= 2 [ ak( n + 1) + j bk( n + 1)] − 2 [ ak( n) + j bk( n)] +
(37)
λ 1 [ y 1( n − k) + j y 2( n − k)] − j λ 2 [ y 1( n − k) + j y 2( n − k)]
= 2 [ ak( n + 1) + j bk( n + 1)] − 2 [ ak( n) + j bk( n)] +
( λ 1 − j λ 2) [ y 1( n − k) + j y 2( n − k)]
= 0 .
Thus, we get
2 [w k( n + 1) − w k( n)] + λ∗ wy( n − k) = 0, f or k = 0, 1, . . . , M − 1
(38)
where λw is a complex Lagrange multiplier for TEQ as
λw = λ 1 + j λ 2 .
(39)
In order to find the unknown λ∗ w, we multiply both sides of Eq.(38) by y∗( n − k) and then sum
over all integer values of k for 0 to M − 1. Thus, we have
2 [w k( n + 1) − w k( n)] y∗( n − k) = − λ∗ w y( n − k) y∗( n − k) M−1
M−1
2 ∑ [ w k( n + 1)y∗( n − k) − w k( n)y∗( n − k) ] = − λ∗ w ∑ |y( n − k)|2
k=0
k=0
2 w T( n + 1) y∗( n) − w T( n) y∗( n) = − λ∗ w y( n) 2
Therefore, the complex conjugate Lagrange multiplier λ∗ w can be formulated as
−
λ∗
2
w =
w T( n + 1) y∗( n) − w T( n) y∗( n)
,
(40)
y( n) 2
Adaptive Step-size Order Statistic LMS-based
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391
where y( n) 2 is the Euclidean norm of the tap-input vector y( n).
From the definition of the estimation error e( n) in Eq.(18), the conjugate of e( n) is written as
e∗( n) = w T( n + 1) y∗( n) − b T( n + 1) d∗( n) .
(41)
The mean-square error | e( n)| 2 is minimised by the derivative of | e( n)| 2 with respect to w( n +
1) be equal to zero.
∂| e( n)| 2 =
∂
w H ( n + 1) y( n) − b H( n + 1) d( n) y∗( n) = 0 .
(42)
w( n + 1)
Hence, we have
w H ( n + 1) y( n) = b H( n + 1) d( n) ,
(43)
and the conjugate of Eq.(43) may expressed as
w T ( n + 1) y∗( n) = b T( n + 1) d∗( n) .
(44)
To substitute Eq.(44) and Eq.(41) into Eq.(40) and then formulate λ∗ w as
λ∗ w =
2
e∗( n) .
(45)
y( n) 2
We rewrite Eq.(38) using Eq.(14) by writing,
2 δw( n + 1) = − λ∗ w y( n)
(46)
The change δw( n + 1) is redefined by substituting Eq.(45) in Eq.(46). We thus have
−
δ
1
w( n + 1) =
y( n) e∗( n) .
(47)
y( n) 2
To introduce a step-size for TEQ denoted by μw and then we may express the change δw( n +
1) as
− μ
δw( n + 1) =
w
y( n) e∗( n) .
(48)
y( n) 2
We rewrite the tap-weight vector of TEQ w( n + 1) as
w( n + 1) = w( n) + δw( n + 1) .
(49)
Finally, we may obtain the tap-weight vector of TEQ w( n + 1) in the well-known NLMS
algorithm.
w( n + 1) = w( n) −
μw
y( n) e∗( n) .
(50)
y( n) 2
where e∗( n) is described in Eq.(41).
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Discrete Time Systems
4.2 The proposed normalised least mean square-target impulse response (NLMS-TIR)
We formulate the real-valued cost function J 2( n) for the constrained optimisation problem
using Lagrange multiplier.
J 2( n) = δb( n + 1) 2 + λ 3 { [ uk( n + 1) d 1( n − k) + vk( n + 1) d 2( n − k)] − g 2 a( n)}
+ λ 4 { [ uk( n + 1) d 2( n − k) − vk( n + 1) d 1( n − k)] − g 2 b( n)}
M−1
= ∑ { [ uk( n + 1) − uk( n)] 2 + [ vk( n + 1) − vk( n)] 2}
k=0
(51)
M−1
+ λ 3 { ∑ [ uk( n + 1) d 1( n − k) + vk( n + 1) d 2( n − k)] − g 2 a( n)}
k=0
M−1
+ λ 4 { ∑ [ uk( n + 1) d 2( n − k) − vk( n + 1) d 1( n − k)] − g 2 b( n)} , k=0
where λ 3 and λ 4 are Lagrange multipliers. We find the optimum values of uk( n + 1) and
vk( n + 1) by differentiating the cost function J 2( n) with respect to these parameters and then
set the results equal to zero. Hence,
∂J 2( n)
=
∂
0 ,
uk( n + 1)
and
∂J 2( n) =
∂
0 .
vk( n + 1)
The results are
2 [ uk( n + 1) − uk( n)] + λ 3 d 1( n − k) + λ 4 d 2( n − k) = 0 , (52)
2 [ vk( n + 1) − vk( n)] + λ 3 d 2( n − k) − λ 4 d 1( n − k) = 0 .
