Communication Systems, 5e

Transcription

1 Communicaion Sysems, 5e Chaper 3: Signal Transmission and Filering A. Bruce Carlson Paul B. Crilly The McGraw-Hill Companies

2 Chaper 3: Signal Transmission and Filering esponse of LTI sysems Signal disorion Transmission Loss and decibels Filers and filering Quadraure filers and Hilber ransform Correlaion and specral densiy The McGraw-Hill Companies

3 Filer ypes Low pass: rejecs high frequencies High pass: rejecs low frequencies Band pass: rejecs frequencies above and below some limis Noch: rejecs one frequency Band rejec: rejecs frequencies beween wo limis The McGraw-Hill Companies

4 Filer Design Noes Buerworh Filer Definiion Poles on he uni circle Frequency Scaling Acive Audio Filer Implemenaions One Pole Op Amp design Sallen-Key LPF Acive Filer -pole filer implemenaion per Op Amp (non-invering) Muliple Feedback (MFB) Circui Lowpass Filer Alernae -pole design (invering) Cascading sages for higher order filers Texas Insrumens, Acive Low-Pass Filer Design, Applicaion epor, SLOA49B Passive LC filer T and Pi Filers Buy i from Coilcraf 4

5 Buerworh Low Pass Filer H jw H jw n H s s j w j n n w n s w s w w n n Aenuaion (db) s order nd order 3rd order 4h order Buerworh Filer Family 5h order Frequency (normalized) Maximally Fla, Smooh oll-off, Consan 3dB poin for all orders M.E. Van Valkenburg, Analog Filer Design, Oxford Univ. Press, 98. SBN:

6 Buerworh Filer PSD Buerworh Filer Family - Aenuaion (db) s order nd order 3rd order 4h order 5h order Frequency (normalized) 6

7 Buerworh Filer PSD () Buerworh Filer Family - - Aenuaion (db) s order nd order 3rd order 4h order 5h order -9 - Frequency (normalized) 7

8 Malab Scrip: BuerPlo.m % % Buerworh filer plos % freqrange = logspace(-,3,4)'; wrange=*pi*freqrange; [B,A]=buer(,*pi,'s'); [H] = freqs(b,a,wrange); [B,A]=buer(,*pi,'s'); [H] = freqs(b,a,wrange); [B3,A3]=buer(3,*pi,'s'); [H3] = freqs(b3,a3,wrange); [B4,A4]=buer(4,*pi,'s'); [H4] = freqs(b4,a4,wrange); figure() semilogx(freqrange,db(psdg(hmarix))); grid ile('buerworh Filer Family'); xlabel('frequency (normalized)'); ylabel('aenuaion (db)'); legend('s order','nd order','3rd order','4h order','5h order','locaion','souhwes'); axis([^- ^3-3]); figure() semilogx(freqrange,db(psdg(hmarix))); grid ile('buerworh Filer Family'); xlabel('frequency (normalized)'); ylabel('aenuaion (db)'); legend('s order','nd order','3rd order','4h order','5h order','locaion','souhwes'); axis([^- 3-9 ]); [B5,A5]=buer(5,*pi,'s'); [H5] = freqs(b5,a5,wrange); Hmarix=[H H H3 H4 H5]; 8

9 Chebyshev Type I Filer PSD (ChebyPlo.m) Chebyshev Type I Filer Family - Aenuaion (db) s order nd order - 3rd order 4h order 5h order Frequency (normalized) 9

10 Chebyshev Type I Filer PSD () Chebyshev Type I Filer Family - - Aenuaion (db) s order -7 nd order 3rd order -8 4h order 5h order -9 - Frequency (normalized)

11 Available MATLAB Filers (Signal Proc. TB) Analog or Digial Buerworh Chebyshev Type I Chebyshev Type II Ellipic or Cauer Bessel Digial barhannwin barle blackman blackmanharris bohmanwin chebwin flaopwin gausswin hamming hann kaiser nuallwin parzenwin recwin riang ukeywin

