Digital Signal Processing 70 (2017) 105–113
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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 b
Graduate Institute of Applied Science and Engineering, Fu Jen Catholic University, New Taipei City 24250, Taiwan Department of Electrical Engineering, Fu Jen Catholic University, New Taipei City 24250, Taiwan
a r t i c l e
i n f o
Article history: Available online 7 August 2017 Keywords: Conventional decision feedback equalizer (CDFE) Decision feedback equalizer (DFE) Minimum mean-squared error (MMSE) Multipath channels Predictive decision feedback equalizer (PDFE)
a b s t r a c t 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 communications [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 derivation of the infinite-length CDFE by Salz [13] under the minimummean-squared-error (MMSE) criterion (MMSE–CDFE) and its subsequent 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 LópezValcarce 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 infinitelength MMSE–CDFE. As discussed in [18], the feedforward filter (FFF) and feedback filter (FBF) of the PDFE are optimized independently, whereas those of the CDFE are optimized jointly. Notably, deriving the FFF and the FBF independently renders the finite-
*
Corresponding author. E-mail address:
[email protected] (W.-C. Chang).
http://dx.doi.org/10.1016/j.dsp.2017.08.001 1051-2004/© 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 decoder can afford delayed reliable decisions for the FBF in the coding 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 singleinput multiple-output (SIMO) system. Although several techniques have used adaptive algorithms [23–27] based on the stochastic gradient 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 optimized 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 analyzes 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–
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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 (upper case) and vectors (lower case). Furthermore, (·)∗ , (·) T , (·) H , and (·)−1 represent complex conjugate, transpose, Hermitian, and inverse terms, respectively; and E {.} and . p denote the expectation operation and l p -norm, respectively. Additionally, I M and 0 M × N are used to represent an M × M identity matrix and M × N all-zero matrix, respectively. In particular, the column vector δ k has only one nonzero coefficient in the kth position. The variable C M × N is used to represent an M × N complex-valued matrix, and diag{.} denotes a diagonal matrix.
C
2.1. System model
LN f ×1
(0)
( L −1)
(0)
= [ w n , . . . , w n−N f +1 | · · · | w n Δ
, rn:n− N f +1 =
⎡
( L −1)
, . . . , w n−N f +1 ] T
.. .
⎣
∈
⎤
C(0) ⎢ C(1) ⎥ Δ⎢ ⎥
C=⎢
∈
[rn , . . . , rn(0−)N f +1 | · · · |rn(L −1) , . . . , rn(L−−N1f)+1 ] T (0)
⎥ ∈ C LN f × P ⎦
(4)
C( L −1) Here, C(l) ∈ C N f × P is a Toeplitz matrix corresponding to the lth subchannel and is defined as
⎡
⎢ ⎢ ⎢ =⎢ ⎢ ⎣
c0
(l)
c1
0
c0
.. . 0
(l) (l)
.. .
0
···
(l)
c M −1
0
···
0
⎤
⎥ · · · c (Ml)−2 c (Ml)−1 · · · 0 ⎥ ⎥ .. .. .. ⎥ .. .. ⎥ . . . . . ⎦ (l) · · · c 0(l) c1 · · · c (Ml)−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
Rss = E sn:n− P +1 snH:n− P +1 ∈ C P × P
(5)
R w w = E wn:n− N f +1 wnH:n− N f +1 ∈ C LN f × LN f
(6)
Δ
Δ
where Rss and R w w are assumed to be nonsingular and positivedefinite, and the transmitted symbol sn is assumed to be uncorrelated with the channel noise w n .
A continuous-time received signal can be expressed as
sk c (t − kT ) + w (t )
Δ
Δ
C LN f ×1 , and
C
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 second 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.
