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Optimal MSE solution for a decision feedback equalizer
 Veeraruna Kavitha^{1}Email author and
 Vinod Sharma^{2}
https://doi.org/10.1186/168761802012172
© Kavitha and Sharma; licensee Springer. 2012
Received: 19 February 2012
Accepted: 8 July 2012
Published: 16 August 2012
Abstract
Due to the inherent feedback in a decision feedback equalizer (DFE) the minimum mean 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 LMSDFE 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 LMSDFE and show the proximity of LMSDFE 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 LMSDFE 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 LMSDFE.
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 a more 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 (LMSLE), a computationally efficient iterative algorithm, is used extensively (see [4–6]) to 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 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 DFEWF 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 DFEWF, 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 DFE WF. 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 DFEWF 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 LMSDFE is not well understood even for a fixed channel, again due to the complexity introduced by the feedback. Trajectory of the LMSDFE 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 LMSDFE with a hard decoder. Beneveniste et al. [6] have shown the ODE approximation of an LMSDFE 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 DFEWF.
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 LMSDFE 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 DFEWF (the minimum MSE solution) for every channel state (whenever the domain of optimization is compact). We then show that the DFEWF and an LMSDFE 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 DFEWF.
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 LMSDFE and DFEWF” obtains an ODE approximation and then the analysis of the attractors of LMSDFE. 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.
The model and notation
We provide below the assumptions made and the notations used in this article. Most of these assumptions can be generalized as discussed at the end of this section.

Sequences {s_{ k }}and {n_{ k }}are independent and identically distributed (i.i.d) sequences and are independent of each other. The inputs {s_{ k }}are uniformly distributed over { + 1,−1}(BPSK modulation).

${f}_{\mathcal{N}}\left(y\right)$ is the N dimensional standard i.i.d Gaussian density, where N is the dimension of the vector y, i.e., ${f}_{\mathcal{N}}\left(y\right)={\left(2\Pi \right)}^{N/2}\mathit{ex}{p}^{\frac{{\lefty\right}^{2}}{2}}$. Whenever not mentioned, integrability is with respect to ${f}_{\mathcal{N}}\left(y\right)\mathit{dy}$.

The equalizer forward, feedback filters are given by ${\left\{{\theta}_{{f}_{l}}\right\}}_{l=0}^{{N}_{f}1}$, ${\left\{{\theta}_{{b}_{l}}\right\}}_{l=1}^{{N}_{b}}$ respectively. Also, let ${N}_{L}\triangleq {N}_{f}+L1$.

We assume that the symbols are modulated using BPSK and so the hard decoder equals, Q(x):=1_{{x≥0}}−1_{{x<0}}in (1).

For any vector, x, x_{ l }represents its l^{ th }component and ${x}_{l}^{k}$, l≤k, represents the vector [ x_{ k }x_{k−1} … x_{ l } ]^{ T }.
The following vector notations are used:$\begin{array}{ll}{S}_{k}& \triangleq \phantom{\rule{2em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{s}_{k{N}_{L}+1}^{k},\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}{N}_{k}\triangleq \phantom{\rule{1em}{0ex}}{n}_{k{N}_{f}+1}^{k},\\ {U}_{k}& \triangleq \phantom{\rule{2em}{0ex}}\phantom{\rule{0.3em}{0ex}}{u}_{k{N}_{f}+1}^{k},\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}{\u015c}_{k}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\triangleq \phantom{\rule{1em}{0ex}}{\u015d}_{k{N}_{b}+1}^{k},\\ {X}_{k}& \triangleq \phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\left[\phantom{\rule{2.77695pt}{0ex}}{U}_{k}^{T}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\u015c}_{k1}^{T}\phantom{\rule{2.77695pt}{0ex}}\right]}^{T},\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}{G}_{k}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\triangleq \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\left[\phantom{\rule{2.77695pt}{0ex}}{S}_{k}^{T}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{X}_{k}^{T}\phantom{\rule{2.77695pt}{0ex}}\right]}^{T},\\ {\theta}_{f}& \triangleq \phantom{\rule{2em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{{\theta}_{f}}_{0}^{{N}_{f}1},\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\theta}_{b}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\triangleq \phantom{\rule{2em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{{\theta}_{b}}_{1}^{{N}_{b}},\\ {J}_{k}& \triangleq \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\left[\phantom{\rule{2.77695pt}{0ex}}{S}_{k}^{T}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\u015c}_{k1}^{T}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{N}_{k}^{T}\phantom{\rule{2.77695pt}{0ex}}\right]}^{T},\phantom{\rule{2em}{0ex}}\Theta \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\triangleq \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\left[{{\theta}_{f}}^{T}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{{\theta}_{b}}^{T}\right]}^{T},\\ Z& \triangleq \phantom{\rule{0.3em}{0ex}}\left[\phantom{\rule{2.77695pt}{0ex}}{z}_{0},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{z}_{1}\phantom{\rule{2.77695pt}{0ex}}\dots \phantom{\rule{2.77695pt}{0ex}}{z}_{L1}\phantom{\rule{2.77695pt}{0ex}}\right].\end{array}$In the above, S_{ k },U_{ k },N_{ k }and ${\u015c}_{k1}$, respectively represent the vector of input symbols, channel outputs, noise samples and the decoder decisions that influence the equalizer output at time k. Vector X_{ k } forms input to the equalizer at time k while G_{ k }, J_{ k } are the two alternate representations of the system state at time k. Vector Z is the vector form of the channel while θ_{ f }, θ_{ b } are that of the equalizer feedforward and feedback filters.

