Digital Signal Processing, Fall 2010

Transcription

1 Digital Sigal Processig, Fall 2 Lecture : Itroductio, Discrete-time sigals ad sstems Zheg-Hua Ta Departmet of Electroic Sstems alborg Uiversit, Demar zt@es.aau.d Digital Sigal Processig, I, Zheg-Hua Ta Part I: Itroductio Itroductio (Course overview) Discrete-time sigals Discrete-time sstems Liear time-ivariat sstems Sstem impulse respose Liear costat-coefficiet differece equatios 2 Digital Sigal Processig, I, Zheg-Hua Ta

2 Geeral iformatio Course website Tetboo: Oppeheim,.V., Schafer, R.W, "Discrete-Time Sigal Processig", 2d Editio, Pretice-Hall, 999. Readigs: 3 Steve W. Smith, The Scietist ad Egieer's Guide to Digital Sigal Processig, Califoria Techical Publishig, / dfb (You ca dowload the etire boo!) Kermit Sigmo, "Matlab Primer", Third Editio, Departmet of Mathematics, Uiversit of Florida. V.K. Igle ad J.G. Proais, "Digital Sigal Processig usig MTLB", Booware Compaio Series, 2. Digital Sigal Processig, I, Zheg-Hua Ta Geeral iformatio Duratio ECTS (5 Lectures) Prerequisites: Bacgroud i advaced calculus icludig comple variables, Laplace- ad Fourier trasforms. Course tpe: Stud programme course (SE-course), ruig evaluatio Lecturer: ssociate Professor, Ph.D., Zheg-Hua Ta Niels Jeres Vej 2, 6-39 zt@es.aau.d, Digital Sigal Processig, I, Zheg-Hua Ta 2

3 Course at a glace MM Discrete-time sigals ad sstems Sstem MM2 Fourier trasform ad Z-trasform Samplig ad recostructio Sstem aalsis DFT/FFT MM5 MM3 Filter desig MM4 5 Digital Sigal Processig, I, Zheg-Hua Ta Course objectives To uderstad the cocepts of discrete-time sigals ad sstems To uderstad the Z- ad the Fourier trasform ad their iverse To uderstad the relatio betwee digital filters, differece equatios ad sstem fuctios To uderstad the priciples of samplig ad recostructio To be able to appl digital filters accordig to ow filter specificatios To ow the priciples behid the discrete Fourier trasform (DFT) ad its fast computatio To be able to appl MTLB to DSP problems 6 Digital Sigal Processig, I, Zheg-Hua Ta 3

4 What is a sigal? flow of iformatio. (mathematicall represeted as) a fuctio of idepedet variables such as time (e.g. speech sigal), positio (e.g. image), etc. commo covetio is to refer to the idepedet variable as time, although ma i fact ot. 7 Digital Sigal Processig, I, Zheg-Hua Ta Eample sigals Speech: -Dimesio sigal as a fuctio of time s(t);. Gre-scale image: 2-Dimesio sigal as a fuctio of space i(,) Video: 3 3-Dimesio sigal as a fuctio of space ad time {r(,,t), g(,,t), b(,,t)}. 8 Digital Sigal Processig, I, Zheg-Hua Ta 4

5 Tpes of sigals The idepedet variable ma be either cotiuous or discrete Cotiuous-time sigals Discrete-time sigals are defied at discrete times ad represeted as sequeces of umbers The sigal amplitude ma be either cotiuous or discrete alog sigals: both time ad amplitude are cotiuous Digital sigals: both are discrete Computers ad other digital devices are restricted to discrete time Sigal processig sstems classificatio follows the same lies 9 Digital Sigal Processig, I, Zheg-Hua Ta Tpes of sigals From Digital Sigal Processig, I, Zheg-Hua Ta 5

6 Digital sigal processig Modifig ad aalzig iformatio with computers so beig measured as sequeces of umbers. Represetatio, trasformatio ad maipulatio of sigals ad iformatio the cotai Digital Sigal Processig, I, Zheg-Hua Ta Tpical DSP sstem compoets Iput lowpass filter to avoid aliasig alog to digital coverter (DC) Computer or DSP processor Digital to aalog coverter (DC) Output lowpass filter to avoid imagig 2 Digital Sigal Processig, I, Zheg-Hua Ta 6