wn:n− N f +1
(l) Δ
2. System description and review of MMSE–CDFE
r (t ) =
Δ
where P = M + N f − 1, sn:n− P +1 = [sn , . . . , sn− P +1 ] T ∈ C P ×1 ,
(1)
2.2. MMSE–CDFE
k
where sk is the complex transmitted symbol with symbol rate 1/ T ; c (t ) is the continuous-time impulse response of a linear timeinvariant 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 expressed as
r (nT + lΔ) =
sk c (nT − kT + lΔ) + w (nT + lΔ),
k
l = 0, . . . , L − 1
(2)
(l) Δ
(l) Δ
(l) Δ
By defining rn = r (nT + lΔ), cn = c (nT + lΔ), and w n = w (nT + lΔ), and by considering a finite impulse response causal chan(l) nel with ck = 0 for k < 0 or k > M − 1, we may express (2) as (l)
M −1
(l)
As shown in Fig. 1, the SIMO CDFE consists of a fractionally spaced FFF, an FBF, and a memoryless decision device whose output 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 simultaneously 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
(l)
rn = k=0 ck sn−k + w n . A block of LN f received samples may be formulated as the LN f -by-1 vector
rn:n− N f +1 = Csn:n− P +1 + wn:n− N f +1
(3)
In this section, a finite-length MMSE–PDFE in a SIMO system is derived. In contrast to the CDFE, the proposed finite-length PDFE Δ
uses tn = xn − sˆn rather than the decision device output sˆn 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− N b , 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 fractionally spaced MMSE linear equalizer (MMSE–LE) (see [29]), such that Δ
( 0)
( 0)
( L −1 )
foH = f 0 , . . . , f N −1 | · · · | f 0 f
( L −1 ) ∗
, . . . , f N f −1
−1 = δ αT Rsr Rrr
∈ C1× LN f
Δ
where Rrr =
−1 δkT Rss − Rsr Rrr Rrs δk
E {rn:n− N f +1 rnH:n− N +1 }, f
Δ Δ By defining b˜ T = [b T , 1] and tn = [tn− N b , . . . , tn ] T , (12) can be rewritten as ∗
b t . k=1 k n−k
en = b˜ H tn
When the FBF is long enough to reduce the correlation of tn , en becomes approximately white; consequently, sˆn becomes sufficiently reliable. According to (12) and (13), the MSE of the PDFE, which is essentially the mean-squared prediction error based on the predictor b, can be computed to be
(8)
Rsr =
E {sn:n− P +1 rnH:n− N +1 }, f
xn = fo 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
b = [b N b , . . . , b2 , b1 ] T ∈ C N b ×1 Δ
(10)
bk∗ (xn−k − sˆn−k )
Δ
en = yn − sn−α
(i , j )
Δ
random variable with variance σs2 and the noise is AWGN with 2 variance σ w , the FFFo in (7) becomes
foH = δ αT C H E H
(16)
with the following optimal delay:
δkT I P − C H EC δk
(17)
2 where E = [CC H + (σ w /σs2 ) · I N f ]−1 , and (15) can be rewritten as
H
2 H S(k) C H fo − δ α + σ w fo W(k) fo
(18) (i , j ) Δ
(i , j ) Δ
2 where Rss = σs2 S(k) , R w w = σ 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
(k)
bk∗ (xn−k − sn−α −k )
E tn−i tn∗− j = σs2 C H fo − δ α
S ( p ,q) =
k =1
1 for p − q = k, and 0 ≤ p , q ≤ P − 1 0 otherwise
(19)
⎧ ⎨ 1 for p − q = k, iN f ≤ p , q ≤ iN f + N f − 1, (k) and 0 ≤ i ≤ L − 1 W ( p ,q) = ⎩
bk∗ tn−k
0
k =1
= tn + tˆn
Δ
