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Comparison of Finite Interval Constant Modulus Algorithm and Fast-ica for Block Siso Blind Equalization
| Content Provider | Semantic Scholar |
|---|---|
| Author | Tichavský, Petr Koldovský, Zbynek Hermanek, Antonin Regalia, Phillip A. |
| Abstract | This paper shows similarity between the finite interval constant modulus algorithm (CMA) [1] and the so called ”Fast-ICA” algorithm, originally designed to solve the independent component analysis problem [2,3] in application to blind SISO deconvolution. The latter algorithm was found to be a generalization of the former algorithm. The two algorithms may optimize the same criterion, but they differ in numerical methods used for optimization. While CMA is based on the steepest descent procedure, Fast-ICA uses approximate Newton method. However, for certain choice of the step in the steepest descent procedure, both algorithms are shown to be identical. Advantages of both of the algorithm are demonstrated in simulations in blind deconvolution of QAM, and V29 signals passed through a realworld microwave channel and distorted by additive (complex) Gaussian noise. The methods exhibit good performance compared to that of a popular super-exponential algorithm [4]. 1. THE BLIND EQUALIZATION PROBLEM FORMULATION We consider the blind deconvolution problem in which we observe a noisy output of an unknown possibly nonminimum phase linear system from which we want to recover its input using an adjustable linear filter (equalizer). This is a well known problem, see [1,4,5]. Let the unknown linear system be denoted and let it be fed by unobserved input which is a sample function from a discrete-time white (iid) random process. In particular, we shall assume that are uniformly distributed in a finite alphabet . The observed data is (1) This work was supported by the Grant Agency of the Czech Republic through grant 102/01/0021. where denotes the convolution, is an additive noise which is assumed to be independent of , white and Gaussian. The channel coefficients and the source signal can be both complex-valued. Examples of real-world channels that can be found in microwave applications can be found in [6]. Note that these channels are complex-valued; in this case we assume that the additive noise is complex circular Gaussian. We want to adjust an equalizer ! so that its output " approximates the input up to a constant delay and possibly a constant phase shift. Thus, if we denote by # $ % & ' $ %(*)+ ) & , ) (2) the unit sample response of the unknown equalizer, then we want to set so that # $ .-0/ 1 , 23 54 -0/*687 %9 : 6 7<; %9 (3) As a measure of equalization performance we take the intersymbolinterference (ISI) defined by ISI =?> @ BA DC # C EGFHC # IGJ K C E C # IGJ K C E (4) where # IGJ K is the component of # having the maximal absolute value (the leading tap or cursor). Clearly, a small value of ISI indicates the proximity to the desired solution. 2. REVIEW OF THE METHODS 2.1. Super-Exponential Algorithm In this paper we consider the version of the super-exponential algorithm [1] which utilizes fourth-order cumulants and is suitable for symmetrically distributed input sources . The algorithm starts with an initial estimate = : 6 '6 : 6 6 : 6 6 : @ where ”1” is placed somewhere around the middle of the vector, and iterates I � , I (5) I I I for : 6 6 6 until the convergence is achieved; In (5), is the ! matrix whose elements are 32#" $ % ( '& , 2 )( ,)" (6) and I is the vector whose elements are I * cum = " 6 " 6 " ( 6 )( , @ $ % ( '& C "! C E " ( , F $ % ( '& C " C E $ % ( '& "! ( , F $ % ( '& " E $ % ( '& " ( ( , (7) and " " I I is the equalizer output at iteration (the superscript “(m)” of " is omitted in (7) for brevity). 2.2. Constant-Modulus Algorithm (CMA) This algorithm is based on minimizing the constant modulus criterion + = @ A % '& = F C " C E @ E where again, " is the equalizer output, " . An equivalent formulation of the algorithm is that the equalizer should minimize the criterion [1] , = @ .% ( '& C " C /10 % ( '& C " C E 0 E (8) This algorithm is tailored for the case when the whole alphabet of information symbols lies on the unit circle in the complex plane, C C for all 2 . However, the algorithm was shown to perform well even in cases when do not have all the same modulus, such as the V29 sources. The algorithm implementation proposed in [4] is based on gradient descent technique. It begins with forming the data matrix 3 which consists of delayed versions of the input data , such that the equalizer output can be written as 4 65 " 6 '6 " % 7 98 (9) The algorithm proceeds in 3 steps. 1. Run the QR decomposition of 8 , 8 ;:=< so that columns of : provide an orthonormal basis of the column space of 8 . 2. Starting with an > initial vector ? with a unit norm (as in the super-exponential algorithm), iterate ? I ? I FA@ I : = 4CBD4EBF4 ( @ (10) ? I ? I ) ? I until convergence is achieved, where B denotes the elementwise product, and 4 stands for the vector of equalizer output at iteration , 4 4 I G: ? I (11) and @ I IH A = C " I C /1J , is a step-size. 3. 9< , ?LK 2.3. Fast-ICA This algorithm can be viewed as a generalized version of the CM algorithm, where the gradient descent minimization is replaced by approximate Newton method [2,3]. First, the data are “whitened”, i.e. the data matrix 8 is decomposed as 8 M:N< , where, aqain, columns of : provide an orthonormal basis of the column space of 8 . It is not a must, but usually < is determined as the Hermitian symmetric square root of 8POQ8 . Below we summarize complex valued version of the algorithm [3]. The algorithm searches for such a linear combination of the data that minimizes the expected value of certain suitable nonlinear function of data ( : ), E 5 R = C : ? C E @ 7 subject to constrant ? , where the nonlinear function R = CTS C E @ is applied elementwise. The idea behind that is that probability distribution of separated sources (estimated original sources) should have different from Gaussian distribution as much as possible, and the non-gaussianity can be mesured by expectation of suitable nonlinear function of the data. In this paper, we prefer the option R ='U @ U E which makes the algorithm equivalent to CMA. Let V = S @ and V = S @ denote the first and second derivative of R = S @ . The algorithm proceeds recursively, starting with an W initial vector ? with a unit norm iterates ? I X: 5 4 ( B V = C 4 C E @ 7'Y $ (12) F[Z 5 V = C 4 C E @ C 4 C E B V = C 4 C E @ 7 ? I ? I ? I ? I |
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| Alternate Webpage(s) | http://itakura.ite.tul.cz/zbynek/pubs/2004.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
