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.

In modern cities, more and more people drive the vehicles, equipped with the GPS devices, which create a large scale of trajectories. Gathering and analyzing these large-scale trajectory data provide a new opportunity to understand the city dynamics and to reveal the hidden social and economic phenomena. This paper designs and implements a tool, named as PGWinFunc, to analyze trajectory data by extending a traditional relational database. Firstly we introduce some efficient query process and optimization methods for SQL Window Aggregate Functions in PostgreSQL. Secondly, we present how to mine the LBS (Location-Based Service) patterns, such as the average speed and traffic flow, from the large-scale trajectories with SQL expression with Window Aggregate Functions. Finally, the effectiveness and efficiency of the PGWinFunc tool are demonstrated and we also visualized the results by BAIDU MAP.