Digital Signal Processing 70 (2017) 105–113

Contents lists available at ScienceDirect

Digital Signal Processing

www.elsevier.com/locate/dsp

Finite-length predictive decision feedback equalizer design

for multipath channels with large delay spread

Wei-Chieh Chang a,∗, Jenq-Tay Yuan b

a Graduate Institute of Applied Science and Engineering, Fu Jen Catholic University, New Taipei City 24250, Taiwanb Department of Electrical Engineering, Fu Jen Catholic University, New Taipei City 24250, Taiwan

a r t i c l e i n f o a b s t r a c t

Article history:Available online 7 August 2017

Keywords:Conventional decision feedback equalizer (CDFE)Decision feedback equalizer (DFE)Minimum mean-squared error (MMSE)Multipath channelsPredictive decision feedback equalizer (PDFE)

Decision feedback equalizers (DFEs) have been widely used to mitigate the effect of intersymbol interference. Most previous studies have focused on conventional DFEs (CDFEs), and relatively little research has addressed predictive DFEs (PDFEs). A finite-length minimum-mean-squared-error predictive DFE (MMSE–PDFE) was developed herein in the presence of multipath channels with large delay spread. We found that the MMSE–PDFE may have lower computational complexity and achieve a better symbol error rate performance than the existing MMSE–CDFE. Therefore, the proposed MMSE–PDFE may offer a viable alternative to the MMSE–CDFE. Computer simulations were conducted to verify our results using highly dispersive multipath channels.

© 2017 Elsevier Inc. All rights reserved.

1. Introduction

Decision feedback equalizers (DFEs) are equalization structures that are widely used to eliminate severe intersymbol interference (ISI), and their computational complexity is considerably lower than that of nonlinear maximum-likelihood receivers. DFEs are used in digital television (DTV) systems, high-speed chip-to-chip links, underwater acoustics (UWA), and optical fiber communica-tions [1–11], and can be found in many practical channels that are characterized by long and sparse channel impulse responses (CIRs) spanning more than 100 symbol periods because of large delay spreads. The structure of an infinite-length conventional DFE (CDFE) was first proposed by Austin [12], followed by the deriva-tion of the infinite-length CDFE by Salz [13] under the minimum-mean-squared-error (MMSE) criterion (MMSE–CDFE) and its sub-sequent finite-length counterpart proposed by Al-Dhahir and Cioffi [14] and Casas et al. [15]. A thorough analysis of the structure and properties of the MMSE–CDFE has been presented by López-Valcarce in [16]. In addition, Belfiore and Park [17] introduced the infinite-length predictive DFE (PDFE) under the MMSE criterion (MMSE–PDFE), which was shown to be identical to the infinite-length MMSE–CDFE. As discussed in [18], the feedforward filter (FFF) and feedback filter (FBF) of the PDFE are optimized indepen-dently, whereas those of the CDFE are optimized jointly. Notably, deriving the FFF and the FBF independently renders the finite-

* Corresponding author.E-mail address: ja0613ck@gmail.com (W.-C. Chang).

http://dx.doi.org/10.1016/j.dsp.2017.08.0011051-2004/© 2017 Elsevier Inc. All rights reserved.

length PDFE attractive in coding systems [19,20], as well as in hybrid DFE designs [21,22]. This is because the PDFE with a de-coder can afford delayed reliable decisions for the FBF in the cod-ing systems, whereas the CDFE with a decoder requires delay-free decisions for the FBF whose results may not be sufficiently reliable [19]. Furthermore, in the hybrid DFE designs, the FFF and the FBF are implemented in the frequency-domain and the time-domain, respectively. Consequently, the PDFE is more suitable for the hybrid DFE designs without losing optimality than the CDFE, because the FFF and the FBF of the PDFE can be optimized independently [22].

This paper develops a finite-length MMSE–PDFE in a single-input multiple-output (SIMO) system. Although several techniques have used adaptive algorithms [23–27] based on the stochastic gra-dient descent (SGD) method to approximate the tap weights of a finite-length MMSE–PDFE, the proposed MMSE–PDFE algorithm is based on channel estimation by exploiting the channel outputs and a known pilot signal to estimate the CIR, which is subsequently employed to estimate the optimal tap weights of the MMSE–PDFE. Because the FFF and the FBF of the proposed MMSE–PDFE are op-timized separately, their derivations vary from the two existing MMSE–CDFEs [14,15]; moreover, the computational complexity of the MMSE–PDFE tends to be lower than that of the MMSE–CDFEs, especially when the CIR is long. This study also theoretically an-alyzes the ISI performance of the MMSE–PDFE and compares the analytic results with those of the MMSE–CDFE [15]. Our results demonstrate that in order to achieve the desired ISI performance, the MMSE–PDFE may have a much lower norm of FBF tap weights than that of the MMSE–CDFE and, consequently, the MMSE–PDFE may be much less vulnerable to error propagation than the MMSE–

106 W.-C. Chang, J.-T. Yuan / Digital Signal Processing 70 (2017) 105–113

Fig. 1. Block diagram of the finite-length SIMO CDFE.

