Radio Science Bulletin 313 - June 2005 - URSI

Bandwidths f l / f c f h / f c B F Classification E relB 3dB 0.410 1.30 1.04 UWB / SHB 79.05 %B 10dB 0.170 1.65 1.63 UWB / HB 97.18 %B 20dB 0.050 2.62 1.93 UWB / HB 99.58 %B RMS 0.000 2.24 2.00 UWB / HB 99.04 %B 90EB 0.305 1.445 1.30 UWB / SHB 90 %B 99EB 0.051 2.215 1.91 UWB / HB 99 %Table 5. Classification of the one-cycle sine for six bandwidth definitions.UWB signals provides incomplete and fragmentarycharacterizations of the signals. Therefore, additionalparameters, like the waviness (ripple), which characterizesthe variation of the amplitude in the specified band, areneeded for a more complete understanding of signalcharacterizations.5.4 Linear Frequency-ModulatedSine (Chirp)2⎧ ( f − fc)− jπ⎪ fcµ⎨e ⎡⎣−C( x1) − jS( x1) + C( x2) + jS( x2)⎤⎦⎪⎩2( f + fc)⎫jπfcµ⎪−e ⎡⎣C( x3) − jS( x3) − C( x4) + jS( x4)⎤⎬ ⎦ ,(21b) ⎪⎭A classical radar signal is the linear frequencymodulated(LFM) waveform, which is implemented bymodulating the sine function to increase the signalbandwidth. This waveform and pulse compression in aradar receiver are used to simultaneously obtain the energyof a long-duration signal and the resolution of a high-energyshort-duration signal. Pulse compression “is implementedin high-power radar applications that are limited by voltagebreakdown if a short-pulse were to be used” [13]. The linearfrequency-modulated waveform, also called chirpmodulation, sweeps linearly over a frequency band duringa given time interval, and is given by [14]⎡ ⎛ 1 2 ⎞⎤s() t = s0sin ⎢2π fc⎜t+ µ t ⎟ ⎡σ() t −σ( t−Tn)⎤2⎥⎣ ⎦,⎣ ⎝ ⎠⎦(21a)with the pulse durationTn1 ⎛ nµ⎞= − 1+ 1+µ ⎜f ⎟,⎝c ⎠where n is the number of half cycles, s 0 is the nominalamplitude, and m is the chirp rate. For 0 ≤t≤ Tn, theinstantaneous frequency, f () t , is f c + µ t , which variesfrom f c (the carrier and starting frequency) to fc+ µ Tn.Evaluating the Fourier transform yields1Sˆ( f ) =2j 2f c µwith2cf − f + nfcµx1 = 2,f µxx3 2cf − f= ,f µ2 2cc2cf + f + nfcµ= ,f µxcf + f= ,f µ4 2ccwhere C()x and S()x are the Fresnel integralsx⎛π2 ⎞C() x = ∫ cos⎜ y ⎟dy⎝ 2 ⎠,0x⎛π2 ⎞S( x) = ∫ sin ⎜ y ⎟dy.⎝ 2 ⎠0The relatively constant run of the in-band magnitudespectrum is the main advantage of the linear frequencymodulatedwaveform for practical applications. TheTheRadio Science Bulletin No 313 (June, 2005) 23