  Secondary spectrum auction is widely applied in wireless networks for
mitigating the spectrum scarcity. In a realistic spectrum trading market, the
requests from secondary users often specify the usage of a fixed spectrum
frequency band in a certain geographical region and require a duration time in
a fixed available time interval. Considering the selfish behaviors of secondary
users, it is imperative to design a truthful auction which matches the
available spectrums and requests of secondary users optimally. Unfortunately,
existing designs either do not consider spectrum heterogeneity or ignore the
differences of required time among secondary users. In this paper, we address
this problem by investigating how to use auction mechanisms to allocate and
price spectrum resources so that the social efficiency can be maximized. We
begin by classifying the spectrums and requests from secondary users into
different local markets which ensures there is no interference between local
markets, and then we can focus on the auction in a single local market. We
first design an optimal auction based on the Vickrey-Clarke-Groves (VCG)
mechanism to maximize the social efficiency while enforcing truthfulness. To
reduce the computational complexity, we further propose a truthful sub-optimal
auction with polynomial time complexity, which yields an approximation factor
6+4\surd2. Our extensive simulation results using real spectrum availability
data show that the social efficiency ratio of the sub-optimal auction is always
above 70% compared with the optimal auction.