CDFE, especially at medium-low signal-to-noise ratio (SNR). As a result, the MMSE–PDFE may outperform the MMSE–CDFE in terms of the symbol error rate (SER) at medium-low SNR in the presence of symbol detection errors. These results may lend confidence to the use of the MMSE–PDFE as a viable alternative to the MMSE–CDFE.

In this paper, boldface letters are used to denote matrices (up-per case) and vectors (lower case). Furthermore, (·)∗ , (·)T , (·)H , and (·)−1 represent complex conjugate, transpose, Hermitian, and in-verse terms, respectively; and E{.} and ‖.‖p denote the expectation operation and lp -norm, respectively. Additionally, IM and 0M×N are used to represent an M × M identity matrix and M × N all-zero matrix, respectively. In particular, the column vector δk has only one nonzero coefficient in the kth position. The variable CM×N is used to represent an M × N complex-valued matrix, and diag{.}denotes a diagonal matrix.

2. System description and review of MMSE–CDFE

In this section, the system model is formulated first, followed by a brief introduction of two existing schemes for the design of the finite-length MMSE–CDFE (Fig. 1). The first MMSE–CDFE scheme, proposed by Al-Dhahir and Cioffi [14], constrains the FFF length only and is referred to as the MMSE–CDFE–FC. The sec-ond scheme, proposed by Casas et al. [15], constrains both the FFF length and the FBF length, and is referred to as the MMSE–CDFE–FBC.

2.1. System model

A continuous-time received signal can be expressed as

r(t) =∑

k

skc(t − kT ) + w(t) (1)

where sk is the complex transmitted symbol with symbol rate 1/T ; c(t) is the continuous-time impulse response of a linear time-invariant causal communication channel, and w(t) is the complex channel noise. To retain the channel diversity and suppress the timing phase sensitivity [28], a multirate system model is used at the receiver. If Δ is the sampling interval given by Δ = T /L, where L is a positive integer denoting the oversampling factor, then the lth output of the oversampled received signal at time n can be ex-pressed as

r(nT + lΔ) =∑

k

skc(nT − kT + lΔ) + w(nT + lΔ),

l = 0, . . . , L − 1 (2)

By defining r(l)n

Δ= r(nT + lΔ), c(l)n

Δ= c(nT + lΔ), and w(l)n

Δ= w(nT +lΔ), and by considering a finite impulse response causal chan-nel with c(l)

k = 0 for k < 0 or k > M − 1, we may express (2) as r(l)

n = ∑M−1k=0 c(l)

k sn−k + w(l)n . A block of LN f received samples may

be formulated as the LN f -by-1 vector

rn:n−N +1 = Csn:n−P+1 + wn:n−N +1 (3)

f f

where PΔ= M + N f − 1, sn:n−P+1

Δ= [sn, . . . , sn−P+1]T ∈ CP×1,

wn:n−N f +1Δ= [w(0)

n , . . . , w(0)n−N f +1| · · · |w(L−1)

n , . . . , w(L−1)n−N f +1]T ∈

CLN f ×1, rn:n−N f +1

Δ= [r(0)n , . . . , r(0)

n−N f +1| · · · |r(L−1)n , . . . , r(L−1)

n−N f +1]T ∈C

LN f ×1, and

CΔ=

⎡⎢⎢⎢⎣

C(0)

C(1)

...

C(L−1)

⎤⎥⎥⎥⎦ ∈C

LN f ×P (4)

Here, C(l) ∈ CN f ×P is a Toeplitz matrix corresponding to the lth

subchannel and is defined as

C(l) Δ=

⎡⎢⎢⎢⎢⎢⎣

c(l)0 c(l)

1 · · · c(l)M−1 0 · · · 0

0 c(l)0 · · · c(l)

M−2 c(l)M−1 · · · 0

......

. . ....

.... . .

...

0 0 · · · c(l)0 c(l)

1 · · · c(l)M−1

⎤⎥⎥⎥⎥⎥⎦

Channel convolutional matrix, C, in the SIMO system is assumed to be of full rank. The autocorrelation matrices of the transmitted symbol and noise are respectively defined as

RssΔ= E

{sn:n−P+1sH

n:n−P+1

} ∈CP×P (5)

Rw wΔ= E

{wn:n−N f +1wH

n:n−N f +1

} ∈CLN f ×LN f (6)

where Rss and Rw w are assumed to be nonsingular and positive-definite, and the transmitted symbol sn is assumed to be uncorre-lated with the channel noise wn .

2.2. MMSE–CDFE

As shown in Fig. 1, the SIMO CDFE consists of a fractionally spaced FFF, an FBF, and a memoryless decision device whose out-put is used as the FBF input. The design of the MMSE–CDFE–FC constrains the optimal FFF (FFFo) to a fixed length, and therefore the optimal FBF (FBFo) needs to be long enough to eliminate the post-cursor at the FFFo output [14]. If the FBFo is not sufficiently long, the MMSE–CDFE–FC may result in high mean-squared error (MSE). In contrast to MMSE–CDFE–FC, the design of the MMSE–CDFE–FBC constrains both the FFFo and the FBFo to have fixed lengths, and both the FFFo and FBFo lengths are determined si-multaneously at the beginning of the derivation [15]. In summary, if the design of the MMSE–CDFE–FC satisfies the condition of a sufficiently long FBFo, then it is identical to that of the MMSE–CDFE–FBC for the same number of tap weights. Therefore, only the MMSE–CDFE–FBC is considered in our computer simulations.