= E {sn−i :n−i − P +1 snH− j :n− j − P +1 } and R w w = E {wn−i :n−i − N f +1 wnH− j :n− j − N +1 }. If the transmitted symbol is i.i.d. f Rss
where xn−k = foH 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
= tn +
(i , j )
(15)
Δ
(11)
Δ
H (i , j ) (i , j ) = C H fo − δ α Rss C H fo − δ α + foH R w w fo
k 0≤k≤ P −1
k =1
Nb
α = arg min
and the PDFE output can be expressed as
= (xn − sn−α ) +
(14)
E tn−i tn∗− j = E (xn−i − sn−α −i )(xn− j − sn−α − j )∗
where
Nb
where Rtt = E {tn tnH } 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
Δ
Δ
Δ H
Nb
J MSE = E |en |2 = b˜ H Rtt b˜
H and Rrs = Rsr . Therefore, the FFFo output at time n may be computed as
yn = xn +
(13)
Δ
k 0≤k≤ P −1
Δ
employing tn−1 , . . . , tn− N b to compute the predicted value tˆn =
Nb
(7)
with the optimal decision delay being
α = arg min
the previous estimation errors of the MMSE–LE. Therefore, (12) reveals that the FBF is used as a linear predictor that predicts tn by
(12)
where tn = xn − sˆn = xn − sn−α denotes the estimation error of the MMSE–LE at time n, and tn−k = xn − sn−α −k for 1 ≤ k ≤ N b denote
(20)
otherwise
where the channel noise is assumed to satisfy E { w n ( w n )∗ } = 0 if i = j. For the PDFE with an oversampling factor L = 2, S(k) and W(k) can be expressed as (i )
( j)
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Table 1 Comparison of equalizers according to the number of complex multiplications per iteration. Equalizer MMSE–LE [29]
Tap weight calculation O ( L 3 N 3f ) + 2P ( LN f )2
Equalization LN f
O (( LN f + N b )3 ) + 2P ( LN f + N b )2 O ( L 3 N 3f ) + 2P ( LN f )2 + 2O (( N b + 1)3 ) + P ( LN f )
LN f + N b LN f + N b
O ( L 3 N 3f ) + 2P ( LN f )2 + 2O ( P 3 )
MMSE–CDFE–FC [14] MMSE–CDFE–FBC [15] MMSE–PDFE
LN f + N b
+ ( N b + 1)( P 2 + L 2 N 2f + 2)
⎧ 0( P −|k|)×|k| I( P −|k|) ⎪ ⎪ if − P < k < 0 ⎪ ⎪ 0|k|×|k| 0|k|×( P −|k|) ⎪ ⎪ ⎪ ⎨I if k = 0 P S(k) = ⎪ 0 0 k×k ⎪ ⎪ k×( P −k) if 0 < k < P ⎪ ⎪ I( P −k) 0( P −k)×k ⎪ ⎪ ⎩
(1) Computation of α in (8) requires the inversion of Rrr with complexity O ( L 3 N 3f ) and the determination of the matrix
(21)
otherwise
0P ×P
⎧⎡ ⎤ 0( N f −|k|)×|k| I( N f −|k|) ⎪ ⎪ 0 ⎪ N f ×N f ⎥ ⎢ ⎪ ⎪ ⎥ ⎪ ⎢ 0|k|×|k| 0|k|×( N f −|k|) ⎪ ⎪ ⎣ 0( N f −|k|)×|k| I( N f −|k|) ⎦ ⎪ ⎪ 0 ⎪ N f ×N f ⎪ 0|k|×|k| 0|k|×( N f −|k|) ⎪ ⎪ ⎪ ⎪ ⎪ if − Nf P − 1 and w n−k = 0 for k > N f − 1. We then have
W.-C. Chang, J.-T. Yuan / Digital Signal Processing 70 (2017) 105–113
sn−k:n−k− P +1 = snT:n− P +1 S(k) = S(−k) sn:n− P +1 wn−k:n−k− N f +1 =
wnT:n− N f +1 W(k)
(−k)
=W
wn:n− N f +1
rn−k:n−k− N f +1 = CS
(−k)
sn:n− P +1 + W
wn:n− N f +1
Δ
(27)
hLE = C H fo
(28)
and hLE − δ α denotes the residual ISI associated with the FFFo . By using (13) and (29), the MMSE–PDFE output can be expressed as
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 (−k)
Nb
bk∗ tn−k
= sn−α +
(29)
Nb ∗
H
s
desired symbol
+ foH wn:n− N f +1
+ (hCDFE − δ α ) sn:n− P +1 + ISI term
filtered channel noise
hCDFE = C H fo − (b¯ o − δ α ) Δ
(31)
