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The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

RECURSIVE LMMSE-BASED EQUALIZATION IN WIRELESS MOBILE CODED OFDM Daniel N. Liu Department of Electrical Engineering University of California Los Angeles Los Angeles, CA 90095, U.S.A. A BSTRACT Orthogonal frequency division multiplexing (OFDM) system suffers extra performance degradation in fast fading channels due to intercarrier interference (ICI). Combining frequency domain equalization and bit-interleaved coded modulation (BICM), the iterative receiver is able to harvest both temporal and frequency diversity. Using the fact that ICI energy is clustered in adjacent subcarriers, frequency domain equalization is made localized. A computational efficient recursive q-tap SIC-LMMSE equalizer is derived. Comparing to conventional method which requires matrix inversion using O(32 · q 2 K 2 ), where K is number of subcarriers, the proposed recursive equalizer only uses O(20 · qK 2 ) but without sacrificing performance. I.

I NTRODUCTION

Over the past decade, there has been tremendous effort to improve the performance and robustness of orthogonal frequency division multiplexing (OFDM) system in wireless mobile environment. OFDM transmission enjoys a simple one-tap equalization in time-invariant channels due to the fact that the Fourier basis forms an orthogonal eigenbasis for the channels. Provided that guard interval (GI) is inserted to the beginning of each OFDM symbol to eliminate intersymbol-interference (ISI), orthogonality among the subcarriers ensures inexpensive hardware implementations and makes OFDM the prominent candidate for high data rates applications such as: digital video broadcasting (DVB) [1], wireless local area networks (WLAN) [2] and worldwide interoperability for microwave access (WiMAX) [3]. However, OFDM transmission in a wireless mobile environment with rapid channel variation over each symbol period severely corrupts the orthogonality among each subcarrier and give rises to intercarrier interference (ICI) [4–7] (and the references therein). The information theoretic studies of ICI bounds [4–7] spawned two lines of work: one which considered ICI suppression in the un-coded system [7,8] and one which considered ICI suppression in the coded system [9–11]. This work is concerned with the latter research paradigm where suppressing ICI in the coded system. The optimal receiver calls for maximum-likelihood (ML) joint equalization and decoding using a hypertrellis which is constructed by taking into account of both outer channel code and the ICI channel structure. This is clearly computational infeasible. Owning to the concept of iterative turbo processing [12], a practical decoding strategy performs iterative processing between two separate entities: front-end soft-in softout (SISO) equalizer and outer channel decoder. Huang et c 1-4244-1144-0/07/$25.00°2007 IEEE

Michael P. Fitz Department of Electrical Engineering University of California Los Angeles Los Angeles, CA 90095, U.S.A. al. [13] proposed a reduced complexity ML equalizer which works for relative low normalized Doppler frequency by considering only the main tap. Kim and Pottie [10] presents an one-shot q-tap linear minimum mean square error (LMMSE) equalizer with a (2q + 1) × K observation window, where K is the total number of subcarriers. While the iterative counterpart Soft Interference Cancellation-LMMSE (SIC-LMMSE) equalizer can be found in [11] but with overwhelming complexity due to matrix inversion. In this paper, following the footsteps of Cai and Giannakis [7], recursive algorithms to update LMMSE equalizer coefficients for q-tap SIC-LMMSE equalizer is derived which further reduce complexity without compromising performance. Based on the fact that most of a symbol’s energy is distributed over neighboring subcarriers, a low complexity recursive q-tap SIC-LMMSE equalizer is derived. A direct computation of the LMMSE equalizer coefficient bears the complexity of O(K 3 ) by inverting a K × K channel matrix H has justified a localized LMMSE equalizer implementation [7, 10]. As shown in [7], the q-tap equalizer coefficient can be computed recursively by traversing the diagonal of H with a (2q + 1) × K observation window. Also incorporating the a priori information feed back from outer channel decoder, the q-tap SIC-LMMSE equalizer is really just a generalization of q-tap LMMSE equalizer. In this paper, it’s shown that the q-tap SIC-LMMSE equalizer can also be updated recursively as done in the un-coded system. In Section II. system model and notations are introduced. Section III. derives the low complexity recursive q-tap SICLMMSE equalizer analyzes the computational complexity of proposed algorithm. Section IV. shows numerical example about deployment of proposed algorithms over WiMAX system [3]. Section V. concludes the paper. II.

