2.2 Fourier transform and Spectra Dirac delta function Definition 1 a(to(t)dt=O(0 Definition2both∫。6(t)lt=1and x 6(x)= 0 x+o need to be satisfied .. Definition 3 8(x)=[0 etJ2x dy e Sifting property: ∫。o(x)6(x-x0)akx=0(x0) 16
16 2.2 Fourier Transform and Spectra • Dirac delta function • Definition 1 ( ) ( ) =(0) − t t dt • Definition 2 both ( ) =1 − t dt and = = 0 0 0 ( ) x x x need to be satisfied • Definition 3 − x = e dy j2xy ( ) Sifting property: ( ) ( ) ( ) 0 0 x x − x dx = x −
2.2 Fourier transform and Spectra Unit step function t>0 l(t)= 0t<0 s· Relationship between u(t)andδ(t) ∫(x)dr=() and du(t) =(t) 17
17 2.2 Fourier Transform and Spectra • Unit step function = 0 0 1 0 ( ) t t u t • Relationship between u(t) and δ(t) (x)dx u(t) t = − and ( ) ( ) t dt du t =
2.2 Fourier Transform and Spectra Time Domain 叫() T Sa(nrn Some useful pulses 1.0 ∏() J I 1sT/2 (a) Rectangular Pulse and Its Spectrum 0|>T/2 wSA(oWn) SInx Sa(x) x 1/7≤7m- (e)Trangular Pulse and Its Spectrum Figure 2-6 Spectra of rectangular, (sin x)/x, and triangular pulses
18 2.2 Fourier Transform and Spectra • Some useful pulses − = = = t T t T t T T t x x Sa x t T t T T t 0 1 | | / ( ) sin ( ) 0 / 2 1 / 2 ( ) • Fig.2-6 p56
2.3 Power Spectral Density and Autocorrelation Function Power Spectral Density(PSD) The normalized power of a waveform will now be related to its frequency domain description by the use of a function called the psd. The psd is very useful in describing how the power content of signals and noise is affected by filters and other devices in communication system 19
19 2.3 Power Spectral Density and Autocorrelation Function • Power Spectral Density(PSD) The normalized power of a waveform will now be related to its frequency domain description by the use of a function called the PSD. The PSD is very useful in describing how the power content of signals and noise is affected by filters and other devices in communication system
2.3 Power Spectral Density and Autocorrelation Function The average normalized power in time domain dc escription iS T/2 1→+oTm/2o2(t)od=lm OT (tdt T→∞T Where ot(t=o(tII(t/T)is the truncated version of the waveform. By the use of parseval theorem. we have. w(f P= lim Im J(2-65) T→∞T 20
20 2.3 Power Spectral Density and Autocorrelation Function • The average normalized power in time domain description is: − → − → = = t dt T t dt T P T T T T T ( ) 1 ( ) lim 1 lim / 2 2 / 2 2 Where ωT(t)= ω(t)Π(t/T) is the truncated version of the waveform. By the use of Parseval’s theorem, we have: − → → − = = (2 - 65) ( ) ( ) lim 1 lim 2 2 d f T W f W f d f T P T T