Chapter 3: Signal Transmission and Filtering. A. Bruce Carlson Paul B. Crilly 2010 The McGraw-Hill Companies

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1 Communicaion Sysems, 5e Chaper 3: Signal Transmission and Filering A. Bruce Carlson Paul B. Crilly 00 The McGraw-Hill Companies

2 Chaper 3: Signal Transmission and Filering Response of LTI sysems Signal disorion Transmission Loss and decibels Filers and filering Quadraure filers and Hilber ransform Correlaion and specral densiy 00 The McGraw-Hill Companies

3 Pulse Response and Riseime 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 d( (wide bandwidh) sinc funcion in frequency a wide rec funcion in ime resuls in a narrow (small bandwidh) sinc funcion in frequency 3

4 Filer Sep Response Hz and 0 Hz 4 h order Buerworh LPF Filers The sep response can be used o help define he bandwidh required for pulse signals. Buerworh Filers 0.4 Sep Response enuaion (db) A Ampliude Hz 0 Hz Frequency (normalized) 0. Hz 0 Hz Time (sec) 4

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

6 Pulse Filering Buerworh Filers.5 Hz 50Hz Hz 0. Hz Four one-sided BW filers 0. sec pulse responses Aenuaio on (db) Frequency (fs = 00 Hz) 0.6 Ampliu ude (db) 0.4 Buerworh Filers Tes Signal.5 Hz 5.0 Hz 0. Hz 0. Hz PulseTes.m (digial filers) Time (fs=00hz) 6

7 Tex Comparison Char (.5, 5.0 and 0 Hz Plos) Pulse response of an ideal LPF Figure See PulseTes.m or PulseTes3.m (digial filers) Copyrigh The McGraw-Hill Companies, Inc. Permission required for reproducion or display. 7

8 Copyrigh The McGraw-Hill Companies, Inc. Permission required for reproducion or display. Pulse resoluion of an ideal LPF. B = /τ See Fig03_4_.m (Buerworh filers) The inheren ime delay has been removed from he oupu 8

9 Pulse Resoluion: Malab Using a s order and 6 h order Buerworh Filer The ex uses an ideal filer Filers have group and phase delay! s Order Linear Sim e Ampliude Time (sec) 6h Order Linear Sim Ampliude Time (sec) 9

10 Hilber Transform I is a useful mahemaical ool o describe he complex envelope of a real-valued carrier modulaed signal (i.e. make a complex signal from a real one) The precise definiion is as follows: xˆ ( ) x( ) π = π ( λ ) x dλ λ h Q( ) π H Q ( f ) j sgn( f ) H j, f > 0 = 0, f = 0 j 0 > f Q * ( f ) = H ( f ) H ( f ) = Q Q hp://en.wikipedia.org/wiki/hilber_ransform 0

11 Quadraure filer Allpass nework ha shifs he phase of posiive frequencies by -90 and negaive frequencies by +90 { j f > 0 H ( f) = jsgn f = h( ) = Q + j f < 0 π 00 The McGraw-Hill Companies

12 Original Real Hilber Transform Real o Complex Conversion ( ) X ( f ) Hilber Transform Complex hc x () = [ x( ) + j xˆ ( ) ] X ( f ) j j sgn( f ) X ( f ) ( ) = [ x ( ) + j xˆ ( ) ] X ( f ) + sgn ( f ) X ( f ) hc + hc( ( ) = [ x ( ) + j xˆ ( ) ] ( f ) X, for f > 0 0, for f < 0 The Hilber Transform can be used o creae a single sided specrum! The complex represenaion of a real signal.

