Experiment 3. Filters II Filter Design with MATLAB

Transcription

1 Experiment 3 Filter II Filter Deign with MATLAB The objective o thi experiment i to gain ome experience in deigning ilter with deired peciication. You will work with a number o tool helping you in deigning ilter in MATLAB. Prelab P1. Review o Filter Frequency Domain The eect o a ytem on a ignal input to it can be decribed in both time and requency domain. We oberved the time repreentation o the ytem in lat experiment. Here we will tudy the requency domain repreentation o the ytem a a ilter. Dierent requency component o the input ignal experience dierent attenuation/ampliication and delay a they pa through the ilter thu a ilter alter the ignal input to it in requency domain (Fig. P1). One o the main application o a ilter i to ilter out the unwanted part o an input ignal. The unwanted part might be noie preent out o the requency band occupied by the deired ignal (high requency noie) or other interering ignal that are unwanted, e.g. when you tune to a radio tation, the radio receiver will ilter out other tation o you would hear a clear ound (Fig. P2). Figure P1 The alteration o requency component o a ignal input to a ilter Simple ilter can be categorized into our major categorie: lowpa, highpa, bandpa and bandtop (or band-reject). Figure P3 how the ideal orm o thee our ilter in the requency domain. A the name ugget a lowpa ilter or intance, pae the low requency component o a ignal while blocking the high requency one. A you have learned the convolution operation in time domain i equivalent to multiplication in requency domain. Thu, multiplying the requency domain repreentation o the input ignal with that o the ilter, high requency component o the input ignal will be multiplied in zero and would thu vanih, while the lower requency component will be multiplied by one and would pa without any change. 1

2 Noie Interering tation Deired tation Figure P2 Some baic application o ilter: top) noie reduction by removing out o band (high requency noie), bottom) removing interering ignal: paing o a tuned channel in a radio. In thi experiment you will learn how to ue ome tool in MATLAB ignal proceing toolbox to deign ilter with your deired characteritic. Figure P3 The ideal requency repone o the our baic ilter type: 1) lowpa, 2) highpa, 3) bandpa and 4) bandtop (or band-reject) 2

3 P2. Filter Deign Beore deigning a ilter, you hould know what parameter need to be known or the deign. The ideal ilter a hown in igure P3, cannot be phyically realized, ince they are noncaual, i.e. to generate the output at intance n in time, they require the knowledge o uture ample o the input ignal, i.e. at n+ 1, n+ 2,..., which i impoible. The deign technique are a a reult, approximation o the ideal cae. The digital ilter deign method all into two main categorie, FIR ilter deign and IIR ilter deign. You will learn more about thee type in your uture coure. In thi experiment you will be introduced to ome tool or baic IIR ilter deign. Firt o all you hould know the parameter o the ilter that you are going to deign. Some o thee parameter are decribed in thi ection. db Notation Something you will very oten encounter when dealing with ilter i the deci-bell (db) notation. The reaon i that when you re dealing with a wide range o variation in the magnitude o a quantity, ay 5 order o magnitude dierence between maximum and minimum, it i hard to work with the number. I you ue the logarithm o the magnitude, the variation will be too mall! So Alexander Graham Bell, invented thi new notation named ater him by uing 20 time the logarithm o the magnitude. Thu x 20 xdb = 20log 10 x x = 10 db Some igure that you hould know without uing your calculator are 3 db and it multiple. 3 db i almot equal to 2, thu i you have a quantity that i 3 db greater than another, it i actually two time greater than it. (Note that you are uing logarithm, thu addition o the logarithm i equivalent to multiplication o the number themelve.) By analogy, -3 db, 6 db and -6 db are equivalent to 0.5, 4 and 0.25 repectively. Paband Parameter The paband o the ideal ilter i the band o requencie where the ilter requency repone i not zero. I a ignal requency content lie in thi band, the ignal will pa without any alteration. In cae o practical ilter, the requency repone doe not have an abrupt change rom zero to one or the value in paband (Fig P4). Intead, there i a tranition rom a higher level to a lower level. Hence, or a practical ilter the band o requencie over which the magnitude o requency repone i greater than a threhold i called the paband. Thi threhold i uually et at -3 db o the maximum magnitude in paband. The requency at which the ilter requency repone magnitude reache thi threhold i called the cut-o requency o the ilter ( c ). I the requency repone doe not decreae monotonically in the paband the ilter i aid to have ripple in the paband. The dierence between the maximum and minimum magnitude value in the paband deine the paband ripple actor R p uually given in db. The maximum value o the magnitude in paband i called the gain o the ilter. Stopband Parameter The topband o the ideal ilter i the band o requencie where the ilter requency repone i zero. Thu the requency component o the ignal lying in thi band will not 3

