Robust Turbo Equalization A Minimax Perspective Muhammad Danish Nisar and Wolfgang Utschick, Associate Institute for Signal Processing (MSV), Technical University Munich, Munich, Germany. [email protected], [email protected]

Abstract—The linear equalization based turbo detection schemes often employ the conventional linear equalizers inherently designed under the assumption of perfect channel knowledge. In practical systems, where the knowledge of channel and the knowledge of interference correlations is available only as estimates, the use of such equalizers may lead to an uncontrolled degradation in system performance. In this work, we focus on the design of a priori information aware channel equalizer under the presence of an uncertainty about the channel knowledge and the knowledge of interference plus noise correlations. To this end, we pursue a minimax optimization procedure to design the equalizer such that it offers the best worst-case performance. We study the performance of the proposed minimax robust equalizer in comparison to the conventional equalizer, and analyze its convergence behavior, as a building block of the turbo equalization system, via EXIT charts.

I. I NTRODUCTION The idea of iterative turbo processing between receiver blocks dates back to 1993, when Berrou et al. [1] proposed an exchange of soft information between two soft-in soft-out (SISO) channel decoders operating in parallel. It was shown that the exchange of independent information at each iteration results into significant BER improvements even though the constituent blocks are conventional SISO decoders. Further insights into the turbo principle were later gained by several authors who investigated the concept form different perspectives [2]–[4]. Most notably, Douillard et al. [2] extended the concept of turbo processing between two SISO decoders to a similar exchange of information between other blocks in the receiver chain. Precisely speaking, an exchange of independent soft information between a MAP based trellis detector and a SISO channel decoder was demonstrated to yield BER improvements for an inter-symbol interference channel. The MAP based detectors, employed in early turbo equalization literature, suffer from an exponential increase in complexity and this led to the proposal of linear equalization based turbo detection systems [5], [6]. The proposed receiver effectively replaced the exponential complexity MAP detector by a soft interference canceler, a linear MMSE equalizer and an a priori information aware demodulator, as the detection unit. The extrinsic Log Likelihood Ratios (LLRs) generated by the channel decoder are transformed first into soft estimates of the transmit symbols, whose interfering contribution to the

received signal is subtracted by the interference canceler block. A linear equalization of the residual system followed by an a priori information aware demodulation leads finally to the bit LLRs which are then fed to the channel decoder. An iterative process of exchange of extrinsic (new) information between the equalizer and the decoder continues, and brings significant performance improvements. An interesting interpretation of the linear equalization based turbo detector was offered in [6] and further refined in [7], where turbo equalization is presented as a concatenation of an a priori information aware MMSE equalizer and a channel decoder. In this context, the extrinsic LLRs from the decoder are mapped to the a priori information about the transmit symbols in terms of their means and correlations and the equalizer is designed to incorporate this a priori information directly. In this paper, we follow this later interpretation of the turbo equalization systems. Most work on turbo equalization, including those that incorporate channel estimation into the iterative process such as [8], [9], often employ the a priori information aware equalizers that are inherently designed under the assumption of perfect channel knowledge. Moreover, a perfect knowledge of interference plus noise correlations is also typically assumed. In this paper, we explicitly handle the imperfectness of the channel estimates and the interference plus noise correlation estimates, while designing the a priori information aware equalizer. We analyze the performance of the resulting robust turbo equalization in comparison to the conventional nonrobust design. It is worth mentioning that our work differs from a recent work [10] which considers a perfect knowledge of noise covariance, a specialized structure of channel matrix, and a norm based uncertainty class. II. S YSTEM M ODEL We consider a typical baseband communication system with a matrix channel, r = Hs + ε.

(1)

The channel matrix H ∈ CP ×Q may represent for instance a flat fading MIMO transmission (with P receive and Q transmit antennas), or it may correspond to a block based transmission over a multipath channel possibly with the use of a cyclic prefix. The vector s ∈ CQ contains independent

ˆ H

r

ˆε R

first symbol can be expressed as

Apr-Info-Aware Equalizer (w, a) & Demodulator

Lapr

(ms , Rs )

Channel Decoder

Lapo

Lext Transform to a priori info.

