Abstract—The power spectral density (PSD) of time-hopping (TH) Ultra Wide Band (UWB) signals plays a major role in key aspects like coexistence with conventional radio systems. In the past, several papers have been published, presenting on the one hand the effects of modulation and timing jitter on the PSD assuming random TH codes. On the other hand, papers have been presented dealing with the PSD of specific TH codes without modulation. This paper presents a mathematical frame work that enables the evaluation of the PSD of a modulated impulse radio signal using a deterministic TH code. Besides being of theoretical interest, our results can be the starting point for the development of TH code design criteria aimed at the spectral shaping of the UWB signal. Index terms— PSD, UWB, Impulse Radio, Time-hopping 1. INTRODUCTION Almost all communications systems in use today employ a sinusoid as an elementary waveform, on which information is mapped via some sort of modulation. The result is that signal energy is concentrated in a well defined frequency band, which makes noise and interference suppression relatively easy, e.g. by means of band-pass filtering. Unfortunately, such narrow band systems are inherently sensitive to fading. To obtain a robust wireless communication system, a fading margin has to be respected resulting in a lower capacity. Furthermore, spectral resources are divided into many narrow frequency bands causing the spectral resources to be fragmented. In the last ten years the interest in ultra wide band (UWB) technology has grown [1], [2] and [3]. Not only due to its promise to re-use rare spectrum, but also due to its inherent resilience against fading leading to increased capacity in multipath environments. Additionally, generation of UWB signals requires low complexity, if ultra-short pulses are transmitted. A system deploying this technique is often referred to as an impulse radio (IR). Due to its high bandwidth a UWB signal is able to resolve its surrounding with high resolution, which in principle allows a single UWB device to be used for communication and radar applications. Currently, regulation authorities in both Europe and the US are in the process of developing legislation for UWB signals. Clearly, no gigahertz bandwidth at the lower frequencies will be allocated for UWB exclusively. The US telecommunications regulator (FCC) has indicated that UWB devices most probably have to operate within limits described in Part 15 of FCC regulation. These limitations set boundaries on the transmit signal in both the frequency domain (limited power in a 10 MHz bandwidth) and the time domain (limited peak to mean, depending on the signal bandwidth). European regulatory bodies will most likely set similar limitations on the transmit signal. Although UWB signals are alike in the frequency domain, they are diverse in the time domain. An important type of UWB signals (e.g.[3]) is constituted by a sequence of very short pulses, which occur pseudo-randomly in time. This pseudo-randomness is generated by a time hopping (TH) code. These signals can be modulated in several ways, including pulse amplitude modulation (PAM) and pulse position modulation (PPM). For this type of UWB signal, the TH code and the modulation scheme shall be designed such that reliability and throughput of the UWB system are maximized, without violating regulation. Furthermore, interference with narrowband systems should be kept to a minimum to accelerate the acceptance of UWB technology. A good understanding of the power spectral density (PSD) of UWB signals and how it is influenced by both the TH code and the modulation is mandatory to achieve these goals. In the past, papers have been published on the PSD of UWB signals. In [4], the PSD of a modulated TH UWB signal is computed assuming a random TH code. However, the role of the TH code is not explicitly identified. In [5], the PSD of an IR employing a finite TH code is investigated without addressing the effect of modulation. In this paper we will derive the PSD of a modulated TH UWB signal in a form that allows to explicitly study the effects of the TH code and of the modulation on the PSD shape. Besides being of theoretical interest, our results are the starting point for the development of code design criteria aimed at the spectral shaping of the UWB signal. 2. SIGNAL DEFINITION In this section a format for the transmitted signal is introduced. A similar format is presented in [3], but some modifications are required. Namely, in an IR the ultra-short pulses are randomized by a pseudo-random TH code, which inevitable will repeat itself. Therefore, it is convenient to describe firstly the waveform s p (t) transmitted in a single repetition period T TH . In order to construct the waveform, the total period T TH contains N b equally sized time intervals of length T b , which is the symbol duration. Note that N b T b may be smaller as T TH . On its turn, the symbol duration T b contains N s equally sized time intervals named frame T f , again T f N s may be smaller as T f . To form a waveform, a pulse is allocated inside each frame. Its position is dictated by the TH code. Specifically, each frame is divided into N h chips of duration T c . Logically, the TH code is a sequence of elements with an integer value between 0 and N h -1. Again N h T c is only upper bounded by T f . As a result, the Jac Romme 1 and Lorenzo Piazzo 2 1 IMST GmbH, Carl-Friedrich-Gauß-Str. 2, D-47475, Kamp-Lintfort, Germany 2 INFOCOM dept., University of Rome “La Sapienza”, V. Eudossiana 18, I-00184 Rome, Italy On the Power Spectral Density of Time- Hopping Impulse Radio