Digital Signal Processing Handbook P11

Karam, L.J.; McClellan, J.H.; Selesnick, E.W. & Burrus, C.S. “Digital Filtering” Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton: CRC Press LLC, 1999 c  1999byCRCPressLLC 11 Digital Filtering Lina J. Karam Arizona State University James H. McClellan Georgia Intitute of Technology Ivan W. Selesnick Polytechnic University C. Sidney Burrus Rice University 11.1 Introduction 11.2 Steps in Filter Design Creating the Design Specifications • Specs Derived from Ana- logFiltering • Specifying an Error Measure • SelectingtheFilter Type and Order • Designing the Filter • Realizing the Designed Filter 11.3 Classical Filter Design Methods FIR Design Methods • IIR Design Methods 11.4 Other Developments in Digital Filter Design FIR Filter Design • IIR Filter Design 11.5 Software Tools Filter Design: Graphical User Interface (GUI) • Filter Imple- mentation References 11.1 Introduction Digital filters are widely used in processing digital signals of many diverse applications, including speech processing and data communications, image and video processing, sonar, radar, seismic and oil exploration, and consumer electronics. One class of digital filters, the linear shift-invariant (LSI) type, are the most frequently used because they are simple to analyze, design, and implement. This chapter treats the LSI case only; other filter types, such as adaptive filters, require quite different design methodologies. An LSI digital filter can be uniquely identified in the time/space domain by its impulse response h(n) (where n is an integer index). Alternatively, the LSI digital filter can be uniquely characterized in the frequency domain by its frequency response H(ω)(where ω is a real-valued frequency variable in radians), which is also the Discrete-Time Fourier Transform (DTFT) of the sequence h(n). LSI digital filters are of two main types: Finite-duration Impulse Response (FIR) filters for which the impulse response h(n) is non-zero for only a finite number of samples, and Infinite-duration Impulse Response (IIR) filters for which h(n) has an infinite number of non-zero samples. In the FIR case, the samples of the sequence h(n) are commonly referred to as the filter coefficients; for the IIR case, the filter coefficients include feedback terms in a difference equation. Digital filter design has been extensively addressed within the last 25 years. The design and realization of digital filters involve a blend of theory, applications, and technologies. For most applications, it isdesirable to design frequency-selective filters whichalter orpass unchanged different frequency components. In this case, the desired design specifications are given in the frequency domain by specifying a desired frequency response D(f ). Note that D(f ) is, in general, complex valued, consisting of a desired magnitude response |D(f )| and a desired phase response  D(f ). c  1999 by CRC Press LLC One of the most important problems is the design of a highly frequency-selective filter with sharp cutoff edges (short transition bands). However, ideal sharp edges correspond mathematically to discontinuities and cannot be realized in practice. Therefore, the filter design problem consists in finding an implementable filter whose order is low and whose frequency response H(f) best approximates the specified ideal magnitude and phase responses which are given as the desired design specifications or constraints. The design of digital filters is typically done by performing the following steps: 1. Convert the desireddesign constraints intoprecisespecifications of thedesired magnitude and phase responses, designed filter type (FIR or IIR), filter order, error tolerance, or criteria. 2. Approximate the design specifications (of Step 1) by finding the implementable FIR or IIR filter such that the obtained filter frequency response best meets the design specs according to a mathematical error criterion. 