## 1 Introduction  
Broadband wireless communications, electronics and sensor networks technologies have evolved dramatically in the past twenty years. The growth tendency of wireless sensor networks and the electronics devices market is expected to continue in the following few decades, as the demand for information services is increasing. With more and more communication system and electronic devices going to wireless, the next generation of wireless sensor networks will be confronted with frequency spectral crowding, and coexistence with the pre-existing wireless devices will be one of the most important challenges [1; 2; 3; 4]. Cognitive radio (CR) techniques [2; 5; 6] and ultra-wide band (UWB) radio communication techniques are considered as a part of the most important solutions for friendly coexistence. In recent years, ultra-wide band (UWB) wireless communication and sensor network technologies have attracted considerable interests in academia and industry [7; 8; 9; 10], due to their promise to provide precision positioning and reliable communication ability at low cost with extremely low energy consumption [11; 12]. Ultra-wide band wireless sensor networks are characterized as the transmitted signals with instantaneous frequency spectral occupancy greater than 500 MHz, or with a fractional bandwidth of more than 20% [13]. UWB radio techniques originally started with the spark-gap transmitter of Hertz and Marconi. However, it was not until the 1990s that the interest was renewed. The pioneering work of developing the concept of time-hopping (TH) impulse-based ultra-wide band radio systems can be found in [8; 9]. In recent years, the impulse-based UWB techniques have been considered as a competitive candidate for the low-cost high performance wireless sensor networks. UWB wireless sensors promise to coexist with other licensed and unlicensed narrow band or broadband wireless signals [14]. To ensure the UWB signal without causing significant interferences to the incumbent systems, a number of very strict regulations are imposed on UWB wireless sensor networks [15]. According to the frequency spectrum regulations of the Federal Communications Commission (FCC), UWB wireless sensors must transmit below specified power levels in order not to cause notable interferences with the pre-existing wireless devices in the same frequency spectrum. The detailed average power spectral density (PSD) limitation of the transmitted UWB radios for the indoor environment and the outdoor environment can be found in [13]. However, from the narrow band interferences' side point of view, the effects of pre-existing narrow band radios on the UWB wireless sensor networks can still be notable. In some extreme situations, the strong interference signals may jam the reception module of the UWB wireless sensors thoroughly [16; 17]. The unifying mathematical framework is developed to characterize the aggregate interference in wireless heterogeneous networks [18; 19; 20]. The aggregate multiple access interferences and the narrow band interferences are modeled as Gaussian random variables from which the variance is determined [21; 22; 23]. Even if the bandwidth of narrow band interferences is only a small fraction of the huge frequency spectrum UWB radio occupied, due to their relatively high PSD with respect to the primary ultra-wide band radios, the reception performance of the UWB wireless sensors can be influenced enormously. The work in [24] indicated that the average probability of error (PE) of the ultra-wide band sensors will be observably degraded due to the influence of in-band pre-existing strong narrow band interferences (NBIs). The inherent processing gain of the impulse-based UWB wireless signal can resist the narrow band interferences to a certain level [9; 25]. However, on many practical occasions, even the inherent processing gain is insufficient enough to compensate the impacts of in-band pre-existing strong NBIs [26; 27].