数字信号处理英文版课件：Chap 9 IIR Digital Filter Design

1、DSP group 2007 chap9-ed1 1 Chapter 9 IIR Digital Filter DesignSpecifications for DT filtersBilinear transform method for lowpass digital filterOther types of IIR Filters designComputer-aided Design of IIR digital filtersDSP group 2007 chap9-ed1 2 IntroductionAn important research on digital filter:

2、Determine realizable transfer function approximating the given frequency specifications; The process of deriving the transfer function is called digital filter design;Another important research on digital filter: Realization of digital filter (IIR/FIR)DSP group 2007 chap9-ed1 3 9.1 Preliminary Consi

3、derationsTwo major issues on digital filter: reasonable filter frequency response specifications from the requirements; FIR or IIR digital filter is needed;3 steps on digital filter design: step1: specification of desired system Hd(e j); step 2: approximation H(e j) Hd(e j ) () stage 3: realization

4、(digital computation)DSP group 2007 chap9-ed1 4 9.1.1 Digital filter specifications0PassbandStopband Specifications: Transition Tolerance (peak ripple value)Edge frequencySpecifications in terms of loss function: DSP group 2007 chap9-ed1 5 9.1.1 Digital filter specifications peak passband ripple min

5、imum stopband attenuationExample 9.1: p = 0.1dB, s= 35 dB.Solution: DSP group 2007 chap9-ed1 6 9.1.1 Digital filter specificationsSpecifications in normalized form: 0PassbandStopband Transition maximum passband deviation maximum stopband magnitude edge frequency usually specified in HzDSP group 2007

6、 chap9-ed1 7 9.1.1 Digital filter specificationsExample9.2 Highpass: Fp = 7kHz, Fs= 3kHz. FT = 25kHzSolution: 9.1.2 Selection of the filter type IIR digital filterP133 3.65DSP group 2007 chap9-ed1 8 9.1.2 Selection of the filter typeCausal;Stable (an important factor needs to be considered);Lowest o

7、rder FIR digital filterDSP group 2007 chap9-ed1 9 9.1.2 Selection of the filter type Advantages and properties of FIR digital filterLinear phase;Always Stable;Order NFIR NIIR; multiplications per output sample: FIR filter: NFIR+1 or half, IIR filter: 2NIIR+1; The IIR filter is usually computationall

8、y more efficient.DSP group 2007 chap9-ed1 10 9.1.3 Basic approach to IIR filter design Transform the analog filter into the desired digital filter. Reasons for IIR design method Analog filter design techniques are highly advanced; They usually yield closed-form solutionsExtensive tables are availabl

9、e for analog filter design;Many applications require the digital simulation of analog filters. DSP group 2007 chap9-ed1 11 9.1.3 Basic approach to IIR filter design Transform rules: the imaginary (j) axis in the s-plane be mapped onto the unit circle of the z-plane; A stable analog transfer function

10、 be transformed into a stable digital transfer function. DSP group 2007 chap9-ed1 12 9.1.3 Basic approach to IIR filter design Steps for this transforming: Step 1: Specifications Conversion;Step 2: analog filter design; prototype analog filterStep 3: system function transforming (From s-plane to z-p

11、lane) .DSP group 2007 chap9-ed1 13 9.1.5 Scaling the Digital Transfer Function For implementation: Ht(z)=KH(z), so that the maximum magnitude in the passband is unity. K=1/Hmax.DSP group 2007 chap9-ed1 14 9.2 Bilinear Transformation Method of IIR Filter Design9.2.1 Bilinear TransformationOne-to-one

12、mapping, it maps a single point in the s-plane to a unique point in the z-plane, and vice versa.DSP group 2007 chap9-ed1 15 9.2.1 Bilinear Transformation Mapping relationsDSP group 2007 chap9-ed1 16 9.2.1 Bilinear Transformation Mapping relationsjs - planez =e jRe zjIm zz - planeDSP group 2007 chap9

13、-ed1 17 9.2.1 Bilinear Transformation Mapping relationsThe exact relation between the imaginary axis (j) in the s-plane and the unit circle (e j) in the z-plane is of interest:DSP group 2007 chap9-ed1 18 Mapping is highly nonlinear;the negative imaginary axis in the s-plane from = to =0 is mapped in

14、to the lower half of the unit circle from = (z= 1) to =0 (z=1);the positive imaginary axis in the s-plane from =0 to =+ is mapped into the upper half of the unit circle from =0 (z=1) to = (z= 1); 9.2.1 Bilinear Transformation Mapping relations19Frequency warping is introduced due to the distortion i

15、n frequency;Steps for IIR filter design using bilinear transform;Prewarp the critical band edge frequencies (p and s) to find their equivalents (p and s);Using eq.(9.20) 9.2.1 Bilinear Transformation Mapping relationsDSP group 2007 chap9-ed1 20 9.2.1 Bilinear Transformation steps for designDesign th

