EELE Lecture 8 Example of Fourier Series for a Triangle from the Fourier Transform. Homework password is: 14445

Transcription

1 EELE445-4 Lecure 8 Eample o Fourier Series or a riangle rom he Fourier ransorm Homework password is: 4445

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3 EELE445-4 Lecure 8 LI Sysems and Filers 5 LI Sysem 6 3

4 Linear ime-invarian Sysem Deiniion o a Linear Sysem: or L y [], he linear dierenial equaion L[ a + bz ] al[ ] + al[ z ] sysem operaor : ime Invarian : he delayed oupu y by he same amoun as he inpu, he shape o he oupu waveorm is independan o when he inpu waveorm is applied o he sysem.. is delayed 7 LI Sysem y Ry τ lim Ry τ lim P F y τ h τ dτ y y τ dτ ime avg or h u u du [ R τ ] PSD o he oupu specrum y power auocor h v v dv d 8 4

5 5 LI Sysem oupu Power and Volage Specrum [ ] H X y h y H P P R F P y y y τ 9 Couch, Digial and Analog Communicaion Sysems, Sevenh Ediion 7 Pearson Educaion, Inc. All righs reserved Figure 7 Convoluion o a recangle and an eponenial. λ λ λ d w w w w w 3

6 Power Auocorrelaion: R power Power Signal hrough a Filer: 6

7 7 Periodic Signal hrough a Filer: ime Average Auocorrelaion a periodic signal or o o o o o o o o k k k o k d R d k k Lim d k Lim d Lim R τ τ τ τ τ τ 3 Periodic Signal hrough a Filer: ime Average Auocorrelaion 4

8 Periodic Signal hrough a Filer ime Average Auocorrelaion hrough a LI Sysem 5 EELE445-4 Lecure 9 LI Sysems and Filers 6 8

9 Filers-Applicaions Conrol he Modulaion Signal Bandwidh E. Null bandwidh o recangular pulse Limis ransmission Bandwidh or Specrum Conservaion Opimize he SNR a he receiver inpu Waveorm Shaping: y*h Mached deecion 7 Ideal Filer Response X h, H y Y H H Linear phase: θ ω θ τ d ω 8 9

10 Ideal Filer-Magniude, Phase/Delay ω θ π θ ω π π j j j e G e j H j H For a ranser uncion Hs, a real requencies, wih sjω, Where Gω and θω are he gain and phase componens. 9 Ideal Filer-Magniude, Phase, Delay Pd d d π τ θ ω ω θ π θ τ ω ω θ π θ Phase Linear Group Delay ime delay Phase or

11 Ideal Filer-Magniude, Phase/Delay Boh Pd and τ d are uncions o requency Phase delay Pd is he absolue delay and is o lile signiicance Group Delay τ d is used as he crierion o evaluae phase nonlineariy. Group Delay is consan or all requencies in an ideal iler. Ideal Filer-Magniude, Phase/Delay Linear phase variaion wih requency over a band o requencies implies a consan Group Delay no phase disorion in ha band o requencies In order o preserve he inegriy o a pulse, i is mandaory ha he Group Delay o he sysem be consan up o he maimum requency componen o he pulse. his implies equal ime delay or all requencies o ineres.

12 hp://en.wikipedia.org/wiki/buerworh_iler Linear analog elecronic ilers Buerworh iler Chebyshev iler Ellipic Cauer iler Bessel iler Gaussian iler Opimum "L" Legendre iler 3 Filer Eample, ime Domain: Hs Hs V Vy : + RC s h RC e RC τ e τ 4

13 Filer Eample, Frequency Domain: Le sjπjω H ω H an ω R C an θω + + j ω ω o j o θ an o ω o : RC o : π R C ω ω o 5 Filer Eample: he Group delay o he RC low pass is: dτ d ω ω τ d ω dω ω + ω τ d + 6 3

14 Figure 5 Characerisics o an RC low-pass iler. Couch, Digial and Analog Communicaion Sysems, Sevenh Ediion 7 Pearson Educaion, Inc. All righs reserved Figure 6 Disorion caused by an RC low-pass iler. Couch, Digial and Analog Communicaion Sysems, Sevenh Ediion 7 Pearson Educaion, Inc. All righs reserved

15 EELE445-3 Lecure Filers con d and Noise 9 Figure 6 Disorion caused by an RC low-pass iler. Couch, Digial and Analog Communicaion Sysems, Sevenh Ediion 7 Pearson Educaion, Inc. All righs reserved

16 Figure 6 Disorion caused by an RC low-pass iler. Couch, Digial and Analog Communicaion Sysems, Sevenh Ediion 7 Pearson Educaion, Inc. All righs reserved A RC Filer Disorion Problem: Assume we wan he ampliude Lineariy <% and he group delay variaion lineariy <5% Find he usable bandwidh o he s order Buerworh iler i he 3dB bandwidh is MHz 3 6

17 A LPF Disorion Problem: Consrains: H H a H τ d τ d φ τ d ε a. ε φ.5 % Volage ampliude error 5% delay variaion o 6 : τ : π o H : + j o τ d : o π + o 33 Ampliude Error: ε a a A LPF Disorion Problem: H a : ε H φ p..95 a + o o Phase Error: : o τ d p τ d p + o a : o.98 o p :.95 o a.3 5 Hz p.94 5 Hz 34 7

18 A LPF Disorion Problem: So he ampliude error will limi he usable bandwidh o 3 KHz V/V H Filer Magniude Response Hz 35 A LPF Disorion Problem: Degrees 9 arg H 8 π Filer Phase Response Hz 36 8

19 A LPF Disorion Problem: usec.59 τ.d Filer Group Delay Response Hz 37 A LPF Disorion Problem: Filer Ampliude and Phase Error..3 5 Error ε.a ε.φ Hz Ampliude Error Group Delay Error 38 9

20 W noise 39 Filer Noise Equivalen Bandwidh We oen equae he -3dB bandwidh o a real Filer o he bandwidh o an ideal iler ha would Pass he same noise power. W 3dB -3dB Whie Noise Filer Noise Equivalen Bandwidh Real Filer 3 db S n Ideal Filer W Find W For Equal Powers From Filers S n is a Whie Noise Power Densiy N o Was/Hz

21 P Filer Noise Equivalen Bandwidh S n H d N o H d P S n Π W eq d W eq W eq N o d N o W eq W eq H d Buerworh lowpass ilers H + o n Where n is he iler order number o poles Noe ha n is he RC lowpass

22 Buerworh lowpass ilers Buerworh lowpass ilers Passband n 6 db per ocive n db per decade

23 Buerworh lowpass ilers EQUIVELEN NOISE BANDWIH FOR BUERWORH FILERS B Noise_Power_From_Real_Filer_Wih_3db_Bandwidh_o_ Noise_Power_From_Ideal_Filer_Wih_Bandwidh_o_ n :.. 6 iler orders rom o 6 n d + B n : Π, d Firs order iler wih a 3dB bandwidh o passes 57% more noise power when compared wih an ideal iler wih a bandwidh o. A 3rd order iler only passes 4.7% more noise power when compared wih an ideal iler. noe ha: n B n d π π.57 3