3. Realize the filter using the digital technology most suitable for the considered application. While Step 2 is performed using mathematical optimization and approximation methods, Step 1 is highly dependent on the application and the detail provided by the user. Step 3 depends on the technology or software used to build the filter. Nowadays, the optimization needed in Step 2 is usually done with computer software that im- plements sophisticated numerical optimization routines. In addition, these design packages usually have a convenient graphical user interface to aid in the conversion of specs needed in Step 1. With such software, a filter design can be carried out quickly so that many designs can be tried in the process of getting the best filter. Since most filter design techniques involve the trade-off among competing parameters, the software can also incorporate design rules that allow the user to predict the order needed for certain specs without actually designing the filter, for example. This chapter is organized as follows. Section 11.2 provides a discussion of Steps 1 and 3, including creating the design specifications, selecting the filter type and order, specifying the error tolerances and criteria, and realizing the designed filter. Step 2 is treated in Sections 11.3 and 11.4. Section 11.3 describes the classical FIR and IIR design methods. Section 11.4 presents nonclassic and more recentlydeveloped design methods with added efficiency and/or flexibility. Finally, Section 11.5 gives examples of some of the currently available software design tools and describes the characteristics that a user can expect from such tools. 11.2 Steps in Filter Design Lina J. Karam The general filter design problem can be briefly stated as follows. Given some ideal frequency re- sponse, D(ω), finda realizableIIR or FIRdigital filter whose frequencyresponse, H(ω), approximates D(ω). The realizable filter is found by optimizing some measure of the filter’s performance, e.g., minimizing the filter order (IIR) or the filter length (FIR), or minimizing the width of the transition bands, or reducing the passband error and/or stopband error. Setting up the specifications for the general filter design problem will define these parameters and show which trade-offs are possible. 11.2.1 Creating the Design Specifications Since the frequency response of a digital filter is always periodic in the frequency variable ω with aperiodof2π, the design specifications need only be specified for one period; usually, over the frequency region [−π, π]. Furthermore, when the frequency response is conjugate-symmetric (i.e., c  1999 by CRC Press LLC D ∗ (ω) = D(−ω)), then it is sufficient to specify the response only on the positive frequency interval [0,π]. The conjugate-symmetric case is the most common, because it corresponds to filters with real coefficients. The simplestcase is thatof an ideal low-passdigital filterwith zerophase, whose frequencyresponse can be expressed as: D(ω) =  1, |ω| <ω c 0,ω c < |ω| <π (11.1) where ω c is the cutoff frequency corresponding to the location of a sharp cutoff edge, as shown in Fig. 11.1(a). In this case, the frequency response, D(ω), is real-valued and, therefore, corresponds also to the magnitude response of the filter (since the phase is zero). Ideal frequency responses of other commonly used frequency-selective filters are shown in Fig. 11.1. FIGURE 11.1: Common ideal digital filter types. These ideal filters have frequency responseswith sharp cutoff edges (discontinuities) and cannot be implemented directly. They must be approximated with a realizable system—the sharp cutoff edges need to be replaced with transition bands in which the designed frequency response would change smoothly in going from one band to the other. So, design templates need to be provided where the sharp cutoff edges are replaced with non-zero width transition bands located around the ideal cutoff edges. A typical design template for a lowpass filter is shown in Fig. 11.2, where: • ω p is the passband cutoff frequency. • ω s is thestopband cutofffrequency. The cutofffrequency ω c is usuallytaken tobe midway between the passband and stopband cutoff frequencies. • The open interval (ω p ,ω s ) is the transition