Multi-Dimensional Graph-Based Soft Iterative Receiver for MIMO-OFDM

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Multi-Dimensional Graph-Based Soft Iterative Receiver for MIMO-OFDM Christopher Knievel, Student Member, IEEE, Peter Adam Hoeher, Senior Member, IEEE, Alexander Tyrrell, Member, IEEE, and Gunther Auer, Member, IEEE

Abstract—A graph-based receiver is presented that iteratively performs soft channel estimation and soft data detection. Reliability information of data symbols is utilized to improve channel estimation, and in turn, soft channel estimates refine data symbol estimates. The proposed multi-dimensional factor graph introduces transfer nodes that exploit correlation of adjacent channel coefficients in an arbitrary number of dimensions (e.g. time, frequency, and spatial domain). This establishes a simple and flexible receiver structure that facilitate soft channel estimation and data detection in multi-dimensional dispersive channels, and supports arbitrary modulation and channel coding schemes. Simulation results demonstrate that the proposed multidimensional graph-based receiver outperforms iterative and noniterative state-of-the-art receivers. Index Terms—Channel estimation, factor graph, MIMOOFDM, iterative decoding.

I. I NTRODUCTION ULTIPLE-input multiple-output (MIMO) transmission in conjunction with orthogonal frequency-division multiplexing (OFDM) is considered a key technology for emerging wireless radio systems such as 3GPP long term evolution (LTE). To facilitate coherent detection, knowledge of the channel response is required at the receiver end. The most common method to provide the receiver with channel state information (CSI), is to embed training symbols, known at the receiver, within the transmitted signal stream [1], [2]. To reconstruct the OFDM channel impulse response at the positions of the unknown data symbols, two-dimensional interpolation and filtering over time (OFDM symbols) and frequency (subcarriers) is often adopted [3]–[5]. With the growing popularity of MIMO transmission, channel estimators operating not only over time and/or frequency, but also the spatial domain emerged [6]–[9]. While MIMO-OFDM promises substantial diversity and/or capacity gains [10], [11], the required training overhead grows in proportion to the number of transmit antennas [12]. Iterative receivers utilizing the turbo principle that jointly carry out channel estimation and data detection are a potential enabler to reduce the required training overhead for MIMO-OFDM.

M

Paper approved by N. Al-Dhahir, the Editor for Space-Time, OFDM and Equalization of the IEEE Communications Society. Manuscript received February 14, 2011; revised July 25, 2011, October 31, 2011, and January 24, 2012. C. Knievel and P. A. Hoeher are with the Information and Coding Theory Lab, University of Kiel, Germany (e-mail: {chk, ph}@tf.uni-kiel.de). A. Tyrrell and G. Auer are with DOCOMO Euro-Labs, Munich, Germany (e-mail: [email protected]). Preliminary results of this work were presented at the IEEE Int. Conf. on Communications (ICC), Cape Town, South Africa, June 2010. Digital Object Identifier 10.1109/TCOMM.2012.042712.110108

These iterative receivers refine channel estimates by generating pseudo-training symbols by previously detected data symbols [13]–[16]. Unfortunately, the complexity of the performance optimal maximum-likelihood (ML) and mean squared error (MMSE) estimators grows exponentially with the modulation order and the number of transmit antennas [14]. The class of suboptimal iterative receivers based on the expectation maximization (EM) algorithm significantly reduce the computational cost and have attracted considerable interest recently [17]–[20]. Unfortunately, the general structure of the EM based receiver prohibits the use of reliability information in terms of loglikelihood ratios (LLR) for iterative channel estimation, but relies on hard decisions instead. Moreover, initialization of an EM based iterative receiver is susceptible to estimation errors. For initialization EM therefore requires either a preamble or a computationally complex algorithm, such as a linear MMSE estimator. Graph-based algorithms pose a viable alternative for iterative receivers [21]. Factor graphs [22], [23] constitute a versatile framework that has been applied to a variety of signal processing problems. Several graph-based receivers for iterative detection and/or channel estimation have been published in recent years [24]–[28]. An graph based OFDM receiver for joint channel estimation and data detection algorithm using belief propagation has been presented for a single antenna system in [24] and for a MIMO system in [25]. The former approach considers a frequency-selective channel and calculates the channel estimates vector-wise, whereas the latter assumes a time-variant channel and uses first-order auto-regressive relations to model the fading characteristics. Both graphbased receivers cannot be extended to multiple dimensions (time-varying channels in [24] or frequency-selective channels in [25]) without a significant increase in complexity. Other factor graph based receivers, such as [26] are only applicable to data detection, while [27], [28] apply a factor graph to refine channel estimates, which are previously estimated by a conventional training based channel estimation algorithm. Graph-based iterative receivers that introduce the concept of soft channel estimates have been developed in [29]–[31]. Soft channel estimates are generated and refined by iterative message-passing through a factor graph for a MIMO system with BPSK modulation over a block-fading channel in [31]. Furthermore, a message passing algorithm that produces soft channel estimates for single antenna systems over a timevarying channel has been presented in [32]. In this paper, we expand the principle of soft channel estimates of [31] to a graph-based MIMO-OFDM receiver over

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a time-variant frequency-selective channel. Transfer nodes utilize channel correlation information of multiple dimensions in the time, frequency and spatial domains. On the basis of the sum-product algorithm, a symbol-wise message exchange between nodes of the factor graph is established. The proposed multi-dimensional graph-based soft iterative receiver (MDGSIR) integrates channel estimation and data detection in a consistent way where both tasks process soft information, so as to iteratively improve performance. With a well designed message passing schedule, the effects of short cycles are mitigated. To our best knowledge, a complete description of a graph-based receiver incorporating multiple antennas, higher-order modulation, and a realistic channel model, while maintaining a low computational complexity, has not been presented before in the literature. The performance of MD-GSIR is evaluated for a theoretical channel model with uniform distributed parameters in all domains, time, frequency and space; as well as for a more realistic channel model derived from measurement campaigns [33]. The convergence of MD-GSIR with respect to (w.r.t.) the training density is analyzed by means of extrinsic information transfer (EXIT) charts. The MD-GSIR performance is compared to non-iterative and iterative stateof-the-art algorithms, namely to a non-iterative symbol-wise ML detector in combination with a 2D Wiener filter, and an iterative a posteriori probability (APP) detector with EM based channel estimation. The presented simulation results provide a comprehensive overview of the achievable performance of advanced MIMO-OFDM receivers operating in various channel conditions, with different modulation formats and code rates. The remainder of this paper is organized as follows: Section II defines the system and channel model. The MDGSIR and the transfer nodes are introduced in Section III. A convergence analysis of MD-GSIR based on EXIT charts as well as numerical results for two different channel models are presented in Section IV. Finally, Section V draws the conclusions. Throughout the paper, we adopt the following notation conventions: (·)∗ denotes the conjugate complex, P (x) and p(x) are the probability and the probability density function (pdf) of a random variable x, respectively. The expectation operator is given by E {·}. II. S YSTEM AND C HANNEL M ODEL The MIMO-OFDM system under consideration comprises L OFDM subcarriers and K OFDM symbols per frame. The equivalent discrete-time model of a MIMO channel with NT transmit (Tx) and NR receive (Rx) antennas can be represented in the frequency domain (after OFDM demodulation) as yn [l, k] =

