Characerisics o Linear Sysem h g h : Impulse response F G : Frequency ranser uncion Represenaion o Sysem in ime an requency. Low-pass iler g h G F he requency ranser uncion is he Fourier ransorm o he impulse response o a linear sysem. g F G ERG3A-I p.i-39 I-39
Banwih o he Sysem he banwih BW sys o a sysem is eine as he inerval o posiive requencies over which he magniue remains wihin a given numerical acor. A popular acor is - 3B ha is, in volage or in power, he corresponing banwih is calle he -3B banwih or hal-power banwih. W W - ERG3A-I p. I-4
Banwih o he Signal he banwih BW sig o a signal is eine as he range o posiive requencies over which he magniue F remains wihin a given numerical acor. 3-B or hal-power banwih: posiive requency range a which he ampliue specrum F rops o a value equals o F or F in power. A signal is calle ban-limie signal i F or > M. ERG3A-I p. I-4
Disorionless ransmission Oupu signal shape is exacly he same as he inpu signal shape hough heir respecive ampliues may be ieren an may be elaye in ime. g K G θ Ke Ke K F K >: consan, : ime elay e θ consan over enire requency range A phase o is linear wih requency K h g g θ Η KA Slope- ERG3A-I p. I-4
Disorionless ransmission I K is no a consan i.e. K is requency epenen Frequency componens o inpu signal are ransmie wih ieren amoun o gain aenuaion. Ampliue isorion I he phase o is no linear wih requency Frequency componens o inpu signal are ransmie wih ieren amoun o ime elay. Phase isorion Disore oupu signal ERG3A-I p. I-43
Filers F G Aer he inpu signal passes hrough he sysem, relaive ampliues o he requency componens in he inpu signal are change or some o is requency componens are suppresse ilering requency-selecive ranser uncion requency response o a sysem acs as a iler on he inpu signal ERG3A-I p. I-44
Low-Pass igh-pass Ban-Pass Ban-Sop Ieal Frequency-Selecive Filers < c > c : cuo requency < < oherwise ERG3A-I p. I-45 c < c > < < oherwise c Η c c Η c c Η Η
Non-ieal Frequency-Selecive Filers Consier a RC iler, he oupu y an he inpu x is relae by x Ri y R x y RC y ake Fourier ransorm on boh sies, gives X Y RC Y X θ an o o RC [ ] o o where o RC ERG3A-I p. I-46 x i C y o o θ
ERG3A-I p. I-47 LI Sysem escribe by Dierenial Equaions A general N-h orer linear ierenial equaion is given by N M x b y a M k k k k N k k k k where a k, b k are real consans ake Fourier ransorm on boh sies, gives N k k k M k k k M k k k N k k k a b X Y X b Y a F n n n Q
ERG3A-I p. I-48 LI Sysem escribe by Dierenial Equaions Example: A sysem is escribe by y yxx. Deermine he impulse response h. ' ' x x y y X Y X X Y Y X Y ake Fourier ransorm on boh sies, gives F n n n Q ake Inverse Fourier ransorm on boh sies, gives u e h δ a u e -a Q < > or or u
ERG3A-I p. I-49 LI Sysem escribe by Dierenial Equaions Example: Consier a coninuous-ime LI sysem escribe by x y y Fin he oupu y i inpu xe - u. ake Fourier ransorm on boh sies, gives X Y Y X Y X X Y u e e y As aking inverse Fourier ransorm, gives a u e -a Q F n n n Q
Energy Specral Densiy Parseval s heorem gives a relaion beween a ime signal an is Fourier ransorm F. F. F where is he energy per uni o requency normalize o a resisance o ohm. I is calle he energy specral ensiy o he signal represens is energy per uni o requency isplays he relaive energy conribuions o various requency componens. ERG3A-I p. I-5
Energy Specral Densiy Consier a signal is applie o he inpu o a linear ime-invarian sysem whose requency ranser uncion. he specral ensiy o he oupu is G F. he normalize energy ensiy o G is G F, an he normalize energy in he oupu signal is E g F. All phase inormaion in boh he inpu signal an he sysem ranser uncion is los in he calculaion o energy an energy ensiy. ERG3A-I p. I-5
ERG3A-I p. I-5 Consier a signal applie o he inpu o a very narrow banpass iler whose requency ranser uncion is as ollows. he energy o he oupu g o he narrow-ban iler is, F F F F F E g Energy Specral Densiy
I he signal is real-value [ * ], hen an hus Energy Specral Densiy E g F F, F * F F F hereore, E g F Noe: al o energy is conribue by he negaive requency componens an hal by he posiive requency componens i he signal is real-value. ERG3A-I p. I-53
Energy Specral Densiy: Measuremen he energy specral ensiy can be measure by a evice calle a muli-channel specrum analyzer. Noe: he area uner he energy specral ensiy gives he energy wihin a given ban o requencies. ERG3A-I p. I-54
Energy Specral Densiy: Example a A signal e u is applie o he inpu o a low-pass iler wih a magniue requency ranser uncion. b b Deermine he require relaions beween he consans a, b such ha exacly 5% o he inpu signal energy, on a -ohm basis, is ranserre o he oupu. Soluion: he energy in he inpu signal is across ohm a E e ERG3A-I p. I-55 a he energy in he oupu signal g is across ohm E g b G a b a b b. F.