band of width ω t = ω s − ω p . In the commondesign methods, no design specifications are givenin the transition bands which are therefore commonly known as “don’t care bands.” However, it is usually desirable to have the frequency response change smoothly (i.e., no fluctuations or overshoots) in the c  1999 by CRC Press LLC transition bands; this requirement might not be satisfied by a design method that places no design constraints on the frequency response in the transition bands. • δ p is known as the passband ripple and is the maximum allowable error in the passband. • δ s is known as the stopband ripple and is the maximum allowable error in the stopband. FIGURE 11.2: Design template for a lowpass filter. The objective of filter design then is to find a realizable FIR or IIR filter whose frequency response H(ω)approximates the specified design constraints given by the design template. Ideally, the filter design process would make each of the following parameters as small as possible: δ p , δ s , ω t , IIR filter order (number of poles of H(z) which is a rational function) or FIR filter length (number of zeros of H(z) which is a finite polynomial). Practically, the filter design process minimizes one of these parameters while holding the others fixed. Traditionally, many of the filters designed in practice are specified in terms of constraints on the magnitude response and no constraints on the phase response other than those imposed implicitly by stability and/or causality requirements (e.g., poles inside unit circle in the complex Z-plane for IIR, and linear-phase for FIR [1]). More recently, design methods that include phase design specifications have been presented [2, 3, 4, 5]. In this latter case, two design templates must be provided, one for the magnitude response and another for the (passband) phase response. An ideal phase response is most likely a constant slope phase function:  D(ω) =−Mω The parameter M is equivalent to the desired delay of the filter (in samples). An error template for the phase would be a tolerance about the desired phase, e.g., δ φ would denote the maximum allowable phase ripple, so that we require |  H(ω)−  D(ω)| <δ φ 11.2.2 Specs Derived from Analog Filtering Often, the desired design specifications are not given directly in the digital domain. Instead, an equivalent analog filtering operation is desired but is to be performed using an embedded digital filter. Figure 11.3 shows a standard system for processing continuous-time (-space) signals using a digital filter. The analog input signal is first transformed into a digital signal through an analog-to- digital (A/D)conversionoperation; then, filteringis carriedoutusinga digitalfilter; finally, thefiltered digital output is converted back to the analog domain using a digital-to-analog (D/A) converter. For c  1999 by CRC Press LLC this system, if the sampling period T s of the A/D and D/A converters is chosen appropriately to avoid aliasing of the input spectrum, the overall system (consisting of the A/D converter, the digital filter, and the D/A converter) behaves as an equivalent analog filter. In this case, the frequency response H a () of the equivalent analog filter is related to the frequency response H(ω) of the digital filter through a simple linear scaling relation between the digital frequency ω and the analog frequency . This linear scaling relation is given by ω = T s (11.2) leading to the following expression of the analog H a () in terms of the digital H(ω): H a () =        H (T s ), || < π T s 0, ||≥ π T s (11.3) FIGURE 11.3: Standard system for processing analog signals using a digital (discrete-time) filter. Equivalently, H(ω)can also be expressed in terms of H a () as follows: H(ω)= H a (ω/T s ), |ω| <π. (11.4) A typical filter design problem corresponding to this system is to design the digital filter such that the overall equivalent analog filter best approximates some ideal analog specifications. So, if we are given the desired analog specifications of the overall analog system, these can be turned into specifications for the desired digital filter by using Eq. (11.4). Then, a digital filter H(ω) can be designed to approximate the derived desired digital specifications. Finally, the resulting analog frequency response of the overall system can be found using Eq. (11.3), for example, to compare with the ideal analog response. 