Θ_{ k }represent the time varying equalizer at time k.

Let $\mathcal{S}:=\{+1,1\}$. Under the above assumptions, {G_{ k }} and {J_{ k }} are Markov chains for a fixed channel, equalizer pair at (Z,Θ). These two Markov chains take values in ${\mathcal{S}}^{{N}_{L}}\times {\mathcal{S}}^{{N}_{b}}\times {\mathcal{R}}^{{N}_{f}}$, where $\mathcal{R}$ is the set of real numbers. The current and the previous states of both these Markov chain are represented by the ordered pairs (i,y), (j,y^{′}) respectively. Here i,j take values from the discrete part of the state part of the state space, ${\mathcal{S}}^{{N}_{L}}\times {\mathcal{S}}^{{N}_{b}}$, while y,y^{′} take values in ${\mathcal{R}}^{{N}_{f}}$.

Ψ={ψ_{ l }}l=0N_{ L }−1 represents the convolution of the channel {z_{ l }} and the forward filter θ_{ f }.

The input to the hard decoder for a given state of the Markov chain is represented by,${e}_{\Theta}(i,y):=\sum _{l=0}^{{N}_{L}1}{\psi}_{l}{s}_{kl}+\sum _{l=0}^{{N}_{f}}{{\theta}_{f}}_{l}{n}_{kl}+\sum _{l=1}^{{N}_{b}}{{\theta}_{b}}_{l}{\u015d}_{kl}.$
Note that ${\u015d}_{k1}=Q\left({e}_{\Theta}\right(j,{y}^{\prime}\left)\right)$.

B(Θ,δ), $\stackrel{\u0304}{B}(\Theta ,\delta )$ are the open and closed balls respectively with center Θ and radius δ.

