Cramér-Rao Lower Bounds for Unconstrained and Constrained Quaternion Parameters

The Cramér-Rao lower bound (CRLB) is a fundamental result in statistical signal processing, however, the CRLB for quaternion parameters is not yet established. To this end, we develop the theory of quaternion Cramér-Rao lower bound (QCRLB), based on the generalized Hamilton-real (GHR) calculus. For generality, this is achieved in a way that conforms with the real and complex CRLB. We first provide the properties of the quaternion covariance matrix and the quaternion Fisher information matrix (FIM), paving the way for the derivation of the QCRLB. This serves as a basis for the formulation of the QCRLB without constraints and a criterion for determining whether the QCRLB is attained. We also establish the QCRLB for constrained quaternion parameters, including both nonsingular and singular cases of the quaternion FIM. These broaden the theoretical framework and enhance its applicability to diverse practical scenarios. The practical efficacy of the QCRLB is demonstrated through two illustrative examples. Numerical validations confirm that the maximum-likelihood estimator (MLE) attains the QCRLB for the linear model, and the quaternion gradient ascent (QGA) algorithm achieves the QCRLB at each iteration with theoretical guarantees. We also propose the quaternion constrained scoring (QCS) algorithm, which converges in one step in the linear constrained MLE case, for the linear model. These results significantly contribute to both the theory and practical application of quaternion signal processing, bringing valuable insights into the quaternion parameter estimation.