A fractionally spaced blind equalization algorithm ... - Semantic Scholar
ARTICLE IN PRESS
Signal Processing 88 (2008) 200–209 www.elsevier.com/locate/sigpro
A fractionally spaced blind equalization algorithm with global convergence$ Alper T. Erdogan! EE Department, Koc University, Sariyer, 34450 Istanbul, Turkey Received 1 September 2006; received in revised form 4 April 2007; accepted 23 April 2007 Available online 6 May 2007
Abstract Two different fractionally spaced extensions of the SubGradient based Blind equalization Algorithm (SGBA) are provided. The first one is the direct extension of the linearly constrained SGBA for the symbol spaced setting. The second extension is the weighted and the 2-norm constrained fractionally spaced SGBA (FS-SGBA) algorithm. It is proven that the latter algorithm is globally convergent to a perfect equalization point under the well-known equalizability conditions for the fractionally spaced setting. The simulation results provided illustrates the relative merit of the proposed algorithm in comparison to the state of the art algorithms. r 2007 Elsevier B.V. All rights reserved. Keywords: Adaptive filtering; Blind equalization; Fractionally spaced; Subgradient
1. Introduction In the area of blind equalization, convex cost functions play an important role due to their surface structures which are free of local and false minima and saddle points. The pioneering work in this field is due to Vembu and Verdu [1] who cast the blind equalization problem as a convex infinity norm minimization problem under a linear constraint. This approach exploits the magnitude bounded structure of the PAM constellations used in digital communications. They also proposed the use of large-p
$ This work was supported in part by TUBITAK Career Award, Contract No.: 104E073. !Tel.: +90 212 338 1490; fax: +90 212 338 1548. E-mail addresses: [email protected], [email protected] (A.T. Erdogan).
norm approximation of the proposed cost function to obtain a gradient search based iterative algorithm. In [2], Ding and Luo posed the infinity norm minimization of the affine function corresponding to blind equalization as a linear programming problem. The same reference proposes the modification of the infinity norm based cost function for handling complex QAM constellations. The extension of this work for the fractionally spaced equalizers is proposed in [3]. Although the linear programming based approaches have better performance than p-norm based approximation in [1], they are computationally expensive. Recently, a subgradient based framework for direct iterative minimization of infinity norm based blind equalization cost function was proposed [4] as an alternative. The proposed framework enables the development of iterative algorithms with low complexity for the convex problem posed in [1] and its variations.
0165-1684/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2007.04.017
ARTICLE IN PRESS 201
A.T. Erdogan / Signal Processing 88 (2008) 200–209
In this article, we present the extension of the approach in [4] for the fractionally spaced equalization problem. We will first provide the simple generalization of the Sub Gradient based Blind equalisation Algorithm (SGBA) algorithm based on the linear constraint. We later replace the linear constraint on the equalizer with the quadratic constraint and provide an algorithm for this case. In particular, we show that the resulting algorithm is globally convergent to a perfect equalizer point under the well-known equalizability assumptions for the fractionally spaced case and under a generically true assumption about the initial search vector. The organization of the article is as follows: In Section 2.2, we provide the blind equalization setup and the convex formulation proposed in [1]. Section 3, is the main part of the article where we provide the fractionally spaced SGBA algorithms. Section 4 focuses on the convergence of the proposed algorithms. The simulation examples illustrating the performance of these algorithms are provided in Section 5. Finally, Section 6 is the conclusion. 2. Notation and blind equalization setup 2.1. Notation
from a square QAM constellation where maxðRefxn gÞ ¼ % minðRefxn gÞ,
! !
!
Use
A (bold-capital letters) x (bold-lowercase letters) s (normal-lowercase letters) ¯ A
matrices vectors scalars
AT AH Ay A:;n Am;: Ref:g Imf:g
ð2Þ
Although we will assume complex QAM constellation for the rest of the article, the presented algorithms are trivially applicable to real baseband case with PAM constellations. We assume a uniform distribution for the constellation points with a variance s2x . M is the oversampling factor of the fractionally spaced equalization. We will assume that M ¼ 2 without loss of generality. fhn ; n 2 f0; . . . ; N H % 1gg is the effective impulse response of the overall communication channel, where we assume N H to beP even without loss of generality. H %1 %n If we define HðzÞ ¼ N and write n¼0 hn z HðzÞ ¼ H ðeÞ ðz2 Þ þ H ðoÞ ðz2 Þz%1
(3)
Y ðzÞ ¼ Y ðeÞ ðz2 Þ þ Y ðoÞ ðz2 Þz%1
(4)
ðeÞ
We use the following notation throughout the paper: Symbol
ð1Þ
¼ maxðImfxn gÞ ¼ % minðImfxn gÞ ¼ Q.