S YSTEM M ODEL

Fig. 1 depicts a single-input single-ouput coded OFDM system. A set of K-coded QAM “frequency-domain” symbols T d = [d(1) . . . d(K)] forms the input to IFFT. This paper further assumes the average symbol energy Es ≡ E|d(k)|2 = 1 and symbols are equally likely chosen from a complex constellation D with cardinality |D| = 2Mc . The time domain waveform at the output of IFFT x(t) is then given by K 2π 1 X d(k)ej Ts kt , x(t) = √ Ts k=1

−Tg ≤ t ≤ Ts

(1)

where Ts , and Tg are the OFDM symbol duration, and guard interval length, respectively. Thus, one OFDM symbol block

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

x(t)

GI insertion

IFFT

d

Channel h(t, τ )

mapper µ

π(LA )

y(t) AWGN

˜ d

GI removal

FFT

Equalizer

c

π

π −1

b

LA

π

LE

BCC encoder

π −1 (LE )

SISO BCC decoder

˜ b

n(t) Figure 1: A baseband equivalent model for coded OFDM system

in time is Tb = Tg + Ts . This paper assumes a time-varying (TV) wireless multipath channel with a impulse response: mp −1

X

h(t, τ ) =

h(t, τn )δ(τ − τn ),

(2)

n=0

where τ0 ≤ τ1 ≤ · · · ≤ τmp −1 with τn being the tap-delay on nth tap, and h(t, τn ) is the randomly time-varying tap gains: h(t, τn ) = αn (t)ejθn (t) , repectively. Moreover, h(t, τn ) is modeled as wide sense stationary uncorrelated-scattering (WSSUS) channel and the tap gains {h(t, τn )} are complex GausPmp −1 2 sian with zero mean and variance σn2 , where n=0 σn = 1. The autocorrelation of the WSSUS channel is E[h(t, τ1 )h(t + ∆t, τ2 )† ] = Rh (∆t)φτ (τ )δ(τ1 − τ2 ),

(3)

with the assumption that the angle of arrival of the received signal waveform is a uniformly distributed random variable, E[h(t, τ1 )h(t + ∆t, τ2 )† ] is separable in time and delay. In (3), Rh (∆t) is the time-correlation function and φτ (τ ) refers to as the delay power spectrum with σn2 ≡ φτ (τn ). The received waveform y(t) depends on x(t) via, mp −1

y(t) =

X

h(t, τn )x(t − τn ) + n(t)

(4)

and n ˜ m is a complex Gaussian random variables with zero meanPand variance N0 /2 per dimension. One can easily see that k6=m hm,k d(k) = 0 if h(t, τ ) remains constant over Ts . On the other hand, ICI is arised when h(t, τ ) is varying. In a mobile OFDM environment over a TV channel, ICI can be characterized by normalized Doppler frequency fd Ts , where fd is the maximum Doppler frequency. Intuitively, one can think of the normalized Doppler frequency as a maximum cycle change of the TV channel per OFDM symbol in a statistical sense. In [4–7], the explicit mathematical expression for ICI power is derived. By further assuming RH (∆t) = J0 (2πfd ∆t) where J0 (·) is the zeroth-order Bessel function of first kind, the normalized ICI power of subcarrier m caused by subcarrier k is [7] 1 γm,k ≡ 2 π ·

Z

1

Sn (f )sin2 (πfd Ts f )×

0

¸ 1 1 + df, (fd Ts f + (m − k))2 (fd Ts f − (m − k))2 (8)

where Sn (f ) is the normalized Doppler spectrum, and total ICI power for subcarrier m is

n=0

where n(t) is AWGN. To recover the original frequency domain message on mth subcarrier, FFT is performed Z Ts 2π 1 ˜ d(m) = √ y(t)e−j Ts mt dt. (5) Ts 0 Clearly, (5) can be rewritten as ˜ d(m) = hm,m d(m) +

X k6=m

|

hm,k d(k) +˜ nm {z

(6)

}

ICI

where hm,k is the (m, k)th element of H ∈ CK×K and defined as mp −1 −j 2π kτn Z Ts X e Ts 2π h(t, τn )e−j Ts (m−k)t dt, (7) hm,k ≡ Ts 0 n=0

γm =

K X

γm,k .