7 DC ad DC Trasducers e.g. microphoes alog-to-digital coverters Phsical sigals alog sigals Digital sigals Output devices Digital-to-alog coverters 3 Digital Sigal Processig, I, Zheg-Hua Ta Pros ad cos of DSP Pros Cos Eas to duplicate Stable ad robust: ot varig with temperature, storage without deterioratio Fleibilit ad upgrade: use a geeral computer or microprocessor Limitatios of DC ad DC High power cosumptio ad compleit of a DSP implemetatio: usuitable for simple, low-power applicatios Limited to sigals with relativel low badwidths 4 Digital Sigal Processig, I, Zheg-Hua Ta 7

8 pplicatios of DSP Speech processig Ehacemet oise filterig Codig, sthesis ad recogitio Image processig Ehacemet, codig, patter recogitio (e.g. OCR) Multimedia processig Media trasmissio, digital TV, video coferecig Commuicatios Biomedical egieerig Navigatio, radar, GPS Cotrol, robotics, machie visio 5 Digital Sigal Processig, I, Zheg-Hua Ta Histor of DSP Prior to 95 s: aalog sigal processig usig electroic circuits or mechaical devices 95 s: computer simulatio before aalog implemetatio, thus cheap to tr out 965: Fast Fourier Trasforms (FFTs) b Coole ad Tue mae real time DSP possible 98 s: IC techolog boostig DSP 6 Digital Sigal Processig, I, Zheg-Hua Ta 8

9 Part II: Discrete-time sigals Itroductio Discrete-time sigals Discrete-time sstems Liear time-ivariat sstems Sstem impulse respose Liear costat-coefficiet differece equatios 7 Digital Sigal Processig, I, Zheg-Hua Ta Discrete-time sigals Sequeces of umbers { }, - where is a iteger Periodic samplig of a aalog sigal a ( T ), wheret is called the samplig period. 2 8 Digital Sigal Processig, I, Zheg-Hua Ta 9

10 Sequece operatios The product ad sum of two sequeces ad : sample-b-sample productio ad sum, respectivel. Multiplicatio of a sequece b a umber : multiplicatio of each sample value b. Dela or shift of a sequece where is a iteger 9 Digital Sigal Processig, I, Zheg-Hua Ta Basic sequeces Uit sample sequece (discrete-time impulse, impulse),,,, sequece ca be represeted as a sum of scaled, delaed impulses a3 3 a a5 5 More geerall 2 Digital Sigal Processig, I, Zheg-Hua Ta

11 Uit step sequece Defied as, u,,, Related to the impulse b Coversel, u 2... or u u u u 2 Digital Sigal Processig, I, Zheg-Hua Ta Epoetial sequeces Etremel importat i represetig ad aalzig LTI sstems. Defied as If ad are real umbers, the sequece is real. If ad is positive, the sequece values are positive ad decrease with icreasig. If, the sequece values alterate t i sig, but agai decrease i magitude with icreasig. If, the sequece values icrease with 2(.5) icreasig. 22 Digital Sigal Processig, I, Zheg-Hua Ta 2(.5) 2 2

12 2 Combiig basic sequeces epoetial sequece that is zero for <,,, u Digital Sigal Processig, I, Zheg-Hua Ta 23 Siusoidal sequeces ith ad real costats for all ), cos( with ad real costats. The with comple has real ad imagiar parts that are epoetiall weighted siusoids. the, ad If e e e e j j j j Digital Sigal Processig, I, Zheg-Hua Ta 24 ) si( ) cos( ) ( j e e e j

