Solving Toeplitz-related systems has been of interest for their ubiquitous applications, particularly in image science and the numerical treatment of differential equations. Extensive study has been carried out for Toeplitz matrices as well as Toeplitz-function matrices , where h(z) is a certain function. Owing to its importance in developing effective preconditioning approaches, their spectral distribution associated with Lebesgue integrable generating functions f has been well investigated. While the spectral result concerning is largely known, such a study is not complete when considering with being the anti-identity matrix. In this book chapter, we attempt to provide numerical evidence for showing that the eigenvalues of can be described by a spectral symbol which is precisely identified.