CHAPTER 2 Signals And Spectra

Transcription

1 CHAPER Signals And Specra Properies of Signals and Noise In communicaion sysems he received waveform is usually caegorized ino he desired par conaining he informaion, and he undesired par. he desired par is called he signal Undesired par is called he noise. he chaper develops mahemaical ools ha are used o describe signals and noise from a deerminisic waveform poin of view. Properies for characerizing hese waveforms such as DC value, roo mean square (rms) value, normalized power, magniude specrum, phase specrum, power specral densiy and bandwih will be sudied. Physically realizable waveforms saisfy he following properies:. significan nonzero values over a composie ime inerval which is finie. he specrum has significan nonzero values over a composie frequency inerval which is finie 3. a coninuous funcion of ime 4. finie peak value 5. has only real values (no complex values for any ime) Someime we develop mahemaical models o approximae he physical signals. (power v.s. energy signal) Summer 3

2 ime Average Operaor he ime average operaor is given by / [] = lim [] / he ime average operaor is linear since he average of he sum of wo quaniies is he same as he sum of heir averages. a w ( + a w ( = a w ( + a w ( () If he waveform is periodic wih period hen ( ) for all w( = w + () For example a sinusoidal waveform of frequency f Hz is periodic since i saisfies eq.(). From his definiion i is clear ha periodic waveform will have significan values over an infinie ime inerval (-,). herefore physical waveforms can no be ruly periodic (finie ime inerval!). However hey can have periodic values over a finie ime inerval. If he waveform is periodic, he ime average operaor can be reduced o: / + a [] = [] / + a (3) where a is an arbirary real consan which may be aken as zero also. Summer 3

3 DC Value he dc value of a waveform w( is given by is ime average w ( Hence W dc = lim / / w( For a physical waveform we are ineresed in evaluaing he dc value only over a finie inerval of ineres, say from o, so ha he dc value is w( Power In communicaion sysems, if he received (average) signal power is sufficienly large compared o he (average) noise power, informaion may be recovered. his concep was demonsraed by he Shannon channel capaciy formula. Hence average power is an imporan concep ha needs o be clearly undersood. If we assume ha v( is he volage across a se of circui erminals and i( is he curren hrough he erminal, he insananeous power in he circui is given by: p( = v(i( he average power is given by : P = p( = v( i( Summer 3

4 Example : Evaluaion of power he circui in Fig.. conains -V, 6Hz fluorescen lamp wired in a high-power-facor configuraion. Assume ha he volage and curren are boh sinusoids and in phase (uniy power facor) as shown in Fig..3. he DC value of his (periodic) volage waveform is V dc / = v( = V cosw = V cosw = / where; w = π / and f = / 6Hz = / Similarly, I = i( = I cosw = cos = I w dc / he insananeous power is: p( = [Vcos(w ][Icos(w ] = /VI [+cos w ] he average power is / VI P = VI w VI = ( + cow = ( + cosw / Hin : Use figure.3 Summer 3

5 RMS Value and Normalized Power he roo mean square ( rms ) value of w( is given by W rms = w ( Average Normalized power is given by: P = w / () = lim w () / If a load is resisive (i.e. uniy power facor) he average power is given by v ( V R rms P = = i ( R = = I R = rms R here R is he value of he resisive load. V rms I rms Summer 3

6 Energy and Power Waveforms If a signal w( is classified as a power waveform hen he normalized average power P, is finie and nonzero (i.e. < P < ) If he signal w( is an energy waveform hen he oal normalized energy is finie and nonzero ( < E < ) he oal normalized energy is given by E = lim / w / () If a waveform is classified as eiher one of hese ypes i can no be he oher ype. I is also possible o have boh infinie power and energy : his implies he signal will be neiher power nor energy signal. Summer 3

