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Misspecified Cramer–Rao Bounds for Blind Channel Estimation Under Channel Order Misspecification
In estimation, the misspecified Cramer–Rao bound (MCRB), which is an extension of the well-known Cramer–Rao bound (CRB) when the underlying system model is misspecified, has recently attracted much attention. In this paper, we introduce a new interpretation of the MCRB, called the generalized MCRB (GMCRB), via the Moore–Penrose inverse operator. This bound is useful for singular problems and particularly blind channel estimation problems in which the Hessian matrix is noninvertible. Two closed-form expressions of the GMCRB are derived for unbiased blind estimators when the channel order is misspecified. The first bound deals with deterministic models where both the channel and unknown symbols are deterministic. The second one is devoted to stochastic models where we assume that transmitted symbols are unknown random variables i.i.d. drawn from a Gaussian distribution. Two case studies of channel order misspecification are investigated to demonstrate the effectiveness of the proposed GMCRBs over the classical CRBs. When the channel order is known or accurately estimated, both generalized bounds reduce to the classical bounds. Besides, the stochastic GMCRB is lower than the deterministic one, especially at high SNR.
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Electronics and Communications
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