IIR FM pre-emphasis filter investigations and a digital implementation.
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- Alaina Armstrong
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1 II FM pr-mph fltr vtgto gtl mplmtto. Itl rmlg. So, lk I w, you'r wtg to mk compct gtl rcurv pr-mph fltr. Wll, th how I got o. A pr-mph fltr, fltr tht oot th hgh frquc. A II mplmtto ft mpul rpo mplmtto, thy r rcurv tur. II fltr rprt wht c m wth lctroc compot. A gtl mplmtto pproxmt om ort of log g. Frt of ll you to kow o or two thg. h mot mportt g trfr fucto. A trfr fucto fucto tht rlt put gl wth output gl uully ot wth H. If you wr to multply H y put gl you woul gt wht th fltr woul output. Hr' xmpl, fgur how voltg vr twork. h put voltg th output voltg r rlt through th fucto V H. h th. h m th trfr fucto qul to out V quto complx frqucy, whr gulr frqucy r ( f ) th rt of xpotl chg ( or for mor fo). A th voltg vr twork ot ffct y frqucy, h o ffct t trfr fucto th c. V k V out V k Vout Fgur : Voltg vr twork.
2 Now wht' th g l wth trfr fucto? h g l you c how th H f you wll gt fltr wll rpo wh you chg th frqucy. If you plot plot of lr g vru frqucy hrt, whlt, rg umth f ph rpo. Of cour for our prt xmpl H f, rg umth f wll gv you prouc trght l t ro. So t' o to mor trtg trfr fucto. Now, my gol w to mk pr-mph fltr for th w FM t MH. Wht th offcl tr? I ll my lookg o th Itrt I coul ot f o plc tht woul gv m offcl trght wr. All I coul f wr mor lk rumor, o offcl ocumt. h rlly gr my gr. Why c't thr ly fou wt tht cot ll th offcl tr for ll coutr of th worl. h wht I fou, ut I coul't f y offcl ocumt t t to tk th prgrph wth pch of lt. It w vry commo for popl to quot th 3 B corr whch rumor h t 75 µ oly for th USA 50 µ for th rt of th worl. A fw plc woul lo y tht ftr th 3 B corr th g woul cr t 6 B pr octv. No o v mto trfr fucto. It lo lv lot of quto uwr; wht o th g o tw 0 B 3 B, f th cl g wth rpct to octv ot lr t 6 B pr octv vrywhr, th how o th g chg wh t' ot movg t 6 B pr octv? Alo, wht th ph rpo? Ufortutly th ll I h to go o. A mpl trfr fucto t coffct gv woul hv ll I, woul hv tol m vrythg pol out th fltr tht I w ftr. It th wht I h to o. rlmry vtgto. Frt I om mpl trfr fucto to vtgt. h woul rprt om log lctroc. h mplt o I coul thk of w lr o. H c () Formul jut uch fucto. From wht w kow of pr-mph, wh w put 0 H w woul lk to gt lr g of. h g w f our 0 B powr mrk. h g th c mpl tht c. H () A w r mly trt whr 0 c w c rwrt formul for th pcl H (3)
3 Whr th ph rpo of th fltr. A powr proportol to lr g qur, w c rv th followg quto of th powr g rpo of th fltr. H l By formul 3 () 0 l 0log B 0 By formul l B 0log (5) 0 Wth pr-mph w hv th tm cott, whch I rlr pprtly 75 µ for th US 50 µ for th rt of th worl. A I lv th rt of th worl, from ow o I'm trtg t g 50 µ. h o-cll pr-mph tm cott th rcprocl of th gulr frqucy wh th powr g 3 B. h m th followg. B B 3 Dfto of tm cott. By formul 5 (6) H By formul (7) ht olv th ukow for our trfr fucto formul. Howvr, w hv't look to th rt of chg of th frqucy rpo; rmmr, om wt hv tht th rt of chg of th g woul cr t 6 B pr octv ftr th 3 B corr. h rg m to othr o of my pt pv, th cl cl. Frt w oct to crt om ort of octv cl, o lt. h o th jo whr oct th octv. Icr oct y o th frqucy oul.. octv. 0 B y formul 5. (8) l 0 oct 0 l 0 0 l 0 B B oct oct y formul 8. oct 0l l 0 l oct (9) Clrly formul 9 ot lr wth rpct to frqucy. But wh, w c mk pproxmto whch th followg.