where b¯ o = [01×α , 1, bTo , 01×( P −α − N b −1) ] T ∈ C P ×1 , and bo ∈ C N b ×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: Δ
hCDFE − δ α = C fo − δ α − (b¯ o − δ α ) H
Nb
(32)
where C fo − δ α denotes the residual ISI terms in the FFFo output
and b¯ o − δ α is the reconstruction of the FBFo . By defining d = Δ Δ Δ [d0 , . . . , d P −1 ] T = C H fo − δ α and d¯ = [d¯ 0 , . . . , d¯ P −1 ] T = hCDFE − δ α , we can write (32) as
tn−k = xn−k − sn−α −k
= foH C − δ αT S(−k) sn:n− P +1 + foH 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 =
s
n −α desired symbol
⎤
⎢ . ⎥ ⎢ . ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎥ ⎡ ⎤ ⎢¯ ⎥ d α −1 ⎥ 0(α +1)×1 ⎢ d α −1 ⎥ ⎢ ⎥ ⎢ ⎢ ¯ ⎥ ⎢ ⎦ = d − (b¯ o − δ α ) ⎣ bo ⎢ dα ⎥ = ⎢ dα ⎥ ⎥− ⎢¯ ⎥ ⎥ d 0 ⎢ d α +1 ⎥ ⎢ α + 1 ( P − α − N − 1 )× 1 b ⎢ ⎥ ⎢ . ⎥ ⎥ ⎢ .. ⎥ ⎢ ⎣ . ⎦ ⎣ .. ⎦ d P −1 d¯ P −1 (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 FFFo length, setting the FBFo length such that N b < P − α − 1 may yield an unsatisfactory performance because the post-cursor ISI from the FFFo output (i.e., dα +1 , . . . , d P −1 ) can only be partially eliminated by the FBFo . To entirely eliminate the post-cursor ISI, the optimal MMSE–CDFE–FC requires a long enough FBFo such that N b = 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 N b < P − α − 1. For the MMSE–PDFE, the FFFo output (i.e., the MMSE–LE output) in (9) can be rewritten as
xn = sn−α + (hLE − δ α ) H sn:n− P +1 + foH wn:n− N f +1
+ (hPDFE − δ α ) H sn:n− P +1 + f¯oH wn:n− N f +1 (37) ISI term
(34)
where hLE denotes the vector-valued impulse response of the combined channel-FFFo system
filtered channel noise
where the last term on the right-hand side denotes the filtered
N
channel noise with ¯fo = k=b 0 (bk W(k) fo ), and hPDFE denotes the vector-valued impulse response of the combined channel-FFFo -FBFo system such that
Δ
d0
(36)
where bk , 0 ≤ k ≤ N b are the elements of b˜ o with b0 = 1. Additionally,
Δ
H
⎡
bk∗ foH W(−k) wn:n− N f +1
k =0
(30)
where fo ∈ C LN 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
⎤
k =0
n −α
d¯ 0
bk foH C − δ αT S(−k) sn:n− P +1
= sn−α +
H = hCDFE sn:n− P +1 + foH wn:n− N f +1
⎡
k =0
yn = en + sn−α
(35)
yn = en + sn−α
We first consider the MMSE–CDFE, whose output can be computed using (3), as
=
109
Δ
hPDFE = δ α +
Nb
bk S(k) C H fo − δ α
(38)
k =0
The residual ISI resulting from the MMSE–PDFE in (37) can therefore be represented in terms of hLE defined in (35) as
hPDFE − δ α =
Nb
bk S(k) (hLE − δ α )
(39)
k =0
Δ
Δ
Δ
By defining a = [a0 , . . . , a P −1 ] T = hLE − δ α and a¯ = [¯a0 , . . . , Δ
a¯ P −1 ] T = hPDFE − δ α , and by letting N b = P − 1, we can express (39) as
⎡
a¯ 0
⎤
⎡
1
⎢ .. ⎥ ⎢ . ⎥ ⎢ ⎢ ⎥ b1 ⎢ a¯ α −1 ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ a¯ α ⎥ = ⎢ ⎢ ⎥ ⎢ b2 ⎢ a¯ α +1 ⎥ ⎢ ⎢ ⎥ ⎢ . ⎢ . ⎥ ⎣ .. ⎣ .. ⎦ a¯ P −1
b P −1
0
..
.
..
.
..
. ···
··· .. . .. . .. .
··· .. . .. . .. .
b2
b1
0
⎤
⎡
a0
⎤
⎢ .. ⎥ . ⎥ ⎥⎢ ⎥ ⎥⎢ ⎥ a ⎥⎢ α −1 ⎥ ⎥⎢ ⎢ ⎥ ⎢ aα ⎥ ⎥ = Ba ⎥⎢ ⎥ ⎢ a α +1 ⎥ ⎥ 0 ⎦ ⎢ .. ⎥ ⎣ . ⎦ .. . .. .