(9)

k=1,k6=m

Examining (8) and (9) reveals the symbol energy leakage caused by ICI is highly concentrated in the neighboring subcarriers. Fig. 2 demonstrates the distribution of normalized ICI power as a function of frequency spacing (i.e. ∆f = |m − k|/Ts ) for different values of normalized Doppler frequency fd Ts . As a result of increased fd Ts , more and more symbol energy leak into it’s neighboring subcarriers contributes to the increment of normalized ICI power level. It is evident from Fig. 2 that most of the symbol energy concentrate around the neighborhood of the desired subcarrier. For example, more than 98% of the symbol energy cluster in the neighboring 5 subcarriers for fd Ts = 13.64%. Collecting (6) into matrix notation, we arrived at ˜ = Hd + n ˜. d (10)

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

column of the channel matrix Hm ∈ C(2q+1)×K respectively. Notice that Hm is certainly a sub-matrix of H in (7) and defined as   hm−q,1 · · · hm−q,K   .. .. ..   . . .   . h · · · h Hm =  (13) m,1 m,K     .. .. ..   . . . hm+q,1 · · · hm+q,K

0 fdTs = 2.27% fdTs = 4.55% fdTs = 6.82% f T = 13.64% d s fdTs = 40.92%

Normalized ICI Power, dB

−10

−20

−30

−40

−50

The optimal LMMSE equalizer coefficient wm is obtained by solving (11) and is given by µ ¶−1 N0 wm = I2q+1 + Hm ∆m H†m hm , (14) Es

−60

−70

−80 −150

−100

−50

0

50

100

150

Frequency Spacing

Figure 2: Normalized ICI power distribution with K = 256 Approaching ML performance with reasonable complexity relies on iterative processing between equalization and decoding. Analogous to a turbo decoder, the inner soft-in soft-out (SISO) equalizer and outer channel decoder can be regarded as two elementary “decoders” [14] in a serial concatenation archi˜ tecture. Log-likelihood ratio (LLR) such as: LA (cl ), LD (cl |d) and LE (cl ) can be computed efficiently as shown in [15, 16]. III.

I TERATIVE E QUALIZATION AND D ECODING

A. Low Complexity Recursive q-tap SIC-LMMSE Equalizer Exploiting the fact that most of the ICI caused symbol energy leakage is only distributed over the neighboring subcarriers, the SIC-LMMSE equalizer requires a filter length much less than the number of total subcarriers K. In general, the SIC-LMMSE equalizer/detector consists of three distinct stages of processing. This three three stages include: soft interference cancellation, LMMSE equalizing/filter and LLR computation respectively. This paper focuses on the development of LMMSE equalizer stage and leaves details of other two stages to references [15, 16]. The q-tap SIC-LMMSE equalizer only takes observation of ±q subcarriers around the desired subcarrier m. Given the observation model, the LMMSE equalization process is precisely stated in the following optimization problem: minimize subject to

2 ˆ E|d(m) − d(m)| † ˆ d(m) = w ~dm ,

(11)

2 and σd(k) , k = 1, 2, . . . , K with k 6= m, is the transmit symbol variance and generally can be computed as, X 2 ¯ 2 P [d(k) = d]. σd(k) = |d − d(k)| (16) d∈D

In (16), P [d(k) = d] is a priori symbol probability can be calculated from a priori information fed back from outer channel decoder [16]. There is an interesting way to recursive update the q-tap LMMSE filter wm in (14). In (13), it is crucial to observe that the channel matrix Hm is only different from the previous one Hm−1 by last row. That means: Hm−1 = £ ¤† £ † ¤† ¯† ¯ and Hm = vm−1 H H vm . Thus, m−1 m−1 there ought to be a way to recursive update the current equalizer coefficients wm from the previous one wm−1 . The recursive update relationship between wm−1 and wm hinges upon rewriting wm as µ ¶−1 N0 † wm = I2q+1 + Hm (∆m−1 + Ψm ) Hm hm Es  −1 N   0  =  I2q+1 + Hm ∆m−1 H†m +Hm Ψm H†m   Es  | {z }

