数字信号处理教学课件：Chapter9 IIR Digital Filter Design

1、Chapter 9 IIR Digital Filter DesignPreliminary ConsiderationIIR Digital Filter Design MethodsMATLAB Functions in IIR Filter DesignFilter Design ObjectiveObjective - Determination of a realizable transfer function G(z) approximating a given frequency response specification is an important step in the

2、 development of a digital filter.If an IIR filter is desired, G(z) should be a stable real rational function.Digital filter design is the process of deriving the transfer function G(z).Preliminary ConsiderationsDigital Filter SpecificationsSelection of the Filter TypeBasic Approach to IIR Digital Fi

3、lter DesignIIR Digital Filter Order EstimationScaling the Digital Transfer Function9.1.1 Digital Filter SpecificationsFor example, the magnitude response |G(ej)| of a digital lowpass filter may be given as indicated below:Digital Filter SpecificationsAs indicated in the figure, in the passband, defi

4、ned by 0p, we require that |G(ej)|1 with an error p, i.e., 1- p |G(ej)| 1+ p, | | pIn the stopband, defined by s , we require that |G(ej)|0 with an error s i.e., |G(ej)| s, s |Digital Filter Specificationsp - passband edge frequency.s - stopband edge frequency.p - peak ripple value in the passband.

5、s - peak ripple value in the stopband.Since G(ej) is a periodic function of , and |G(ej)| of a real-coefficient digital filter is an even function of .As a result, filter specifications are given only for the frequency range 0 |.Digital Filter SpecificationsSpecifications are often given in terms of

6、 loss function A()=-20log10 |G(ej)| in dBPeak passband ripple p= -20log10 (1- p ) dBMinimum stopband attenuation s= -20log10 (s ) dBDigital Filter SpecificationsMagnitude specifications may alternately be given in a normalized form as indicated below:Digital Filter SpecificationsHere, the maximum va

7、lue of the magnitude in the passband is assumed to be unity.1/(1+2) - Maximum passband deviation, given by the minimum value of the magnitude in the passband. 1/A - Maximum stopband magnitude.Digital Filter SpecificationsFor the normalized specification, maximum value of the gain function or the min

8、imum value of the loss function is 0 dB.Maximum passband attenuation:dB For p1, it can be shown that: dBDigital Filter SpecificationsIn practice, passband edge frequency Fpand stopband edge frequency Fs are specified in Hz.For digital filter design, normalized bandedge frequencies need to be compute

9、d from specifications in Hz using:Digital Filter SpecificationsExample: Let Fp=7 kHz, Fs= 3 kHz, and FT=25 kHzThen9.1.2 Selection of the Filter TypeThe objective of digital filter design is to develop a casual,stable transfer function G(z) which meets the FR specifications.Whether IIR or FIR should

10、be selected?IIR: lower order to reduce computational complexity; must be stable.FIR: higher order; linear phase.9.1.3 Basic Approach to IIR Digital Filter DesignMost common approach to IIR filter design-The most common practice is to transform Ha(s) into the desired digital transfer function G(z) .

11、The implement steps are:Transform it into the desied digital filter transfer function G(z).Convert the digital filter specifications into analog lowpass prototype one.Determine the analog lowpass filter Ha(s) meeting these specifications.Basic Approach to IIR Digital Filter DesignWhy this approach h

12、as been widely used:(1) Analog approximation techniques are highly advanced.(2) They usually yield closed-form solutions.(3) Extensive tables are available for analog filter design.(4) Many applications require digital simulation of analog systems.Basic Approach to IIR Digital Filter DesignAn analog

13、 transfer function to be denoted as Ha(s)= Pa(s) / Da(s) where the subscript “a” specifically indicates the analog domainA digital transfer function derived from Ha(s) shall be denoted as G(z)=P(z)/D(z)Basic Approach to IIR Digital Filter DesignApplying a mapping from s-plane to the z-plane so that

14、the essential properties of the analog frequency response are preserved.The mapping implies:(1)The imaginary axis in the s-plane be mapped onto the unit circle of the z-plane.(2) A stable analog transfer function be transferred into a stable digital transfer function.4.4 Analog Filter DesignBasic ja

15、rgons:Passband , Stopband , Passband Edge Frequency,Stopband Edge Frequency, Ripples, Peak Passband Ripples,Minimum Stopband AttenuationTransition Ration / Selectivity ParameterDiscrimination Parameter Analog Filter DesignButterworth approximationChebyshev approximation Type1 Type2Elliptic approxima

