s 2.1 Properties of signal and Noise Phasors A complex number c is said te be phasor if it is used to represent a sinusoidal waveform. That is (t)= ccos(ot+∠c) Where c=X+jy=clejo 25sin(2n500t+450)could be denoted by the phasor 25∠-45 0 10cos(ot+35) could be denoted by the phasor10∠35 0
16 2.1 Properties of signal and Noise • Phasors A complex number c is said to be phasor if it is used to represent a sinusoidal waveform. That is ( ) | | cos( ) 0 t c t c Where c=x+jy=|c|e jΦ 25sin(2π500t+450) could be denoted by the phasor 0 25 45 10cos(ωt+350) could be denoted by the phasor 0 1035
2.2 Fourier transform and Spectra The fourier transform of a waveform o(t) is: W()=o()= o(t)e j2nft Where f is the frequency parameter with units of Hz(1/s), w(f is also called a two sided spectrum of o(t). 17
17 2.2 Fourier Transform and Spectra • The Fourier Transform of a waveform ω(t) is: W f t t e dt j ft 2 ( ) F[ ( )] ( ) Where f is the frequency parameter with units of Hz(1/s), W(f) is also called a two- sided spectrum of ω(t)
2.2 Fourier transform and Spectra It should be clear that the spectrum of a voltage(or current waveform is obtained by a mathematical calculation and that it does not appear physically in an actual circuit, f is just a parameter that determines which point of the spectral function is to be evaluated
18 2.2 Fourier Transform and Spectra It should be clear that the spectrum of a voltage(or current)waveform is obtained by a mathematical calculation and that it does not appear physically in an actual circuit, f is just a parameter that determines which point of the spectral function is to be evaluated
2.2 Fourier transform and Spectra Parseval's theorem ∫∞o1(0o2(t=W1()W2()d where W(f)=[01(t)],W2(=[02(t) when Rayleigh's energy theorem is obtained: E=∫o()2h=W()2 Energy spectrum density is: E(O)=
19 2.2 Fourier Transform and Spectra • Parseval’s theorem (t) (t)dt W ( f )W ( f )df * 1 2 * 1 2 ( ) [ ( )], ( ) [ ( )] 1 1 2 2 where W f F t W f F t when ( ) ( ) ( ) 1 2 t t t Rayleigh’s energy theorem is obtained: E t dt W f df 2 2 1 ( ) ( ) Energy spectrum density is: 2 E( f ) W ( f )
2.2 Fourier transform and Spectra Dirac delta function · Definition 1∫o()()dr=o() Definition 2 both o &(tdt=1 and 8(x)-0 x*o need to be satisfied · Definition3o(x)=∫e/by Sifting property ∫。o(x)8(x-x0)dx=o(x0)
20 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