provide the full spatial diversity advantage, . The example code shown in Fig. 2 is designed for transmit antennas using BPSK symbols and has 4 trellis states. B. MIMO-DFE with Embedded STTC Decoder for Frequency Selective Fading Channels The STTC described above provides full spatial diversity of for flat, Rayleigh fading channels. It has been argued in [9] that the above codes are capable of achieving a diversity order of at least , even in the presence of frequency selectivity. However, with proper code design, a larger diversity order can be obtained (due to the additional multipath diversity available). To investigate the STTC performance over UWA channels, we propose a receiver structure for joint equalization and decoding of the STTC. The proposed
(12) (13)
3
PSfrag replacements
)
$ !
section for Transmitter
section for Transmitter
$# !
section for Transmitter
Fig. 4.
$ !
-
= /10
2 +nFKG
Symbol mapper
57i +(-
/ 9M5J+-
= / [
Layered space-time encoder.
the entire block may not be necessary and tentative hard decisions can be made with a smaller delay). However, for a MIMO, UWA channel, which is characterized by rapid temporal variations, coupled with long ISI and severe CCI, this delay can be unacceptable. In other words, performing equalization and decoding separately can cause the DFE to suffer from excessive error propagation due to the unreliable decisions (obtained by a symbol-by-symbol slicer) being fed back to it during the equalization process. To overcome this problem, we propose a joint equalization/decoding strategy in which we feed back instantaneous, but reliable symbol decisions to the equalizer, by making use of the trellis structure of the STTC. The proposed algorithm works as follows: At every time instant , after the Viterbi algorithm computes the surviving branches, we compute the joint probability of each outgoing state , and all the equalizer outputs from time to time . This is similar to what is done in the BCJR algorithm [10]. 4 4 This joint probability is defined as 4 4 4 - / c:- 7 = G /e0 cK- G /`c:- G / (18)
Encoder
structure consists of a DFE with an embedded space-time trellis decoder, implemented using the Viterbi algorithm as shown in Fig. 3. However, the standard Viterbi algorithm produces symbol decisions with a certain delay, and hence is not suitable for use in the decision feedback process. Therefore, the Viterbi algorithm structure is modified to facilitate the generation of instantaneous (but tentative) symbol decisions, which are then fed back to the equalizer. Since these decisions are made using the powerful trellis structure of the code, they are much more reliable than the decisions made by a symbol-by-symbol slicer, and hence helps reduce the DFE error propagation effects. In the rest of this section, we outline the Viterbi algorithm for the sake of completeness, and then explain the proposed joint equalization/decoding algorithm in detail. i = =Na aNa>=
, denote the soft equalizer Let 5 + - />= 0 outputs at time instant , and 5 +(- = / denote the symbol transmitted from the transmit antenna, corresponding to the state transition of the trellis, at time . The branch 4 metric for each trellis branch is first computed as
Modified Viterbi
$# !
Symbol mapper
S/P
Information stream
Fig. 3. MIMO DFE with embedded STTC decoder using modified Viterbi algorithm.
4
Encoder
section for Transmitter 1
where i
^ 5 G - / = aNa a>= 5 i
4
0
^
4
G =Na aNa>=!
0 + and the vector is4 the vector at the equalizer output at - / is computed 4 time . The variable recursively as 4 -/ 0 c:- =! G / 4 G
2 0 2
c:-7=
0
s"
(16)
b
- / b
4
s 4
#
4
-
= = /
G / 4 k
G - /
(19)
4 where # - = /10 c:- =! / is the state transition probability for the transition $ % . The variable # -=/ is computed as 4 4 # - =/ 0 cK-7= / 4 0 cK-& 7= 4 /`c:- / )
i / [ , ' O7 ( c 9 2 [ 5J+- / 9 57+(- = + " + : F G +
* 4
The overall metric corresponding to each outgoing state at time is then computed recursively as 4 4 4 - / 0 ^ - =/ T G - / b (17) k s where f - / denotes the overall metric accumulated at state at time instant o . Thus, we have a candidate sequence (survivor) for each outgoing state, at every time instant, obtained by the above Viterbi algorithm. The hard decisions on the information symbols are then made by retracing the trellis from state once the entire block is processed, thereby introducing a delay of one code block (in practice, processing
4
c:-*
/
(20)
is the information symbol corresponding to the where * and * + [ is the variance of the residual ISI transition 4
)#%
!"