13 Hilber ransform properies. x( ) and xˆ ( ) have same ampliude specrum. Energy and power in a signal and is Hilber ranform are equal 3. x ( ) and x ˆ( ) are orhogonal xxd ( ) ˆ( ) = 0 (energy) T lim xxd ( ) ˆ( ) = 0 (power) T T T 00 The McGraw-Hill Companies

14 Copyrigh The McGraw-Hill Companies, Inc. Permission required for reproducion or display. Hilber ransform of a recangular pulse (a) Convoluion; (b) Resul in ime domain y ( ) = x( ) + j xˆ ( ) 4

15 Hilber Transform of Cosine Xˆ x ( ) = A cos( π f ) 0 A 0 0 A = j δ f f + δ f + () f = [ δ( f f ) + δ( f + f )] H () f xˆ [ ( ) ( )] 0 f 0 ( ) = A sin( π f ) 0 Q This is useful in generaing a complex signal from a real inpu signal as follows 5

16 Real o Complex Conversion () δ ( ) x( ) x y( ) = x( ) + j xˆ ( ) π xˆ ( ) 6

17 Real o Complex Mixing Analog Devices AD8347: 0.8 GHz o.7 GHz Direc Conversion Quadraure Demodulaor RF inpu and LO inpu Quadraure reoupu 7

18 Quadraic Filers We may wan o process real signals using complex filering or ranslaed ino he complex domain. Quadraure Signal lprocessing 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 () exp( j π f ) x( ) cos( π f ) + j sin( π f ) in phase + j quadraure phase 8

19 Correlaion and specral densiy Correlaion of power signals Correlaion of energy signals Specral densiy funcions 00 The McGraw-Hill Companies

20 Correlaion and Specral Densiy Using Probabiliy and he s and nd momens Assuming an ergodic, WSS process we use he ime average ( ) ( ) ( ) 0 v v v P = Properies: ( ) ( ) ( ) 0 v v v P v = * * ( ) ( ) ( ) () ( ) ( ) ( ) ( ) z a z a z a z a z z z z 0 * * + = + = = Schwarz s Inequaliy ( ) ( ) ( ) ( ) z a z a z a z a () () w v w v P P

21 Auocorrelaion and Power Auocorrelaion Funcion Properies R vv ( τ) v( ) v( τ) = v( + τ) v( ) R R vv( 0) = Pv vv( 0) R vv( τ) ( τ ) = R ( τ ) R vv R vv

22 Sum and difference signals z () = v () ± w () R () τ = R () τ + R () τ ± R () τ + R () τ ± ±[ ] z v w vw wv If v () and w () are uncorrelaed τ R () τ = R () τ = 0 R () τ = R () τ + R () τ z v w wih τ= 0 Pz = Pv + Pw vw wv Useful when compuing signal plus noise. 00 The McGraw-Hill Companies

23 Crosscorrelaion Crosscorrelaion Funcion Properies R vw ( τ) v( ) w( τ) = v( + τ) w( ) R vv ( 0) R ww ( 0) R vw ( τ) R () τ = R () τ vw wv 3

24 Energy Specral Densiy The Fourier Transform of he Auocorrelaion G R v v v v ( f ) I[ R ( τ )] = R ( τ ) exp( j π f τ ) = dτ ( ) [ ( )] ( ) ( ) τ = I G f = G f exp j π f τ df When Noise is a random variable: Describe filered oupu v Compare deerminisic and saisical signals v Deerminisic have no informaion conen; herefore, only signals wih varying, random messages are acually of ineres! 4

25 Sysem analysis in τ domain x() R ( τ) x h() y ( τ ) ( τ ) R y y () = x () h () = h( λ) x ( λ) dλ R () τ = h() τ R () τ = h( λ) R ( τ λ) dλ yx x x and wih μ= λ oupu auocorrelaion is: * * Ry () h ( ) Ryx () h ( ) Ryx ( ) d τ = τ τ= μ τ μ μ R τ = h τ h τ R τ * y() ( ) () x() 00 The McGraw-Hill Companies

26 Sysem Analysis in f domain G x() y τ G x ( f ) h() H ( f ) G ( f ) R ( τ) y ( f ) = I[ ( τ )] x x R x G y ( f ) = H ( f ) G ( f ) x ( ) y ( τ ) ( f ) = I ( τ ) R y G y R y [ ] 00 The McGraw-Hill Companies

27 Copyrigh The McGraw-Hill Companies, Inc. Permission required for reproducion or display. Inerpreaion of specral densiy funcions 7