Chapter 2 Systems ad Sigals 1 Itroductio Discrete-Time Sigals: Sequeces Discrete-Time Systems Properties of Liear Time-Ivariat Systems Liear Costat-Coefficiet Differece Equatios Frequecy-Domai Represetatio of Discrete-Time Sigals ad Systems Represetatio of Sequece by Fourier Trasforms Symmetry Properties of the Fourier Trasform Fourier Trasform Theorems
1. Itroductio-- Defiitio 2 Sigals Ay physical quatity that varies with time, space or ay other idepedet variable. Commuicatio bewee humas ad machies. Systems mathematically a trasformatio or a operator that maps a iput sigal ito a output sigal. ca be either hardware or software. such operatios are usually referred as sigal processig. Digital Sigal Processig The represetatio of sigals by sequeces of umbers or symbols ad the processig of these sequeces. S
3 1. Itroductio-- Defiitio
1. Itroductio-- Classificatio 4 Cotiuous-Time versus Discrete-Time Sigals Cotiuous-time sigals are defied for every value of time. Discrete -time sigals are defied at discrete values of time. Cotiuous-Valued versus Discrete-Valued Sigals A sigal which taes o all possible values o a fiite rage or ifiite rage is said to be a cotiuous-valued sigal. A sigal taes o values from a fiite set of possible values is said to be a discrete-valued sigal. Multichael versus Multidimesioal Sigals Sigals may be geerated by multiple sources or multiple sesors. Such sigals are multi-chael sigals. A sigal which is a fuctio of M idepedet variables is called multi-dimesioal sigals.
1. Itroductio-- Basic Elemets 5 A/D Coverter Coverts a aalog sigal ito a sequece of digits D/A Coverter Coverts a sequece of digits ito a aalog sigal Aalog Iput Sigal 0 Aalog Output Sigal t A/D Coverter D/A Coverter Digital Iput Sigal {3, 5, 4, 6...} Digital Output Sigal Digital Sigal Processig
6 1. Itroductio-- Advatages of Digital over Aalog Processig Better cotrol of accuracy Easily stored o magetic media Allow for more sophisticated sigal processig Cheaper i some cases
Examples 7 A picture is a two-dimesioal sigal I(x,y) is a fuctio of two variables. A blac-ad-white televisio picture is a threedimesioal sigal I(x,y,t) is a fuctio of three variables. A color TV picture is a three-chael, three-dimesioal sigals Ir(x,y,t), Ig(x,y,t), ad Ib(x,y,t)...
2. Discrete-Time Sigals: Sequeces 8 Cotiuous sigals f(x), u(x), ad s(t), ad so o. Sampled sigals s, f, u(). Shiftig by uits of time y x Uit sample sequece (Kroecer Delta) ( ) 0, 0 1, 0 Liear combiatio form x x Uit step sequece u 0, 0 1, 0 Siusoidal Fuctio si( ) si( 2f ) x a (t) x t Time
2. Discrete-Time Sigals: Sequeces (c.1) 9 Defiitio X a (t) = A cos( W t+ ), - <t< A is the amplitude of the siusoid. W is the frequecy i radias per secod. is the phase i radias. F=W/2 is the frequecy i cycles per secod or hertz. Remars The fudametal period is T=1/F. For every fixed value of F, f(t) is periodic f(t+t) = f(t), T=1/F Cotiuous-time siusoidal sigals with distict frequecies are themselves distict. Icreasig the frequecy F results i a icrease i the rate of oscillatio. Time
2. Discrete-Time Sigals: Sequeces (c.2) 10 Discrete-Time Siusoidal Sigals X() = A cos( + ), =1, 2,... A is the amplitude of the siusoid is the frequecy i radias per sample is the phase i radias f=/2 is the frequecy i cycles per sample. X() = A cos( + )
