EELE445 Monana Sae Universiy Lecure 3,4 EE445 - Oucomes EELE445-4 Lecure 3,4 Poer, Energy, ime average operaor secion. In his lecure you: be able o use he ime average operaor [] for finie ime duraion signals and periodic signals Be able o find he value of a aveform be able o find he poer or energy of a aveform in he ime domain Parseval s heorem, find he poer or energy of a aveform in he frequency domain Concep of PSD an ESD Physically ealizable Wavefo Figure Physical and mahemaical avefo. measurable in he laboraory. aveform has significan nonzero values over a composie ime inerval for a finie ime duraion. coninuous funcion of ime 3. finie peak ampliude 4. real funcion, no complex values 3 4
EELE445 Monana Sae Universiy 5 ime average operaor ime verage Operaor mean, of : lim dc W 6 ime average operaor ime verage Operaor for signal of finie ime duraion: N n n n N a a a a a o o o any real for any real for + + + For a Periodic Signal: 7 oo Mean Square- MS roo mean square of a signal of duraion : W Poer meansquare W 8 MS o simplify he inegrals! Choose is an arbirary ime shif. here value of periodic avefo : For + +
EELE445 Monana Sae Universiy MS example MS of a sinave : [ cos ω o ] appendix 4 use : cos x + cosx MS of a SquareWave 9 Figure 3 Seady-sae aveshapes for Example. 3
EELE445 Monana Sae Universiy insananious Poer averagepoer Poer of a signal p v i v p v i eal Zload v hen Z + j Z+j Figure Polariy convenion used for volage and curren. 3 Couch, Digial and nalog Communicaion Sysems, Sevenh Ediion 7 Pearson Educaion, Inc. ll righs reserved. -3-449- verage Poer in P E P P v i i Poer of a signal v v i signal of duraion : p p is he insananeous poer, P is he average poer over ime For Normalized Poer: ohm 4 P E E v Energy of a Signal Energy in signal of duraion : v i E P We ypically calulae normalized Energy or Poer, S N V Log V communicaion meric: Signal o Noise aio in db P Log P signal noise Log s signal / db noise n 5 6 4
EELE445 Monana Sae Universiy Figure Physical and mahemaical avefo. EELE445-4 Lecure 4 he Fourier ransform and he Frequency Domain 8 Couch, Digial and nalog Communicaion Sysems, Sevenh Ediion 7 Pearson Educaion, Inc. ll righs reserved. -3-449- he Fourier ransform We use he Fourier ransform o deermine he frequencies of he sinusoids presen in a non repeiive aveform. he fourier series may be considered he fourier ransform of a non-repeiive aveform ha is repeaed coninuously. W f F Wha is : [ ] xy??? Fourier ransform j e πf Correlaion coeff C xy x y < C < 9 5
EELE445 Monana Sae Universiy Fourier ransform Figure 4 Specrum of a sine ave. W f F j πf [ ] [ ] e So he Fourier ransform is ansering he quesion: ho much of e jπf is in? e jx cos x j sin x Couch, Digial and nalog Communicaion Sysems, Sevenh Ediion 7 Pearson Educaion, Inc. ll righs reserved. -3-449- Figure 5 Waveshapes and corresponding symbolic noaion. Figure 5 Waveshapes and corresponding symbolic noaion. 3 Couch, Digial and nalog Communicaion Sysems, Sevenh Ediion 7 Pearson Educaion, Inc. ll righs reserved. -3-449- 4 Couch, Digial and nalog Communicaion Sysems, Sevenh Ediion 7 Pearson Educaion, Inc. ll righs reserved. -3-449- 6
EELE445 Monana Sae Universiy Figure 6 Specra of recangular, sin x/x, and riangular pulses. Figure 6 Specra of recangular, sin x/x, and riangular pulses. ime - bandih produc or bandih- duraion produc : B 4π he Gaussian pulse, e equaliy sign. π saifies he condiion ih he 5 Couch, Digial and nalog Communicaion Sysems, Sevenh Ediion 7 Pearson Educaion, Inc. ll righs reserved. -3-449- 6 Couch, Digial and nalog Communicaion Sysems, Sevenh Ediion 7 Pearson Educaion, Inc. ll righs reserved. -3-449- Figure 6 Specra of recangular, sin x/x, and riangular pulses. Figure 6 Specra of recangular, sin x/x, and riangular pulses. ime Domain Frequency Domain ime Domain Frequency Domain 7 Couch, Digial and nalog Communicaion Sysems, Sevenh Ediion 7 Pearson Educaion, Inc. ll righs reserved. -3-449- 8 Couch, Digial and nalog Communicaion Sysems, Sevenh Ediion 7 Pearson Educaion, Inc. ll righs reserved. -3-449- 7
EELE445 Monana Sae Universiy Fourier ransform heorems Fourier ransform heorems hese heorems are useful for he homeork and he exams! 9 3 Fourier ransform Pairs Fourier ransform Pairs he recangular funcion is used o mahemaically runcae an infinie ime aveform. I is he ime indo ha is analyzed over. Wha is he effec on he measuremen of as approaches infiniy? Ho abou as approaches? compare he Phasor and he sinusoid ransfo. 3 3 8
EELE445 Monana Sae Universiy Energy and Poer in a Waveform Energy and Poer in a Waveform Energy Specral Densiy, ESD E f W f W f e jπf Poer Specral Densiy, PSD, for a runcaed aveform of duraion 33 34 Energy and Poer in a Waveform Energy and Poer in a Waveform Figure 4 Specrum of a sine ave. Figure 9 Poer specrum of a sinusoid. 35 36 9
EELE445 Monana Sae Universiy Example- Poer and Energy in ime. EELE445-3 LECUE 5 Examples, Fourier Series from he Fourier ransform KHz sineave source is applied o a 5 ohm resisor. fine he poer delivered o he load. v 5 Ω v cosπf vol f Hz 3 sec o 38 P o o o Example- Poer and Energy in ime. For a periodic Signal v o o o o o o v + o o o o : cosπf cos4πf o o so Mean Squared vols vols P.Wa dbm 5 Fourier Series from Fourier ransfo. Find he runcaed signal period V o. Deermine he Fourier ransform of he aveform V o 3. Find he value of he Fourier Series coefficiens by:. eplacing f in he ransform ih fn/ o. Scaling he magniude of he ransform by / o Duy Cycle,D o 39 o 4
EELE445 Monana Sae Universiy Fourier Series from Fourier ransfo f Fourier Series from Fourier ransfo no make a repeiive pulse ih period o From Fourier ransform able: F Π Sa πf Le f n o and scale by o c n nπ Sa o o 4 4 Properies of a epeiive ecangular Pulse he null bandih, B n of he specrum is equal o he inverse of he pulse ih,. B n he harmonic frequencies are n/ o epeiive ecangular Pulse he number of harmonics, n, from DC o he specral null is: o n D pproximaely 9 percen of he poer of he signal is in he specrum from DC o he null bandih frequency, only % of he signal poer is in he specrum frequency range from he null bandih frequency o infiniy! 43 44
EELE445 Monana Sae Universiy epeiive ecangular Pulse Single ecangular Pulse 45 46 epeiive ecangular Pulse epeiive ecangular Pulse 47 48
EELE445 Monana Sae Universiy epeiive ecangular Pulse ecangular Pulse Poer o he firs null bandih sinc x dx 9.8 x Px sinc x : dx 49 5 epeiive ecangular Pulse Poer vs Bandih 97.475 Percen of Poer vs Bandih 9 9 8 P k 7 6 5 5.5.5.5 3 3.5 4 k 4 5 3