IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 9, SEPTEMBER 2005 1851 Equivalent System Model and Equalization of Differential Impulse Radio UWB Systems Klaus Witrisal, Member, IEEE, Geert Leus, Member, IEEE, Marco Pausini, Student Member, IEEE, and Christoph Krall, Student Member, IEEE Abstract—A discrete-time equivalent system model is derived for differential and transmitted reference (TR) ultra-wideband (UWB) impulse radio (IR) systems, operating under heavy in- tersymbol-interference (ISI) caused by multipath propagation. In the systems discussed, data is transmitted using differential modulation on a frame-level, i.e., among UWB pulses. Multiple pulses (frames) are used to convey a single bit. Time hopping and amplitude codes are applied for multi user communications, em- ploying a receiver front-end that consists of a bank of pulse-pair correlators. It is shown that these UWB systems are accurately modeled by second-order discrete-time Volterra systems. This proposed non- linear equivalent system model is the basis for developing optimal and suboptimal receivers for differential UWB communications systems under ISI. As an example, we describe a maximum like- lihood sequence detector with decision feedback, to be applied at the output of the receiver front-end sampled at symbol rate, and an adaptive inverse modeling equalizer. Both methods significantly increase the robustness in presence of multipath interference at tractable complexity. Index Terms—Differential receivers, equalization, impulse radio (IR), transmitted reference, ultra-wideband (UWB) communica- tions, Volterra systems. I. INTRODUCTION I N autocorrelation receiver (AcR) front-ends for impulse radio (IR) ultra-wideband (UWB) communications systems, the received signal consisting of a train of pulses is delayed and correlated with itself. The basic idea is to use the delayed signal as a template in the demodulation block, without requiring any kind of channel estimation. Thus, one pulse is used to “sound” the channel and another one is used to convey data. Data can be applied, for instance, by differentially modulating the polarity of the “data” pulse with respect to the “reference” pulse. The delay line in an AcR front-end, or “pulse-pair” correlator, is matched to the lag between the reference and the data pulses. The overall system is often called a transmitted reference (TR) AcR [1]–[13]. Manuscript received March 31, 2004; revised January 22, 2005 and March 20, 2005. The work of G. Leus was supported in part by NWO-STW under the VICI Program (DTC.5893). The work of M. Pausini was supported in part by the Dutch Ministry of Economic Affairs/Ministry of Education Freeband-Impulse Project Airlink. This paper was presented in part at the IEEE Global Telecommu- nications Conference (GLOBECOM), Dallas, TX, November/December 2004. K. Witrisal and C. Krall are with the Signal Processing and Speech Commu- nication Laboratory and the Christian Doppler Laboratory for Nonlinear Signal Processing, Graz University of Technology, A-8010 Graz, Austria (e-mail: Witrisal@tugraz.at; Christoph.Krall@tugraz.at). G. Leus and M. Pausini are with the Circuits and Systems Group and the Wireless and Mobile Communications Group, Delft University of Tech- nology, NL-2628 CD Delft, The Netherlands (e-mail: Leus@cas.et.tudelft.nl; M.Pausini@ewi.tudelft.nl). Digital Object Identifier 10.1109/JSAC.2005.853876 AcRs have the big advantage of capturing energy from all multipath components at low implementation complexity, com- pared with coherent receivers. Unfortunately the reference pulse is corrupted by noise and interference, which is an inherent dis- advantage. If the data rate of such systems is increased, inter- ference among multiple pulses becomes one of the most funda- mental deteriorating effects, due to the multipath propagation. This effect is termed interframe-interference (IFI), if interfer- ence between multiple pulses of one data symbol is referred to, and intersymbol-interference (ISI) for the interference among consecutive data symbols. The characterization, modeling, and suppression of IFI and ISI is still a young topic of research. A few important basic results are found, e.g., in [2], [4], and [14]. However, most of them are limited to rather simple cases, for ex- ample considering IFI among a pair of reference and data pulses only [2], [4]. A more sophisticated model has been presented in [12], where both IFI and ISI are taken into consideration. Based on this work, linear weighting of an oversampled AcR output is suggested in [13] to improve the receiver performance. Most other previous works assume pulse spacings sufficiently long such that IFI is completely avoided, which ultimately limits the transmission rates [5]–[11]. In this paper, we derive an equivalent system model for AcR- based UWB systems, describing the IFI and ISI in a multipath channel. That is, high-rate differential transmission schemes are studied. The system model accurately relates the transmitted data bits to the test statistics at the decision device .A nonlinear second-order Volterra model [15] is found to describe the data dependency, whereby the nonlinearity is caused by the multiplication in the pulse-pair correlators. The data model can be written as (1) Besides a linear finite impulse response (FIR) component specified by the coefficients , it comprises an additive bias term and product terms of the transmitted data symbols, weighted by the second-order coefficients . The memory depth is expressed by , which is determined by the ratio of the maximum channel excess delay and the symbol duration. Ad- ditive noise samples are denoted by , whose second-order moments are studied in this paper. We will demonstrate the suit- ability of this model for the differential UWB scheme described in [14]. This scheme achieves higher efficiency and potentially 0733-8716/$20.00 © 2005 IEEE