Experimental Simulation of Digital IIR Filter Design Technique Based on Butterworth and Impulse Invariance Concepts

Transcription

1 Exerimental Simulation of Digital IIR Filter Deign Tehnique Bae on Butterworth an Imule Invariane Conet Vorao Patanaviit Aumtion Univerity of Thailan Bangkok, Thailan Abtrat - The igital IIR (Infinite Imule Reone) filter i one of the mot imortant art in teleommuniation an Digital Signal Proeing (DSP) framework an it eign tehnique have been invetigate for auray, omutational omlexity an ratial imlementation. In thi aer we invetigate a novel hybri IIR filter eign tehnique uing Butterworth an imule invariane onet, with the imlementation obetive of le omlexity. We invetigate two ae tuie of the low a IIR filter eign tehnique uing both mathematial moelling an omuter imulation. By uing Butterworth onet, the magnitue in eibel (B) an the hae of the frequeny reone between the analog filter an the igital filter, whih are eire by imule invariane onet, have been omaratively invetigate in term of the eign erformane an auray. Keywor - igital IIR (Infinite Imule Reone) filter, Butterworth Filter Deiring Tehnique, Imule Invariane Conet, Digital Signal Proeing (DSP) I. ITRODUCTIO In general, a frequeny-eletive filter i efine a a ytem [-5] within the grou of LTI (Linear Time- Invariant) ytem [6, 7] that ae artial frequeny ban an omletely eliminate all other ban. More general efinition of a filter i a ytem that reform artial frequeny ban while omletely aing all other ban. Due to it frequeny reone harateriti, the igital IIR (Infinite Imule Reone) filter [8, 9, ] ha beome the rimary art of teleommuniation an Digital Signal Proeing (DSP) framework [,, 5, 6] leaing to many eign tehnique of a igital IIR filter, whih have been reearhe over the lat two eae, from tehnial requirement eretive inluing auray, omutational omlexity, ratial imlementation, et. In thi aer, tarting from a theoretial eretive, the Butterworth onet [] i firt ue to etermine the ytem funtion of an analog filter. Seonly, the imulive invariant onet [] i ue to onvert from the ytem funtion of analog filter to the ytem funtion of igital filter. Then we invetigate in etail uing omuter imulation exeriment the igital IIR filter eign tehnique of the ombine Butterworth an the imule invariane onet. II. TE BUTTERWORT FILTER The roerty of the Butterworth low a filter [] i that the magnitue reone i flat harateriti (or monotonially ereaing) in both aban an toban. The magnitue quare reone ( ) of thi Butterworth low a filter an be mathematially exree a () where i the orer of the Butterworth low a filter an i the utoff frequeny (ra/e) The eign of the Butterworth low a filter an be exree a following te.. Determine the orer of Butterworth low a filter from the eifiation: R (aban rile arameter) an A (toban attenuation arameter) R A log () log where i the roun u oerator. R i the aban rile arameter (B) or R log when i the aban rile arameter (, ). A i the toban attenuation arameter (B) or A log when A i the toban A attenuation arameter: (, ). A DOI.5/IJSSST.a ISS: 7-8x online, 7-8 rint

2 . Determine the filter arameter (or the utoff frequeny of the Butterworth CT filter) from the eifiation:, R, A, an. (.) R or A (.). Determine the ole of the ytem funtion of the Butterworth CT filter (from the filter arameter an ) k k ex, k,,,, for i o. (.) k k ex, k,,,, for i even. (.) 5 -lane Figure. The ole lot of the ytem funtion of the Butterworth CT filter for lane Figure. The ole lot of the ytem funtion of the Butterworth CT filter for The table an aual filter an be efine by limiting ole in the left half-lain. ) of the Butterworth CT filter (from the etimate ole) (5). Determine the ytem funtion ( LP k III. IMPULSE IVARIACE The imule invariane onet [] i that the imule reone of the DT ytem ( hn ) i etimate from amling the imule reone of the CT ytem ( h t ). Therefore, the imule reone of the DT ytem ( hn) an be mathematially exree a hn Th nt where T i a amle erio. (6) By uing both the CT Fourier analyi an the DT Fourier analyi, the ytem funtion of the DT ytem an be mathematially exree a e k (7) k T T If the CT filter i banlimite (or, T ) then the ytem funtion of the DT ytem an be mathematially imlifie a follow: e, (8) k T Therefore, the obetive of thi imule invariane onet i for reerving the hae of the imule reone of the CT ytem ( h t ) by uing the amling tehnique. Conequently, the CT ytem mut be banlimite (or DOI.5/IJSSST.a ISS: 7-8x online, 7-8 rint

