Radio Science Bulletin 313 - June 2005 - URSI

Bandwidths f l / GHz f h / GHz B F Classification E relB 3dB 0.998 1.002 0.003 NB 50 %B 10dB 0.995 1.005 0.010 NB 80 %B 20dB 0.984 1.016 0.032 MB 94 %B RMS 0.859 1.141 0.280 UWB / SHB 99 %B 90EB 0.990 1.010 0.020 MB 90 %B 99EB 0.870 1.080 0.220 MB 99 %Table 2. Classification of the exponentially damped sine with Q = 314.47 ,β = 10 MHz, and f = 1GHz, for six bandwidth definitions.c5.1 Exponentially Damped SineA classical representative function for narrowbandand wideband waveforms is the damped sine,−βt() ( π ) σ()s t = s0e sin 2 f t t , (18a)Sˆ( f)=s02πfβ + j2π f + 4πf( )cc2 2 2c, (18b)where σ () t is the unit step function and f c is the carrierfrequency. The behavior of the power spectral densitydepends on the ratio, β f c , of the damping factor to thecarrier frequency. Specifically, for β f c less than thethreshold ⎡ 2 24π( 8π−1) ⎤1/2≅0.712, the maximum,⎣⎦f M ,is approximately equal to f c , and the spectrum for positivefrequencies is essentially symmetric about f M . As β f cincreases to this threshold, the spectrum becomesincreasingly asymmetric over the positive frequencies, andf M migrates towards dc (0 Hz), which it reaches whenβ f c = 0.712 . To illustrate these behaviors, the waveformand power spectral density are plotted for two representativevalues of β f c : one much less than the threshold (0.01),and one less than but near to 0.712 (0.60).In the case of a low damping factor relative to f c( β = 10 MHz and f c = 1GHz, that is, a medium qualityfactor of Q = 314.47 ), the time-domain representation, s,consists of a large number of cycles (Figure 1a), and thespectrum is essentially symmetric about fc ≅ fM(Figure 1b). Consequently, all bandwidth definitions arewell defined for this signal (Table 2). Many engineersclassify the damped sinusoidal waveform that is depicted inFigure 1 as wideband. With the exception of classificationsthat use B 3dB or B 10dB , which identify the signal asnarrowband, the bandwidth definitions agree with thispopular view.Increasing β f c (decreasing Q) reduces the numberof effective cycles in the time domain and increases the lowfrequencycontent of the spectrum. By taking this to theextreme, the spectrum of the damped sine can be madeUWB with a significant power level at dc. For example,2when β = 0.6 GHz and f c = 1GHz,S ˆ >−20dB for0 Hz ≤ f ≤ 1.7 f c and fM ≅ fc(Figure 2). For this type ofwaveform, the spectrum is no longer symmetric about thepeak frequency, f M . For some bandwidth definitions( B 20dB , B RMS , B 99EB ), the lower frequency, f l , goes tozero, and the fractional bandwidth is two (Table 3). Sincethe calculation of b r would require dividing by zero in thatsituation, b would not be defined for this waveform,rFigure 1a. A time-domain representation of an exponentiallydamped sine for β = 10 MHz and f c = 1GHz.Figure 1b. A frequency-domain representation of anexponentially damped sine for β = 10 MHz and f c = 1GHz.TheRadio Science Bulletin No 313 (June, 2005) 19