Presentation on theme: "CHAPTER 7 filter design techniques"— Presentation transcript:
1CHAPTER 7 filter design techniques 7.0 introduction7.1design of discrete-time IIR filters from continuous-time filters7.1.1 filter design by impulse invariance7.1.2 filter design by bilinear transformnot low pass filter design and other method7.2 design of FIR filters by windowing. 7.3 summary
2ideal frequency selective filter 7.0 introductionideal frequency selective filterband-stop filter：notch filter；quality factor of band-pass filter=pass-band width/center frequency
3impulse response of ideal low-pass filter，noncausal，unrealizable Figure 7.1impulse response of ideal low-pass filter，noncausal，unrealizableSpecifications for filter design ：given in frequency domainphase：linear?magnitude：given by a tolerance schemeanalog or digital，absolute or relative
4Figure 7.2 absolute specification magnitude response of equivalent analog systemmonotonous descentpassband tolerancestopband tolerancepassband cutoff frequencystopband cutoff frequency3dB cutoff frequencyabsolute specification
5digital specification, finally: relative specifications：maximum magnitude in passband is normalized to 1, viz. 0dBmaximum attenuation in passbandminimum attenuation in stopband3dB cutoff frequency：magnitude response of equivalent analog system:digital specification, finally:
6Specifications for bandpass and bandstop filters： up and down passband cutoff frequency，up and down stopband cutoff frequencyDesign steps：（1）decide specifications according to application（2）decide type according to specification：generally , if the phase is required , choose FIR.（3）approach specifications using causal and stable discrete-time system：viz. design H(z0) or h[n]，nonuniform（4）choose a software or hardware realization structure, take effects of limited word length into considerationH(z) or h[n]
77.1 design of discrete-time IIR filters from continuous-time filters attention：original analog filter and equivalent analog filter is different, their frequency response is not always the same7.1.0 introduction of analogy filter7.1.1filter design by impulse invariance7.1.2 filter design by bilinear transform
87.1.0 introduction of analog filter comparison:1.wave2.the same order,increase performance3.increase design complexity(A)（C）small aliasing in the impulse invariance design technique
9magnitude frequency function: BW design formula: specificationsystem functionmagnitude frequency function:take the specifications into system function and get the results from equation group:ORconfirm the poles of system function:（in the left half plane）get system function：
13design a low pass cheby analogy filter: EXAMPLEdesign a low pass cheby analogy filter:[N,Wc]=cheb1ord(2000*pi,4000*pi,1,15, 's')[Bs,As]=cheby1(N,1,Wc, 's')[H,W]=freqs(Bs,As);plot(W/2/pi,20*(log10(abs(H))))axis([0,4000,-30,0])grid on
14design a low pass cheby analogy filter: EXAMPLEdesign a low pass cheby analogy filter:[N,Wc]=cheb2ord(2000*pi,4000*pi,1,15, 's')[Bs,As]=cheby2(N,15,Wc, 's')[H,W]=freqs(Bs,As);plot(W/2/pi,20*(log10(abs(H))))axis([0,4000,-30,0])grid on
15design a high pass analogy filter: EXAMPLEdesign a high pass analogy filter:[N,Wc]=buttord(4000*pi,2000*pi,1,15, 's')[Bs,As]=butter(N,Wc, 'high', 's')[H,W]=freqs(Bs,As);plot(W/2/pi,20*(log10(abs(H))))axis([0,4000,-16,0])grid on
16transformation from analog filter to digital filter: H(s)H(z)，viz. mapping from S plane to Z plane, must satisfy:the same frequency response（map imaginary axis to the unit circle）；causality and stability is preserved（map poles from left half plane to the inside circle）。imaginary axis the unit circlepoles in left half plane inside the unit circleS plane Z plane
177.1.1 filter design by impulse invariance according to ：conversion formula：
18Relationship between poles（causal and stable): relation between frequencies:S planeZ plane
19relation between frequency response: when aliasing is small, the frequency response is the same. strongpoint：linear frequency mapping; shortcoming：aliasing in frequency response。restriction in application：can not used in high-pass and bandstop filter
20Design steps： （3） about Td: independent of T; do not influence aliasing;arbitrary value，generally, take 1（attention（1）and（3）have the same value）.