NT 

hn,m [l, k] xm [l, k] + wn [l, k] NT

 i=1

i=m



hn,i [l, k] xi [l, k] + wn [l, k],    

Possible scenarios for mutually independent transmit signals are single-user MIMO systems using spatial multiplexing and/or multi-user MIMO systems. The principle of the graphbased receiver can also be applied to the case of spatial precoding, though this is beyond the scope of this paper. The channel coefficients are assumed to be wide-sense stationary (WSS), complex Gaussian variables with zero mean, and can be modeled as [34] NP   1  hn,m [l, k]= lim √ exp j θi +2πfD,i kTs −2πτi lF NP →∞ NP i=1

 dRx dTx + 2πm sin (ϕi ) +2πn sin (ϑi ) , (3) λ λ

where Ts and F denote the OFDM symbol duration and subcarrier spacing, respectively. Each of the NP multipath components consists of an individual random-phase θi ∈ [0, 2π), a propagation delay τi ∈ [0, τmax ], and a Doppler frequency fD,i ∈ [−fD,max , +fD,max], where τmax and fD,max denote the maximum propagation delay and maximum Doppler frequency. The latter two components determine the fading in frequency and time, respectively. Moreover, a linear antenna array is assumed, with a spacing of dTx at the transmitter side and dRx at the receiver side. A multipath component departs with an angle of ϕi and is received with ϑi . Training symbols are inserted in time and frequency domain, according to Fig. 1, with an equidistant spacing in time Dt and frequency Df . Furthermore, training symbols associated to different Tx antennas are orthogonally multiplexed in time and frequency, i.e., when one transmit antenna is transmitting a training symbol, all other antennas remain silent (antenna muting). Hence, multi-antenna interference at training symbol positions is mitigated, which simplifies the generation of initial channel estimates. III. MD-GSIR: M ULTI -D IMENSIONAL G RAPH -BASED S OFT I TERATIVE R ECEIVER

m=1

= hn,m [l, k] xm [l, k] +

where l ∈ {0, 1, . . . , L−1} and k ∈ {0, 1, . . . , K−1} represent the OFDM subcarrier and OFDM symbol index. The channel coefficient hn,m [l, k] ∈ C relates Tx antenna m with Rx antenna n, and is normalized such . that E{|hn,m [l, k]|2 } = 1. The observation at Rx antenna n, yn [l, k] ∈ C, consists of NT superimposed signals. In order to recover the channel input at Tx antenna m, denoted by xm [l, k] ∈ C, the signals originating from other Tx antennas i=m are observed as multi-antenna interference (MAI). Together with MAI an additive white Gaussian noise (AWGN) 2 corrupts the reterm wn [l, k] with zero mean and variance σw ceived signal. The composite interference may be represented by the effective noise term vn,m [l, k]: . yn [l, k] = hn,m [l, k]xm [l, k] + vn,m [l, k]. (2)



AWGN

MAI

(1)

Factor graphs are powerful graphical tools that have been applied to a large variety of problems in digital communications [22], [23]. Generally, a factor graph is a bipartite graph, i.e., the nodes of a graph are partitioned into two disjoint sets U and V such that every edge connects a node u ∈ U with a node of v ∈ V. An exchange of information between

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Fig. 1. Training grid with a periodic spacing between adjacent training symbols of Df in frequency and and Dt in time.

Fig. 3. Factor graph structure of a 2×2 MIMO-OFDM system and QPSK modulation. Without loss of generality, the transfer nodes connect adjacent coefficient nodes in the time domain.

Fig. 2. decoder.

Message exchange between the GSIR receiver and the channel

nodes of the same set is forbidden. Message exchange between these sets is facilitated, e.g., according to the sum-product algorithm [22]. The following sections explain the proposed multidimensional graph-based soft iterative receiver (MD-GSIR) in detail. The general factor graph structure of MIMO-OFDM and the generation of soft channel estimates are introduced in Section III-A and III-B. Section III-C derives the transfer nodes that leverage multi-dimensional channel estimation. The information exchange at coefficient nodes and soft data detection are presented in Section III-D and III-E. Finally, the message scheduling achieving best performance is detailed in Section III-F. A. Receiver Structure and Associated Factor Graph Bit-interleaved coded modulation (BICM) is considered. After OFDM demodulation, the MD-GSIR jointly estimates the channel coefficients and data symbols as illustrated in Fig. 2. The log-likelihood ratios (LLRs) of the data bits are passed to the deinterleaver and descrambler. The channel decoder processes the deinterleaved and descrambled LLRs and feeds back extrinsic information of the decoded data symbols. One global iteration comprises one MD-GSIR and one Turbo decoding iteration. After interleaving and scrambling, the MDGSIR uses the extrinsic information as a priori information in subsequent iterations. The objective of MD-GSIR is to perform joint channel estimation and data detection, which involves the estimation of two types of variables, namely data bits bm,ν [l, k] and channel coefficients hn,m [l, k]. The factor graph for a 2×2 MIMOOFDM system employing QPSK modulation is illustrated in Fig. 3. The two disjoint sets U and V are visualized by (i) circles for the unknown data bits (bit nodes), data symbols (symbol nodes) and channel coefficients (coefficient nodes) and (ii) rectangles for the received samples (observation nodes). Mapping nodes connect bit nodes with symbol nodes

Fig. 4. 3D structure of the factor graph connecting channel coefficient nodes in time, frequency and space.

according to the modulation format, depicted as a black dot within a square. All messages exchanged between nodes are instances of the sum-product algorithm. Fig. 4 shows the connections of the channel coefficients with the transfer nodes in three dimension; time, frequency and space. In the spatial domain the transfer nodes utilize the channel correlation between transmit antennas. We note that the number of dimension of the factor graph is not limited to three; by also taking into account the channel correlations between receive antenna elements, a four-dimensional factor graph is obtained. An MD-GSIR iteration is outlined below, with each step being explained in detail in subsequent sections. 1. At the bit nodes bit probabilities P (bm,ν =0) and P (bm,ν =1) are sent to mapping nodes. 2. At mapping nodes symbol probabilities N b P (bm,ν ) are calculated, where Nb denotes P (xm )= ν=1