(53)
From Eq.(22) and Eq.(24), we combine these two real results into a single complex one as
∂J 2( n)
=
∂J 2( n) + ∂J 2( n) =
∂
j
0 .
(54)
b k( n + 1)
∂uk( n + 1)
∂vk( n + 1)
Therefore,
∂J 2( n)
= {
∂
2 [ u
b
k( n + 1) − uk( n)] + λ 3 d 1 ( n − k) + λ 4 d 2( n − k)}+
k ( n + 1)
j {2 [ vk( n + 1) − vk( n)] + λ 3 d 2( n − k) − λ 4 d 1( n − k)}
= 2 [ uk( n + 1) + j vk( n + 1)] − 2 [ uk( n) + j vk( n)] +
λ
(55)
3 [ d 1( n − k) + j d 2( n − k)] − j λ 4 [ d 1( n − k) + j d 2( n − k)]
= 2 [ uk( n + 1) + j vk( n + 1)] − 2 [ uk( n) + j vk( n)] +
( λ 3 − j λ 4) [ d 1( n − k) + j d 2( n − k)]
= 0 .
Adaptive Step-size Order Statistic LMS-based
Time-domain Equalisation in Discrete Multitone Systems
393
Thus, we have
2 [b k( n + 1) − b k( n)] + λ∗ bd( n − k) = 0, f or k = 0, 1, . . . , M − 1
(56)
where λb is a complex Lagrange multiplier for TIR
λb = λ 3 + j λ 4
(57)
To multiply both side of Eq.(56) by d∗( n − k) to find the unknown λ∗ and then sum over all
b
possible integer values of k for 0 to M − 1. Thus, we get
2 [b k( n + 1) − b k( n)] d∗( n − k) = − λ∗ b d( n − k) d∗( n − k) M−1
M−1
2 ∑ [ b k( n + 1)d∗( n − k) − b k( n)d∗( n − k) ] = − λ∗
|
b ∑
d( n − k)|2
k=0
k=0
2 b T( n + 1) d∗( n) − b T( n) d∗( n) = − λ∗ b d( n) 2
Therefore,
−
λ∗ =
2
b
b T( n + 1) d∗( n) − b T( n) d∗( n)
.
(58)
d( n) 2
where d( n) 2 is the Euclidean norm of the tap-input vector d( n).
To substitute Eq.(41) and Eq.(44) into Eq.(58) and then formulate λ∗ as
b
λ∗ =
2
b
e∗( n) .
(59)
d( n) 2
We rewrite Eq.(56) using Eq.(15) by
2 δb( n + 1) = λ∗ b d( n)
(60)
To redefine the change δb( n + 1) by substituting Eq.(59) in Eq.(60). We thus get,
δb( n + 1) =
1
d( n) e∗( n) .
(61)
d( n)|2
To introduce a step-size for TIR μb and then we redefine the change δb( n + 1) simply as
δb( n + 1) =
μb
d( n) e∗( n) ,
(62)
d( n) 2
where μb is the step-size for the NLMS-TIR.
We rewrite the tap-weight vector of TIR b( n + 1) as
b( n + 1) = b( n) + δb( n + 1) .
(63)
Finally, we may formulate the tap-weight vector of TIR b( n + 1) in the normalised LMS
algorithm.
b( n + 1) = b( n) +
μb
d( n) e∗( n) ,
(64)
d( n) 2
where e∗( n) is given in Eq.(41).
To comply with the Euclidean norm constraint, the tap-weight vector of TIR b( n + 1) is
normalised as
b( n + 1) = b( n + 1)
b( n + 1) .
(65)
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Discrete Time Systems
5. Adaptive step-size order statistic-normalised averaged least mean
square-based time-domain equalisation
Based on least mean square (LMS) algorithm, a class of adaptive algorihtms employing order
statistic filtering of the sampled gradient estimates has been presented in (Haweel & Clarkson,
1992), which can provide with the development of simple and robust adaptive filter across a
wide range of input environments. This section is therefore concerned with the development
of simple and robust adaptive time-domain equalisation by defining normalised least mean
square (NLMS) algorithm.
Following (Haweel & Clarkson, 1992), we present the NLMS algorithm which replaces linear
smoothing of gradient estimates by order statistic averaged LMS filter. A class of order statistic
normalised averaged LMS algorithm with the adaptive step-size scheme for the proposed
NLMS algorithm in Eq.(50) and Eq.(64) that are shown as (Sitjongsataporn & Yuvapoositanon,
2007).
w( n + 1) = w( n) − μw( n) M w aw ,
(66)
y( n) 2
b( n + 1) = b( n) + μb( n) M
d( n) 2
b ab ,
(67)
with
M w = ˜ T{ ˜ e∗( n)y( n), ˜ e∗( n − 1)y( n − 1), . . . , ˜ e∗( n − Nw + 1)y( n − Nw + 1)} , (68)
M b = ˜ T{ ˜ e∗( n)d( n), ˜ e∗( n − 1)d( n − 1), . . . , ˜ e∗( n − Nb + 1)d( n − Nb + 1)} , (69)
H
˜ e( n) = w H( n)y( n) − b ( n)d( n) ,
(70)
and
aw = [ aw(1), aw(2), . . . , aw(