3 , T ) beaue if the CT ytem i not banlimite then the aliaing roblem our. The tranformation between CT omlex lain an DT omlex lain, whih an be illutrate a following figure, an be mathematially exree a The Re i mae into Rez (or inie the unit irle). The Re i mae into Rez (or inie the unit irle). The Re i mae into Rez (or outie the unit irle). -lane T T T T Many-to-One Tranformation z e T z-lane Unit irle Figure. The imule invariane maing of the omlex lane from the -lane to z-lane. The eign of the DT IIR lowa filter by uing the imule invariane an be exree a following te.. Determine the ontinuou frequeny of aban ( ) an toban ( ) from the eifiation:, an T (9) an () T. Determine the ytem funtion of ontinuoutime lowa filter (Butterworth filter, Chebyhev filter or Elliti filter) from the eifiation:,, R an A (the etail of the ontinuou-time filter eign i exree in the reeing etion) of ontinuoutime lowa filter in the artial fration exanion form, whih an be mathematially exree a (). Determine the ytem funtion k k. Determine the ytem funtion z of irete-time lowa IIR filter from the ytem funtion of T ontinuou-time lowa filter by uing imule invariane tranformation for onverting the ontinuou-time ole to be irete-time lowa k ole T k e. Conequently, the ytem funtion z of irete-time lowa IIR filter an be mathematially exree a following z () T k k e z The next etion reent numerou examle of the eigning of CT Butterworth lowa filter by uing the mathematial analyi for emontrating the ytem funtion, the magnitue reone an the hae reone for examining the erformane of thi filter. IV. EXPERIMETAL SIMULATIO All imulation outome are omute by the MATLAB oftware, whih are oerate by PC with CPU: Intel i7-67q an RAM Memory: 6 GB. A. Exerimental Simulation Reult for Cae By uing the imule invariane tehnique, eign the Butterworth IIR igital filter where the aban gain (. ) between B an -7 B, an toban (. ) ha attenuation of -6 B where T. Sketh the magnitue in eibel (B), the magnitue an the hae of thi frequeny reone of thi Butterworth IIR igital filter The eign of the DT IIR lowa filter by uing the imule invariane an be exree a following te. Ste : Determine the ontinuou frequeny of aban ( ) an toban ( ) from the eifiation:, an T (9). T T. (.) an ().. (.) Ste : Determine the ytem funtion of ontinuou-time lowa filter (Butterworth filter) from the eifiation:,, R an A (the etail of the ontinuou-time filter eign i exree in the reeing etion) Ste.: Determine the filter arameter (or the orer of the Butterworth CT filter). The loe form equation for etermining the arameter of the CT Butterworth filter an be mathematially exree a R A log () log DOI.5/IJSSST.a ISS: 7-8x online, 7-8 rint

4 6 7 log. log or (.) Ste.: Determine the filter arameter (or the utoff frequeny of the Butterworth CT filter). By ubtituting arameter in Eq.(.), the filter arameter an be mathematially exree a (.) R..985 (.) 6.67 By ubtituting arameter in the Eq.(.), the filter arameter an be mathematially exree a (.) A..5 (.5) From the reult of the filter arameter in the above equation, if the filter arameter then the filter arameter an be et between therefore the filter arameter an be et to be.5 in thi ae for imliity eretive. Ste.: Determine the ole of the ytem funtion of the Butterworth CT filter lane Figure. The 6 ole lot of the magnitue of quare funtion for th orer Butterworth filter From the filter arameter an.5, the magnitue of quare funtion omrie of 6 (or ) ole, whih are uniformly loate on the irle with.5 raiu a illutrate in the following figure. From the tability ontrain of the DT-LTI ytem, the ole of the table ytem funtion mut be loate in the left half of the -lain (, an ) a illutrate in the following figure Figure 5. The ole lot of the table ytem funtion orer -lane for th From the above figure, the loation of eah ole an be mathematially exree a o.5.5, o in, ,.5. From the tability ontrain of the DT-LTI ytem, the ole of the table ytem funtion mut be loate in the left half of the -lain (, an ) a illutrate in the following figure..5ex,.5ex,.5ex.5+.,.5,.5. (.6) Ste.: Determine the ytem funtion ( ) of the Butterworth CT filter DOI.5/IJSSST.a ISS: 7-8x online, 7-8 rint