4 oct 0l l0 0l B l 0 Wh y formul 9. oct B Wh B 6.0 B octv Wh (0) oct W c lo formul 9 mootoc crg fucto. h m tht th 6.0 B octv th ggt th cl pr octv powr g c hv t hpp t frqucy of fty. h I prum whr th 6B octv com from tht om wt mto. Now our trfr fucto prtty much mtch wht wt mto out FM pr-mph fltr (rlly th ot hr cu thy o't tll u much). Now lt' grph th frqucy rpo ph rpo of th trfr fucto. Fgur : G ph rpo of formul 7. So wht o th lctroc twork tht rprt th trfr fucto look lk? h wr fgur 3.
5 V C C Fgur 3: Schmtc rprtto of formul 7. o tht fgur 3 rlt to formul, lt, V rm voltg t thr rpctv poto. Clcult th trfr fucto of ch compot, th put g th currt through th compot th output g voltg cro th compot. For th rtor th mply th rtor vlu V I. For th cpctor C w to clcult th Lplc trformto of th voltg ovr th cpctor v th y th Lplc trformto of th currt through th cpctor. h ffrtl quto tht rlt currt voltg wth th cpctor t C t vt. Ug th ffrtto proprty of th Lplc trformto th mpl tht I CV hc H C C. Oc g for mor tl C C, h, w c clcult how th voltg rlt to voltg through th followg. G V Wht op-mp o. () C V C C voltg vr twork. V C () G C By formul. G C
6 G C Aumg tht th g of th op mp vry lrg. C C H C (3) h jut formul 7 wth C. h fly th otmt. hr prolm wth th trfr fucto w hv rv. h prolm th frqucy cr o o th g, wth th trfr fucto th vr top. h m t ft frqucy w hv ft g. h c olv ut frt w to hv look rou th -pl. So fr w hv ot look to ythg mor th th pot lyg o th mgry x of th -pl. Now lt' look ovr th -pl mor. Fgur how cotour plot of th mgtu of th trfr fucto w hv rv o fr. hk of color hght, purpl lu r low hght whl yllow r hgh hght. h pot t,0 (purpl color) h mgtu of ro cll ot urprgly ro. h vrtcl lck l th frqucy rpo of th fltr. Fgur : cotour plot of th mgtu of th trfr fucto o fr.
7 B g prtty pctur wht o fgur how u? Wll, t how u f w r r ro mov wy from t, our g go up. But thr' mor to lf th jut ro. hr' omthg cll pol. A pol t tht go up t of ow lk ro you pproch t. I fct mthmtclly pol jut th rcprocl of ro. h m f you'r tg t pol, you r t fty, wh you go wy from t, you'll g go ow. So f w wr to put pol lghtly to th lft of th ro w hv fgur, w t o th lck l t pot wth vry hgh frqucy ( log wy from oth th pol th ro), th pol th ro wll m to mor or l th m tc from u, hc wll ccl ch othr out o our g top crg. But, t low frquc to ro tll much clor to u th th pol, mg th ro omt th frqucy rpo, w houl tll hv roly mlr frqucy rpo to tht fgur for low frquc. ht' th ywy, lt' wht rlly hpp. ol cg. Ag pol to th trfr fucto formul gv u th w trfr fucto of formul. H () h o wll work, ut lt tk t tp-y-tp frt. A for w clcult th cl powr g of th fltr. H B 0log 0 (5) h for w rv formul tht rlt th 3 B corr th tm cott togthr. B B 3 Dfto of tm cott. By formul 5. (6) I fct cu mut mll, formul 6 pproxmtly qul to formul 6. ht gt out of th wy, ut wht o w o out? Wll, I t, I c thk of fw
8 wy of og t. h frt wy woul ttg th mxmum cl powr g mu 3 B to om frqucy lk 0 kh (th ov th mxmum frqucy w c trmt uo. h co wy woul to tpult th mxmum rt of chg of th powr g of th fltr t ccorgly. hrly you coul tpult mxmum powr g t ccorgly. I'll trt off y og t th frt wy. Lt th rcprocl of th gulr frqucy tht prouc 3 B l th th mxmum powr g. 0log 0 B By formul log log A th 3 B l th mxmum powr. 3 log 0 0 By formul 6. 8 (7) 8 6 A (8) For th co mtho quto gt k of my, ut thr' othg rlly ffcult out thm. A th frt trfr fucto w look t, w clcult th rt of chg of g of powr B wth rpct to octv low.
