# Estimation of the Feedback Reliability for IB-DFE Receivers

Authors: Fábio Silva; Rui Dinis; Paulo Montezuma
Journal: ISRN Communications and Networking
 (2011)

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## Abstract

single-carrier (SC) modulation  with frequency-domain equalization (FDE)  combined with iterative (turbo) FDE schemes has been shown to be suitable for cyclic-prefix (CP)-assisted block transmission within broadband wireless systems. Its excellent performance in severely time-dispersive channels, makes it a promising candidate for future broadband wireless systems. In this paper we investigated the impact of the correlation factor estimation in the performance of iterative block decision feedback equalization (IB-DFE)  receivers. Since the correlation factor represents a key parameter to ensure the good performance of these receivers, reliable estimates of this factor are needed in the feedback loop. We present several methods to estimate the correlation coefficient.
We also propose a technique to compensate the inaccuracy of the estimation.


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## Body

## 1. Introduction

Future wireless systems are required to support high quality of service at high data rates, for which we can have severely time-dispersion effects associated to the multipath propagation. It is known that block transmission techniques, with appropriate cyclic extensions and employing FDE techniques, are excellent candidates for severely time-dispersive channels [1, 2]. The most popular techniques based on this concept are orthogonal frequency division multiplexing (OFDM) and single-carrier with frequency-domain equalization (SC-FDE). However, the OFDM signals have high envelope fluctuations and a high peak-to-mean envelope power ratio (PMEPR) leading to amplification difficulties. An alternative to OFDM modulations are SC modulations using FDE and CP-assisted block transmission techniques, especially for the uplink of broadband wireless systems [1, 2]. The performances and overall implementation complexities are similar to SC-FDE and OFDM [1, 2].Typically the receiver for SC-FDE schemes is a linear FDE. However, it is known that nonlinear equalizers outperform linear equalizers [3]. iterative block decision feedback equalizer (IB-DFE) [4] is a promising iterative FDE technique for SC-FDE that was first proposed in [5] and extended to diversity scenarios [6] and layered space-time schemes [7]. IB-DFE receivers can be regarded as iterative DFE receivers with the feedforward and the feedback operations implemented in the frequency domain. Since the feedback loop takes into account not just the hard decisions for each block, but also the overall block reliability we have small error propagation. Consequently, the IB-DFE techniques offer much better performances than the noniterative methods, with performances that can be close to the matched filter bound (MFB) [8] and can be regarded as low complexity turbo equalization schemes [9, 10], since the feedback loop uses the equalizer outputs instead of the channel decoder outputs. True turbo FDE schemes can also be designed based on the IB-DFE concept [11, 12].In this paper, we study the impact of the correlation coefficient estimation on the asymptotic performance of IB-DFE schemes. Several methods to estimate the correlation coefficient are derived as well as a proposed technique to compensate the inaccuracy of the estimations.This paper is organized as follows: conventional and turbo IB-DFE receivers are described in Section2. Various methods to estimate the correlation coefficient are presented in Section 3. The compensation technique is also included. A set of performance results is presented in Section 4, and Section 5 is concerned with the conclusion of this paper.


## 2. System Description

### 2.1. Transmitted and Received Signals

We consider an SC-FDE modulation where the data is transmitted in blocks ofN-useful modulation symbols {sn;n=0,…,N-1}, resulting from a direct mapping of the original data bits into a selected signal constellation, for example, QPSK.  A cyclic prefix with length longer than the channel impulse response is posteriorly appended, resulting the transmitted signal {sn;n=-NG,…,N-1}, where NG denotes the number of guard samples. This signal is transmitted over a time-dispersive channel. The received signal is sampled at the receiver where the CP samples are removed, leading in the time domain the samples {yn;n=0,…,N-1}. The corresponding frequency-domain block, obtained after an appropriate size-N discrete fourier transform (DFT) operation, is {Yk;k=0,1,…,N-1}, where Yk can be written as (1)Yk=SkHk+Nk,
where Hk denotes the overall channel frequency response for the kth frequency of the mth time block, and Nk represents channel noise term in frequency-domain.


