Iterative Decision Feedback Equalization Using Online Prediction

In this article, a new category of soft-input soft-output (SISO) minimum-mean square error (MMSE) ﬁnite-impulse response (FIR) decision feedback equalizers (DFEs) with iteration-wise static ﬁlters (i.e. iteration variant) is investigated. It has been recently shown that SISO MMSE DFE with dynamic ﬁlters (i.e. time-varying) reaches very attractive operating points for high-data rate applications, when compared to alternative turbo-equalizers of the same category, thanks to sequential estimation of data symbols. However the dependence of ﬁlters on the feedback incurs high amount of latency and computational costs, hence SISO MMSE DFEs with static ﬁlters provide an attractive alternative for computational complexity-performance trade-off. However, the latter category of receivers faces a fundamental design issue on the estimation of the decision feedback reliability for ﬁlter computation. To address this issue, a novel approach to decision feedback reliability estimation through online prediction is proposed and applied for SISO FIR DFE with either a posteriori probability (APP) or expectation propagation (EP) based soft feedback. This novel method for ﬁlter computation is shown to improve detection performance compared to previously known alternative methods, and ﬁnite-length and asymptotic analysis show that DFE with static ﬁlters still remains well-suited for high-spectral efﬁciency applications.


I. INTRODUCTION
Joint detection and decoding through iterative exchange of extrinsic information between a soft-input soft-output (SISO) detector and a SISO decoder can achieve near capacity performance with a well-designed coding scheme. In particular, turbo equalization seeks to provide robust high-throughput links over strongly frequency-selective channels, whose frequency responses' incorporate spectral nulls.
However, unlike SISO finite-impulse response (FIR) minimum mean squared error (MMSE) linear equalizers (LE) [2], practical SISO MMSE FIR DFE structures have not been thoroughly investigated, and only gathered attention in recent years [1], [3], [4]. In this article, a novel filter computation approach for such structures is proposed, through a predictive estimation of the decision feedback reliability. In the following, we only consider SISO MMSE FIR receiver structures.
A widespread and widely accepted nomenclature for categorizing such equalizers has not been established in the literature. In our view, the work in [5] provides an accurate categorization based on the occurrence of SISO adaptive filter updates with prior information. This nomenclature is attractive as it is directly related to the assumptions used for the derivation of the equalizer, and it also gives some insights on both the decoding performance and the computational complexity.
Time varying (TV) FIR filters are updated at each single symbol, by fully exploiting prior information, and they are well-suited for doubly-selective channels. Iteration varying (IV) FIR filters are static and updated only at the beginning of each turbo iteration, by using the knowledge of overall quality of the feedback, thus reducing the involved computational costs.
The nature of feedback "decisions" (or rather "estimates") also impacts the DFE error-rate performance. Hard decisions can be taken, as in conventional DFE [6], but this results in a significant amount of error propagation and unpredictable behavior [7], unless complex heuristics are used [8]. Alternatively, a widespread category of DFE with soft feedback use a posteriori probability (APP) distributions, due to its relative simplicity and fair performance [3], [4], [9].