Feed-forward section for Transmitter 1
From DFE
!"
$# % &
From DFE
Feedback section for Transmitter
MAP
)#'
Feed-forward section for Transmitter
From DFE
!"
From DFE
MAP
# ' $(
!"
Feedback section for Transmitter
Fig. 5.
Iterative MIMO DFE Structure.
and AWGN for the 4 stream. Assuming that all the symbols are equally likely (c:-* / does not depend on a specific state transition), the proportionality 4 in (20) can be re-written as )
4 4 # - = /10+* O( c 9 5 i + - / 9M5 + - = / [ , = (21) 2 [ n + K F G
+*
At the transmitter, the incoming bit stream is spatially multiplexed across the transmit antennas, whereby each substream, so formed is independently encoded, interleaved and mapped into symbols before being transmitted over the channel as shown in Fig. 4. Since independent streams are transmitted from each transmit antenna, the system’s spectral efficiency grows linearly with the number of transmit antennas. At the receiver, each receive antenna observes a superposition of all the streams corrupted by AWGN. For a flat fading channel, each stream can be successively decoded by using layered successive interference cancellation and nulling techniques [4]. For a frequency selective channel, however, each stream also experiences ISI, in addition to the CCI from the other streams. It has been shown [12] that for a coded system, significant performance improvement can be obtained by performing iterative (turbo) equalization. In other words, the equalizer and the channel decoder exchange soft information in an iterative fashion, thereby reducing the bit error rate of the system gradually. Several low-complexity alternatives to the above algorithm have been proposed, including the iterative DFE structure in [7]. In the rest of this section, we outline the extension of the structure in [7] for MIMO systems. The iterative MIMO DFE block diagram is shown in Fig. 5. The equalizer outputs for each stream are de-interleaved, and given to the soft-input, soft-output channel decoder. The decoder for each substream computes the log-likelihood ratios (LLR) for the corresponding information sub-stream, and the extrinsic information for the coded bit stream. This extrinsic information is fed back to the DFE for the next iteration and this iterative procedure is repeated several times. At each iteration, the DFE combines the soft information provided by the decoder from the previous iteration with the equalizer
4
where * is a constant, independent of $ or . Using (19) and (21), the state probability is computed for every outgoing state at time , and the state with the highest value of is chosen for the instantaneous trellis traceback. In other words, the most likely state at time is computed 4 as 4
! 0
0
0
,.-/ 0,1 s ,.-/ 0,1 s ,.-/ 0,1 s
c:-
G4 /
c:- 7= G / 4 - /
(22)
4
The trellis is then read back from state ! at each time instant and the coded symbols, so obtained, are fed back to the feedback portion of the equalizer. Clearly, by making use of the trellis structure in the equalization process, the tentative hard decisions for the DFE are much more reliable than the symbol-by-symbol decisions made by the memoryless slicer, thereby reducing the effects of error propagation considerably. IV. J OINTLY O PTIMIZED MIMO-DFE WITH L AYERED S PACE -T IME C ODES Layered space-time codes were introduced by Foschini et al. [4], [11], for rich scattering, flat Rayleigh fading environments. Unlike STTC, which tries to exploit the full spatial diversity of the system, these codes aim at achieving the very high spectral efficiencies possible in a system with a large number of transmit antennas. 5
TABLE I R ESULTS FOR LOW
1 1 2
33 LSTC
LSTC
SISO
Test number 1 33 1
STTC
BAND SINGLE CARRIER ,
MIMO DATA SETS . R ANGE :
1 1 2
Information rate (bps) 1000 1000 2000
Packet length (sec) 4.8 4.8 4.8
No. of bits 4800 4800 9600
2
2
2000
4.8
9600
33
4
4
4000
1.2
4800
33
4
1
4000
1.2
4800
Iter. 1 2 3 4 5 6 Iter. 1 2 3 4 5 6
[
*
[ +
5i+ =
y?
