s 2.1 Properties of signal and Noise Decibel signal-to-noise ratio is (s N)dB=10 logl signa noise S 10 logl 20lo rms-signa rs-nolse
11 2.1 Properties of signal and Noise • Decibel signal-to-noise ratio is = noise signal dB P P (S / N) 10 log = ( ) ( ) 10 log 2 2 n t s t = − − rms noise rms signal V V 20 log
2.1 Properties of signal and Noise Decibel measure may also be used to indicate absolute levels of power with respect to some reference level for example, when ImW reference level is used d Bm=10 log/ actual power level(watts) 30+10 log actual power level (watts) When 1w reference level is used the db level is denoted dBw when lkW reference level is used, the dB level is denoted dBk When 1 millivolt rms level across a 75 Ohm load is used as a reference. db level is called dBmy and is defined as dBm=2010g(3) 10
12 2.1 Properties of signal and Noise • Decibel measure may also be used to indicate absolute levels of power with respect to some reference level ，for example, when 1mW reference level is used 30 10 log(actualpower level(watts)) 10 actualpower level(watts) 10 log 3 = + = − dBm • When 1W reference level is used, the dB level is denoted dBW, • when 1kW reference level is used , the dB level is denoted dBk • When 1 millivolt rms level across a 75 Ohm load is used as a reference, dB level is called dBmV, and is defined as ) 10 20 log( −3 = Vrms dBmV
as 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 O(t)= CCOS(o0t+∠c) Where c=xtjy=cel 25sin(2500t+45)could be denoted by the 0 f=, phasor25∠-45 10cos(at+35)could be denoted by the phasor10∠350 13
13 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|ejΦ 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()=()-=mo(l2 at Where f'is the frequency parameter with units of Hz(1/s), w(f is also called a two-sided spectrum of o(t). 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
14 2.2 Fourier Transform and Spectra • The Fourier Transform of a waveform ω(t) is: − − W f = t = t e dt j f t 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). 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 theoren 。1(1)o2()dt=JW1(2()f where WI(=Flo,tlw2o=Flo,(t) when 0(1)=O1(1)=02(t) Rayleigh's energy theorem is obtained: E=」o1(o)at=W() Energy spectrum density is: E(f)=|( 15
15 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 )