More on FT. Lecture 10 4CT.5 3CT.3-5,7,8. BME 333 Biomedical Signals and Systems - J.Schesser

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1 Mr n FT Lcur 4CT.5 3CT.3-5,7,8 BME 333 Bimdicl Signls nd Sysms - J.Schssr 43

2 Highr Ordr Diffrniin d y d x, m b Y b X N n M m N M n n n m m n m n d m d n m Y n d f n [ n ] F d M m bm m X N n n n n n m p n n p p y bp m x B p y x Ap B Y X Η X A BME 333 Bimdicl Signls nd Sysms - J.Schssr 44

3 Frquncy Rspns W us hv sn h h FT f h diffrnil quin dscribing sysm yilds h Frquncy Rspns, Hω W hv ls sn h cnvlving h impuls rspns, h, wih ny inpu, x will yild is upu, y. Is h Frquncy Rspns rld h impuls rspns? BME 333 Bimdicl Signls nd Sysms - J.Schssr 45

4 Frquncy Rspns nd h Impuls Rspns Using cnvluin, fr nly sinusidl inpus: x A y h x d h A d y x h d Tking h FT f bh sids y d x dh d h A d A h d x h d nd ning h h d is n funcin f : Y Y X h d H h d X Bu h d is h FT f h nd, hrfr, h frquncy rspns is us h FT f h impuls rspns. W sy h H nd h r FT pirs BME 333 Bimdicl Signls nd Sysms - J.Schssr 46

5 Anhr Inrsing Exmpl f fr - ; [ u F u d f ] d sin S FT f rcngulr puls is h Smpling Funcin Wh is h FT f u? BME 333 Bimdicl Signls nd Sysms - J.Schssr 47

6 FT f h Uni Impuls [ ] Frm [ ] If x=δ, hn y=h h rspns du uni impuls rspns nd Hω is h nwrk rspns r sysm funcin in phsr frm B B Y X ; X ; H A A d f x f x d This implis h h FT f h impuls rspns is h nwrk rspns r frquncy rspns H [ h ] h d BME 333 Bimdicl Signls nd Sysms - J.Schssr 48

7 FT f δ cninud [] d cs sin d csd sind cs d cs d; using h prpris f vn nd dd funcins. cs d BME 333 Bimdicl Signls nd Sysms - J.Schssr 49

8 Odd nd Evn Prpris f Signl N h h sin is dd nd lim sind lim{ sind sin d} If sin d g ; hn sin d g sinc L x, sin x dx sin x dx sin x dx sin x dx g lim{ g g } N h h csin is vn nd lim csd lim{ csd cs d} If cs d g ; hn cs d g sinc L x, cs x dx cs x dx cs x dx cs x dx g lim{ g g } lim{ g } lim csd BME 333 Bimdicl Signls nd Sysms - J.Schssr 5

9 Clculin f h Frquncy Rspns Givn h =δ nd i is knwn h = - u wh is FT f [ - u ] which mus qul b Hω? W cn clcul FT f [ - u ] by vluing h ingrl f - u -ω r clcul h sysm rspns funcin: / H [ u ] BME 333 Bimdicl Signls nd Sysms - J.Schssr 5

10 Furir Trnsfrm f A Cnsn [] d cs sin d Sinc cnsn is n vn funcin vr : [] csd Bu w sid h csd Our firs Duliy: A cnsn in h im dmin is uni impuls funcin in h frquncy dmin A uni impuls funcin in h im dmin is cnsn in h frquncy dmin BME 333 Bimdicl Signls nd Sysms - J.Schssr 5

11 FT f Uni Sp Funcin [ ] u u d d, which is undfind L's us nhr chniqu : df [ ] F d du [ ] [ ] [ u ] d [ u ], fr BME 333 Bimdicl Signls nd Sysms - J.Schssr 53

12 BME 333 Bimdicl Signls nd Sysms - J.Schssr 54 Tim & Frquncy Displcmn d d d T x T T x d d F dx x f dx x f d T f T f F f - T l x, ] [ is hn wh, ] [ If Tim Displcmn d S sin / sin S