### 2.2. IB-DFE Basic Receiver Structure

For the sake of simplicity, we will assume in this section perfect synchronization conditions.In SC-FDE to deal with the channel effects, we can employ a linear FDE. However, the performance can be improved if the linear FDE is replaced with an IB-DFE [5], whose basic structure is depicted in Figure 1. In this case, for the ith iteration, the frequency-domain block at the output of the equalizer is {S̃k(i);k=0,1,…,N-1}, with (2)S̃k(i)=Fk(i)Yk-Bk(i)Ŝk(i-1),
where {Fk(i);k=0,1,…,N-1} are the feedforward coefficients, and {Bk(i);k=0,1,…,N-1} are the feedback coefficients. {Ŝk(i-1);k=0,1,…,N-1} is the DFT of the block {ŝn(i-1);n=0,1,…,N-1}, with ŝn denoting the “hard decision” of sn from the previous FDE iteration. It can be shown that the optimum coefficients Bk and Fk that maximize the overall SNR (instantaneous signal-to-noise ratio) associated to the samples S̃k are [6](3)Bk(i)=ρ(Fk(i)Hk-1),(4)Fk(i)=κHk*α+(1-(ρm(i-1))2)|Hk|2;
respectively, where α=E[|Nk|2]/E[|Sk|2] (common to all data blocks), and κ selected to guarantee that 1/N∑k=0N-1Fk(i)Hk=1.Figure 1
Basic receiver structure of an IB-DFE.As we can see from (3) and (4), the correlation factor ρ(i-1) is a key parameter for the good performance of IB-DFE receivers, since it supplies a blockwise reliability measure of the estimates, associated to the previous iteration, that are employed in the feedback loop. This is done in the feedback loop by taking into account the “hard-decisions” for each block plus the overall block reliability, which reduces error propagation problems. The correlation factor ρ(i-1) is defined as (5)ρm(i-1)=E[ŝn(i-1)sn*]E[|sn|2]=E[Ŝk(i-1)Sk*]E[|Sk|2],
where the block {ŝn(i-1);n=0,1,…,N-1} denotes the data estimates associated to the previous iteration, that is, the hard decisions associated to the time-domain block at the output of the FDE, {s̃n(i);n=0,1,…,N-1} = IDFT {S̃k(i);k=0,1,…,N-1}.It should be noted that (2) can be written as (6)S̃k(i)=Fk(i)Yk-Bk′(i)S¯k,Block(i-1),
where Bk′(i)=Bk(i)/ρm(i-1) and S¯k,Block(i-1)=ρm(i-1)Ŝk(i-1) (as stated before, ρ(i-1) can be considered as the blockwise reliability of the estimates {Ŝk(i-1);k=0,1,…,N-1}).To improve the IB-DFE performance, it is possible to use “soft decisions,”s¯n(i), instead of “hard decisions,” ŝn(i). Under these conditions, the “blockwise average” is substituted by “symbol averages” [11]. This can be done by using {S¯k,Symbol(i-1);k=0,1,…,N-1} instead of {S¯k,Block(i-1);k=0,1,…,N-1} = DFT {s¯n,Symbol(i-1);n=0,1,…,N-1}, where s¯n,Symbol(i-1) denotes the average symbol values conditioned to the FDE output from previous iteration, s̃n(i-1). To simplify the notation, we will replace s¯n,Symbol(i-1) with s¯n(i-1) in the following equations.For QPSK constellations, the conditional expectations associated to the data symbols for theith iteration are given by(7)s¯n(i)=tanh(LnI(i)2)+jtanh(LnQ(i)2)=ρnIŝnI+jρnQŝnQ,
with the loglikelihood ratios (LLRs) of the “in-phase bit” and the “quadrature bit,” associated to snI and snQ, given by(8)LnI(i)=2σi2s̃nI(i),(9)LnQ(i)=2σi2s̃nQ(i),
respectively, with(10)σi2=12E[|sn-s̃n(i)|2]≈12N∑n=0N-1|ŝn(i)-s̃n(i)|2
with the signs of LnI and LnQ defining the hard decisions ŝnI=±1 and ŝnQ=±1, respectively. In (7), ρnI and ρnQ denote the reliabilities related to the “in-phase bit" and the “quadrature bit” of the nth symbol and are given by(11)ρnI(i)=E[snIŝnI]E[|snI|2]=|tanh(LnI(i)2)|,ρnQ(i)=E[snQŝnQ]E[|snQ|2]=|tanh(LnQ(i)2)|.
Therefore, the correlation coefficient employed in the feedforward coefficients will be given by (12)ρ(i)=12N∑n=0N-1(ρnI(i)+ρnQ(i)).
Obviously, for the first iteration ρnI(0)=ρnQ(0)=0, and consequently s¯n=0.We may note that the receiver that employs “blockwise reliabilities” is referred as IB-DFE with “hard decisions,” while the receiver that employs “symbol reliabilities” is referred as IB-DFE with “soft decisions.” The feedforward coefficients used in both types of IB-DFE receivers are still given by (4), but the feedback loop of the IB-DFE with “hard decisions” uses the estimated data block, weighted by a reliability coefficient common to the entire block, while for IB-DFE with “soft decisions,” the feedback loop uses a different reliability coefficient for each symbol (changes from bit to bit).With a conventional IB-DFE receiver, the log-likelihood values are computed on a symbol-by-symbol basis (i.e., we do not need to perform the channel decoding in the feedback loop). As an alternative, we can define a turbo IB-DFE that employs the channel decoder outputs instead of the uncoded “soft decisions” in the feedback loop. The main difference between conventional IB-DFE and turbo IB-DFE is in the decision device; in the first case, the decision device is a symbol-by-symbol soft decision (for QPSK constellation this corresponds to the hyperbolic tangent, as in (7)); for the turbo IB-DFE, a soft-in, soft-out (SISO) channel decoder is employed in the feedback loop. The SISO block can be implemented as defined in [13] and provides the LLRs of both the “information bits” and the “coded bits.” The input of the SISO block is LLRs of the “coded bits” at the FDE output, given by LnI(i) and LnQ(i).


