Tangential-Sphere Bound Revisited

With the recently proposed parameterized Gallager's first bounding technique (GFBT), the tangential-sphere bound (TSB) of Poltyrev, which was considered as one of the tightest upper bounds, is revisited by defining nested Gallager regions. The nested Gallager regions are chosen to be a family of the boundary surface of the circular cone whose central line passes through the origin and the transmitted signal. The parameterized TSB proposed in this paper can be proved to be equivalent to the TSB proposed by Poltyrev. The equation for the optimal parameter can be obtained in an intuitive manner without resorting to the derivatives. The new derivation reveals that the optimal region of TSB is a cone rather than a double cone with a geometrical explanation, which is considered to be a double cone in the original literature.