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the number of bits per symbol. Subsequently the symbol probabilities are sent to observation nodes. At the observation nodes the symbol probabilities P (xm ) are utilized in combination with the received samples, to calculate channel coefficient estimates for each symbol (cf. Section III-B). In order to facilitate message exchange throughout the entire frame, channel coefficients of neighboring OFDM subcarriers, OFDM symbols, transmit and receive antennas are connected via transfer nodes. A message of a coefficient node is distributed to all other coefficient nodes (cf. Section III-C). Combining the messages of neighboring coefficient nodes (cf. Section III-D), refined coefficient estimates are sent back to the observation nodes, where in turn refined symbol probabilities are generated (cf. Section III-E). The mapping nodes finally calculate LLR values for the bit nodes, which are passed to the channel decoder for further processing.

B. Soft Channel Estimation The task of channel estimation in the observation nodes is to compute the conditional pdf p(yn |hn,m ). During initialization, only the information of training symbols is utilized, while additional information of data symbols is exploited in subsequent iterations. Without loss of generality, the OFDM symbol and subcarrier indices k and l are omitted in the following to enhance readability. In order to reduce computational cost, the effective noise term vn,m in (2) is approximated by a Gaussian variable, 2 which is characterized by vn,m ∼ CN (μv,n,m , σv,n,m ). The conditional pdf p(yn |hn,m ) is calculated as follows:  p (yn |hn,m )= P (xm ) p (yn |hn,m , xm ) xm ∈X

=



xm ∈X

P (xm ) 

exp −

1 · 2 πσv,n,m

|hn,m −(yn −μv,n,m )/xm |2 2 σv,n,m /|xm |2

2 is interpreted as the coefficient, whereas the variance σh,n,m reliability information. Thus, a large variance results in a less reliable estimate of the channel coefficient and vice versa. In the high signal-to-noise ratio (SNR) region, the performance of factor graph based receivers is improved by scaling the variance of the coefficient message slightly. This effect was observed in [35], where short cycles led to an overestimation of the quality of the estimates, resulted in LLR values that were too large. For MD-GSIR the variance of the channel coefficients is therefore increased by a factor β>1. The message sent from a coefficient node to a transfer node   2 . The value therefore yields p(hn,m ) ∼ CN μh,n,m , βσh,n,m of β = 1.02 is found by empirical measures and attains good performance for all studied cases.

C. Transfer Node The channel estimation accuracy may be improved by utilizing the reliability information of neighboring channel coefficients. This is accomplished by transfer nodes, which distribute messages from channel coefficients throughout all dimensions, as shown in Fig. 4, for a 3D factor graph. Specifically, a transfer node describes the deviation between channel coefficients hn,m [l, k] and hn+n ,m+m [l+l , k+k  ], which are denoted by h and h to simplify the notation: . |ω| = 1. (7) Δn ,m [l , k  ] = h − ωh , The tuning factor ω ∈ C depends on the correlation properties between adjacent transfer nodes. For a symmetrically distributed spectrum, the correlation function is real valued such that ω=1, whereas the tuning factor ω = ejϕ is complex valued for non-symmetric distributions. Given a Rayleigh fading channel, a transfer node is approximated by a zeromean Gaussian pdf:   2   Δn ,m [l , k  ] ∼ N 0, σΔ,n (8)  ,m [l , k ] . According to (7), information between adjacent channel coefficients is exchanged as follows:

 , (4)

μh = ω·μh ,

2   σh2  = σh2 +σΔ,n  ,m [l , k ].

(9)

where X is the symbol alphabet. In order to reduce the computational complexity of calculating the mixed Gaussian distribution, (4) is approximated by a Gaussian distribution: p(yn |hn,m ) ∝ p(hn,m ). If transmitted symbols are reliably detected during iterations, we will have one symbol xi with P (xi )  P (xj ), i = j. This justifies the Gaussian approximation. The mean and variance of p (hn,m ) are thus given by [32]:  P (xm ) μh,n,m =α(yn −μv,n,m ) , (5) xm |xm |2 xm ∈X   P (xm )  2 2 =α σv,n,m +|yn −μv,n,m |2 −|μh,n,m |2 , σh,n,m |xm |4

A message exchanged within a transfer node scales its mean μh by a factor of ω, whereas its variance σh2  is increased by the variance of the domain-specific transfer node. Since the variance of a channel coefficient is interpreted as reliability information (cf. Section III-B), the transfer function reduces the reliability of the message with each node; inherently decreasing the influence of this message on the overall message generation. The variance of the transfer node is calculated as follows

2   |h − ωh |2 σΔ,n  ,m [l , k ] = E



= E |h|2 + E |h |2 −E {ωhh∗ } − E {ωh∗ h }     . . =1 =1

(6)

(10)

xm ∈X

where α is a normalization factor defined as  2 . The mean value μh,n,m P (x )/|x | α=1/ m m xm ∈X of the channel coefficient represents the hard estimate of

= 2 (1 − Re [ωE {h∗ h }]) ,

where E{h∗ h } corresponds to the multi-dimensional autocorrelation function between the two channel coefficients h and h .

KNIEVEL et al.: MULTI-DIMENSIONAL GRAPH-BASED SOFT ITERATIVE RECEIVER FOR MIMO-OFDM

In order to prevent short cycles and the inherent exchange of intrinsic information, a transfer node only connects two neighboring channel coefficients of one domain, i.e., l = ± 1→k  =n =m =0, k  = ± 1→l =n =m =0, etc., as illustrated in Fig. 4. Provided a WSS channel, the correlation function E{h∗ h } therefore only depends on the considered channel dimension, so that the transfer nodes can be calculated independently for each domain; either time, frequency or space. 1) Frequency Domain: The variance for each domain depends on the distribution of the domain specific dispersion parameters. In case the exact distribution of the power delay profile is not known, a common approximation is to assume that propagation delays in (3) are uniformly distributed within [0, τmax ]. Then the variance of the transfer node (10) in the frequency domain between adjacent subcarriers amounts to    τmax 2 , (11) = 2 1 − sinc (τmax F ) Re ωf ej2π 2 F σΔ,f where τmax and F denote the maximum propagation delay and the OFDM subcarrier spacing. The variance (11) is minimized when the real part is maximized. By setting the value of the tuning factor ωf as ωf = e−j2π

