Analog and Digital Signals. Introduction to Digital Signal Processing. Discrete-time Sinusoids. Analog and Digital Signals

Transcription

1 Itroductio to Digital Sigal Processig Chapter : Itroductio Aalog ad Digital Sigals aalog = cotiuous-time cotiuous amplitude digital = discrete-time discrete amplitude cotiuous amplitude discrete amplitude cotiuous-time x(t) x(t) Professor Deepa Kudur t t Uiversity of Toroto - - discrete-time x[] x[] Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig / 58 Chapter : Itroductio Aalog ad Digital Sigals aalog system = aalog iput aalog output digital system = digital iput digital output Chapter : Itroductio Siusoids x() = A cos(ω θ) = A cos(πf θ), Z discrete-time (ot digital), A x a (t) A ad Z A = amplitude ω = frequecy i rad/sample f = frequecy i cycles/sample; ote: ω = πf θ = phase i rad Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 3 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 4 / 58

2 Chapter : Itroductio Chapter : Itroductio MINIMUM OSCILLATION MAXIMUM OSCILLATION MINIMUM OSCILLATION ENVELOPE CYCLES Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 5 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 6 / 58 Chapter : Itroductio Chapter : Itroductio NOT PERIODIC ENVELOPE CYCLES Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 7 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 8 / 58

3 Chapter : Itroductio Uiqueess: Cotiuous-time Chapter : Itroductio Uiqueess: Cotiuous-time F F : For F F, A cos(πf t θ) A cos(πf t θ) except at discrete poits i time. Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 9 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 0 / 58 Chapter : Itroductio Uiqueess: Chapter : Itroductio Let f = f 0 k where k Z, x () = A e j(πf θ) = A e j(π(f 0k)θ) = A e j(πf 0θ) e j(πk) = x 0 () = x 0 () Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig / 58

4 Chapter : Itroductio Harmoically Related Complex Expoetials Harmoically related s k (t) = e jkω0t = e jπkf0t, (cts-time) k = 0, ±, ±,... Scietific Desigatio Frequecy (Hz) k for F 0 = 8.76 C C C C C C C Chapter : Itroductio Harmoically Related Complex Expoetials Scietific Desigatio Frequecy (Hz) k for F 0 = 8.76 C C C C4 (middle C) C C C C C C C3 C4 C5 C6 C7 C8 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 3 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 4 / 58 Chapter : Itroductio Harmoically Related Complex Expoetials Chapter : Itroductio Harmoically Related Complex Expoetials What does the family of harmoically related siusoids s k (t) have i commo? Harmoically related s k (t) = e jkω0t = e jπ(kf0)t, (cts-time) k = 0, ±, ±,... fud. period: T 0,k = cyclic frequecy = kf 0 period: T k = ay iteger multiple of T 0 commo period: T = k T 0,k = F 0 Case: For periodicity, select f 0 = N where N Z: Harmoically related s k () = e jπkf0 = e jπk/n, (dts-time) k = 0, ±, ±,... There are oly N distict dst-time harmoics: s k (), k = 0,,,..., N. Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 5 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 6 / 58

5 Chapter : Itroductio Aalog-to-Digital Coversio Chapter : Itroductio Aalog-to-Digital Coversio A/D coverter A/D coverter x (t) a Sampler x() Quatizer x () q Coder x (t) a Sampler x() Quatizer x () q Coder Aalog Quatized Digital Aalog Quatized Digital Samplig: coversio from cts-time to dst-time by takig samples at discrete time istats E.g., uiform samplig: x() = x a (T ) where T is the samplig period ad Z Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 7 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 8 / 58 Chapter : Itroductio Aalog-to-Digital Coversio Chapter : Itroductio Aalog-to-Digital Coversio A/D coverter A/D coverter x (t) a Sampler x() Quatizer x () q Coder x (t) a Sampler x() Quatizer x () q Coder Aalog Quatized Digital Aalog Quatized Digital Quatizatio: coversio from dst-time cts-valued to a dst-time dst-valued quatizatio error: e q () = x q () x() for all Z Codig: represetatio of each dst-value x q () by a b-bit biary sequece e.g., if for ay, x q () {0,,..., 6, 7}, the the coder may use the followig mappig to code the quatized amplitude: Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 9 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 0 / 58

6 Chapter : Itroductio Aalog-to-Digital Coversio Samplig Theorem Chapter : Itroductio x (t) a A/D coverter Sampler x() Quatizer x () q Coder If the highest frequecy cotaied i a aalog x a (t) is F max = B ad the is sampled at a rate Aalog Example coder: Quatized Digital F s > F max = B the x a (t) ca be exactly recovered from its sample values usig the iterpolatio fuctio g(t) = si(πbt) πbt Note: F N = B = F max is called the Nyquist rate. Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig / 58 Samplig Theorem Chapter : Itroductio Chapter : Itroductio Badlimited Iterpolatio Samplig Period = T = F s = Samplig Frequecy Therefore, give the iterpolatio relatio, x a (t) ca be writte as x a (t) = x a (t) = = = x a (T )g(t T ) x() g(t T ) where x a (T ) = x(); called badlimited iterpolatio. badlimited iterpolatio fuctio -- sic x() samples 0 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 3 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 4 / 58