5 LP k Ste : Determine the ytem funtion (5) (.7) of ontinuou-time lowa filter in the artial fration exanion form, whih an be mathematially exree a () k k Ste : Determine the ytem funtion ( z ) of the DT filter (from the ytem funtion ( ) of the Butterworth CT filter) for T z () T k k e z z e z e z e z z.665 z z z..7z z (.8).z.6z.679z Firt, the magnitue in eibel (B), the magnitue an the hae of thi frequeny reone of the analog filter an be illutrate a figure 6. Later, By uing imule invariant onet, the magnitue an the hae of e an be thi frequeny reone of the igital filter illutrate a figure 7. From thee exerimental imulation reult, the imule invariane onet an erfetly onvere from the analog filter to the igital filter for magnitue eretive a hown in Fig. 6(a) an Fig. 7(a). owever, the hae of the frequeny reone of the igital filter, whih i onverte from analog filter, i everely itorte from original analog filter a hown in Fig. 6() an Fig. 7(). B. Exerimental Simulation Reult for Cae By uing the imule invariane tehnique, eign the Butterworth IIR igital filter where the aban gain (. ) between B an - B, an toban (. ) ha attenuation of -5 B where T. Sketh the magnitue in eibel (B), the magnitue, the hae an the grou elay of thi frequeny reone of thi Butterworth IIR igital filter The eign of the DT IIR lowa filter by uing the imule invariane an be exree a following te. Ste : Determine the ontinuou frequeny of aban ( ) an toban ( ) from the eifiation:, an T (9). T T. (.) an ().. (.) Ste : Determine the ytem funtion of ontinuou-time lowa filter (Butterworth filter) from the eifiation:,, R an A (the etail of the ontinuou-time filter eign i exree in the reeing etion) Ste.: Determine the filter arameter (or the orer of the Butterworth CT filter). The loe form equation for etermining the arameter of the CT Butterworth filter an be mathematially exree a R A log () log 5 log. log or 6 (.) Ste.: Determine the filter arameter (or the utoff frequeny of the Butterworth CT filter). DOI.5/IJSSST.a ISS: 7-8x online, 7-8 rint

6 Figure 6 (a) The relationhi between the magnitue in eibel (B) of the e, an. frequeny reone of the analog filter, log Figure 7 (a) The relationhi between the magnitue in eibel (B) of the e, an. frequeny reone of the igital filter, log Figure 6 (b) The relationhi between the magnitue of the frequeny reone of the analog filter,, an analog frequeny. Figure 7 (b) The relationhi between the magnitue of the frequeny reone of the igital filter, e, an igital frequeny. Figure 6 () The relationhi between the hae of the frequeny reone, an analog frequeny. of the analog filter, By ubtituting arameter 6 in Eq.(.), the filter arameter an be mathematially exree a (.) R 6.7 (.) By ubtituting arameter 6 in the Eq.(.), the filter arameter an be mathematially exree a Figure 7 () The relationhi between the hae of the frequeny reone e, an igital frequeny. of the igital filter, A 6 5 (.).787 (.) From the reult of the filter arameter in the above equation, if the filter arameter 6 then the filter arameter an be et between therefore, the filter arameter an be et to be.7 in thi ae. Ste.: Determine the ole of the ytem funtion of the Butterworth CT filter. From the filter arameter 6 an.7, the magnitue of quare funtion: DOI.5/IJSSST.a ISS: 7-8x online, 7-8 rint

7 whih omrie (or ) ole, whih are uniformly loate on the irle with.7 raiu a illutrate in the following figure. From the tability ontrain of the DT-LTI ytem, the ole of the table ytem funtion mut be loate in the left half of the -lain (,,, 5, 6 an 7 ) a 8 illutrate in the following figure lane Figure 8. The ole lot of the magnitue of quare funtion for 6 th orer Butterworth filter Figure 9. The 6 ole lot of the table ytem funtion -lane for 6 th orer From the above figure, the loation of eah ole an be mathematially exree a, o 5 in 5 8, , , 7 o in, , , 6 o in 5, , (.6) (.7) (.8) From the tability ontrain of the DT-LTI ytem, the ole of the table ytem funtion mut be loate in the left half of the -lain (, an ) a illutrate in the following figure..5ex,.5ex,.5ex.5+.,.5,.5. Ste.: Determine the ytem funtion ( ) of the Butterworth CT filter (5) LP k (.9) DOI.5/IJSSST.a ISS: 7-8x online, 7-8 rint