Energy Specral Densiy: Example Using a able o inegrals, Since we require so ha E b b g ab a b a a E g E b. a a b a b. Solving i, we in ha he require relaion is a b. ERG3A-I p. I-56
Power Specral Densiy he ime-average power o a signal is given by P lim. assuming a -ohm resisance. his quaniy is calle he mean-square value o he signal, esignae simply as. Le s eine a uncion calle he power specral ensiy uncion, S, where P S. he power specral-ensiy uncion S is in unis o power i.e. was per z. I escribes he isribuion o power versus requency. ERG3A-I p. I-57
Power Specral Densiy Assuming is inie over he inerval -,. hen he runcae uncion rec has inie energy an is Fourier ransorm is F I{ rec }. Parseval s heorem or his runcae uncion is F. ERG3A-I p. I-58
ERG3A-I p. I-59 hus, we have. lim lim F P ence he average power P across a -ohm resisor is given by. lim F S Power Specral Densiy. lim F S
Power Specral Densiy: Perioic Funcion Assume is perioic an ha i is represene by he exponenial Fourier series n F n e. P lim lim n F n n F e n n o n he corresponing power specrum is Fn δ n n S. ERG3A-I p. I-6
Power Specral Densiy:Example Develop an expression or he power specral ensiy o A exp rec, boh or inie an as. Soluion: From able, we have I{ A rec } A Sa. Using he requency ranslaion propery o he Fourier ransorm, we ge I{ A exp rec } A Sa[ ]. hus, or a inie observaion inerval, we have S F A Sa [ ]. ERG3A-I p. I-6
Power Specral Densiy:Example As he observaion inerval is mae increasingly longer, he Sa paern becomes more an more concenrae abou an in he limi, lim F A A lim{ Sa [ lim Sa [ ]} ] A δ. ERG3A-I p. I-6
ERG3A-I p. I-63 F G F G A linear ime-invarian sysem he runcae response uncion is he power specral ensiy o he oupu signal is hen, lim lim F F S g. S S g Power Specrum o Sysem Oupu
Average Power o Sysem Oupu he average power in he oupu signal is given by P g S. Noe ha all phase inormaion in boh he inpu signal an he sysem ranser uncion is los. ERG3A-I p. I-64
ERG3A-I p. I-65 he inverse Fourier ransorm o S is calle he auocorrelaion uncion R τ o. I * * * * * * lim lim lim lim lim lim } { e e e e e F F e F S R τ τ δ τ τ τ τ τ Auocorrelaion { } τ R S I
ERG3A-I p. I-66 Deermine an skech he auocorrelaion uncion o a perioic square wave wih peak-o-peak ampliue A, perio, an mean value A. Soluion: : For < τ < τ τ τ 4 4, A A R : For < τ < τ τ τ 4 4, A A R Auocorrelaion: Example
Auocorrelaion: Example Fin he auocorrelaion uncion o cos θ. Soluion: R τ cos cos cos τ. θ cos θ cos τ θ τ θ Noe ha he auocorrelaion uncion is inepenen o he phase θ. ERG3A-I p. I-67
Auocorrelaion: Applicaion Deec or recognize signals ha are maske by aiive noise. Drawbacks: Presence o noise in he oupu an los o relaive ime shi. ERG3A-I p. I-68
Crosscorrelaion he cross-correlaion uncion R g τ is eine as * R g τ lim g γ. Consier he waveorm g g n. ERG3A-I p. I-69
Correlaion: Properies Symmery R τ R * τ. Perioiciy R τ R τ Mean-Square Value R Maximum Value R τ R. Average Value Consier x m m R g τ m m, g y x an y are uncions wih zero average value. hen, he average value o he cross-correlaion uncion is ERG3A-I p. I-7