2. Discrete-Time Sigals: Sequeces (c.3) 11 Discrete-Time Siusoidal Sigals A discrete-time sisoidal is periodic oly if its frequecy f is a ratioal umber X(+N) = X(), N=p/f, where p is a iteger Discrete-time siusoidals where frequecies are separated by a iteger multiple of 2 are idetical X1() = A cos( 0 ) X2() = A cos( ( 0 2) ) The highest rate of oscillatio i a discrete-time siusoidal is attaied whe = or (=-), or equivaletly f=1/2. X() = A cos(( 0 +)) = -A cos(( 0 +) X() = A cos( + )
2. Discrete-Time Sigals: Sequeces (c.4) 12 Exercise: Fid the periods. si0.1, cos 10.1, si0.1, cos3/7 A observatio For the samplig process f= f(t), the mappig betwee discrete-frequecy ad aalog-frequecy is oe-to may. Example si1.1t, si3.1t, si3.1t, si(-0.9)t, si5.1pt, si (-2.9)t have the sample digital evelop for samplig time T=1. Reaso : 1 T - 2 T = x(2)
2. Discrete-Time Sigals: Sequeces (c.5) 13 The Digital Frequecy If 1 T ad 2 T differ by a multiple of 2, they are still cosidered equal. The digital frequecy ca be restricted to lie i a iterval of 2 to elimiate this ouiqueess. (-,, 0, 2) or, 3 If (-, is selected, the Digital 3/Ts /Ts 0 Example: Fid the frequecies of the followig sequeces for T=1 si 4.2, cos5.2, si 10, si(-2.1), cos20 /T /T /Ts 3/Ts 5/Ts Aalog
3. Discrete-Time Systems 14 Systems Mathematically a trasformatio or a operator that maps a iput sigal ito a output sigal Ca be either hardware or software. Such operatios are usually referred as sigal processig. E.x. y( ) x( ) x( 1) x( ) y( 1) x( ) Discrete-Time System H
3. Discrete-Time Systems-- Classificatio 15 Time-Ivariat versus Time-Variat Systems A system H is time-ivariat or shift ivariat if ad oly if x() ---> y() implies that x(-) --> y(-) for every iput sigal x() x() ad every time shift. Causal versus Nocausal Systems The output of a causal system satisfies a equatio y() = Fx(), x(-1), x(-2),... where F. is some arbitrary fuctio. Memory versus Memoryless Systems A sysetm is referred to as memoryless system if the output y() at every value of depeds oly o the iput x() at the same value of.
16 3. Discrete-Time Systems-- Classificatio (c.1) Liear versus Noliear Systems A system H is liear if ad oly if Ha 1 x 1 ()+ a 2 x 2 () = a 1 Hx 1 () + a 2 Hx 2 () for ay arbitrary iput sequeces x 1 () ad x 2 (), ad ay arbitrary costats a 1 ad a 2. Multiplicative or Scalig Property Hax() = a Hx() Additivity Property Hx 1 () + x 2 () = Hx 1 () + Hx 2 () Stable versus Ustable Systems A arbitrary relaxed system is said to be boudediput-bouded-output (BIBO) stable if ad oly if every bouded iput produces a bouded output. u1() u2() u1() u2() a b + Liear systems Liear systems Liear systems a b + y() y()
4. Liear Time-Ivariat Systems 17 The Importace of LTI Systems Powerful aalysis techiques exist for such systems. May real-world systems ca be closely approximated for liear, time-ivariat systems. Aalysis techiques for LTI systems suggest approaches for that of oliear systems. Liear Systems Time-Ivariace y()=hx() ==> Hx(-) = y(-) Liear superpositio Ha 1 x 1 ()+a 2 x 2 () = a 1 Hx 1 ()+a 2 Hx2() = a 1 y 1 () + a 2 y 2 () Specificatio for LSI Systems H x ( ) H x ( ) ( ) x ( ) H ( ) x ( ) h ( ) Impulse Respose of the System
18 4. Liear Time-Ivariat Systems
4. Liear Time-Ivariat Systems (c.1) 19 Liear Systems (c.1) Fiite Impulse Respose (FIR)or Ifiite Impulse Respose (IIR) Depeds o the the fiite ad ifiite umber of terms of h() Covolutio Formula Ex. y ( ) x ( ) h ( ) x ( ) h ( ) A savig accout with mothly iterest rate 0.5%. The iterest is added to the pricipal at the first day of each moth. If we deposit u0 = $100.00, u1 = -$50.00, u2 = 200.00 what is the total amout of moey. What is the total amout of moey o the first day of the fifth moth?