Fig. 1. Structure of the iterative receiver under consideration. To simplify presentation, interleaver and de-interleaver are not shown in this block diagram. The LLRs are labeled w.r.t. channel decoder and Lext = Lapo − Lapr .

complex valued transmit symbols, for instance from a M-QAM constellation, while r ∈ CP contains the received symbols corrupted by the possibly correlated interference and noise term ε ∈ CP , with covariance matrix Rε . We assume that the transmit vector s comes from the mapping of an interleaved coded bit stream. At the receiver, we consider an iterative turbo detector consisting of an a priori information aware equalizer, a demodulator and a channel decoder. This is illustrated in the block diagram in Fig. 1. The channel decoder accepts the demodulator output and performs decoding via a SISO (e.g. BCJR or Viterbi) algorithm. It outputs the a posteriori bit LLRs for detection, and also produces the extrinsic LLRs, that represent the new information that it gained while exploiting the code structure. This extrinsic information is then employed to produce the a priori information for the equalizer in terms of symbol means msi and variances csi [7], [11]. Under the assumption of independent symbols, the symbol covariance matrix is obtained as Cs = diag([cs1 , cs2 , . . . , csQ ]). This leads to the correlation matrix Rs = Cs + ms mH s where ms = [ms1 , ms2 , . . . , msQ ]T denotes the symbol mean vector. The a priori information aware equalizer performs the channel equalization w.r.t. the provided a priori information (ms , Rs ) about the transmit symbols, and feeds the equalized output to the demodulator along with the knowledge of constellation bias and variance. The bit LLRs produced by the demodulator are passed on to the decoder. The iterative process continues until a predefined criterion is met or a fixed number of iterations is reached. In the following, we focus our attention on the design of the a priori information aware equalizer. The main challenge that we tackle here is to incorporate the fact that our knowledge of channel and that of interference plus noise correlations comes from some estimators, and is therefore imperfect. We pursue the minimax optimization technique [12], [13] to arrive at an equalizer that offers the best worst-case performance under this uncertainty of channel and interference plus noise covariance matrix. III. A

PRIORI I NFORMATION

AWARE E QUALIZER D ESIGN

We consider the design of a linear (affine) equalizer for the matrix channel model in (1) in presence of an a priori information (ms , Rs ) about the symbol vector s. In this context, the affine transformation to recover for instance the

sˆ1 = wH r + a,

(2)

where w performs a linear wighted combination, while a accounts for the non-zero mean of the symbol. Our objective here is to make an optimal choice of the a priori information aware equalizer (w, a), where optimality is measured in terms of the equalization MSE, ρ´(H, Rε , w, a) = E[|eH s − sˆ1 |2 ] H = q H Rs q + aaH − q H ms aH − amH s q + w Rε w, (3)

with q = e − H H w, where e ∈ CQ is the first column of IQ , the identity matrix of dimensions Q. Before turning our attention to the design of robust equalizer under uncertainty of channel matrix and the interference plus noise covariance matrix, we first consider the standard equalizer design. A. Perfect Knowledge – Conventional Design With MSE as our cost function, the equalizer design problem, under a perfect knowledge of the channel matrix H and the interference plus noise covariance matrix Rε , reads as min

(w,a)

ρ´(H, Rε , w, a),

(4)

which leads to the following closed form expression for the optimal equalizer [6], [7]
−1 wCV = HCs H H + Rε HCs e, (5)
H H aCV = e − wCV H ms , (6)

where the subscript (•)CV signifies the conventional design approach. We note, that besides the knowledge of a priori information (ms , Rs ), the equalizer needs knowledge of the channel matrix and the interference plus noise covariance matrix. B. Imperfect Knowledge – Minimax Robust Design

