Discrete-Time Signals and Systems. Signals and Systems. Digital Signals. Discrete-Time Signals. Operations on Sequences: Basic Operations

Transcription

1 -6.3 Digital Sigal Processig ad Filterig..8 Discrete-ime Sigals ad Systems ime-domai Represetatios of Discrete-ime Sigals ad Systems ime-domai represetatio of a discrete-time sigal as a sequece of umbers Basic sequeces ad operatios o sequeces Discrete-time systems i processig of discrete-time sigals Liear ad time-ivariat systems 8 Olli Simula Discrete-ime Sigals Sequece {} ca be cosidered as a periodically sampled cotiuous-time sigal x a (t) x ( t) x ( )...,,,,,,... a t a Digital Sigals Digital sigal Discrete-time ad discrete-valued sequece of umbers Digital sigal processig he sequece is trasformed to aother sequece by meas of arithmetic operatios Samplig iterval: Samplig frequecy: F / 8 Olli Simula 3 8 Olli Simula 4 ypes of Sequece Fiite-duratio or fiite-legth sequece: Defied i the iterval N <<N, where N ad N are fiite ad N >N Legth (duratio): N N -N + Ifiite-duratio or ifiite-legth sequece: a) Right-sided sequece:, <N b) Left-sided sequece:, >N 8 Olli Simula 5 Operatios o Sequeces: Basic Operatios Product (modulatio) operatio: Modulator w[ y [ x [ w [ A applicatio is i formig a fiite-legth sequece from a ifiite-legth sequece by multiplyig the latter with a fiite-legth sequece called a widow sequece Process called widowig 8 Olli Simula 6 Mitra 3rd Editio: Chapter ; 8 Olli Simula

2 -6.3 Digital Sigal Processig ad Filterig..8 Operatios o Sequeces: Basic Operatios Additio operatio: Adder + y [ + w[ Multiplicatio operatio: w[ Operatios o Sequeces: Basic Operatios ime-shiftig operatio: where N is a iteger If N >, it is delayig operatio Uit delay If N <, it is a advace operatio N z y [ x [ Multiplier A A Uit advace z + 8 Olli Simula 7 8 Olli Simula 8 Operatios o Sequeces: Basic Operatios ime-reversal (foldig) operatio: Brachig operatio: Used to provide multiple copies of a sequece Example: Averagig filter Combiatios of Basic Operatios αx [ + α + α3 + α4 3 8 Olli Simula 9 8 Olli Simula Samplig Rate Alteratio: Basic Operatios Employed to geerate a ew sequece with a samplig rate F higher or lower tha that of the samplig rate F of a give sequece ' F Samplig rate alteratio ratio is: R F If R >, the process called iterpolatio If R <, the process called decimatio Samplig Rate Alteratio: Basic Operatios I up-samplig by a iteger factor L >, L - equidistat zero-valued samples are iserted by the up-sampler betwee each two cosecutive samples of the iput sequece : / L,, ± L, ± L, L x u [, otherwise x [ L x u [ 8 Olli Simula 8 Olli Simula Mitra 3rd Editio: Chapter ; 8 Olli Simula

3 -6.3 Digital Sigal Processig ad Filterig..8 Samplig Rate Alteratio: Basic Operatios A example of the up-samplig operatio.5 Iput Sequece.5 Output sequece up-sampled by 3 Samplig Rate Alteratio: Basic Operatios I dow-samplig by a iteger factor M >, every M-th samples of the iput sequece are ept ad M - i-betwee samples are removed: Amplitude -.5 Amplitude -.5 y [ M ime idex ime idex x [ M 8 Olli Simula 3 8 Olli Simula 4 Samplig Rate Alteratio: Basic Operatios Periodic Sequeces A example of the dow-samplig operatio Amplitude Iput Sequece ime idex Amplitude Output sequece dow-sampled by ime idex 8 Olli Simula Periodicity: x p [x p [+N, for all he sequece x p [ is periodic with period N where N is a positive iteger ad is ay iteger he fudametal period N f is the smallest N for which the above equatio holds Notice! Samplig of a periodic cotiuous-time sigal does ot guaratee the periodicity of the sampled sequece 8 Olli Simula 6 Example: Siusoidal Sequeces cos(π /) Periodic, N x [ cos(8π / 3) Periodic, N3 x [ cos( / 6) Not periodic 8 Olli Simula 7 Classificatio of Sequeces A sequece is bouded if x [ B A sequece is absolutely summable if x [ < A sequece is square- summable if < he eergy of a sequece is x < 8 Olli Simula 8 E Mitra 3rd Editio: Chapter ; 8 Olli Simula 3

