Signal Recovery - Prof. S. Cova - Exam 2016/02/16 - P1 pag.1

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1 gal Recovery - Pro.. Cova - Exam 06/0/6 - P pag. PROBEM Data ad Note Appled orce F rt cae: tep ple ecod cae: rectaglar ple wth drato p = 5m Pezoelectrc orce eor A q =0pC/N orce-to-charge covero C = 500pF total capactae o eor ad coected crct Preampler Ipt retae R A very hgh > 500 MΩ, to be treated a pa =0MHz bad lmt v, = 0 / Hz wde-bad (lateral), = 0, pa / Hz wde-bad (lateral); where t tated, take to accot alo a / compoet wth corer reqey c =KHz (A) Optmm Flterg Noe ot whte, thereore: optmm lter = oe-whteg lter ollowed by matched lter. oltage oe at the preamp otpt + ω = v+ = v ω C ω wth C v = = 50µ Noe-whteg lter = CR deretator wth tme cotat RC= H B = H ( ω) + B ω = +ω whteed oe B = v Cae : tep appled orce A tep orce F appled to the eor geerate the capactor a pezoelectrc charge Q = A q F ad thereore a tep voltage gal

2 gal Recovery - Pro.. Cova - Exam 06/0/6 - P pag. F A q = C F he orce-to-voltage covero actor th Aq 0 m / Newto C = he otpt gal o the whteg lter ( ) ( ) v = t exp t / B F he matched lter weghtg to ha the ame hape a th gal w m t = exp Deotg by v,b the blateral dety ad by v, the lateral dety o the whte oe at the pt o the matched lter, the /N ( ) ( ) v t w t dt ( ) N B m 0 F F F = = wm t dt = = 0 op ( ) v, b v, b v, v, b wm t dt 0 he mmm mearable ampltde o the voltage gal, whch correpod to /N=, v F m, op =,8 µ he correpodg mmm orce F N µ N F m, op 3 m, op = 0,4 0 = 40 Aq C Cae : appled orce rectaglar wth drato P he whteg lter ad t otpt oe are the ame o cae. he rectaglar gal the m o a potve tep wth ampltde F apled at t=0 ad a egatve tep wth eqal ampltde F apled at a t= P. hereore, the gal at the otpt o the whteg lter compoed by two eqal expoetal ple, a potve oe a t=0 ad a egatve oe at t= P. { ( ) [ ] ( ) ( ) } v = t exp t / t exp t / B F P P hereore, th cae the optmm weghtg to

3 gal Recovery - Pro.. Cova - Exam 06/0/6 - P pag.3 w = ( t) exp [ t / ] ( t ) exp ( t ) / m P P It worth otg that there eglgble perpoto o the two expoetal ple becae P We ee thereore that the cae the meare ca be obtaed by btractg rom the meare o the rt expoetal ple (a made cae ) the meare o the ecod expoetal ple (egatve ple). We ote that. he gal ampltde doble. he mea qare oe doble 3. the /N reaed by the actor ad the mmm ampltde redced by th actor h relt ca be obtaed alo wthot coderg the compoto o two meare, bt jt by comptg the /N wth the correct to v B ad w m ad takg to accot that P. (B) Approxmato o the matched lter wth a low-pa lter wth cotat parameter Cae : tep appled orce he weghtg to o the matched lter how that t a low-pa lter. hereore, alo the lter to be employed a approxmato o the matched lter mt be a low-pa lter. We ote that the reqey doma the matched lter ha modle o the weghtg to eqal to that o a mple RC low-pa lter wth RC=. hereore, wth ch a lter the ltered whte oe eqal to that o the matched lter. = v,,4µ 4 However, the relt obtaed or the ltered gal deret ad obvoly le avorable. he otpt gal o the whteg lter tme doma v ( t) ( t ) = exp / aplace doma = B F he acto o the RC low-pa lter wth RC= obtaed by mea o B F + δ repoe tme h t = exp traer to aplace doma H = + he otpt o the RC low-pa lter t = exp / aplace doma tme v ( t) ( t ) U F = ( + ) F F