ðoÞ
then H ðzÞ and H ðzÞ represent the Z-transforms of the even and odd subsamples of the channel impulse response. fyn g is the oversampled received sequence at the receiver. If we define Y ðzÞ as the Z-transform of fyn g and write ðeÞ
ðoÞ
then Y ðzÞ and Y ðzÞ represent the Z-transforms of the even and odd subsamples of fyn g. Since Y ðzÞ ¼ HðzÞX ðz2 Þ ¼ H ðeÞ ðz2 ÞX ðz2 Þ þ H ðoÞ ðz2 ÞX ðz2 Þ z%1 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl {zfflfflfflfflfflfflfflfflffl} Y ðeÞ ðz2 Þ
A with conjugated elements transpose of A Hermitian transpose of A pseudoinverse of A nth column of A mth row of A real Part operator imaginary Part operator
Y ðoÞ ðz2 Þ
we can write " # " # H ðeÞ ðzÞ Y ðeÞ ðzÞ ¼ X ðzÞ, Y ðoÞ ðzÞ H ðoÞ ðzÞ |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} YðzÞ
(5)
HðzÞ
which is nothing but the multichannel representation of the oversampled setup. For the adaptive implementation we assume that a window of channel output samples fyk : k ¼ 1; . . . ; Og, where O is the length of the window, is available for the adaptation of the equalizer.
2.2. Blind equalization setup The setup for fractionally spaced equalization is shown in Fig. 1 where
!
fxn g is the transmitted digital communication sequence. We assume that xn takes its values
xn
M
h
yn
w
M
Fig. 1. The fractionally spaced equalization setup.
on
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! !
A.T. Erdogan / Signal Processing 88 (2008) 200–209
fwn ; n 2 f0; . . . ; N W % 1gg is the impulse response of the equalizer, where we assume N W is even without loss of generality. fon g is the overall output which can be written as h i on ¼ w0 w1 w2 w3 . . . wN W %2 wN W %1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} y2n 7 6 7 6 y 2n%1 7 6 7 6 7 6 y 2n%2 7 6 7 6 7 6 y 2n%3 7 6 7 6 '6 7 . .. 7 6 7 6 7 6 7 6 7 6 7 6 7 6y 6 2n%Nwþ2 7 5 4 y2n%N W þ1 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} yn
2
6 6 6 6 T ¼ w TðHÞ 6 |fflfflfflfflffl{zfflfflfflfflffl} 6 6 gT 4
where ed is a standard basis vector with single non-zero entry located at the index d. 3. Fractionally spaced SGBA In [2], the fractionally spaced blind equalization problem is posed as the optimization problem
ð6Þ
xn
xn%1 .. .
3
7 7 7 7 7, 7 7 5
xn
minimize subject to
max jRefon gj n
Refw0 þ w1 g þ Imfw0 þ w1 g ¼ 1, NH NW ;...; , w2k%1 ¼ 0; k ¼ 2 2
where the linear programming is proposed to obtain the solution of the problem. The linear programming solution is computationally involved. Instead, we propose a simple modification to SGBA proposed in [4], which yields the following iterations: ð7Þ
xn%N W þN H þ2 2 |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
where TðHÞ is the N W ' L block convolution matrix corresponding to the transfer function HðzÞ where L ¼ ðN H þ N W Þ=2 % 1. For equalizability, we assume [5], ( H ðeÞ ðzÞ and H ð0Þ ðzÞ do not have any common zeros, ( N W =2 þ 1XN H =2.
wðkþ1Þ l
wT TðHÞ ¼ eTd ,
wT
3
2
Under these assumptions, TðHÞ would be a fullrank tall (or square) matrix such that we can find a vector w for which
wðkþ1Þ ¼ wðkÞ % mðkÞ signðRefoðkÞ gÞ¯ynðkÞ , nðkÞ ðkþ1Þ
w
ðkþ1Þ
¼ Pfw
g,
ð8Þ ð9Þ
where
! ! !