Finally, soft feedback based on expectation propagation (EP) [10] improves the performance of TV DFE [1], in addition to being more predictable, due to relatively lowered correlations with the equalized estimates [11].
The design of optimum IV DFE receivers is however non-trivial. Indeed, static filters should depend on the decision feedback reliability, and the decision feedback naturally depends on the filters. As a consequence, there does not exist closed-form expression of the optimal filter due to this non-linear "chicken-and-egg" inter-dependence.
IV DFE proposals in the literature use a variety of sub-optimal heuristics for filter computation [3], [4], [9]. For DFE with hard decisions, the conventional approach is to assume a perfect feedback, causing error propagation and performance degradation in real operation [7]. Using soft APP decisions while still assuming perfect feedback partially mitigates error propagation, as soft symbols' magnitudes scale down with unreliability [12]. The first reference to incorporate APP soft feedback reliability in filter computations is the receiver proposed in [9] for the special case of binary-phase-shift-keying (BPSK) modulation. The direct dependencies between soft symbols and log-likehood ratios (LLRs) for the BPSK constellation enable the use of a tractable density evolution on the APP LLR distribution, given a prior LLR distribution from the decoder. This property is used by the BPSK receiver of [9] to estimate the decision feedback reliability.
However, this scheme cannot be directly generalized to high-order constellations; hence [3] proposed a receiver that implements a LE at the first turbo iteration, and then it uses previous turbo iteration's demapper APP LLRs to estimate soft-symbol statistics for IV DFE. Note that this approach is only possible with Gray mapped constellations. More recently, [4] proposed pre-equalization with LE over a few symbols, and then to use APP distribution of these symbols to estimate APP soft-feedback statistics for IV DFE.
Considering the previous developments, our contributions can be summarized as follows: • An IV DFE with soft APP feedback based on online prediction is formulated. The technique in [9] is extended with models from the field of semi-analytic performance prediction [13]- [15], used both for physical layer abstraction or link layer adaptation.
• A low complexity IV DFE with soft EP-based feedback is proposed, as an IV extension of [1], by removing the bias from predicted APP estimates' reliability.
• A novel approach for estimating soft decision feedback reliability is proposed, based on online binary and symbol-wise semi-analytic performance prediction.
• The accuracy and the convergence of these prediction schemes are evaluated and the performance of IV equalizers are compared. To our knowledge, the accuracy of different IV DFE heuristics is not compared elsewhere, despite their direct impact on DFE error propagation.
The remainder of this paper is organized as follows. Section II presents the system model and the general structure of proposed IV DFE receivers. The involved semi-analytic prediction scheme June 3, 2021 DRAFT is developed and analyzed in Section III. Finally, before concluding, the proposed equalizers with online prediction are numerically analyzed in Section IV, in the finite-length and the asymptotic regimes.