4 -6.3 Digital Sigal Processig ad Filterig..8 Some Basic Sequeces Uit sample sequece, δ[, Uit step sequece Relatios betwee Basic Sequeces Uit sample ad uit step sequeces are related as follows: μ[ δ[ δ[ μ[ μ[, μ[, <... he above relatios ca be implemeted with simple computatioal structures cosistig of basic arithmetic operatios 8 Olli Simula 9 8 Olli Simula Relatios betwee Basic Sequeces he uit sample is the first differece of the uit step: δ[ μ[ μ[ μ[ μ[ δ[... μ [ δ[ Olli Simula... D μ[ Realizatio Relatios betwee Basic Sequeces Uit step is the ruig sum of the uit sample: μ [ δ[ m δ [ m + δ[ μ[ + δ[ m m δ[ 8 Olli Simula + D Realizatio μ[ μ[ Basic Operatios o Sequeces x [ + x Additio: [ + x[ x [ a Multiplicatio: x [ a Expoetial ad Siusoidal Sequeces Complex expoetial sequece Aα where A ad α are complex jφ ( σ + jω ) σ j( ω+ φ ) Ae e Ae e σ Ae [ cos( ω + φ ) + j si( i( ω + φ ) Uit delay: x [ D 8 Olli Simula 3 8 Olli Simula 4 Mitra 3rd Editio: Chapter ; 8 Olli Simula 4

5 -6.3 Digital Sigal Processig ad Filterig..8 Real Expoetial Sequeces With both A ad α real, the sequece reduces to a real expoetial sequece A Family of Siusoidal Sequeces A real siusoidal sequece: Acos(ω +φ) 8 Olli Simula 5 8 Olli Simula 6 he Samplig Process A discrete-time sequece is developed by uiformly samplig the cotiuous-time sigal x a (t) x ( t) x ( ) a t a he time variable -time t is related to the discrete time variable oly at discrete-time istats t π t F Ω with ad F / Ω πf (samplig frequecy) (samplig agular frequecy) 8 Olli Simula 7 he Samplig Process Cosider ( t) Acos( Ωt + φ) Now where x a Acos( Ω + φ) πω Acos + φ Acos( ω + φ) Ω ω πω Ω Ω ω is the ormalized agular frequecy 8 Olli Simula 8 Example: hree Siusoidal Sequeces g [ cos(.6π) g [ cos(.4π ) g [ cos(.6π) 3 g [ cos((π.6π ) ) cos(.6π) g[ g 3[ cos((π +.6π ) ) cos(.6π) g[ Amplitude time he Aliasig Pheomeo I geeral, the family of cotiuous-time siusoids xa, ( t) Acos(( Ω + Ω ) t + φ), ±, ±,... lead to idetical sampled sigals x a, ( ) Acos(( Ω + Ω ) + φ ) π ( Ω + Ω ) πω Acos + φ Acos + φ Ω Ω Acos( ω + φ) he pheomeo is called aliasig 8 Olli Simula 9 8 Olli Simula 3 Mitra 3rd Editio: Chapter ; 8 Olli Simula 5

6 -6.3 Digital Sigal Processig ad Filterig..8 Arbitrary Sequece -3 Arbitrary Sequece A arbitrary sequece ca be expressed as a superpositio of scaled versios of shifted uit impulses, δ[- 8 Olli Simula 3 3 δ [ + 3 δ [ 4 δ[ 4 x [ + - I geeral: + x [ δ[ 8 Olli Simula 3 Discrete-ime Systems Discrete-time system x [ Iput sequece Output sequece Sigle-iput sigle-output system Output sequece is geerated sequetially, begiig with a certai time idex value A certai class of discrete-time systems, liear ad time ivariat (LI) systems will be discussed 8 Olli Simula 33 Liearity A liear system is a system that possesses the importat property of superpositio Additivity: he respose to x [+x [ is y [+y [ Scalig or homogeeity: he respose to ax [ is ay [ where a is ay complex costat 8 Olli Simula 34 Liearity Combiig the two properties of superpositio ito a sigle statemet Discrete-time: ax [ + bx[ a + b where a ad b are ay complex costats he superpositio property holds for liear systems 8 Olli Simula 35 Liearity a x [ [ y [ [ ay x [ [ y [ b by [ + ay [ + b a x ax [ [ b + [ ay [ + b ax x [ [ + bx[ bx [ 8 Olli Simula 36 Mitra 3rd Editio: Chapter ; 8 Olli Simula 6