τmax 2

F

= e−jπτmax F ,

(12)

the imaginary part in (11) diminishes. By substituting (12) into (11), the minimum variance for the frequency domain transfer node results in

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Assuming a uniform Doppler power spectral density defined 2 over the interval [−fD,max, fD,max ], the variance of σΔ,t yields   2 = 2 1 − sinc (2fD,maxTs ) , (17) σΔ,t where fD,max and Ts denote the maximum Doppler frequency and the OFDM symbol duration. Due to the symmetric distribution of the Doppler frequencies, the time domain correlation function is real valued and thus, the tuning factor is set to ωt =1. A commonly used distribution of the Doppler frequencies is given by the Jakes power spectral density [36], for which 2 results in the variance of σΔ,t   2 σΔ,t = 2 1 − J0 (2πfD,max Ts ) , (18) where J0 (·) is the Bessel function of the first kind and order zero. 3) Spatial Domain: The variance of the transfer node in the spatial domain between neighboring transmit antennas is dependent on multiple parameters: Namely, the spacing of transmit antennas dTx , the wavelength λ, and the distribution of the azimuth angle of departure ϕ. Starting from (10), the variance of the spatial domain (considering only transmit antennas) can be expressed by ⎞ ⎛ !  "⎟  ⎜ dTx ⎟ ⎜ 2 sin(ϕ) = 2 ⎜1− Re E ωs exp j2π σΔ,s ⎟. ⎠ ⎝ λ   C

(13)

(19)

Note that shifting the tuning factor ωf = e−jπτmax F is equivalent to shifting the power delay profile by −τmax /2 in the delay domain [3], [4]. A commonly adopted distribution of the power delay profile is an exponentially decaying function:  1 στ exp(−τn /στ ) 0 ≤ τn ≤ τmax (14) p(τn ) = 0 else,

In the following, a uniform distribution of the angular spread ϕ Θ within the interval [φ+ Θ 2 , φ− 2 ] is assumed. With the spatial autocorrelation function given in [37], C can be rewritten as ⎡ ⎛

2 σΔ,f = 2 (1 − sinc (τmax F )) .

where στ denotes the root mean square (RMS) delay spread. The variance of the frequency domain transfer node is given by ⎧ ⎫ ⎨ ω ∞ ⎬ f 2 = 2 − 2 · Re · exp(−2jπF τ ) exp (−τ /στ ) dτ σΔ,f ⎩ στ ⎭ 0   ωf = 2 − 2 · Re . (15) 1 + j2πστ F By setting ωf =

1 1−j2πστ F

2 σΔ,f

, (15) is minimized and results in   1 =2 1− . (16) 1 + 4π 2 στ2 F 2

2) Time Domain: Analogous to the frequency domain, channel coefficients of adjacent OFDM symbols are connected with a transfer node, modeling the time-variant channel variations. The variance of adjacent channel coefficients in the time domain is determined by calculating (10) between two adjacent OFDM symbols.

⎢ ⎜ ∞  ⎢ ⎜ sin(2mΘ) ⎜J0 (z) + 2 · ω C=Re ⎢ · J2m (z) cos(2mφ) + s ⎢ ⎜ 2mΘ ⎣ ⎝ m=1   A ⎞⎤ j2 

∞  m=0

J2m+1 (z) sin((2m + 1)φ)

⎟⎥ ⎥ sin((2m + 1)Θ) ⎟ ⎟⎥ , ⎟ (2m + 1)Θ ⎠⎥ ⎦ 

B

(20) with z = 2π dλTx . To determine the spatial tuning factor ωs = exp(jx), which maximizes C, the derivative dC/dx is set to zero:   B x = tan−1 − , (21) A which results in the tuning factor    B ωs = exp j tan − . A Inserting (22) into (19) yields   √ |A| A2 + B 2 2 . σΔ,s = 2 1− A

(22)

(23)

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E. Soft Data Detection Utilizing the  updated message received from a coefficient  node p hin,m , an observation node yn calculates      (27) p (yn |xm ) = p yn |hin,m , xm p hin,m dhn,m Fig. 5.

=

Message exchange at a coefficient node.

For an azimuth angle of φ = 0◦ , ωs = 1 and the calculation of the variance in spatial domain can further be simplified as 2 σΔ,s

   dTx = 2 1 − sinc Θ . λ

(24)

Apart from the uniform distribution, typical distributions of the azimuth angle of departure are the von Mises distribution or the Laplacian distribution [38]. Given identical distributions, the rule to calculate the variance is independent of the domain, as can be seen from (13), (17), and (24). In case that fading is present in one domain only, the same uncoded BER and/or mean squared error (MSE) performance is achieved, independent in which domain the fading occurs. This property is elaborated further in Section IV-A.

D. Information Exchange at Coefficient Nodes In general, a message generated at a node of a factor graph needs to consider all adjacent nodes, except the node for which the message is generated, so to ensure that only extrinsic messages are exchanged. For a node that is connected to N other nodes, N −1 incoming messages have to be processed in order to generate one outgoing message. For the MD-GSIR, a coefficient node is connected to two transfer nodes in each domain (time, frequency or space) and one symbol node, as illustrated in the left part of Fig. 5. Suppose a channel coefficient receives the messages  pj (h) ∼ CN μj , σj2 , j ∈ {1, . . . , N } from N adjacent transfer nodes. The product of Gaussian pdfs results in a complexvalued normal distribution ⎞ ⎛ N , ⎝ pj (h)⎠ = pi (h) , (25) j=1,j=i

with mean and variance N 

μi =

j=1,j=i N  j=1,j=i

μj σj2

, 1 σj2

σi2 =

1 N  j=1,j=i

.

(26)

1 σj2

The exchange of extrinsic messages is depicted in the right part of Fig. 5. The combined message, denoted  2 [l, k] , is subsequently by p(hin,m ) ∼ CN μi,n,m [l, k], σi,n,m sent to an observation node.

π



1 2 2 σi,n,m |xm |2 +σv,n,m



·

|yn −μi,n,m xm −μv,n,m |2 exp − 2 2 σi,n,m |xm |2 +σv,n,m

 .