K X

à ˜ −1 + R m

= |

¯ d(k)h k

hm

˜ −1 R m

(12)

k=1,k6=m

h

(15)

m

where ~dm ∈ C(2q+1)×1 represents the channel observation after “parallel” soft interference cancellation. Thus, ~dm linearly ˜ m ∈ C(2q+1)×1 via depends on d ~dm = d ˜m −

where the covariance matrix ∆m is " 2 # 2 2 2 σd(1) σd(m−1) σd(m+1) σd(K) ∆m = diag ,··· , , 1, ,··· , , Es Es Es Es

iT ˜ m = d(m ˜ − q), . . . , d(m ˜ + q) , d(k) ¯ is the symbol where d mean calculated from a priori information and hm is the mth

m X

(17)

!−1 ψk,k hk h†k

k=m−1

{z

hm

(18)

}

Rm

· σ2 where Ψm = diag 0, . . . , 0, d(m−1) − 1, 1 − Es

¸

2 σd(m) Es , 0, . . . , 0

Examining (18) reveals that R−1 m is a series of rank-one mod˜ −1 . Applying the degenerate matrix inversion ifications to R m

.

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

lemma, it has been shown that Rm can be obtained through ˜ m [16] via the following relationship: recursive updates from R

(i−1) Rm −

³ ´³ ´† ψi,i ³ ´ R(i−1) hi R(i−1) hi , m m (i−1) 1 + ψi,i h†i Rm hi (19)

(0) ˜ m . Thus, wm can be recursively with initialization of Rm = R ˜ m . Obviously, now the question becomes: calculated from R ˜ m from Rm−1 ? Since R−1 , which is defined how to obtain R m−1 as N0 R−1 I2q+1 + Hm−1 ∆m−1 H†m−1 (20) m−1 = Es

˜ −1 by the first row and first column, there only differs from R m ˜ m from Rm−1 recursively as should be a way to compute R well. With this observation, it is useful to partition the Hermitian matrix R−1 m−1 into · R−1 m−1

=

θm−1 bm−1

b†m−1 Am−1

¸

3500

,

3000

where Am−1 ∈ C2q×2q , bm−1 ∈ C2q×1 and θm−1 is scalar ˜ −1 is partitioned into respectively. Similarly, R m · ¸ ˜m Am−1 b −1 ˜ Rm = , (22) ˜† b θ˜m m ˜m = H ¯ m−1 ∆m−1 vm and θ˜m = v† ∆m−1 vm + with b m Moreover, let the inverse of R−1 be partitioned the same way m−1 as in (21) · ¸ p p†m−1 Rm−1 = , (23) pm−1 Pm−1 N0 Es .

2q×2q

pm−1 p†m−1 . p

(24)

˜ m is the just inverse of R ˜ −1 by To this end, it is clear that R m definition and can be directly constructed from (22) · −1 ¸ † ˜ m = Am−1 + ηm um um ηm um , R (25) ηm u†m ηm ³ ´−1 ˜ m and ηm = θ˜m − b ˜ † A−1 b ˜m with um = −A−1 b m m−1 m−1 ˜ m can be computed from Rm−1 respectively. Therefore, R through A−1 m−1 as shown in (25). Table 1 summarizes the recursive q-tap SIC-LMMSE equalizer algorithm. Being able to recursively update the LMMSE filter coefficients, the computational complexity for q-tap SIC-LMMSE equalizer remains manageable. Retaining only the dominant terms, the computational complexity of recursive q-tap SICLMMSE equalizer is shown in right column of Table 1 in terms

2500

2000

1500

1000

500

0

2q×1

where Pm−1 ∈ C , pm−1 ∈ C and p is a scalar. It can be shown that the inverse of Am−1 is the Schur complement of p in Rm−1 [7, 17]. That is: A−1 m−1 = Pm−1 −

5−tap SIC−LMMSE Equalizer with Matrix Inversion Recursive 5−tap SIC−LMMSE Equalizer 10−tap SIC−LMMSE Equalizer with Matrix Inversion Recursive 10−tap SIC−LMMSE Equalizer

(21)