## 2.1. Transmitted and Received Signals

We consider an SC-FDE modulation where the data is transmitted in blocks ofN-useful modulation symbols {sn;n=0,…,N-1}, resulting from a direct mapping of the original data bits into a selected signal constellation, for example, QPSK.  A cyclic prefix with length longer than the channel impulse response is posteriorly appended, resulting the transmitted signal {sn;n=-NG,…,N-1}, where NG denotes the number of guard samples. This signal is transmitted over a time-dispersive channel. The received signal is sampled at the receiver where the CP samples are removed, leading in the time domain the samples {yn;n=0,…,N-1}. The corresponding frequency-domain block, obtained after an appropriate size-N discrete fourier transform (DFT) operation, is {Yk;k=0,1,…,N-1}, where Yk can be written as (1)Yk=SkHk+Nk,
where Hk denotes the overall channel frequency response for the kth frequency of the mth time block, and Nk represents channel noise term in frequency-domain.


## 2.2. IB-DFE Basic Receiver Structure

For the sake of simplicity, we will assume in this section perfect synchronization conditions.In SC-FDE to deal with the channel effects, we can employ a linear FDE. However, the performance can be improved if the linear FDE is replaced with an IB-DFE [5], whose basic structure is depicted in Figure 1. In this case, for the ith iteration, the frequency-domain block at the output of the equalizer is {S̃k(i);k=0,1,…,N-1}, with (2)S̃k(i)=Fk(i)Yk-Bk(i)Ŝk(i-1),
where {Fk(i);k=0,1,…,N-1} are the feedforward coefficients, and {Bk(i);k=0,1,…,N-1} are the feedback coefficients. {Ŝk(i-1);k=0,1,…,N-1} is the DFT of the block {ŝn(i-1);n=0,1,…,N-1}, with ŝn denoting the “hard decision” of sn from the previous FDE iteration. It can be shown that the optimum coefficients Bk and Fk that maximize the overall SNR (instantaneous signal-to-noise ratio) associated to the samples S̃k are [6](3)Bk(i)=ρ(Fk(i)Hk-1),(4)Fk(i)=κHk*α+(1-(ρm(i-1))2)|Hk|2;
respectively, where α=E[|Nk|2]/E[|Sk|2] (common to all data blocks), and κ selected to guarantee that 1/N∑k=0N-1Fk(i)Hk=1.Figure 1
Basic receiver structure of an IB-DFE.As we can see from (3) and (4), the correlation factor ρ(i-1) is a key parameter for the good performance of IB-DFE receivers, since it supplies a blockwise reliability measure of the estimates, associated to the previous iteration, that are employed in the feedback loop. This is done in the feedback loop by taking into account the “hard-decisions” for each block plus the overall block reliability, which reduces error propagation problems. The correlation factor ρ(i-1) is defined as (5)ρm(i-1)=E[ŝn(i-1)sn*]E[|sn|2]=E[Ŝk(i-1)Sk*]E[|Sk|2],
where the block {ŝn(i-1);n=0,1,…,N-1} denotes the data estimates associated to the previous iteration, that is, the hard decisions associated to the time-domain block at the output of the FDE, {s̃n(i);n=0,1,…,N-1} = IDFT {S̃k(i);k=0,1,…,N-1}.It should be noted that (2) can be written as (6)S̃k(i)=Fk(i)Yk-Bk′(i)S¯k,Block(i-1),
where Bk′(i)=Bk(i)/ρm(i-1) and S¯k,Block(i-1)=ρm(i-1)Ŝk(i-1) (as stated before, ρ(i-1) can be considered as the blockwise reliability of the estimates {Ŝk(i-1);k=0,1,…,N-1}).To improve the IB-DFE performance, it is possible to use “soft decisions,”s¯n(i), instead of “hard decisions,” ŝn(i). Under these conditions, the “blockwise average” is substituted by “symbol averages” [11]. This can be done by using {S¯k,Symbol(i-1);k=0,1,…,N-1} instead of {S¯k,Block(i-1);k=0,1,…,N-1} = DFT {s¯n,Symbol(i-1);n=0,1,…,N-1}, where s¯n,Symbol(i-1) denotes the average symbol values conditioned to the FDE output from previous iteration, s̃n(i-1). To simplify the notation, we will replace s¯n,Symbol(i-1) with s¯n(i-1) in the following equations.For QPSK constellations, the conditional expectations associated to the data symbols for theith iteration are given by(7)s¯n(i)=tanh(LnI(i)2)+jtanh(LnQ(i)2)=ρnIŝnI+jρnQŝnQ,
with the loglikelihood ratios (LLRs) of the “in-phase bit” and the “quadrature bit,” associated to snI and snQ, given by(8)LnI(i)=2σi2s̃nI(i),(9)LnQ(i)=2σi2s̃nQ(i),
respectively, with(10)σi2=12E[|sn-s̃n(i)|2]≈12N∑n=0N-1|ŝn(i)-s̃n(i)|2
with the signs of LnI and LnQ defining the hard decisions ŝnI=±1 and ŝnQ=±1, respectively. In (7), ρnI and ρnQ denote the reliabilities related to the “in-phase bit" and the “quadrature bit” of the nth symbol and are given by(11)ρnI(i)=E[snIŝnI]E[|snI|2]=|tanh(LnI(i)2)|,ρnQ(i)=E[snQŝnQ]E[|snQ|2]=|tanh(LnQ(i)2)|.
Therefore, the correlation coefficient employed in the feedforward coefficients will be given by (12)ρ(i)=12N∑n=0N-1(ρnI(i)+ρnQ(i)).
Obviously, for the first iteration ρnI(0)=ρnQ(0)=0, and consequently s¯n=0.We may note that the receiver that employs “blockwise reliabilities” is referred as IB-DFE with “hard decisions,” while the receiver that employs “symbol reliabilities” is referred as IB-DFE with “soft decisions.” The feedforward coefficients used in both types of IB-DFE receivers are still given by (4), but the feedback loop of the IB-DFE with “hard decisions” uses the estimated data block, weighted by a reliability coefficient common to the entire block, while for IB-DFE with “soft decisions,” the feedback loop uses a different reliability coefficient for each symbol (changes from bit to bit).With a conventional IB-DFE receiver, the log-likelihood values are computed on a symbol-by-symbol basis (i.e., we do not need to perform the channel decoding in the feedback loop). As an alternative, we can define a turbo IB-DFE that employs the channel decoder outputs instead of the uncoded “soft decisions” in the feedback loop. The main difference between conventional IB-DFE and turbo IB-DFE is in the decision device; in the first case, the decision device is a symbol-by-symbol soft decision (for QPSK constellation this corresponds to the hyperbolic tangent, as in (7)); for the turbo IB-DFE, a soft-in, soft-out (SISO) channel decoder is employed in the feedback loop. The SISO block can be implemented as defined in [13] and provides the LLRs of both the “information bits” and the “coded bits.” The input of the SISO block is LLRs of the “coded bits” at the FDE output, given by LnI(i) and LnQ(i).