7 -6.3 Digital Sigal Processig ad Filterig..8 ime Ivariace A system is time-ivariat (or shift-ivariat) if a time shift i the iput sigal results i a idetical time shift i the output sigal y [ x [ ( ) ( ) For time-ivariat systems the system properties do ot chage with time ime Ivariace A time ivariat discrete-time system [ y [ si A time variat discrete-time system y [ Coefficiet is chagig with time 8 Olli Simula 37 8 Olli Simula 38 Causality I a causal discrete-time system the output sample at time istat depeds oly o the iput samples for < ad does ot deped o iput samples for > If y [ ad y [ are the resposes of a causal system to two iputs u [ ad u [, respectively, the u u [, for < N implies that [ y [, for < N 8 Olli Simula 39 Stability A discrete-time system is stable if ad oly if, for every bouded iput, the output is also bouded If the respose to is the sequece, ad if for all values of, the 8 Olli Simula 4 B x B y for all values of, where B x ad B y are fiite costats Bouded-iput bouded-output (BIBO) stability Impulse ad Step Respose x [ h[ Uit sample respose or impulse respose is the respose of the system to a uit impulse x [ δ[ ; h[ Uit step respose or step respose is the output sequece whe the iput sequece is the uit step x [ μ[ ; s[ Covolutio Liearity: he respose of a liear system to will be the superpositio of the scaled resposes of the system to each of these shifted impulses ime ivariace: he resposes of a time-ivariat system to time-shifted uit impulses are the time-shifted versios of oe aother 8 Olli Simula 4 8 Olli Simula 4 Mitra 3rd Editio: Chapter ; 8 Olli Simula 7

8 -6.3 Digital Sigal Processig ad Filterig..8 Covolutio he uit impulse respose of a system is h[ ( ) δ [ h[ he uit impulse respose h[ is the respose of the system to a uit impulse Covolutio y [ ( ) δ[ Additivity : Homogeeity : Shift ivariace: ( δ[ ) y [ ( δ[ ) y [ y [ h[ 8 Olli Simula 43 8 Olli Simula 44 Basic Properties of LI Systems he Commutative Property he Distributive Property he Associative Property he Commutative Property x [ * h[ h[ * Let r- or -r; substitutig to covolutio sum: x [ * h[ h[ r x [ r h[ r h[ * 8 Olli Simula 45 8 Olli Simula 46 he Commutative Property x [ h[ h[ [ x [ he output of a LI system with iput ad uit impulse respose h[ is idetical to the output of a LI system with iput h[ ad uit impulse respose he Distributive Property ( h [ + h [ ) * x [ * h[ + * h[ he distributive property has a useful iterpretatio i terms of system itercoectios > PARALLEL INERCONNECION 8 Olli Simula 47 8 Olli Simula 48 Mitra 3rd Editio: Chapter ; 8 Olli Simula 8

9 -6.3 Digital Sigal Processig ad Filterig..8 he Distributive Property he Associative Property x [ h + h [ [ y [ h [ x [ + h [ y [ ( h [ * h [ ) * ( x * h [ )* h [ [ As a cosequece of associative property the followig expressio is uambiguous y [ * h[ * h[ 8 Olli Simula 49 8 Olli Simula 5 he Associative Property x [ h * h [ ( h [ * h [ ) y [ * [ ( * h [ )* h [ y [ * h [ y [ y [ x [ h [ h [ he Associative Property he associative property ca be iterpreted as > SERIES (OR CASCADE) INERCONNECION OF SYSEMS 8 Olli Simula 5 8 Olli Simula 5 he Associative ad Commutative Property * ( h [ * h [ ) * ( h [ * h [ ) y [ x [ h * h [ [ ( * h [ )* h [ y [ * h [ y [ y [ x [ h [ h [ he Properties of Cascade Coectio of Systems he order of the systems i cascade ca be iterchaged he itermediate sigal values, w i [, betwee the systems are differet Differet structures have differet properties whe implemeted usig fiite precisio arithmetic 8 Olli Simula 53 8 Olli Simula 54 Mitra 3rd Editio: Chapter ; 8 Olli Simula 9