MegaFLOPs/Second

R(i) m =

Table 1: Recursive q-tap SIC-LMMSE Equalizer Recursive Algorithm Computational Complexity 1. Form R−1 as in (20). ∼ 16q 2 KRA + 16q 2 KRM m−1 2. Compute Rm−1 by directly inverting (20). ∼ 48q 3 RA + 48q 3 RM 3. Partition Rm−1 as shown in (23). 0 4. Calculate the Schur complement of p in Rm−1 as in (24). ∼ 16q 2 RA + 24q 2 RM ˜ m and θ˜m 5. Compute b as in (22). ∼ 8qKRA + 12qKRM ˜ 6. Construct Rm directly ˜ through A−1 m−1 , bm ˜ and θm as in (25). ∼ 48q 2 RA + 56q 2 RM 7. Obtain Rm by recursive ˜ m as in (19). update on R ∼ 64q 2 RA + 80q 2 RM

0

200

400

600

800

1000

1200

K, Number of subcarrier

Figure 3: Complexity of proposed Recursive q-tap SICLMMSE Equalizer vs. number of subcarrier of real-number addition (RA) and real-number multiplication (RM). Realizing steps 1 and 2 in Table 1 only need to compute once for initialization, the recursive algorithm cycles through step 3 to 7 updating equalizer coefficients from one to another. Instead performing matrix inversion for each subcarrier k which bears a computational complexity of O(32 · q 2 K 2 ), the recursive q-tap SIC-LMMSE equalizer only spends O(20 · qK 2 ) without compromising performance. B.

Complexity Analysis

To demonstrate the effectiveness of the proposed equalizers, the actual number of floating-point operations (FLOP) per one OFDM block is measured and plotted. Fig. 3 illustrates the complexities of varies proposed front-end equalizers in terms of MegaFLOPs per second versus different number of frequency subcarriers. In particular, at K = 512, recursive 10-tap SIC-LMMSE equalizer only spends 67.2 MegaFLOPS

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

V.

0

10

MMSE w/ 1 tap, Un−Coded QPSK:[7] DFE + PIC w/ 1 tap, Un−Coded QPSK:[7] MMSE w/ 5 tap, Un−Coded QPSK:[7] DFE + PIC w/ 5 tap, Un−Coded QPSK:[7] MMSE w/ 25 tap, Un−Coded QPSK:[7] DFE + PIC w/ 25 tap, Un−Coded QPSK:[7] no ICI, Coded OFDM 16QAM w/ SO RECURSIVE 5−tap SIC−LMMSE, 1 Iter RECURSIVE 5−tap SIC−LMMSE, 2 Iter RECURSIVE 5−tap SIC−LMMSE, 3 Iter RECURSIVE 5−tap SIC−LMMSE, 4 Iter

−1

10

−2

BER

10

C ONCLUSION

Iterative equalization and decoding of the wireless mobile coded OFDM system is considered. A recursive q-tap SICLMMSE equalizer is derived which provides extra computational complexity reduction without sacrifice performance. The performance of the proposed equalizer has been evaluated over the COST-TU channel model with application to WiMAX. The results show that the iterative processing allows full exploration of both temporal and frequency diversities available in the system.

−3

10

−4

10

R EFERENCES [1] ETSI EN 300 744 v1.5.1 (2004-2006), “Digital Video Broadcasting (DVB); Framing structure, channel coding and modulation for digital terrestrial television ,” European Broadcasting Union, Tech. Rep., 2006.

−5

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−6

10

0

5

10

15

20

25

30

Eb/N0, dB

Figure 4: BER Performance for Recursive 5-tap SIC-LMMSE Equalizer, K = 256, fd Ts = 6.82%, 16QAM, 64-State Rate1/2 BCC with COST-TU Channel which is only 7.5% of computational effort of conventional SIC-LMMSE equalizer. IV.

E XAMPLE OF A PPLICATION T O T HE W I MAX S YSTEM

This section presents computer simulation results of the proposed front-end equalizers with application to IEEE 802.16e mobile WiMAX standard. The number of subcarriers is assumed to be K = 256 and the length of guard interval is Tg = Ts /4. The outer channel code is the de facto standard 64state rate-1/2 binary convolutional code (BCC) with polynomials in octal notation (133, 171)8 . With different BCC code rate and constellation mappings (i.e. QPSK, 16QAM and 64QAM), variety of spectral efficiency which range from R = 1 bit per channel use (BPCU) to R = 5 BPCU can be achieved. This paper considers the COST typical urban (TU) channel model [18] with independent Rayleigh faded rays.The ray’s relative power and delay are: PdB = [−4, −3, 0, −2.6, −3, −5, −7, −5, −6.5, −8.6, −11, −10] , τµs = [0, 0.1, 0.3, 0.5, 0.8, 1.1, 1.3, 1.7, 2.3, 3.1, 3.2, 5] . It is further assumed that perfect timing synchronization and perfect channel state information (PCSI) for the iterative receiver. Fig. 4 illustrates the performance of purposed Recursive qtap SIC-LMMSE Equalizer in the coded OFDM system. In the simulation the number of subcarriers K = 256, the normalized Doppler frequency fd Ts = 6.82%, 16QAM modulation, and the standard 64-state BCC with rate-1/2 are used. Curves with up to 4 iterations are shown. Being able to harvest diversity provided by both of the ICI-channel and outer channel code, the iterative receiver even provides a gain of ∼ 1 dB at BER of 10−4 compared to no ICI.