8 Ste : Determine the ytem funtion of ontinuou-time lowa filter in the artial fration exanion form, whih an be mathematially exree a () k k z.8.55z.97 z.699 z.69 z.699 z z z.997 z.57z () Firt, the magnitue in eibel (B), the magnitue an the hae of thi frequeny reone of the analog filter an be illutrate a figure. Later, the magnitue an the hae of thi frequeny reone of the igital filter e an be illutrate a figure. From thee exerimental imulation reult, the imule invariane onet an erfetly onvere from the analog filter to the igital filter for magnitue eretive a hown in Fig. (a) an Fig. (a). owever, the hae of the frequeny reone of the igital filter, whih i onverte from analog filter, i everely itorte from original analog filter a hown in Fig. () an Fig. (). Ste : Determine the ytem funtion ( z ) of the DT filter (from the ytem funtion ( ) of the Butterworth CT filter) for T z () T k k e z e z e z e z z e z e z.679 e.8 z z z z.56.9z.56.9z z z Figure (a) The relationhi between the magnitue in eibel (B) of log e, an. the frequeny reone of the analog filter, Figure (b) The relationhi between the magnitue of the frequeny reone of the analog filter,, an analog frequeny. DOI.5/IJSSST.a ISS: 7-8x online, 7-8 rint

9 V. COCLUSIO We invetigate the aaity of the igital IIR filter eign tehnique uing Butterworth an imule invariane onet. We emontrate the filter eire roertie in both mathematial an omuter imulation. The reult how the eign tehnique ha goo erformane for magnitue reone requirement but oor erformane for hae reone requirement, eeially at high frequeny. Figure () The relationhi between the hae of the frequeny reone, an analog frequeny. of the analog filter, ACKOWLEDGMET The reearh roet wa fune by Aumtion Univerity. REFERECES Figure (a) The relationhi between the magnitue in eibel (B) of log e, an. the frequeny reone of the igital filter, Figure (b) The relationhi between the magnitue of the frequeny reone of the igital filter, e, an igital frequeny. Figure () The relationhi between the hae of the frequeny reone e, an igital frequeny. of the igital filter, [] A. V. Oenhiem, A. S. Willky, an S.. awab, Signal an Sytem, Prentie-all, n e., 997. [] A. V. Oenheim, an R.W. Shafer, Direte-Time Signal Proeing, Prentie all, n e., 998. [] A. V. Oenheim, an R.W. Shafer, Direte-Time Signal Proeing, Prentie all, n e.,. [] C. Ray Wylie & Loui C. Barrett, Avane Engineering Mathemati, MGraw-ill Comanie, In., 6th e., 995 [5] Erwin Kreyzig, Avane Engineering Mathemati, John Wiley & Son, In., th e.,. [6] wei P. u, SCAUM'S OUTLIES OF Theory an Problem of Signal an Sytem SCAUM' Outline, MGraw-ill Comanie, In., 995. [7] L. Philli, J. M. Parr, an E. A. Rikin, Signal, Sytem, an Tranform, Prentie-all, th e., 7. [8] Monon. aye, "Shaum' Outline of Theory an Problem of Digital Signal Proeing" SCAUM' Outline, MGraw-ill Comanie, In., 999 [9] Rafael C. Gonzalez & Rihar E. Woo, Digital Image Proeing, Prentie-all, th e.,. [] S. aykin an B. V. Veen, Signal an Sytem, John Wiley & Son, In., n e.,. [] Shibenu Mahata, Suman Kumar Saha, Raib Kar, Durbaal Manal, Aurate integer-orer rational aroximation of frational-orer low-a Butterworth filter uing a metaheuriti otimiation aroah, IET Signal Proeing, 8. [] T.J. Cavihi, Imule invariane an multile-orer ole, IEEE Tranation on Signal Proeing, Volume:, Iue: 9, 996 [] Vinay K. Ingle an John G. Proaki, Digital Signal Proeing uing Matlab, Brook/Cole Thomon Learning,. [] Vorao Patanaviit an Kornkamol Thakulukanant, Mathematial Tutorial of Direte-Time Analyi of Aliaing an on-aliaing Perioi Samling Conet with Fourier Analyi for Digital Signal Proeing an Digital Communiation Proetive, SDU Reearh Journal Siene an Tehnology, Suan Duit Raabhat Univerity, Vol. 9, o., Se.-De. 6. (inexe by TCI Grou an ACI) [5] Vorao Patanaviit, Mathematial Tutorial of Direte-Time Analyi of Samling Rate Changing Conet for Digital Signal Proeing an Digital Communiation Proetive, RMUTT Journal Siene an Tehnology, RMUTT, Vol. 6, o., July.- De. 7. (inexe by TCI Grou an ACI) [6] Vorao Patanaviit, Conetual Framework of Suer Reolution Reontrution Bae on Frequeny Domain From Aliae Multi- Low Reolution Image: Theory Part, Panyaiwat Journal, Panyaiwat Intitute of Management (PIM), Thailan, Vol. 8, o., May. Aug. 6. (inexe by TCI Grou an ACI). DOI.5/IJSSST.a ISS: 7-8x online, 7-8 rint