1
Determining the Optimal Decision Delay
Parameter for the Linear Equalizer
E.S. Chng and S. Chen
School of Computer Engineering
Nanyang Technological University, Nanyang Avenue
Singapore 639798.
School of Electronics and Computer Science
University of Southampton, Highfield
Southampton SO17 1BJ, U.K.
n(k)
A BSTRACT
s(k)
The achievable bit error rate of a linear equalizer is crucially determined by the choice of decision delay parameter.
This brief paper presents a simple method for the efficient
determination of the optimal decision delay parameter that
results the best bit error rate performance for the linear equal-
channel h
^
x(k)
+
x(k)
yd (k)
linear
equalizer wd
^
s(k−d)
Fig. 1. Baseband model of communication system with linear equalizer
(Qureshi 1985; Proakis 1995)
izer.
"!#
$
(1)
I. I NTRODUCTION
where denotes symbol index, !#%
is a white Gaussian
noise with variance &() ' , are the taps of the channel
impulse
The equalization technique plays an ever-increasing role in response (CIR) which has a memory * , and
is a binary
combating distortion and interference in modern communica- input drawn from the set +-,.-/ with equal probability. AlKeywords: Linear equalizer, decision delay, bit error rate
tion links (Qureshi 1985; Proakis 1995). It is well-known that
though the analysis presented assumes the binary phase shift
the choice of equalizer decision delay parameter critically
determines the achievable bit error rate (BER) performance
keying (BPSK) modulation, the results can be generalized.
(Chng et al 1995; Li & Fan 2001). We presents a simple
and effective method of determining the optimal decision de-
The linear equalizer with length
0 and decision delay 1
uses the vector of noisy observations
2#
$43 5%
55
6 . 7878795
6 0
. :<;
(2)
%
> 1 of the transmitted symbol
relative measure for each decision delay value that charac- to provide an estimate =
6 1 , and is specified by
terizes the degree of linear separability between the different
lay parameter that results the best bit error rate performance
for the linear equalizer. The proposed technique compute a
signal classes for the given decision delay value. From the
resulting set of the measures for every decision delay value,
it is straightforward to choose the optimal decision delay that
provides the best achievable BER performance.
We consider the baseband digital communication system
depicted in Fig. 1. The received signal after the communication channel and sampled at symbol rate is modeled by
Corresponding author, E-mail: [email protected], Tel./fax: +44 (0)23
8059 6660/4508
and
?A@B
#DCE;@ 2#%
GFI H(J @NM5%
6
LK
= %
6 1 # sgn O?A@B
$9
(3)
(4)
where sgn P- denotes the sign function and
CE@IQ3 N@ R @TS787U7 @WV5XBS9: ;
K K
K
is the equalizer weight coefficient vector.
(5)
2
Given the equalizer’s weight vector C @ for a fixed decision
delay 1 , define the normalized decision
variable for 2 as
The equalizer input vector 2#%
is given by
2#%
#
B
( 2
$ $
(6)
where B
$#43 %
%
. 7U787 %
. : ; is the vector
of *
0 transmitted digital symbols, $ 3 !#%
where C
!#%
6 . 7U787!#%
6 0
. : ; is the noise vector, 2#
is the
9 9;
:
vector of noise-free input signal called the channel state, and
is the 0
J
787U7
..
787U7
78 7U7
... 78. .78.7 . .J .
2#
$ It is obvious that
. .
.
J
.
..
.
..
.
787U7
depends only on
... (7)
symbols in
B
and hence the valid range of decision delay is
+
1
78787
. . /
(8)
With given decision delay 1 , various designs for the equalizer
weight vector C @ can be considered. The best known one is
= @ , which
the minimum mean square error (MMSE)
solution CE
minimizes the mean square error 3 1 ? @ %
9 ' : and
is given by (Qureshi 1985; Proakis 1995)
>;
! F#" HJ%$
C= @ $
if and only if
@ is the 1
& )'
. th column of
@
(9)
9 9
C @; 2
CE@
OC @ #
(12)
C ; C . Then 2 is correctly classified by C @
E
C
@
9
sgn
sgn @ (13)
8
channel convolution matrix given by
8
76
The BER of this equalizer is evaluated by (Mulgrew & Chen
2001; Yeh & Barry 2000)
>? 8
(*)
CE@ #
0
=< >@? 8
. F
+
OC @ (14)
J
where C @ 9 denotes the probability of error due to the
received channel state being 2 , and is evaluated
by
> ? 8
C D EGFIH J MLR7KMOSMUKVNQMWP H NQT P 2 is correctly classified
BA
. EGFIH J R S H T otherwise
(15)
C @ 9
X P X denotes the absolute value and
E : .-ZY\[^`] _ba!cd e -g' f 1 e
(16)
8
Note that X CE@ X is the distance of the channel state 2 where
denotes the to the decision boundary specified by C ;@ 2 . For CE@
F
0
0 identity matrix. Alternatively, the minimum BER to achieve the desired linear separability, all the 2 must be
(MBER) solution CE@ is obtained by directly minimizing the correctly classified by C @ , that is, the condition (13) holds
OC @ of the equalizer with decision delay 1 (Chen et
787U7 0 . Since P- decays exponentially,
BER
for .