7 Chapter : Itroductio Digital-to-Aalog Coversio Chapter : Itroductio Digital-to-Aalog Coversio origial/badlimited iterpolated x() origial/badlimited iterpolated 0 zero-order hold -3T t -T -T 0 T T 3T Commo iterpolatio approaches: badlimited iterpolatio, zero-order hold, liear iterpolatio, higher-order iterpolatio techiques, e.g., usig splies I practice, cheap iterpolatio alog with a smoothig filter is employed. Commo iterpolatio approaches: badlimited iterpolatio, zero-order hold, liear iterpolatio, higher-order iterpolatio techiques, e.g., usig splies I practice, cheap iterpolatio alog with a smoothig filter is employed. Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 5 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 6 / 58 Chapter : Itroductio Digital-to-Aalog Coversio Chapter : Dst-Time Sigals & Systems Elemetary Discrete-Time Sigals. uit sample sequece (a.k.a. Kroecker delta fuctio): origial/badlimited iterpolated -3T liear iterpolatio t -T -T 0 T T 3T. uit step : δ() = u() = {, for = 0 0, for 0 {, for 0 0, for < 0 Commo iterpolatio approaches: badlimited iterpolatio, zero-order hold, liear iterpolatio, higher-order iterpolatio techiques, e.g., usig splies I practice, cheap iterpolatio alog with a smoothig filter is employed. 3. uit ramp : Note: u r () = {, for 0 0, for < 0 δ() = u() u( )= u r ( ) u r () u r ( ) u() = u r ( ) u r () Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 7 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 8 / 58

8 Chapter : Dst-Time Sigals & Systems Sigal Symmetry Chapter : Dst-Time Sigals & Systems Sigal Symmetry Eve : x( ) = x() Odd : x( ) = x() Eve compoet: x e () = [x() x( )] Odd compoet: x o () = [x() x( )] x() x() x() Note: x() = x e () x o () Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 9 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 30 / 58 Chapter : Dst-Time Sigals & Systems Sigal Symmetry Chapter : Dst-Time Sigals & Systems Simple Maipulatio of Discrete-Time Sigals x() x(-) (x()x(-))/ (x()-x(-))/ eve part odd part Trasformatio of idepedet variable: time shift: k, k Z Questio: what if k Z? time scale: α, α Z Questio: what if α Z? Additioal, multiplicatio ad scalig: amplitude scalig: y() = Ax(), < < sum: y() = x () x (), < < product: y() = x ()x (), < < - - Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 3 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 3 / 58

9 Chapter : Dst-Time Sigals & Systems Chapter : Dst-Time Sigals & Systems Simple Maipulatio of Discrete-Time Sigals I Simple Maipulatio of Discrete-Time Sigals II Fid x() x( ). 3 x() Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig / x() 34 / 58 Simple Maipulatio of Discrete-Time Sigals Graph of x( 3 ). ). < > Chapter : Dst-Time Sigals & Systems Simple Maipulatio of Discrete-Time Sigals I Fid 3 4 x()-x() Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig Chapter : Dst-Time Sigals & Systems x( x() 3 3 < 0 if > 9 0 if 3 x( 3 ) is a iteger; udefied otherwise udefied x() = udefied x(4) = udefied x(7) = 3 udefied x(0) = udefied x(3) = udefied x(6) = udefied x(9) = is a iteger; udefied otherwise Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig / This is udefied for values of that are ot eve itegers ad zero for eve itegers ot show o this sketch Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 36 / 58

10 Chapter : Dst-Time Sigals & Systems Iput-Output Descriptio of Dst-Time Systems iput/ excitatio x() System y() output/ respose Chapter : Dst-Time Sigals & Systems Classificatio of Discrete-Time Systems Commo System Properties: static vs. dyamic time-ivariat vs. time-variat Iput-output descriptio (exact structure of system is ukow or igored): y() = T [x()] black box represetatio: liear vs. oliear causal vs. o-causal stable vs. ustable systems x() T y().. Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 37 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 38 / 58 Chapter : Dst-Time Sigals & Systems Static vs. Dyamic Chapter : Dst-Time Sigals & Systems Static vs. Dyamic Static system (a.k.a. memoryless): the output at time depeds oly o the iput sample at time ; otherwise the system is said to be dyamic Cosider the geeral system: y() = T [x( N), x( N ),, x( ), x(), x( ),, x( M ), x( M)], N, M > 0 a system is static iff (if ad oly if) For N = M = 0, y() = T [x()], the system is static. for every time istat. y() = T [x(), ] For 0 < N, M <, the system is said to be dyamic with fiite memory of duratio N M. For either N ad/or M equal to ifiite, the system is said to have ifiite memory. Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 39 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 40 / 58

