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

Transcription

1 Frquncy Rspns Lcur # Chapr BME 3 Bimdical Cmpuing - J.Schssr 99

2 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, filr which hav idal prfrmanc bu ar vry difficul cnsruc. BME 3 Bimdical Cmpuing - J.Schssr 3

3 BME 3 Bimdical Cmpuing - J.Schssr 3 A simpl Filr Idal Dlay Idal Dlay Filr => y = x d : h upu is sam as h inpu xcp shifd in im by an amun d scnds. Th impuls rspns is us h = δ d Th frquncy rspns is hn Th Frquncy Rspns f an Idal Dlay filr has a cnsan magniud wih a phas ha is linar wih frquncy Thrfr, i ds n affc h magniud f h inpu. I nly ffcs h phas by an amun f -ω d d d d d H H y x y x And hrfr, Bu hn, Alrnaivly, if ] [ d d d d h H

4 A signal f.. y H x. 4 Exampl h frm x 4 Or rwrin as : y 4 Thn h upu signal bcms : H dlay filr wih dlay f.sc. Th frquncy rspns is : H H 4..5 [. ] 4 is inpu an idal. 4 BME 3 Bimdical Cmpuing - J.Schssr 3

5 Idal Lw Pass Filr This filr nly passs frquncis blw a valu ω c and anuas all frquncis abv ω c. W call ω c h cuff frquncy. Thrfr, h frquncy rspns f a lw pass filr is: H lp c c ω c ω c BME 3 Bimdical Cmpuing - J.Schssr 33

6 Idal High Pass Filr This filr nly passs frquncis abv a valu ω c and anuas all frquncis blw ω c. W call ω c h cuff frquncy. Thrfr, h frquncy rspns f a high pass filr is: H hp c c ω c ω c BME 3 Bimdical Cmpuing - J.Schssr 34

7 Idal Band Pass Filr This filr nly passs frquncis abv a valu ω c and blw a valu ω c and anuas all hr frquncis usid his rang. W call ω c h lwr r lw cuff frquncy and ω c h uppr r high cuff frquncy. Thrfr, h frquncy rspns f a bandpass filr is: c H bp c c c -ω c ω c ω c ω c BME 3 Bimdical Cmpuing - J.Schssr 35

8 Applicain f Idal Filrs W will apply a band pass filr a pridic squar wav filr u is fundamnal frquncy. L ur inpu signal hav a prid f T = 5µs r f =khz => ω =π rad/sc and is frm vr n prid is: x < T / x = T / < T -T -T T / T / T T BME 3 Bimdical Cmpuing - J.Schssr 36

9 Applicain f Idal Filrs a Sinc x is a prid, l s calcula h Furir sris fr dcmps h inpu in is frquncy cmpnns. x k r T k k k T a T k [ k Rcall ha T k x k k k k k ] d T T cs k T [ k k d T k T k T sin k fr vn valus f fr dd valus f ] a k k [ k ] k [ k ] k k / a T k k T xd d T T T fr dd valus f k fr vn valus f k k BME 3 Bimdical Cmpuing - J.Schssr 37

10 Applicain f Idal Filrs Nw l s apply an idal band pass filr wih lw frquncy cuff f,5 Hz and high frquncy cuff f,75 Hz which has a bandwidh f 5 Hz and is cnrd arund Hz which is h fundamnal frquncy f his squar wav Hz BME 3 Bimdical Cmpuing - J.Schssr 38

11 Applicain f Idal Filrs If h filr is LTI, hn h upu signal is als pridic wih sam fundamnal frquncy. Thrfr, y can b wrin as a Furir Sris. y k b k k By suprpsiin f hs cmplx xpnnial signals a k b k H k a k Bu sinc H is nly dfind fr 5 75, hn nly h rms fr which k will b lf upn his muliplicain b k H k a k k / fr k BME 3 Bimdical Cmpuing - J.Schssr y b b / / / / / / / / 4 cs / 39

12 Tim Dmain r Frquncy Dmain W hav sn ha a LTI can b rprsnd by is impuls rspns in h im dmain and by is frquncy rspns in h frquncy dmain. In gnral whn wrking wih sinusids r cmplx xpnnials ihr singl r summd signals, i is asir wrk in h Frquncy Dmain. If h signal cnsiss f impulss, sp funcins, r hr nn-sinusidal signals.g., signals which ar prgrssiv ingrains f h impuls funcin, cnvluin f h impuls rspns Tim Dmain is usually asis. BME 3 Bimdical Cmpuing - J.Schssr 3

13 An Exampl An LTI sysm has an impuls rspns f h = δ - π -π u Th fllwing signal is applid: x=+δ -.+4csπ +.3π fr all Th inpu has 3 pars: a cnsan, an impuls and a csin wav. W will ak ach par sparaly and us h asis mhd find h sluin. BME 3 Bimdical Cmpuing - J.Schssr 3

14 BME 3 Bimdical Cmpuing - J.Schssr 3 An Exampl L s firs find h frquncy rspns f h sysm frm h impuls rspns: d d u d d u H ] [

15 An Exampl Nw l s ak h firs cnsan, ω = par and h hird csin par and valua h sluin using h frquncy rspns: Th firs par f h inpu : H H Th hird par f h inpu : 4cs.3 H 4cs.3 4 H cs[.3 H ] H 4 4 4cs.3 4 cs[.3.5 ] BME 3 Bimdical Cmpuing - J.Schssr 4 cs[.55 ] 33

16 An Exampl Nw fr h scnd par f h inpu h impuls funcin, w will apply h impuls rspns: Th scnd par f h inpu :. h.. [ u.. u.] Th Cmpl sluin by suprpsiin is: y.. 4 u. 4 cs.55 BME 3 Bimdical Cmpuing - J.Schssr 34

17 Hmwrk Exrciss:.4-.7 Prblms:.5,.6,.7 Us Malab pl x; shw yur cd BME 3 Bimdical Cmpuing - J.Schssr 35