the
C @ is dominated by the largest C @ 9 when
al 1998; Mulgrew & Chen 2001; Yeh & Barry 2000). The BER
main purpose of this brief paper is to determine the opti- & )
. Thus an upper bound of the BER
is given by
mal decision delay for the given weight vector design that
OC @ #
+ CE@ 9 /
(17)
achieves the smallest BER.
J
F
where
&
'
(*)
-
,+
()
h
(*)jilk
II. T HE BIT
0
+
and
,+ .-0/
B $ 32
Denote the 0
, i.e.
ERROR RATE
combinations of A%
as
+
.#1 1 0 + /
E
mn >@? 8
Qo o a <
III. T HE OPTIMAL
, .,1 1
(10)
>? 8
DECISION DELAY
The optimal decision delay can in theory be defined by
lpLqbr snZtvuWmg@7yZwVz x ( ) i{k C @ (18)
From the definition of >|? %P in (15), it is obvious that this
1
2%
only takes values from the channel state set deoptimal decision delay can alternatively be determined by
fined by
CE@ 2#
$
+2 (19)
1
@
. 0 /
(11)
J
F
The set of channel states can be divided into two subsets To derive a computationally simpler way of evaluating 1
,
1 . Specifically, denote the define
conditioned on the value of
@
@
; CE@
1 th element of as . Then
specifies which class ( .
@ 3 @ @ 787U7
@ :; (20)
2
E
C
@
or . ) that belongs to.
J
HJ
Then
4
51 1 +
4
76
76
ƒ
pUqbr n0t}uWm7yZn za ~ mgwx 8
Qo o *<
€
„ 6 „ 6 v„ / 6
9 9
‚pUqbr
9 CE@ 9
C ;@
OC @ OC @ #
-@ @ @
„ 76 76
„ 76
ƒ
@;
0
(21)
@ N „ }6 }6
S (22)
Furthermore, @ @ is the main tap of the combined impulse
7U787 : ; and the norresponse of the channel 3 $
9 9
„ 76
@ J CE@ , and -@ @
malized equalizer weight vector C
.
-1
log10(Bit Error Rate)
8
or
8
Then
3
@ @
@ @
(23)
S
into account of Taking
+ ,. / , for (13) to hold for all
the 2 , it is sufficient that
„ 76
N „ v6 v6 76
}6 X „-@76 @ X
N X „ }6 X
8
mgwVx 8
o o =<
=<
X „ 76 X
N X „ }6 X
(25)
The minimum distance measure
metric interpretation. A positive
-4
0
25
s;
decision delay 1
were calculated and the correspond
ing minimum distance measures C = @ were evaluated using
C @ = for 1 787U7
.
were
‚
$
. . !@‚}A @B@!CvEDFBF}A C6DF!vED:F!F!v @G@!C G@
The above results indicate that 1 or 1 results in
a nonlinearly separable equalization problem with the worst
(25). The computed
BER performance, and the best BER performance is achieved
with the optimal decision delay 1 . To verify these pre-
C @ @
has a clear geoto evaluate the BERs of
C @ indicates that CE@ dictions, simulation was conducted
= @ with various decision delays
the linear MMSE equalizer CE
0
nitude means that nonlinear separability is more severe, and
-1
a larger positive value of OC @ indicates that the channel states are located further away from the linear decision
boundary which implies a better BER performance. From
(19), the optimal decision delay is determined as
OC @ /
(26)
log10(Bit Error Rate)
arability quantitatively. A negative C @ with a larger mag-
pUqbr n0t}uOm@7yZn z a + 5
10
15
20
Signal to Noise Ratio (dB)
Fig. %'
2.&)(+Bit
of decision delay for-0
channel
*,.error
-0/21 34rate
3658performance
741 /4*:9 &!; /21 as3434a*:function
9=< with equalizer
? . The
length >
MMSE design is employed
achieves linear separability and a negative value otherwise.