[2] IEEE Std. 802.11a-1999, “Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specification: high speed physical layer in the 5 GHz band,” IEEE-SA Standards Board(1999-09-16), Tech. Rep., 1999. [3] IEEE Std. 802.16e-2005 and IEEE 802.16-2004/Cor1-2005, “Part 16: Air Interface For Fixed and Mobile Broadband Wireless Access Systems,” IEEE-SA Standards Board(2006-02-28), Tech. Rep., 2006. [4] M. Russell and G. L. St¨uber, “Interchannel interference analysis of OFDM in a mobile environment,” in Proc. IEEE Vehicular Tech. Conf., 1995, pp. 820–824. [5] P. Robertson and S. Kaiser, “The effects of doppler spreads in OFDM(A) mobile radio systems,” in Proc. IEEE Vehicular Tech. Conf., 1999, pp. 329–333. [6] Y. G. Li and L. J. Cimini, “Bounds on the interchannl interference of OFDM in time-varying impairments,” IEEE Trans. Commun., vol. 49, pp. 401–404, Mar. 2001. [7] X. Cai and G. B. Giannakis, “Bounding performance and suppressing intercarrier interference in wireless mobile OFDM,” IEEE Trans. Commun., vol. 51, pp. 2047–2056, Dec. 2003. [8] Y.-S. Choi, P. J. Voltz, and F. A. Cassara, “On channel estimation and detection for multicarrier signals in fast and selective Rayleigh fading channels,” IEEE Trans. Commun., vol. 49, pp. 1375–1387, Aug. 2001. [9] Y. H. Kim, I. Song, H. G. Kim, T. Chang, and H. M. Kim, “Performance analysis of a coded OFDM system in time-varying multipath Rayleigh fading channels,” IEEE Trans. Veh. Tech., vol. 48, pp. 1610–1615, Sep. 1999. [10] S. Kim and G. J. Pottie, “Robust OFDM in fast fading channels,” in Proc. IEEE Global Telecommunications Conf., 2003, pp. 1074–1078. [11] R. Chen, Y. Xu, H. Zhang, and H. Luo, “Iterative ICI mitigation method for MIMO OFDM systems,” Institute of Electronics, Info. and Comm. Engineers Trans. Comm., vol. E89-B, No. 3, pp. 859–866, Mar. 2006. [12] J. Hagenauer, “The turbo principle: Tutorial introduction and state of the art,” in Proc. International Symposium on Turbo Codes and Related Topics, Brest, France, Sep. 1997, pp. 1–11. [13] D. Huang, K. B. Letaief, and J. Lu, “Bit-interleaved time-frequency coded modulation for OFDM systems over time-varying channels,” IEEE Trans. Commun., vol. 7, pp. 1191–1199, July 2005. [14] C. Berrou and A. Glavieux, “Near optimum error correcting coding and decoding: Turbo-codes,” IEEE Trans. Commun., vol. 44, no. 10, pp. 1261–1271, Oct. 1996. [15] X. Wang and H. V. Poor, “Iterative(turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, pp. 1046–1061, July 1999. [16] D. N. Liu and M. P. Fitz, “Low complexity Affine MMSE detector for iterative detection-decoding MIMO-OFDM systems,” submitted to IEEE Trans. Commun. [17] G. H. Golub and C. F. Van Loan, Matrix Computations. The John Hopkins University Press, 1989.

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)

[18] Commission of the European Communities, “Digital Land Mobile Radio Communications - COST 207, Final Report ,” Office for Official Publications of the European Communities, Luxembourg, Tech. Rep., 1989.