Notations
Bold lowercase letters are used for vectors: let u be a N × 1 vector, then u n , n = 1, . . . , N are its entries, unless specified otherwise. Capital bold letters denote matrices: for a N × M matrix A, [A] n,: and [A] :,m respectively denote its n th row and m th column, and a n,m = [A] n,m is the entry (n, m). I N is the N × N identity matrix, 0 N,M and 1 N,M are respectively all zeros and all ones N × M matrices. e n is the N × 1 indicator whose only non-zero entry is e n = 1.
Operator Diag(u) denotes the diagonal matrix whose diagonal is defined by u. R, C, and F k are respectively real, complex and k th order Galois fields. Let x and y be two random variables, µ x = E[x] is the expected value, σ 2 x = Var[x] is the variance and σ x,y = Cov[x, y] is the covariance. The probability of the discrete random variable x taking the value α is P[x = α]. For random vectors x and y, we define µ µ µ x = E[x] and covariance matrices Σ x,y = Cov[x, y] and Σ x = Cov[x, x]. CN (µ x , σ 2 x ) denotes the circularly-symmetric complex Gaussian distribution of mean µ x and variance σ 2 x .

A. Single Carrier BICM Transmission
Single carrier transmission using a bit-interleaved coded modulation (BICM) scheme is considered.
Let b ∈ F K b 2 be a K b -bit information packet. b is encoded and then interleaved into a codeword d ∈ F K d 2 . A memoryless modulator ϕ then maps d to the symbol block x ∈ X K , where X is the M th order complex constellation, with zero mean and average power σ 2 x = 1, and with Q = log 2 M . The Q-word associated to x k is denoted d k = [d] Q(k−1)+1:Qk , and both d k,q and ϕ −1 q (x k ) denote the value of the q th bit labelling the x k , i.e. d Q(k−1)+q , with q = 1, . . . , Q.
We consider an equivalent baseband frequency selective channel with the impulse response , of delay spread L. Thus the received samples are for k = 1, . . . , K, where the noise w k is modelled as an additive white Gaussian noise (AWGN), with CN (0, σ 2 w ), i.e. a zero mean Gaussian process with variance σ 2 w . The receiver is assumed to be ideally synchronized in time and frequency, and perfect channel state information is available. We consider an iterative BICM receiver where a SISO channel decoder and a SISO symbol receiver exchange extrinsic information for iterative detection and decoding, as shown in Fig. 1. A priori, extrinsic and a posteriori log likelihood ratios (LLRs) of coded bits d are respectively denoted L p (·), L e (·) and L(·), with respect to the SISO receiver. The considered SISO symbol detector consists of a SISO channel equalizer and a symbol-wise SISO demapper module, as shown in Fig. 1. The latter consits in a soft-output maximum a posteriori (MAP) demapper, and a soft-mapping unit, an APP distribution estimator, and the eventual use of the so-called "Gaussian division" operation for computing extrinsic symbol feedback (see discussion below and [1]).
The SISO equalizer computes an estimate x e k of x k , affected by a residual noise of variance v e x,k , whereas the SISO demapper uses these estimates to compute L e (d), and to deliver soft feedback x d k to the equalizer for additional interference cancellation, such that v d x,k is the variance of residual interference and noise of the feedback (this will be discussed in more detail afterwards).
In short, soft mapper uses LLRs from the decoder to estimate a prior distribution on x k = α, Soft demapper estimates a posteriori symbol distribution which allows computing extrinsic LLRs towards the decoder

Exact TV DFE receivers carry out interference cancellation with anti-causal estimatesx
Causal estimates are directly dependent on the values of (x e k , v e x,k ) from previous turbo iterations, and they depend on mapping constraints. The equalized estimates are [1] x IV DFE is obtained whenv dfe k is independent of k, withv dfe k =v dfe , all filters being invariant, f , g c and g a , as in [4]: The variances of soft interference cancellation estimates arē wherev a x andv c x are respectively the overall reliability of anti-causal and causal estimates. For interference cancellation, the set of anti-causal estimates are available before equalization, and an accurate estimate of their reliability is given by the least-squares estimation;v a x = K −1 K−1 k=0v a x,k . In most SISO DFE structures, the anti-causal estimates are the prior estimates given by the decoder 1 As stated in the introduction, the core of the problem lies in the computation ofv c x . A simple, but inaccurate solution to this is the "perfect decision assumption" :x c k are all assumed to be equal to x k , yieldingv c x = 0. This approach is sufficient at high SNR operating points, but as shown in [1], [7], it degrades performance in moderately or severely selective channels.
Hence, with these notations, this paper's main objective is to evaluate novel prediction methods to computex c k for optimizing the inter-symbol interference (ISI) mitigation performance of IV DFE. In the following the cases of IV DFE based on APP-based soft feedback (see Fig. 2), and EP-based soft feedback (see Fig. 3) will be discussed.

C. APP Soft Feedback Computation
A common approach to compute soft feedback for DFE is to use APP estimates, as in IV structures of [3], [4], [9]. Derivation of a DFE APP for TV FIR is available in [1].
These soft estimates are given by the mean and the variance of the posterior symbol distribution For the IV DFE APP filter computation, an invariant variance γ d x is needed, as with the causal reliabilityv c x . Unlike the anti-causal reliability, γ d x cannot be estimated using the causal estimates , as causal estimates are only available once the filter is computed, and the equalization is being carried out. Thus, a predictive estimation is required. This receiver corresponds to the Fig. 2

