engleski

Time-domain synthesis of pulse shapers for ultra- wideband impulse radio

Ultra-wideband (UWB) impulse radio uses very short pulses that meet spectral mask released by Federal Communications Commission (FCC). The PSD of each UWB pulse is regularized by the FCC mask, which is provided separately for indoor and outdoor communications. However, both masks have the same UWB passband. In addition, to eliminate inter-symbol interference in multiple-access applications, these pulses should be orthogonal. Other important measure for UWB pulses is spectral efficiency. The spectral efficiency is defined as the average power of the pulse normalized to the total power allowed by the FCC constraints. Various techniques for utilization of the UWB passband have been developed. One technique to generate these pulses is shaping. The shaping is realized with bandpass filters called pulse shapers. The most popular FCC-compliant orthogonal pulses utilize prolate spheroidal wave functions and Hilbert pairs of Gaussian derivatives. Prolate spheroidal pulses utilize the idea that a pulse filtered by an ideal bandpass filter determined by FCC UWB passband is not distorted. In practice, Gaussian derivatives, also called Gaussian monocycles, up to the seventh order are employed. Hilbert transforms of Gaussian derivatives are related to Dawson's integral. Since this integral cannot be expressed in a closed form, various methods for the approximation of the derivatives have been developed. Popular methods are based on numerical integration as well as on series expansions via sets of polynomials. The numerical integration should be performed for each point of the derivative's transform, what makes the former approach computationally demanding. A closed-form approximation can be obtained by using the weighted polynomial fitting in which Gaussian function is used as the weighting function. Such an approach results in explicit approximation formulas. In literature, they are available only for the derivatives of the second, third, and fourth order. Furthermore, they utilize only low-order polynomials. Additionally, fractional Hilbert transform is the generalization of Hilbert transform. It introduces an arbitrary phase shift. The approximation of fractional Hilbert transform of the Gaussian derivative can be easily obtained by using Hilbert-transform pair. Here, we present analog pulse shapers whose impulse responses approximate the prolate spheroidal pulses and Gaussian monocycles in the least squares sense. Furthermore, we provide their FCC-compliant transfer functions that yield highly orthogonal impulse responses. Since the orthogonality is defined in the time domain, to obtain the optimum shapers we use the time- domain synthesis. Furthermore, by using the zero-pole-gain model of the shapers, we express the measure for orthogonality of their impulse responses. Orthogonality is defined via normalized cross-correlation of shapers’ impulse responses. The transfer functions are represented with the zero-pole-gain model having simple poles only. For each pulse, the sixth-, eighth-, and tenth-order transfer functions are obtained. Additionally, we give the spectral efficiency of each impulse response, measured within the FCC UWB passband. Finally, we present an approximation of the Hilbert transforms of the Gaussian derivatives of arbitrary orders, which utilize high-order polynomials. The coefficients of these polynomials are obtained by using the least- squares error criterion.

analog filters ; multiple orthogonal pulses ; prolate spheroidal wave functions ; Gaussian monocycles ; pulse-shaping filters ; impulse response approximation ; time-domain synthesis ; ultra-wideband impulse radio (UWB-IR)

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