20 4. Liear Time-Ivariat Systems (c.2) Discrete Covolutio-- Table Lists Cosider the covolutio of {1, 2, 3, 4, -1,...} ad {-1, 1, -3, 2,...} Discrete Covolutio-- -1 1-3 2 y0 y1 y2 y3... 1 2 3 4-1 -1-2 -3-4 1 1 2 3 4-1 -3-6 -9-12 3 2 4 6 8-2 Graphical Computatio Flippig h i to yield h -i Shiftig h -i to yield h -i Multiplicatio of h -i ad u i Summatio of h -iu i from i =0, 1, 2,... y h i u i h i u i i0 i0
5. Properties of the LTI Systems 21 Commuicative y h i u i h * u i h i u i u * h i Parallel Sum x *{ h h } 1 2 x * h x * h 1 2 x h x x y h y x y h1+h2 y h1 + h2 Cascade Form x *{ h 1 { x * h * h 1 2 } }* h 2 x h1 h2 y x y h1*h2
5. Properties of the LTI Systems (c.1) 22 Stability of LTI Systems (BIBO, Bouded-Iput-Bouded Output System) <pf> Liear time-ivariat systems are stable if ad oly if the impulse respose is absolutely summable, i.e., if q -> p Sice that S y h If x is bouded so that the h h x x y B h x x B x ~q -> ~p If S= the the bouded iput x will geerate h * h h, 0 0, h 0 h y 0 x h S h 2
5. Properties of the LTI Systems (c.2) Ex. Fid the impulse respose of the followig systems 1 1 1 2 1 2 1 x x y x y x M M y x y M M d 23
24 6. Liear Costat-Coefficiet Differece Equatios Iput-Output Descriptio From Differece Equatio to Impulse Respose Not every covolutio ca be trasformed ito a simple differece equatio If the differece equatio descriptio of a system is ow, the the impulse respose of the system ca be readily obtaied. Example N y( ) a y( ) b x( ) M 1 0 y 1 1005. y u 1 or y 1005. y 1 u
25 6. Liear Costat-Coefficiet Differece Equatios Ex 2.16
6. Liear Costat-Coefficiet Differece 26 Equatios (c.1) Solutio Specificatio for the LCCDE The solutio ca be obtaied recursively for either positive time or egative time. Liear, time-ivariat, ad causal ==> the solutio is uique. Iitial Coditio Iitial-rest coditios if the iitial coditio is zero. Iitial rest coditio ==> LTI ad causal x() Liear Costat Coefficiet Equatios by {a, b, N, M} y()
27 7. Frequecy-Domai Represetatio of Discrete-Time Sigals ad Systems Eigefuctios of LTI Systems Respose to Expoetial Fuctios j ( ) j j j y( ) h( ) e e h( ) e H( e ) e Eigevalues of the System Frequecy Respose-- Complex Magitude ad Phase j j j j H ( e ) H ( e ) jh ( e ) R I Ex. y x d M 2 1 y x M M 1 1 2 M 1 X() = A cos( + ) j j j H ( e j ) H ( e ) H ( e ) e
7. Frequecy-Domai Represetatio of 28 Discrete-Time Sigals ad Systems Ex. 2.17 Method 1: Frequecy Respose Method 2: Derive from Impulse Respose.
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30 8. Represetatio of Sequeces by Fourier Trasforms Fourier Trasform Pair Trasform j X ( e ) x ( )exp j 1 x X e j j ( ) ( ) e d 2 Iverse Trasform (Sythesis Formula) j j j X ( e j ) X ( e ) X ( e ) e Magitude & Phase Spectrum Questio: 1. Periodic Fuctios for the Trasform. 2. Relatioship with the Frequecy respose. 3. Iverse of each other? 4. Existece of the trasform for fuctios.
