Performance of Iterative MAP Receiver for MIMO-OFDM Channels with Anti-Gray Mapping S. Ahmed, T. Ratnarajah, M. Sellathurai, and C. F. N. Cowan, ECIT, Queen’s University Belfast Queen’s Road, Queen’s Island, Belfast BT3 9DT, UK Email:
[email protected]
Abstract— In recent years, due to the low complexity nature of the turbo processing and excellent bit-error-rate (BER) performance, designing turbo-like receivers for frequency-selective MIMO channels has been of great research interest. The performance gain in turbo decoders is due to an extrinsic information transfer (EXIT) process between the detection and the decoding stages as compare to a traditional system that treats these processes in isolation. However, the challenge faced with these iterative receivers is the understanding of their performance and convergence behaviour. In this paper, we study an iterative maximum a posteriori (MAP) receiver for MIMO orthogonal frequency division multiplexing (OFDM) channels and its convergence behaviour. We analyze the performance of the proposed transceiver system with Gray and anti-Gray mapping using EXIT chart and study the effects of various settings of transmit and receive antenna on the turbo cliff in BER performance. Index Terms— MIMO frequency-selective channels, iterative receiver, OFDM, MAP, anti-Gray and Gray modulation, EXIT chart
I. I NTRODUCTION
T
HE new generation of wireless communication systems are providing multimedia services that require very high data rates [1]. Communication theory suggests that high spectral efficiency can be achieved by using multiple antennas at both the transmitter and receiver ends, so-called MIMO systems [2], [3]. In this context, we may mention a strategy for MIMO flat fading channels which offers tremendous potential to increase the information capacity of single user wireless communication systems, namely, the Bell-Labs Layered Space-Time (BLAST) architecture, see [4] and references therein. For high data rate applications, such as high-speed downlink packet access (HSDPA), it might be necessary to utilize signals whose bandwidth exceeds the coherence bandwidth of the channel, and this brings the issue of frequencyselective channels. Designing practical low complexity equalizers for MIMO frequency-selective channels is a challenging task due to inter-symbol-interference (ISI) caused by multipath and co-antenna interference (CAI) caused by the signal received from number of transmit antennas. Several equalization techniques have been studied in the literature for MIMO frequency-selective channels. In [5], [6] time domain decision feedback equalizers (DFE) have been proposed. In these algorithms, hard decisions are input to the feedback filter due to which, in many cases the phenomenon of error propagation may occur which degrades the BER performance. The work in [7] proposed soft decision feedback equalization algorithms. Here, the computational complexity is depended on the number of transmit antenna and channel ISI length, which limits its use only in short channel ISI length environments. Low complexity frequency domain equalization (FDE) methods are also proposed to combat severe time dispersive channels [8], [9]. Moreover, it has been
This work was supported by the UK Engineering and Physical Sciences Research Council under grant number EP/D07827X/1.