10 -6.3 Digital Sigal Processig ad Filterig..8 he Cascade Coectio of Systems y [ x [ h [ h [ x [ h * h [ [ x [ h * h [ [ y [ x [ h [ h [ he Cascade Coectio of Systems he properties of the cascade system deped o the sequetial order of cascaded blocs he behavior of discrete-time systems with fiite wordlegth is sesitive to sigal values, w i [, betwee the blocs What is the optimal sequetial order of cascaded blocs? 8 Olli Simula 55 8 Olli Simula 56 Stability for LI Systems Cosider a iput that is bouded i magitude < B for all he output is give by the covolutio sum y [ h[ y [ h[ Stability for LI Systems For bouded iput - < B B h[ he output [ is bouded if the the impulse respose is absolutely summable h [ < A SUFFICIEN CONDIION FOR SABILIY! for all 8 Olli Simula 57 8 Olli Simula 58 Causality Coditio Let x [ ad x [ be two iput sequeces with x [ x [ for the the correspodig output sequece of a causal system y [ y [ for he system is causal if ad oly if h[ for < 8 Olli Simula 59 Fiite-Dimesioal LI Discrete-ime Systems A importat subclass of LI discrete-time is characterized by a liear costat coefficiet differece equatio N d y [ M p x [ where ad are, respectively, the iput ad output of the system ad {d } ad {p } are costats he order of the system is give by max{n,m} 8 Olli Simula 6 Mitra 3rd Editio: Chapter ; 8 Olli Simula

11 -6.3 Digital Sigal Processig ad Filterig..8 Fiite-Dimesioal LI Discrete-ime Systems he output ca be computed recursively by solvig N d p y [ + d d M d provided that d. he output ca be computed for all, owig the iput ad the iitial coditios -, -,..., -N Classificatio of LI Discrete-ime Systems LI discrete-time are usually classified either accordig to the legth of the their impulse resposes or accordig to the method of calculatio employed to determie the output samples Impulse respose classificatio: Fiite impulse respose (FIR) systems Ifiite impulse respose (IIR) systems 8 Olli Simula 6 8 Olli Simula 6 Classificatio Based o Impulse Respose If h[ is of fiite legth, i.e., h[, for < N ad > N, with N < N the it is ow as a fiite impulse respose (FIR) discrete-time system he covolutio sum reduces to N N h[ ca be calculated directly from the fiite sum 8 Olli Simula 63 Classificatio Based o Impulse Respose If h[ is of ifiite legth the the system is ow as a ifiite impulse respose (IIR) discrete-time system For a causal IIR discrete-time time system with causal iput, the covolutio sum ca be expressed as h[ ca ow be calculated sample by sample 8 Olli Simula 64 Classificatio Based o Output Calculatio Process If the output sample ca be calculated sequetially, owig oly the preset ad past iput samples, the filter is said to be orecursive discrete-time system If, o the other had, the computatio of the output ivolves past output samples i additio to the preset ad past iput samples, the filter is ow as recursive discrete-time system N d d [ + y 8 Olli Simula 65 M p d Classificatio Based o Output Calculatio Process A differet termiology is used to classify causal fiite-dimesioal LI systems i differet applicatios, such as model-based spectral aalysis he classes assiged here are based o the form of the liear costat coefficiet differece equatio modelig the system 8 Olli Simula 66 Mitra 3rd Editio: Chapter ; 8 Olli Simula