13 Comple epoetial sequece Whe, e ( ) cos( ) j si( ) j B aalog with the cotiuous-time case, the quatit is called the frequec of the comple siusoid or comple epoetial ad is call the phase. is alwas a iteger differeces betwee discrete-time ad cotiuous-time 25 Digital Sigal Processig, I, Zheg-Hua Ta importat differece frequec Cosider a frequec ( 2 ) j( j j j e 2 ) e e 2 e ( 2r ), r More geerall beig a iteger, j( r j j e 2 ) e e 2 Same for siusoidal sequeces r j e cos( 2 r ) cos( ) 2 So, ol cosider frequecies i a iterval of such as or 2 26 Digital Sigal Processig, I, Zheg-Hua Ta 3

14 importat differece frequec For a cotiuous-time siusoidal sigal ( t) cos( t ), as icreases, ( t ) oscillates more ad more rapidl For the discrete-time siusoidal sigal cos( ), as icreases from towards, oscillates more ad more rapidl as icreases from towards 2, the oscillatios become slower. 27 Digital Sigal Processig, I, Zheg-Hua Ta other importat differece periodicit I the cotiuous-time case, a siusoidal sigal ad a comple epoetial sigal are both periodic. I the discrete-time case, a periodic sequece is defied as where the period N is ecessaril a iteger. For siusoid, cos( ) N, which requires that where is a iteger. for all cos( N ) N 2 or N 2 / 28 Digital Sigal Processig, I, Zheg-Hua Ta 4

15 other importat differece periodicit Same for comple epoetial sequece j ( N ) j e e, which h is true ol for N 2 So, comple epoetial ad siusoidal sequeces are ot ecessaril periodic i with period 2 / ) ad, depedig o the value of, ma ot be periodic at all. Cosider cos( / 4), 2 with a period of N 8 ( cos(3 / 8), with a period of N 6 Icreasig frequec icreasig period! 29 Digital Sigal Processig, I, Zheg-Hua Ta Frequec 3 Digital Sigal Processig, I, Zheg-Hua Ta 5

16 Frequec 2 4 or 8 8 N=4 N=8 6 or 8 8 N=8 N=2 3 Digital Sigal Processig, I, Zheg-Hua Ta Part III: Discrete-time sstems Itroductio Discrete-time sigals Discrete-time sstems Liear time-ivariat sstems Sstem impulse respose Liear costat-coefficiet differece equatios 32 Digital Sigal Processig, I, Zheg-Hua Ta 6

17 Discrete-time sstems trasformatio or operator that maps iput ito output T { } T{.} Eamples: The ideal dela sstem d, d memorless sstem 2 ( ), 33 Digital Sigal Processig, I, Zheg-Hua Ta Liear sstems sstem is liear if ad ol if additivit propert T{ 2 } T{ } T{ 2 } 2 ad T{ a } at{ } a scalig propert where a is a arbitrar costat Combied ito superpositio T{ a b2 } at{ } at{ 2 } a a2 34 Digital Sigal Processig, I, Zheg-Hua Ta 7

18 8 Eamples ccumulator sstem a liear sstem oliear sstem ) (, b a b a Digital Sigal Processig, I, Zheg-Hua Ta 35 oliear sstem ad Cosider ) ( log 2 Time-ivariat sstems For which a time shift or dela of the iput sequece causes a correspodig shift i the output sequece. ccumulator sstem Digital Sigal Processig, I, Zheg-Hua Ta 36

19 Causalit The output sequece value at the ide = depeds ol o the iput sequece values for <=. Eample d, Causal for d >= Nocausal for d < 37 Digital Sigal Processig, I, Zheg-Hua Ta Stabilit sstem is stable i the BIBO sese if ad ol if ever bouded iput sequece produces a bouded output sequece. Eample stable ( ) 2, 38 Digital Sigal Processig, I, Zheg-Hua Ta 9

20 Part IV: Liear time-ivariat sstems Course overview Discrete-time sigals Discrete-time sstems Liear time-ivariat sstems Sstem impulse respose Liear costat-coefficiet differece equatios 39 Digital Sigal Processig, I, Zheg-Hua Ta Liear time-ivariat sstems Importat due to coveiet represetatios ad sigificat applicatios liear sstem is completel characterised b its impulse respose T{ } T{ } Time ivariace LTI T{ } h h h h * h h Covolutio sum 4 Digital Sigal Processig, I, Zheg-Hua Ta 2

