Radio Science Bulletin 313 - June 2005 - URSI

of the spectrum is required to completely specify the signalwaveform. Consequently, the inverse Fourier transformation(Equation (3)) can be rewritten as [9]⎧⎪⎫2() 2 ˆ js t = R⎨ S( f) e df ⎬⎪⎩⎭∞π ft ⎪∫ , (7)0⎪where R {} x denotes the real part of the argument x. Sincethe signals under consideration have finite energy, themagnitude of Ŝ must have a maximum value that isattained for at least one frequency, f M , that is,for every f.( ) ˆ( )Sˆf ≤ S f M(8)4.2.1 General AxiomsBefore discussing various definitions of signalbandwidth B, three axioms associated with the notion ofspectral width are stated because every appropriate measureof the nominal width of the spectrum Ŝ must comply withthis minimal set of axioms.4.2.1.1 Bandwidth is non-negative:{ ( )}B S ˆ f ≥ 0 . (9)4.2.1.2 Bandwidth is independent of height:{ˆ( )} ˆ( ){ }B kS f = B S f . (10)4.2.1.3 Stretching the argument of Ŝ by a constantfactor c stretches the bandwidth by c:{ˆ( )} ˆ( ){ }B S cf = cB S f . (11)Axiom 4.2.1.2 permits normalization of Ŝ by a constant k,so that kSˆ( f ) has unit area without affecting the bandwidth.Axiom 4.2.1.3 is useful in determining the bandwidth of amodulated signal.4.2.2 Root Mean SquareBandwidthA bandwidth definition that is often used in radarsignal theory is the rms bandwidth, B RMS . The Institute ofElectrical and Electronics Engineers (IEEE) has a standard(IEEE Std 686-1997) [10] that used moment theory todefine B RMS as the second moment of the square magnitudeof Ŝ about a designated frequency, typically 0 or thespectral center, where the integration is taken over allfrequencies ( −∞ < f < +∞ ). Unfortunately, this definitionis ambiguous, and has difficulties when used with UWBsignals. In particular, since the first moment about 0 Hz (themean frequency) is 0 Hz for every real-valued waveform,the notion of spectral center is not very meaningful, even for2narrowband radar signals, since Ŝ (the power spectraldensity or energy density) has a maximum value at nonzerofrequencies. To address this shortcoming, Rihaczek [9]defined the moments over the positive spectrum. Althoughthe IEEE standard is not ambiguous for signals with spectraldensities that have a unique absolute maximum at 0 Hz, theauthors followed Rihaczek’s definitions:BRMS=∞∫0( − ) ⎤ ˆmp ( )∫0( )2 2⎡2πf f S f df⎣⎦∞2 , (12)Sˆf dfwhere the denominator of the radicand is half of the signalenergy, the mean frequency, f , over positive frequenciesmpis given byfmp=∞∫0∞∫0( )( )22f Sˆf dfSˆf df, (13)and the rms band is ⎡max{ 0, fmp− 0.5BRMS}⎣ ,fmp+ 0.5BRMS⎤⎦ , because fmp− 0.5BRMScould benegative. The dimensions of B RMS are radians per second.For well-behaved spectra, f mp is usually very near to thefrequency associated with the maximum value of the energydensity, which is usually the carrier frequency for radiatednarrowband signals.4.2.3 X dB Power-Level Bandwidth[5]The X dB power-level bandwidth, B XdB , is given byBXdB = fh − fl,(14)which is implicitly defined by the lowest ( f l ) and highestf ) frequencies that solve( h( ) ˆ( )Sˆf = S f − X (15)20log10 20log10M( f)2ˆXS−⇔ = 10 102 ,Sˆ( f )MTheRadio Science Bulletin No 313 (June, 2005) 17