Analysis of Digital IIR Filter with LabVIEW

Transcription

1 Anlysis of Digitl IIR Filter with LbVIEW Yduvir Singh Associte professor Thpr University, Ptil Punjb, Indi Swet Tripthi Assistnt professor Mewr University, Chittorgrh Rjsthn, Indi Mnoj Pndey Assistnt professor St. Mrgret Engineering College Neemrn, Rjsthn, Indi ABSTRACT Aiming t importnce of virtul instruments in the field of Digitl Signl Processing, digitl IIR Filter system is developed using Ntionl Instruments (NI) dt LbVIEW softwre pckge. All the types of IIR filters like Butterworth filters, Chebyshev filters, inverse Chebyshev filters, nd Elliptic filters re designed to generte their mgnitude response nd filter coefficients. The LbVIEW softwre is used to develop virtul instrument (VI) tht includes front pnel nd functionl digrm. The VI reds the desired prmeters of the filters entered by the user on the front pnel nd determines its mgnitude response nd filter coefficients. Keywords Digitl IIR filter, LbVIEW, Virtul Instruments.. INTRODUCTION The Digitl Filter Design problem involves the determintion of set of filter coefficients to meet set of design specifictions. These specifictions typiclly consist of the width of the pssbnd nd the corresponding gin, the width of the stopbnd(s) nd the ttenution therein; the bnd edge frequencies (which give n indiction of the trnsition bnd) nd the pek ripple tolerble in the pssbnd nd stopbnd(s) [7]. The LbVIEW bsed digitl filter system involves the concept of Virtul Instrumenttion. A virtul instrumenttion system is computer softwre tht user would employ to develop computerized test nd mesurement system, for controlling from computer desktop n externl mesurement hrdwre device, nd for displying test or mesurement dt collected by the externl device on instrumentlike pnels on computer screen. Two types of digitl filters exist the IIR (Infinite Impulse Response) nd the FIR (Finite Impulse Response). IIR filter possess certin properties, which mke them the preferred design choices in numerous situtions over FIR filters. Most notbly, FIR filters (ll zero system function) re lwys stble, with reliztion existing for ech FIR filter. Another feture exclusive to FIR filters is tht of liner phse response [9]. The design of IIR filters proceeds through vstly different set of steps thn those followed by FIR filter design lgorithms. The design of IIR filters is closely relted to the design of nlog filters, which is widely studied topic. An nlog filter is usully designed nd trnsformtion is crried out into the digitl domin. Two trnsformtions exist the impulse invrint trnsformtion nd the biliner trnsformtion. In this pper, the focus is on designing minimum order IIR filters to meet set of specifictions using LbVIEW functions. Ech design is ccompnied by plot of its frequency response, impulse response nd pole-zero digrms. The responses of IIR filters using LbVIEW re compred with the responses from MATLAB with the sme specifictions. The min gol of this pper is to obtin n optimized filter response long with the filter coefficients. 2. DIGITAL IIR FILTERS In signl processing, the function of filter is to remove unwnted prts of the signl, such s rndom noise, or to extrct useful prts of the signl, such s the components lying within certin frequency rnge. There re two min kinds of filter, nlog nd digitl. There re some considerble dvntges of digitl over nlog filters which mke digitl filters unvoidble [8]. Some of these re s follows. A digitl filter is progrmmble, i.e. its opertion is determined by progrm stored in the processor's memory. This mens the digitl filter cn esily be chnged without ffecting the circuitry (hrdwre). 2. Digitl filters re esily designed, tested nd implemented on generl-purpose computer or worksttion. 3. Unlike their nlog counterprts, digitl filters cn hndle low frequency signls ccurtely. As the speed of DSP technology continues to increse, digitl filters re being pplied to high frequency signls in the RF (rdio frequency) domin, which in the pst ws the exclusive preserve of nlog technology. Further digitl filters cn be clssified s FIR (Finite impulse response) filters nd IIR filters. Finite impulse response (FIR) filter, lso known s non-recursive filters (in non-recursive filter the current output is clculted solely from the current nd previous input vlues). Infinite impulse response (IIR) filter, lso known s recursive filter ( recursive filter is one which in ddition to input vlues lso uses previous output vlues). IIR filters hve the dvntges of providing the higher selectivity for prticulr order. IIR filters cn chieve the sme level of ttenution s FIR filters but with fr fewer coefficients. Therefore, n IIR filter cn provide significntly fster nd more efficient filtering opertion thn n FIR filter. FIR filters provide liner-phse response []. IIR filters provide nonliner-phse response. FIR filters re used for pplictions tht require liner-phse responses like high qulity udio systems. IIR filters re used for pplictions tht do not require phse informtion, such s signl monitoring pplictions. Infinite impulse response (IIR) filters opertes on current nd pst input vlues nd current nd pst output vlues. Theoreticlly, the impulse response of n IIR filter never reches zero nd is n infinite response. A recursive filter is one which in ddition to 23