D. Predictive EP-based Soft Feedback Computation
In [1], an expectation propagation (EP) based soft feedback is used within a TV DFE which proved to bring several performance improvements. Unlike APP estimates, EP-based estimates carry only the extrinsic information brought by the demapper, and prevents DFE from relying on its own bias, and improves the asymptotic predictability of the receiver. For TV DFE, these estimates are obtained by the division of two Gaussian PDFs, which yields another Gaussian PDF with mean and variance given by For the sake of simplicity, this operation is referred to as "Gaussian division".
For IV DFE EP structure, this feedback is not adapted, as the invariant filter is unable to adapt its coefficients to handle the strong variations of v d x,k , which depends on the instantaneous APP variance γ d x,k . Hence we propose to use a feedback based on overall APP reliability, with where a predicted invariant APP variance γ d x is used to generate the feedback. Moreover EPbased estimates have an invariant variance, i.e. v d x,k , ∀k, as the causal reliability is directly related to the predicted APP variance. The receiver based on this structure is illustrated in the

III. SEMI-ANALYTIC ABSTRACTION OF SISO FIR DFE
In this section, a prediction model for the turbo DFE structure of Equation (7) is exposed, without loss of generality, for the case where anti-causal estimates are given by decoder's extrinsic LLRs, i.e.v a x = v p x . Such models are usually used for handling physical layer link quality prediction that is necessary to enable link adaptation with low computational complexity. The originality is that, in our context, it will be exploited for online estimation of the reliability of causal estimates for SISO DFE filter computation.

A. General Structure and Analytical Equalizer Model
SISO DFE is modelled with two independent components; an analytical model for the equalizer, and a numerical model for the soft demapper. Unlike asymptotic transfer models (K → +∞) used in extrinsic information transfer (EXIT) analysis [16], finite-length transfer models are used for characterizing the demapper, as prior works on performance prediction noted their positive impact on accuracy [13], [14]. Fig. 4. Block diagrams for bit-wise (left) and symbol-wise (right) causal reliability prediction schemes.
Following Eq. (7), the IV DFE-IC output reliability v e x is modelled by a function φ REC as wherev dfe x is given by Eq. (8), withv a x = v p x . This function is strictly increasing withv c x ∈ [0, σ 2 x ]. As an analytical model is unavailable for characterizing the demapper, it is modelled with a look-up table (LUT) φ DEM , given byv wherev c x is the expected value of causal estimates' variance, taken over realizations of the channel noise, the equalizer outputs and the prior LLRs. The second argument '·' in Equation (13) models prior information, and the exact nature of the argument depends on the selected prediction approach. Improvements proposed in the upcoming subsection concern this module.
Since equalizer and demapper iteratively exchange reliabilities, the two functions representing their model must be composed to yield a recursive equation onv c x . Hence by using n = 0, . . . , N pred for indexing recursions, we havē If f pred admits a unique fixed-point onv c x , then the desired predicted reliability estimate is this fixed-point. Moreover, the optimality of IV DFE-IC strongly depends onv c x and hence on the accuracy of Φ DEM . Fig. 4 illustrates reliability prediction structures with the two semi-analytical models that will be introduced below.

B. Numerical Demapping Models for APP/EP
Modelling the demapper with prior information is challenging due to its highly non-linear behavior, and due to strong simplifying assumptions. The main focus of the numerical model DRAFT June 3, 2021 will be the model of the posterior symbol distribution's variance γ d x , required for using the DFE with APP feedback, discussed in Subsection II-C. Besides, as the variance v d x of the proposed predictive EP feedback, in Subsection II-D, is analytically linked to γ d x , numerical model of APP estimates is common for both types of feedback.