nðkÞ is the index for which the maximum magnitude RefoðkÞ n g is achieved, mðkÞ is the step size, P is the minimum distance projection operator which projects its argument to the affine set defined by the constraints on the equalizer coefficients. The relation between wðkþ1Þ and wðkþ1Þ is simply given by
8 > Ref3w0ðkþ1Þ % w1ðkþ1Þ g % Imfw0ðkþ1Þ þ w1ðkþ1Þ g þ 1 > > > > 4 > > > > ðkþ1Þ ðkþ1Þ > > % w1 g % Refwðkþ1Þ þ w1ðkþ1Þ g þ 1 > 0 > þj Imf3w0 ; > > > 4 > > > > < Ref3wðkþ1Þ % wðkþ1Þ g % Imfwðkþ1Þ þ wðkþ1Þ g þ 1 1 0 0 1 ¼ 4 > > > > > > Imf3w1ðkþ1Þ % wðkþ1Þ g % Refwðkþ1Þ þ w1ðkþ1Þ g þ 1 > 0 0 > þj ; > > > 4 > > > > > 0; > > > > : wðkþ1Þ l
l ¼ 0;
l ¼ 1; l ¼ N H % 1; N H þ 1; N H þ 3; . . . ; N W % 1; otherwise:
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Note that since the corresponding optimization problem is convex (with convex cost function and convex constrained set), the algorithm defined by (8) and (9) converges to the global optimal point. However, this approach requires the exact knowledge of the channel length N H , which limits the practicality of the algorithm. We obtain an alternative version of this algorithm by removing (9) and using a weighting y
¯ y y¯ nðkÞ , wðkþ1Þ ¼ wðkÞ % mðkÞ signðRefoðkÞ gÞP nðkÞ
(10)
where mðkÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi pffiffiffi qffiP ðkÞ 2 O jRefoðkÞ gj % ð 2 Q=s O Þ x l¼1 jRefol gj nðkÞ 2yH Py y nðkÞ y nðkÞ
(11) and Py is the covariance of y which is equal to Py ¼ s2x TðHÞTðHÞH .
(12)
Note that in applications, this covariance matrix should actually be estimated from the observations. However, to simplify our analysis later, we’ll assume that our estimate of the covariance is equal to the true covariance. In order to avoid all zeros solution and to fix a numerical range for w we introduce the normalization wðkþ1Þ ¼
wðkÞ , k w kP¯ y =s2x
(13)
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¯ H Py k w kP¯ y =s2x 9 wðkÞ 2 w¯ ðkÞ , sx
(14)
which is a scaling by a weighted norm of wðkÞ such that the average equalizer output power is fixed as s2x . One clear advantage of this variation over the linearly constrained version is that, the exact knowledge of the channel length is not required. 4. Convergence analysis of the fractionally spaced blind equalization algorithm Among the two alternative fractionally spaced SGBA algorithms presented in the previous section, the former algorithm corresponds to a conventional subgradient search algorithm for a convex cost function with a convex constraint. The corresponding cost function was shown to have a perfect
equalization point as its optimal point in [3] and a smart choice of step size rule satisfying zero-limitdivergent-sum (ZLDS) rule, i.e., lim mðkÞ ! 0
zero limit,
k!1
(15)
and lim
O!1
O X k¼0
jmðkÞ j ! 1 divergent sum
(16)
would guarantee convergence of the algorithm to this perfect equalization point [4,6,7]. Note that the choice mð0Þ (17) kþ1 would satisfy the requirements of the ZLDS rule. The convergence of the latter algorithm is less trivial due to the fact that the iterations correspond to the non-convex optimization problem mðkÞ ¼
minimize
kok1 ,
subject to kwkP¯ y =s2x ¼ 1, for which the cost function is convex but the constraint set is not. Therefore, a direct assessment of convergence based on subgradient optimization literature is not possible. In this section, we are going to show that the second fractionally spaced SGBA algorithm introduced above is globally convergent under some mild assumptions. We start by multiplying both sides of (10) by TðHÞT from left TðHÞT wðkþ1Þ ¼ TðHÞT wðkÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} gðkÞ
gðkþ1Þ
y
¯ y y¯ nðkÞ , % mðkÞ signðRefoðkÞ gÞTðHÞT P nðkÞ ð18Þ
ðkÞ
where g would be the overall effective impulse response at the kth step. If we write down a (full) singular value decomposition of TðHÞ as # $ Rr H TðHÞ ¼ U V , (19) 0 where 2
s1
6 6 0 6 Rr ¼ 6 6 .. 6 . 4 0
0
...
s2
...