## 3. Correlation Coefficient Estimation

As shown in the previous section, the correlation factor is a key parameter for the good performance of IB-DFE receivers, since it supplies a blockwise reliability measure of the estimates employed in the feedback loop. However, contrarily to the channel frequency response, it changes from block to block and iteration to iteration. Therefore, it cannot be computed using reference blocks and needs to be obtained from the equalized output.In this section, we present several methods to estimate the correlation coefficient. The correlation coefficientρ(i) was defined in (5), and for the sake of simplicity, the iteration number i will be ignored in the following equations. Therefore, (5) can be rewritten as (13)ρ=E[ŝnsn*]E[|sn|2]=E[ŜkSk*]E[|Sk|2].
Let us consider the transmitted symbols {sn;n=0,…,N-1} corresponding to a QPSK constellation, under a Gray mapping rule. Consequently, the sn may be written as (14)sn=snI+snQ=±d±jd,
in which snI=Re{sn} is the “in-phase” component of sn, and snQ=Im{sn} is the “quadrature” component of sn. Also, d is given by d=D/2, where D denotes the minimum Euclidean distance between two constellation symbols. Under these conditions, E[|sn|2]=D2/4. The time-domain samples at the FDE's output are given by (15)s̃n=s̃nI+s̃nQ=sn+θn,
where θn denotes the global error term, which is Gaussian-distributed, with zeromean. The symbol estimates are then given by (16)ŝn=sn+ϑnI+jϑnQ,
where ϑnI denotes the error coefficient in ŝnI, and ϑnQ denotes the error coefficient in ŝnQ. Clearly, if ŝnI and ŝnQ have no errors, then ϑnI and ϑnQ are null. On the other hand, if there are errors, then the value of ϑnI and/or ϑnQ will be ±D. Consequently, ϑnI and ϑnQ are random variables, with values 0, +D and -D with probabilities 1-2Pe, Pe and Pe respectively, where Pe denotes the bit error rate. Therefore, it can be shown that (17)ρ=1-2Pe.
Naturally, in practice, we do not know the transmitted symbols {sn;n=0,…,N-1}.


### 3.1. Method I: Estimation Based on the BER Estimate

The total variance of the overall noise plus residual ISI,σEq2, is given by (18)σEq2=12E[|sn-s̃n|2].
The exact value of σEq2 can not be used in practice, because we do not know the transmitted symbols {sn;n=0,…,N-1}. Therefore, we may use an approximated value of σEq2, given by (19)σ̂Eq2=12N∑n=0N-1|ŝn-s̃n|2.
Provided that the numbers of decision errors in ŝn are small and N is high, we have (20)σEq2≈σ̂Eq2.
For a QPSK constellation, the estimated BER, denoted by P̂e, can be approximated by (21)P̂e≈Q(1σ̂Eq),
and, from (17), the estimated value of ρ will be(22)ρ̂=1-2P̂e.