?
(23)
where * + is the variance of the channel noise and the residual ISI, at the output of the equalizer (for the data stream). Let the extrinsic information provided by the channel decoder be ext -5J+/ . The overall likelihood of the transmitted symbols are formed as
overall
- 5i+ / T
-5 + /10
ext
-5 + /
K M FOR TEST 1,
No. of errors at the equalizer output 176 2 Transmitter : 3 Transmitter : 6 Transmitter : 0 Transmitter : 0 Tx Tx Tx 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Tx Tx Tx 389 151 523 156 0 406 21 0 153 7 0 4 1 0 0 0 0 0
*
K M FOR TEST 33.
No. of information bit errors 0 0 3
0 Tx
464 428 304 124 18 2 Tx
822 875 908 920 910 926
Tx 0 0 0 0 0 0 Tx 144 7 3 0 0 0
Tx 0 0 0 0 0 0 Tx 0 0 0 0 0 0
Tx 0 0 0 0 0 0 Tx 342 132 11 0 0 0
Tx
310 219 129 10 2 0 Tx
550 564 563 569 599 562
Total 310 219 129 10 2 0 Total 1036 703 577 569 599 562
Data was transmitted in bursts of 4 a seconds. Thus each burst contained 9600 symbols. The interleaver size for LSTC was )4 coded bits, which corresponds to a transmission length of a seconds. Ideally, for a MIMO system the power transmitted by each transmit antenna should be normalized, so as to keep the total power of the system constant. However, in this experiment each transmit antenna transmitted with the same power level as the SISO system. At the receiver, a -element receiver array was deployed at a depth of
meters, with a -m hydrophone spacing. Before processing the data, the impulse response at each receiver was observed, using linear frequency modulated (LFM) probes, which were transmitted prior to each data set. The hydrophones having nearly equal strength response to all the transmit elements were chosen for decoding. The decoding results for several data sets are tabulated in Table I. We can make several observations for comparison of the SISO and the MIMO data sets. Comparing SISO with
-transmitter STTC, we observe that both the data sets are decoded with zero or very few errors. However, the data rate achieved by STTC is twice that of SISO, indicating that, for similar error-rate performance, STTC is capable of transmitting twice as fast as SISO. On the other hand, if we compare the performance of STTC, with the uncoded SISO performance, i.e., the error-rate performance at the SISO equalizer output, we find that the data rates are the same for both (since we are observing the uncoded SISO performance, ) there is no rate penalty), but the error-rate for STTC ( is roughly two orders of magnitude lower than that of SISO ). Thus, for the same data rate, STTC is able to ( achieve much lower error-rate performance than SISO. Next, we compare SISO with the 4 -transmitter LSTC case. Looking at the error-rate performance of the 4 4 LSTC, we observe that as the number of turbo iterations increase, the
output of the current iteration, to form more reliable hard symbol estimates. The equalizer outputs are converted into corresponding LLR values by approximating their distribution to a normal distribution. Under this assumption, the equivalent LLR corresponding to the soft equalizer output can be computed as [7]: - 5 i + /10
(24)
-5 + /
and a hard decision on overall is made to form the symbols which are fed back to the DFE. V. E XPERIMENTAL R ESULTS In October 2003, a shallow-water experiment was conducted in the Mediterranean sea, off the coast of Elba island to demonstrate: (1) the feasibility of space-time coded MIMO UWA communication, and (2) to illustrate the performance of several space-time codes over real, UWA channels. The space-time codes used were the -transmitter STTC (Fig. 2), and the 4 -transmitter LSTC. Several SISO data sets were also transmitted for performance comparison. The SISO and the LSTC data sets were encoded with a rate convolutional ^ = b code, with a generator matrix of octal. The bandwidth a available was kHz. The center frequency was kHz and the symbol rate was + symbols per second. Both BPSK and QPSK modulation schemes were used; although, the results presented here are for the BPSK data sets only. 6
TABLE II
Interestingly, these initial results suggest that, even with such modest choice of transmission parameters (excess band width of , a rate code and BPSK modulation), the MIMO systems are capable of achieving better or similar spectral efficiencies than some of the existing SISO systems, for similar range and depth conditions. As an example, data rates of a kbps and a kbps, over a bandwidth of and kHz, respectively are reported in [13] for a code rate of approximately . Comparing the results of [13] with our results, we note that we achieve better spectral efficiencies with MIMO systems, even with BPSK modulation and lower code rates, thereby demonstrating that MIMO systems are very promising candidates for high data-rate, shallow-water UWA communication.