3. Proposed MMSE–PDFE

In this section, a finite-length MMSE–PDFE in a SIMO system is derived. In contrast to the CDFE, the proposed finite-length PDFE uses tn

Δ= xn − sn rather than the decision device output sn as the

W.-C. Chang, J.-T. Yuan / Digital Signal Processing 70 (2017) 105–113 107

Fig. 2. Block diagram of the finite-length SIMO PDFE.

input of the FBF, which then predicts tn by using the previous samples tn−1, . . . , tn−Nb , as depicted in Fig. 2. The computational complexities required by the use of the MMSE–PDFE, the MMSE–CDFE–FC and the MMSE–CDFE–FBC are then compared.

3.1. MMSE–PDFE

In contrast to the MMSE–CDFE presented in Section 2, the FFF and FBF of the MMSE–PDFE are optimized separately; therefore the fractionally spaced FFFo of the MMSE–PDFE is identical to the frac-tionally spaced MMSE linear equalizer (MMSE–LE) (see [29]), such that

fHo

Δ= [f (0)0 , . . . , f (0)

N f −1| · · · | f (L−1)0 , . . . , f (L−1)

N f −1

]∗ ∈C1×LN f

= δTαRsrR−1

rr (7)

with the optimal decision delay being

α = argk

min0≤k≤P−1

{δT

k

(Rss − RsrR−1

rr Rrs)δk

}(8)

where RrrΔ= E{rn:n−N f +1rH

n:n−N f +1}, RsrΔ= E{sn:n−P+1rH

n:n−N f +1}, and Rrs = RH

sr . Therefore, the FFFo output at time n may be com-puted as

xnΔ= fH

o rn:n−N f +1 (9)

Note that the optimal decision delay of the MMSE–PDFE is equal to that of the MMSE–LE in (8) because the FBFo of the MMSE–PDFE is known to be a predictor that does not cause any filter delay [17].

The FBF vector of the PDFE is defined as

bΔ= [bNb , . . . ,b2,b1]T ∈ C

Nb×1 (10)

and the PDFE output can be expressed as

yn = xn +Nb∑

k=1

b∗k(xn−k − sn−k) (11)

where xn−kΔ= fH

o rn−k:n−k−N f +1 denotes the MMSE–LE output at time n − k. Under the assumption of correct previous decisions, the estimation error of the PDFE at time n can be expressed as

enΔ= yn − sn−α

= (xn − sn−α) +Nb∑

k=1

b∗k(xn−k − sn−α−k)

= tn +Nb∑

k=1

b∗ktn−k

= tn + tn (12)

where tn = xn − sn = xn − sn−α denotes the estimation error of the MMSE–LE at time n, and tn−k = xn − sn−α−k for 1 ≤ k ≤ Nb denote

the previous estimation errors of the MMSE–LE. Therefore, (12) re-veals that the FBF is used as a linear predictor that predicts tn by employing tn−1, . . . , tn−Nb to compute the predicted value tn

Δ=∑Nbk=1 b∗

ktn−k . By defining bT Δ= [bT , 1] and tnΔ= [tn−Nb , . . . , tn]T ,

(12) can be rewritten as

en = bH tn (13)

When the FBF is long enough to reduce the correlation of tn , en be-comes approximately white; consequently, sn becomes sufficiently reliable. According to (12) and (13), the MSE of the PDFE, which is essentially the mean-squared prediction error based on the predic-tor b, can be computed to be

JMSEΔ= E

{|en|2} = bH Rtt b (14)

where RttΔ= E{tntH

n } is the correlation matrix of the estimation error of the MMSE–LE, and the element in the ith row and jth column of Rtt is given by

E{

tn−it∗n− j

} = E{(xn−i − sn−α−i)(xn− j − sn−α− j)

∗}= (

CH fo − δα

)HR(i, j)

ss(CH fo − δα

) + fHo R(i, j)

w w fo (15)

where R(i, j)ss

Δ= E{sn−i:n−i−P+1sHn− j:n− j−P+1} and R(i, j)

w wΔ=

E{wn−i:n−i−N f +1wHn− j:n− j−N f +1}. If the transmitted symbol is i.i.d.

random variable with variance σ 2s and the noise is AWGN with

variance σ 2w , the FFFo in (7) becomes

fHo = δT

αCH EH (16)

with the following optimal delay:

α = argk

min0≤k≤P−1

{δT

k

(IP − CH EC

)δk

}(17)

where E Δ= [CCH + (σ 2w/σ 2

s ) · IN f ]−1, and (15) can be rewritten as

E{

tn−it∗n− j

} = σ 2s

(CH fo − δα

)HS(k)

(CH fo − δα

) + σ 2w fH

o W(k)fo

(18)

where R(i, j)ss

Δ= σ 2s S(k) , R(i, j)

w wΔ= σ 2

w W(k) , and k = j − i. The elements in the pth row and qth column of S(k) and W(k) are given by

S(k)(p,q) =

{1 for p − q = k, and 0 ≤ p,q ≤ P − 10 otherwise

(19)

W (k)(p,q) =

⎧⎨⎩

1 for p − q = k, iN f ≤ p,q ≤ iN f + N f − 1,

and 0 ≤ i ≤ L − 10 otherwise

(20)

where the channel noise is assumed to satisfy E{w(i)n (w( j)

n )∗} = 0if i �= j. For the PDFE with an oversampling factor L = 2, S(k) and W(k) can be expressed as

108 W.-C. Chang, J.-T. Yuan / Digital Signal Processing 70 (2017) 105–113

Table 1Comparison of equalizers according to the number of complex multiplications per iteration.