978-1-4244-1645-5/08/$25.00 ©2008 IEEE
shown that the BER performance can be improved significantly by using practical low complexity near optimal iterative decoding [10]–[13]. In this paper, we study an iterative maximum a posteriori (MAP) receiver for MIMO frequency-selective channels. For ISI channels the computational complexity of MAP increases exponentially with the channel ISI length. In this work, to combat the ISI, orthogonal frequency division multiplexing (OFDM) is exploited, which simplifies the problem of detection in the presence of ISI and its computational complexity is independent of channel ISI length. It should be noted that the computational complexity of MAP detector can be further reduced by using sphere decoding methods, see [14]. The major challenge faced with iterative receivers is the understanding of their convergence behavior with iterations. To investigate the performance and convergence of an iterative system, EXIT charts were initially introduced by Stephan ten Brink as an approximate technique to predict the convergence of an iterative system, see [15], [16]. In an EXIT chart, transfer characteristics based on mutual information are used to describe the flow of extrinsic information through the constituent decoders of an iterative system. By knowing the EXIT characteristics of the component decoders one can easily predict the convergence behavior of an iterative system [16], [17]. Moreover, EXIT characteristics have analytical properties that have useful implications for designing codes and iterative processors. Here, to better understand the convergence behaviour of the proposed iterative receiver, we study the notion of EXIT characteristics. Using simulations, we derive the extrinsic information trajectory on the EXIT chart at various Eb /No ranges to confirm the convergence of the proposed MAP receiver along with the soft-output Viterbi decoder (SOVD). Moreover, we show that the proposed iterative receiver with anti-Gray mapping of bits on quadrature amplitude modulation (QAM) symbols offer better BER performance compared to Gray mapping of bits. This paper is organized as follows. In section II, the signal model is presented. In section III discussed the demapping of transmitted bits from the received samples. Section IV discussed the complexity of the demapper and in section V derived the EXIT chart. Simulation results are given in section VI, finally, concluding remarks are given in section VII. Notation: Bold upper case letters, X(k), and lower case letters, x(k), with indices respectively denote matrices and vectors having all the elements at frequency k, while without indices denote general matrices and vectors. Conjugate transposition of a matrix is denoted by (.)H . E{.} and .N denote respectively the statistical expectation and the modulo-N operation. II. S YSTEM M ODEL The basic baseband MIMO-OFDM transmission and reception model is given in Figure 1. Here, the signals are transmitted with nT and received by nR antennas. On each transmit antenna, first
719
IFFT & CP
x1 ( N − L + 1)
PTS
x1 ( N − 1)
r1 (0)
r1 (1) STP Discard r 1 ( N − 1) CP
H12 H nT1
s1 ( N − 1)
Tx n x nT ( N − L + 1)
IFFT & CP
STP Disard CP
H nT 2
xnT ( N − 1)
sˆ1 (0)
y1 ( 0) y1 (1)
FFT
k=0 MAP
y1 ( N − 1)
sˆnT (0)
Rx nR
T
snT (0)
Bit LLR
Rx1 H 11
PTS
sˆ1 ( N − 1)
y nR (0) y nR (1)
rnR ( 0) rnR (1) rnR ( N − 1)
FFT
y nR ( N − 1)
k = N-1 MAP sˆnT ( N − 1)
De-Interleaver SOVD Interleaver
Data
Convolutional Encoder Interleaver Mapping
Tx 1 s1 (0)
snT ( N − 1)
Bit LLR Fig. 1.
Basic baseband model of the transmitter, the channel and the iterative receiver (or transceiver).
of all a data block of N symbols is converted into the time domain by applying an inverse fast Fourier transform (IFFT) operation and a cyclic prefix (CP) of length L − 1 is appended at the beginning of the time samples. The whole block of data after adding CP is termed as an OFDM symbol. The block of symbols is transmitted serially with parallel-to-serial (PTS) converter. Similarly, at each receive antenna the first L−1 symbols are ignored in each received OFDM symbol. If the sampling rate at the receiver is equal to the symbol transmission rate then the received baseband signal at the receive antenna p at sample time n after removing the CP can be written as rp (n) =
nT L−1 t=1 l=0
htp (l)xt (n − lN ) + vp (n),
=
N −1 2π 1 √ st (k)ej N kn , N k=0
n = 0, 1, . . . , N − 1.
