IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 11, NOVEMBER 2004 875 Cramer–Rao Lower Bound for Constrained Complex Parameters Aditya K. Jagannatham, Member, IEEE, and Bhaskar D. Rao, Fellow, IEEE Abstract—An expression for the Cramer–Rao lower bound (CRB) on the covariance of unbiased estimators of a constrained complex parameter vector is derived. The application and useful- ness of the result is demonstrated through its use in the context of a semiblind channel estimation problem. Index Terms—Channel estimation, constrained parameters, Cramer–Rao bound (CRB), MIMO, semiblind. I. INTRODUCTION T HE CRAMER–RAO lower bound (CRB) serves as an important tool in the performance evaluation of estima- tors which arise frequently in the fields of communications and signal processing. Most problems involving the CRB are formulated in terms of unconstrained real parameters [1]. Two useful developments of the CRB theory have been presented in later research. The first being a CRB formulation for uncon- strained complex parameters given in [2]. This treatment has valuable applications in studying the base-band performance of modern communication systems where the problem of estimating complex parameters arises frequently. A second result is the development of the CRB theory for constrained real parameters [3]–[5]. However, in applications such as semiblind channel estimation one is faced with the estimation of constrained complex parameters. Though one can reduce the problem to that of estimating constrained real parameters by considering the real and imaginary components of the complex parameter vector, the complicated resulting expressions result in loss of insight. Using the calculus of complex derivatives as is often done in signal processing applications, considerable insight and simplicity can be achieved by working with the complex vector parameter as a single entity [1], [6], [7]. We thus present an extension of the result in [3]–[5] inspired by the theory in [2] for the case of constrained complex parameters. To conclude, we illustrate its usefulness by an example of a semiblind channel estimation problem. II. CRB FOR COMPLEX PARAMETERS WITH CONSTRAINTS Consider the complex parameter vector . Let such that the real and imaginary parameter vectors Manuscript received February 2, 2004; revised April 7, 2004. This work was supported by CoRe Research Grant Cor00-10074. The associate editor coordi- nating the review of this manuscript and approving it for publication was Prof. Steven M. Kay The authors are with the Center for Wireless Communications (CWC), University of California at San Diego, La Jolla CA 92093 USA (e-mail: jak@ucsd.edu). Digital Object Identifier 10.1109/LSP.2004.836948 and . Assume that the likelihood func- tion of the (possibly complex) observation vector pa- rameterized by is . Let be given as , where are unbiased estimators of , respectively. In the foregoing analysis, we define the gradient of a scalar function as a row vector: (1) Let be defined as in [2] by (2) Suppose now that the complex constraints on are given as (3) i.e., . We then construct an extended constraint set (of possibly redundant constraints) as (4) An important observation from (4) above is that symmetric complex constraints on these parameters are treated as disjoint. For instance, given the orthogonality of complex parameter vectors , i.e., , the symmetric constraint is to be treated as an additional complex con- straint and hence . The extension of the constraints is akin to the extension of the parameter set from to called for when dealing with complex parameters, and the need will become evident from the proof of lemma (1). Reparameterizing in terms of , let the set of parameter constraints for be given by . Employing notation defined in [3] and borrowing the notion of a complex derivative from [1], [6], we define as (5) It then follows from the properties of the complex derivative [6] that (6) 1070-9908/04$20.00 © 2004 IEEE