### 3.2. Method II: Estimation Based on the LLR

This technique consists in the estimation of the correlation coefficient with resort to the log-likelihood ratio (LLRs), as derived in Section2.2. However, as in method I, an approximated value for the variance of channel and interference noise is considered, instead of the optimum value (23)σ̂Eq2=12N∑n=0N-1|ŝn-s̃n|2.
The calculation of the LLRs of the “in-phase bit” and the “quadrature bit”, associated to snI and snQ, are based in (8) and (9), respectively, but with σ̂Eq2 in place of σEq2.Consequently, the reliabilities related to the “in-phase bit” and the “quadrature bit” of thenth symbol, are based on (11), respectively, but with L̂nI and L̂nQ in place of LnI and LnQ.Therefore, the estimated value of the correlation coefficient will be given by(24)ρ̂=12N∑n=0N-1(ρ̂nI+ρ̂nQ).


### 3.3. Method III: Estimation Based on the MSE

This estimation method is based on the same technique employed in method I, given by (22). However, unlike method I, the BER calculation employs the mean-squared error (MSE) of the equalized samples, instead of an approximated value for the variance of the channel and interference noise. Therefore, in method III, the estimated bit error rate denoted by P̂e is given by (25)P̂e=Q(|γ|2⋅E|Sk|2σMSE2).
Once again, assuming a QPSK constellation with a gray mapping and sn=±1±j, then (25) can be written as (26)P̂e=Q(|γ|σMSE2),
where σMSE2 corresponds to the MSE and is defined as(27)σMSE2=1N2∑k=0N-1Θk,
with(28)Θk=E[|S̃k-Sk|2].
By combining (1) and (6), and assuming E[|Nk·Δk*|]=0, E[|Nk·Sk*|]=0, and E[|Sk·Δk*|]=0, we can rewrite (28) as(29)Θk=E[|S̃k-Sk|2]=E[|FkSkHk+FkNk-BkρŜk-Sk|2]=E[|FkHk-Bkρ2-1|2]⋅2⋅σS2+E[|Fk|2]⋅2⋅σN2+E[|Bk|2]ρ2⋅(1-ρ2)⋅2⋅σS2,
where the variance of the transmitted frequency-domain data symbols is given by(30)σS2=E[|Sk|2]2,
the variance of the channel noise is obtained by(31)σN2=E[|Nk|2]2,
the expected value of the error term for the kth frequency-domain “hard decision” estimate is defined as(32)E[|Δk|2]=σS2(1-ρ2),
and the estimated value of ρ can be expressed by(33)ρ̂=1-2P̂e,
with P̂e given by (26).


## 3.1. Method I: Estimation Based on the BER Estimate

The total variance of the overall noise plus residual ISI,σEq2, is given by (18)σEq2=12E[|sn-s̃n|2].
The exact value of σEq2 can not be used in practice, because we do not know the transmitted symbols {sn;n=0,…,N-1}. Therefore, we may use an approximated value of σEq2, given by (19)σ̂Eq2=12N∑n=0N-1|ŝn-s̃n|2.
Provided that the numbers of decision errors in ŝn are small and N is high, we have (20)σEq2≈σ̂Eq2.
For a QPSK constellation, the estimated BER, denoted by P̂e, can be approximated by (21)P̂e≈Q(1σ̂Eq),
and, from (17), the estimated value of ρ will be(22)ρ̂=1-2P̂e.


## 3.2. Method II: Estimation Based on the LLR

This technique consists in the estimation of the correlation coefficient with resort to the log-likelihood ratio (LLRs), as derived in Section2.2. However, as in method I, an approximated value for the variance of channel and interference noise is considered, instead of the optimum value (23)σ̂Eq2=12N∑n=0N-1|ŝn-s̃n|2.
The calculation of the LLRs of the “in-phase bit” and the “quadrature bit”, associated to snI and snQ, are based in (8) and (9), respectively, but with σ̂Eq2 in place of σEq2.Consequently, the reliabilities related to the “in-phase bit” and the “quadrature bit” of thenth symbol, are based on (11), respectively, but with L̂nI and L̂nQ in place of LnI and LnQ.Therefore, the estimated value of the correlation coefficient will be given by(24)ρ̂=12N∑n=0N-1(ρ̂nI+ρ̂nQ).