The N time domain samples of the sequence {xt (n)} can be written in vector form as xt = FH st . Here F is the unitary DFT matrix of dimension N ×N , xt = [xt (0)xt (1) · · · xt (N −1)]T and st = [st (0)st (1) · · · st (N − 1)]T . Note that each receive antenna signal not only experiences ISI due to multipaths but also CAI. The presence of CAI and ISI makes the equalization computationally expensive and degrades the BER performance. The use of OFDM can eliminate ISI due to multi-paths and convert the problem into a system with CAI only. The received signal vector in the frequency domain at antenna p can be written as yp =
nT
FHpt FH st + wp ,
y(k) = H(k)s(k) + w(k), k = 0, 1, . . . , N − 1,
p = 1, 2, . . . , nR ,
t=1
where Hpt is the channel convolution matrix between the receive antenna p and the transmit antenna t, yp = [yp (0) yp (1) · · · yp (N − 1)]T is the vector of frequency domain samples at antenna p and wp = [wp (0) wp (1) · · · wp (N − 1)]T is the vector of frequency domain noise samples. A low
(2)
where y(k)
(1)
where htp (l) is the unknown complex channel gain between the transmit antenna t and receive antenna p for the lth multi-path, xt (n) is the transmitted signal from the antenna t after applying the IFFT on samples {st (k)} and vp (n) is the noise at receive antenna p. At each receive antenna, the received serial data is again converted into blocks each of length N with serial-to-parallel (STP) converter. If {st (k)} are the frequency domain symbols to be transmitted by antenna t and {xt (n)} are the corresponding time domain samples after an IFFT operation then the relationship between them can be described by the following N -point discrete Fourier transform (DFT) operation xt (n)
complexity iterative receiver can be designed by collecting all the samples received at frequency k. Therefore, as shown in Figure 1, the collection of samples received at frequency k, from receive antennas l = 1, 2, . . . , nR , can be written in vector form as
[ y1 (k) y2 (k) · · · ynR (k) ⎡ H11 (k) H12 (k) · · · ⎢ H21 (k) H22 (k) · · · ⎢ ⎢ .. .. .. ⎣ . . . HnR 1 (k) HnR 2 (k) · · ·
=
H(k)
=
s(k)
=
[ s1 (k)
s2 (k)
···
]T , H1nT (k) H2nT (k) .. . HnR nT (k)
⎤ ⎥ ⎥ ⎥, ⎦
snT (k) ]T
and w(k)
=
[ w1 (k)
w2 (k)
···
wnR (k) ]T .
Here, Htr (k) is the frequency response of the channel between the transmit antenna antenna r at frequency k √ t and the receive −j 2πlk N . In (2), it can be noted h (l)e and Htr (k) = N L−1 tr l=0 that the transmitted symbols at frequency k from nT antennas contribute only in received signal at nR antenna at frequency k. Therefore, the transmitted symbols at frequency k can be easily estimated using the received signal collected from nR receive antennas at frequency k.
III. D E - MAPPING OF TRANSMITTED BITS In this section, we present a symbol-by-symbol MAP algorithm for MIMO-OFDM channel to detect the bits mapped on QAM symbols. In the received signal model in (2) the transmitted signal vector s(k) contains the transmitted symbols from antenna t = 1, 2, ..., nT at frequency k. The symbol st (k) can be a QAM symbol with M coded bits and one of possible 2M constellation points. If the transmitted symbol st (k) = map{bo , b1 , ..., bM −1 }|bm ∈{0, 1} and the signal is transmitted from nT antenna then s(k) is mapped with nT M bits and s(k) = map{bo , b1 , ..., bnT M −1 }. The a posteriori log-likelihood-ratio (LLR) of mapped bit bm can be written as
720
L(bm |y(k))
=
ln
Pr(bm = 1|y(k)) . Pr(bm = 0|y(k))
(3)
To simplify (3), we define I+m ⎛ 0 ⎜ 0 ⎜ ⎜ I−0 = ⎜ ... ⎜ ⎝ 0 0 ⎛ =
I+0
⎜ ⎜ ⎜ ⎜ ⎜ ⎝
and I−m for m = 0 as ⎞ 0 ··· 0 0 0 ··· 0 1 ⎟ ⎟ .. .. .. ⎟ . ··· . . ⎟ ⎟ 0 ··· 1 1 ⎠
1
···
0
1
1 0 .. . 0 1
··· ··· ··· ··· ···
1 0 .. . 1 0
(4) V. F ORMULATION OF EXIT CHART
nT M ×2nT M −1
1 0 .. . 0 0
demapping of the bits from the received signals, the number of multiplications required to find the LLR per transmitted bit per iteration are P (nT + 1)nR 2nT , where P = 2M is the number of possible constellation points in the modulation scheme.