The equalizer output without noise, e_{ Θ }(i,0)≠0for all values of i at the LMS attractor. Without this assumption the LMS algorithm makes more errors than the correct decisions.
a computationally efficient iterative algorithm, is expected to provide the MMSE solution. However, with a feedback structure inserted, the convergence behavior of LMS is not understood properly. In fact, it is not even clear if the minimum mean square problem is well posed neither is it clear if an MMSE solution exists. Even prior to these questions, one first needs to define the expectation in (2) appropriately. One is often interested in optimizing a stationary performance, i.e., expectation in (2) is with respect to the stationary distribution of the system. However the stationary distribution depends upon the parameter Θ. The existence of the stationary distribution for any given Θ is not known. We take up these issues one by one and our final goal is to show that the above iterative algorithm (3) indeed converges close to the MMSE solution.
One can easily extend the theory of this article to any finite alphabet (complex) input source with any arbitrary distribution and to a complex channel. However we stick to BPSK modulation and to a real channel to keep the explanations simple. Also, the theory to follow, considers an optimal equalizer for delay 0. The entire theory will go through for any arbitrary delay. Indeed in Section “Numerical examples”, an example with an optimal equalizer for delay 1, is presented. This is once again done to simplify the explanations.
The issues and our approach
where the expectation on the right hand side is defined under stationarity for a given Θ. Vector ${X}_{k}={\left[{U}_{k}^{T},\phantom{\rule{2.77695pt}{0ex}}{\u015c}_{k1}^{T}\right]}^{T}$, includes previous decisions ${\u015c}_{k1}$ and hence its stationary distribution depends upon the parameter Θ. Thus this is a complex case of optimization in which, the stationary distribution defining the average cost also depends upon the parameter to be optimized. There is no known technique to compute a WF, Θ^{∗} of (4), even for a fixed channel.
This computation may be expensive because of matrix inversion and LMS (3) is actually used as an alternative [4, 5]. Our claim is that in case of a DFE, apart from being computationally efficient the LMS algorithm also outperforms the IDFE, Θ_{IDFE}. This is because we will see briefly that the LMS attractors are close to DFEWF while IDFE is away from DFEWF. We achieve this goal by showing that the LMSDFE attractors are close to that of the DFEWF at high SNRs (later in Section “Numerical examples” we show that this covers the practically used SNRs). Further, LMS can also be used to track the channel variations. We first study an LMSDFE on a fixed channel and later on briefly discuss its tracking behavior.
Another issue related to (4) is that we should take the expectation in the right hand side under stationarity. However, it appears that the existence of stationary distribution of {X_{ k }} for a given Θ is not known. Thus, first, in Theorem 2, we show the existence of a unique stationary distribution (and stationary density w.r.t. ${f}_{\mathcal{N}}\left(y\right)\mathit{dy}$) for {X_{ k }} for any Θ.
where Π_{ Θ } is the stationary density of the Markov chain, {J_{ k }}, w.r.t. the Lebesgue measure, when the DFE Θ is used. One can expect the LMSDFE attractors to be close to the DFEWF, if the second term in the RHS of (6) is close to zero. However, we could not even get differentiability of Π_{ Θ }. Nevertheless, we achieve the required differentiability (Theorem 3) by considering a hard decoder that is a slightly perturbed version of the original hard decoder. We also show that the DFEWF and an LMSDFE attractor of this perturbed decoder converge to that of the original hard decoder as the level of perturbation tends to zero (Theorem 4). We then analyze this perturbed decoder and show that the LMSDFE attractors of this decoder are close to its DFEWFs at high SNR (Theorem 5). This suggests that at high SNR an LMS attractor for the original decoder is close to its DFEWF.
Analysis of LMSDFE and DFEWF
We provide a step by step analysis of LMSDFE and its connection to DFEWF in this section, while addressing the issues raised in Section “The issues and our approach” one after the other.
Previous ODE approximation result
with initial condition Θ(0)=a, where ${P}_{\Theta}^{n}$ is the nstep transition function of the Markov chain J_{ k }with DFE Θ, and ${P}_{\Theta}^{n}{H}_{\Theta}(j,{y}^{\prime})$ is the expectation of the function H_{ Θ }(G) (defined in (3)) using the conditional measure P Θ n(.j,y^{′})(Note G_{ k } is a fixed function of J_{ k }). The limit in (7) will be independent of the initial condition (j,y^{′}) ([6], p. 252).
It is easy to see that the LMS algorithm satisfies all the required hypothesis of ([6], Theorem 13, p. 278) and hence one can approximate its trajectory on any finite time scale with the solution of the ODE (7), the precise result is:
Theorem 1
whenever μ_{ k }≤1 for all k and if $lim\underset{k}{inf}\frac{{\mu}_{k+r}}{k}>0$for every integer r. ▀
Stationary distribution and a simplified ODE
We will show below that the RHS of the ODE (7) is same as that of the ODE (5) and hence equate the ODE (7) with a more tractable ODE (5). As a first step, we prove that the Markov chain {J_{ k }} has a stationary distribution for any given DFE, channel pair (Z,Θ). In the following, at many places we do not include channel value Z for notation, as this article mainly works with fixed channel behavior. However the proofs are applicable for any pair (Z,Θ) and the notation includes Z, when required to be specific.
Theorem 2
 (i)
For every fixed (Z,Θ), Markov chain {J _{ k }}has a unique stationary distribution π _{Z,Θ}.
 (ii)
Starting from any initial condition (i,y), the nstep transition measure (${P}_{\Theta}^{n}\left(.\righti,y)$) of the Markov chain converges geometrically to the stationary distribution, Π _{ Θ }, in total variation norm.
 (iii)
The continuous part of the stationary distribution has a density, Π _{ Θ }that is continuous with respect to (Z,Θ)in L _{1}norm.
 (iv)
The MSE under stationarity is continuous in (Z,Θ).
Proof: Please see Appendix 1. ▀
where $\stackrel{\u0304}{\delta}(y,{y}^{\prime})$ equals 1 when the vector formed from all but the last component of the vector y^{′}equals the vector formed from all but the first component of the vector y and otherwise zero and $\stackrel{~}{\delta}(i,j)=\stackrel{\u0304}{\delta}\left({i}_{1}^{{N}_{L}},{j}_{1}^{{N}_{L}}\right)\stackrel{\u0304}{\delta}\left({i}_{{N}_{L}+1}^{{N}_{b}+{N}_{L}},{j}_{{N}_{L}+1}^{{N}_{b}+{N}_{L}}\right)$ (note that the first component, ${i}_{1}^{{N}_{L}}$, represents the sample value of S_{ k }, while the second one, ${i}_{{N}_{L}+1}^{{N}_{b}+{N}_{L}}$, represents the sample value of ${\u015c}_{k1}$). The only component of the transition function (8) that depends upon Θ is ${P}_{\Theta}({i}_{{N}_{L}+1}=1j,{y}^{\prime})={1}_{\left\{{e}_{\Theta}(j,{y}^{\prime})>0\right\}}$.