(28)

It is important to emphasize that besides the variance of the effective noise also the variance of the channel coefficient is considered in the denominator. Hence, an unreliable channel estimate reduces the log-likelihood ratio for the corresponding data symbol. For the calculation of the soft outputs, in the numerical results we choose the joint Gaussian detector [39]. The LLR of bit bm,ν contained in data symbol xm is approximated as   |yn − μi,n,m xm − μv,n,m |2 LLR (bm,ν ) ≈ max − − 2 2 xm,0 |xm |2 σi,n,m + σv,n,m   |yn − μi,n,m xm − μv,n,m |2 max − , 2 2 xm,1 |xm |2 σi,n,m + σv,n,m (29) with xm,0 , xm,1 ∈ xm (bm,ν = 0), xm (bm,ν = 1). Apart from the joint Gaussian detector, there exist many alternatives to calculate the exact or approximate LLRs of bm,ν , e.g. optimum MAP detection or the max-log approximation [40]. F. Message Exchange Scheduling The structure of the underlying factor graph is a trade-off between complexity and performance. It is well known that short cycles deteriorate the performance of the sum-product algorithm. Short cycles occur when a message, which leaves a node, ’travels’ only a few nodes until it is send back to its origin. Strategies have been developed that merge several nodes in order to prevent short cycles in the message exchange. This inherently involves higher computational complexity w.r.t. the message generation. The presented symbol-wise factor graph structure exhibits a low computational complexity. Hence, message scheduling plays an important role to achieve a good performance. For a factor graph with cycles, message scheduling should ensure that the message a node receives contains as few information of the node itself. Related to MD-GSIR, this means that messages of different domains should be exchanged successively, e.g., messages are exchanged in the time domain first, afterwards messages are exchanged in the frequency domain, or vice versa. The underlying principle is dubbed “two-way schedule” in [41]. A message is sent from a coefficient node to a transfer node and from there to the next coefficient node in the same domain; thus producing a burst that traverses in the direction of the desired domain. If the last coefficient node of the selected domain is reached, messages are propagated in the reversed direction. The two-way schedule is finished when the coefficient node that started the message exchange receives a

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TABLE I PARAMETERS OF THE S YMMETRIC AND THE WINNER C2 NLOS C HANNEL

Time domain Frequency domain Transmitter topology Azimuth angle of departure Angular spread Carrier frequency OFDM subcarrier spacing OFDM symbol duration

Symmetric channel model Normalized fading Distribution 0.01 uniform 0.01 uniform Uniform linear array 10 λ spacing 0◦ 14◦ fc = 4 GHz F = 15 kHz Ts = 71.43μs

message. The schedule hereby ensures that incoming messages are the combination of the maximum number of outgoing messages of one domain. An alternative scheduling is given by the “flooding” schedule [41], which distributes the messages simultaneously in all domains. However, messages arriving at a node may contain intrinsic information with this schedule, which leads to a degradation w.r.t. performance. The scheduling also determines which domain exchanges its information first. During initialization, all messages are set to zero mean and a large variance, except for the training symbols where the mean value is known and the variance is determined by AWGN. If messages of one domain with a low variance have already been exchanged, messages with a larger variance contribute less to the overall message generation. It is therefore recommended that the domain with the largest variance should be exchanged first; unless the variance exceeds a certain threshold, in which case messages in this domain should not be sent first. To this end, in case dTx  λ, the variance of the spatial domain between transmit antenna elements is typically beyond such a threshold. Although these messages are considered to be unreliable, they nevertheless contribute to a performance improvement in subsequent iterations as demonstrated in Section IV. IV. N UMERICAL R ESULTS The performance of MD-GSIR is evaluated by means of Monte Carlo simulations for two different channel models. First, a symmetric channel model with a uniform distribution of all dispersion parameters in time, frequency and space. The symmetric channel model allows to analyze the possible improvements of each additional dimension that is utilized by the graph based receiver. Second, a macro cellular channel model that represents realistic channel conditions of an urban environment [33]. The parameters chosen for the symmetric as well as the WINNER channel model are given in Table I. Three variants of MD-GSIR are investigated, which are characterized by the number of utilized channel dimensions. 1D-GSIR takes into account only channel variations in the frequency domain, whereas 2D-GSIR utilizes channel variations in both time and frequency domains. Finally, 3DGSIR exploits channel correlation in three dimensions; time, frequency and space. The system parameters are summarized in Table II. The performance of MD-GSIR is compared with two stateof-the-art receivers: (i) a non-iterative receiver with ML detection and 2D channel estimation by Wiener filtering; and

WINNER channel model Normalized fading Distribution 0.0106 Jakes 0.0277 exp Uniform linear array 10 λ spacing Distributed according to [33] 35◦

TABLE II PARAMETERS OF THE S IMULATION S ETUP

No. of OFDM subcarriers No. of OFDM symbols No. of transmit antennas No. of receive antennas Training spacing

Modulation Channel code (code rate) No. of iterations

Symmetric channel model L = 64 K = 64 NT = 8 NR = 8 Dt =16/32 Df =16/32 Ds = 1 BPSK Repetition code R=1/4 10

WINNER channel model L = 64 K = 32 NT = 2/4 NR = 4 Dt = 16 Df = 16 Ds = 1 QPSK / 16-QAM Turbo code R=1/3,1/2,3/4 10

(ii) an iterative receiver with APP detection and EM-based channel estimation. Specifically, for channel estimation an extension of the EM termed space alternating generalized EM (SAGE) algorithm has been implemented [20], which offers faster convergence. Note that the computational complexity of the comparison algorithms based on ML or APP detection increases exponentially with the modulation order and the number of transmit antennas. Thus, these algorithms can become infeasible for higher-order modulation and multiple transmit antennas. On the other hand, MD-GSIR employs a Gaussian detector and, hence, the complexity is significantly lower than ML and/or APP detection. Furthermore, the complexity is linear w.r.t. the number of transmit and receive antennas, the number of dimensions, as well as the modulation order. A. Symmetric Channel Model Apart from a symmetric channel model with equivalent dispersion parameters in each dimension, a symmetric frame structure in time and frequency domain is adopted. The frame structure as well as the channel code was chosen in such a way that simulation effects, e.g. limited sequence length and/or edge effects, affect the time, frequency, and spatial domain similarly. An 8×8 MIMO system with BPSK modulation is implemented and the number of OFDM symbols and OFDM subcarriers are set to K=L=64. As forward error correction a repetition code was chosen as its performance is independent of the system parameters, unlike convolutional/turbo/LDPC codes which improve e.g. with increased sequence length. The BER curves for the 1D, 2D and 3D-GSIR are shown in Fig. 6. Although the variance of the transfer nodes of the spatial domain is larger than that of the time and/or

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Fig. 6. BER performance of the MD-GSIR with eight transmit and eight receive antennas for different training spacings in a symmetric channel model.

