Outline Chapter 2: Signals and Systems

Transcription

1 Ouline Chaper 2: Signals and Sysems Signals Basics abou Signal Descripion Fourier Transform Harmonic Decomposiion of Periodic Waveforms (Fourier Analysis) Definiion and Properies of Fourier Transform Imporan Correspondences Linear Time-Invarian (LTI) Sysems Convoluion and Frequency Response Bandwidh Definiions for Time Duraion and Bandwidh Relaion of Time and Frequency Descripion of a Signal: Time-Bandwidh Produc Discree-Time Signals Sampling Theorem

2 Signals Physical represenaion of daa as funcion of ime: x() Speech, music, picure, measuremen daa, , coninuous ime & ampliude con. ime, disc. ampliude Classificaion coninuous or discree in ime and / or coninuous or discree in ampliude Analog signal coninuous ime & coninuous ampliude Digial signal discree ime & discree ampliude (sampling and quanizaion) disc. ime, con. ampliude discree ime & ampliude

3 Signals Parameers of he signal represen he value of he daa Parameer of periodic signals, i.e. x() = x(+ T), = ±, ±2, period T frequency f =/T in Hz = /s ampliude A phase shif ϕ Example: x() = A sin(2πf + ϕ) = A sin( + ϕ) angular frequency =2π f Differen represenaions of signals x() Ampliude X(f) Frequency Specrum Phase sae diagram Q = A sin ϕ ϕ I = A cos ϕ ϕ T = /f f 3

4 Harmonic Decomposiion of Periodic Waveforms (Fourier Analysis) A periodic signal componens: x() = a P x() =x( + T ) n= can be decomposed ino harmonic an cos(2πn T )+b n sin(2πn T ) Fourier coefficiens: a n = 2 T T/2 R T/2 x() cos(2πn T ) d; b n = 2 T T/2 R T/2 x() sin(2πn T ) d Compac complex formulaion: x() = P n= c n exp(j2πn T ); c n = T T/2 R T/2 x() exp( j2πn T ) d; 4

5 Properies of Fourier Coefficiens if real signals: x() R c n = c n ; c 0 R x() = c 0 + = c 0 + X cn e j2πn T + c n e j2πn T n= X h Re{c n } e j2πn T +e j2πn T {z } 2cos(2πn/T ) n= + jim{cn } e j2πn i T e j2πn T {z } 2j sin(2πn/t ) a n =2Re{c n }; b n = 2Im{c n }; a 0 =2c 0 conjugae even symmery: x() =x ( ) c n = a n /2 R conjugae odd symmery: x() = x ( ) jc n = b n /2 R 5

6 Recangular Waveform c n = T = ZT/2 T/2 Example x()e j2πn T d = T h sin(2πn 2πn T i +T/4 T/4 ZT/4 T/4 - = πn sin π 2 n = x() 0 cos(2πn T ) d /T /2 n =0 ±/πn n odd,n6= 0 0 n even 6

7 Fourier Transformaion (FT) Le x() be a deerminisic signal (no necessarily periodic) Fourier Transformaion of x() wih angular frequency =2π f X(j) = F {x()} = x() e j d x() =F {X(j)} = 2π X(j) e j d Inegral exiss if x() is absoluely inegrable: Decomposiion of x() ino is frequency componens In general X(j) is complex-valued Terms and definiions: X(j) : specrum of he signal phase specrum X(j) : magniude specrum Lineariy a x ()+a 2 x 2 () R x() d < X(j) = X(j) e jϕ(j) ϕ(j) =arg{x(j)} =arcan a X (j)+a 2 X 2 (j) ½ ¾ Im {X(j)} Re {X(j()} 7