12 Analog Lowpass Filer Design Buerworh Monoonic Decreasing Magniude All poles Chebyshev (Cheby Type ) Passband ipple All poles Inverse Chebyshev (Cheby Type) Sopband ipple Ellipical or Cauer Filer Passband ipple Sopband ipple Bessel Filer Linear Phase Maximized Buerworh Order Predicaion Filer Order = 4 3dB BW = Hz Bessel Order Predicaion Filer Order = 4 3dB BW = Hz Chebyshev Type I Order Predicaion Filer Order = 3 3dB BW = Hz Filer Comparison: Magniude Chebyshev Type II Order Predicaion Filer Order = 3 3dB BW = Hz Ellipical or Cauer Order Predicaion Filer Order = 3 3dB BW = Hz Buer Bessel Cheby Cheby Ellip Spec

13 Malab Filer Generaion () Passband Sopband Passband ipple (db) Sopband ipple (db) fpass=4; fsop=6; AlphaPass=.5; AlphaSop=6; w#### = x pi x f#### Filer Order and oher design parameers [Nbuer, Wnbuer] = buord(wpass, wsop, AlphaPass, AlphaSop,'s'); [Ncheby, Wncheby] = chebord(wpass, wsop, AlphaPass, AlphaSop,'s'); [Ncheby, Wncheby] = chebord(wpass, wsop, AlphaPass, AlphaSop,'s'); [Nellip, Wnellip] = ellipord(wpass, wsop, AlphaPass, AlphaSop,'s'); 3

14 Malab Filer Generaion () Filer Transfer Funcion Generaion [numbuer,denbuer] = buer(nbuer,wnbuer,'low','s') [numbesself,denbesself] = besself(nbuer,wnbuer) [numcheby,dencheby] = cheby(ncheby,alphapass, Wncheby,'low','s') [numcheby,dencheby] = cheby(ncheby,alphasop, Wncheby,'low','s') [numellip,denellip] = ellip(nellip,alphapass,alphasop, Wnellip,'low','s'); Specral esponse from Transfer Funcion [Specbuer]=freqs(numbuer,denbuer,wspace); [Specbesself]=freqs(numbesself,denbesself,wspace); [Speccheby]=freqs(numcheby,dencheby,wspace); [Speccheby]=freqs(numcheby,dencheby,wspace); [Specellip]=freqs(numellip,denellip,wspace); 4

15 Malab Filer Generaion (3) figure() semilogx((fspace),db(psdg([specbuer Specbesself Speccheby Speccheby Specellip])),... specfreq,specmag,'k-.',specfreq,specmag,'k-.',specfreq3,specmag3,'k-.'); ile('filer Comparison: Magniude') legend('buer','bessel','cheby','cheby','ellip','spec') - -4 Filer Comparison: Magniude Buer Bessel Cheby Cheby Ellip Spec

16 Using he resuls un using AnalogFilerCompare %% % Analog Filer Comparisons % clear close all clc fprinf('\nfiler Comparison\n'); ffsize=496; fpass= 4; fsop= 6; AlphaPass=.5; AlphaSop=6; Phase (deg) Magniude (db) Buerworh Filer Bode Plo Frequency (rad/s) Buerworh Order Predicaion Filer Order = 6 3dB BW = Hz hbuer =.33e (s^ + 6.4e4s +.e9) (s^ e4s +.e9) (s^ +.646e4s +.e9) 6

17 Using he esuls Buerworh Order Predicaion Filer Order = 6 3dB BW = Hz Bessel Order Predicaion Filer Order = 6 3dB BW = Hz Chebyshev Type I Order Predicaion Filer Order = 5 3dB BW = 4 Hz Build a 6 h Order Sallen-Key Buerworh Low Pass Filer 3 db cuoff khz [numbuer,denbuer] = buer(nbuer,wnbuer,'low','s') roos(denbuer) % Ge he roos of he denomenaor ans = Chebyshev Type II Order Predicaion Filer Order = 5 3dB BW = 64.8 Hz Ellipical or Cauer Order Predicaion Filer Order = 4 3dB BW = 4 Hz.e+4 * i i i i i i 7