..
..
0
.
.
...
0
3
7 0 7 7 7 .. 7 .7 5 sr
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with s1 ; s2 ; . . . ; sr 40, and U; V are unitary matrices. Since yn ¼ TðHÞxn it is easy to show that Py ¼ TðHÞTðHÞH s2x " 2 # Rr 0 2 UH , ¼ sx U 0 0
ð20Þ ð21Þ
and consequently, " %2 # 0 Rr y %2 ¯ y ¼ sx U ¯ P UT : 0 0
ð22Þ
¼ Q signð~gðkÞ Þ þ signðRefoðkÞ gÞx" nðkÞ . nðkÞ
As a result, by plugging (31) in (27),we obtain mðkÞ Q g~ ðkþ1Þ ¼ g~ ðkÞ % 2 signð~gðkÞ Þ sx %
#
Refxg , %Imfxg
(25)
mðkÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi qffiP pffiffi ðkÞ 2 O O Þ jRefoðkÞ gj % ðQ 2 =s x l¼1 jRefol gj nðkÞ 2yH Py y nðkÞ y nðkÞ
¼
then we can rewrite (24) as (27)
Note that Refon g ¼ g~ T x! n
(28)
x! nðkÞ ¼ +Q signð~g Þ þ x" nðkÞ ,
!
AðkÞ l
for all l 2 ½1; 2L*, where is a random value from the set of real and imaginary components of the constellation. Note that since x" nðkÞ is orthogonal to gðkÞ , it has no effect at the output. Therefore,
%2 2LðkÞ g~ Qsx
,
ð36Þ
the fact that gj ¼ jx! TnðkÞ g~ ðkÞ j, jRefoðkÞ nðkÞ
(37)
and since x! nðkÞ ¼ +Q signð~gðkÞ Þ þ x" nðkÞ , gj ¼ k~gðkÞ k1 Q jRefoðkÞ nðkÞ
(29)
where x" nðkÞ is the vector that has zero values for the indexes where g~ ðkÞ is non-zero, and has arbitrary values from the set of real and imaginary components of the constellation points, i.e., the lth component of x" nðkÞ is given by 8
Signal Processing 88 (2008) 200–209 www.elsevier.com/locate/sigpro
A fractionally spaced blind equalization algorithm with global convergence$ Alper T. Erdogan! EE Department, Koc University, Sariyer, 34450 Istanbul, Turkey Received 1 September 2006; received in revised form 4 April 2007; accepted 23 April 2007 Available online 6 May 2007
Abstract Two different fractionally spaced extensions of the SubGradient based Blind equalization Algorithm (SGBA) are provided. The first one is the direct extension of the linearly constrained SGBA for the symbol spaced setting. The second extension is the weighted and the 2-norm constrained fractionally spaced SGBA (FS-SGBA) algorithm. It is proven that the latter algorithm is globally convergent to a perfect equalization point under the well-known equalizability conditions for the fractionally spaced setting. The simulation results provided illustrates the relative merit of the proposed algorithm in comparison to the state of the art algorithms. r 2007 Elsevier B.V. All rights reserved. Keywords: Adaptive filtering; Blind equalization; Fractionally spaced; Subgradient
1. Introduction In the area of blind equalization, convex cost functions play an important role due to their surface structures which are free of local and false minima and saddle points. The pioneering work in this field is due to Vembu and Verdu [1] who cast the blind equalization problem as a convex infinity norm minimization problem under a linear constraint. This approach exploits the magnitude bounded structure of the PAM constellations used in digital communications. They also proposed the use of large-p
$ This work was supported in part by TUBITAK Career Award, Contract No.: 104E073. !Tel.: +90 212 338 1490; fax: +90 212 338 1548. E-mail addresses: [email protected], [email protected] (A.T. Erdogan).
norm approximation of the proposed cost function to obtain a gradient search based iterative algorithm. In [2], Ding and Luo posed the infinity norm minimization of the affine function corresponding to blind equalization as a linear programming problem. The same reference proposes the modification of the infinity norm based cost function for handling complex QAM constellations. The extension of this work for the fractionally spaced equalizers is proposed in [3]. Although the linear programming based approaches have better performance than p-norm based approximation in [1], they are computationally expensive. Recently, a subgradient based framework for direct iterative minimization of infinity norm based blind equalization cost function was proposed [4] as an alternative. The proposed framework enables the development of iterative algorithms with low complexity for the convex problem posed in [1] and its variations.