4 pa and will be ully removed. Again, ince the tranition rom high to low i not abrupt in practical ilter, a threhold level deine thi region. So, the band o requencie, where the magnitude o the requency repone o the ilter drop below a threhold level i called topband. The threhold value i dependent on the application o the ilter uually about 40 db and i uually denoted by A or R. I there are ripple in the topband, the maximum value o the magnitude o requency repone in the topband hall lie below the threhold. The requency at which the magnitude o the ilter repone irt all below the threhold value i called the topband requency denoted by. Figure P4 The ilter parameter: 1) paband, 2) topband, 3) cut-o requencie, 4) topband requencie, 5) paband ripple, 6) topband level, 7) ilter gain. P3. Four Claical IIR Filter Type In thi ection I will introduce and compare the our claical IIR ilter type, namely Butterworth, Chebychev Type I and II, and Elliptic ilter. You will encounter thee ilter and probably their deign methodologie in your uture coure. Since traightorward algorithm or deign and implementation o thee ilter exit, many otware program help you in deigning thee ilter by imply peciying their parameter. Hence, or the ake o thi experiment you need not know all the mathematical detail involved in the deign o thee ilter 1. All ilter here reer to the lowpa cae. You can imply generalize the reult to the other type by analogy. 1 I you are intereted in urther tudy o thee detail you may reer to Digital Signal Proceing, Sytem Analyi and Deign, by P.S.R Diniz, E.A.B. da Silva and S.L. Netto, Cambridge Univerity Pre,

5 Butterworth Filter The Butterworth ilter o order N, alo called the maximally lat ilter, i an approximation o the ideal ilter, which the irt 2N 1derivative o it magnitude quared are zero. A a reult the requency repone o thi ilter decreae monotonically with requency and H( = c ) = 1 2. The decreae i very low in the paband and quick in the topband. In a deign problem where no ripple i acceptable in paband and topband, Butterworth ilter i a good choice. Figure P5 The requency repone o a Butterworth ilter with ω c = 1 Chebychev Type I Filter A Chebychev type I approximate the ideal ilter by minimizing the abolute dierence between the reulting ilter and the ideal ilter, over the paband. Thi ilter reult in appearance o ripple in the paband. The ripple have a contant magnitude o R p which i one o the deign parameter o thi ilter. The topband repone i maximally lat. The tranition rom paband to topband occur ater than the Butterworth ilter: H( = c ) = R p. I the ripple in the paband are acceptable, a Chebychev ilter uually require a lower-order traner unction than a Butterworth ilter or the ame peciication. Figure P6 The requency repone o a Chebychev type I ilter with ω c = 1 5

6 Chebychev Type II Filter A Chebychev type I approximate the ideal ilter by minimizing the abolute dierence between the reulting ilter and the ideal ilter, over the topband. Thi ilter reult in appearance o ripple in the topband. The ripple have a contant magnitude o R which i one o the deign parameter o thi ilter. The paband repone i maximally lat and H( = c ) = R. The tranition rom paband to topband occur lower than the type I ilter and or even ilter order, it never reache zero. It advantage over type I i the abence o ripple in the paband. Figure P7 The requency repone o a Chebychev type II ilter with ω = 1 Elliptic Filter The elliptic ilter reult in the teepet tranition rom paband to topband by allowing ripple in both paband and topband. Thi ilter uually atiie the deired ilter peciication with the lowet order among the our type tudied. The paband and topband ripple R and R are alo required a deign parameter. p Figure P8 The requency repone o an elliptic ilter with ω c = 1 6