12 -6.3 Digital Sigal Processig ad Filterig..8 Movig Average (MA) Model he simplest model is described by the iputoutput relatio M [ y p A movig average (MA) model is a FIR discrete-time system It ca be cosidered as a geeralizatio of the M-poit movig average filter with differet weights assiged to iput samples 8 Olli Simula 67 Autoregressive Models he simplest IIR, called a autoregresive (AR) model is characterized by the iput-output relatio N y x d [ [ he secod type of IIR system, called a autoregresive movig average (ARMA) model is described by the iput-output relatio M [ p y d 8 Olli Simula 68 N Correlatio of Sigals ad Matched Filters Correlatio of Sigals here are applicatios where it is ecessary to compare oe referece sigal with oe or more sigals to determie the similarity betwee the pair ad to determie additioal iformatio based o the similarity 8 Olli Simula 7 Example: Commuicatios I digital commuicatios, a set of data symbols are represeted by a set of uique discrete-time sequeces If oe of these sequeces has bee trasmitted, the receiver has to determie which particular sequece has bee received he received sigal is compared with every member of possible sequeces from the set Correlatio Example: Radar Applicatios Similarly, i radar ad soar applicatios, the received sigal reflected from the target is a delayed versio of the trasmitted sigal By measurig the delay, oe ca determie the locatio of the target he detectio problem gets more complicated i practice, as ofte the received sigal is corrupted by additive radom oise 8 Olli Simula 7 8 Olli Simula 7 Mitra 3rd Editio: Chapter ; 8 Olli Simula

13 -6.3 Digital Sigal Processig ad Filterig..8 Correlatio of Sigals Defiitios A measure of similarity betwee a pair of eergy sigals, ad, is give by the crosscorrelatio sequece r xy [l defied by [ rxy l l, l, ±, ±,... he parameter l called lag, idicates the time-shift betwee the pair of sigals Correlatio of Sigals Sequece is said to be shifted by l samples to the right with respect to the referece sequece for positive values of l, ad shifted by l samples to the left for egative values of l he orderig of the subscripts xy i the defiitio of r xy [l specifies that is the referece sequece which remais fixed i time while is beig shifted with respect to 8 Olli Simula 73 8 Olli Simula 74 Correlatio of Sigals If is made the referece sigal ad shift with respect to, the the correspodig cross-correlatio sequece is give by y [ x [ r [ l l yx m + l m rx l m hus, r yx [l is obtaied by time-reversig r xy [l Correlatio of Sigals he autocorrelatio sequece of is give by r [ l l xx obtaied by settig [ i the defiitio of the cross-correlatio sequece r xy [l Note: he eergy of the sigal is r xx x [ [ E x r xy [l 8 Olli Simula 75 8 Olli Simula 76 Correlatio ad Covolutio From the relatio r yx [l r xy [-l it follows that r xx [l r xx [-l implyig that r xx [l is a eve fuctio for real A examiatio of r [ l l xy reveals that the expressio for the crosscorrelatio loos quite similar to that of the liear covolutio 8 Olli Simula 77 Covolutio Revisited he covolutio of m ad h[m was defied as y [ m h[ m Compare to correlatio r [ l l xy Replacig ow m by l ad by, we obtai r [ l ( l ) xy 8 Olli Simula 78 Mitra 3rd Editio: Chapter ; 8 Olli Simula 3

14 -6.3 Digital Sigal Processig ad Filterig..8 Correlatio ad Covolutio he expressio for the cross-correlatio is ow similar to the covolutio, i.e., rx l ( l ) l l he equatios of correlatio ad covolutio are the same, except the mius sig iside the summatio I step-by-step calculatio of the covolutio, the other sequece is time-reversed; i correlatio, it is ot 8 Olli Simula 79 Matched Filter he cross-correlatio of with the referece sigal ca be computed by processig with a LI discrete-time system of impulse respose - x [ r xy [ he impulse respose, h[, of the matched filter is the time-reversed versio of the of referece sigal, i.e., h[ - 8 Olli Simula 8 Applicatios of Matched Filters I matched filters, the impulse respose of the filter is matched to the sigal, or sigal patter of iterest Applicatios: Radar, the impulse respose of the filter is the timereversed versio of the sigal to be detected Patter recogitio emplate matchig i image aalysis, i.e., sub-areas of the image are correlated with the desired template Autocorrelatio Liewise, the autocorrelatio of ca be computed by processig with a LI discrete-time system of impulse respose - x [ r xx [ 8 Olli Simula 8 8 Olli Simula 8 Mitra 3rd Editio: Chapter ; 8 Olli Simula 4