Equalizer Tap weight calculation EqualizationMMSE–LE [29] O (L3 N3

f ) + 2P (LN f )2 LN f

MMSE–CDFE–FC [14] O (L3 N3f ) + 2P (LN f )

2 + 2O (P 3) LN f + Nb

MMSE–CDFE–FBC [15] O ((LN f + Nb)3) + 2P (LN f + Nb)2 LN f + Nb

MMSE–PDFE O (L3 N3f ) + 2P (LN f )

2 + 2O ((Nb + 1)3) + P (LN f )

+ (Nb + 1)(P 2 + L2 N2f + 2)

LN f + Nb

S(k) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

[0(P−|k|)×|k| I(P−|k|)

0|k|×|k| 0|k|×(P−|k|)

]if − P < k < 0

IP if k = 0[0k×(P−k) 0k×k

I(P−k) 0(P−k)×k

]if 0 < k < P

0P×P otherwise

(21)

W(k) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎡⎢⎢⎣

0(N f −|k|)×|k| I(N f −|k|)0|k|×|k| 0|k|×(N f −|k|)

0N f ×N f

0N f ×N f

0(N f −|k|)×|k| I(N f −|k|)0|k|×|k| 0|k|×(N f −|k|)

⎤⎥⎥⎦

if − N f < k < 0

I(2N f ) if k = 0⎡⎢⎢⎣

0k×(N f −k) 0k×k

I(N f −k) 0(N f −k)×k0N f ×N f

0N f ×N f

0k×(N f −k) 0k×k

I(N f −k) 0(N f −k)×k

⎤⎥⎥⎦

if 0 < k < N f

0(2N f )×(2N f ) otherwise

(22)

Using the Cholesky decomposition RttΔ= LDLH , we can write

(14) as

JMSE = bH LDLH b (23)

where L ∈ C(Nb+1)×(Nb+1) is a lower triangular matrix, and D Δ=

diag{D0, . . . , DNb }. It can be readily shown that (23) can be min-

imized by choosing θ ∈ {0, 1, . . . , Nb} such that bH matches the θ th row of L−1, whose index is equal to that of min{D0, . . . , DNb }. Therefore, the (Nb + 1)-by-1 FBF vector can be expressed as

bH = δTθ L−1 (24)

The optimal value of θ and the MMSE can then be derived to be

β = argθ

min0≤θ≤Nb

{δTθ Dδθ

}(25)

JMMSE = δTβDδβ = Dβ (26)

It is worth noting that the choice of β may affect the MSE per-formance of the MMSE–PDFE. On the basis of the Gram–Schmidt orthogonalization [30], Dβ indicates the backward prediction er-ror power corresponding to the βth row of L−1, whose elements are the coefficients of the backward prediction error filter (i.e., bH

o = δTβ L−1). Because D0 ≥ D1 ≥ · · · ≥ D Nb , choosing β = Nb

yields JMMSE = D Nb and bHo = δT

NbL−1, both of which along with

(16) and (17) are referred to as the MMSE–PDFE.

3.2. Computational complexity

The computational complexity of the proposed MMSE–PDFE is measured according to the number of complex multiplications per iteration by considering the case of an i.i.d. transmitted symbol and white noise.

(1) Computation of α in (8) requires the inversion of Rrr with complexity O (L3N3

f ) and the determination of the matrix product RsrR−1

rr Rrs , which requires P (LN f )2 complex multipli-

cations.(2) Computation of fH

o by using (7) involves P (LN f )2 complex

multiplications.(3) Computation of E{tn−it∗

n− j} by using (18) requires (P 2 +L2N2

f + 2) complex multiplications. Consequently, the compu-

tation of Rtt requires (Nb + 1)(P 2 + L2N2f + 2) complex multi-

plications because a) E{tn−it∗n− j} = E{tn−pt∗

n−q} if j − i = q − pand b) E{tn−it∗

n− j} = E∗{tn−it∗n− j}.

(4) Computation of bHo by using (24) requires the eigendecompo-

sition of Rtt with complexity O ((Nb + 1)3) and the inversion of L with complexity O ((Nb + 1)3).

