Optimal MSE Solution for a Decision Feedback Equalizer

Due to the inherent feedback in a decision feedback equalizer (DFE) the minimummean square error (MMSE) or Wiener solution is not known exactly. The main difficulty in such analysis is due to the propagation of the decision errors, which occur because of the feedback. Thus in literature, these errors are neglected while designing and/or analyzing the DFEs. Then a closed form expression is obtained for Wiener solution and we refer this as ideal DFE (IDFE). DFE has also been designed using an iterative and computationally efficient alternative called least mean square (LMS) algorithm. However, again due to the feedback involved, the analysis of an LMS-DFE is not known so far. In this paper we theoretically analyze a DFE taking into account the decision errors. We study its performance at steady state. We then study an LMS-DFE and show the proximity of LMS-DFE attractors to that of the optimal DFE Wiener filter (obtained after considering the decision errors) at high signal to noise ratios (SNR). Further, via simulations we demonstrate that, even at moderate SNRs, an LMS-DFE is close to the MSE optimal DFE. Finally, we compare the LMS DFE attractors with IDFE via simulations. We show that an LMS equalizer outperforms the IDFE. In fact, the performance improvement is very significant even at high SNRs (up to 33%), where an IDFE is believed to be closer to the optimal one. Towards the end, we briefly discuss the tracking properties of the LMS-DFE. Introduction A channel equalizer is an important component of a communication system and is used to mitigate the inter symbol interference (ISI) introduced by the channel. The equalizer depends upon the channel characteristics. A variety of equalizers have been proposed and utilized in communication systems [1-3] Usually simple linear equalizers (LE) would suffice (see for e.g., [1-3]) but for a channel with deep spectral nulls one would require amore complex, non LE like a decision feed back equalizer (DFE). A LE is a linear filter that is used to mitigate ISI while a Wiener filter (WF) equalizer is an optimal filter that minimizes the mean square error (MSE) between the input symbols and the decoded symbols (decoded after the equalizer). Closed form expression for WF LE is available ([4,5] etc). This closed form expression involves a matrix inverse which can be computationally intensive if the filter has a large dimension. Alternatively, least mean square linear equalizer (LMS-LE), a computationally efficient iterative algorithm, is used extensively (see [4-6]) to *Correspondence: [email protected] 1Indian Institute of Technology, Mumbai, India Full list of author information is available at the end of the article obtain the WF equalizer. It can also track the time variations in the WF, if required, as in the case of Wireless channels. For a fixed channel its convergence to the WF has been studied in [6,7] (see also the references therein). Its performance on a wireless (time varying) channel has been studied theoretically in [8,9] (see also [4,5,10] and the references there in, where the performance has been studied via simulations, approximations and upper bounds on probability of error). Decision feedback are nonlinear equalizers (a pair of linear filters one in the forward path and another in the feedback path), which can provide significantly better performance than LE [3,11,12], especially for ‘bad’ channels. A DFE feeds back the previous decisions of the transmitted symbols, to nullify the ISI due to them (which can now happen without amplifying the thermal noise) and makes a better decision about the current symbol. Although these equalizers have also been used for quite sometime, due to feedback their behavior is much more complex than that of the LEs. Hence their performance is not well understood. Existence of a hard decoder inside the feedback loop, due to its nonlinearity, makes the study all the more difficult. A DFE mainly exploits the finite alphabet structure of the hard decoder output [2,13] and hence the © 2012 Kavitha and Sharma; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Kavitha and Sharma EURASIP Journal on Advances in Signal Processing 2012, 2012:172 Page 2 of 16 http://asp.eurasipjournals.com/content/2012/1/172 hard decoder cannot be ignored (i.e., its performance is better than a system with a soft decoder). Since the statistics of the previous decisions in a DFE are not known, there is no known technique available that provides an minimum MSE (MMSE) DFE (we will call it as DFE-WF in the rest of the article) even for a fixed channel [2,3,14]. Thus an MMSE DFE is commonly designed by assuming perfect decisions (see, e.g., [2,15]). For convenience, for the rest of the article, we will call such a DFE as ideal DFE (IDFE). In this article IDFE is also computed using perfect channel estimates. The IDFE often outperforms the Linear WF significantly [3,11,12]. But it is generally believed that DFE-WF, the true MSE optimal DFE (designed considering the decision errors), can outperform even this. Another way to obtain an optimal filter is to replace the feedback filter at the receiver by a precoder at the transmitter [3,14]. This way one can indeed obtain the optimal filter but this requires the knowledge of the channel at the transmitter. For wireless channels, which are time varying, this is often not an attractive solution [2,3]. Some research has been done to deal with the decision errors in a DFE. Sternad et al. [16] approximated the errors in decisions with an additive white Gaussian noise (AWGN) independent of the input sequence and obtained a DFEWF. But as is stated in the article this approximation is not realistic. Erdogan et al. [13] obtain an H∞ optimal DFE taking into account the decision errors. However no comparison to DFE-WF was provided. Ideal DFE also contains a matrix inverse for which LMS is again used as a computationally efficient alternative in practical communications systems. However, convergence of LMS-DFE is not well understood even for a fixed channel, again due to the complexity introduced by the feedback. Trajectory of the LMS-DFE algorithm, on a fixed channel, with a soft decoder in the feedback loop has been approximated by an ODE in [17]. But this ODE does not approximate the LMS-DFE with a hard decoder. Beneveniste et al. [6] have shown the ODE approximation of an LMS-DFE with a hard decoder. But the ODE obtained by them is not explicit enough. Furthermore, they do not relate the attractors of this ODE to the DFE-WF. Our conjuncture is that LMS can actually converge to the true DFE WF (obtained considering the decision errors) and one of the main goals of this article is to prove the same. In this article, we study an LMS-DFE on a fixed channel using an ODE approximation. Towards this, we first obtain the stationary performance of the system with DFE and prove the existence of DFE-WF (the minimumMSE solution) for every channel state (whenever the domain of optimization is compact). We then show that the DFE-WF and an LMS-DFE attractor are close to each other at high signal to noise ratios (SNRs). We show the same is true for nominal values of SNRs via simulations. Further we demonstrate via simulations, that the LMSDFE can outperform the commonly used IDFE, at all practical SNRs. An interesting observation is that, the improvement is significant even at high SNRs where an IDFE does not suffer from error propagation and is believed to be close to the true DFE-WF. The article is organized as follows. Our system model, notations and assumptions are discussed in Section “The model and notation”. In Section “The issues and our approach” we discuss our approach. Section “Analysis of LMS-DFE and DFE-WF” obtains an ODE approximation and then the analysis of the attractors of LMS-DFE. Section “Numerical examples” provides some examples. Section “Tracking analysis” briefs the tracking behavior while Section “Conclusions” concludes the article. The sections Appendices 1 to 5 provide proofs for our theorems. Themodel and notation We consider a communication system with a DFE (see Figure 1). Inputs {sk} enter a finite impulse response channel {zl}L−1 l=0 , and are corrupted by an additive zero mean white Gaussian noise {nk} with variance σ 2. The channel output, uk , at time k, is given by,