S PECTRAL EFFICIENCIES ACHIEVED BY VARIOUS MIMO SYSTEMS . Modulation Spectral Prob. of method efficiency (bps/Hz) bit error SISO BPSK
STTC BPSK * LSTC BPSK SISO [13] QPSK NA
error-rate decreases gradually, ultimately clearing the frame in the iteration. Interestingly, we note that the error-rate per formance is dominated by transmitter . This effect becomes more prominent for the 4 LSTC case where we observe that transmitter failed to converge. This phenomenon was also observed for several other data sets, wherein the link from transmitter to all the receivers exhibited considerably lower average SNR than the other links, which may be due to the positioning of that particular transmitter in the water column or a malfunctioning transmitter. Since LSTC does not have transmit diversity and has to rely on receive and temporal diversity (coding), the lack of sufficient receive diversity caused the receiver to fail for the 4 case. Nevertheless, with sufficient number of receive antennas (the 4 4 case), the turbo equalization process is able to achieve error free performance (with an increased complexity), and achieve a data rate 4 times that of SISO. In order to get an intuitive idea of the order of magnitude of the error-rates, let us compare the uncoded performance of SISO and LSTC. Note that, to observe the uncoded LSTC performance, we should look at the number of equalizer errors, only after the first iteration (since subsequent iterations utilize feedback from the decoder). For reasons described above, if we disregard transmitter (a 4 system), we find that the SISO system has an uncoded error-rate of
, whereas, the uncoded error-rate of LSTC is , (since each transmitter transmits 4 uncoded symbols in a
seconds, and we are considering only transmitters) with LSTC transmitting three times as fast as SISO. Thus, we can conclude that for comparable error rates, LSTC can achieve an fold higher spectral efficiency than SISO. Finally, we note that the primary experimental aim was to demonstrate the feasibility of MIMO systems for shallowwater, UWA channels, as opposed to achieving very high spectral efficiencies. Consequently, the design of the data sets and the transmission parameters were carried out rather conservatively. For instance, the shaping pulse at the transmitter had an excess bandwidth of , which immediately reduces the spectral efficiency by a factor of . Similarly the symbol rates and the code rates were chosen to be somewhat lower than what one would normally implement. However, we observe that, even with such non-aggressive data sets, the spectral efficiencies achieved by the MIMO systems are significant. Table II lists the spectral efficiencies (bps/Hz) achieved by SISO, STTC and LSTC, with BPSK modulation (probability of error implies that no errors were observed in a block of 9600 symbols).