18 Deermine Poles 6 poles: hree complex pole pairs Deermine he roos on he uni circle. Muliply by he cuoff frequency Deermine nd order coefficiens w and LPFPoles =.e+4 * Vou Vin s Carbirary K w s s a j bs a j b s w s w i i i i i i w a w a b 8

19 The Pole Pairs LPFPoles =.e+4 * i i i i i i H H H 3 C K w + 6.4e4 s +.e9 s w s w arbirary s s C K w e4 s +.e9 s w s w arbirary s s C K w +.646e4 s +.e9 s w s w arbirary 3 s s 3 Solve for he damping facors ζ, ζ, and ζ 3 The value of w is given. 9

20 Deermine s and Cs + - +Vdc -Vdc V OP- Amp Vou C 3 4 C w s w s w K C C C C C s s C C s V s Vou K MaxGain C C C K C w 3 K Le = kohm Find C and 4

21 Collec esuls Sallen-Key LPF Design Summary Filer Gain = 4.476, Filer Gain (db) =.4748 Sage C 3 4 Gain. kohm pf. kohm.685 kohm.7. kohm pf. kohm kohm kohm pf. kohm kohm.48

22 The Easier Way see BW_SKLPF_Gen.m %% % BW Filer generaion demonsraion % close all clear all clc BWn=6; % Buerworh Filer Order PB_Hz= ; % 3dB BW Derived from AnalogLPFCompare PBfreq=*pi*PB_Hz; Afer execuion you ge: Sallen-Key LPF Design Summary Filer Gain = 4.476, Filer Gain (db) =.4748 Sage C 3 4 Gain. kohm pf. kohm.685 kohm.7. kohm pf. kohm kohm kohm pf. kohm kohm.48 Magniude (db) Buerworh SK LPF wih Fc = khz and Gain =.47 db BW Goal Final Freq (Hz)

23 Check he Spice Design 3

24 The Specral Plo 4

25 Taking i o he nex level Wha resisors are acually available? Wha Capaciors are acually available? Sandard E-Series Values for resisors and capaciors: seleced o be evenly spaced logarihmically across one decade ypically relaed o value olerance (+/- % accuracy) where a value is available wihin he olerance Example E3 has hree values available (,, 47) Normally hink in %, 5%, or % olerance E for %, E4 for 5%, and E96for % hp:// 5

26 Malab Code AnalogFilerCompare.m BW_SKLPG_Gen.m subrouines Addiional esources Dr. Bazuin s Filer Noes on web sie Dr. Bazuin s Draf Filer Manual for ECE 48 folks (or pre 48) see me 6

27 Pulse esponse and iseime Low Pass Filers cause sharp signal edges o be smoohed. The amoun of smoohing is based on he bandwidh of he filer More smoohing smaller bandwidh Fourier relaionship: a narrow rec funcion in ime resuls in a broad (wide bandwidh) sinc funcion in frequency a wide rec funcion in ime resuls in a narrow (small bandwidh) sinc funcion in frequency 7

28 Filer Sep esponse Hz and Hz 4 h order Buerworh LPF Filers The sep response can be used o help define he bandwidh required for pulse signals. Buerworh Filers.4 Sep esponse Aenuaion (db) -6-8 Ampliude Hz Hz Frequency (normalized). Hz Hz Time (sec) 8

29 Filer Bandwidh for Pulses Pulse of lengh T rec T T sin c f T.5 Null-o-null BW of null o null Single Sided BW desired B T B/ may be accepable in some cases T See exbook discussion

30 Pulse Filering - -4 Buerworh Filers.5 Hz 5. Hz. Hz. Hz Four one-sided BW filers. sec pulse responses Aenuaion (db) Frequency (fs = Hz).6 Ampliude (db).4 Buerworh Filers Tes Signal.5 Hz 5. Hz. Hz. Hz PulseTes.m (digial filers) Time (fs=hz) 3