0165-1684/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2007.04.017
ARTICLE IN PRESS 201
A.T. Erdogan / Signal Processing 88 (2008) 200–209
In this article, we present the extension of the approach in [4] for the fractionally spaced equalization problem. We will first provide the simple generalization of the Sub Gradient based Blind equalisation Algorithm (SGBA) algorithm based on the linear constraint. We later replace the linear constraint on the equalizer with the quadratic constraint and provide an algorithm for this case. In particular, we show that the resulting algorithm is globally convergent to a perfect equalizer point under the well-known equalizability assumptions for the fractionally spaced case and under a generically true assumption about the initial search vector. The organization of the article is as follows: In Section 2.2, we provide the blind equalization setup and the convex formulation proposed in [1]. Section 3, is the main part of the article where we provide the fractionally spaced SGBA algorithms. Section 4 focuses on the convergence of the proposed algorithms. The simulation examples illustrating the performance of these algorithms are provided in Section 5. Finally, Section 6 is the conclusion. 2. Notation and blind equalization setup 2.1. Notation
from a square QAM constellation where maxðRefxn gÞ ¼ % minðRefxn gÞ,
! !
!
Use
A (bold-capital letters) x (bold-lowercase letters) s (normal-lowercase letters) ¯ A
matrices vectors scalars
AT AH Ay A:;n Am;: Ref:g Imf:g
ð2Þ
Although we will assume complex QAM constellation for the rest of the article, the presented algorithms are trivially applicable to real baseband case with PAM constellations. We assume a uniform distribution for the constellation points with a variance s2x . M is the oversampling factor of the fractionally spaced equalization. We will assume that M ¼ 2 without loss of generality. fhn ; n 2 f0; . . . ; N H % 1gg is the effective impulse response of the overall communication channel, where we assume N H to beP even without loss of generality. H %1 %n If we define HðzÞ ¼ N and write n¼0 hn z HðzÞ ¼ H ðeÞ ðz2 Þ þ H ðoÞ ðz2 Þz%1
(3)
Y ðzÞ ¼ Y ðeÞ ðz2 Þ þ Y ðoÞ ðz2 Þz%1
(4)
ðeÞ
We use the following notation throughout the paper: Symbol
ð1Þ
¼ maxðImfxn gÞ ¼ % minðImfxn gÞ ¼ Q.
ðoÞ
then H ðzÞ and H ðzÞ represent the Z-transforms of the even and odd subsamples of the channel impulse response. fyn g is the oversampled received sequence at the receiver. If we define Y ðzÞ as the Z-transform of fyn g and write ðeÞ
ðoÞ
then Y ðzÞ and Y ðzÞ represent the Z-transforms of the even and odd subsamples of fyn g. Since Y ðzÞ ¼ HðzÞX ðz2 Þ ¼ H ðeÞ ðz2 ÞX ðz2 Þ þ H ðoÞ ðz2 ÞX ðz2 Þ z%1 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl {zfflfflfflfflfflfflfflfflffl} Y ðeÞ ðz2 Þ
A with conjugated elements transpose of A Hermitian transpose of A pseudoinverse of A nth column of A mth row of A real Part operator imaginary Part operator
Y ðoÞ ðz2 Þ
we can write " # " # H ðeÞ ðzÞ Y ðeÞ ðzÞ ¼ X ðzÞ, Y ðoÞ ðzÞ H ðoÞ ðzÞ |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} YðzÞ
(5)
HðzÞ
which is nothing but the multichannel representation of the oversampled setup. For the adaptive implementation we assume that a window of channel output samples fyk : k ¼ 1; . . . ; Og, where O is the length of the window, is available for the adaptation of the equalizer.
2.2. Blind equalization setup The setup for fractionally spaced equalization is shown in Fig. 1 where
!
fxn g is the transmitted digital communication sequence. We assume that xn takes its values
xn
M
h
yn
w
M
Fig. 1. The fractionally spaced equalization setup.
on
ARTICLE IN PRESS 202
! !