13 Tim & Frquncy Displcmn Frquncy Displcmn If [ f ] F, hn wh is F Assum hr xis duliy bwn Tim nd Frquncy displcmn nd ry [ f ] f d f d F BME 333 Bimdicl Signls nd Sysms - J.Schssr Our nx Duliy: A shif in h im dmin quls muliplicin by -ωt h frquncy dmin A shif in h frquncy dmin quls muliplicin by - h im dmin 55

14 BME 333 Bimdicl Signls nd Sysms - J.Schssr 56 FT Symmris ]is dd Im[ ]is vn, R[ Which mns : And : And : N : * * F F F F d f F d f F d f F

15 Sm Inrsing FTs Expnnil Funcins [ [ u ] u ] du sinc [ ] d,sinc [ u ] [ u ],sinc [ u ] [ u ], BME 333 Bimdicl Signls nd Sysms - J.Schssr 57

16 Mr Inrsing FTs [ f f ] m cs m m M[ ] M[ ] This sys h w hv md w cpis f h spcrum f m n shifd h righ f h ω= xis nd n h lf. Bh f hs cpis r rducd by ½. M S / sin L s lk h cs whr m is puls nd hs h spcrum shwn bv. BME 333 Bimdicl Signls nd Sysms - J.Schssr 58

17 Frquncy Shifing Exmpl / M S sin F M[ ] M[ ] sin[ ] sin[ ] { } Mω..8.6 F =τ =,ω = BME 333 Bimdicl Signls nd Sysms - J.Schssr 59

18 BME 333 Bimdicl Signls nd Sysms - J.Schssr 6 Furir Trnsfrm nd Furir Sris FT FS Tim Apridic Pridic Frquncy Cninuus Discr ] [ ] [ ] [sin ] [ ] [ ] [cs Frquncy Shif ] [ ] [ [] ] [ Nw wh bu FT f pridic funcin Rcll:

19 FT f Pridic Funcin FT f Pridic Funcin f A pridic funcin cn frmuld in FS: f [ f ] F k k T k T FT f pridic funcin is sris f uni impuls funcins nd is discr in h frquncy dmin. N : FT f rin f uni impulss f [ f ] F A n nt FT f rin f uni impulss is FSin h frquncy dmin nd is cninuus in h frquncy dmin k A nt n BME 333 Bimdicl Signls nd Sysms - J.Schssr 6

20 Enrgy f f Enrgy dissipd by, sy, hm rsisnc vr prid T T is v d T W ls clld his h qudric cnn f signl I cn b shwn h h qudric cnn f signl in rms f is FT is: T f d * F F d T And T T f d F d Th lr quin is clld Prsvl s Thrm BME 333 Bimdicl Signls nd Sysms - J.Schssr 6

21 Hmwrk Prblm Shw h Fω f n vn funcin f=f- is rl Shw h Fω f n dd funcin f=-f- is imginry Prblm Find Fω fr h fllwing funcins nd skch hir mpliud nd phs spcr: f = - u; f b = u-+ - u; f c = - u-+ - u; f d = sin u- BME 333 Bimdicl Signls nd Sysms - J.Schssr 63

22 Hmwrk Prblm 3 Clcul h Fω fr h wvfrms blw. N h h scnd is h ingrl f h firs CT.5., 4CT.5. 3CT.5., 3CT.5., 3CT.5.3 BME 333 Bimdicl Signls nd Sysms - J.Schssr 64

Frequency Response. Lecture #12 Chapter 10. BME 310 Biomedical Computing - J.Schesser

Frequency Response. Lecture #12 Chapter 10. BME 310 Biomedical Computing - J.Schesser Frquncy Rspns Lcur # Chapr BME 3 Bimdical Cmpuing - J.Schssr 99 Idal Filrs W wan sudy Hω funcins which prvid frquncy slciviy such as: Lw Pass High Pass Band Pass Hwvr, w will lk a idal filring, ha is,