Table 1 presents a comparison of the total numbers of complex multiplications per iteration when the MMSE–LE, MMSE–CDFE–FC, MMSE–CDFE–FBC, and MMSE–PDFE are adopted in a SIMO sys-tem. The computational complexity of the channel estimation is not considered in the comparison because the four equalizers re-quire the same number of complex multiplications for fractionally spaced channel estimations. The MMSE–CDFE–FC, MMSE–CDFE–FBC, and MMSE–PDFE are compared according to the number of complex multiplications per iteration by using an example in which M = 100, L = 2, N f = 50, and Nb = 50. The MMSE–CDFE–FC, MMSE–CDFE–FBC, and MMSE–PDFE respectively require 1.06 ×107, 1.01 × 107, and 5.9 × 106 complex multiplications per itera-tion, whereas they require 2.07 × 107, 1.23 × 107, and 7.79 × 106

complex multiplications per iteration, respectively, when the sub-channel length increases such that M becomes 150, and L, N f , and Nb remain unchanged. Thus, for the three DFEs, both the MMSE–CDFE–FBC and MMSE–CDFE–FC require larger numbers of complex multiplications than the MMSE–PDFE does when the CIR is long. However, both CDFEs may require much smaller numbers of complex multiplications per iteration than those shown previ-ously when using some simple constellations such as binary or quadrature phase shift keying (BPSK or QPSK). This is due to the fact that the FBF outputs of CDFE require only additions and sub-tractions because the real and imaginary parts of the FBF input are either +1 or −1, owing to the decision device, which is not shared by the PDFE.

4. Performance analysis

To investigate the ISI performance of the four equalizers listed in Table 1, we explicitly evaluated the residual ISI at their outputs based on the following two assumptions:

(1) The transmitted symbol is an i.i.d. random variable, with Rss =σ 2

s IP , and(2) The channel noise is AWGN, with Rw w = σ 2

w ILN f .

Consider that the LN f received symbols in (3) such that r(l)

n−k = 0 if k > N f − 1, which in turn results in sn−k = 0 for k > P − 1 and w(l) = 0 for k > N f − 1. We then have

n−k

W.-C. Chang, J.-T. Yuan / Digital Signal Processing 70 (2017) 105–113 109

sn−k:n−k−P+1 = sTn:n−P+1S(k) = S(−k)sn:n−P+1 (27)

wn−k:n−k−N f +1 = wTn:n−N f +1W(k) = W(−k)wn:n−N f +1 (28)

where the elements of S(k) and W(k) are defined in (19) and (20), respectively, and S(−k) = (S(k))T and W(−k) = (W(k))T . Hence, the LN f received samples at time n − k can be expressed as

rn−k:n−k−N f +1 = CS(−k)sn:n−P+1 + W(−k)wn:n−N f +1 (29)

We first consider the MMSE–CDFE, whose output can be com-puted using (3), as

yn = en + sn−α

= hHCDFEsn:n−P+1 + fH

o wn:n−N f +1

= sn−α︸︷︷︸desired symbol

+ (hCDFE − δα)H sn:n−P+1︸ ︷︷ ︸ISI term

+ fHo wn:n−N f +1︸ ︷︷ ︸

filtered channel noise

(30)

where fo ∈ CLN f ×1 is the FFFo; α is the optimal system delay,

and hCDFE is the vector-valued impulse response of the combined channel-FFFo–FBFo system such that

hCDFEΔ= CH fo − (bo − δα) (31)

where boΔ= [01×α, 1, bT

o, 01×(P−α−Nb−1)]T ∈ CP×1, and bo ∈ C

Nb×1

is the FBFo. In passing, it should be noted that the first term on the right-hand side of (30) is the desired symbol, the middle term is the ISI term, and the last term is the filtered channel noise. The ISI term at the MMSE–CDFE output in (30) can therefore be expressed as follows:

hCDFE − δα = (CH fo − δα

) − (bo − δα) (32)

where CH fo − δα denotes the residual ISI terms in the FFFo output and bo − δα is the reconstruction of the FBFo. By defining d Δ=[d0, . . . , dP−1]T Δ= CH fo − δα and d

Δ= [d0, . . . , dP−1]T Δ= hCDFE − δα , we can write (32) as⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

d0...

dα−1

dα

dα+1...

dP−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

d0...

dα−1dα

dα+1...

dP−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

−⎡⎣ 0(α+1)×1

bo0(P−α−Nb−1)×1

⎤⎦ = d − (bo − δα)

(33)

Equation (33) reveals that the post-cursor ISI components dα+1,

. . . , dα+Nb in the FFFo output can be eliminated by using the FBFo bo. However, for MMSE–CDFE–FC, which constrains the FFFolength, setting the FBFo length such that Nb < P −α − 1 may yield an unsatisfactory performance because the post-cursor ISI from the FFFo output (i.e., dα+1, . . . , dP−1) can only be partially elim-inated by the FBFo. To entirely eliminate the post-cursor ISI, the optimal MMSE–CDFE–FC requires a long enough FBFo such that Nb = P − α − 1 (as described in [14]). By contrast, the FBFo of the MMSE–CDFE–FBC can entirely eliminate the post-cursor ISI, even if Nb < P − α − 1.