8 Example Specrum of a recangular impulse rec T () x() = Definiion: sinc T 2 << T 2 = T else sinc(x) = sin(x) x x() X(j) = sinc(x) = 0 for x = 2π, = ±, ±2, x() e j d = ZT/2 T/2 e j d = e j j = e jt/2 e +jt/2 = sin(t/2) j /2 µ T = T sinc 2 X(j) T equally spaced zeros sinc T 2 T/2 T/2 T/2=`π =0 -T/2 T/2-6π/T -4π/T 2π/T -2π/T 4π/T 6π/T 8

9 Example Impulse response of an ideal low pass X(j) = g << g = 2 g 0 else X(j) x() =F {X(j)} = 2π = 2π Z g g e j d = g /π x() g π X(j) e j d sin( g ) g = g π sinc ( g) equally spaced zeros sinc( g ) g =`π =0 - g Dualiy: g -4π/ g -2π/ g x() X(j) X(j) 2π x( ) -π/ g π/ g 2π/ g 4π/ g 9

10 Symmery of Specrum for real-valued Signals Assumpion: x() R is a real-valued signal in general X(j) C Fourier Transform of real-valued signal X(j) = x() e j d = x() cos()d j x() sin()d wih e +j =cos()+j sin() {z } Re{X(j)} and {z } Im{X(j)} e j =cos() j sin() Symmery condiions for real/imaginary par of X(j) and magniude/angle specrum Re {X(j)} =Re{X( j)} even w.r.. X(j) = X( j) Im {X(j)} = Im {X( j)} odd w.r.. arg X(j) = arg X( j) specrum is conjugae even X(j) =X ( j) conjugae complex If symmery condiion is no saisfied he ime signal is complex-valued 0

11 Modulaion, Time Delay Modulaion FT of ime signal x() muliplied wih exponenial funcion exp(j 0 ) x() e j 0 F x() e j 0 ª = x() e j 0 e j d = x() e j( 0) d Muliplicaion wih exponenial funcion corresponds o shif in frequency domain Time delay x( 0 ) F{x( 0 )} = = X(j j 0 ) X(j) x( 0 ) e j d = X(j(- 0 )) 0 x(τ) e j(τ+ 0) dτ = e j0 x(τ) e jτ dτ = X(j) e j 0 Time delay corresponds o phase shif in frequency domain

12 Properies of Fourier Transform () ime domain x () X ( j) frequency domain lineariy ax() + ax() 2 2 ax( j ) + ax ( j) 2 2 lineariy real x * () = x() X j X j * ( ) = ( ) conjugae even even (real) x () = x( ) X j X j * ( ) = ( ) real (even) odd (real) x () = x( ) [ jx j ] * jx ( j ) = ( ) imaginary (odd) ime delay x () = x ( ) 0 e j 0 X( j) mul. wih exp. modulaion x () j 0 e X ( j j ) 0 shif by 0 convoluion x () x () 2 X ( j) X ( j) 2 muliplicaion muliplicaion x () x () 2 2 π X( j) X2( j) convoluion differeniaion inegraion d x d () x() τ dτ j X( j) [ ] j X( j ) + πx( j0) δ( ) muliplicaion wih j muliplicaion wih j 2

13 Properies of Fourier Transform (2) ime domain x () X ( j) frequency domain inverse ime axis x( ) X ( j) inverse frequency axis conjugae complex x * () X * ( j) conjugae + inverse frequency axis inverse ime axis + conjugae x * ( ) X * ( j) conjugae complex Theorem of Parseval: Similariy: x() 2 d = 2π X(j) 2 d x() X(j) x(a) a X j a for a R \{0} Dualiy: x() X(j) X(j) 2π x( ) 3

14 Dirac δ() =0 if 6= 0 R and δ()d = Consan x() = Example X(j) = X(j) =2πδ() δ() e j d = 2π X(j) X(j) Exponenial Funcion x() =e j 0 X(j) =F e j 0 ª =2πδ( 0 ) 2π X(j) Cosine cos( 0 )= 2 e j 0 +e j 0 X(j) =F ½ 2 e j 0 + e j 0 ¾ = 2 [2πδ( 0)+2πδ( + 0 )] π X(j) 0 Sine sin( 0 )= 2j = π [δ( 0 )+δ( + 0 )] e j 0 e j 0 X(j) =jπ [ δ( 0 )+δ( + 0 )] - 0 jπ X(j) 0 4