In case, we do not have a perfect knowledge of H and Rε , the equalizer (wCV , aCV ) ceases to be optimal. In fact, since it ignores the uncertainty of parameters it could possibly lead to a severe degradation of performance in practice. In this sub-section, we consider the design of the equalizer while incorporating the fact that we only have estimated versions of ˆ and R ˆ ε. H and Rε available to us, as H We pursue the deterministic approach to optimize the equalizer performance under uncertainty, i.e. we would like to design the equalizer (w, a) such that the equalization MSE over the unknown actual transmission scenario is guaranteed to stay below a certain maximum level. To this end, we let ˆ + H∆ , the actual channel matrix be represented as H = H and then constrain the set of possibly encountered channels and the interference plus noise covariance matrices (H∆ , Rε ) via a Kullback-Leibler (KL) divergence [14] based distance measure. Thus, we define the uncertainty set as 2 UKL = {(H∆ , Rε ) : KL(H∆ , Rε ) ≤ δK , Rε  0},

(7)

where KL(H∆ , Rε ) denotes the KL divergence between the estimated and the actual system model. Under the assumption

of Gaussian interference plus noise signals, the KL divergence between the conditional probability densities associated with the two models1 , Actual Model : f (r | s) = N (Hs, Rε ), ˆ R ˆ ε ), Estimated Model : fˆ(r | s) = N (Hs,

(8) (9)

can be interpreted as the degree of mismatch and its expected value w.r.t. the transmit vector s can be obtained as "Z ! # f (r | s) dr KL(H∆ , Rε ) = Es f (r | s) log fˆ(r | s) ˆ −1 H Rs ) + tr(R ˆ −1 Rε − IP ) = tr(H H R ∆

ε

∆

ε

ˆ −1 Rε )). − log(det(R ε

(10)

This measure has been used for instance in [15], [16] to define the uncertainty class for the design of a robust LS estimator and a conventional MMSE equalizer. With the uncertainty set characterized, the design problem of minimax robust equalizer can now be posed as follows, min

max

(w,a)

(H∆ ,Rε )∈UKL

ρ(H∆ , Rε , w, a),

(11)

ˆ + H∆ , Rε , w, a) in (3). We where ρ(H∆ , Rε , w, a) = ρ´(H note here that the cost function ρ(H∆ , Rε , w, a) is jointly convex in H∆ and Rε , so that the inner maximization problem is non-convex. Nevertheless, we show in [17] that a form of strong Lagrangian duality holds. This owes primarily to the fact that since the constraint set UKL is convex (it is defined as sub-level set of a convex function), the constraint is always ⋆ active. This allows us to conclude [17, Proposition 1] that H∆ , ⋆ Rε constitute a strict local maximum of ρ(H∆ , Rε , w, a), if the KKT conditions are satisfied and the associated Lagrangian 2 multiplier λ⋆ > kwkRˆ ε . A crucial observation here is that 2 with λ⋆ > kwkRˆ ε , the Lagrangian function becomes globally concave in H∆ and Rε which implies the global optimality ⋆ of H∆ , R⋆ε [17, Proposition 2]. In summary, the existence of strong Lagrangian duality allows us to write the problem equivalently as min

(w,a)

max

min

(H∆ ,Rε ) λ>kwk2R ˆ

ρ(H∆ , Rε , w, a)

ε

2 + λ(δK − KL(H∆ , Rε )). (12)

The KKT optimality conditions for inner maximization lead to the following worst case uncertainties, ˆ εw
R −1 ⋆ ˆH , (13) α amH H∆ = s Rs − q λ   −1 wwH ˆ −1 , (14) R⋆ε = R ε − λ where ˆ H w, qˆ = e − H

α=

1 , 1−β

β=

ˆ εw wH R . λ

(15)

1 Here N (b, C) denotes the Gaussian distribution with mean vector b and covariance matrix C.

⋆ Plugging H∆ and R⋆ε back into the Lagrangian function, we get the dual function

H Rs qˆ − ms aH + aaH θ(w, a, λ) =α Rs qˆ − ms aH R−1 s −1 H 2 − amH s Rs ms a + λ log(α) + λδK ,

(16)

and the remaining robust equalizer design problem from (12) can be equivalently expressed as min

(w,a)

min

λ>kwk2R ˆ

θ(w, a, λ).