2 input vlues lso uses previous output vlues [5]. The expression for recursive filter therefore contins not only terms involving x x, x,l but lso terms involving n, n n 2 yn, y n 2,L. the input vlues ( ) the pst output vlues The following generl difference eqution chrcterizes IIR filters Ν b Ν = b jx i j 0 j= 0 k= y i k i k ( ) where y b j the set of forwrd coefficients is, N b is the number of forwrd coefficients, k is the set of reverse coefficients, nd N is the number of reverse coefficients. Where xi is the current input, xi j is the pst inputs, nd y i k is the pst outputs. From this explntion, recursive filters require more clcultions to be performed, since there re previous output terms in the filter expression s well s input terms. In fct, the reverse is usully the cse: to chieve given frequency response chrcteristic using recursive filter generlly requires much lower order filter (nd therefore fewer terms to be evluted by the processor) thn the equivlent nonrecursive filter. IIR filters might hve ripple in the pssbnd, the stopbnd, or both. IIR filters hve nonliner-phse response [9]. Eqution 2 defines the direct-form trnsfer function of n IIR filter. ( Νb) ( b0 + b z bν z b H z) = ( Ν ) (2) + z Ν A filter implemented by directly using the structure defined by Eqution 3.2. Where n b nd n re the reverse nd forwrd coefficients of the IIR filter. Digitl IIR filter designs come from the clssicl nlog designs nd include the following filter types: Butterworth filters Chebyshev filters Chebyshev II filters, lso known s inverse Chebyshev nd Type II Chebyshev filters Elliptic filters, lso known s Cuer filters The IIR filter designs differ in the shrpness of the trnsition between the pssbnd nd the stopbnd nd where they exhibit their vrious chrcteristics in the pssbnd or the stopbnd [9]. z 2. Butterworth Filters Butterworth filters hve the following chrcteristics: Smooth response t ll frequencies Monotonic decrese from the specified cut-off frequencies Mximl fltness, with the idel response of unity in the pssbnd nd zero in the stopbnd Hlf-power frequency, or 3 db down frequency, tht corresponds to the specified cut-off frequencies. The trnsfer function for Butterworth filter is given by B Where n is the order of filter [6]. Butterworth filters do not lwys provide good pproximtion of the idel filter response becuse of the slow rolloff between the pssbnd nd the stopbnd. 2.2 Chebyshev Filters Chebyshev filters hve the following chrcteristics: Minimiztion of pek error in the pssbnd Equiripple mgnitude response in the pssbnd Monotoniclly decresing mgnitude response in the stopbnd Shrper rolloff thn Butterworth filters Compred to Butterworth filter, Chebyshev filter cn chieve shrper trnsition between the pssbnd nd the stopbnd with lower order filter. The shrp trnsition between the pssbnd nd the stopbnd of Chebyshev filter produces smller bsolute errors nd fster execution speeds thn Butterworth filter. The frequency response of the filter is given by H ( Ω) = + ε Τ Ν Ω Ω (4) Ρ Where ε prmeter of the filter is relted to ripple present in T N x is the Nth- order Chebyshev the pssbnd nd ( ) polynomil defined s T Ν ( ω ) = = cos( N cos + cos( N cosh x) x) 2 n x x 2.3 Chebyshev II Filters or Inverse Chebyshev Filters Chebyshev II filters hve the following chrcteristics: Minimiztion of pek error in the stopbnd ω ω 0 2 (3) (5) 24