For the MMSE–PDFE, the FFFo output (i.e., the MMSE–LE out-put) in (9) can be rewritten as

xn = sn−α + (hLE − δα)H sn:n−P+1 + fHo wn:n−N f +1 (34)

where hLE denotes the vector-valued impulse response of the com-bined channel-FFFo system

hLEΔ= CH fo (35)

and hLE − δα denotes the residual ISI associated with the FFFo. By using (13) and (29), the MMSE–PDFE output can be expressed as

yn = en + sn−α

= sn−α +Nb∑

k=0

(b∗

ktn−k)

= sn−α +Nb∑

k=0

[b∗

k

(fHo C − δT

α

)S(−k)

]sn:n−P+1

+Nb∑

k=0

[b∗

k fHo W(−k)

]wn:n−N f +1 (36)

where bk , 0 ≤ k ≤ Nb are the elements of bo with b0 = 1. Addition-ally,

tn−k = xn−k − sn−α−k

= (fHo C − δT

α

)S(−k)sn:n−P+1 + fH

o W(−k)wn:n−N f +1

denotes the previous estimation error of the MMSE–LE at time n −k. Hence, the MMSE–PDFE output can be separated into three parts as follows:

yn = sn−α︸︷︷︸desired symbol

+ (hPDFE − δα)H sn:n−P+1︸ ︷︷ ︸ISI term

+ fHo wn:n−N f +1︸ ︷︷ ︸

filtered channel noise

(37)

where the last term on the right-hand side denotes the filtered channel noise with fo

Δ= ∑Nbk=0(bkW(k)fo), and hPDFE denotes the

vector-valued impulse response of the combined channel-FFFo-FBFosystem such that

hPDFEΔ= δα +

Nb∑k=0

[bkS(k)

(CH fo − δα

)](38)

The residual ISI resulting from the MMSE–PDFE in (37) can there-fore be represented in terms of hLE defined in (35) as

hPDFE − δα =Nb∑

k=0

[bkS(k)(hLE − δα)

](39)

By defining a Δ= [a0, . . . , aP−1]T Δ= hLE − δα and aΔ= [a0, . . . ,

aP−1]T Δ= hPDFE − δα , and by letting Nb = P − 1, we can express (39) as⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

a0...

aα−1aα

aα+1...

aP−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1 0 · · · · · · 0

b1. . .

. . .. . .

...

b2. . .

. . .. . .

......

. . .. . .

. . . 0bP−1 · · · b2 b1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

a0...

aα−1aα

aα+1...

aP−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

= Ba (40)

where B ∈ CP×P is defined in terms of the elements of bo. Equa-

tion (40) also reveals that the FBFo, B, generates backward predic-tion errors of various orders, a0, . . . , aP−1, on the basis of linear prediction theory with tap weights bk . In other words, the pre-cursor ISI (i.e., a0, . . . , aα−1), cursor ISI (i.e., aα ), and post-cursor ISI (i.e., aα+1, . . . , aP−1) resulting from the use of the MMSE–LE have all been further suppressed by the FBFo. In contrast to the MMSE–CDFE–FC [14], the MMSE–PDFE may yield a satisfac-tory performance even when Nb < P − α − 1. However, when the

110 W.-C. Chang, J.-T. Yuan / Digital Signal Processing 70 (2017) 105–113

Fig. 3. Channel Impulse response for SIMO system with oversampling factor of 2: (a) Channel 1 and (b) Channel 2.

FBFo in the MMSE–CDFE–FC has a large number of tap weights, the MMSE–CDFE–FC is likely to result in severe error propaga-tion, which in turn may appreciably increase the SER. Therefore, compared with the MMSE–CDFE–FC, the MMSE–PDFE may be con-siderably less vulnerable to error propagation because a shorter FBFo may be sufficient to mitigate the residual ISI at the FFFo out-put of the MMSE–PDFE. Furthermore, the proposed MMSE–PDFE performs as well as the MMSE–CDFE–FBC [15], in which the FFFoand the FBFo can mitigate the precursor ISI and the post-cursor ISI, respectively. These results were verified in the present study through computer simulations, which are discussed in Section 5.

5. Simulation results

This section reports that computer simulations were conducted by adopting two multipath microwave channel models. Channel 1 is the long and sparse Brazil D channel [3,31] with a carrier fre-quency of 473 MHz and a symbol rate of 10.76 MHz as shown in Fig. 3(a), and Channel 2 was adopted by Casas et al. [15] as shown in Fig. 3(b). The CIR lengths of Channel 1 and Channel 2 are M = 90 and M = 24, respectively, and their oversampling factors are equal to 2. The transmitted symbol and channel noise were modeled as an i.i.d. random variable of variance σ 2

s and AWGN with variance σ 2

w , respectively, such that the input SNR was given by SNRI

Δ= σ 2s /σ 2

w . The residual ISI of the MMSE–LE, MMSE–CDFE, and MMSE–PDFE can be determined by substituting (35), (31), and (38) into the following equation, respectively:

ISI(h)(dB)Δ= 10 log10

{∑k |hk|2 − |max(hk)|2

|max(hk)|2}

(41)

where max(hk) denotes the maximum element of h.

5.1. ISI performance

Fig. 4 shows the theoretical results for the MMSE–CDFE–FBC and the proposed MMSE–PDFE in terms of the residual ISI as a function of N f and Nb , ranging from 1 to 70, for Channel 1 in a SIMO system where L = 2, σ 2

s = 1, and SNRI = 15 dB. Notably, the residual ISI depends on both the FFFo and FBFo lengths, such that it decreases when N f and Nb increase. For the MMSE–CDFE–FBC, if N f < 3, Nb must be larger than 60 to achieve a desirable resid-ual ISI performance, as indicated in Fig. 4(a). However, if N f ≥ 3, Nb = 30 would be large enough to achieve a desirable residual ISI performance. For the MMSE–PDFE, the residual ISI gradually decreases as N f increases, provided that Nb ≥ 5, as shown in Fig. 4(b).