(17)

ε

It can be shown [17] that owing to the special structure of the uncertainties, we can reduce the problem dimensions by ˆ ε−1/2 , where a pre-processing of the system model via U1H R U1 ∈ CP ×Q comes from the following SVD,     ˆ −1/2 H ˆ = U1 U2 Σ1 V H ∈ CP ×Q , (18) R ε 0

with Σ1 ∈ CQ×Q being the diagonal matrix containing the Q singular values. Solving the reduced dimensional problem, we finally arrive at the following expressions for the minimax robust equalizer, ˆ ε− H /2 U1 g ⋆ , wMR = R

(19)

H

(20)

aMR = γ qˆ ms ,

where g ⋆ and λ⋆ are obtained via numerical solution of the following convex optimization problem,
2 ˆ +λ log(α)+λδK . {g ⋆ , λ⋆} = argmin αqˆH Rs − γms mH s q g,λ>g H g

(21)

ˆ and Here qˆ is defined in the new notations as qˆH = eH −g H G H ˆ G = Σ1 V , while the scalars α, β and γ can be given as, α=

1 , 1−β

β=

gH g , λ

γ=

α −1 1 + αβmH s Rs ms

(22)

This completes the design of the maximally robust a priori information aware spatial MMSE equalizer (wMR , aMR ) in comparison to the conventional non-robust design (wCV , aCV ). We note that the associated worst-case channel uncertainty ⋆ H∆ and the worst-case interference plus noise covariance matrix R⋆ε for the minimax robust equalizer can be obtained via (13) and (14) respectively by substituting w = wMR , a = aMR and λ = λ⋆ . For an arbitrary equalizer (w, a), the worst-case uncertainty can be obtained by plugging in w and a in (13) and (14) and then optimizing the dual function over λ. IV. S IMULATION R ESULTS We provide simulation results for the transmission over a 5×3 MIMO transmission channel (i.e. P = 5 and Q = 3) with i.i.d. complex Gaussian distributed channel coefficients. The transmit symbols s is assumed to come from an encoded and interleaved bit stream that is mapped to a 16-QAM constellation. Besides the uncorrelated Gaussian noise, the received signal vector r is impaired by the spatially correlated interference term with a given Carrier-to-Interference (C/I) level.

KL(H∆ , Rε ) ≤ δ, ˆ −1 ˆ ˆ HR σs2 eH H ε He

0

−2

WC Equalization MSE (dB)

The a priori information about s is provided to the equalizer in terms of the mean ms and the correlation matrix Rs . For the sake of simulations, these a priori information are derived under the common assumption of Gaussian LLRs [18] for the bits that constitute a symbol. Thus, the LLRs are assumed to have the distribution N (±µ, σ 2 = 2µ), i.e. their variance is equal to twice the absolute value of their mean. With the sign of mean chosen w.r.t. the actual bit being ±1, the quality of the a priori information is solely controlled via the mean µ. Note further that for easier interpretation of simulation results we use a normalized KL measure

−4

−6

−8

−10

−12 Conventional Equalizer Minimax Robust Equalizer

(23)

−14

0

5

2 δK ,

A. Equalization MSE Analysis In Fig. 2, we compare the performance of the two a priori information aware equalizers, in terms of the equalization MSEs for their respective worst-case channels. We consider two interference levels namely C/I = 20 dB and 30 dB respectively in the two sub-figures. In each subfigure, we show the comparison at four uncertainty levels, δ = −5, −10, −15 and −20 dBs. It is observed that the performance of the proposed equalizer is superior at all uncertainty levels, and as expected the performance in terms of the worst-case equalization MSEs converges as the uncertainty magnitudes decreases. Additionally, it can be seen that at high uncertainties, while the MSE of proposed equalizer only saturates with increasing SNR, the MSE performance of the conventional design worsens with increasing SNR. This results from the fact that the conventional equalizer ignores the presence uncertainty in its design. B. Convergence Analysis via EXIT Charts S. t. Brink proposed in [18] a useful tool for analyzing the performance of turbo processing based receivers. The so called EXtrinsic Information Transfer (EXIT) chart plots the mutual information at the input of a block vs the mutual information at its output. Thus it basically plots the ability of the constituent block of a turbo system in isolation. Since the output of one block forms the input to the other block, the EXIT curves of the two blocks can be drawn into a single plot with axes swapped for one of the blocks. Such an EXIT chart can be used to analyze the performance of turbo processing system without