11.2.3 Specifying an Error Measure An error measure is needed to assess how much the designed filter H(ω) deviates from the desired filter D(ω). Defining the pointwise error E(ω) as: E(ω) =[D(ω)− H(ω)], (11.5) we must reduce E(ω) to a scalar error measure (also called an error norm). With a correctly chosen norm, there are many possible optimization algorithms that will compute the best filter parameters to minimize the chosen error norm. The following error norms are the most commonly used in filter design: c  1999 by CRC Press LLC • Mean Squared Error (MSE) or L 2 norm E 2 =  1 2π  B | E(ω) | 2 dω  1/2 (11.6) • L p norm which is a generalization of the L 2 norm and where p is a non-zero integer E p =  1 2π  B | E(ω) | p dω  1/p (11.7) • Chebyshev or L ∞ norm E ∞ = max ω∈B |E(ω)| (11.8) The Chebyshev error norm limits the worst case deviation from the ideal specifications. In the above definitions, |·|denotes the complex error magnitude and B is the frequency region of interest over which the error norm is to be minimized. The frequency subset B ⊂[−π, π) is taken to be the union of the desired passbands and stopbands. A more selective controlof the approximation accuracy can be achieved by introducing a weighting function W(ω)in Eq. (11.5) as follows: E(ω) = W(ω)[D(ω)− H(ω)]. (11.9) The weighting function W(ω) must be a real, strictly positive and continuous function on B.It can force a better match over selected regions or frequency points relative to other regions in B. Alternatively, note that Eq. (11.5) reduces to Eq. (11.9)ifwereplaceD(ω) with W (ω)D(ω) and H(ω)with W (ω)H (ω). 11.2.4 Selecting the Filter Type and Order As mentioned in Section 11.1, there are two main types of filters, namely FIR and IIR. These differ in their characteristics and in the way they are designed. Since the design algorithm depends strongly on the choice of IIR vs. FIR filter, the designer should make this decision as early as possible. Although the desired frequency response specifications can be approximated with either type of filter, deciding which of the two filter types to use depends on many factors including the implementation hardware, as well as the magnitude and phase characteristics of the resulting filter. To aid in this decision, the main characteristics of FIR and IIR filters are discussed below. 11.2.4.1 FIR Characteristics 1. The impulse response h(n) has a finite length, i.e., h(n) is non-zero only for a finite range of indices n. For a general N-length FIR system, h(n) = 0 only for N 1 ≤ n ≤ N 2 = (N 1 + N − 1). When N 1 ≥ 0, the filter is also causal. 2. The FIR frequency response H(ω)is a finite-degree polynomial in e jω of the form H(ω)= N 2  n=N 1 h(n)(e jω ) −n (11.10) where N 1 and N 2 are (negative or positive) integers corresponding to the indices of the first and last samples of h(n), respectively. The N impulse response samples are the free parameters of the design procedure. This form is general enough to represent non-causal filters such as zero-phase filters. c  1999 by CRC Press LLC 3.DesigninganFIRfilterconsistsinfindingthepolynomialH(ω)thatbestapproximatesthe designspecifications.Thisisdonebycomputingthe“optimal”(relativetosomecriteria) impulseresponsesamples{h(n)} N 2 n=N 1 ,whichcorrespondtotheunknowncoefficientsof thepolynomialH(ω).TheimpulseresponselengthNisusuallyfixed,butitcouldalso beconsideredasafreeparametertobeoptimized.ProceduresfordesigningFIRfilters aregiveninSections11.3.1and11.4.1. 4.Thefiltertransferfunction,denotedbyH(z),isthez-transformofh(n)andisusefulfor studyingthestabilityofthesystem.ForFIRfilters,H(z)isafinite-degreepolynomialin thecomplexvariablezandisgivenby H(z)=H(e jω )| e jω =z = N 2  n=N 1 h(n)z −n . (11.11) ItfollowsthatthefunctionH(z)hasnopolesexceptpossiblyat0or∞,i.e.,itcannotbe infiniteforanypointzwith0<|z|<∞.Ithasonlyzeros(pointszatwhichH(z)=0). Therefore,anFIRfilterisalwaysstable. 