4 gal Recovery - Pro.. Cova - Exam 06/0/6 - P pag.4 Wth cotat parameter lter the ple mearemet obtaed by mearg the peak ampltde o the otpt gal tme. I the preet cae the maxmm o the gal od at t= F U = 0,37 F e We have th F = = 0,74 N e e N N,36 N v, op op op e v F m, = 3,8 µ Cae : appled orce rectaglar wth drato P We ca employ alo th cae the ame procedre a (A) or comptg the optmm relt. he mearemet obtaed by mmg (algebrcally) the mearemet o the (potve) tep gal at t=0 ad the mearemet o the (egatve) tep gal gal at t= P. It thereore colded that alo employg a approxmate matched lter a mple cotatparameter low-pa lter, wth a rectaglar ple the /N ad the mmm mearable ampltde are mproved by a actor,4 wth repect to the cae o tep ple (C) Mearemet preee o / compoet the crret oe We take ow to accot alo a / compoet wth corer reqey c the oe crret c = he correpodg oe compoet added to the otpt o the whteg lter a / compoet ltered by the capactae C ad by the hgh-pa CR whteg lter. ω ω ω ω ω ω ω ω c c = v C + + It thereore a / oe compoet ltered oly by a low-pa lter (a mple pole wth tme cotat ), wthot ay hgh-pa lterg. Cae : tep appled orce he lter employed a a approxmato o the matched lter( ee B) a mple low-pa lter wth bad lmt (eqal to that o the matched lter). We kow (ee B) that the whte oe compoet gve a cotrbto = v,,4µ 4 to the ltered oe.

5 gal Recovery - Pro.. Cova - Exam 06/0/6 - P pag.5 A coer the / oe, the low-pa lter doe ot provde ay hgh-pa lterg acto that lmt the / cotrbto, whch th domat wth repect to the whte oe U. For lmtg t th eceary to trodce ater the low-pa lter a table hgh-pa lter that trodce a bad lmt at low reqey. At hgh reqey the bad lmt gve by the pole o the low-pa lter wth tme cotat., thereore ~ / e t >>, we ca e the harp bad-cto approxmato or evalatg the ltered / oe = v c l 0, 63µ l et coder ome deret type o hgh pa lter that ca be employed or lmtg the / oe. he mplet lterg approach to maally et to zero the baele at the tart o a cycle o mearemet. I the cycle ha a drato m, the lower bad-lmt ~ / m. he pper bad-lmt et by the low-pa lter pole ~ / For a mearemet cycle o 5m= 000 we ca cot o a hgh-pa cto wth m 5m= 000 = mhz We have th hereore t m m = = 0 7 0, 63µ 4,=,6µ almot doble o the whte oe A better relt ca be obtaed by employg a cotat parameter CR hgh-pa lter. he deretato tme cotat D mt be log wth repect to the drato o the gal, order to avod a redcto o the gal ampltde. For tae, we ca e D D = 00 = 5m th obtag = = 00 ad thereore 0, 63µ,4=,35µ almot eqal to the whte oe h relt qte atactory, bt t poble to obta a rther mprovemet by employg a Baele Retorer BR. e the BR a wtched-parameter lter, t poble to employ a horter deretato tme cotat B wthot redcg the gal ampltde. For tae we may employ B B = 0 = 500µ whch gve = = 0 ad thereore 0, 63µ, 4= 0,9µ lower tha the whte oe cotrbto

6 gal Recovery - Pro.. Cova - Exam 06/0/6 - P pag.6 Cae : appled orce rectaglar wth drato P he mearemet obtaed a deree o two beqet mearemet, paced by a tme terval P. It a Correlated Doble Flterg CDF, whch add to the low-pa lterg (wth tme cotat ) a hgh-pa lterg, wth lower bad-lmt eqvalet to that o a cotat-parameter CR lter wth deretato tme cotat eqal to the terval P. I or cae t P P = 5m thereore = = 00 A low reqey cto prodced by th CDF lterg wthot eedg to trodce ay other lterg. However, t mt be oted that the CDF doble the mea qare oe the bad deed by the lmt ad v c l = 0, 63µ,4=,9µ he CDF alo doble the mea qare oe de to the whte oe the lterg bad v, µ 4 We ee that the two cotrbto o the / oe ad o the whte oe th cae are almot eqal. We ca colde that the mearemet o the rectaglar gal t ot eceary to trodce a rther hgh-pa lter or lmtg the cotrbto o the / oe. I act, the lterg that gve a optmzed (or approxmately optmzed) mearemet wth whte oe heretly lde a hgh-pa lterg that provde adeqate redcto o the / oe cotrbto.