Quasi-Convexity of the Asymptotic Channel MSE in Regularized Semi Blind Estimation

In this paper, the quasi-convexity of a sum of quadratic fractions in the form Σi=1n [(1+ci x2)/((1+dix)2)] is demonstrated where ci and di are strictly positive scalars, when defined on the positive real axis R+. It will be shown that this quasi-convexity guarantees it has a unique local (and hence global) minimum. Indeed, this problem arises when considering the optimization of the weighting coefficient in regularized semi-blind channel identification problem, and more generally, is of interest in other contexts where we combine two different estimation criteria. Note that V. Buchoux have noticed by simulations that the considered function has no local minima except its unique global minimum but this is the first time this result, as well as the quasi-convexity of the function is proved theoretically.

[1]  Karim Abed-Meraim,et al.  Blind channel shortening in OFDM system using nulltones and cyclic prefix , 2008, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing.

[2]  J. C. Burkill,et al.  Complex Functions , 1968, Nature.

[3]  Julian Havil.  Gamma: Exploring Euler's Constant , 2003 .

[4]  Eric Moulines,et al.  Minimum constrast estimation with applications to array processing , 1993, 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[5]  Eric Moulines,et al.  On the performance of semi-blind subspace-based channel estimation , 2000, IEEE Trans. Signal Process..

[6]  Zhi Ding,et al.  Semi-blind channel identification for individual data bursts in GSM wireless systems , 2000, Signal Process..

[7]  Karim Abed-Meraim,et al.  Blind channel equalization and carrier frequency offset estimation for MC-CDMA systems using guard interval redundancy and excess codes , 2009 .

[8]  R. Steele.  Optimization , 2005 .

[9]  Eric Moulines,et al.  Estimation semi-autodidacte par méthode sous-espace pour systèmes de réception multicapteurs , 1999 .

[10]  Jacques Palicot,et al.  Efficient semi-blind estimation of multipath channel parameters via a delay decoupling optimization approach , 2005, Signal Process..

[11]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[12]  Ales Pultr,et al.  Introduction to Mathematical Analysis , 2013 .

[13]  Franck Wielonsky.  A Rolle's theorem for real exponential polynomials in the complex domain , 2001 .

[14]  R. Remmert,et al.  Theory of Complex Functions , 1990 .

[15]  Anthony G. Constantinides,et al.  A weighted mixed statistics algorithm for blind source separation , 2002, 2002 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[16]  P. Duhamel,et al.  A combination of statistical and structural approaches in blind equalization algorithms , 2000, Conference Record of the Thirty-Fourth Asilomar Conference on Signals, Systems and Computers (Cat. No.00CH37154).

[17]  Karim Abed-Meraim,et al.  Blind identification of sparse SIMO channels using maximum a posteriori approach , 2008, 2008 16th European Signal Processing Conference.

[18]  Yonghong Zeng,et al.  A blind MIMO channel estimation method robust to order overestimation , 2004, Signal Process..