frequency domain, an almost constant improvement with each additional domain can be observed. For a training spacing of Dt = Df = 16 the performance improves by approximately 3.5 dB from 1D to 2D to 3D-GSIR at a BER of 10−3 . Increasing the training spacing to Dt = Df = 32 leads to a severe performance degradation for 1D-GSIR whereas 2DGSIR and 3D-GSIR achieve similar performance as for Dt = Df = 16. The performance degradation of 3D-GSIR w.r.t. 2DGSIR with perfect channel knowledge is roughly 2 dB. The uniformly distributed dispersion parameters of the symmetric channel model are a challenging environment for channel estimation and may be viewed as the worst-case scenario. Improving the quality of the initial channel estimates further improves performance of MD-GSIR, which is examined in a companion paper [42]. B. WINNER Channel Model The macro cellular channel model WINNER C2 without line-of-sight (LOS) propagation path represents more realistic channel conditions [33]. The number of OFDM symbols and subcarriers is set to K=32 and L=64, and a fixed number of 10 global iterations is used for both MD-GSIR, as well as the SAGE estimator with APP detection. The observed Doppler frequencies fD are distributed according to the Jakes spectrum. Perfect knowledge about the exponentially decaying power delay profile specified in [33] is assumed for the calculation of the transfer node variance (16) in the frequency domain. The tuning factors in (7) for the time and frequency domains are set to ωt =1 and ωf =τmax /8, respectively. The performance of the GSIR depends on several parameters including training density as well as the training pattern. Due to the structure of the factor graph, training symbols are treated the same way as data symbols. Although arbitrary training patterns are possible, due to the message scheduling, it is beneficial to employ a distributed training pattern, since the variance of the estimated channel coefficients is kept at a comparable level throughout. Therefore, a training pattern with diamond structure (cf. Fig. 1) is adopted here for all numerical evaluations. 1) EXIT Chart Analysis: EXIT charts [43], [44] are a powerful tool to analyze the convergence behavior of iterative

Fig. 7. EXIT chart analysis of the MD-GSIR at an SNR of 6 dB, and a varying spacing of training symbols in time and frequency domain. Additionally the transfer characteristics of MD-GSIR with perfect channel knowledge and a turbo code with code rate R=1/3 are included.

receivers. Initially developed for the convergence analysis of iterative forward error codes, it has been adopted to a variety of applications. The transfer characteristic of MD-GSIR for an SNR of 6 dB is examined in Fig. 7. The density of the training symbols determines the starting point on the left side, as well as the slope of the transfer characteristic. The slope of the transfer characteristic gets steeper as the training density decreases (i.e. larger training spacings Dt and Df ). If the transfer characteristic is steep, the BER improves with iterations, since the output extrinsic information IE becomes more reliable as the input a priori information IA increases. Hence, the lower the training density, the more is MD-GSIR able to gain from reliably detected data symbols. The flat section of MD-GSIR in the range 0 ≤ IA ≤ 0.6 indicates that data symbols contribute only marginally to the message generation. With increasing reliability, and in turn decreasing variance, the contribution of data symbols improve performance. The transfer characteristic of 2D-GSIR in Fig. 7 is superior to that of 1D-GSIR, which leverages convergence at lower SNR values. Alternatively, 2D-GSIR retains the same transfer characteristic as 1D-GSIR at lower training density. The difference between 2D-GSIR and 3D-GSIR is only marginal. The curves of the MD-GSIR with imperfect channel knowledge do not converge in the upper right corner since full a priori information (IA = 1) is only achieved for data symbols but not for channel coefficients. For comparison the transfer characteristics of the MD-GSIR with perfect channel knowledge and a turbo code with code rate R=1/3 are included in Fig. 7. A crossing of the characteristics of the turbo code and 1D-GSIR is observed at an early stage, which prevents convergence. On the other hand, a tunnel opens between the transfer characteristics of 2D and 3D-GSIR and that of the turbo code, and hence they are able to converge. This observation is exemplified by the BER performance discussed

KNIEVEL et al.: MULTI-DIMENSIONAL GRAPH-BASED SOFT ITERATIVE RECEIVER FOR MIMO-OFDM

Fig. 8. BER performance of the MD-GSIR for 2 × 4 MIMO with QPSK and 16-QAM modulation, a turbo code with rate R=1/3, and training symbol spacings Dt =Df =16. For comparison a GSIR with perfect channel state information (p. CSI), a non-iterative ML detector as well as an iterative APP detector are included.

below. 2) Bit Error Performance: Fig. 8 shows BER results for MD-GSIR with QPSK and 16-QAM modulation employing two transmit and four receive antennas. A turbo code with rate R=1/3 is applied in combination with the training symbol spacings Dt =Df =16 in time and frequency domain. The curves for ML and APP detection as well as 2D-GSIR with perfect channel knowledge are included for comparison. For QPSK the difference between 2D and 3D-GSIR is only marginal, as predicted by the EXIT chart analysis in Fig. 7. The comparatively small gain of 3D-GSIR w.r.t. 2D-GSIR is reasoned in the low correlation of the transmit antennas. On the other hand, 2D and 3D-GSIR outperform the ML and APP detectors by about 2.5 dB. For 16-QAM modulation 3D-GSIR converges 0.5 dB earlier than 2D-GSIR at an SNR of 8 dB. Also for 16-QAM modulation 3D-GSIR outperforms the noniterative ML detector and converges 5.5 dB earlier. While the iterative APP detector is able to improve BER performance by roughly 1.5 dB compared to the non-iterative ML detector, 3D-GSIR attains a 3.5 dB gain over the APP detector. The performance loss due to channel estimation is about 2 dB for 3D-GSIR with QPSK and 16-QAM modulation, respectively. Fig. 9 plots BER results of 2D and 3D-GSIR for 4×4 MIMO with QPSK modulation and different code rates (R=1/2 and R=3/4). In Fig. 9 the robustness of the proposed receiver is examined, by comparing the results of a ‘matched’ GSIR, which has perfect knowledge of all dispersion parameters of the transfer function (7), with a ‘mismatched’ GSIR, where channel correlations are approximated by a uniform distribution. 3D-GSIR outperforms 2D-GSIR for both code rates. Additionally, the difference between matched and mismatched receiver is less for 3D-GSIR than for the 2D GSIR. Hence, additional correlation information enhances performance, as well as robustness of the graph-based receiver. 3D-GSIR is about 2.5 dB and 3 dB worse compared to the GSIR with perfect channel knowledge (p. CSI) given a code rate R=1/2 and R=3/4, respectively. Higher code rates require additional iterations and/or enhanced initialization for the MD-GSIR to improve its performance.

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Fig. 9. BER performance of MD-GSIR for 4 × 4 MIMO with QPSK modulation and different code rates. The solid and dashed lines denote a matched and mismatched calculation of transfer variances, respectively. For comparison a GSIR with perfect channel state information is included.