## 3.3. Method III: Estimation Based on the MSE

This estimation method is based on the same technique employed in method I, given by (22). However, unlike method I, the BER calculation employs the mean-squared error (MSE) of the equalized samples, instead of an approximated value for the variance of the channel and interference noise. Therefore, in method III, the estimated bit error rate denoted by P̂e is given by (25)P̂e=Q(|γ|2⋅E|Sk|2σMSE2).
Once again, assuming a QPSK constellation with a gray mapping and sn=±1±j, then (25) can be written as (26)P̂e=Q(|γ|σMSE2),
where σMSE2 corresponds to the MSE and is defined as(27)σMSE2=1N2∑k=0N-1Θk,
with(28)Θk=E[|S̃k-Sk|2].
By combining (1) and (6), and assuming E[|Nk·Δk*|]=0, E[|Nk·Sk*|]=0, and E[|Sk·Δk*|]=0, we can rewrite (28) as(29)Θk=E[|S̃k-Sk|2]=E[|FkSkHk+FkNk-BkρŜk-Sk|2]=E[|FkHk-Bkρ2-1|2]⋅2⋅σS2+E[|Fk|2]⋅2⋅σN2+E[|Bk|2]ρ2⋅(1-ρ2)⋅2⋅σS2,
where the variance of the transmitted frequency-domain data symbols is given by(30)σS2=E[|Sk|2]2,
the variance of the channel noise is obtained by(31)σN2=E[|Nk|2]2,
the expected value of the error term for the kth frequency-domain “hard decision” estimate is defined as(32)E[|Δk|2]=σS2(1-ρ2),
and the estimated value of ρ can be expressed by(33)ρ̂=1-2P̂e,
with P̂e given by (26).


## 4. Performance Results

In the following, we present a set of results obtained during the research of the impact of the correlation coefficient estimation on the performance of IB-DFE receivers, for 10 iterations, for a given channel realization. We consider uncoded transmissions of FFT blocks withN=1024 data symbols, selected from a QPSK constellation under a gray mapping rule. Similar results were observed for other values of N, provided that N≫1.


### 4.1. Results without Correlation Coefficient Compensation

Figure2 presents the evolution of the correlation coefficient for values of Eb/N0. The results show the differences between the ideal value of the correlation coefficient (obtained with (13)), and the estimated value for methods I, II, and III (obtained by (22), (24) and (33), resp.). As we can see, for methods I and II, the ρ̂ curve suffered a deviation from the curve corresponding to the evolution of the optimum (true) correlation coefficient. This is caused by the use of the approximated value σ̂Eq in the correlation coefficient calculation of both methods, which leads to an optimistic Pe derivation for method I, while as for method II results in optimistic versions of the LLRs. Consequently, these effects will originate optimistic estimates of the transmitted symbols in the feedback loop. In the case of method III, the ρ̂ curve is very close to the optimum (true) ρ curve. This means that the reliability measure estimation, with resort to the MSE, used in the feedback loop is very accurate. It is important to note that the curve related with method III is presented for comparison purposes only, since the variance of the channel and interference noise estimation, σ̂Eq, is not considered in the estimation process.Figure 2
Evolution ofρ as function of the Eb/N0 for methods I, II, and III.Next we present the BER performance results. Since that the results were similar for all methods, as an example, we present in Figure3 the results related with method I. These results are expressed as function of Eb/N0, where N0 is the one-sided power spectral density of the noise, and Eb is the energy of the transmitted bits (i.e., the degradation due to the useless power spent on the cyclic prefix is not included). These results also refer to a scenario where the IB-DFE suffers from error propagation.Figure 3
BER performance for method I.Figure4 presents the required BER to achieve Eb/N0=6 (dB) as a function of the iterations number. Clearly, there is a performance degradation after a few iterations. Although the approximation in (20) may seem reliable, the true value of σEq2 is in fact higher than the estimated one due to decision errors. Consequently, the estimated BER, P̂e, is lower than the true bit error rate, Pe, leading to ρ̂≥ρ, that is, we are assuming that the estimates used in the feedback loop are more reliable than what they are in fact. It should be pointed out that while underestimating ρ (i.e., using ρ̂<ρ) leads to a slower convergence of the IB-DFE, overestimating ρ (i.e., using ρ̂>ρ) leads to a fast convergence but worse BER values.Figure 4
Required BER to achieveEb/N0=6 (dB) as a function of the iterations number for methods I, II, and III without compensation.


### 4.2. Results with Correlation Coefficient Compensation

#### 4.2.1. Technique Description

To overcome the problem of using an optimistic version of transmitted symbols estimates in the feedback loop, we propose a technique to compensate the inaccuracy of the correlation coefficient estimation.The compensation factor denoted, byχ(ρ̂), can be expressed as χ(ρ̂)=ρ̂/ρ, where ρ̂ is the estimated correlation coefficient derived from a given estimation method. The curves in Figure 5, obtained by simulation, show the relation between the correlation coefficient estimation, ρ̂, and the compensation factor, χ(ρ̂), for methods I, II, and III. Clearly for each method, we can determine the value of the compensation factor χ(ρ̂) for each correspondent value of ρ̂, by a simple interpretation of the plot presented in Figure 5. Thus, by knowing χ(ρ̂), we can determine a very precise estimated version of the optimum correlation factor given by (34)ρ≈ρEst+Comp=ρ̂χ(ρ̂),
where ρEst+Comp denotes the compensated correlation coefficient. Therefore, ρEst+Comp can then be used in the derivation of the feedback and feedforward coefficients parameters of the IB-DFE receiver.Figure 5
Relation between the correlation coefficient estimation and the compensation factor for methods I, II, and III.


