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We provide a rigorous account of the equivalence between the complex-valued widely linear estimation method and the quaternion involution widely linear estimation method with their vector-valued real linear estimation counterparts. This is achieved by an account of degrees of freedom and by providing matrix mappings between a complex variable and an isomorphic bivariate real vector, and a quaternion variable versus a quadri-variate real vector. Furthermore, we show that the parameters in the complex-valued linear estimation method, the complex-valued widely linear estimation method, the quaternion linear estimation method, the quaternion semi-widely linear estimation method, and the quaternion involution widely linear estimation method include distinct geometric structures imposed on complex numbers and quaternions, respectively, whereas the real-valued linear estimation methods do not exhibit any structure. This key difference explains, both in theoretical and practical terms, the advantage of estimation in division algebras (complex, quaternion) over their multivariate real vector counterparts. In addition, we discuss the computational complexities of the estimators of the hypercomplex widely linear estimation methods. [ABSTRACT FROM AUTHOR]

Smith charts are an extremely powerful visualization tool used by microwave engineers for designing impedance-matching circuits, filters, amplifiers and transmission lines. But, their usefulness extends beyond traditional analog and microwave analysis and design. They can be powerful tools for high-frequency digital designers as well. The basics of Smith charts are covered in this article and it is shown how easy they are to use [ABSTRACT FROM PUBLISHER]

Orthogonal-design-based space-time (ODST) codes of size (n×n) offer maximum diversity gain advantage and a simple yet optimal decoding algorithm under an arbitrary signal alphabet or constellation A. However, these designs only exist for n=2, 4, 8 when A is real and for n=2 when A is complex. In this letter, we address the question of the existence of ODST codes of other sizes when A is restricted to be a proper subset of either real or complex numbers. We refer to these as restricted-alphabet ODST (RA-ODST) codes. We show that real RA-ODST codes of size greater than 8 that also guarantee maximum diversity advantage do not exist. Without the diversity requirement, RA-ODST codes exist only when A={a,-a}, 0

Fast and accurate methods for computing the arctangent of a complex number x = I + jQ have been the subject of extensive study because estimating the angle θ of a complex value has so many applications in the field of signal processing. Practitioners interested in high-speed arctangent computations typically use look-up tables where the values of I and Q specify an address in read-only memory (ROM) containing an approximation of angle θ. In this paper, another method to compute the arctangent approximation is proposed that uses neither the look-up table nor high order polynomials. [ABSTRACT FROM PUBLISHER]