11 Chapter : Dst-Time Sigals & Systems Static vs. Dyamic Examples: memoryless or ot? Chapter : Dst-Time Sigals & Systems Discrete-Time Bouded Sigals x[] x[] y() = A x(), A 0 y() = A x() B, A, B, 0 y() = x() cos( π ( 5)) 5 y() = x( ) x[] x[] y() = x( ) y() = x() y() = e 3x() y() = k= x(k) As: Y, Y, Y, N, N, N, Y, N x[] BOUNDED SIGNAL x[] UNBOUNDED SIGNAL Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 4 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 4 / 58 Chapter : Dst-Time Sigals & Systems The Covolutio Sum Chapter : Dst-Time Sigals & Systems The Covolutio Sum Let the respose of a liear time-ivariat (LTI) system to the uit sample iput δ() be h(). Recall: x() = x(k)δ( k) k= δ() δ( k) α δ( k) x(k) δ( k) x(k)δ( k) T h() T h( k) T α h( k) T x(k) h( k) T x(k)h( k) k= x() T k= y() Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 43 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 44 / 58

12 Chapter : Dst-Time Sigals & Systems The Covolutio Sum Chapter : Dst-Time Sigals & Systems Fiite vs. Ifiite Impulse Respose Therefore, y() = x(k)h( k) = x() h() for ay LTI system. k= Implemetatio: Two classes Fiite impulse respose (FIR): y() = M k=0 h(k)x( k) Ifiite impulse respose (IIR): y() = h(k)x( k) k=0 } orecursive systems } recursive systems Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 45 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 46 / 58 Chapter : Dst-Time Sigals & Systems System Realizatio Chapter : Dst-Time Sigals & Systems Buildig Block Elemets Geeral expressio for Nth-order LCCDE: N M a k y( k) = b k x( k) a 0 k=0 k=0 Iitial coditios: y( ), y( ), y( 3),..., y( N). Adder: Costat multiplier: Uit delay: Uit advace: Need: () costat scale, () additio, (3) delay elemets. Sigal multiplier: Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 47 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 48 / 58

13 Chapter : Dst-Time Sigals & Systems Direct Form I vs. Direct Form II Realizatios Chapter : Dst-Time Sigals & Systems Direct Form I IIR Filter Implemetatio y() = N a k y( k) k= M b k x( k) k=0 is equivalet to the cascade of the followig systems: v() }{{} output y() }{{} output M = b k x( k) }{{} k=0 iput N = k= a k y( k) v() }{{} iput orecursive recursive LTI All-zero system v() LTI All-pole system Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 49 / 58 Requires: M N multiplicatios, M N additios, M N memory locatios Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 50 / 58 Chapter : Dst-Time Sigals & Systems Direct Form II IIR Filter Implemetatio Chapter : Dst-Time Sigals & Systems Direct Form II IIR Filter Implemetatio Adder: Costat multiplier: Sigal multiplier: Uit delay: Uit advace: LTI All-pole system LTI All-zero system Requires: M N multiplicatios, M N additios, M N memory locatios Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 5 / 58 For N>M Requires: M N multiplicatios, M N additios, max(m, N) memory locatios Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 5 / 58

14 Chapter 3: The z-trasform ad Its Applicatios The Direct z-trasform Chapter 3: The z-trasform ad Its Applicatios Regio of Covergece Direct z-trasform: Notatio: X (z) = x()z = X (z) Z{x()} the regio of covergece (ROC) of X (z) is the set of all values of z for which X (z) attais a fiite value The z-trasform is, therefore, uiquely characterized by:. expressio for X (z). ROC of X (z) x() Z X (z) Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 53 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 54 / 58 Chapter 3: The z-trasform ad Its Applicatios ROC Families: Fiite Duratio Sigals Chapter 3: The z-trasform ad Its Applicatios ROC Families: Ifiite Duratio Sigals Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 55 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 56 / 58

15 Chapter 3: The z-trasform ad Its Applicatios z-trasform Properties Chapter 3: The z-trasform ad Its Applicatios Commo Trasform Pairs Property Time Domai z-domai ROC Notatio: x() X (z) ROC: r < z < r x () X (z) ROC x () X (z) ROC Liearity: a x () a x () a X (z) a X (z) At least ROC ROC Time shiftig: x( k) z k X (z) At least ROC, except z = 0 (if k > 0) ad z = (if k < 0) z-scalig: a x() X (a z) a r < z < a r Time reversal x( ) X (z ) r < z < r Cojugatio: x () X (z ) ROC dx (z) z-differetiatio: x() z r dz < z < r Covolutio: x () x () X (z)x (z) At least ROC ROC amog others... Sigal, x() z-trasform, X (z) ROC δ() All z u() z z > 3 a u() az 4 a u() az ( az ) 5 a u( ) az 6 a az u( ) 7 cos(ω 0 )u() 8 si(ω 0 )u() 9 a cos(ω 0 )u() 0 a si(ω 0 )u() ( az ) z cos ω 0 z > a z > a z < a z < a z cos ω 0 z z > z si ω 0 z cos ω 0 z z > az cos ω 0 z > a az cos ω 0 a z az si ω 0 az cos ω 0 a z z > a Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 57 / 58 Professor Deepa Kudur (Uiversity of Toroto)Itroductio to Digital Sigal Processig 58 / 58