16、tionLinear-phase approximationAnalog Filter DesignButterworth ApproximationThe magnitude-square response of an N-th order analog lowpass Butterworth filter is given byThe Butterworth lowpass filter thus is said to have a maximally-flat magnitude at = 0.Butterworth ApproximationGain in dB is G()=10lo

17、g10|Ha(j)|2 As G(0)=0 and G(c)=10log10(0.5)=-3.0103-3 dB c is called 3-dB cutoff frequencyTypical magnitude responses with c =1Butterworth ApproximationTwo parameters completely characterizing a Butterworth lowpass filter are c and N. These are determined from the specified bandedges p and s , and m

18、inimum passband magnitude 1/(1 + 2) , and maximum stopband ripple 1/A.Butterworth Approximationc and N are thus determined from:Solving the above, we get:Butterworth ApproximationTransfer function of an analog Butterworth lowpass filter is given by:Where:Denominator DN(s) is known as the Butterworth

19、 polynomial of order N.Butterworth ApproximationExample: Determine the lowest order of a Butterworth lowpass filter with a 1-dB cutoff frequency at 1 kHz and minimum attenuation of 40 dB at 5 kHz.Now 10log101/(1+2)=-1Which yields 2=0.25895And 10log10(1/A2)=-40Which yields A2=10000Butterworth Approxi

20、mationTherefore 1/k1=(A2-1)/=196.51334And 1/k=s/p=5Hence N=log10(1/k1)/log10(1/k)=3.2811Choose N=4Analog Filter DesignChebyshev ApproximationThe magnitude-square response of an N-th order analog lowpass Type 1 Chebyshev filter is given bywhere TN() is the Chebyshev polynomial of order N:Chebyshev Ap

21、proximationTypical magnitude response plots of the analog lowpass Type 1 Chebyshev filter are shown belowChebyshev ApproximationIf at = s the magnitude is equal to 1/A, then Order N is chosen as the nearest integer greater than or equal to the above value.Solving the above we get:Chebyshev Approxima

22、tionThe magnitude-square response of an N-th order analog lowpass Type 2 Chebyshev (also called inverse Chebyshev) filter is given by:where TN() is the Chebyshev polynomial of order N.Chebyshev ApproximationTypical magnitude response plots of the analog lowpass Type 2 Chebyshev filter are shown belo

23、wChebyshev ApproximationThe order N of the Type 2 Chebyshev filter is determined from given , s, and A usingExample - Determine the lowest order of a Chebyshev lowpass filter with a 1-dB cutoff frequency at 1 kHz and a minimum attenuation of 40 dB at 5 kHz.Analog Filter DesignElliptic ApproximationT

24、he square-magnitude response of an elliptic lowpass filter is given by:where RN() is a rational function of order N satisfying RN(1/)=1/ RN() , with the roots of its numerator lying in the interval 0 1 and the roots of its denominator lying in the interval 1 .Elliptic ApproximationFor given p, s, ,

25、and A, the filter order can be estimated usingwhereElliptic ApproximationExample - Determine the lowest order of a elliptic lowpass filter with a 1-dB cutoff frequency at 1 kHz and a minimum attenuation of 40 dB at 5 kHz Note: k = 0.2 and 1/k1=196.5134Substituting these values we get k=0.979796, 0=0

26、.00255135, =0.0025513525and hence N = 2.23308Choose N = 3 Elliptic ApproximationTypical magnitude response plots are shown below:Analog Lowpass Filter DesignExample: Design an elliptic lowpass filter of lowest order with a 1-dB cutoff frequency at 1 kHz and a minimum attenuation of 40 dB at 5 hHzCod

27、e fragments usedN, Wn=ellipord(Wp, Ws, Rp, Rs, s);b, a=ellip(N, Rp, Rs, Wn, s);Rp=1;Rs=40;Analog Lowpass Filter DesignGain plot9.1.4 Estimation of the Filter OrderIIR lowpass-filter:Based on use what type of analog filter Ha(s) to perform the conversion to get G(z) :Look at equations(4.35)(4.43)(4.5

28、4)(the order estimations of each analog filter)Then the order of G(z) can be determined automatically from the transformation being used to convert Ha(s) into G(z). 9.1.5 Scaling the Digital Transfer FunctionThe scaled transfer function Gt(z)=KG(z)Where: (1) K=1/Gmax (2) For a stable G(z) with real

29、coefficients, KG(z) is a bounded real(BR) function.Discuss the scaling factor K of several common digital filter:(1)LPF:if ,(2)HPF:if ,(3)BPF:if ，9.2 The Design of IIR FilterThe core idea of design this kind of filter is to convert an analog transfer function into a digital one G(z).There are two co