15 SNR (dB)

20

25

30

(a) C/I = 20 dB 0 −2

WC Equalization MSE (dB)

in place of KL(H∆ , Rε ) ≤ such that a design value of δ = 0 corresponds to no uncertainty, while a value of δ = 1 corresponds to a maximum uncertainty that leads to an all zero equalizer being optimal having a MSE equal to σs2 . We analyze the equalizer performance in two scenarios. In the first, it is studied in isolation and we compare the performance of the conventional equalizer (wCV , aCV ) and the proposed minimax robust equalizer (wMR , aMR ) in terms of equalization MSE for their respective worst case channels. While in the second scenario, the equalizer is considered as part of a turbo equalization system (i.e. in concatenation with a decoder) and the performance is analyzed via EXIT charts.

10

−4 −6 −8 −10 −12 −14 −16

Conventional Equalizer Minimax Robust Equalizer 0

5

10

15 SNR (dB)

20

25

30

(b) C/I = 30 dB Fig. 2. Performance comparison in terms of equalization MSE for worstcase channels at different uncertainty levels. Solid lines represent curves for δ = −5 dB, dashed lines for δ = −10 dB, dot-dashed lines for δ = −15 dB, and dotted lines for δ = −20 dB.

the need to simulate the whole iterative process. Furthermore it provides useful insights on the convergence behavior of different schemes. In Fig. 3, we plot the EXIT characteristic curves for the two equalizers at two C/I levels in the two sub-figures. In each sub-figure, the curves are plotted for the equalizers at an uncertainty level of δ = −10 dB and two operating SNRs, namely SNR = 15 dB and SNR = 30 dB. Along the xaxis, the mutual information between the a priori information at equalizer input and the actual bits is plotted, while along the y-axis the mutual information between the output of the equalizer/demodulator and the actual bits is recorded. We note that with no a priori information at the input, the output mutual information of the equalizers is solely determined by the operating SNR. As the quality of the input a priori information improves, both the equalizers produce an increasingly better mutual information (MI) at the output. Also shown in figures, is the rate 1/2 channel decoder EXIT curve from [19]. The convergence behaviors of the equalizers can be analyzed by tracing the staircase trajectories starting

1 Mutual Info. at Equalizer Output (Decoder Input)

from equalizers with no a priori information. It can be seen that the iterative performance improves until an intersection between the EXIT curves of the equalizer and the decoder. For instance at SNR of 15 dB, the EXIT curves for both equalizers intersect rather early with the decoder curve at the decoder output MI of around 0.4. However as the SNR increases to 30 dB, we note that while the conventional equalizer’s curve intersects still around an MI of 0.45, the trajectory of the robust equalizer continues as far as 0.96 leading to a significantly better detection quality. Similar comments hold for the sub-figure with C/I = 30 dB, where the tunnel becomes even wider, implying that the convergence is attained with lower number of iterations. Thus in both scenarios, we observe gains of the proposed minimax robust equalizer as compared to the conventional non-robust equalizer.

Conventional Equalizer Minimax Robust Equalizer

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

0

0.2 0.4 0.6 0.8 Mutual Info. at Equalizer Input (Decoder Output)