31 Tex Comparison Char (.5, 5. and Hz Plos) Pulse response of an ideal LPF Figure 3.4- See PulseTes.m or PulseTes3.m (digial filers) Copyrigh The McGraw-Hill Companies, Inc. Permission required for reproducion or display. 3

32 Copyrigh The McGraw-Hill Companies, Inc. Permission required for reproducion or display. Pulse resoluion of an ideal LPF. B = / See Fig3_4_.m (Buerworh filers) The inheren ime delay has been removed from he oupu 3

33 Pulse esoluion: Malab Using a s order and 6 h order Buerworh Filer The ex uses an ideal filer Filers have group and phase delay! Linear Simulaion esuls Ampliude Time (sec) Linear Simulaion esuls Ampliude Time (sec) 33

34 The Hilber Transform I is a useful mahemaical ool o describe he complex envelope of a real-valued carrier modulaed signal. Make a real signal complex complex Posiive frequency The precise definiion is as follows: xˆ x h Q x d H Q f jsgn f H Q x Hilber x j xˆ j,, j f f f * f H f H f Q Q hp://en.wikipedia.org/wiki/hilber_ransform 34

35 Copyrigh The McGraw-Hill Companies, Inc. Permission required for reproducion or display. Hilber ransform of a recangular pulse (a) Convoluion; (b) esul in ime domain y x j xˆ 35

36 Hilber Transform of Cos Xˆ x A A cosf f f f f f H f j xˆ A f f f f A sin f Q This is useful in generaing a complex signal from a real inpu signal as follows 36

37 eal o Complex Conversion x x y x j xˆ xˆ 37

38 eal o Complex Mixing Analog Devices AD8347:.8 GHz o.7 GHz Direc Conversion Quadraure Demodulaor F inpu and LO inpu Quadraure Oupu 38

39 Original eal Hilber Transform eal o Complex Conversion Hilber Transform Complex hc hc hc x X f x j xˆ X f j j sgn f X f x j xˆ X f sgn f X f x j xˆ X, f, for for f f The Hilber Transform can be used o creae a single sided specrum! The complex represenaion of a real signal. 39

40 Quadraic Filers We may wan o process real signals using complex filering or ranslaed ino he complex domain. Quadraure Signal Processing involves creaing an In-Phase and Quadraure-Phase signal represenaion. Usually his is done by quadraure mixing which creaes wo oupus from a real daa sream by mixing one by a cosine wave and he over by a sine wave. x expj f xcosf jsinf in phase jquadraure phase 4

41 4 Correlaion and Specral Densiy Using Probabiliy and he s and nd momens Assuming an ergodic, WSS process we use he ime average Properies: Schwarz s Inequaliy v v v P v z a z a z a z a z z z z * * w v w v P P

42 Auocorrelaion and Power Auocorrelaion Funcion Properies vv v v v v vv vv Pv vv vv vv 4

43 Crosscorrelaion Crosscorrelaion Funcion Properies vw v w v w vv ww vw vw wv 43

44 44 Applicaion Correlaion of phasors T T T d w w j exp T lim w j exp w j exp T w w sin c lim w j exp w j exp T else, w w, w j exp w j exp

45 Power Specral Densiy The Fourier Transform of he Auocorrelaion emember ECE 38! 45

46 Sysem analysis in τ domain x() ( ) x h() y y y () x () h () h( ) x ( ) d () h() () h( ) ( ) d yx x x and wih oupu auocorrelaion is: * * y() ( ) yx() ( ) yx( ) h h d h h * y() ( ) () x() The McGraw-Hill Companies

47 Sysem Analysis in f domain G G x f f x() ( ) x x x h() H f y f y G G y f y y Y G f H f X f y f H f G f x The McGraw-Hill Companies

48 Copyrigh The McGraw-Hill Companies, Inc. Permission required for reproducion or display. Inerpreaion of specral densiy funcions Power or Energy Specral Densiy is energy per uni frequency 48