7 MATLAB ptool: Filter Deign and Signal Proceing Tool 1. Adding Phone Eect to a Sound In thi ection you will learn how to ue a powerul ignal proceing tool in MATLAB, ptool which i part o the MATLAB ignal proceing toolbox and help you with ilter deign and ignal proceing with an eay to ue GUI. In what ollow, you will deign a bandpa ilter that reemble the bandpa ilter in a telephone ytem. You will then apply thi ilter to the French peaker ound ile you ued beore, to create a phone eect to the ound. At MATLAB command prompt type ptool. In the Filter lit click on the New button. Thi create a new ilter deign. In the ampling requency edit box enter a the ampling requency. From the algorithm dropbox, elect Butterworth IIR ilter. Select bandpa in the ilter type dropbox. Enter 200 and 1200 or the low and high topband requencie repectively. Enter 250 and 1000 or the low and high paband requencie repectively. Click the Apply button at the bottom o the let panel. Cloe the ilter deign tool. Your ilter i now deigned and ready or ue! You hould import the French peaker ound ile into ptool, apply your ilter to it: At MATLAB command prompt load the.wav ile rench.wav. Downample the ound rom to 11025Hz (don t orget to update FS). In ptool, in File menu elect Import Set Import a to ignal. Select rom workpace and elect the name o the vector containing the ound data and click the right arrow to import the data Select the ampling requency rom the workpace variable and import it. Enter a name or the imported ignal, the deault i ig1. Click OK to complete import. From the Filter lit elect your ilter deign and click the apply button. Leave the deault a they are and click OK. In the Signal lit, you will ee the reult o iltering, elect it, hold down control key and elect the input ignal. Click the View button. I you can t ee both ignal, click on the elect trace in the toolbar and elect the output ignal. Click the play ound in the toolbar to liten to the output. Switch the elected trace rom output to input, play the ignal and note the dierence. Now go back to ilter deign creen to compare dierent ilter deign method with each other. Make ure the value o R p i et at 3 db and R at 20 db. 7

8 Note the hape o the Butterworth ilter repone and peciically the topband requencie. Now change the ilter type to Chebychev type I. Take note o the actual topband requencie and the hape o the ilter repone. Repeat or the Chebychev type II and elliptic ilter type. What i your deduction? To view the impule repone o each ilter, go back to ptool main window and elect your ilter and click the View button. A new tool window appear. You may ee the phae and amplitude repone o the ilter you have deigned in thi window. From the let panel, check the Impule Repone. Ue the zoom tool, to zoom in and ee the irt portion o the repone. Ue the zoom all tool, to elect another potion or ue the zoom tool to zoom in again. Uncheck the Magnitude and Phae boxe and check the Zero and Pole. Zoom out to ee the ull plot. In the deign window, change the ilter type. What happen to the poition o the pole and zero? Can you compare the our baic deign? 2. Out o Band Noie Removal In thi part you will ue a lowpa ilter to remove the high requency noie in a ignal. At MATLAB command prompt ue ize command to determine the number o ample in the downampled French peaker ound you loaded beore. Generate a random noie vector n with the ame number o ample a the French peaker ignal. Ue MATLAB randn command to generate the noie vector whoe ample have Gauian probability ditribution. Scale down the noie by multiplying your vector by 0.1. Add the noie and your original ound to make a new vector. Make ure you don t detroy your original ound ile. Import the noie and the noiy ound in ptool. View the noie, noiy ound and the original ound uing View button. Liten to the three ignal. Now deign a lowpa ilter with ollowing peciication: o c = 1500 Hz and = 1650 Hz. o No ripple in paband i allowed. o Your ilter order hould not be more than 15. (Check the right panel to ee the order o your ilter) o R hould be at leat 30 db. Apply your ilter to the noie ignal and to the noie. View the reult and ignal beore iltering. Liten to them and compare them. Now try to export your ilter deign to MATLAB: In ptool main window in File menu elect Export Select your ilter name in the let panel To ee the traner unction o the ilter, check the Export Filter a TF Object. 8

9 Click the Export to workapce button. At MATLAB command prompt, type the name o your ilter deign to ee it traner unction. You will ee the name o your ilter in MATLAB workpace panel. Return to ptool and again export your ilter deign now with the Export Filter a TF Object unchecked. Conirm replacement o your old workpace ilter export. At MATLAB command prompt, type the name o your ilter you hould ee omething like thi: ilt2 = t: [1x1 truct] : [] zpk: [1x1 truct] o: [] imp: [] tep: [] t: [] H: [1x4096 double] G: [4096x1 double] : [1x4096 double] pec: [1x1 truct] F: type: 'deign' lineino: [1x1 truct] SPTIdentiier: [1x1 truct] label: 'ilt2' You will ee the name o your ilter intead o ilt2. To acce the traner unction parameter o your ilter ue thi yntax and replace ilt2 with your ilter name: ilt2.t.num to ee the numerator coeicient o your ilter and ilt2.t.den to ee it denominator coeicient. You may acce the other parameter o the ilter, like location o pole and zero and ampling requency in the ame way. 9