V. C ONCLUSION A novel soft iterative channel estimation and data detection scheme for MIMO-OFDM over multi-selective channels is presented in this paper. Messages representing estimates of the channel coefficients and data symbols are efficiently generated according to the sum-product algorithm and distributed throughout the factor graph via transfer nodes. Closed-form derivations for the transfer nodes in time, frequency and space are presented, which are able to accurately model the fading process of a mobile radio channel in three dimensions. Reliability information of both data symbols and channel estimates are used to iteratively refine each other. Simulation results show that a gain of 3 dB with each additional domain can be achieved for a triply-selective channel model and i.i.d. dispersion parameters in all domains. For the more realistic WINNER C2 channel model, the multidimensional graph-based receiver outperforms non-iterative and iterative state of the art algorithms. The complexity of the proposed MD-GSIR is greatly reduced by implementing the joint Gaussian approach compared to ML or APP detection algorithms. The graph can easily be extended to take additional dimensions into account, such as correlation between receive antennas and/or polarization of antennas. R EFERENCES [1] J. K. Cavers, “An analysis of pilot symbol assisted modulation for Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 40, no. 4, pp. 686–693, Nov. 1991. [2] N. W. K. Lo, D. D. Falconer, and A. U. H. Sheikh, “Adaptive equalization and diversity combining for mobile radio using interpolated channel estimates,” IEEE Trans. Veh. Technol., vol. 40, no. 3, pp. 636–645, Aug. 1991. [3] P. Hoeher, “TCM on frequency-selective land-mobile fading channels,” in Proc. 1991 Tirrenia Int. Workshop Digital Commun., E. Biglieri and M. Luise, editors, Coded Modulation and Bandwidth-Efficient Transmission. Elsevier Science Publishers, 1991, pp. 317–328. [4] P. Hoeher, S. Kaiser, and P. Robertson, “Two-dimensional pilot-symbolaided channel estimation by Wiener filtering,” in Proc. 1997 IEEE Int. Conf. Acoustics, Speech, Signal Processing, pp. 1845–1848. [5] G. Auer, “Channel estimation in two dimensions for OFDM systems with multiple transmit antennas,” in Proc. 2003 IEEE Global Commun. Conf., pp. 322–326.

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[6] M. Stege, P. Zillmann, and G. Fettweis, “MIMO channel estimation with dimension reduction,” in Proc. 2002 Int. Symp. Wireless Personal Multimedia Commun., pp. 417–421. [7] H. Miao and M. J. Juntti, “Space-time channel estimation and performance analysis for wireless MIMO-OFDM systems with spatial correlation,” IEEE Trans. Veh. Technol., vol. 54, no. 6, pp. 2003–2016, Nov. 2005. [8] J.-W. Choi and Y.-H. Lee, “Complexity-reduced channel estimation in spatially correlated MIMO-OFDM systems,” IEICE Trans. Commun., vol. E90-B, no. 9, pp. 2609–2612, Sep. 2007. [9] G. Auer, “3D MIMO-OFDM channel estimation,” IEEE Trans. Commun., to be published. [10] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Pers. Commun., vol. 6, no. 3, pp. 311–335, Mar. 1998. [11] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans. Telecommun. (ETT), vol. 10, no. 6, pp. 585–595, Nov./Dec. 1999. [12] B. Hassibi and B. M. Hochwald, “How much training is needed in multiple-antenna wireless links?” IEEE Trans. Inf. Theory, vol. 49, no. 4, pp. 951–963, Apr. 2003. [13] M. C. Valenti and B. D. Woerner, “Iterative channel estimation and decoding of pilot symbol assisted turbo codes over flat-fading channels,” IEEE J. Sel. Areas Commun., vol. 19, no. 9, pp. 1697–1705, Sep. 2001. [14] F. Sanzi, S. Jelting, and J. Speidel, “A comparative study of iterative channel estimators for mobile OFDM systems,” IEEE Trans. Wireless Commun., vol. 5, no. 2, pp. 849–859, Sep. 2003. [15] C. Cozzo and B. Hughes, “Joint channel estimation and data detection in space-time communications,” IEEE Trans. Commun., vol. 51, no. 8, pp. 1266–1270, Aug. 2003. [16] G. Auer and J. Bonnet, “Threshold controlled iterative channel estimation for coded OFDM,” in Proc. 2007 IEEE Veh. Technol. Conf. – Spring, pp. 1737–1741. [17] B. Lu, X. Wang, and Y. Li, “Iterative receivers for space-time block coded OFDM systems in dispersive fading channels,” in Proc. 2001 IEEE Global Commun. Conf., pp. 514–518. [18] Y. Xie and C. N. Georghiades, “Two EM-type channel estimation algorithms for OFDM with transmitter diversity,” IEEE Trans. Commun., vol. 51, no. 1, pp. 106–115, Jan. 2003. [19] M. Khalighi and J. J. Boutros, “Semi-blind channel estimation using the EM algorithm in iterative MIMO APP detectors,” IEEE Trans. Wireless Commun., vol. 5, no. 11, pp. 3165–3173, Nov. 2006. [20] J. Ylioinas and M. Juntti, “Iterative joint detection, decoding, and channel estimation in turbo-coded MIMO-OFDM,” IEEE Trans. Veh. Technol., vol. 58, no. 4, pp. 1784–1796, May 2009. [21] A. P. Worthen and W. E. Stark, “Unified design of iterative receivers using factor graphs,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 843– 849, Feb. 2001. [22] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 498–519, Feb. 2001. [23] H.-A. Loeliger, J. Dauwels, J. Hu, S. Korl, L. Ping, and F. R. Kschischang, “The factor graph approach to model-based signal processing,” Proc. IEEE, vol. 95, no. 6, pp. 1295–1322, June 2007. [24] C. Novak, G. Matz, and F. Hlawatsch, “Factor graph based design of an OFDM-IDMA receiver performing joint data detection, channel estimation, and channel length selection,” in Proc. 2009 IEEE Int. Conf. Acoustics, Speech, Signal Processing, pp. 2561–2564. [25] Y. Zhu, D. Guo, and M. L. Honig, “A message-passing approach for joint channel estimation, interference mitigation and decoding,” IEEE Trans. Wireless Commun., vol. 8, no. 12, pp. 6008–6018, Dec. 2009. [26] D. Fertonani, A. Barbieri, and G. Colavolpe, “Novel graph-based algorithms for soft-output detection over dispersive channels,” in Proc. 2008 IEEE Global Commun. Conf. [27] H. Niu, M. Shen, J. Ritcey, and H. Liu, “A factor graph approach to iterative channel estimation and LDPC decoding over fading channels,” IEEE Trans. Wireless Commun., vol. 4, no. 4, pp. 1345–1350, July 2005. [28] G. E. Kirkelund, C. N. Manchon, L. P. B. Christensen, E. Riegler, and B. H. Fleury, “Variational message-passing for joint channel estimation and decoding in MIMO-OFDM,” in Proc. 2010 IEEE Global Commun. Conf. [29] T. Wo, J. C. Fricke, and P. A. Hoeher, “A graph-based iterative Gaussian detector for frequency-selective MIMO channels,” in Proc. 2006 IEEE Inf. Theory Workshop, pp. 581–585. [30] T. Wo, C. Liu, and P. A. Hoeher, “Graph-based iterative Gaussian detection with soft channel estimation for MIMO systems,” in Proc. 2008 Int. ITG-Conf. Source Channel Coding.