7 Decibel he decibel is a base logarihmic measure of power raios. he decibel gain of a circui is given by : P db = log P ou in Similarly, he decibel signal-o-noise raio is given by: ( S / N ) P = log P signal noise = log db s n ( () Decibel Power Level he decibel power level wih respec o mw is given by acualpowerlevel( was) dbm = log 3 = 3 + log [ acual power level( was) ] he m in he dbm denoes he milliwa reference. Summer 3

8 Fourier ransform & Specra How does one find he frequencies which are presen in a waveform? A: ake he specrum of he signal he Fourier ransform (F) of a waveform w( is given by W [ w( ] = [ w( ] j πf ( f ) = F e () W(f) is he wo sided specrum of w( since boh he negaive and posiive frequencies are obained from he Fourier ransform. Inverse Fourier ransform (IF) he inverse Fourier ransform is given by : w( [ W ( f )] = W ( f ) j πf = F e df A Waveform w( is Fourier ransformable if i saisfies he wo Dirichle condiions (sufficien condiions) :. Over any ime inerval of finie wih, he funcion w( is single valued wih a finie number of maxima and minima, and he number of disconinuiies is finie.. w( is absoluely inegrable ( w ( < ) A weaker sufficien condiion for he exisence of he Fourier ransform is E = w( < where E is he normalized energy. Show Example.: Specrum of an Exponenial Pulse Pages45-46 Summer 3

9 Properies of Fourier ransform If w( is real, hen W(f) is conjugae symmeric: W(-f) = W*(f) he aserisk superscrip implies he conjugae operaion his implies ha he magniude specrum is even abou he origin. ( f ) W ( f ) W = whereas he phase specrum (angle) is odd abou he origin θ ( f ) = θ ( f ) Parseval s heorem and Energy Specral Densiy Parsevals heorem w () w () W ( f ) W ( f ) = df Rayleigh s heorem ( for w ( w ( = w( () W ( f ) w = df Energy Specral Densiy (ESD) ξ ( f ) = W ( f ) = ) ESD has he unis of joules/hers. he area under i gives he oal normalized energy. Summer 3

10 If we use Parseval s heorem he oal normalized energy is given by he area under he ESD funcion : E = ξ ( f ) df able. on page 5 can be used o evaluae oher Fourier ransform problems. Afer obaining he resul, hese properies mus be validaed o ensure correc evaluaion.. W ( f ) = W ( f ) or W ( f ) is even and ( f ). W ( f ) is real when w( is even 3. W ( f ) is imaginary when w( is odd θ is odd Dirac Dela Funcion and Uni Sep Funcion he dirac dela funcion δ(x) is defined by ( x) δ ( x) dx w( ) w = where, w(x) is any funcion which is coninuous a x =. Alernaively, he dela funcion is defined using and δ δ ( x) dx = ( x) he uni sep funcion u( is x = = x u () = > < Summer 3

11 he uni sep and he dela funcion are relaed by he following equaion: du( = δ () Show Example.4 Specrum of a Sinusoid ( on page 5) Recangular, Sa(.), and riangular Pulses he recangular pulse is defined as Π, =, / > / he Sa(.) funcion is defined as Sa ( x) sin x = x he riangular funcion is defined as Λ, > Summer 3

12 he wave-shapes for he above expressions are as shown below. (a) Recangular Pulse Π -/ / (a) riangular Funcion Λ - Summer 3

13 (a) Sa(x) Funcion Sa( x) = sin x x x = sinc π. -3π -π -π π π 3π Show Example -5 : Specrum of a Recangular Pulse (page 54-55) he specra obained in Example.5 is real because ime domain pulses are real and even. If he pulses are offse in ime o desroy he even symmery, he specra will be complex. For example if he pulse is represened as below: v (), =, < < / = Π elsewhere hen using he delay propery he specrum may be wrien as jπ V ( f ) = e Sa( πf ) Summer 3

14 e erms of he quadraure noaion becomes V ( f ) = [ Sa( πf )cos( πf )] + j[ Sa( πf )sin( πf )] X ( f ) Y ( f ) Show Example -6: Specrum of a riangular Pulse (page 57) Summer 3