A.T. Erdogan / Signal Processing 88 (2008) 200–209
fwn ; n 2 f0; . . . ; N W % 1gg is the impulse response of the equalizer, where we assume N W is even without loss of generality. fon g is the overall output which can be written as h i on ¼ w0 w1 w2 w3 . . . wN W %2 wN W %1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} y2n 7 6 7 6 y 2n%1 7 6 7 6 7 6 y 2n%2 7 6 7 6 7 6 y 2n%3 7 6 7 6 '6 7 . .. 7 6 7 6 7 6 7 6 7 6 7 6 7 6y 6 2n%Nwþ2 7 5 4 y2n%N W þ1 |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} yn
2
6 6 6 6 T ¼ w TðHÞ 6 |fflfflfflfflffl{zfflfflfflfflffl} 6 6 gT 4
where ed is a standard basis vector with single non-zero entry located at the index d. 3. Fractionally spaced SGBA In [2], the fractionally spaced blind equalization problem is posed as the optimization problem
ð6Þ
xn
xn%1 .. .
3
7 7 7 7 7, 7 7 5
xn
minimize subject to
max jRefon gj n
Refw0 þ w1 g þ Imfw0 þ w1 g ¼ 1, NH NW ;...; , w2k%1 ¼ 0; k ¼ 2 2
where the linear programming is proposed to obtain the solution of the problem. The linear programming solution is computationally involved. Instead, we propose a simple modification to SGBA proposed in [4], which yields the following iterations: ð7Þ
xn%N W þN H þ2 2 |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
where TðHÞ is the N W ' L block convolution matrix corresponding to the transfer function HðzÞ where L ¼ ðN H þ N W Þ=2 % 1. For equalizability, we assume [5], ( H ðeÞ ðzÞ and H ð0Þ ðzÞ do not have any common zeros, ( N W =2 þ 1XN H =2.
wðkþ1Þ l
wT TðHÞ ¼ eTd ,
wT
3
2
Under these assumptions, TðHÞ would be a fullrank tall (or square) matrix such that we can find a vector w for which
wðkþ1Þ ¼ wðkÞ % mðkÞ signðRefoðkÞ gÞ¯ynðkÞ , nðkÞ ðkþ1Þ
w
ðkþ1Þ
¼ Pfw
g,
ð8Þ ð9Þ
where
! ! !
nðkÞ is the index for which the maximum magnitude RefoðkÞ n g is achieved, mðkÞ is the step size, P is the minimum distance projection operator which projects its argument to the affine set defined by the constraints on the equalizer coefficients. The relation between wðkþ1Þ and wðkþ1Þ is simply given by
8 > Ref3w0ðkþ1Þ % w1ðkþ1Þ g % Imfw0ðkþ1Þ þ w1ðkþ1Þ g þ 1 > > > > 4 > > > > ðkþ1Þ ðkþ1Þ > > % w1 g % Refwðkþ1Þ þ w1ðkþ1Þ g þ 1 > 0 > þj Imf3w0 ; > > > 4 > > > > < Ref3wðkþ1Þ % wðkþ1Þ g % Imfwðkþ1Þ þ wðkþ1Þ g þ 1 1 0 0 1 ¼ 4 > > > > > > Imf3w1ðkþ1Þ % wðkþ1Þ g % Refwðkþ1Þ þ w1ðkþ1Þ g þ 1 > 0 0 > þj ; > > > 4 > > > > > 0; > > > > : wðkþ1Þ l
l ¼ 0;
l ¼ 1; l ¼ N H % 1; N H þ 1; N H þ 3; . . . ; N W % 1; otherwise:
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Note that since the corresponding optimization problem is convex (with convex cost function and convex constrained set), the algorithm defined by (8) and (9) converges to the global optimal point. However, this approach requires the exact knowledge of the channel length N H , which limits the practicality of the algorithm. We obtain an alternative version of this algorithm by removing (9) and using a weighting y
¯ y y¯ nðkÞ , wðkþ1Þ ¼ wðkÞ % mðkÞ signðRefoðkÞ gÞP nðkÞ
(10)
where mðkÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi pffiffiffi qffiP ðkÞ 2 O jRefoðkÞ gj % ð 2 Q=s O Þ x l¼1 jRefol gj nðkÞ 2yH Py y nðkÞ y nðkÞ
(11) and Py is the covariance of y which is equal to Py ¼ s2x TðHÞTðHÞH .