#### 4.2.2. Results

We now present the performance results obtained regarding the correlation coefficient compensation. Figure6 presents the evolution of the correlation coefficient as function of Eb/N0. As we can see, for methods I and II the curves are now very close to the curve corresponding to the optimum (true) correlation coefficient evolution, when compared with the results obtained without compensation given by Figure 2. This demonstrates that the correlation coefficient inaccuracy, due to the use of optimistic estimates of the transmitted symbols in the feedback loop, can be efficiently avoided with resort to the proposed compensation technique. As expected for method III, the impact of the correlation coefficient with compensation in the evolution of the correlation coefficient is not so significative. This can be explained by the fact that correlation coefficient ρ̂ obtained under method III was already close to the curve corresponding to the optimum (true) correlation coefficient evolution.Figure 6
Evolution ofρ as function of the Eb/N0 for methods I, II, and III with compensation.In Figure7, we present the BER performance results. Once again, since that the results were similar to all methods, as an example, we present in Figure 3 the results related with method I. The improvements obtained with compensation are very significative, since that the performance curves obtained with the compensated correlation coefficient (denoted as “ρEst1+Comp”, “ρEst2+Comp,” and “ρEst3+Comp” in the figures) are now very close to those obtained with optimum (true) correlation coefficient (denoted as “ρTrue” in the figures), as the error propagation problem was corrected.Figure 7
BER performance for method I with compensation.Figure8 presents the required BER to achieve Eb/N0=6 (dB) as a function of the iterations number. Clearly, there is a performance enhancement when compared with the correlation coefficient estimates without compensation for all methods.Figure 8
Required BER to achieveEb/N0=6 (dB) as a function of the iterations number for methods I, II, and III with compensation.


## 4.1. Results without Correlation Coefficient Compensation

Figure2 presents the evolution of the correlation coefficient for values of Eb/N0. The results show the differences between the ideal value of the correlation coefficient (obtained with (13)), and the estimated value for methods I, II, and III (obtained by (22), (24) and (33), resp.). As we can see, for methods I and II, the ρ̂ curve suffered a deviation from the curve corresponding to the evolution of the optimum (true) correlation coefficient. This is caused by the use of the approximated value σ̂Eq in the correlation coefficient calculation of both methods, which leads to an optimistic Pe derivation for method I, while as for method II results in optimistic versions of the LLRs. Consequently, these effects will originate optimistic estimates of the transmitted symbols in the feedback loop. In the case of method III, the ρ̂ curve is very close to the optimum (true) ρ curve. This means that the reliability measure estimation, with resort to the MSE, used in the feedback loop is very accurate. It is important to note that the curve related with method III is presented for comparison purposes only, since the variance of the channel and interference noise estimation, σ̂Eq, is not considered in the estimation process.Figure 2
Evolution ofρ as function of the Eb/N0 for methods I, II, and III.Next we present the BER performance results. Since that the results were similar for all methods, as an example, we present in Figure3 the results related with method I. These results are expressed as function of Eb/N0, where N0 is the one-sided power spectral density of the noise, and Eb is the energy of the transmitted bits (i.e., the degradation due to the useless power spent on the cyclic prefix is not included). These results also refer to a scenario where the IB-DFE suffers from error propagation.Figure 3
BER performance for method I.Figure4 presents the required BER to achieve Eb/N0=6 (dB) as a function of the iterations number. Clearly, there is a performance degradation after a few iterations. Although the approximation in (20) may seem reliable, the true value of σEq2 is in fact higher than the estimated one due to decision errors. Consequently, the estimated BER, P̂e, is lower than the true bit error rate, Pe, leading to ρ̂≥ρ, that is, we are assuming that the estimates used in the feedback loop are more reliable than what they are in fact. It should be pointed out that while underestimating ρ (i.e., using ρ̂<ρ) leads to a slower convergence of the IB-DFE, overestimating ρ (i.e., using ρ̂>ρ) leads to a fast convergence but worse BER values.Figure 4
Required BER to achieveEb/N0=6 (dB) as a function of the iterations number for methods I, II, and III without compensation.


## 4.2. Results with Correlation Coefficient Compensation

### 4.2.1. Technique Description

To overcome the problem of using an optimistic version of transmitted symbols estimates in the feedback loop, we propose a technique to compensate the inaccuracy of the correlation coefficient estimation.The compensation factor denoted, byχ(ρ̂), can be expressed as χ(ρ̂)=ρ̂/ρ, where ρ̂ is the estimated correlation coefficient derived from a given estimation method. The curves in Figure 5, obtained by simulation, show the relation between the correlation coefficient estimation, ρ̂, and the compensation factor, χ(ρ̂), for methods I, II, and III. Clearly for each method, we can determine the value of the compensation factor χ(ρ̂) for each correspondent value of ρ̂, by a simple interpretation of the plot presented in Figure 5. Thus, by knowing χ(ρ̂), we can determine a very precise estimated version of the optimum correlation factor given by (34)ρ≈ρEst+Comp=ρ̂χ(ρ̂),
where ρEst+Comp denotes the compensated correlation coefficient. Therefore, ρEst+Comp can then be used in the derivation of the feedback and feedforward coefficients parameters of the IB-DFE receiver.Figure 5
Relation between the correlation coefficient estimation and the compensation factor for methods I, II, and III.