Thus ODE (7) simplifies to ODE (5).
By Theorem 2, MSE is a continuous function of Θ and so by confining our search in (4) to a compact region, we obtain the existence of the WF, DFEWF. Next we consider the LMS attractors which are now the attractors of ODE (5). The ODE attractors will be zeros of the RHS of (5), while the DFEWF will be a zero of the gradient (if it exists) of the MSE (the cost in the RHS of (4)). As discussed in Section “The issues and our approach”, these two can be related as in (6) and for comparison of the two zeros, one needs to study, ∇_{ Θ }Π_{ Θ }, the gradient of the stationary density. That is, to get the connection between an LMSDFE attractor and the DFEWF one needs to consider the differentiability of the stationary density.
Differentiability of the stationary density
One can see from Equation (9) that it is difficult to comment on differentiability of the nstep transition density itself. Thus, it is even more difficult to discuss the differentiability of the stationary density. To proceed further with the analysis, we perturb the hard decoder Q such that the nstep transition density and the stationary density become differentiable. Next we show that the LMS attractors and the DFEWF of this perturbed decoder converge to that of the original decoder as the level of perturbation tends to zero. Finally we study the DFE using these perturbed decoders in Section “LMS attractors versus WF at high SNRs”.
For these perturbed decoders, we show that the stationary density (with respect to ${f}_{\mathcal{N}}\left(y\right)\mathit{dy}$) also becomes differentiable. Furthermore, using an Implicit function theorem, we get a bound on the norm of this gradient.
Theorem 3
Proof: The proof is provided in Appendix 2. ▀
We conclude this section by showing that the DFEWFs and the LMSDFE attractors of the perturbed decoder converge to that of the original decoder. In the following, let ${\Theta}_{n}^{\ast}$ and ${\Theta}_{n}^{\text{LMS}}$ denote the DFEWF and an LMSDFE attractor (whose existence at high SNRs with small ε_{0} is established at the end of Appendix 4 and hence is assumed in the proof of the following theorem) for perturbation ${{\epsilon}_{0}}_{n}$.
Theorem 4
Proof: Please see Appendix 3. ▀
Thus we can always take the perturbation ε_{0}in (10) small enough such that the LMS attractors and the DFEWFs for the perturbed decoder are close enough to the corresponding equalizers for the original decoder. Henceforth, we analyze these perturbed decoders to draw important conclusions.
LMS attractors versus WF at high SNRs
In this section we would like to understand the connection between an LMS attractor and a DFEWF for a perturbed decoder. Since the former is a zero of the RHS of Equation (5) and the later is the zero of the gradient of the MSE (the cost in the RHS of (4)), we study the connection between the two.
Here equality a follows by the existence of the stationary density ${\Pi}_{\Theta}^{\left({\epsilon}_{0}\right)}$ with respect to the Gaussian measure ${f}_{\mathcal{N}}\left(y\right)\mathit{dy}$. Equality b is given by Lemma 2 of Appendix 5. The above equality above equality (14) is true for any ε_{0}>0 and for any σ^{2}. We will show below that the DFEWF will be close to the limiting LMSDFE if the second term on the right hand side of (14) is small.
Theorem 5
Proof: Please see Appendix 4. ▀
Using the above theorem, we obtain the proximity of LMS attractors and the WFs in the following.
Note that q(σ^{2},0) is a zero of s(Θ,σ^{2}) (note w(q(σ^{2},0),σ^{2},0)=s(Θ,σ)) and hence is an LMS attractor at σ^{2}. Similarly from (14), q(σ^{2},η_{ w })is a zero of the gradient of MSE cost and hence is a DFEWF. Thus, for all ${\sigma}^{2}\le {\sigma}_{0}^{2}$, the LMS attractors, q(σ^{2},0), by continuity arguments of Theorem 5 will be close to that of the WFs, q(σ^{2},η_{ w }).
It is clear from the above Theorem that at high SNRs, for very small ε_{0}(close to the practical decoder), the LMS attractor is close to the DFEWF. Since, IDFE Θ_{IDFE}, is designed with an improper assumption (like perfect decisions), there is a good chance of these filters to be inefficient in comparison to the LMS attractors. We will see this in the examples provided in Section “Numerical examples”.
We conclude this section by pointing out another useful consequence of the Theorem 5. This theorem also establishes the existence of the LMS attractors at high SNRs for perturbed decoders with perturbation level ε_{0} small. A Remark at the end of Appendix 4 establishes this point.
One of the uses of the above ODE approximation is that, one can approximately obtain the performance (e.g., Bit error rate, MSE) of LMSDFE at any time by using the trajectory of this ODE. Of course, obtaining bit error rate (BER) theoretically is still a problem because the BER of a system with a fixed known channel and a fixed DFE is still not available. But our ODE approximation is still useful because one can obtain the performance (transient as well as stationary) of the LMSDFE with only one simulation, which would not be possible otherwise. This is because by Theorem 1, the ODE solution approximates the LMSDFE trajectory in probability.
Numerical examples
In this section we reinforce the theory developed so far using some examples. We take a few examples of channels obtained from previous studies and show the proximity of the DFEWF and the LMS attractor for practical values of SNRs. We also show that in many cases, the IDFE performs much worse than the DFEWF but an LMS attractor performs close to the DFEWF. BER and the MSE are used to compare the various equalizers. For every sample of the channel, we have used MonteCarlo simulations to estimate the corresponding BER and MSE using one million samples of data.
Here s_{k,i} are i.i.d with the distribution of the inputs, s_{ k }. Sequences {Δ_{ k }}and {μ_{ k }}are chosen appropriately to reduce to zero. In our simulations we used ${\mu}_{k}=\frac{0.07}{{k}^{0.6}}$, Δ_{ k }=5μ_{ k } and N=4×10^{5}.
Least mean square attractors are obtained as the time limit of the LMS algorithm (3), with similar settings as with DFEWF estimation.
Comparison of DFEs for raised cosine channel with N _{ f } = 5, N _{ b } = 10 and channel fixed at [0.45 0.59 0.43 0.11 −0.22 −0.32 −0.27 0 0.11 0.11]
SNR  Θ ^{∗}  Θ _{IDFE}  Θ _{LMS}  