Interleaving A1
1 1 .. . 1 1
⎟ ⎟ ⎟ ⎟. ⎟ ⎠
{yt(k)}
Pr(B/A)Pr(A) Pr(B)
we can write (3) as 2nT M −1 p=0
p(y(k)|map{i+mp })eLimp
, p(y(k)|map{i−mp })eLimp (6) where i+mp and i−mp are (p + 1)th column vectors of matrices I+m and I−m . Moreover, imp is the (p + 1)th column vector of matrix I+m but here mth entry is set equal to zero, finally L = [L(bo ) L(b1 ) ... L(bnT M −1 )] is a row vector of a posteriori LLRs. The second term in (6) is the extrinsic information of bit bm . If we denote the extrinsic information of mth bit by E(bm ) then we have 2nT M −1 p(y(k)|map{i+mp })eLimp p=0 . (7) E(bm ) = ln nT M −1 2 p(y(k)|map{i−mp })eLimp p=0 p=0
If we assume the noise at each receive antenna is Gaussian with variance σn2 , then the conditional probability density function (PDF) of y(k) can be written as −
p (y(k)|bm =i ) ≈ e
|y(k)−H(k)s(k)|2 2 σn
,
(8)
here, mth mapping bit in s(k) is i ∈ {0, 1}. Therefore, using expression (8) in (6) we can write L(bm |y(k)) = L(bm )+ 2nT M −1 − |y(k)−H(k)×map{i+mp }|2 /σn2 Limp e e p=0 ln n M −1 (9) 2 2 − |y−H(k)×map{i−mp }| /σn Limp 2 T e e p=0 |y(k) − H(k)map{i+mp }|2 ≈ L(bm ) − max + Li mp p σn2 |y(k) − H(k)map{i−mp }|2 (10) + Li +max mp p σn2 IV. C OMPUTATIONAL C OMPLEXITY OF D EMAPPER The use of IFFT at the transmitter and FFT at the receiver requires N log2 N multiplications. Therefore, if we have nT transmit and nR receive antenna then the overall computational complexity for FFT and IFFT operations will be N log2 N (nT + nR ). In
Π −1
+ L2
A2
SOVD
Estimated Data Bits
De-Interleaving
Fig. 2.
Here, it can be noted that for m = 0 the first row of matrix I−0 has all elements equal to 0, while the first row of matrix I+0 has all elements equal to 1. The other matrices for m = 1, 2, ..., nT M −1 can be found by exchanging the 1st row with the corresponding (m + 1)th row. Now, exploiting Bayes’ theorem [18]
L(bm |y(k)) = L(bm )+ln nT M −1 2
+
E1
(5)
nT M ×2nT M −1
Pr(A/B) =
MAP
L1
E2
Π
⎞
Demapping and decoding structure
The iterative demapping and decoding is shown in Figure 2. For each iteration the demapper takes in the frequency domain channel observations and the intrinsic information, A1 = L(bm ), on the mapped bit bm then using (9) finds the a posteriori LLR, L1 . The extrinsic information, E1 = L1 − A1 , is de-interleaved to become intrinsic information, A2 , for soft output Viterbi decoder (SOVD). The SOVD finds the a posteriori LLR, L2 , and calculates the extrinsic information E2 = L2 − A2 . The extrinsic information E2 is interleaved and fedback to become intrinsic information A1 for demapper. This constitute an iterative process with an information transfer between two decoders which is not easy to analyze or describe. One very useful tool is the EXIT chart pioneered by ten Brink [15]. To derive an EXIT chart, in the sequel we omit the subscript on A and E and suppose binary random variable B denotes the transmitted bits with realization bm ∈ {+1, −1}. Furthermore, define IE as the mutual information between the transmitted bits B and the extrinsic information E from the demapper/decoder and IA as the mutual information between the transmitted bits B and the a priori information A at the demapper/decoder. A transfer function relates IE to IA and Eb /No on the channel is given by IE = f (IA , Eb /No ).