### 4.2.2. Results

We now present the performance results obtained regarding the correlation coefficient compensation. Figure6 presents the evolution of the correlation coefficient as function of Eb/N0. As we can see, for methods I and II the curves are now very close to the curve corresponding to the optimum (true) correlation coefficient evolution, when compared with the results obtained without compensation given by Figure 2. This demonstrates that the correlation coefficient inaccuracy, due to the use of optimistic estimates of the transmitted symbols in the feedback loop, can be efficiently avoided with resort to the proposed compensation technique. As expected for method III, the impact of the correlation coefficient with compensation in the evolution of the correlation coefficient is not so significative. This can be explained by the fact that correlation coefficient ρ̂ obtained under method III was already close to the curve corresponding to the optimum (true) correlation coefficient evolution.Figure 6
Evolution ofρ as function of the Eb/N0 for methods I, II, and III with compensation.In Figure7, we present the BER performance results. Once again, since that the results were similar to all methods, as an example, we present in Figure 3 the results related with method I. The improvements obtained with compensation are very significative, since that the performance curves obtained with the compensated correlation coefficient (denoted as “ρEst1+Comp”, “ρEst2+Comp,” and “ρEst3+Comp” in the figures) are now very close to those obtained with optimum (true) correlation coefficient (denoted as “ρTrue” in the figures), as the error propagation problem was corrected.Figure 7
BER performance for method I with compensation.Figure8 presents the required BER to achieve Eb/N0=6 (dB) as a function of the iterations number. Clearly, there is a performance enhancement when compared with the correlation coefficient estimates without compensation for all methods.Figure 8
Required BER to achieveEb/N0=6 (dB) as a function of the iterations number for methods I, II, and III with compensation.


## 4.2.1. Technique Description

To overcome the problem of using an optimistic version of transmitted symbols estimates in the feedback loop, we propose a technique to compensate the inaccuracy of the correlation coefficient estimation.The compensation factor denoted, byχ(ρ̂), can be expressed as χ(ρ̂)=ρ̂/ρ, where ρ̂ is the estimated correlation coefficient derived from a given estimation method. The curves in Figure 5, obtained by simulation, show the relation between the correlation coefficient estimation, ρ̂, and the compensation factor, χ(ρ̂), for methods I, II, and III. Clearly for each method, we can determine the value of the compensation factor χ(ρ̂) for each correspondent value of ρ̂, by a simple interpretation of the plot presented in Figure 5. Thus, by knowing χ(ρ̂), we can determine a very precise estimated version of the optimum correlation factor given by (34)ρ≈ρEst+Comp=ρ̂χ(ρ̂),
where ρEst+Comp denotes the compensated correlation coefficient. Therefore, ρEst+Comp can then be used in the derivation of the feedback and feedforward coefficients parameters of the IB-DFE receiver.Figure 5
Relation between the correlation coefficient estimation and the compensation factor for methods I, II, and III.


## 4.2.2. Results

We now present the performance results obtained regarding the correlation coefficient compensation. Figure6 presents the evolution of the correlation coefficient as function of Eb/N0. As we can see, for methods I and II the curves are now very close to the curve corresponding to the optimum (true) correlation coefficient evolution, when compared with the results obtained without compensation given by Figure 2. This demonstrates that the correlation coefficient inaccuracy, due to the use of optimistic estimates of the transmitted symbols in the feedback loop, can be efficiently avoided with resort to the proposed compensation technique. As expected for method III, the impact of the correlation coefficient with compensation in the evolution of the correlation coefficient is not so significative. This can be explained by the fact that correlation coefficient ρ̂ obtained under method III was already close to the curve corresponding to the optimum (true) correlation coefficient evolution.Figure 6
Evolution ofρ as function of the Eb/N0 for methods I, II, and III with compensation.In Figure7, we present the BER performance results. Once again, since that the results were similar to all methods, as an example, we present in Figure 3 the results related with method I. The improvements obtained with compensation are very significative, since that the performance curves obtained with the compensated correlation coefficient (denoted as “ρEst1+Comp”, “ρEst2+Comp,” and “ρEst3+Comp” in the figures) are now very close to those obtained with optimum (true) correlation coefficient (denoted as “ρTrue” in the figures), as the error propagation problem was corrected.Figure 7
BER performance for method I with compensation.Figure8 presents the required BER to achieve Eb/N0=6 (dB) as a function of the iterations number. Clearly, there is a performance enhancement when compared with the correlation coefficient estimates without compensation for all methods.Figure 8
Required BER to achieveEb/N0=6 (dB) as a function of the iterations number for methods I, II, and III with compensation.


## 5. Conclusions

In this paper, we studied the impact of the correlation factor estimation in the performance of IB-DFE receivers. Several methods to estimate the correlation coefficient were presented as well as a technique to compensate the estimation errors. It was shown that the inaccuracy due to the use of optimistic estimates of the transmitted symbols in the feedback loop can be efficiently avoided with resort to the proposed compensation technique.



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*Source: 980830-2011-08-07.xml*