29  MSE  BER  Dist from Θ^{∗}  MSE  BER  Dist from Θ^{∗}  MSE  BER 
16.7  0.21  0.024  1.1  0.26  0.027  0.035  0.21  0.024 
14.5  0.38  0.089  1.7  0.64  0.10  0.037  0.38  0.091 
12.5  0.49  0.150  1.6  0.94  0.184  0.027  0.49  0.151 
11.5  0.54  0.176  1.5  1.0  0.215  0.023  0.54  0.177 
4.5  0.81  0.311  0.64  0.94  0.33  0.021  0.80  0.311 
1.5  0.87  0.353  0.37  0.93  0.364  0.023  0.87  0.353 
Comparison of DFEs with N _{ f } = N _{ b } = 2, and channel fixed at [ 0.41 .82 0.41]
SNR  Θ ^{∗}  Θ _{IDFE}  Θ _{ LMS }  

MSE  BER  Dist from Θ^{*}  MSE  BER  Dist from Θ^{∗}  MSE  BER  
16.7  0.11  0.0027  0.13  0.12  0.0035  0.014  0.11  0.0028 
14.5  0.16  0.01  0.26  0.18  0.015  0.021  0.16  0.011 
12.5  0.23  0.03  0.35  0.26  0.037  0.025  0.23  0.032 
11.5  0.27  0.047  0.40  0.30  0.055  0.031  0.28  0.050 
4.5  0.54  0.184  0.43  0.59  0.2  0.009  0.54  0.184 
1.5  0.65  0.235  0.32  0.69  0.25  0.008  0.65  0.235 
We have developed the theory for an equalizer with delay zero. One can easily extend these results to the equalizer with any arbitrary delay. In fact, the channel in Table2is one such example. Here the equalizer with delay 1 will be the best one. The channel of Table 2 is very widely used (see [1], p. 165 and [4], p. 414). We can see once again a huge improvement (up to 30%) in BER for the LMSDFE with respect to Θ_{IDFE}. We also see that the LMS attractors are close to the DFEWF, Θ^{∗}for all practical SNRs.
In this section, we are comparing directly the time limits of LMS algorithm (3) with that of the true DFEWF iteration mentioned at the beginning of this section. These two limits are further compared with IDFE, closed form expression. That the LMS trajectory approximates the solution of the ODE (5) is established theoretically (Theorem 1) in this article. In [9, 18] etc., we have demonstrated the same even via numerical simulations, for time varying channels. In Figures two, three and four of [9], it is shown numerically that the LMS trajectory approximates the appropriate ODE solution when the underlying channel is a time varying AR (2) process.
Tracking analysis
LMS being an iterative algorithm can track the channel variations if the update coefficients μ_{ k }, in (3), converge to a non zero value. In [9, 18], we study the tracking behavior of an LMSDFE, while it is operating on a wireless channel characterized by an AR(2) process. We demonstrate that an LMSDFE can also track the time varying DFEWF, whose variations result from the variations in a wireless channel. We also show that LMSDFE can outperform the IDFE, on a time varying channel.
Conclusions
Obtaining MSE optimal filter for DFE is a longstanding problem. Precoding provides one practical solution but may not be feasible with wireless channels. The difficulty in the design and or analysis is because, the analysis of the past decisions (with feedback) is not known so far. To circumvent this, one commonly uses the optimal WF obtained assuming perfect past decisions. LMS, a computationally efficient alternative, is an iterative algorithm designed to converge to the WF. However, once again because of the feedback involved, complete analysis of an LMSDFE is not available.
We show via ODE analysis, that LMS itself can provide/track the optimal WF. This article concentrates on fixed channel behavior and proves that the attractors of the LMS are close to that of the optimal DFE at high SNRs. Proofs become nontrivial partly because of the nondifferentiability of the hard decoder. We circumvent this problem, by studying another hard decoder which is a slightly perturbed version of the original one. We first show that the LMS attractors and the DFEWFs of the perturbed decoder converge to that of the original decoder and then show that the two themselves are close to each other at high SNRs. Next, we show by examples that the SNRs need not be very high, i.e., in fact practically used SNRs (upto 1.5 dB) can be sufficient. We also show that the BER (probability of error) of the commonly used WF, designed assuming perfect past decisions (also using perfect channel estimates), can be up to 33% higher than the optimal WF even at high SNRs (where the former is believed to be closer to the later).