Figs. 5 and 6 compare the norms of the optimal FBFs of the proposed MMSE–PDFE and the MMSE–CDFE–FBC through (38) and (31) when SNRI = 15 dB. Notably, N f = 60 and Nb = 40 were cho-sen for the MMSE–PDFE and MMSE–CDFE–FBC so that the smallest

Fig. 4. Plots of residual ISI versus the FFF length (N f ) and the FBF length (Nb) for Channel 1 and SNRI = 15 dB for the (a) MMSE–CDFE–FBC and (b) MMSE–PDFE.

possible residual ISI could be achieved for the same total num-ber of tap weights with Ntotal = N f + Nb = 100, as shown in Figs. 4(b) and 4(a). The optimal system delay of the MMSE–PDFE can be computed to be α = 78 by using (17), while that of the MMSE–CDFE–FBC can be computed to be α = 59 [15,16]. The cur-sor indices of the MMSE–PDFE and MMSE–CDFE–FBC are identical to their corresponding optimal system delays [cf. Figs. 5(a) and 6(a)]. The precursor ISI is located to the left of the cursor, whereas the post-cursor ISI is located to the right of the cursor. Figs. 5(b) and 6(b) show that the FBFo of the MMSE–CDFE–FBC can only eliminate the post-cursor ISI, whereas that of the MMSE–PDFE can mitigate the cursor ISI, precursor ISI, and post-cursor ISI. Conse-quently, the norm of FBFo tap weights of the MMSE–PDFE, 0.262, is much lower than that of the MMSE–CDFE–FBC, 0.783 [cf. Figs. 5(c) and 6(c)].

Fig. 7 compares the norms of the optimal FBFs of the MMSE–PDFE and the MMSE–CDFE–FBC as a function of N f and Nb under the same conditions as those in Figs. 4–6. For the MMSE–CDFE–FBC, if N f ≥ 3 and Nb ≥ 30, the norm of FBFo tap weights remains higher than 0.5, and it markedly increases as Nb increases. How-ever, for the MMSE–PDFE, the norm of FBFo tap weights with arbi-trarily chosen N f and Nb tend to be lower than 0.5, thus indicating that the proposed MMSE–PDFE may end up having a lower norm of FBFo tap weights than that of the MMSE–CDFE–FBC regardless of the values of N f and Nb . Similar results can be obtained by using other channels. As a result, the MMSE–PDFE may be less vulnerable to error propagation than the MMSE–CDFE–FBC is at medium-low SNR.

5.2. SER performance

Computer simulations were conducted to compare the pro-posed MMSE–PDFE with the MMSE–CDFE–FBC in terms of SER performance as a function of SNRI, ranging from 0 to 25 dB, for the four-ray pulse-amplitude modulation (4-PAM) constellation with Channel 1 and Channel 2. The “CF” and “DF” designators in Figs. 8and 9 are respectively referred to as the cases where the FBFo used

W.-C. Chang, J.-T. Yuan / Digital Signal Processing 70 (2017) 105–113 111

Fig. 5. Impulse responses of (a) the combined channel-FFFo-FBFo system, (b) the difference between channel-FFFo-FBFo system and channel-FFFo system, and (c) the FBFo for the proposed MMSE–PDFE with N f = 60 and Nb = 40 when Channel 1 was used for SNRI = 15 dB.

Fig. 6. Impulse responses of (a) the combined channel-FFFo-FBFo system, (b) the difference between channel-FFFo-FBFo system and channel-FFFo system, and (c) the FBFo for the MMSE–CDFE–FBC with N f = 60 and Nb = 40 when Channel 1 was used for SNRI = 15 dB.

Fig. 7. Plots of the norm of the FBFo tap weights versus the FFF length (N f ) and the FBF length (Nb) for Channel 1 and SNRI = 15 dB for the (a) MMSE–CDFE–FBC and (b) MMSE–PDFE.

the correct decision as its input and where the FBFo used the pre-viously detected symbol as its input.

Notably, the MMSE–PDFE is less vulnerable to error propaga-tion than the MMSE–CDFE–FBC, especially at medium-low SNRI[cf. Figs. 8 and 9]. For example, the performance loss due to er-ror propagation from the MMSE–PDFE(CF) to the MMSE–PDFE(DF) was 1 dB when SER = 10−1.5, whereas that from the MMSE–CDFE–FBC(CF) to the MMSE–PDFE-FBC(DF) was 4 dB with Channel 1 as depicted in Fig. 8. Similarly, the performance loss due to er-ror propagation in the MMSE–PDFE was 1.5 dB when SER = 10−2, whereas that of the MMSE–CDFE–FBC was 3 dB with Channel 2 as depicted in Fig. 9. Because the norm of FBFo tap weights of the MMSE–PDFE with N f = 60 and Nb = 40 was much lower than that of the MMSE–CDFE–FBC with N f = 60 and Nb = 40[cf. Figs. 5(c) and 6(c)], the MMSE–PDFE(DF) significantly outper-formed the MMSE–CDFE–FBC(DF) at SNRI between 5 and 16 dB [cf. Fig. 8]. Fig. 9 shows similar simulated results for Channel 2. These results also confirm that a large FBF tap weight is a major culprit for the error propagation in DFEs, thus likely being respon-sible for raising the SERs [32–34], owing to the result that the error propagation may well emerge at medium-low SNRI. Moreover, the MMSE–PDFE required only approximately 65% (90%) of complex

Fig. 8. Comparison of SER performance of the proposed MMSE–PDFE, MMSE–LE, and MMSE–CDFE–FBC when the 4-PAM and Channel 1 were used.

multiplications per iteration as required by the MMSE–CDFE–FBC for Channel 1 (Channel 2).

