 (1) Some Backgrounds of Newton algorithm in design of blind adaptive receivers:

Recently, many literatures focus on the constant modulus algorithm (CMA) and the multimodulus algorithm (MMA) in blind equalization algorithms. Both the CMA and the MMA are attractive due to their stochastic gradient descent (SGD)-type simplicity to implement with low complexity. However, the rate of convergence of both the CMA and the MMA, which is very important for practical systems, is usually slow. It is well known that the conventional least-squares algorithms such as the recursive least-squares (RLS) algorithm used for trained equalization cannot be directly applied to blind equalization algorithms. This is because the cost functions of blind equalization algorithms are nonquadratic and have many saddle points.

However, the convergence of blind equalization algorithms such as the CMA and the MMA may be significantly improved by premultiplying the gradient by the inverse of the input correlation matrix or more practically by its estimate to preorthogonalize the input correlation matrix (i.e., to make all the eigenvalues identical). The resulting algorithms are referred to as the orthogonalized

CMA (O-CMA) ,  and orthogonalized MMA (O-MMA) when the orthogonalizing algorithm is applied to the CMA and the MMA, respectively.

The Newton algorithm is typically known to be implemented by premultiplying the gradient vector by the inverse of the Hessian (the second derivative of the cost function). Consequently, an improvement in the rate of convergence in blind equalization may also be achieved by the

Newton algorithm employing the stochastic estimate of the inversion of the Hessian of the CMA

(or MMA) cost function. Yan and Fan  proposed a suboptimal version of the Newton-based

CMA, coined in  as the pseudo-Newton (PN) CMA (PN-CMA). Li and Zeng  extended the pseudo-Newton algorithm to MMA, called the modified PN-MMA (MPN-MMA) and demonstrated that the MPN-MMA converged faster than the MMA. Although both the PN-CMA and the MPN-MMA may provide much faster convergence than the CMA and the MMA, respectively, the inversion of the Hessian utilizing the well-known matrix inversion lemma  is computationally demanding and requires O N

2

( ) operations per iteration, where N is the tap length of blind equalizer. Our task is to reduce the computational complexity by another method in instead of exploiting the matrix inversion lemma.

(2) Objectives of this course :

(a) We will implement and demonstrate that the PN-CMA and the MPN-MMA may be closely approximated by the O-CMA and the O-MMA, respectively.

(b) Compare the baud-rate equalization using the PN-CMA and the MPN-MMA with the

O-CMA and the O-MMA, respectively; compare the fractionally spaced equalization (FSE) using the PN-CMA and the MPN-MMA with the O-CMA and the O-MMA, respectively.

(c) Moreover, instead of exploiting the matrix inversion lemma to compute the inverse of the

Hessian as in the PN-CMA and MPN-MMA techniques, which requires O N

2

( ) computational complexity, we will exploit the QR-decomposition-based least-squares lattice (QRD-LSL) interpolation algorithm to implement both the O-CMA and the O-MMA, which requires only

O N

2

N )] operations per iteration, where N is the tap length of blind equalizer.

(d) Analyzing the Hessian in Newton-like blind algorithms, i.e., PN-CMA, PN-MMA, and

O-CMA… etc ., in order to use the computed curvatures to demonstrate the fast convergence speed of the Newton-like blind equalization algorithms.

 R. Gooch, M. Ready, and J. Svoboda, “A lattice-based constant modulus adaptive filter,” in Proc. 20 th Asilomar

Conf. Signals, Syst., Comput.

, Pacific Grove, CA, 1987, pp. 282-286.

 R. Gooch and J. Lundell, “The CM array: An adaptive beamformer for constant modulus signal,” Proc. ICASSP ,

Japan, 1986, pp. 2523-2526.

 G. Yan and H. H. Fan, “A Newton-like algorithm for complex variables with applications in blind equalization,”

IEEE Trans. Signal Process.

, vol. 48, pp. 553-556, Feb. 2000.

 K. Kreutz-Delgado, Y. Isukapalli, “Use of the Newton method for blind adaptive equalization based on the constant modulus algorithm,” IEEE Trans. Signal Process.

, vol. 56, pp. 3983-3995, Aug. 2008.

 X.-L Li and W.-J. Zeng, “Performance analysis and adaptive Newton algorithms of multimodulus blind equalization criterion,” Signal Process ., vol. 89, pp. 2263-2273, Nov. 2009.

 S. Haykin, Adaptive Filter Theory , 4 th ed. Englewood Cliffs, NJ: Prentice-Hall, 2002.