In [18], we show that the LMSDFE converges and then moves close to the instantaneous DFEWF after the initial transience, while it is tracking a DFEWF of a wireless channel modeled by an AR(2) process. We also show in [18] that the performance measures BER and MSE of the LMSDFE are close to that of the DFEWF after the transient period, while that of an IDFE are substantially inferior to that of the DFEWF and the LMSDFE.
Thus we conclude: (1) in case of a DFE, an LMS algorithm (originally designed for computational efficiency) converges and/or tracks a filter close to the Wiener solution; (2) the closed form expression for DFE WF (obtained after approximating the decision errors to zero) is far away from the Wiener solution and its performance can be significantly inferior.
Appendices
Appendix 1
Continuity of the map considered in (16) and compactness of the closed balls $\stackrel{\u0304}{B}({\Theta}_{0},\epsilon )$, $\stackrel{\u0304}{B}({Z}_{0},{\epsilon}_{0})$ ensures M_{1}<∞.
where sets ${C}_{1}\in {\mathcal{R}}^{{N}_{b}}$, ${C}_{2}\in {\mathcal{R}}^{{N}_{f}}$ are selected such that their respective Lebesgue measures are not equal to zero and ∩l=k−N_{ b }−1k{N_{ l }∈C}⊃C_{1}×C_{2}.
where the measure ${\nu}_{{n}_{0}}\left(\right)$ is defined by, ${\nu}_{{n}_{0}}\left(\right[\phantom{\rule{2.77695pt}{0ex}}1\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2.77695pt}{0ex}}.\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}1\phantom{\rule{2.77695pt}{0ex}}]\times [\phantom{\rule{2.77695pt}{0ex}}1\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}.\phantom{\rule{2.77695pt}{0ex}}.\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}1\phantom{\rule{2.77695pt}{0ex}}]\times {B}_{E}):=\mathrm{\alpha P}({N}_{k}\in {B}_{E}\cap {C}_{2})$. Thus the entire state space ${\mathcal{S}}^{{N}_{L}}\times {\mathcal{S}}^{{N}_{b}}\times {\mathcal{R}}^{{N}_{f}}$ is ${\nu}_{{n}_{0}}$small (hence also a petite set) for all the Markov chains {J_{ k }}, parameterized by $\Theta \in \stackrel{\u0304}{B}({\Theta}_{0},\delta )$ and $Z\in \stackrel{\u0304}{B}({Z}_{0},{\epsilon}_{0})$. Then using ([19], Proposition 9.1.7, p. 206 and Theorem 10.01, p. 230) one obtains the existence and uniqueness of the stationary distribution, π_{Z,Θ}for each Z,Θ.
The stationary distribution, π_{Z,Θ}, has discrete and continuous components. The continuous component of π_{Z,Θ}, is absolutely continuous with respect to the measure ${f}_{\mathcal{N}}\left(y\right)\mathit{dy}$ for every (Z,Θ). Hence the stationary density, π_{Z,Θ}for {J_{ k }}exists. Continuity in total variation norm of the stationary distribution implies the continuity of the stationary densities in L_{1}norm ([20], Theorem 8.2, p. 110). It is also easy to see that the stationary density Π_{Z,Θ}(i,y)≤1 for all (i,y).
Now by fixing the channel at some value Z, MSE, the cost in RHS of Equation (4), can be rewritten as, ${E}_{\Theta}{\left[{\Theta}^{t}Xs\right]}^{2}=\sum _{S,\u015c}{E}_{{f}_{\mathcal{N}}}\left[{\left({\Theta}^{t}Xs\right)}^{2}{\Pi}_{\Theta}\right]$. Lemma 1 in Appendix 5, now gives the continuity of the MSE with respect to Θ for any fixed Z.
One can show the same conclusions for the Markov chain, {G_{ k }}, as G_{ k }=Γ(J_{ k }) for some fixed oneone, onto C^{ ∞ } function Γ, whenever the channel and equalizer values are fixed. ▀
Appendix 2
Proof of Theorem 3: The existence and continuity of the stationary density ${\Pi}_{\Theta}^{\left({\epsilon}_{0}\right)}$ for every ε_{0}is achieved in a similar way as in the proof of the Theorem 2. The only difference being, ε_{0} must be added to −M_{1} in the definition of the set (17). We leave superscript ε_{0}to simplify the notations in the rest of this proof.
We use Implicit function theorem to prove differentiability. For that, we will consider the Banach spaces:

$X={\mathcal{R}}^{{N}_{f}+{N}_{b}}$ with Euclidean norm.

$Y=\{g:{\mathcal{S}}^{{N}_{L}+{N}_{b}}\times X\to \mathcal{R};\leftg\right<\infty \}$ with L_{2} norm, ., defined by,$\leftg\right:=\frac{1}{\left\mathcal{S}\right}\sum _{i}{\left({\int}_{y}{\leftg(i,y)\right}^{2}{f}_{\mathcal{N}}\left(y\right)\mathit{dy}\right)}^{1/2},$
where $\left\mathcal{S}\right$ represents the cardinality of set ${\mathcal{S}}^{{N}_{L}+{N}_{b}}$.
where,
By Lemmas 1 and 2 the function f is differentiable with respect to Π and Θ, respectively and further the derivative $\frac{\mathrm{\partial f}}{\mathrm{\partial \Pi}}$ is a homeomorphism. Also, $\left{\left(\frac{\mathrm{\partial f}}{\mathrm{\partial \Pi}}\right)}^{1}\right$ and $\left\frac{\mathrm{\partial f}}{\mathrm{\partial \Theta}}\right$ are upper bounded locally by the RHS of (18) and (24) respectively.
Upper bound 13 is obtained by bounding the above gradient using the upper bounds (18) and (24). ▀
Lemma 1.
for all Θ∈B(Θ_{0},δ), ${\sigma}^{2}\le {\sigma}_{0}^{2}$.
This provides a contradiction as $0<{\nu}_{{n}_{0}}\left(Y\right)<1$ and hence Π_{1}=g(Θ,Π)_{1}<Π_{1}. This proves that the partial derivative (19) is oneone. The inequality is obtained by using the majorizing measure, ${\nu}_{{n}_{0}}\left(.\right)$, defined in the proof of continuity of stationary distribution.
The map g(Θ,Π)is compact integral operator ([22], Example 2, p. 277). The last map of the partial derivative has onedimensional range and hence is compact. Therefore, the partial derivative equals T−I, where T is a compact operator. Then by Riesz–Schauder Theory ([22], Theorem 1, p. 283), the fact that $\frac{\mathrm{\partial f}}{\mathrm{\partial \Pi}}$ is oneone implies that it is onto and also further that the inverse is bounded. Hence $\frac{\mathrm{\partial f}}{\mathrm{\partial \Pi}}$ is a linear homeomorphism.
Furthermore, the mapping $({\sigma}^{2},\Theta )\mapsto \left{\left[{\left.\frac{\mathrm{\partial f}}{\mathrm{\partial \Pi}}\right}_{(\Theta ,{\Pi}_{\Theta})}\right]}^{1}\right$ is continuous. This continuity follows by the joint continuity of the n_{0}step transition function, ${p}_{\Theta}^{{n}_{0}}(i,yj,{y}^{\prime})$ with respect to (σ^{2},Θ) and then by bounded convergence theorem (as ${p}_{\Theta}^{{n}_{0}}(i,yj,{y}^{\prime})+1$ is uniformly bounded) and finally by the continuity of the map x↦x^{−1} ([23], p. 135). Hence the lemma follows for some C_{0}<∞, $\delta >0,{\sigma}_{0}^{2}>0$. ▀
Lemma 2.
f is differentiable with respect to Θ. The partial derivative ${\left.\frac{\mathrm{\partial f}}{\mathrm{\partial \Theta}}\right}_{(\Theta ,{\Pi}_{\Theta})}$ is upper bounded by bound (24).
Proof: We reintroduce the notations that will be used here (notation of Equation (9)).

$i=\left[{S}_{k+{n}_{0}({N}_{f}+L2)}^{k+{n}_{0}},{\u015c}_{k+{n}_{0}1({N}_{b}1)}^{k+{n}_{0}1}\right]$, $y={N}_{k+{n}_{0}({N}_{f}1)}^{k+{n}_{0}}$, represent the current state of the Markov Chain, at k + n_{0}.

$j=\left[{S}_{k({N}_{f}+L2)}^{k},{\u015c}_{k({N}_{b}1)}^{k1}\right],{y}^{\prime}={N}_{k({N}_{f}1)}^{k}$ represent the initial condition for n_{0}−step transition function, which transition function, which is the state of the Markov chain at k.

$l=\left[{S}_{k+1}^{k+{n}_{0}({N}_{f}+L3)},{\u015c}_{k}^{k+{n}_{0}1({N}_{b}2)}\right]$, v=N k + 1k + n_{0}−(N_{ f }−2)represent the intermediate input, decision and noise vectors.

