UWB Signals as Shot Noises A. Ridolﬁ School of Computer and Communication Sciences Ecole Polytechnique F´ ed´ erale de Lausanne 1015 Lausanne, Switzerland e-mail: andrea.ridolfi@epfl.ch M. Z. Win Laboratory for Information and Decision Systems Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA e-mail: moewin@mit.edu Abstract — We present a shot-noise based model for a large family of ultrawide bandwidth (UWB) signals. These include time-hopping and direct- sequence signaling with pulse position, interval and amplitude modulations. Each speciﬁc signal is constructed by adding features to a basic model in a modular, simple, and tractable way. Our work uniﬁes the contributions scattered in the literature and provides a general approach that allows various extensions of previous works. The exact power spectrum is then evaluated using shot-noise spec- tral theory, which provides a simpler, systematic, and rigorous approach to the spectra evaluation of complex UWB signals. The strength of our methodology is that diﬀerent features of the model contribute clearly and separately in the resulting spectral expressions. I. Introduction UWB radio communicates with pulses of very short duration, thereby spreading the energy of the radio signal over several GHz. UWB signals are transmitted with spread spectral content while main- taining the average power level required for reli- able communications. As a consequence, the spectral characteristics (spectral occupancy and composition) of an UWB transmission has a key role in the design of UWB systems. Exact spectral evaluation has already received at- tention in the communications community. Among several contributions, we mention the computation of the power spectrum of a general time-hopping, pulse position modulated signal [1]–[4], and that of the spectral density of the family of pulse interval modulated signals [5]. In these contributions, the sig- nal models are based on Dirac pseudo-functions and lack generality, especially with respect to the type of temporal modulation. The spectrum computation is performed using the classical wide-sense stationary (w.s.s.) approach, i.e., as the Fourier transform of the correlation function. Although this is a common approach, the resulting computations are complex and the introduction of additional random features of the model, such as clock jitter, pulse losses or pulse distortion, requires a new computation from scratch. Here we show that uwb signals are aptly modelled as shot noises with random excitation, i.e., a ﬁltered stream of spikes at random times {T n } n∈ , where the ﬁltering function h (t, Z ) depends on a random parameter Z X(t)=  n∈ h (t − T n ,Z n ) , t ∈ . (1) More precisely, the sequence of random times {T n } n∈ determines the temporal structure of the signal and the random function h (t, Z ) character- izes the shape of the pulses and its random modiﬁ- cations (amplitude, displacement, distortion). Shot- noise processes have received much attention in the applied literature (see for instance [6], [7] and the references therein). Concerning communications sys- tems, they have been widely used in queuing and teletraﬃc theory [8], and a Poisson based model for pulse coded optical transmissions has been proposed in [9], [10]. In the context of uwb signals, a Poisson shot-noise was proposed by [11] to model interference on narrowband systems. As we shall see, there are two considerable ad- vantages in modeling UWB signals as shot-noise processes. Firstly, the model is modular, simple, and tractable. Thus, one can construct diﬀerent UWB signals in a unifying way by simply adding features to a basic model systematically, and can easily take into account random quantities, such as jitter, losses, or pulse distortions, which aﬀect the signals. Secondly, spectra are obtained, in a systematic and rigorous manner from a single general formula. This formula simpliﬁes previous proofs of the existing results and provides spectrum expressions of highly complicated signals where various features of the model appear