VI. C ONCLUSIONS
AND
F UTURE W ORK
We have studied the performance of space-time coding, for MIMO systems over real, frequency selective, underwater acoustic channels. We have extended the phase coherent receiver structure proposed in [6], for MIMO systems and have proposed a new receiver structure for joint equalization and decoding of space-time trellis coded signals. We have also extended the iterative DFE structure proposed in [7] for MIMO systems. By successfully decoding several spacetime coded data sets, namely the space-time trellis codes and the layered space-time codes, we have demonstrated the feasibility of MIMO systems over UWA channels. We have also shown that MIMO systems can be successfully used to achieve considerable performance improvement over the more conventional SISO systems, both in terms of added signaling diversity and improved data rate. SSC San Diego and the Center for Ocean Research (SAIC) will be constructing a 10-element, 9 + kHz transmit array and a 32-element receive array, respectively. This MIMO system will be demonstrated and used during the summer 2005 High Frequency Initiative Experiment off the coast of Kauai Hawaii. Several iterative receiver structures will be developed, including the layered successive interference cancellation based receiver and the channel shortening prefilter based receiver, and their performance will be evaluated using the above system. Finally, in order to make the realtime implementation of the system practically feasible, lowcomplexity versions of the above algorithms will be developed. ACKNOWLEDGEMENT This research was sponsored by the ILIR-04 program at Space and Naval Warfare Systems Center, San Diego. The experimental work was conducted within the ElbaEx Experiment as part of the ONR-supported High Frequency Initiative, which is a joint research effort with the NATO Undersea Research Center. Appreciation is expressed to the Captain and Crew of the R/V Alliance, to scientific and technical personnel from NURC, in particular to Chief Scientists Finn Jensen (first half) and Mark Stevenson (second half). 7
Lastly, we would like to thank Dan Kilfoyle (SAIC) and Lee Frietag (Woods Hole Oceanographic Institute) for including our waveforms as part of their MIMO testing during the ElbaEx Experiment. R EFERENCES [1] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Communications, vol. 6, pp. 311–335, March 1998. [2] I. E. Telatar, “Capacity of a multi-antenna Gaussian channels,” European Transactions on Telecommunications, vol. 10, pp. 585–595, November/December 1999. [3] V. Tarokh, N. Seshadri, and A. Calderbank, “Space-time codes for high data rate wireless communications: Performance criterion and code construction,” IEEE Transactions on Information Theory, vol. 44, pp. 744–765, March 1998. [4] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela, “V-BLAST: An architecture for realizing very high data rates over the rich-scattering wireless channel,” URSI International Symposium on Signals, Systems, and Electronics, pp. 295–300, 29 Sept. - 2 Oct. 1998. [5] A. Lozano and C. Papadias, “Layered space-time receivers for frequency-selective wireless channels,” IEEE Transactions on Communications, vol. 50, pp. 65–73, January 2002. [6] M. Stojanovic, J. Catipovic, and J. Proakis, “Phase coherent digital communications for underwater acoustic channels,” IEEE Journal of Oceanic Engineering, vol. 19, pp. 100–111, January 1994. [7] M. Marandin, M. Salehi, J. Proakis, and F. Blackmon, “Iterative decision-feedback equalizer for time-dispersive channels,” Proceedings of the 2001 Conference on Information Sciences and Systems, Baltimore, MD, pp. 225–229, March 2001. [8] S. Haykin, Adaptive Filter Theory. Englewood Cliffs, NJ; Prentice Hall, 1986. [9] V. Tarokh, A. Naguib, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criteria in the presence of channel estimation errors, mobility, and multiple paths,” IEEE Transactions on Communications, vol. 47, pp. 199–207, February 1999. [10] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Transactions on Information Theory, vol. 20, pp. 284–287, March 1974. [11] G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Technical Journal, Autumn 1996. [12] C. Douillard et al., “Iterative correction of intersymbol interference: Turbo equalization,” European Transactions on Telecommunication, vol. 6, pp. 507–511, September-October 1995. [13] L. Freitag, M. Grund, S. Singh, S. Smith, R. Christenson, L. Marquis, and J. Captipovic, “A bidirectional coherent acoustic communication system for underwater vehicles,” IEEE Oceans Conference Proceedings, vol. 1, pp. 482–486, September-October 1998.
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