1) Mutual Information based Prediction (Bit-wise):
In the BPSK receiver of [9], a prediction scheme is considered, assuming input/output LLRs of the demapper to be consistent Gaussian, i.e. L (·) (d k,q ) ∼ N (d k,q µ (·) , 2µ (·) ), whered k,q = 1−2d k,q , and where (·) is p, e or void, depending on concerned LLRs. Using a semi-analytical density evolution, parameter µ e of extrinsic LLRs is predicted using µ p . The parameter µ (·) is bijectively linked to the average mutual information (MI) between LLRs and the associated coded bits, usable for binary prediction as in [13]. Hence using such formalism, the approach of [9] can be extended to any constellation and mapping.
More specifically, the demapper behaviour is numerically integrated for each γ d x,k , k = 1, . . . , K, under the assumption that prior LLRs are consistent Gaussian with the parameter µ p , and the assumption that the residual ISI and noise affecting the equalized symbols x e k are Gaussiandistributed, i.e. x e k ∼ CN (x k , v e x ). Hence with these conditions, a LUT on µ p and v e x is built withv wherev c x is the variance of residual error on APP/EP soft symbols and the priors' parameter µ p is measured with a maximum-likelihood (ML) estimator (see Fig. 4, left) In the case of APP feedback, i.e. forv c x,k = γ d x,k , the expectation in Equation (15) becomes where the APP probability mass function of a dummy symbol x ∈ X is where Z is the normalization constant such that x∈X D(x, x e , L p ) = 1. Moreover, considering the EP feedback's analytical expression in Equation (11), we have The binary prediction scheme above appeared to yield too optimistic estimates in [9], and instead Lopes et al. resorted to obtain µ e and µ p through BPSK channel estimators, which circumvents consistent Gaussian LLR approximation.
More specifically, this problem ensues from well known issues with regards to performance prediction of turbo iterative systems, for which the consistent Gaussian approximation of LLRs was shown to be only accurate at the zeroth turbo-iteration, and in the asymptotic limit. Otherwise inaccurate estimates propagate across turbo iterations due to the internal non-linear dynamics of channel decoding [17]. To overcome this prediction bias, prediction based on a two-parameter LLRs' model has been shown to be much more accurate [18]. Such models consider L (·) (d k,q ) ∼ N (d k,q µ (·) , η (·) µ (·) ), where η (·) is no longer 2. The ML estimator used for measuring µ p in the binary prediction is very sensitive to η p , which is the reason why the binary prediction is not robust enough in practice. Hence, a single-parameter demapper model with reasonable estimation complexity has been preferred. We searched for the parameter which is the least sensitive to the variations of prior LLRs' variance-to-mean ratio η p .

2) Prior Variance based Prediction
Following a thorough and almost exhaustive study of the different alternative parameters for tracking evolution ofv c x , anti-causal variance v p x has been found to be sensitive to the changes on η p , very similarly tov c x , with the advantage of v p x being directly computable online using a simple least-squares estimation. Hence, we propose the following LUT where both input v p x and outputv c x are numerically integrated using prior LLRs generated for a fixed value of η p . Indeed, as η p cannot be accurately measured online, the conventional consistent approximation [13] is kept with η p = 2. In the following, we will assess its impact on the prediction accuracy.

3) Robustness of demapper prediction:
The sensitivity of the considered prediction schemes to variations in η p is evaluated. This aspect is important for characterizing the robustness of iterative receiver prediction schemes, as the hypothesis η p = 2, used for LUT generation, is only true at the initial turbo-iteration and then it varies [17].
An AWGN channel is simulated with blocks of 16-QAM symbols with K = 1024, to emulate the output x e of the equalizer, for v e varying from −15 to 15 dB, along with Gaussian-distributed prior LLRs generated with prior MI I A varying from 0 to 1 bit (and hence determining µ p ), with η p varying from 1 to 3. The average mean squared error (MSE) between the predicted causal covariance and true causal covariance is measured, and plotted in Fig. 5. The left side of the figure provides results for APP feedback, and the right side for EP-based feedback. The binary approach is seen to be severely impacted by the changes in η p , whereas the symbol-wise approach, although not perfect, remains more robust. Considerable differences are seen at low to medium SNR for high prior information, which suggests that symbol-wise schemes would have an advantage at the decoding threshold in asymptotic behaviour, i.e. when a high number of turbo-iterations are used. Oppositely, without any turbo-iteration, both schemes would perform identically. DRAFT June 3, 2021