1
(40)
a P −1
where B ∈ C P × P is defined in terms of the elements of b˜ o . Equation (40) also reveals that the FBFo , B, generates backward prediction errors of various orders, a¯ 0 , . . . , a¯ P −1 , on the basis of linear prediction theory with tap weights bk . In other words, the precursor ISI (i.e., a0 , . . . , aα −1 ), cursor ISI (i.e., aα ), and post-cursor ISI (i.e., aα +1 , . . . , a P −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 satisfactory performance even when N b < P − α − 1. However, when the
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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 propagation, which in turn may appreciably increase the SER. Therefore, compared with the MMSE–CDFE–FC, the MMSE–PDFE may be considerably less vulnerable to error propagation because a shorter FBFo may be sufficient to mitigate the residual ISI at the FFFo output of the MMSE–PDFE. Furthermore, the proposed MMSE–PDFE performs as well as the MMSE–CDFE–FBC [15], in which the FFFo and 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. Fig. 4. Plots of residual ISI versus the FFF length (N f ) and the FBF length (N b ) for Channel 1 and SNRI = 15 dB for the (a) MMSE–CDFE–FBC and (b) MMSE–PDFE.
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 frequency 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 σs2 and AWGN 2 with variance σ w , respectively, such that the input SNR was given Δ
2 by SNRI = σs2 /σ 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:
Δ
ISI(h)(dB) = 10 log10
k |hk |
− | max(hk )|2 | max(hk )|2 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 N b , ranging from 1 to 70, for Channel 1 in a SIMO system where L = 2, σs2 = 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 N b increase. For the MMSE–CDFE–FBC, if N f < 3, N b must be larger than 60 to achieve a desirable residual ISI performance, as indicated in Fig. 4(a). However, if N f ≥ 3, N b = 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 N b ≥ 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 N b = 40 were chosen for the MMSE–PDFE and MMSE–CDFE–FBC so that the smallest
possible residual ISI could be achieved for the same total number of tap weights with N total = N f + N b = 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 cursor 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. Consequently, 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 N b under the same conditions as those in Figs. 4–6. For the MMSE–CDFE– FBC, if N f ≥ 3 and N b ≥ 30, the norm of FBFo tap weights remains higher than 0.5, and it markedly increases as N b increases. However, for the MMSE–PDFE, the norm of FBFo tap weights with arbitrarily chosen N f and N b 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 N b . 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 proposed 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. 8 and 9 are respectively referred to as the cases where the FBFo used
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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 N b = 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 N b = 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 (N b ) 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 previously detected symbol as its input. Notably, the MMSE–PDFE is less vulnerable to error propagation than the MMSE–CDFE–FBC, especially at medium-low SNRI [cf. Figs. 8 and 9]. For example, the performance loss due to error 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 error 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 N b = 40 was much lower than that of the MMSE–CDFE–FBC with N f = 60 and N b = 40 [cf. Figs. 5(c) and 6(c)], the MMSE–PDFE(DF) significantly outperformed 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 responsible 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 outperformed the MMSE–LE, and the former may achieve its optimal performance when N b = 7. Furthermore, the MMSE–LE (N f = 45), MMSE–PDFE (N f = 45 and N b = 4), and MMSE–PDFE (N f = 45 and N b = 7) require 1.83 × 106 , 1.9 × 106 , and 1.94 × 106 complex multiplications per iteration, respectively (Table 1). Therefore,
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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 independently, 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 viable 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 Technology of Taiwan, R.O.C., under Grants MOST 103-2221-E-030-007MY2 and MOST 105-2221-E-030-003-MY2. References
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 performance 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 obtained 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 error propagation. Our simulation results show that the MMSE–PDFE
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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 include filter design, adaptive signal processing, and compressed sensing.
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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 University (FJCU), Taipei, Taiwan, in 1981, the M.S. and the Ph.D. degrees, both in electrical engineering, from Missouri University of Science and Technology (formerly 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 Formosa 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 Electronic 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 President for Academic Affairs at FJCU. His research interest is in the area of statistical and adaptive signal processing with applications in communication systems and his current work involves research in adaptive algorithms and the design of blind adaptive receivers in communication systems.