5.FIRfiltersallowthedesignofcausallinear-phasesystemswhichareveryimportantand widelyusedinpractice.Infact,inmanysignalprocessingapplications,suchasspeech andimageprocessing,itisdesirabletopasssomeportionofthesignalfrequencyband withminimaldistortion.Forthatpurpose,linear-phasesystemsareparticularlydesirable sincetheeffectofthelinear-phaseisapuretimedelay.Foramoredetaileddiscussionof linear-phasesystems,thereaderisreferredto[1]. 6.Becausetheimpulseresponseisoffinitelength,FIRfiltersarerealizedusingtheconvo- lutionoperation[1]whichcanbeimplementeddirectlyinthetime/spacedomain,orin termsoftheFFTinthefrequencydomain.Moredetailsabouttheimplementationwill begiveninSection11.2.6. 7.SinceFIRfiltershavenofeedbackloops,theyarerelativelyinsensitivetoround-offnoise. Noiseduetocoefficientquantizationcanbeaproblemforverylongfilters,butcanbe mitigatedbyavoidingthedirect-formstructures,andusingspecialstructuressuchasthe cascadeformforimplementation. 8.FIRfilterswithverylongimpulseresponses(N≈500)mightberequiredtomeetcertain designspecifications,e.g.,highaccuracyand/orshorttransitionbands.Longerfilterslead toanincreasedcomplexityforbothdesignandimplementation.Theyrequiresignificant computingtimetooptimizealltheparametersh(n),andalsomanyoperationspersecond intheactualfilterimplementation. 9.Thetrade-offamongthefilterdesignparametershasbeendeterminedempiricallyfor sometypesofFIRdesigns.Thefollowingsimple(approximate)formulashowsthe relationshipamongtheripples,bandedges,andfilterlength(N)foronemethod,the Parks-McClellanalgorithm : (N−1)ω≈ −20log 10  δ p δ s −13 2.324 whereω=ω s −ω p isthetransitionwidth.Thisformulaallowsthedesignertopredict thevalueofNthatwillbeneededtosatisfyspecsgivenfor{ω p ,ω s ,δ p ,δ s }.Other designformulasaregiveninSection11.3.1. c  1999byCRCPressLLC 11.2.4.2 IIRCharacteristics 1.Theimpulseresponseh(n)hasaninfinitenumberofnon-zerosamples(infinitelength). Asanexample,forageneralIIRfilter,h(n)=0onlyforN o ≤n≤∞,whereN o isa non-negativeinteger(commonly,N o istakentobe0;inthiscase,thefilterissaidtobe causal). 2.ThefrequencyresponseH(ω)isarationalfunction,i.e.,aratiooftwofinite-degree polynomialsine jω oftheform H(ω)= B(ω) A(ω) =e −jωN o  M k=0 b k e −jωk  N k=0 a k e −jωk (11.12) whereN o isanintegerconstant.TheorderofanIIRfilterisequaltoN,whichisthe degreeofthedenominatorinEq.(11.12);usuallythedegreeofthenumeratorMisno greaterthanN.TheorderNalsodeterminesthenumberofpreviousoutputsamplesthat needtobestoredandthenfedbacktocomputethecurrentoutputsample.Therefore, IIRsystemsarealsoknownasfeedbacksystems.Thefiltercoefficients{b n }and{a n }in Eq.(11.12)correspondtotheunknown(free)parametersofthedesign. 3.DesigninganIIRfilteramountstofindingtherationalfunctionH(ω)thatbestapprox- imatesthedesignspecifications.Inthefrequencydomain,thisisdonebycomputing the“optimal”(relativetosomecriteria)coefficients{b n }and{a n }inEq.(11.12)forthe rationalfunctionH(ω).ThefilterorderNisusuallyfixed,butcanalsobeconsidered asafreeparametertobeoptimized.ProceduresfordesigningIIRfiltersaregivenin Sections11.3.2and11.4.2. 4.Asmentionedpreviousl y,thefiltertransferfunction,denotedby H(z),isthez-transformofh(n)andisusefulforstudyingthestabilityofthesystem.In thecontextofLSIfilters,stabilityimpliesthataboundedinputtothefilterwillalways resultinaboundedoutput.ForIIRfilters,H(z)isarationalfunctioninthecomplex variablezandisgivenby H(z)=H  e jω  | e jω =z =z −N o  M k=0 b k z −k  N k=0 a k z −k (11.13) TherootsofthedenominatorpolynomialarepolesofthefunctionH(z),i.e.,H(z)is infiniteatpointszwith0≤|z|<∞.Stabilitythenrequiresthatnopoleslieonthe UnitCircle(U.C.)(|z|=1)inthez-plane.Causalityandstabilityrequirethatthepoles lieinsidetheU.C.inthez-plane.So,itispossibletoobtainaresultingIIRfilterthatis unstable.Also,coefficientquantizationnoisemightseverelyaffecttheresponseofthe filteranditsstabilitybydisturbingthepoleslocationsandbydrivingsomeofthepoles closertoorontotheU.C. 5.Itisnotpossibletodesigncausallinear-phaseIIRfilters.TheresultingIIRcausalrealizable filtersmusthaveanon-linearphaseresponse.Forward-backwardfilteringcanbeusedas animplementationtoapproximateazero-phaseresponse[1]. 