15 Fourier Transform of Periodic Signals Fourier ransform of cosine and sine cos( 0 )= 2 sin( 0 )= 2j e j 0 +e j 0 X(j) =π [δ( 0 )+δ( + 0 )] e j 0 e j 0 Fourier ransform of periodic signals Conains sequence of impulses in frequency domain a muliples of he fundamenal frequency 0 of he periodic signal discree specrum X X(j) =F{x()} = x n δ( n 0 ) x n given by Fourier Analysis Example: Infinie Dirac impulse series X δ T () = δ( nt ) 0 n= δ T () -2T -T 0 T 2T n= X(j) =jπ [ δ( 0 )+δ( + 0 )] X n= δ( n 0 )= 0 δ 0 (j) δ 0 (j) π jπ X(j) 0 X(j) 0 Infinie dirac sequence in ime Infinie dirac sequence in frequency 5

16 Muli sine funcion x() Modulaion: Muliplicaion wih in ms Example x() =0.5sin(2πf )+0.3sin(2πf 2 )+0.2sin(2πf 3 ) X(j) m() =cos(2πf M ) f in khz and f M =4kHz f = 0,4 khz f 2 = 0,8 khz f 3 =.2 khz Lineariy of FT 0.5 m() x() m() Y(j) Y(j)= 0.5 X(j-j M ) +0.5 X(j+j M ) in ms f in khz 6

17 Imporan Correspondences of Fourier Transform ime domain x () X ( j) frequency domain Dirac impulse δ() consan consan 2 π δ( ) Dirac impulse exponen. osc. cosine sine recangular pulse sinc cos( 0 ) sin( 0 ) rec ( ) = T j 0 e ( g) { T /2 0 else sin /( ) g 2 π δ( 0) [ ( 0) +δ+ ( 0) ] [ ( ) ( )] π δ π δ δ + j 0 0 T T 2 sinc( ) { π / 0 else X( j= ) g g specral line a 0 lines a ± 0 (even) lines a ± 0 (odd) sinc recangular pulse gaussian pulse exp a 2 2 π/ a exp 4 a gaussian pulse Dirac sequence δ ( = kt) k δ = ( ) 2π 2π T k k T Dirac sequence 7

18 Linear Time-Invarian (LTI) Sysems x() y() h() Oupu signal is given by convoluion of x() wih impulse response h() y() =x() h()= x(τ) h( τ)dτ = h(τ) x( τ)dτ symmeric in x() and h() How o calculae? Reverse impulse response h(-τ) h(τ) h(-τ) Shif h(-τ) by h(-τ) 0 T -T 0 Muliply h(-τ) wih x(τ) and calculae he inegral Oupu y() equals inegraion resuls x(τ) Repea for all h(-τ) -T 0 T τ 8

19 Example Convoluion of wo recangular impulses Cases: <0 h(-τ) x(τ) y() =0 x() =h() =rec T () T y() -T 0 T τ 0 T 2T 0 T -T 0 T y() = Z 0 dτ = T 2T 0 -T T y() = Z T T dτ =2T < 2T y() =0 0 T -T 9

20 Transmission of Exponenial Funcion Consider complex exponenial inpu signal y() =x() h() = h(τ) e j0( τ) dτ = e j 0 x() =e j 0 h(τ) e j 0τ dτ =x() H(j 0 ) Oupu corresponds o inpu signal muliplied by frequency dependen consan H(j 0 ) Shape of exponenial signal (or sinusoidal signal) is no affeced Frequency response of he sysem H(j) = Convoluion in ime domain Muliplicaion of frequency specrum y() =x() h()= x(τ) h( τ)dτ Y (j) =X(j) H(j) Treamen in frequency domain leads (ofen) o much simpler inerpreaion Dualiy: Convoluion in frequency corresponds o muliplicaion in ime X(j) H(j) 2π x() h() h() e j d 20