8. Represetatio of Sequeces by Fourier 31 Trasforms Proof of Fourier Pair Subsistig aalysis equatio ito sythesis equatio
L'Hôpital's rule 32 I its simplest form, l'hôpital's rule states that for fuctios ƒ ad g:
Existece of Fourier Trasform 33 Covergece of the ifiite sum The series ca be show to coverge uiformly to a cotiuous fuctio of
Existece of Fourier Trasform 34 Existece of Fourier trasform of ifiite sequece
Existece of Fourier Trasform 35 Relax of the coditio of uiform covergece of ifiite sum
Existece of Fourier Trasform 36 Example
37 8. Represetatio of Sequeces by Fourier Trasforms (c.1) Example Lowpass Filters 1, H e j c ( ) 0, c Impulse Respose 1 j h e d 2 si c, Causal? Decay Factor? Absolutely Summable? Gibbs pheomeo.
38 8. Represetatio of Sequeces by Fourier Trasforms
39 8. Represetatio of Sequeces by Fourier Trasforms (c.2) Fourier Trasform of the periodic trai x ( ) 1 for all j X ( e ) 2 ( 2r ) r j X ( e ) 2 ( 2r ) r 1 j x ( r ) e d 2 2 2 0 r 1 j j0 ( ) e d e for ay. 2 2 0 0 Absolutely Summable? Square Summable? j0 j e e 2 ( 0 2r ) r Lighthill, 1958
40 8. Represetatio of Sequeces by Fourier Trasforms Fourier trasform of siusoidal sigal j0 j e e 2 ( 0 2r ) Exted the theory r
41 9. Symmetry Properties of the Fourier Trasform Defiitio Cojugate-symmetric sequece x e =x* e - Cojugate-atisymmetric sequece x o =-x* o - Properties A sequece ca be represeted as x = x e + x o 1 xe ( x x * ) 2 1 xo ( x x * ) 2 j j j X ( e ) X ( e ) X ( e ) e j 1 j j X e ( e ) X ( e ) X *( e ) 2 j 1 j j X o ( e ) X ( e ) X *( e ) 2 j * j X ( e ) X ( e ) e j * j X ( e ) X ( e ) o e o o
42 9. Symmetry Properties of the Fourier Trasform (c.1) Example Real sequece x=a u j X ( e ) x ( )exp j 1 x X e j j ( ) ( ) e d 2
43 9. Symmetry Properties of the Fourier Trasform (c.2)
44 9. Symmetry Properties of the Fourier Trasform (c.3)
45 10. Fourier Trasform Theorems
46 Theorem Examples
11. Discrete-Time Radom Sigals Precise descriptio of sigals are so complex Modelig the sigals as a stochastic process. Let the meas of the iput ad output processes are m x { x }, m { y } y If x is statioary, the m x is idepedet of ad m x { x }, m { y } y m y { y } h x mx h
11. Discrete-Time Radom Sigals Autocorrelatio fuctio of the output process If x is statioary Let l=r- } { } {, r r yy r m x x r h h r m x x r h h m y y m r xx yy yy r m r h h m m, l hh xx l xx yy yy l c l m l m l h h m m, hh l h h l c
11. Discrete-Time Radom Sigals Fourier Trasform Power desity spectrum Cross-correlatio ) ( ) ( ) ( j xx j hh j yy e e C e xx xy m h m x h x m y x m } { 2 * ) ( ) ( ) ( ) ( j j j j hh e H e H e H e C ) ( ) ( j xx j j xy e e H e
11. Discrete-Time Radom Sigals Ex. White Noise 2 m xx m Power Spectrum x The average power
11. Cocludig Remars 51 Itroductio Discrete-Time Sigals: Sequeces Discrete-Time Systems Properties of Liear Time-Ivariat Systems Liear Costat-Coefficiet Differece Equatios Frequecy-Domai Represetatio of Discrete-Time Sigals ad Systems Represetatio of Sequece by Fourier Trasforms Symmetry Properties of the Fourier Trasform Fourier Trasform Theorems