6.Becausetheimpulseresponseisinfinitelylong,convolutioncannolongerbeusedto implementtheIIRfilters.Instead,IIRfiltersareefficientlyimplementedusingfeedback differenceequationsasdescribedinSection11.2.6. 7.ThenoisecharacteristicsofanIIRfiltercanbeamajorconsiderationwhendoingan implementation,especiallyinfixed-pointarithmetic.Coefficientquantizationdegrades theactualfilterresponsefromthatdesignedbyhigh-precisionsoftware.Morecriticalis c  1999byCRCPressLLC round-offnoisesensitivitywhichcanbeamplifiedbythefeedbackloopsinthefilter. 8.ComparedtoFIRfilters,IIRfilterscanachievethedesireddesignspecificationswitha relativelyloworder(asfewas4to6poles).So,fewerunknownparametersneedtobe computedandstored,whichmightleadtoalowerdesignandimplementationcomplexity. However,thephaseresponseofIIRfiltersisneverlinear,whichleadstotheuseofall- passfilterstocompensatethegroupdelay,andthusraisestheorderofthefilterandthe complexityofthedesignprocess. 9.IIRfiltersarecommonlydesignedbyusingclosed-formdesignformulascorresponding toclassicalfiltertypes.WhileforFIRfiltersthelength-estimatingformulasareonly approximate,theorder-estimatingformulasforIIRfiltersareexactsincetheyarederived fromthemathematicalpropertiesoftheclassicalprototypes.Theseformulasarevery usefultoobtaintheIIRfilterorderneededtosatisfythedesireddesignspecifications. 11.2.5 DesigningtheFilter Afterthedesignedfiltertype(FIRorIIR)isspecified,asuitabledesignprocedurecanbeselected dependingonthechosenfiltertype.Populardesignproceduresarebasedoncomputingtheunknown filterparametersbyoptimizingoneoftheerrorcriteriaindicatedinSection11.2.3. ForFIRfilters,thetwomainclassicalmethodsarethewindowingmethod[1]andtheParks- McClellan(Remez)algorithm[6].ThewindowingmethodminimizestheMSEwhenarectangular window(correspondingtopuretruncationoftheidealimpulseresponse)isusedattheexpenseof possiblelargeovershootsnearthebandedgesandlargeripplesintheresultingfrequencyresponse.It issuboptimalwhenothergeneralwindowsareused.However,theedgeovershoot,transitionwidth, andrippleheightcanbecontrolledbyusingdifferenttypesofwindowsasdescribedinSection onpage 11 -13.TheParks-McClellan(Remez)algorithmminimizestheChebyshev(L ∞ )error normresultinginoptimalequirippledesigns.However,theoriginalParks-McClellanalgorithmis restrictedtothedesignoflinear-phasefilterswithasymmetricmagnituderesponse.Anextension ofthisalgorithmthatallowsthedesignofoptimalFIRfilterswitharbitrarymagnitudeandphase specificationshasbeenpresentedbyKaramandMcClellanin[2,3].Linear-programming-based[4,7] andConstrainedleastsquare[8]optimizationmethodsalsohavebeenpresentedtoallowtheinclusion ofadditionalimportantdesignconstraints.TheseandotherFIRdesignproceduresaredescribedin Sections11.3.1and11.4.1. WhilethedesignofFIRfiltersistypicallyperformeddirectlyinthedigitaldomain,IIRfiltersare commonlydesignedbytransformingthedigitaldesignspecificationsintoanalogdesignspecifications andperformingthefilterdesignintheanalogdomain.Theresultinganalogfilteristhentransformed intoadigitalfilterusingasuitabletransformation.OneimportantclassicalIIRdesignmethodis theBilinearTransformationmethod.Digital-onlyIIRdesignmethodshavealsobeenpresented.A descriptionofIIRdesignproceduresisgiveninSections11.3.2and11.4.2. 11.2.6 RealizingtheDesignedFilter Realizingthedesigneddigitalfiltercorrespondstocomputingtheoutputofthefilterinresponseto anygiveninput.ForLSIfilters,thisissimplifiedbythefactthattheinputandoutputsignalsare relatedthroughasimpleconvolutionoperationinthetime/spacedomain.Ifx(n)istheinput,y(n) thecorrespondingoutput,andh(n)theimpulseresponseoftheLSIfilter,thenthisrelationisgiven by y(n)=h(n)∗x(n)= N 2  k=N 1 h(k)x(n−k), (11.14) c  1999byCRCPressLLC [...]