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the parameters of H(z) may be obtained directly from H(s)

缺點:It is nonlinear between discrete-time frequency and continuous-time frequency.

2.統計上常用 resolution降一半 main lobe 變寬(trade off)

Main lobe變寬(trade off) side lobe降一半

2.main lobe和HANN差不多但side lobe降了10dB

3.Hamming 常用在語音處理

- 1. Feb.2008 DISP Lab 1 FIR and IIR Filter Design Techniques FIR 與 IIR 濾波器設計技巧 Speaker: Wen-Fu Wang 王文阜 Advisor: Jian-Jiun Ding 丁建均 教授 E-mail: r96942061@ntu.edu.tw Graduate Institute of Communication Engineering National Taiwan University, Taipei, Taiwan, ROC
- 2. Feb.2008 DISP Lab 2 Outline Introduction IIR Filter Design by Impulse invariance method IIR Filter Design by Bilinear transformation method FIR Filter Design by Window function technique
- 3. Feb.2008 DISP Lab 3 Outline FIR Filter Design by Frequency sampling technique FIR Filter Design by MSE Conclusions References
- 4. Feb.2008 DISP Lab 4 Introduction Basic filter classification We put emphasis on the digital filter now, and will introduce to the design method of the FIR filter and IIR filter respectively. Filter Analog Filter Digital Filter IIR Filter FIR Filter
- 5. Feb.2008 DISP Lab 5 Introduction IIR is the infinite impulse response abbreviation. Digital filters by the accumulator, the multiplier, and it constitutes IIR filter the way, generally may divide into three kinds, respectively is Direct form, Cascade form, and Parallel form.
- 6. Feb.2008 DISP Lab 6 Introduction IIR filter design methods include the impulse invariance, bilinear transformation, and step invariance. We must emphasize at impulse invariance and bilinear transformation.
- 7. Feb.2008 DISP Lab 7 Introduction IIR filter design methods Continuous frequency band transformation Impulse Invariance method Bilinear transformation method Step invariance method IIR filter Normalized analog lowpass filter
- 8. Feb.2008 DISP Lab 8 Introduction The structures of IIR filter Direct form 1 Direct form2 b0 b1 b2 b2 b1 b0 -a1 -a2 -a1 -a2 x(n) x(n)Y(n) Y(n) 1 z− 1 z− 1 z− 1 z− 1 z− 1 z−
- 9. Feb.2008 DISP Lab 9 Introduction The structures of IIR filter Cascade form x(n) Y(n) b0 b1 b2 -a1 -a2 -c1 -c2 d1 d2 Parallel form Y(n)x(n) b1 b0 d1 d0 E -c1 -c2 -a1 -a2 1 z− 1 z− 1 z− 1 z− 1 z− 1 z− 1 z− 1 z−
- 10. Feb.2008 DISP Lab 10 Introduction FIR is the finite impulse response abbreviation, because its design construction has not returned to the part which gives. Its construction generally uses Direct form and Cascade form.
- 11. Feb.2008 DISP Lab 11 Introduction FIR filter design methods include the window function, frequency sampling, minimize the maximal error, and MSE. We must emphasize at window function, frequency sampling, and MSE. Window function technique Frequency sampling technique Minimize the maximal error FIR filter Mean square error
- 12. Feb.2008 DISP Lab 12 Introduction The structures of FIR filter x(n) x(n) b1 b2 b3 b4 b0 Y(n) Y(n) Direct form Cascade form b1 b2 d1 d2 b0 1 z− 1 z− 1 z− 1 z− 1 z− 1 z− 1 z− 1 z−
- 13. Feb.2008 DISP Lab 13 IIR Filter Design by Impulse invariance method The most straightforward of these is the impulse invariance transformation Let be the impulse response corresponding to , and define the continuous to discrete time transformation by setting We sample the continuous time impulse response to produce the discrete time filter ( )ch t ( )cH s ( ) ( )ch n h nT=
- 14. Feb.2008 DISP Lab 14 IIR Filter Design by Impulse invariance method The frequency response is the Fourier transform of the continuous time function and hence '( )H ω * ( ) ( ) ( )c c n h t h nT t nTδ ∞ =−∞ = −∑ 1 2 '( ) ( )c k H H j k T T π ω ω ∞ =−∞ = − ∑
- 15. Feb.2008 DISP Lab 15 IIR Filter Design by Impulse invariance method The system function is It is the many-to-one transformation from the s plane to the z plane. 1 2 ( ) | )sT cz e k H z H s jk T T π∞ = =−∞ = − ∑