Moreover, OCE@ measures the relative degree of linear sep-
1
d=0,6
d=1,5
d=2,4
d=3
-6
@
(24)
S E
C
@
In fact, the minimum of +
/ F for decision delay 1 is
J
evaluated as
CE@ #
CE@ -@ @
@ S
J
F
-3
-5
For the correct classification (13) to hold, (22) shows that
it is sufficient to have
-2
-2
-3
-4
d=0,6
d=1,5
d=2,4
d=3
-5
IV. E XAMPLES
-6
Example 1. The transfer function of the CIR was defined by
B ' and the equalizer length
. HJ
! " . The range
! ofH decision delays was thereJ
was given by 0 #
fore + . 787U7 / . With a channel signal to noise ratio
$
(SNR) of 25 dB, the MMSE weight vectors C = @ (9) for each
‚ 0
5
10
15
20
Signal to Noise Ratio (dB)
25
Fig. %'
3.&)(+Bit
of decision delay for-0
channel
*,.error
-0/21 34rate
3658performance
741 /4* 9 & ; /21 as3434a* function
? . The
9=< with equalizer
length >
MBER design is employed
4
0
1
, and the results depicted in Fig. 2 agree with the pre @
= .
dicted relative BER performance using CE
log10(Bit Error Rate)
-1
For the same SNR of 25 dB, the MBER weight vectors
were also calculated numerically us-
'
CE@ for all 1
ing the MBER optimization algorithm given in (Chen et al
2001). The related minimum distance measures C @ for
1 '
. 78787
were
. . .
"{-Z"0- . .-.
D!}A v C!FBF} F v= CBFGFv= ‚ D
and the BER performance of the linear MBER equalizer C ' @
with 1 are illustrated in Fig. 3. The results shown in
Fig. 3 agree with the predictions using C ' @ . Also C ' J
and C ' are significantly larger than C = and C = . InJ
specting Figs. 2 and 3, it can be seen that the corresponding
MBER equalizers have much better BER performance than
the related MMSE equalizers. This further confirms the use-
CE@ as a relative BER performance indicator.
fulness of
- - ' and the equalizer
HJ @ H
!F
B
!
F
!
length was chosen as 0 . The range of decision delays
8
7
U
7
7
was therefore the channel SNR
+ .
‚ )D / . Given
= @ (9) for each decision
of 24 dB, the MMSE weight vectors CE
delay 13 were calculated and the corresponding minimum
Example 2. The transfer function of the CIR was given by
'
B#
distance measures C = @ were
"
-
-.
."
.
-
. . !D A @6D v !@ Dl}A lD }A @ v @BD v = C6D
It can be seen that for the MMSE design 1 ‚ . 4D result
0
d=7
d=0
d=1
d=6
d=2
d=3
d=4
d=5
-3
-4
-5
-6
0
5
10
15
20
Signal to Noise Ratio (dB)
25
; /21 43of4*:decision
Fig. % 5. (+Bit
error rate performance as a& function
for channel
9< withdelay
< -0*,3 - /21 343 50/21 343 34*:9
equalizer length
>
. The MBER design is employed
in nonlinear separable problems with the worst BER perfor. The optimal decision delay is 1 " ,
mance given by 1 D
which has the smallest BER. These predictions using C = @ are confirmed by the actual BER performance depicted in
Fig. 4.
The same process was repeated for the MBER design, and
the corresponding minimum distance measures
- -
.
-
'
C @ were
@!FBCv @!‚}A v @ C } @G }A . }A . D F! . B FBD
It can be seen that for the MBER design only 1
result
4
D
in nonlinear separable problems with the worst BER performance given by
1 . The optimal decision delay for the
D
MBER design is also 1 " . Fig. 5 illustrates the actual BER
performance for the linear MBER equalizers CE@ for 1
.
'
-1
log10(Bit Error Rate)
-2
-2
V. E XTENSION
d=7
d=0
d=1
d=2
d=6
d=3
d=4
d=5
-3
-4
-5
-6
0
5
10
15
20
Signal to Noise Ratio (dB)
3
TO GENERAL MODULATION SCHEME
We now show how to extend the proposed method to
the general modulation scheme. First consider the -level
pulse amplitude modulation ( -PAM) scheme, where
drawn from the symbol
set
25
; /21 43of4* decision
Fig. % 4. (+Bit
error rate performance as a& function
for channel
9< withdelay
< -0*,3 - /21 343 50/21 343 34* 9
equalizer length
>
. The MMSE design is employed
Thus
where 0
s- .51 1 6/
B
$ 2 + .51 1 0 U+ /
+ / , and 2# %
4 + 2 .#1 1 0,+U/
+
. is
(27)
(28)
(29)
5
4
The set can be partitioned into subsets depending on the
value of %
6 1 4
+2
4
1 51 1
/ .