1

(a) C/I = 20 dB

In this paper, we focused on the design of a priori information aware equalizer – a crucial building block of a turbo equalization system – in presence of an uncertainty about the channel knowledge and the knowledge of interference plus noise correlations. We pursued the minimax optimization approach to design the equalizer such that it offers the best worst-case performance thereby guaranteeing a certain (superior) level of performance even in presence of uncertainty. The performance of the proposed minimax robust equalizer is compared with that of the non-robust conventional MMSE equalizer, once the equalizers are placed in concatenation with a channel decoder. It is found that the robust equalizer offers significantly superior worst-case performance as compared to the conventional design approach. R EFERENCES [1] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon Limit Error-Correcting Coding and Decoding: Turbo Codes (1),” in IEEE International Conference on Communications ’93, May 1993, pp. 1064– 1070. [2] C. Douillard, M. Jezequel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, “Iterative Correction of Intersymbol Interference: Turbo Equalization,” European Transactions on Telecommunications and Related Technologies, vol. 6, pp. 507–511, 1995. [3] J. Hagenauer, “The Turbo Principle: Tutorial Introduction and State of the Art,” in International Symposium on Turbo Codes, March 1997, pp. 1–11. [4] X. Wang and H. V. Poor, “Iterative (Turbo) Soft Interference Cancellation and Decoding for Coded CDMA,” IEEE Transactions on Communications, vol. 47, pp. 1046–1062, July 1999. [5] D. Raphaeli and A. Saguy, “Linear Equalizers for Turbo Equalization: A New Optimization Criterion for Determining the Equalizer Taps,” in 2nd International Symposium on Turbo Codes, September 2000, pp. 371–374. [6] M. T¨uchler and A. Singer and R. K¨otter, “Minimum Mean Squared Error Equalization Using A-priori Information,” IEEE Transactions on Signal Processing, vol. 50, pp. 673–683, March 2002. [7] G. Dietl and W. Utschick, “Complexity Reduction of Iterative Receivers Using Low-Rank Equalization,” IEEE Transactions on Signal Processing, vol. 55, pp. 1035–1046, March 2007. [8] M. Sandell and C. Luschi and P. Strauch and R. Yan, “Iterative Channel Estimation using Soft Decision Feedback,” IEEE Global Telecommunications Conference, pp. 3728–3733, 1998. [9] R. Otnes and M. T¨uchler, “Iterative Channel Estimation for Turbo Equalization of Time-Varying Frequency-Selective Channels,” IEEE Transactions on Wireless Communications, vol. 3, pp. 1918–1923, November 2004.

Mutual Info. at Equalizer Output (Decoder Input)

1

V. C ONCLUSION

Conventional Equalizer Minimax Robust Equalizer

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

0

0.2 0.4 0.6 0.8 Mutual Info. at Equalizer Input (Decoder Output)

1

(b) C/I = 30 dB Fig. 3. Performance comparison via EXIT curves at fixed uncertainty level of δ = −10 dB. Thick continuous curve (without markers) represents the EXIT curve for a rate 1/2 convolutional decoder. The equalizer EXIT curves are shown for two SNR values. The dashed curves show the performance at SNR = 15 dB, while the solid curves show the performance at SNR = 30 dB. [10] N. Kalantarova, S. S. Kozat, and A. T. Erdogan, “Robust Turbo Equalization Under Channel Uncertainties,” Under review in IEEE Transactions on Communications, 2010. [11] M. D. Nisar, H. Nottensteiner, and W. Utschick, “Iterative Suppression of Co-Channel Interference,” in Seventeenth European Signal Processing Conference, EUSIPCO, August 2009, pp. 461–465. [12] S. Verdu and H. V. Poor, “On Minimax Robustness: A General Approach and Applications,” IEEE Transactions on Information Theory, vol. 30, pp. 328–340, March 1984. [13] S. A. Kassam and H. V. Poor, “Robust Techniques for Signal Processing: A Survey,” Proceedings of the IEEE, vol. 73, pp. 433–481, March 1985. [14] S. Kullback and R. A. Leibler, “On Information and Sufficiency,” Annals of Mathematical Statistics, vol. 22, pp. 79–86, March 1951. [15] B. C. Levy and R. Nikoukhah, “Robust Least-Squares Estimation with a Relative Entropy Constraint,” IEEE Transactions on Information Theory, vol. 50, pp. 89–104, January 2004. [16] Y. Guo and B. C. Levy, “Robust MSE Equalizer Design for MIMO Communication Systems in the Presence of Model Uncertainties,” IEEE Transactions on Signal Processing, vol. 54, pp. 1840–1852, May 2006. [17] M. D. Nisar and W. Utschick, “Minimax Robust A-priori Information Aware Channel Equalization,” Accepted for publication in IEEE Transactions on Signal Processing, 2011. [18] S. t. Brink, “Convergence Behaviour of Iteratively Decoded Parallel Concatenated Codes,” IEEE Transactions on Communications, vol. 49, pp. 1727–1737, October 2001. [19] M. T¨uchler and R. K¨otter and A. Singer, “Turbo Equalization: Principles and New Results,” IEEE Transactions on Communications, vol. 50, pp. 754–767, May 2002.