(12)
Note that in applications, this covariance matrix should actually be estimated from the observations. However, to simplify our analysis later, we’ll assume that our estimate of the covariance is equal to the true covariance. In order to avoid all zeros solution and to fix a numerical range for w we introduce the normalization wðkþ1Þ ¼
wðkÞ , k w kP¯ y =s2x
(13)
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¯ H Py k w kP¯ y =s2x 9 wðkÞ 2 w¯ ðkÞ , sx
(14)
which is a scaling by a weighted norm of wðkÞ such that the average equalizer output power is fixed as s2x . One clear advantage of this variation over the linearly constrained version is that, the exact knowledge of the channel length is not required. 4. Convergence analysis of the fractionally spaced blind equalization algorithm Among the two alternative fractionally spaced SGBA algorithms presented in the previous section, the former algorithm corresponds to a conventional subgradient search algorithm for a convex cost function with a convex constraint. The corresponding cost function was shown to have a perfect
equalization point as its optimal point in [3] and a smart choice of step size rule satisfying zero-limitdivergent-sum (ZLDS) rule, i.e., lim mðkÞ ! 0
zero limit,
k!1
(15)
and lim
O!1
O X k¼0
jmðkÞ j ! 1 divergent sum
(16)
would guarantee convergence of the algorithm to this perfect equalization point [4,6,7]. Note that the choice mð0Þ (17) kþ1 would satisfy the requirements of the ZLDS rule. The convergence of the latter algorithm is less trivial due to the fact that the iterations correspond to the non-convex optimization problem mðkÞ ¼
minimize
kok1 ,
subject to kwkP¯ y =s2x ¼ 1, for which the cost function is convex but the constraint set is not. Therefore, a direct assessment of convergence based on subgradient optimization literature is not possible. In this section, we are going to show that the second fractionally spaced SGBA algorithm introduced above is globally convergent under some mild assumptions. We start by multiplying both sides of (10) by TðHÞT from left TðHÞT wðkþ1Þ ¼ TðHÞT wðkÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} gðkÞ
gðkþ1Þ
y
¯ y y¯ nðkÞ , % mðkÞ signðRefoðkÞ gÞTðHÞT P nðkÞ ð18Þ
ðkÞ
where g would be the overall effective impulse response at the kth step. If we write down a (full) singular value decomposition of TðHÞ as # $ Rr H TðHÞ ¼ U V , (19) 0 where 2
s1
6 6 0 6 Rr ¼ 6 6 .. 6 . 4 0
0
...
s2
...
..
..
0
.
.
...
0
3
7 0 7 7 7 .. 7 .7 5 sr
ARTICLE IN PRESS 204
A.T. Erdogan / Signal Processing 88 (2008) 200–209
with s1 ; s2 ; . . . ; sr 40, and U; V are unitary matrices. Since yn ¼ TðHÞxn it is easy to show that Py ¼ TðHÞTðHÞH s2x " 2 # Rr 0 2 UH , ¼ sx U 0 0
ð20Þ ð21Þ
and consequently, " %2 # 0 Rr y %2 ¯ y ¼ sx U ¯ P UT : 0 0
ð22Þ
¼ Q signð~gðkÞ Þ þ signðRefoðkÞ gÞx" nðkÞ . nðkÞ
As a result, by plugging (31) in (27),we obtain mðkÞ Q g~ ðkþ1Þ ¼ g~ ðkÞ % 2 signð~gðkÞ Þ sx %
#
Refxg , %Imfxg
(25)
mðkÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi qffiP pffiffi ðkÞ 2 O O Þ jRefoðkÞ gj % ðQ 2 =s x l¼1 jRefol gj nðkÞ 2yH Py y nðkÞ y nðkÞ
¼
then we can rewrite (24) as (27)
Note that Refon g ¼ g~ T x! n
(28)
x! nðkÞ ¼ +Q signð~g Þ þ x" nðkÞ ,
!
AðkÞ l
for all l 2 ½1; 2L*, where is a random value from the set of real and imaginary components of the constellation. Note that since x" nðkÞ is orthogonal to gðkÞ , it has no effect at the output. Therefore,
%2 2LðkÞ g~ Qsx
,
ð36Þ
the fact that gj ¼ jx! TnðkÞ g~ ðkÞ j, jRefoðkÞ nðkÞ
(37)
and since x! nðkÞ ¼ +Q signð~gðkÞ Þ þ x" nðkÞ , gj ¼ k~gðkÞ k1 Q jRefoðkÞ nðkÞ
(29)
where x" nðkÞ is the vector that has zero values for the indexes where g~ ðkÞ is non-zero, and has arbitrary values from the set of real and imaginary components of the constellation points, i.e., the lth component of x" nðkÞ is given by 8