30、mmon methods:(1)The impulse invariance method(Problem9.6P455)(2)The bilinear transformation method(9.2P432)Supplement: The Impulse Invariance MethodTransformation theory:If a prototype causal analog transfer function Ha(s) is wanted, and its impulse response is ha(t), the impulse response of the dig

31、ital transfer function G(z) satisfies:)()(nTxnxa=)()(nTynya=)(nThnga=)(txa)(tya)(than=0,1,2,The Relationship Between ZT and LT LT-used in continuous signal analysis ZT- used in discrete signal analysis Given continuous signal xa(t) , sampled signal , their LT are:So we can see: when , ZT=LT.The rela

32、tionship between them is a kind of mapping from s-plane to z-plane.The ZT of the sampled sequenceFrom the relation,we get G(z) and Ha(s) meet:. Z-planeWjsjTjee=Unit cycleThe frequency response meets:The mapping relationship must be satisfied: or .Discussion:The mapping satisfies the essential proper

33、ties (j-axis mapped onto the unit circle,stable region in s-plane mapped into the stable region in z-plane).The exact relation of the normalized angular frequency w and the analog angular frequency is = T .when Nyquist theorem is satisfied,From above equation:The Impulse Invariance MethodWe can see

34、the FR is inverse proportion of T.So we should amend it use:We can get:The Design Steps(1)Expand Ha(s) into partial-fractional expression as:(2)Based on , we can get:Proof:Discussion:- Ha(s) and G(z)The single pole sk in s-plane mapping into the single pole z=eskT in z-plane.They both have the same

35、coefficients Ak . If the analog filter is stable, that is Resk0 , then after conversion,the poles meet:|eskT|1, so the digital filter is stable.The impulse invariance method only ensure relationship of the poles between s and z-plane, but for zeros, they dont exist. 9.2.1 The Bilinear Transformation

36、 MethodTransformation theory:For conquering the effect of multi-value mapping from s to z-plane, we want to find a new mapping:000-11s-plane-planez-planeThe Bilinear Transformation MethodThe bilinear transformation from s to z-plane:The new mapping is: or .So, The relation between G(z) and the paren

37、t analog transfer function Ha(s) is :Discussion:This mapping also satisfies the essential properties. The exact relation of the normalized angular frequencyand the analog angular frequencyis .This mapping introduces a distortion in frequency axis called frequency wrapping. :From following figure we

38、can see clearly the effect of warping caused by nonlinearity. Solutions: Prewarp the critical bandedge frequencies and to find their analog equivalents ( and ) using equation:The design steps:(1)Prewarp the bandedge frequencies using Eq(9.18)，and find the parent analog filter Ha(s) .(2) Get the digi

39、tal filter by the bilinear transformation.a.First decompose Ha(s) into: b. First decompose Ha(s) into:9.2.2 Design of Low-Order Digital FiltersExample Consider:Applying bilinear transformation to the above we get the transfer function of a first-order digital lowpass Butterworth filterDesign of Low-

40、Order Digital FiltersRearranging terms we get:whereDesign of Low-Order Digital Filtersfor which |Ha(j0)| = 0 |Ha(j0)| = |Ha(j)| = 10 is called the notch frequencyIf |Ha(j2)| = |Ha(j1)| =1/2 then B = 2 - 1 is the 3-dB notch bandwidth Example - Consider the second-order analog notch transfer functionD

41、esign of Low-Order Digital FiltersThenwhereDesign of Low-Order Digital FiltersExample - Design a 2nd-order digital notch filter operating at a sampling rate of 400 Hz with a notch frequency at 60 Hz, 3-dB notch bandwidth of 6 Hz Thus 0 = 2(60/400) = 0.3 Bw = 2(6/400) = 0.03 From the above values we

42、get = 0.90993 = 0.587785Design of Low-Order Digital FiltersThe gain and phase responses are shown belowThus9.3 Design of Lowpass IIR Digital FiltersConsider G(z) with a maximally flat magnitude, , with a passband ripple not exceeding 0.5dB; , with the minimum stopband attenuation at the stopband edg

43、e frequency is 15dB. Base on this requirement, if we get:Firstly, prewarpping:.Based on specifications of ,we get:And we know:So we can estimate the order of the filter:use 3-order filterThen we use a third-order analog Butterworth as the parent transfer function,with 3-dB . cutoff frequency is: (use Eq.4.34a or 4.34b)Use M-function buttap ,obtain the 3-order normalized lowpass Butterworth TF as:Then the denormalized lowpass Butterworth TF as:Applying the bilinear transformation to the above