Fig. 10 depicts the role played by the FBFo in the proposed MMSE–PDFE with the previously detected symbol as its input, when comparing the MMSE–PDFE with the MMSE–LE in terms of their SER performance as a function of SNRI, ranging from 10 to 25 dB when the 4-PAM and Channel 2 were used in a SIMO system where L = 2 and N f = 45. Clearly, the MMSE–PDFE out-performed the MMSE–LE, and the former may achieve its optimal performance when Nb = 7. Furthermore, the MMSE–LE (N f = 45), MMSE–PDFE (N f = 45 and Nb = 4), and MMSE–PDFE (N f = 45and Nb = 7) require 1.83 × 106, 1.9 × 106, and 1.94 × 106 com-plex multiplications per iteration, respectively (Table 1). Therefore,

112 W.-C. Chang, J.-T. Yuan / Digital Signal Processing 70 (2017) 105–113

Fig. 9. Comparison of SER performance of the proposed MMSE–PDFE, MMSE–LE, and MMSE–CDFE–FBC when the 4-PAM and Channel 2 were used.

Fig. 10. Comparison of SER performance between the MMSE–LE and MMSE–PDFE when the 4-PAM and Channel 2 were used.

the choice of the FBFo length involves a tradeoff between the per-formance of the MMSE–PDFE and its computational complexity.

6. Conclusion

In this study, we developed and analyzed a finite-length MMSE–PDFE based on channel estimation where the optimal FFF is identical to the MMSE–LE while the optimal FBF can be ob-tained by using the Cholesky decomposition. The proposed MMSE–PDFE was compared with the existing finite-length MMSE–CDFE, in terms of ISI and SER performance when using two multipath microwave channels. The results show that the FBF of the MMSE–PDFE may mitigate the cursor ISI, precursor ISI, and post-cursor ISI, whereas the FBF of the MMSE–CDFE–FBC only eliminates the post-cursor ISI. Moreover, because the MMSE–PDFE may end up having a much lower norm of the optimal FBF tap weights than the MMSE–CDFE–FBC does regardless of the choice of FFF length and FBF length, the MMSE–PDFE tends to be less vulnerable to er-ror propagation. Our simulation results show that the MMSE–PDFE

outperformed the MMSE–CDFE–FBC in terms of SER performance at medium-low SNR with 4-PAM constellation when their FBFs used the previously detected symbol as their input. Finally, because the FFF and the FBF of the MMSE–PDFE are optimized indepen-dently, its computational complexity may be lower than that of the MMSE–CDFE–FBC, especially when the CIR is long. Therefore, our results lend confidence to the use of the MMSE–PDFE as a vi-able alternative to the MMSE–CDFE, especially in coding systems or in hybrid DFE designs.

Acknowledgments

This work was supported by the Ministry of Science and Tech-nology of Taiwan, R.O.C., under Grants MOST 103-2221-E-030-007-MY2 and MOST 105-2221-E-030-003-MY2.

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Wei-Chieh Chang was born in New Taipei City, Taiwan, R.O.C., on June 13, 1990. He received the B.S. degree in electrical engineering from Fu Jen Catholic University (FJCU), New Taipei City, Taiwan, in 2012. He is currently working towards the Ph.D. degree in electrical engineering with the Graduate Institute of Applied Science and Engineering at FJCU. His research interests include signal processing with applications in digital communication system. Areas of focus in-

clude filter design, adaptive signal processing, and compressed sensing.

Jenq-Tay Yuan was born in Taipei, Taiwan, R.O.C., on September 9, 1957. He received the B.S. degree in electronic engineering from Fu Jen Catholic Uni-versity (FJCU), Taipei, Taiwan, in 1981, the M.S. and the Ph.D. degrees, both in electrical engineering, from Missouri University of Science and Technology (for-merly the University of Missouri-Rolla), MO, in 1986 and 1991, respectively. From 1985 to 1991, he was a Teaching and Research Assistant at the University of

Missouri-Rolla. From 1992 to 1993, he was a system engineer in the For-mosa Plastics Corporation, Point Comfort, TX. He joined the Department of Electronic Engineering, FJCU, in 1993 as an Associate Professor. He was promoted to Professor in 2001 and was Head of the Department of Elec-tronic Engineering at FJCU from 2003 to 2006. He was Dean of the College of Science and Engineering at FJCU from 2011 to 2015. He is currently a Professor with the Department of Electrical Engineering and the Vice Pres-ident for Academic Affairs at FJCU. His research interest is in the area of statistical and adaptive signal processing with applications in communica-tion systems and his current work involves research in adaptive algorithms and the design of blind adaptive receivers in communication systems.