(30)
#1 1
. , 4 is a shifted version of 4 by the amount - $ @ .
J
That is,
(31)
4 J 4 - $ @ .51 1 .
This shift property enables us to use any two adjacent subsets
4 and 4 J when considering
the degree of linear separa
bility of the equalizer. Specifically, let us choose I Z- .
It can readily be proved that (Chen et al 2004) for
'
[6]
[7]
[8]
[9]
. . It is clearly that this is
'
J
equivalent to the BPSK case presented in this paper.
Then
. and
.
[5]
function equaliser. in Proc. 5th IEEE Workshop Neural Networks for Signal Processing (Cambridge, USA),
Aug.31-Sept.2, 1995, pp.593–602.
Li, X., and H.H. Fan (2001). Direct blind equalization with best delay by channel output whitening. IEEE
Trans. Signal Processing, 49 (7), 1556–1563.
Mulgrew, B., and S. Chen (2001). Adaptive minimumBER decision feedback equalisers for binary signalling.
Signal Processing, 81 (7), 1479–1489.
Proakis, J.G. (1995). Digital Communications. 3rd edition, New York: McGraw-Hill.
Qureshi, S.U.H. (1985). Adaptive equalization. Proc.
IEEE, 73 (9), 1349–1387.
Yeh, C.C., and J.R. Barry (2000). Adaptive minimum
bit-error rate equalization for binary signaling. IEEE
Trans. Communications, 48 (7) 1226–1235.
For the general complex-valued modulation scheme, such
as quadrature amplitude modulation (QAM) scheme, the extension is more involved, but the derivation can be carried out
similarly.
VI. C ONCLUSIONS
Eng Siong Chng received university education
at University of Edinburgh, Edinburgh, Scotland
(BEng 1991, PhD 1995). After his PhD, he spent
6 months in Japan, working as a research staff
for Riken. After working in industry in Singapore for 7 years, he jointed the School of Computer Engineering, Nanyang Technological University in 2003. His research interests are in digital
signal processing for communication applications,
speech and handwriting recognition and noise reduction.
A simple but computationally efficient method has been
presented to determine the optimal decision delay parameter that achieves the smallest bit error rate for given linear
equalizer design. The proposed method calculates a minimum distance measure for each feasible decision delay value
and choose the decision delay that achieves the maximum of
this minimum distance measure. The usefulness of this technique has been demonstrated using two examples involving
both the minimum mean square error and minimum bit error
rate equalization designs.
R EFERENCES
[1] Chen, S., L. Hanzo, and B. Mulgrew (2004). Adaptive
minimum symbol-error-rate decision feedback equalization for multilevel pulse-amplitude modulation. IEEE
Trans. Signal Processing, 52 (7) 2092–2101.
[2] Chen, S., B. Mulgrew, E.S. Chng and G. Gibson (1998).
Space translation properties and the minimum-BER
linear-combiner DFE. IEE Proc. Communications, 145
(5), 316–322.
[3] Chen, S., A.K. Samingan, B. Mulgrew and L. Hanzo
(2001). Adaptive minimum-BER linear multiuser detection for DS-CDMA signals in multipath channels. IEEE
Trans. Signal Processing, 49 (6), 1240–1247.
[4] Chng, E.S., B. Mulgrew, S. Chen and G. Gibson (1995).
Optimum lag and subset selection for a radial basis
Sheng Chen obtained a BEng degree in control
engineering from the East China Petroleum Institute, Dongying, China, in 1982, and a PhD degree in control engineering from the City University at London in 1986. He joined the School
of Electronics and Computer Science at the University of Southampton in September 1999. He
previously held research and academic appointments at the Universities of Sheffield, Edinburgh
and Portsmouth. Dr Chen is a Senior Member of
IEEE in the USA. His recent research works include adaptive nonlinear signal processing, modeling and identification of
nonlinear systems, neural network and machine learning, finite-precision
digital controller design, evolutionary computation methods and optimization. He has published over 200 research papers. In the database of the
world’s most highly cited researchers in various disciplines, compiled by Institute for Scientific Information (ISI) of the USA, Dr Chen is on the list
of the highly cited researchers in the category that covers all branches of
engineering subjects, see www.ISIHighlyCited.com.