Discrete Time Systems by Mario A. Jordan and Jorge L. Bustamante - HTML preview

= 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

Time-domain Equalisation in Discrete Multitone Systems

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(