3 Equiripple mgnitude response in the stopbnd Monotoniclly decresing mgnitude response in the pssbnd Shrper rolloff thn Butterworth filters Chebyshev II filters re similr to Chebyshev filters. However, Chebyshev II filters differ from Chebyshev filters in the following wys: Chebyshev II filters minimize pek error in the stopbnd insted of the pssbnd. Minimizing pek error in the stopbnd insted of the pssbnd is n dvntge of Chebyshev II filters over Chebyshev filters [9]. Chebyshev II filters hve n equiripple mgnitude response in the stopbnd insted of the pssbnd. Chebyshev II filters hve monotoniclly decresing mgnitude response in the pssbnd insted of the stopbnd. 2.4 Elliptic Filters Elliptic filters hve the following chrcteristics: Minimiztion of pek error in the pssbnd nd the stopbnd Equiripples in the pssbnd nd the stopbnd Compred with the sme order Butterworth or Chebyshev filters, the elliptic filters provide the shrpest trnsition between the pssbnd nd the stopbnd, which ccounts for their widespred use [9]. The trnsfer function is given by 2 2 H( Ω) = + ε U Ω N (6) Ωc U N (x) Where the Jcobin is elliptic function of order N nd ε is constnt relted to pssbnd ripple. They provide reliztion with the lowest order for prticulr set of conditions. Selection of Digitl filters for ny prticulr ppliction mong ll cn be done by considering following points: Does the nlysis require liner-phse response? Cn the nlysis tolerte ripples? Does the nlysis require nrrow trnsition bnd? 3. DESIGN OF DIGITAL IIR FILTER WITH LABVIEW LbVIEW empowers to build own solutions for scientific nd engineering systems. It gives the flexibility nd performnce of powerful progrmming lnguge without the ssocited difficulty nd complexity. It gives thousnds of successful users fster wy to progrm instrumenttion, dt cquisition, nd digitl signl processing [2]. By using LbVIEW to prototype, design, test, nd implement your instrument systems, system development time cn be reduced nd productivity increses by fctor of 4 to 0 [0]. We design IIR filters by pproximting the desired mgnitude response of discrete-time system. 3. Designing Butterworth Filter Figure shows the block digrm of VI tht returns the mgnitude response of butterworth IIR filter. The VI in Figure completes the following steps to compute the mgnitude response, filter coefficients nd pole-zero plots to find the stbility of the filter. Figure Butterworth.vi. Pss ll the prmeters (cutoff frequency, stop frequency, pssbnd ttenution, stopbnd ttenution, smple rte) with filer type lowpss, highpss, 25

4 bndpss, or bndstop to the Cse Structure to clculte the order of filter [3]. 2. Apply these prmeters to the Butterworth Coefficient Function to generte the coefficients [3]. 3. Divides these coefficient rrys into two seprte prts nmed s forwrd nd reverse coefficients. 4. Disply these forwrd nd reverse coefficients with coefficient size. 5. Pss these coefficients to Complex Polynomil Roots VI nd pply its output to Complex to Re/Im Function to disply the pole-zero plots. 6. Compre the output of Complex to Re/Im Function (only for reverse coefficients) with - to find out the stbility. This is shown with the help of LED. If LED is red then the system is unstble otherwise stble. 7. Pss n impulse signl through the butterworth filter function. 8. Apply ll the prmeters nd filter type lowpss, highpss, bndpss, or bndstop to the butterworth filter function [3]. 9. The signl pssed out from this function is the impulse response of the filter. 0. Pss the filtered signl to the FFT VI. Use the FFT VI to perform Fourier trnsform on the impulse response nd to compute the frequency response of the filter, such tht the impulse response nd the frequency response comprise the Fourier trnsform pir h(t) H(f ). h (t) is the impulse response. H(f) is the frequency response [3].. Use the Arry Subset function to reduce the dt returned by the FFT VI. Hlf of the rel FFT result is redundnt so the VI needs to process only hlf of the dt returned by the FFT VI [4]. 2. Use the Complex To Polr function to obtin the mgnitude-nd-phse form of the dt returned by the FFT VI. The mgnitude-nd-phse form of the complex output from the FFT VI is esier to interpret thn the rectngulr component of the FFT [4]. 3. Convert the mgnitude to decibels. 4. Disply the mgnitude response with the help of wveform grph. 3.2 Designing Chebyshev Filter, Inverse Chebyshev Filter nd Elliptic Filter Becuse the sme mthemticl theory pplies to design other type of IIR filters, the block digrm in Figure 2, 3, nd 4 of VI return the mgnitude response of chebyshev, inverse chebyshev nd elliptic IIR filter respectively [4]. The design procedure is sme for ll type of filter s described for butterworth filter. The min difference between these VIs is tht the chebyshev, inverse chebyshev nd elliptic coefficient functions re used in step 2 in plce of butterworth coefficient function nd the chebyshev, inverse chebyshev nd elliptic filter Figure 2 Chebyshev.vi function in plce of butterworth filter function in step 7 nd step 8 []. 26

5 Figure 3 Inverse Chebyshev.vi 4. RESULTS AND DISCUSSIONS Figure 5 shows the response of 5 th order butterworth lowpss filter. The specifictions for this filter re given s: cutoff frequency 500 Hz Figure 4 Elliptic.vi stop frequency 000Hz pssbnd ttenution- 5 db stopbnd ttenution- 00 db smple rte Hz 27