(11)
The mutual information IA for uniformly distributed binary input symbols can be written as [18] ∞ 1 p(α|bm = b) IA = 2 b∈{+1,−1} −∞ 2p(α|bm = b) dα, × log2 i∈{+1,−1} p(α|bm = i) which on straight forward simplification yields 1 IA = 1 − E log2 1 + e−A(bm ) |bm =+1 2 1 (12) − E log2 1 + eA(bm ) |bm =−1 . 2 Where A(bm ) is the intrinsic information of the transmitted bit bm . For simulation A(bm ) can be modeled by adding white Gaussian noise, vm , of variance σn2 and 0 mean to the transmitted bit bm as rm = bm + vm .
(13)
The corresponding a priori LLR or intrinsic information of bm can be written as p(rm |bm = +1) (14) A(bm ) = ln p(rm |bm = −1) which can be simplified to
721
A(bm ) =
2 (bm + vm ) , σn2
(15)
which yields the distribution of A(bm ) ∼ N 2bm /σn2 , 4/σn2 . This tells us that if the noise has Gaussian PDF then the PDF of A(bm ) will be Gaussian with mean 2bm /σn2 and variance 4/σn2 . Therefore, intrinsic information can be model using (15), for example when σn2 = ∞ the intrinsic information A(bm ) = 0 and for σn2 = 0 the intrinsic information A(bm ) = ∞. Once, we have A(bm ), the mutual information IA can be found using (12). Similarly, the value of E(bm ) can be found by inputting the intrinsic information, A(bm ), into (7) and subsequently IE can be derived as ∞ 1 p(ξ|bm = b) IE = 2 b∈{+1,−1} −∞ 2p(ξ|bm = b) dξ × log2 i∈{+1,−1} p(ξ|bm = i) 1 = 1 − E log2 1 + e−E(bm ) |bm =+1 2 1 (16) − E log2 1 + eE(bm ) |bm =−1 . 2 In order to draw the EXIT chart for constituent de-mapper/decoder, first of all intrinsic information, A(bm ), is generated then corresponding mutual information, IA , is found. The intrinsic information is passed to the constituent decoder, which is working on some particular Eb/No value, to calculate the extrinsic information, E(bm ). By exploiting the extrinsic information the mutual information, IE is obtained. Finally, we draw a plot between IA and IE , this graph is called an EXIT chart. 0111 0101
1101
1111
0111 1100
1011
0010
0110
0100
1100
1110
1001
0000
0110
0101
0010
0000
1000
1010
0100
0011
1101
1010
0011
0001
1001
1011
0001
1110
1000
1111
characteristics of Gray mapping while solid lines (−) show the EXIT characteristics of anti-Gray mapping. In Figure 4, nT = nR = 2, while in Figure 5 nT = 2 and nR = 3. In both figures, it can be noted that the slopes of the curves are higher for anti-Gray and lower for Gray mapping. Also note that the slopes of the curves in Figures 4 and 5 demonstrate the potential improvements in performance by using an iterative demapping and decoding receiver when we use anti-Gray mapping. Moreover, by comparing both figures for anti-Gray mapping, it can be observed that increasing number of receive antenna from 2 to 3 not only increases the extrinsic information for a given Eb/No but increases the slope of extrinsic information. Figure 4 shows the BER performance of 16QAM for Gray, dotted-lines (−−), and anti-Gray, soild-lines (−), mapping for 2 transmit and 2 receive antennas. The interleaver length of 2560 bits is used after convolutional encoder. From the figure, it can be seen that at low Eb/No Gray mapping performs better than that of anti-Gray but there is no significant gain in BER performance after second iteration. The anti-Gray mapping at low Eb/No does not perform better than Gray; however, after the turbo cliff region at Eb/No = 10dB anti-Gray mapping not only improves the BER performance significantly but also yields significant gain in BER performance after each iteration. Figure 5 shows the BER performance of 16-QAM for 2 transmit and 3 receive antennas. Other parameters are kept same as in the previous simulation. It can be noted that increasing the number of receive antennas shift the turbo cliff region from Eb/No = 10dB to Eb/No = 8dB for anti-Gray mapping. For Gray mapping, increasing the number of receive antenna improve the BER performance but there is no effect on the BER performance after second iteration. 1 0.9 0.8 0.7 0.6 IE
b. Anti-Gray
a. Gray
Fig. 3. Gray and anti-Gray mapping of data bits on 16 − QAM constellation points.