21Td=1; Wp=wp/Td; Ws=ws/Td [N,Wc]=buttord(Wp,Ws, ap , as, 's') EXAMPLEwp=0.2*pi; ws=0.4*piap=1; as=12Td=1; Wp=wp/Td; Ws=ws/Td[N,Wc]=buttord(Wp,Ws, ap , as, 's')[Bs,As]=butter(N,Wc, 's')[Bz,Az]=impinvar(Bs,As,1/Td)[H,W]=freqs(Bs,As);plot(W/pi,20*(log10(abs(H))), 'r*')hold on[H,w]=freqz(Bz,Az);plot(w/pi,20*(log10(abs(H))))axis([0.2,0.4,-20,0])grid
237.1.2 filter design by bilinear transform design thought：design formula：
24relation between frequencies: left half s-plane inside the unit circle in the z-plane（causality and stability are preserved）；imaginary axis of the s-plane the unit circle in z-plane (the same frequency response), one-by-one mappingrelation between frequencies:
25relation between frequency response: strongpoint：no aliasing；shortcoming：nonlinear。restriction in application：can not used in differentiator.
26design steps: (1) (3) about Td: independent of T； (1) (3)about Td:independent of T；arbitrary value，generally, take 1（attention（1）and（3）have the same value）.
327.1.3 IIR summary 1.design steps 2. impulse invariance: frequency axis is linear many-to-one mapping , aliasing in frequency response, inapplicable to high-pass filter.bilinear transformation：frequency axis is one-to-one mapping with aberrance, no aliasing in frequency response, inapplicable to differentiator
337.1.4 frequency conversionFrequency conversion in analog domain：
37Gibbs phenomenon: frequency response oscillate at wc. transition width of filer’smainlobe width of window spectrum length and shape of window’spassband-stopband error of filter’ssidelobe amplitude of window spectrum window’s shapeThe smaller mainlobe width, the narrower transition width of filter’s;The smaller sidelobe amplitude of window spectrum, the smaller relative error of filter’s.M increases mainlobe width of window spectrum minishes filter oscillate more quickly, transition width minishes, relative amplitude of oscillation is preserved.In order to improve relative amplitude of oscillation, we need to change the shape of window’s.
41(a)-(e) attenuation of sidelobe increases, width of mainlobe increases.
42As B increases, attenuation of sidelobe increases, width of mainlobe increases. As N increases：attenuation of sidelobe is preserved,width of mainlobe decreases.Figure 7.24(b)(c)
43Transition width is a little less than mainlobe width Table 7.1
447.2.3 effect to frequency response attenuation of window spectrum’s sidelobe attenuation of filter’s stopbandwidth of window spectrum’s mainlobetransition width of filter’scheck the table for Blackman window;formulas of Kaiser window:A：attenuation of stopband, Δω：transition width
457.2.4 design step1. Write the ideal impulse response：2.Comfirm the shape of window’s based on attenuation of stopband：（1）check the table for Blackman window（2）calculation for Kaiser window
46(2) calculation for Kaiser window ： 3.Comfirm length of window’s based on transition width（M is even):(1) check the table for Blackman window: compute the length of window’s based on width of mainlobe:(2) calculation for Kaiser window ：4.Cut the ideal impulse response5.We can think that the 4 specifications istransformed into：The same with IIR, maybe the specification is equivalent analog one.
51（5）validate and correct in MATLAB h=fir1(18,0.662, 'high', kaiser (19,3.2953)); H=fft(h,512);k=0:511;subplot(1,2,1); plot(k/256,20*log10(abs(H))); grid on;subplot(2,2,2); plot(k/256,20*log10(abs(H)));axis([0.5,0.55,-50,-0]); grid onsubplot(2,2,4); plot(k/256,20*log10(abs(H)));axis([0.65,0.75,-5,0]); grid on
55FIR summaryshape attenuation of sidelobe attenuation of stopband（the first row in the table）（ the third row in the table）length width of mainlobe transition width(the second row) （the second row /2）
567.3 brief introduction of other filter design techniques using matlab IIR: minimum mean square error in frequency domain: invfreqz( )FIR: frequency sampling ：fir2( )FIR: minimum mean square error ：firls( )FIR: Parks-McCellan/Remez arithmetic：remez( )
577.4 summary FIR digital filter IIR digital filter finite-length IR infinite-length IRno non-zero poles, stable non-zero poles, stable?realization: convolution /recursion recursionrealize linear phase easily can not realize linear phase in deedstable maybe instable because of roundinghigher order lower order because of recursionFFT no fast arithmetic
58summary 7.1 design of IIR（continuous-time filter） 7.1.1 impulse invariance7.1.2 bilinear transform7.2 design of FIR（windowing）7.2.1 design ideas7.2.2 properties of commonly used windows7.2.3 effect to frequency response7.2.4 design step7.3 comparison between IIR and FIR
59requirements：understand the principles of impulse invariance and bilinear transform and their mapping characteristic, implication;understand design ideas of windowing;design various filters using MATLAB.difficulties：relationship among prototype analog filter, digital filter and equivalent analog filter;effects of shape and length of window’s to system characteristic；why impulse invariance and bilinear transform can not be used in high-pass filter design。
60exercise and experiment (a)(b) 7.23the second experiment5052565759