21 Formig the sequece h- 4 Digital Sigal Processig, I, Zheg-Hua Ta Computatio of the covolutio sum * h h Obtai the sequece h- Reflectig h about the origi to get h- Shiftig the origi of the reflected sequece to = Multipl ad h- for Sum the products to compute the output sample 42 Digital Sigal Processig, I, Zheg-Hua Ta 2

22 Computig a discrete covolutio Eample 2.3 pp.26 Impulse respose h u u N, N,, otherwise. iput a u,, a, N, a N N a a ( ), N. a 43 Digital Sigal Processig, I, Zheg-Hua Ta Properties of LTI sstems Defied b discrete-time covolutio Commutative * h h * Liear *( h h2 ) * h * h2 Cascade coectio (Fig. 2. pp.29) h h * h 2 Parallel coectio (Fig. 2.2 pp.3) h h h2 44 Digital Sigal Processig, I, Zheg-Hua Ta 22

23 Properties of LTI sstems Defied b the impulse respose Stable Causalit S h h, 45 Digital Sigal Processig, I, Zheg-Hua Ta Part V: Sstem impulse respose Course overview Discrete-time sigals Discrete-time sstems Liear time-ivariat sstems Sstem impulse respose Liear costat-coefficiet differece equatios 46 Digital Sigal Processig, I, Zheg-Hua Ta 23

24 FIR sstems reflected i the h Ideal dela d, h d, d a positive iteger. Forward differece h Bacward differece h Fiite-duratio impulse respose (FIR) sstem The impulse respose has ol a fiite umber of ozero samples. - d 47 Digital Sigal Processig, I, Zheg-Hua Ta IIR sstems reflected i the h ccumulator Ifiite-duratio impulse respose (IIR) sstem The impulse respose is ifiitive i duratio. Stabilit h u S h? FIR sstems alwas are stable, if each of h values is fiite i magitude. IIR sstems ca be stable, e.g. h a u with a S a ( a ) 48 Digital Sigal Processig, I, Zheg-Hua Ta 24

25 Cascadig sstems Causalit? h, Ideal dela h d Forward differece h Oe - sameple dela h Bacward differece h ocausal FIR sstem ca be made causal b cascadig it with a sufficietl log dela! 49 Digital Sigal Processig, I, Zheg-Hua Ta Cascadig sstems ccumulator + Bacward differece sstem ccumulator sstem h u Bacward differece h Iverse sstem: h h * h h * h i i 5 Digital Sigal Processig, I, Zheg-Hua Ta 25

26 Part VI: LCCD equatios Course overview Discrete-time sigals Discrete-time sstems Liear time-ivariat sstems Sstem impulse respose Liear costat-coefficiet differece equatios 5 Digital Sigal Processig, I, Zheg-Hua Ta LCCD equatios importat class of LTI sstems: iput ad output satisf a Nth-order LCCD equatios N M Differece equatio represetatio of the accumulator 52 a m m b m Digital Sigal Processig, I, Zheg-Hua Ta + Oe-sample dela - Recursive represetatio 26

27 MTLB iteractive, matri-based sstem for umeric computatio ad visualizatio Kermit Sigmo, "Matlab Primer", Third Editio, Departmet of Mathematics, Uiversit of Florida. Matlab Help (>> doc) 53 Digital Sigal Processig, I, Zheg-Hua Ta Summar Course overview Discrete-time sigals Discrete-time sstems Liear time-ivariat sstems Sstem impulse respose Liear costat-coefficiet differece equatios 54 Digital Sigal Processig, I, Zheg-Hua Ta 27

28 Course at a glace MM Discrete-time sigals ad sstems Sstem MM2 Fourier trasform ad Z-trasform Samplig ad recostructio Sstem aalsis DFT/FFT MM5 MM3 Filter desig MM4 55 Digital Sigal Processig, I, Zheg-Hua Ta 28