C. Convergence Analysis
The convergence of the proposed iterative semi-analytical prediction schemes could be assessed formally through fixed-point analysis of Eq. (14). However, due to the untractable non-linear expression of Φ DEM , an analytic approach is not possible, and we resort to numerical evaluations.
In the following, the convergence of the symbol-wise prediction scheme is evaluated. The convergence speed of the prediction scheme is also evaluated numerically in order to determine optimal parameters of the algorithm. Indeed, the fixed-pointv c x =v c x [∞] is reached more or less quickly depending on if the initial valuev c . In particular, due to the near flat evolution of f pred forv c x close to σ 2 x = 1, initializing withv c x [0] = 1 results in fast convergence at low SNRs, and high anti-causal covariance, but slow convergence otherwise. Oppositely withv c x [0] = 0 faster convergence is achieved for high SNRs and low anti-causal covariance. This behaviour is illustrated for Proakis-C 16-QAM APP covariance in Figure 7.
We propose to use the heuristicv c x [0] = min(1, σ w ), when v p x > 0.5, where the standard deviation of the channel noise is experimentally shown to serve as a convergence accelerating heuristic, when the channel is normalized. Otherwise usingv c x [0] = 0 is preferable for faster convergence.

A. Uncoded equalization behaviour
In this paragraph, the uncoded finite-length behaviour of the proposed IV DFE with online prediction is evaluated. Exact TV DFE counterparts are used as lower-bound references on biterror rate (BER), to assess the prediction accuracy. Note IV FIR receivers might outperform TV FIRs in some cases [19], as the latter are more sensitive to the convergence errors committed by the SISO decoder. of TV counterparts, especially at high SNR, but it is seen that a gap remains at medium BER for some constellations, due to dynamic filtering capabilities of TV receivers. EP feedback is shown to be mostly equivalent to APP feedback in this uncoded use case, but at high BER, EP has an advantage over APP both for TV and IV receivers, which suggests that improved decoding thresholds can be obtained with channel coding.

B. On the Operating Regions of FIR Receivers
A previous work on TV FIR turbo equalizers concluded that TV DFE significantly outperforms TV LE at high data rates [1], whereas TV LE remains preferable at very low rates, as it achieves same performance with less complexity. In the following, the asymptotic behaviour and the computational complexity of the proposed receiver along with IV FIRs is evaluated in a similar manner.
Through the extrinsic information transfer (EXIT) analysis of a SISO module, a mutual information (MI) based transfer function model, [16], where I A and I E denote respectively the MI between coded bits and the prior LLRs and the extrinsic LLRs.
EXIT functions notably allow to numerically predict the achievable rates of SISO receivers, through the area theorem of EXIT charts [20]. Indeed, MAP detector's EXIT chart's area yields an accurate prediction of the channel symmetric information rate (SIR) [21], the highest possible transmission rate for practical constellations, without channel knowledge at the transmitter.
However, for approximate receivers which violate the extrinsic message principle of turbo detection, the rates predicted by EXIT can be too optimistic (e.g. SIR of APP DFE appears to surpass MAP, which is impossible). This has been observed for APP-based receivers in [1], [11], but the EP-based DFE does not suffer from this phenomenon. Hence in the following, the proposed predictive EP-based IV DFE is evaluated.
IV DFE EP with symbol-wise prediction scheme is used for 8-PSK transmissions in the Proakis C channel, and numerically obtained achievable rates are plotted in solid lines in Fig. 9.
Dotted plots illustrate the achievable rates without turbo-iterations, for each receiver. IV receivers are shown to follow the behaviour of their TV counterpart within a gap of about 0.1 bits/s/Hz for both LE and DFE, but IV DFE still keep a significant upper hand over TV LE at medium and high spectral efficiency operating points. Using IV FIR receivers to operate at a given rate requires about 1.5 dB more energy than TV FIR, but with significant complexity savings.
Approximate computational complexity per turbo-iteration of considered FIR receivers is given in the Table I. TV LE and DFE receivers use the reduced-complexity TV matrix inversion algorithms in [1], and IV receivers exploit Cholesky decomposition for matrix inversion. The filter computation cost of the proposed IV DFE increases linearly with the number of prediction iterations N pred .