6 Figure 5 Response of Butterworth Lowpss Filter for 5 th order Figure 6 shows the response of 3rd order Chebyshev lowpss filter while ll other specifictions re kept sme s bove. Chebyshev filters re nlog or digitl filters hving steeper roll-off nd more pssbnd ripple (type I) or stopbnd ripple (type II) thn Butterworth filters. Chebyshev filters hve the property tht they minimize the error between the idelized nd the ctul filter chrcteristic over the rnge of the filter, but with ripples in the pssbnd. The pole-zero plot of Chebyshev lowpss filter for Figure 6 Response of Chebyshev Lowpss Filter for 3 rd order bove specifiction is lso shown in figure 6. If we increse the order of the filter keeping the sme specifictions, the mgnitude response plot slope becomes steep. Figure 7 shows the response of 3rd order Inverse Chebyshev lowpss filter while ll other specifictions re kept sme s bove. If we increse the order of the filter keeping the sme specifictions, the mgnitude response plot slope becomes steep. 28

7 Figure 8 shows the response of 3rd order Elliptic lowpss filter while ll other specifictions re kept sme s bove. If we increse the order of the filter keeping the sme specifictions, the mgnitude response plot slope becomes steep. From the bove results we nlyze tht the trnsition bnd become shrper between pssbnd nd stopbnd on incresing the order of filter with other prmeters hving sme vlue. Compred to Butterworth filter s shown in figure 5, Chebyshev filter s shown in figure 6, Inverse Chebyshev of figure 7 cn chieve shrper trnsition between the pssbnd nd the stopbnd with lower order filter. The shrp trnsition between the pssbnd nd the stopbnd of these filter produces smller bsolute errors nd fster execution speeds thn Butterworth filter. Compred with the sme order Butterworth or Chebyshev filters, the elliptic filters figure 8 provide the shrpest trnsition between the pssbnd nd the stopbnd, which ccounts for their widespred use. 5. CONCLUSIONS In this pper, the design of IIR filters ws considered. Severl results from theory were verified in the design. The chrcteristics of number of importnt pproximtions Butterworth, Chebyshev, nd Elliptic were ffirmed from the results obtined. Experimentl results re very enthusistic. In LbVIEW the prmeters cn be chnged t the time of execution of the progrm but in cse of MATLAB it is not possible. There is smooth Trnsition Bnd in LbVIEW Design nd Lest Squre error in cse of LbVIEW is lso less thn MATLAB Design. LbVIEW Design is bsed on G-Progrmming so tht the nlysis of the performnce cn be done very esily. In LbVIEW nlysis of ll types of Filters (LP, HP, BP, nd BS) is possible in single progrm. In MATLAB ll hve seprte progrms. Design Constrints re more ccurte in cse of LbVIEW becuse we cn tke good pproximtion on LbVIEW but in MATLAB ll rel things re implemented there is no pproximtion in progrmming if requiring thn dded on progrm (there is no need of user definition on pproximtion). It is lso verified tht on incresing the order of ny filter the trnsition bnd decreses for the sme prmeters. Figure 7 Response of Inverse Chebyshev Lowpss Filter for 3 rd order 29

8 Figure 8 Response of Elliptic Lowpss Filter for 3 rd order 6. REFERENCES [] Aimin Jing nd Hon Keung Kwn, IIR Digitl Filter Design with Novel Stbility Criterion Bsed on Argument Principle, Deprtment of Electricl nd Computer Engineering,University of Windsor, vol., pp. 26-3, 2007 [2] Beyon, J. Y., Hnds-On Exercise Mnul for LbVIEW Progrmming, Dt Acquisition nd Anlysis, Prentice Hll, Inc., New Jersey, 200 [3] Chugni, M. L., LbVIEW Signl Processing, Prentice Hll, Inc., Upper Sddle River, New Jersey, 998 [4] Clrk C. L., LbVIEW Digitl Signl Processing nd Digitl Communictions, Tt McGRAW-HILL, 2005 [5] Fhmy M.F., Abo-Zhhd M. nd Shoby M.., "Design of Selective Liner Phse Switched-Cpcitor Filters with Equiripple Pssbnd Amplitude Responses", IEEE Trns. on Circuits nd Systems, CAS-35, no. 0, pp , 988 [6] Jckson L. B., Digitl Filters nd Signl Processing, 3 rd ed., Kluwer Acdemic Publishers, 996 [7] Nmjin Kim, Digitl Signl Processing System-Level Design Using LbVIEW, Elsevier Inc.,vol., 22-27, 2005 [8] Oppenheim, A.V., R.W. Schfer, Discrete Time Signl Processing, 2 nd ed., Person Eduction, 2005 [9] Prokis J.G., nd D.G. Mnolkis, Digitl Signl Processing, Principles Algorithms, nd Applictions, 3 rd ed., Person Eduction, Inc., 996 [0] Wells, L. K. nd Trvis, J., LbVIEW for Everyone Grphicl Progrmming Mde Even Esier, Prentice Hll, Inc., Upper Sddle River, New Jersey,