9 oct 0l B By formul 5. (9) l 0 Dffrtt wth rpct to t to ro to gt th mxmum rt of chg th olv for. oct oct B B 0l l 0 0 mx 8 mx (0) h thr mtho rlly y cu lookg t formul 5 th mxmum powr th followg. BMA 0 log 0 By formul 5. () h olv for. 0 BMA BMA By formul 6. () Now w c clcult om vlu for. Evlutg A B. I FM tro ro, t 9 kh thr plot. Bcu of th plot you mut lmt th uo frquc to omwht l. 5 kh m to commo toppg pot. A I for for my clculto I'm gog to u 50 µ pr-mph tm cott. Ug th frt mtho cr ov (formul 7) I omthg hghr th th frquc I'm trt. h g th c lt th frqucy y 0 kh
10 r r By formul 7. 6 & 5.90 By formul 6. A formul 8 y, th commt jut ur formul 6 tht. It' lwy trtg wh thg tur out to o mpl. For th co mtho w to choo frqucy whr w wt th tpt chg of powr B wth rpct to octv to hpp. A our frt trfr fucto coul o o ttr th 6 B octv, w wll o wor wth th trfr fucto cu of th pol. So lt' lct th frqucy to omwhr th ml, y 8 kh. mx r r By formul 0. 6 & 5.80 By formul 6. & oct B. B octv By formul 9. MA Notc w gt mor or l th m vlu ug th mtho th lt. A flly for th lt mtho w choo th mxmum g tht w r prpr to cop wth, lt' y 7 B. BMA 7 B By formul. 6 & By formul 6. So thr w hv t, thr ffrt wy of vlutg th pky cott. Mtho o proly th mot uful cu of th pproxmto tht c m. Howvr, I'm ur th othr two mtho hv thr ow mrt. h lctroc of th trfr fucto. Now com th tm to hv look t wht lctroc th trfr fucto rlt to. Wth t of thkg pol oluto fgur 5.
11 C V C C Fgur 5: chmtc rprtto of our w trfr fucto. I clcult th trfr fucto th m wy I formul to 3. V G Wht op-mp o. (3) C C V Voltg vr twork. C C V () C C G By formul 3. C C G C C G C C Aumg tht th g of th op mp vry lrg. C C C C H (5) h formul wth C, C. A my m gol to mplmt th trfr fucto gtlly, th log mplmtto ot of grt cocr to m. Howvr, whl w hr w mght wll mol t o th computr.
12 Choog th cpctor C. 7F from th frt mtho cr prvouly, I gt 9. K. 6K. A I kow 0 K.5 K rtor xt I'll u th pproxmto. Ug th followg compot vlu grlpurpo uo opmp, I gt th plot fgur 6. C. 7F 0K. 5K Fgur 6: Computr mult frqucy rpo of th chmtc fgur 5. h look roly ccptl. Wll, ough of th log tuff lt' covrt wht w kow out th fltr to gtl mplmtto. From S pl to Z pl. h grt v. So fr w hv lg wth th S pl th Lplc trformto. For crt ytm th quvlt th Z pl t trformto. h Z pl polr whlt S pl Crt. gv goo ght out th Z pl, lo try Fgur 7 how how th two pl r rlt.
13 0, f Img Img S l Z l 0,0 l, l,0 Fgur 7: How th S th Z pl r rlt wth rpct to th Lplc Z trformto rpctvly. ltohp tw th pot r whr mpl pro. Wht I m y th rltohp tw th two pl, th followg. If w hv tm cotuou fucto th tm om prform th Lplc trform o t, w gt th pot of th S pl. Now, f w mpl th m tm cotuou fucto th tm om t trvl f prform th Z trformto o th, w woul gt th pot th Z pl. h wht I m y th pot r rlt y. h rltohp c how y th followg. Frt w clcult th Lplc trform of mpl fucto x t. Oc mpl th fucto com t xt. Whr t Drc com tht pluck out th mpl vlu t 0 t Drc com. t xt t t t xt 0 t xt 0 t x t. By fto th Lplc trform of th mpl fucto. t t t By f of Drc com.
14 x 0 Drc lt pluckg out o vlu. x Dfg x 0 x 0 0 x x crt t. By fto of th Z trform. A th rltohp fll out. Alo ot th followg. & r Whr th gl th pot th Z pl mk wth th rl x, r th tc from th ctr of th crcl. h mot mportt l th mgry x l th S pl th ut crcl th Z pl. h l or th crcl cot vrythg you to kow out th frqucy rpo of th fltr. Alo ot frquc lk f, f, tc r gt mpp to th m pot th Z pl. h ow to trform our trfr fucto from th S pl to th Z pl, th prform th vr Z trform to gt crt fucto. h th fucto tht ctully prform th fltrg o th computr. W hv prolm though; how o w trform our trfr fucto? hr o gl mtho. If you look rou o th t you wll m lk lr trform, mtch Z, vrt mpul, tp vrt, othr. Shortly t wll com clr tht w ftr th trformto to hv trfr fucto tht rtol fucto th vrl. ht rul out ug l rctly, ut tll llow to u pproxmto for t. ht th mtho I m gog to choo to o t. Kow rltohp.