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[31] ——, “Graph-based soft channel and data estimation for MIMO systems with asymmetric LDPC codes,” in Proc. 2008 IEEE Int. Conf. Commun., pp. 620–624. [32] Z. Shi, T. Wo, P. A. Hoeher, and G. Auer, “Graph-based soft iterative receiver for higher-order modulation,” in Proc. 2010 IEEE Int. Conf. Commun. Technol. [33] IST-4-027756 WINNER II, “D1.1.2 WINNER II channel models,” Sep. 2007. [34] P. Hoeher, “A statistical discrete-time model for the WSSUS multipath channel,” IEEE Trans. Veh. Technol., vol. 41, no. 4, pp. 461–468, Nov. 1992. [35] M. R. Yazdani, S. Hemati, and A. H. Banihashemi, “Improving belief propagation on graphs with cycles,” IEEE Commun. Lett., vol. 8, no. 1, p. 57, Jan. 2004. [36] W. C. Jakes, Microwave Mobile Communications. John Wiley & Sons Inc., 1975. [37] J. Salz and J. Winters, “Effect of fading correlation on adaptive arrays in digital mobile radio,” IEEE Trans. Veh. Technol., vol. 43, no. 4, pp. 1049–1057, Nov. 1994. [38] B. H. Fleury, “First and second-order characterization of direction dispersion and space selectivity in the radio channel,” IEEE Trans. Inf. Theory, vol. 46, no. 6, pp. 2027–2044, Sep. 2000. [39] L. Liu, W. K. Leung, and L. Ping, “Simple iterative chip-by-chip multiuser detection for CDMA systems,” in Proc. 2003 IEEE Veh. Technol. Conf. – Spring, pp. 2157–2161. [40] P. Robertson, P. Hoeher, and E. Villebrun, “Optimal and sub-optimal maximum a posteriori algorithms suitable for turbo decoding,” European Trans. Telecommun., vol. 8, no. 2, pp. 119–125, Mar./Apr. 1997. [41] F. R. Kschischang and B. J. Frey, “Iterative decoding of compound codes by probability propagation in graphical models,” IEEE J. Sel. Areas Commun., vol. 16, no. 2, pp. 219–230, Feb. 1998. [42] C. Knievel, P. A. Hoeher, G. Auer, and A. Tyrrell, “Particle swarm enhanced graph-based channel estimation for MIMO-OFDM,” in Proc. 2011 IEEE Veh. Technol. Conf. – Spring. [43] S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun., vol. 49, pp. 1727–1737, Oct. 2001. [44] S. ten Brink, G. Kramer, and A. Ashikhmin, “Design of low-density parity-check codes for modulation and detection,” IEEE Trans. Commun., vol. 52, no. 4, pp. 670–678, Apr. 2004.

Christopher Knievel (S’11) received the Dipl.-Ing. degree in electrical engineering from the University of Kiel, Germany, in 2009. He is currently working towards his Dr.-Ing. (Ph.D.) degree at the Information and Coding Theory Lab, University of Kiel. His research interests include iterative receiver algorithms in multiple-input multiple-output multicarrier systems, and evolutionary algorithms and their application to channel estimation.

Peter Adam Hoeher (SM’97) received Dipl.-Ing. (M.Sc.) and Dr.-Ing. (Ph.D.) degrees in electrical engineering from RWTH Aachen University, Aachen, Germany, and the University of Kaiserslautern, Kaiserslautern, Germany, in 1986 and 1990, respectively. From October 1986 to September 1998, he has been with the German Aerospace Center (DLR), Oberpfaffenhofen, Germany. From December 1991 to November 1992, he was on leave at AT&T Bell Laboratories, Murray Hill, NJ. In October 1998, he joined the University of Kiel, Germany, where he is a professor of electrical and information engineering. His research interests are in the general area of communication theory and applied information theory with applications in wireless communications and underwater communications, including advanced digital modulation techniques, channel coding, iterative processing, equalization, multiuser detection, interference cancellation, channel estimation, and joint communication and navigation. Dr. Hoeher received the Hugo-Denkmeier-Award (’90) and the ITG award (’07). Between 1999 and 2006, he served as an Associated Editor for the IEEE T RANSACTIONS ON C OMMUNICATIONS.

KNIEVEL et al.: MULTI-DIMENSIONAL GRAPH-BASED SOFT ITERATIVE RECEIVER FOR MIMO-OFDM

Alexander Tyrrell studied at the Ecole Supérieure d’Ingénieurs en Electronique et Electrotechnique (ESIEE) in Paris, France, where he received a master’s degree in electrical engineering, with a major in signal processing and telecommunications, in 2005. Parallel to his final year, he did a master’s of research in digital telecommunications systems at the Ecole Nationale Supérieure des Télécommunications (ENST) in Paris, France. From 2005 to 2009, he worked towards his Ph.D. on the topic of decentralized synchronization for wireless networks in DOCOMO Euro-Labs in Munich, Germany, together with Klagenfurt University, Austria. Afterwards, he worked as a researcher in the area of wireless communications with DOCOMO Euro-Labs until May 2011, and is now working at Rohde & Schwarz in Munich, Germany, as a protocol stack developer for 3GPP LTE systems.

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Gunther Auer (M’02) received the Dipl.-Ing. degree in electrical engineering from the Universität Ulm, Germany, in 1996, and the Ph.D. degree from the University of Edinburgh, UK, in 2000. From 2000 to 2001, he was a research and teaching assistant with the Universität Karlsruhe (TH), Germany. Since 2001, he has been with NTT DOCOMO Euro-Labs, Munich, Germany, where he is a team leader and research manager in wireless technologies research. His research interests include green radio, self-organized networks, and multi-carrier based communication systems, in particular, medium access, cross-layer design, channel estimation, and synchronization techniques.