21 Convoluion wih Dirac Impulse Convoluion of ime signal x() wih dirac impulse x() δ()= x(τ) δ( τ)dτ = x() X(j) =X(j) Signal is no affeced when convolved wih dirac impulse Time shif corresponds o convoluion of x() wih δ(- 0 ) δ(-τ) x(τ) 0 T τ x( 0 )=x() δ( 0 ) X(j) e j 0 Convoluion wih finie sequence of dirac impulses x() [δ( )+δ()+δ( 2 )] = x( )+x()+x( 2 ) X(j) e j ++e 2 2

22 Bandwidh Time-domain x() and frequency-domain X(j) of a signal are inversely relaed If x() is changed han X(j) is changed in an inverse manner and vice versa If x() is sricly limied in ime, han X(j) will ail on indefiniely and vice versa Time-Bandwidh produc is consan B T =cons. How o define ime duraion T or bandwidh B for infinie signals? Definiion : (x()=x(-) real, even signal) Definiion of equivalen recangle of heigh x(0) and widh T / heigh X(0) and widh B T = x(0) Time-Bandwidh produc x() B T =2π x()d B = X(0) X(j)d T X(j) B T -T/2 T/2-2π/T 2π/T 22

23 Bandwidh Definiion requires several resricions (e.g. even signals) more general definiion is desired Definiion 2: 2 nd -order definiion (roo mean square) T = v u 2 x 2 ()d, B = v u 2π Uncerainy Relaion of Communicaion Technology Gauss impulse achieves minimum ime-bandwidh produc B T = q 2 ; b T = 4π x() =Ae α2 X(j) = 4α 2 x() and X(j) have same form π α e Opimum impulse wih respec o bandwidh efficiency Applicaion in GSM: Gaussian Minimum Shif Keying (GMSK) Oher definiions: 3-dB bandwidh, null-o-null bandwidh, 2 X(j) 2 d ; B =2πb = angular frequency B T 2 ; b T 4π 23

24 Ideal Lowpass Ideal Pass Filer Signals / Sysems X(j) - g g Ideal Highpass Ideal Bandpass - g X(j) g X(j)

25 Time-Discree Signals x() k T s x[k] =x(k T s ) T s : sampling ime f s =/T s : sampling frequency Non-periodic ime-coninuous signal is sampled a equally disan ime insances T s (sampling ime) Sequence of samples X X x() δ Ts () = x() δ( kt s )= x[k] δ( kt s )=x d [k] k= x() x[k] =x(k T s ) k= T s 2T s 4T s 6T s 8T s 0T s 2T s 4T s 6T s 25

26 Specrum of sampled signal X d (j) =F{x d [k]} = T s X k= Specrum X(j( k s )) Discree signal has a periodic specrum ( periodic signal has discree specrum) Periodic repeiion of he original specrum X(j) wih disance f s = / T s X d (j) X(j + j s ) X(j) X(j j s ) aliasing 2π/Τ s π/τ s π/t s 2π/Τ s Aliasing: Overlapping of periodic specrum In order o avoid aliasing, he sampling frequency f s = / T s has o be chosen efficienly large 26

27 Sampling Theorem Band limied specrum: X(j) =0 for > 2πB No corrupion of original specrum if f s > 2b X d (j) X(j( + s )) X(j) X(j( s )) 2π/Τ s π/τ s π/t s 2π/Τ s Sampling Theorem of Shannon: If a ime-coninuous signal of bandwidh b is sampled wih sampling frequency f s >2b hen he original analog signal can perfecly be reconsruced by filering wih an ideal lowpass filer wih cu-off frequency f s /2. X µ x() = x[k] sinc π kt s T s k= 27