16、e analog prototype Ha(s) using the prewarped critical frequencies; Using the bilinear transformation to obtain the desired digital filter transfer function H(z) using Eq.(9.15) 9.2.2 Design of Low-Order Digital FilterFirst-order Butterworth lowpass and highpass digital filtersDSP group 2007 chap9-ed

17、1 21 9.2.2 Design of Low-Order Digital FilterExample: Lowpass filterRearranging terms we getDSP group 2007 chap9-ed1 22 9.2.2 Design of Low-Order Digital FilterExample: Highpass filterDSP group 2007 chap9-ed1 23 9.3 Design of Lowpass IIR Digital FiltersExample: Design a lowpass IIR digital filter wi

18、th a maximally flat magnitude characteristic. The specifications are:p =0.25, the passband ripple 0.5dB, s =0.55, the stopband attenuation 15dB.Solution:24Step 1: specifications transform from DT to CT (pre-warping) using Eq.(9.18): =(2/T) tan( /2) p=tan(p/2) = tan(0.25/2) = 0.4142136; s= tan(s/2) =

19、 tan(0.55/2) = 1.1708496Step 2: CT filter design:1/A0.414| H(j )| 1.171Assume T=2 P43425Step 3: bilinear transformation:DSP group 2007 chap9-ed1 26Magnitude and gain response of H(ej) 9.3 Design of Lowpass IIR Digital Filters27s-plane: j z-plane: e j Bilinear transformation Spec. from DT to CT prewa

20、rping CT filter design (Butterworth) NcajH22)(11|)(|WW+=W, ()()-=-W=WWW+=WWW+=W=NkkaNkckcNcssaNcppassAsHjsNjHjH12/2222)()()1( :poles)/(1/1|)(|)/(1/1|)(| Transformationfrom CT to DT )112(|)()(1111211-+-=+-=-zzTHsHzHazzTsa 9.3 Design of Lowpass IIR Digital FiltersDSP group 2007 chap9-ed1 28 9.4 Design

21、 of highpass IIR Digital Filters Steps for approach A:Step 1: Prewarp the specifications:Step 2: Obtain the specification of lowpass prototype:Step 3: Design the filter HLP(s); DSP group 2007 chap9-ed1 29 9.4 Design of highpass IIR Digital FiltersStep 4: Frequency transformation again;Step 5: Biline

22、ar transformation.DSP group 2007 chap9-ed1 30 9.4 Design of highpass IIR Digital Filters Steps for approach B The first three steps are the same with in approach A .Step 4: bilinear transformation;Step 5:Frequency transformation.DSP group 2007 chap9-ed1 31 9.4 Design of highpass IIR Digital FiltersE

23、xample 9.3 Design of highpass IIR digital filter: Specifications: Fs=500Hz , Fp=700Hz; p=1dB, s=32dB sampling frequency: FT =2kHz.Solution:Step 0: Obtain the digital angular frequency:DSP group 2007 chap9-ed1 32 9.4 Design of highpass IIR Digital FiltersStep 1: Prewarp the specifications:Step 2: Obt

24、ain the specification of lowpass prototype:Let the passband edge frequency of the prototype be p=1; then the stopband edge frequencyAssume T=2DSP group 2007 chap9-ed1 33 9.4 Design of highpass IIR Digital Filtersthe specifications of prototype analog lowpass filter HLP(s) :Step 3: Design the filter

25、HLP(s) (Butterworth); Order: N=7; Cutoff frequency: c=0.3691;(% buttord(wp,ws,rp,rs)DSP group 2007 chap9-ed1 34 9.4 Design of highpass IIR Digital FiltersTransfer function of HLP(s) :Step 4: Frequency transformation again;Step 5: Bilinear transformation.DSP group 2007 chap9-ed1 35 9.4 Design of high

26、pass IIR Digital FiltersmagnitudephaseDSP group 2007 chap9-ed1 36 9.5 Spectral Transformations of IIR Digital FiltersLowpass prototype digital filter HLP(z),Frequency transfrom Desired other types of digital filter HD(z),z=F(z)I.e., HD(z) = HLP ,F(z)See Table 9.1 in page 382DSP group 2007 chap9-ed1

27、37Example 9.4: Consider the lowpass digital filterRedesign the lowpass filter to move the passband edge frequency from 0.25 to 0.35. 9.5.1 Lowpass-to-Lowpass TransformationSolution :According to the requirement:38 9.5.1 Lowpass-to-Lowpass TransformationDSP group 2007 chap9-ed1 39Example 9.5: Consider the lowpass digital filterRedesign a highpass filter with: 9.5.2 Lowpas