15 ylor xpo. Frt orr pproxmto. : Blr trformto ( h pproxmto jut wht w. It' cll th lr trform. Frtly t' rltvly mpl pproxmto of l wth othg ut coupl of lr polyoml. Scoly, clrly y pot th Z pl oly com from o pot th S pl, ulk l. Lt' hv look how frquc ow ppr ch pl. For prog frquc ur th Z trform w u whlt prog frquc ur th Lplc trform w u. Now, how o our wly cqur gtl fltr rpo to frqucy of f gv uch gl? k th pot th Z pl tht l o th ut crcl h gl of,... o trm whr th pot cm from for th lr trformto w o th followg. Lt Mkg th lr trformto pproxmto. Evlutg o th ut crcl.
16 t t A w th form of prog th log frqucy. (6) t h m ur th trformto th gtl fltr rpo t th log fltr rpo t. Bcu of th tgt fucto, low frquc wll t to frly lr ut th hgh frquc wll compr hc tort wrp. W c corrct for th t frquc tht w c cotrol th g off th log fltr. It' cll pr-wrpg. Fgur 8 how tutv wy how w o th. p Fgur 8: Itutv how pr-wrpg work. So to lct y frqucy our log fltr w o ot choo t rctly, ut u formul 6 plug for our r frqucy u our log fltr g. Now w prform th lr trform o our trfr fucto formul. H From formul. H H H Blr trform.
17 H H (7) h our trfr fucto of our gtl fltr. Ug th Z trform tmhftg proprty, fto of th trfr fucto, w c wrt th gtl fltr rcuro rlto y prformg vr Z trform. From formul 7. x x y y ] [ ] [ ] [ ] [ Ivr Z trform. ] [ ] [ ] [ ] [ y x x y (8) ] [ ] [ ] [ ] [ 0 y x x y (9) 0 H Formul 8 (or 9) th rcuro rlto tht ctully o th fltrg o th computr. uttg t ll togthr. cot r-wrpg. cot r-wrpg. 8 By formul 7. & By formul 6. 0 By formul 8 & 9. & By formul 8 & 9. & By formul 8 & 9.
18 For my mplmtto o th computr, I wll ug oucr mplg t 9000 tm co. I thk tht corrpog to 0 kh houl ccptl my tup. luggg th umr, tl wht I gt. f 9,000 mpl r r mpl r r l : Clcult vlu for tm cott of 50 µ, corrpog to 0 kh, mpl rt of 9,000 mpl co. Fgur 9 how th g ph rpo of th gtl fltr. Fgur 9: G ph rpo of th gtl fltr.
19 Fgur 0 how th cotour plot of th mgtu of th gtl trfr fucto. ou c th ro fgur, ut ow w lo hv pol clo to th lft of th ro. H Im Fgur 0: Cotour plot of th mgtu of th gtl trfr fucto. All tht rm to wrt rout to mplmt th. Ltg how C++ progrm co ppt for og th fltrg. h upt prt of th co mut ru oc mpl. Ltg : C++ co ppt of fltr mplmtto.
20 Amg, ll tht hr work for co tht oly fw l. Implmtg th co upplyg th oucr wth wht o ug othr oucr to lt to th rpo w gt fgur. hr low p FI fltr wth cutoff frqucy of 7 kh t jut pror to th II pr-mph fltr (hc th rol of rou 6 kh). h ctul rl-lf powr g rpo to th C++ co ltg o. Fgur : l-lf wht o tt of th II pr-mph fltr. All thg g qul fgur rltvly goo mtch wth fgur 9. A lwy wth mplmtto, wh ctul rult corrlt wll wth thortcl mol, th, tht' goo. A fw fl wor. Of cour thr r myr of wy tht you coul crt pr-mph fltr. h mtho jut o, fr I c tll t' ccptl. FI fltr coul u, ut r lrg low. II fltr prov vtg tht, mot lkly t th rcvg thr r rl-lf lctroc compot og th -mph, II fltr r quvlt to rl-lf lctroc compot o t mk to puru th II rout th hop tht th pr-mph fltr th -mph fltr hv ttr mtch. Alo II fltr r vry quck. Othr opto woul to vtgt th u of mor pol ro wll o mor rl-lf ttg wth ro trmttr rcvr.
21 h fltr u my M tro cor oftwr for tro trmo o th MH. S for th progrm. Jot. 0
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Chem 115 POGIL Worksheet - Week 4 Moles & Stoichiometry
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