$x\left(q\right):=\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[{S}_{k+{n}_{0}q({N}_{f}+L1)}^{k+{n}_{0}q},\phantom{\rule{2.77695pt}{0ex}}{\u015c}_{k+{n}_{0}q{N}_{b}}^{k+{n}_{0}q1},\phantom{\rule{2.77695pt}{0ex}}{N}_{k+{n}_{0}q{N}_{f}}^{k+{n}_{0}q}\right]$ represent the intermediate state of the Markov chain at k + n_{0}−q.
For obtaining the above upper bound, the mean value theorem is used as explained below for a two dimensional function, which can easily be generalized to any ndimensional function. Say f is any function of two variables. One can write, f(a + h,b + k)−f(a,b) as sum of terms f(a + h,b + k)−f(a + h,b)−f(a,b + k) + f(a,b), f(a + h,b)−f(a,b) and f(a,b + k)−f(a,b). The first term is bounded by mean value theorem for two variables, ([23], Theorem 9.40, p.235), while the remaining two terms can be bounded using mean value theorem for one variable, ([23], Theorem 5.10, p.108).
where the constant c^{ ′′ }depends on Π.
where the constants b_{ l }take values ${S}_{k}^{t}\Psi +{\theta}_{b}^{t}{\u015c}_{k1}$. ▀
Appendix 3
Proof of Theorem 4: Let ${f}_{1}(\Theta ,{\epsilon}_{0}):={E}_{{J}_{k}\left(\Theta \right)}^{\left({\epsilon}_{0}\right)}\left[{\text{err}}_{\Theta}{\left({J}_{k}\right)}^{2}\right],$ and ${f}_{2}(\Theta ,{\epsilon}_{0}):=\left{E}_{{J}_{k}\left(\Theta \right)}^{\left({\epsilon}_{0}\right)}{\nabla}_{\Theta}\left[{\text{err}}_{\Theta}{\left({J}_{k}\right)}^{2}\right]\right.$ Note that for any fixed ε_{0}, LMS attractors will be the zeros, i.e., minima of f_{2}(.,ε_{0}) while the DFEWFs are the minima of the MSE cost, f_{1}(.,ε_{0}).Also note that ε_{0}=0corresponds to the original decoder.
Let $\left\{{{\epsilon}_{0}}_{n}\right\}$ be any sequence converging to 0. Let $\Omega =\left\{{{\epsilon}_{0}}_{n}\right\}$. Take a compact set C large enough such that the WF is inside it (as Θ is increased to infinity, eventually MSE will start increasing and will tend to infinity). One can follow steps as in Theorem 2 and show that the stationary density ${\Pi}_{{\Theta}_{n}}^{{{\epsilon}_{0}}_{n}}$ converges to ${\Pi}_{\Theta}^{0}$ as $({{\epsilon}_{0}}_{n},{\Theta}_{n})\to (0,\Theta )$. Similarly, one can also show that both functions f_{1},f_{2}are jointly continuous in (Θ,ε_{0})∈C×Ω.
The domain of the parameter Θ for every ε_{0}, say D(ε_{0}), is the same compact set C and hence the correspondence ε_{0}↦D(ε_{0})is compact and continuous [25]. Then by the maximum theorem ([25], p. 235), ${{D}_{1}}_{n}^{\ast}:=\text{arg}\phantom{\rule{0.3em}{0ex}}{\text{min}}_{\Theta \in C}{f}_{1}(\Theta ,{{\epsilon}_{0}}_{n})$ and ${{D}_{2}}_{n}^{\ast}:=\text{arg}\phantom{\rule{0.3em}{0ex}}{\text{min}}_{\Theta \in C}{f}_{2}(\Theta ,{{\epsilon}_{0}}_{n})$ are compact valued upper semicontinuous correspondences on Ω. Thus by ([25], Proposition 9.8, p. 231) there exists a subsequence of LMS attractors ${\Theta}_{{n}_{k}}^{\text{LMS}}$ converging to an LMS attractor of the original decoder, ${\Theta}_{0}^{\text{LMS}}$. Once again by the same proposition there exists a further subsequence such that the DFEWFs ${\Theta}_{{{n}_{k}}_{l}}^{\ast}$ converge to a DFEWF of the original decoder, ${\Theta}_{0}^{\ast}$. Thus there exists a sequence (after renaming) ${{\epsilon}_{0}}_{n}\to 0$ such that ${\Theta}_{n}^{\text{LMS}}\to {\Theta}_{0}^{\text{LMS}}$ and ${\Theta}_{n}^{\ast}\to {\Theta}_{0}^{\ast}$. ▀
Appendix 4
Therefore in this case, the DFEWF coincides with the LMSDFE attractor, ${\Theta}_{0}^{\ast}$.
Choose ${{\epsilon}_{0}}_{2}>0$ such that, ${\epsilon}_{{0}_{2}}<\left({S}_{k}^{t}{\Psi}_{0}^{\ast}+{\u015c}_{k1}^{t}{\Theta}_{{0}_{b}}^{\ast}\right$ for all values of s_{ k } and ${\u015c}_{k1}$. The DFEWF (${\Theta}^{\ast ,{\epsilon}_{0}}$) and the LMSDFE attractor (${\Theta}^{\text{LMS},{\epsilon}_{0}}$) coincide and equal ${\Theta}_{0}^{\ast}$ for a noiseless system having a perturbed decoder, with ${\epsilon}_{0}\le {{\epsilon}_{0}}_{2}$. This happens because when there is no noise the perturbed decoder coincides with the original decoder for ${\epsilon}_{0}\le {\epsilon}_{{0}_{2}}$.
where ${R}_{\mathit{\text{xx}}}\left({\Theta}^{\ast ,{\epsilon}_{0}}\right)$ is the autocorrelation matrix of the vector X_{ k }(Θ), under stationarity, at ${\Theta}^{\ast ,{\epsilon}_{0}}$. As ${\Theta}^{\ast ,{\epsilon}_{0}}$ is a WF, the above partial derivative will be invertible (all the eigenvalues of the derivative should be negative for the equilibrium point to be an attractor).
Continuity of the above partial derivative with respect to σ^{2},η,Θ can be seen as before. Applying Implicit function theorem at $({\Theta}^{\ast ,{\epsilon}_{0}},0,0)$, one gets a δ>0, and a continuous function q(σ^{2},η)such that $q(0,0)={\Theta}^{\ast ,{\epsilon}_{0}}$ and w(q(σ^{2},η),σ^{2},η)=0, for all (σ^{2},η) with (σ^{2},η)≤δ. ▀
Remark on existence of LMS attractors
The above theorem also provides the following useful conclusion. For all σ^{2}≤δ, the zeros of w(.,σ^{2},0) exist and equal q(σ^{2},0). These zeros are continuous in σ^{2}. One can see that these zeros will indeed be LMS attractors as invertibility of the derivative of the function f() at σ^{2}=0 guarantees its invertibility in a small neighborhood of σ^{2}=0.
Appendix 5
In this appendix we state and prove the lemmas, which are used in this article.