... input, the size of the DFTs in Eq (11.16) and, therefore, the needed storage vary with the size of the input signal To overcome this problem and to handle the processing of large-size signals, block-based convolution (also known as sectioned or high-speed convolution) is used where the input signal is divided into blocks (sections) of fixed equal size; then, the convolution of each input block with h(n)... NDFT is the size of the DFT and corresponds to the number of sample points within the period 2π It is a known fact that the time/space digital signal can be exactly recovered from its DFT if NDFT is chosen to be greater than or equal to the length of the time/space signal Using the DFT, Eq (11.15) becomes Y (k) = H (k)X(k), k = 0, , NDFT (11.16) where NDFT ≥ max{length of x(n)+ length of h(n) −... “discrete-time/space” (digital) filter design problem into a “continuous-time/space” (analog) filter design problem, which can be solved using well-developed and relatively simple design procedures based on closed-form design formulas Then, a transformation is used to map the designed analog filter into a digital filter meeting the desired specifications Let H (z) denote the transfer function of a digital filter... function Ha (s) can be converted into a digital filter whose transfer function is equal to H (z) = Ha (s)| −1 s=K( 1−z−1 ) (11.66) 1+z Alternatively, the mapping can be used to convert a digital filter into an analog filter by expressing z in function of s Note that the analog frequency variable corresponds to the imaginary part of s (i.e., s = σ + j ), while the digital frequency variable ω (in radians)... s-plane [11] 3 Obtain the transfer function H (z) for the digital filter by applying the bilinear transformation (11.65) to Ha (s) The design parameter K can be fixed or chosen to map one analog frequency point (e.g., the passband or stopband cutoff) into a desired digital frequency point ω 4 The frequency response H (ω) of the resulting stable digital filter can be obtained from the transfer function... of the designed digital IIR filter equals the denominator degree.5 For the design of digital IIR filters with unequal numerator and denominator degree, analytic techniques are available only for special cases (see Section 11.4.2) For other cases, iterative numerical methods are required Highpass, bandpass, and band-reject filters can also be obtained from analog prototypes (or from the digital versions)... δs ) (11.61) These formulas assume that δs < δp If otherwise, then interchange δp and δs Equation (11.60) is the one used in the Matlab implementation (remezord() function) as part of the Matlab Signal Processing toolbox To use the PM algorithm for lowpass filter design, the user specifies N, ωp , ωs , δp /δs The PM algorithm can be modified so that the user specifies other parameter sets [38] For example,... the change of variables, s = K z−1 This mapping z+1 preserves the optimality of the four classical filter types Another method for obtaining IIR digital filters from analog prototypes is the impulse-invariant method [11] In this method, the impulse response of a digital filter is obtained by sampling the continuous-time/space impulse response of the analog prototype However, the impulse invariance method... z-plane) The bilinear transformation design procedure can be summarized as follows: 1 Transform the digital frequency domain specifications to the analog domain using Eq (11.67) The frequency domain specs are given typically in terms of magnitude response specs as shown in Fig 11.2 After the transformation, the digital magnitude response specs are converted into specs on the analog magnitude response 2 Design... important properties: 1 The left-half plane (LHP) of the s-plane maps into the inside of the unit circle in the z-plane As a result, a stable and causal analog filter will always result in a stable and causal digital filter 2 The  axis (imaginary axis) in the s-plane maps into the U.C in the z-plane (i.e, z = e ω ) This results in a direct relationship between the continuous-time frequency and the discrete-time . standard system for processing continuous-time (-space) signals using a digital filter. The analog input signal is first transformed into a digital signal through. References 11.1 Introduction Digital filters are widely used in processing digital signals of many diverse applications, including speech processing and data communications,