0.5 0.4 0.3
VI. S IMULATION For the simulations, it is assumed that the MIMO channel is frequency-selective and block fading, i.e., it is time invariant within each frame of N + L symbol periods but changes independently from one frame to the next. We assume perfect knowledge of the channel and the noise variance at the receiver. For all simulations, the number of sub-carriers is N = 32 and the length of the CP is kept equal to the length of the channel. We use an 8-tap wireless fading channel model in which each channel tap is represented by a complex Gaussian random process independently generated and 2 2 = 1, where σltr is the variance of the lth path between the σltr transmit antenna t and receive antenna r. For encoding, a simple 1/2 rate convolution encoder of polynomial G = [5 7] is used. The benefits and performance trends of using Gray and anti-Gray mapping for various setting of transmit and receive antennas can be seen by exploiting EXIT charts. Gray and anti-Gray mapping used for the simulations are shown in Figure 3. Figure 4 and 5 show the EXIT chart of 16-QAM Gray and anti-Gray demapper at different Eb/No values for different settings of transmit and receive antennas. In the figures dotted lines (−−) show the EXIT
Eb/No=4dB Eb/No=8dB Eb/No=10dB Eb/No=12dB
0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5 I
0.6
0.7
0.8
0.9
1
A
Fig. 4. EXIT characteristics of demapper for Gray(−−) and anti-Gray (−) 16-QAM mapping at different Eb/No values. The number of transmit antenna is 2 and the number of receive antenna is 2.
VII. C ONCLUSIONS In this paper, we studied an iterative receiver for MIMOISI channel, which was based on maximum a posteriori (MAP) principle and OFDM modulation. For our analysis, a convolutional coded MIMO system in a block fading frequency-selective channel environment was considered. The EXIT function of demapper using monte-carlo simulations was derived at various Eb/No for 16-QAM schemes incorporating Gray and anti-Gray mappings. In
722
0
1
10
1st−Iter 2nd−Iter 3rd−Iter 4th−Iter 5th−Iter
0.9 −1
0.8
10
0.7 −2
0.6 I
E
BER
10
0.5
−3
0.4
10
0.3 Eb/No=4dB Eb/No=8dB Eb/No=10dB Eb/No=12dB
0.2
−4
10
0.1 −5
0 0
0.1
0.2
0.3
0.4
0.5 I
0.6
0.7
0.8
0.9
10
1
A
Fig. 5. EXIT characteristics of demapper for Gray(−−) and anti-Gray (−) 16-QAM mapping at different Eb/No values. The number of transmit antenna is 2 and the number of receive antenna is 3.
0
2
4
6
8
10 Eb/No
12
14
16
18
20
Fig. 7. BER performance of 16-QAM coded system for Gray (−) and antiGray (−−) mapping. The number of transmit antennas is 2, the number of receive antennas is 3 and the OFDM block length is 32.