Correlaion: Properies Aiiviy Consier z x y he corresponing auocorrelaion uncion is * * Rz τ lim [ x y ][ x τ R τ R τ R τ R τ. x y xy yx y τ] I x an y are uncorrelae, R z τ R τ R τ. x y ERG3A-I p. I-7
Correlaion or Finie Energy Signal he auocorrelaion uncion R τ or a signal o inie energy is R τ * τ. For signals an g, boh o inie energy, he cross-correlaion uncion R g τ is eine as R g * τ g τ. he Fourier ransorm o he auocorrelaion uncion or inie-energy signals is he energy specral ensiy I{ R τ } F. ERG3A-I p. I-7
ime-average Noise Represenaions Flucuaions in volage or curren en o obscure he original signals are commonly calle noise. Le n be a noise volage or curren again, assume a -ohm resisive loa.. Mean value, n : n lim n.. Mean-square value, n : n lim n. ERG3A-I p. I-73
Ban-Limie Whie Noise Whie noise: power specral ensiy is uniorm over enire requency range S n η or all, Wasz S n given ha n has zero mean value. η Because our measuremen evices are resrice o inie banwihs, he whie noise presene a he oupu is ban-limie. Across a banwih B, he noise power P n is B Pn η ηb Wa B Assuming ha his is evelope across a resisor R, he mean-square noise volage is n RPn ηrb Vol I n is a curren, hen n Pn R ηb R Amp ERG3A-I p. I-74
ERG3A-I p. I-75 Ban-Limie Whie Noise Suppose ha he whie noise n i is ransmie hrough a linear imeinvarian sysem wih ranser uncion. he rms o oupu noise is given by. S S n i o n n o Since he power specral ensiy o he noise is whie, we have. η η n o
Equivalen Noise Banwih he equivalen noise banwih B N is eine such ha a he power specral ensiy a he iler oupu is whie wihin he banwih B N an zero elsewhere, orming an equivalen recangular specral ensiy; b he area uner his recangular specral ensiy is equal o he area uner he specral ensiy a he iler oupu. ERG3A-I p. I-76
ERG3A-I p. I-77 Equivalen Noise Banwih Le be he miban requency o he sysem,. N B B o B n N N η η B N η η Qn o o Η o Β Ν Η o or low-pass iler
Equivalen Noise Banwih Example: Compuer he 3B banwih an he equivalen noise banwih o a iler wih he ollowing magniue ranser characerisic: 4 As, he 3B banwih is oun by solving 3B or 3B.59z 3 B Using B N 4 8.77 z ERG3A-I p. I-78
Signal-o-Noise Raio SNR I is a imensionless raio o signal power o noise power S N s n. I is quie common o express he signal-o-noise raio in ecibels [ ] [ S N B log s n ]. * Boh o hese mean-square values assume zero mean value unless sae oherwise. SNR is commonly use o escribe he qualiy o a signal. ERG3A-I p. I-79
Noise Figure Le he inpu an oupu signal volages or currens in a given sysem be s i, s o respecively an he inpu an oupu noise volages or currens be n i, n o. he inpu signal-o-noise raio is S N i si ni, he oupu signal-o-noise raio is S N s n, o o o SN i h SN o he noise igure F is eine o be he raio o he inpu signal-onoise raio o he oupu signal-o-noise raio: S N i F. S N Noise igure expresse in ecibel, i.e. logf is commonly use o escribe he noise perormance o a sysem. For ieal sysem no noise F ERG3A-I p. I-8 o