Radio Science Bulletin 313 - June 2005 - URSI

Figure 3a. A time-domain representation of aGaussian pulse for arbitrary t p .Figure 3b. A frequency-domain representationof a Gaussian pulse for arbitrary t p .far more technologically challenging. With this in mind, theIEC defined b r to distinguish between technologicallychallenging UWB signals and other UWB signals. Themodified classification scheme of Table 1 retains this featureof sub-categorizing UWB signals.Even though the Gaussian pulse is a non-causalsignal, the monotonicity and significant asymmetry of theGaussian spectrum over the positive frequencies (Figure 3b)cause problems for the B RMS definition. Equations (12)and (13) yield a mean frequency, f mp , of 1 ( 2t p π π ),which differs substantially from the peak frequency, f M ,of 0 Hz, and an rms band of0, f ( ) ( ) 1/2mp + ⎡⎣ π − 2 2 π ⎤⎦ ( 2 tp) .{ }5.3 Finite Half-Cycle SineA waveform that has been discussed in connectionwith UWB radar is n half cycles of the sine function [11].The mathematical representation of this waveform and thecorresponding Fourier transform are() 0 sin ( 2 ⎡ ⎛ ns t = s π fct) ⎢σ()t −σ⎜t−⎣ ⎝ 2 f⎡ˆ s0fnS( f) =−c ⎢1−( −1)e2π2 2f − f ⎢c ⎢⎣cf− jnπf⎞⎤⎟⎥, (20a)⎠⎦c⎤⎥⎥. (20b)⎥⎦In contrast to the previous two waveforms, the spectrum ofthis waveform is characterized by the nulls of its powerspectral density,2 20 c2 22ˆ s f ⎡ n ⎛ fS( f)= ⎢1−( −1)cos⎜nπ2π⎣⎝ fwhich occur at2 2( f − fc)c⎞⎤⎟⎥,⎠⎦Bandwidths f l t p f h t p B F Classification E relB 3dB 0.00 0.132 2.00 UWB / HB 76.0 %B 10dB 0.00 0.242 2.00 UWB / HB 96.8 %B 20dB 0.00 0.342 2.00 UWB / HB 99.8 %B RMS 0.00 0.303 2.00 UWB / HB 99.3 %B 90EB 0.00 0.185 2.00 UWB / HB 90 %B 99EB 0.00 0.290 2.00 UWB / HB 99 %Table 4. Classification of a Gaussian pulse for six bandwidth definitions.TheRadio Science Bulletin No 313 (June, 2005) 21