0
10
1st−Iter 2nd−Iter 3rd−Iter 4th−Iter 5th−Iter
−1
10
−2
BER
10
−3
10
−4
10
−5
10
0
2
4
6
8
10 Eb/No
12
14
16
18
20
Fig. 6. BER performance of 16-QAM coded system for Gray (−) and antiGray (−−) mapping. The number of transmit antennas is 2, the number of receive antennas is 2 and the OFDM block length is 32.
particular, it has been shown that by using anti-Gray mapping in the frame work of iterative demapping and decoding, an excellent performance can be achieved with practical decoding complexity. R EFERENCES [1] S. Ohmori, Y. Yamao, and N. Nakajima, “The future generation of mobile communications based on broadband access technologies,” IEEE Commun. Magazines, vol. 38, pp. 134–149, Dec. 2000. [2] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans. on Telecommunications, pp. 585–595, Nov. 1999. [3] J. G. Foschini and M. G. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Communications, pp. 311–335, Mar. 1998. [4] M. Sellathurai and J. G. Foschini, “Stratified diagonal layered spacetime architecture: Signal processing and information theoretic aspects,” IEEE Trans. on Signal Processing, Nov. 2003, Vol. 51, pp. 2943–2954
[5] N. Al-Dhahir and A. H. Sayed, “The finite-length multi-input multioutput MMSE-DFE,” IEEE Trans. Signal Processing, pp. 2921 – 2936, Oct. 2000. [6] A. Lozano and C. Papadias, “Layered space-time receivers for frequency-selective wireless channels,” IEEE Trans. Commun., pp. 65–73, Jan. 2002. [7] S. Liu and Z. Tian, “Near-optimum soft decision equalization for frequency selective MIMO channels,” IEEE Trans. Signal Processing, pp. 721–733, Mar. 2004. [8] J. L. J.Cimini, “Analysis and simulation of a digital mobile channel using orthogonal frequency division multiplexing,” IEEE Trans Commun., pp. 665–675, Jul. 1985. [9] H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Communications Magazine, vol. 33, pp. 100–109, Feb. 1995. [10] J. Hagenauer and E. Offer and L. Papke, “Iterative decoding of binary block and convolutional codes”, IEEE Transaction on Information Theory, vol. 42, pp. 429–445, Mar. 1996. [11] B. Dong and X. Wang, “Sampling-based soft equalization for frequency-selective MIMO channels”, IEEE Trans. on Wireless Communications, vol. 53, pp. 278–288, Feb. 2005. [12] T. Ratnarajah and M. Sellathurai, “Iterative layered space-time transceiver for ISI wireless channels”, In Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing, Toulouse, France, May. 2006. [13] M. Sellathurai and S. Haykin, “TURBO-BLAST for wireless communications: theory and experiments”, IEEE Transaction on Signal Processing, vol. 50, pp. 2538–2546, Feb. 2002. [14] B. M. Hochwald and S. ten Brink, “Achieving Near-Capacity on a Multiple-Antenna Channel”, IEEE Trans. on Communications, vol. 51, pp. 389–398, Mar. 2003. [15] S. ten Brink, “Convergence of iterative decoding”, Electronics Letter, vol. 35, pp. 806–808, May 1999. [16] S. ten Brink, “Convergence behavior of iteratively decoded parrallel concatenated codes”, IEEE Transaction on Communications, vol. 49, pp. 1727–1737, Oct. 2001. [17] A. Ashikhmin and G. Kramer and S. ten Brink, “Extrinsic information transfer functions: Model and erasure channel properties”, IEEE Transaction on Information Theory, vol. 50, pp. 2657–2673, Nov. 2004. [18] T. M. Cover and J. A. Thomas, “Elements of Information Theory”, John Wiley and Sons, Inc. New York, 1991. [19] A. Boronka and J. Speidel, “A low complexity MIMO system based on BLAST and iterative anti-gray de-mapping”, IEEE Personnel, Indoor and Mobile Radio Communication Conference, vol. 50, pp. 1400–1404, Sep. 2003.
723