C. Finite-Length Turbo-Equalization Performance
In this section, the prediction accuracy is assessed for transmissions encoded with nonrecursive non-systematic convolutional code (NRNSCC) of polynomial [7,5] First, the impact of choosing a symbol-wise or a binary prediction scheme is assessed through finite-length BER evaluations. The block size is kept at K = 256, similarly to the uncoded case, and a MAP decoder based on the BCJR algorithm is used as a SISO decoder [22]. Fig. 10   mechanism is applied to correct the predicted prior covariance with a multiplicative penalty factor. After some ad hoc optimization, this scheme yields more pessimistic predictions that ends up improving the BER prediction accuracy.
Here, a related mechanism is adapted to the proposed online prediction. To avoid overestimation of the causal covariance, the anti-causal covariance can be exploited to derive a "lower-bound" to estimated causal covariances. Empirically, turbo detection systems bring most of the improvements at the initial iterations, hence the improvements after a certain number of iterations can no longer be substantial. Thus, after some trial-and-error tests, we have selected the predicted causal covariancev c x of the current turbo iteration to be modified with the heuristic v c x = max(v c x , βv a x ), with 0 < β ≤ 1. The proposed heuristic is integrated with the symbol-wise prediction, with 3 prediction iterations and β = 0.2, and the IV DFE-EP performance is displayed in turquoise in Fig. 10. Finally, to compare our proposal to the prior work and to evaluate its behavior in different operating regimes, the previously used rate-1/2 encoding with NRNSCC [7,5] Fig. 11, for 8-PSK transmissions in Proakis C channel, with above mentioned codes of rate 1/2, 2/3 and 5/6, and for 0, 1 and 4 turbo-iterations. Proposed predictive IV DFE receivers use symbol-wise prediction with 3 iterations, and the heuristic parameter is β = 0.2.
IV DFE APP significantly outperforms other APP-based DFE receivers when there are no turbo iterations, as this is the operating point where the prediction scheme is the most accurate. In Figure 12, the evolution of BER is plotted as the number of turbo-iterations increases. During intermediary iterations of the rate 2/3 system, previous works of Tao et al. [4] and Lou et al. [3] close most of the gap of iteration zero, with the receiver of [3] slowly converging to the same limit as the proposed receiver. At high rate systems (rate 5/6) the gap between them and our proposal increases, even for 4 turbo-iterations, and from Figure 12 it is seen that the receiver of [4] cannot converge to the same asymptotic limits, probably due to the usage of only a few samples for covariance estimation heuristic. The use of EP-feedback instead of APP does not bring significant improvement for high-rates, or without turbo-iterations, but at medium and low rates, it allows for an additional asymptotic gain over 0.5 dB. However, the predictability of the EP feedback over a wider set of configurations makes it a more attractive solution.

V. CONCLUSION
This paper carries out an original approach to the design of turbo DFE receivers with static filters, through the use of online prediction, based on semi-analytic performance prediction tech-niques as used in physical layer abstraction methods. Due to the lack of a closed-form solution for such receivers, various heuristics are used throughout the literature. However, discussion on the optimality of such approaches was lacking and it is one of the contribution of this paper.
Here, semi-analytical performance prediction of exact time-varying turbo DFE with dynamic filters is exploited to derive static DFE filters. This approach has been carried out for DFE with APP-based or EP-based soft feedback and their detection performance has been evaluated in various configurations. This framework could also be applied to self-iterated FIR DFE [1] for further improved performance, by updating anti-causal variances with causal EP variance of the last self-iteration.
Our analysis shows that significant complexity savings can be achieved with respect to TV DFE, while offering reasonably close performance. Moreover, our method is compatible with any constellation, and spectrally efficient on a large interval of coding rates and with or without turbo-iterations.