- 16. Feb.2008 DISP Lab 16 IIR Filter Design by Impulse invariance method The impulse invariance transformation does map the -axis and the left-half s plane into the unit circle and its interior, respectively jω Re(Z) Im(Z) 1 S domain Z domain sT e jω σ
- 17. Feb.2008 DISP Lab 17 IIR Filter Design by Impulse invariance method is an aliased version of The stop-band characteristics are maintained adequately in the discrete time frequency response only if the aliased tails of are sufficiently small. '( )H ω ( )cH jω 0 ω '( )H ω /Tπ 2 /Tπ ( )cH jω
- 18. Feb.2008 DISP Lab 18 IIR Filter Design by Impulse invariance method The Butterworth and Chebyshev-I lowpass designs are more appropriate for impulse invariant transformation than are the Chebyshev-II and elliptic designs. This transformation cannot be applied directly to highpass and bandstop designs.
- 19. Feb.2008 DISP Lab 19 IIR Filter Design by Impulse invariance method is expanded a partial fraction expansion to produce We have assumed that there are no multiple poles And thus ( )cH s 1 ( ) N k c k k A H s s s= = − ∑ 1 ( ) ( )k N s t c k k h t A e u t = = ∑ 1 ( ) ( )k N s nT k k h n A e u n = = ∑ 1 1 ( ) 1 k N k s T k A H z e z− = = − ∑
- 20. Feb.2008 DISP Lab 20 IIR Filter Design by Impulse invariance method Example: Expanding in a partial fraction expansion, it produce The impulse invariant transformation yields a discrete time design with the system function 2 2 ( ) ( ) c s a H s s a b + = + + 1/ 2 1/ 2 ( )cH s s a jb s a jb = + + + + − ( ) 1 ( ) 1 1/ 2 1/ 2 ( ) 1 1a jb T a jb T H z e z e z− + − − − − = + − −
- 21. Feb.2008 DISP Lab 21 IIR Filter Design by Bilinear transformation method The most generally useful is the bilinear transformation. To avoid aliasing of the frequency response as encountered with the impulse invariance transformation. We need a one-to-one mapping from the s plane to the z plane. The problem with the transformation is many-to-one.sT z e=
- 22. Feb.2008 DISP Lab 22 IIR Filter Design by Bilinear transformation method We could first use a one-to-one transformation from to , which compresses the entire s plane into the strip Then could be transformed to z by with no effect from aliasing. s 's Im( ')s T T π π − ≤ ≤ 's 's T z e= σ jω 'σ jω /Tπ− /Tπ s domain s’ domain
- 23. Feb.2008 DISP Lab 23 IIR Filter Design by Bilinear transformation method The transformation from to is given by The characteristic of this transformation is seen most readily from its effect on the axis. Substituting and , we obtain s 's 12 ' tanh ( ) 2 sT s T − = jω s jω= ' 's jω= 12 ' tan ( ) 2 T T ω ω − =
- 24. Feb.2008 DISP Lab 24 IIR Filter Design by Bilinear transformation method The axis is compressed into the interval for in a one-to- one method The relationship between and is nonlinear, but it is approximately linear at small . ( , ) T T π π − 'ω ω ω 'ω 'ω ω≈ - ω 'ω /Tπ /Tπ−
- 25. Feb.2008 DISP Lab 25 IIR Filter Design by Bilinear transformation method The desired transformation to is now obtained by inverting to produce And setting , which yields 12 ' tanh ( ) 2 sT s T − = 2 ' tanh( ) 2 s T s T = s z 1 ' ( )lns z T = 2 ln tanh( ) 2 z s T = 1 1 2 1 ( ) 1 z T z − − − = + Re(Z) Im(Z) 1 S domain Z domain 1 2 1 2 T s z T s + = − jω σ
- 26. Feb.2008 DISP Lab 26 IIR Filter Design by Bilinear transformation method The discrete-time filter design is obtained from the continuous-time design by means of the bilinear transformation Unlike the impulse invariant transformation, the bilinear transformation is one-to-one, and invertible. 1 1 (2/ )(1 )/(1 ) ( ) ( ) |c s T z z H z H s − − = − + =
- 27. Feb.2008 DISP Lab 27 FIR Filter Design by Window function technique Simplest FIR the filter design is window function technique A supposition ideal frequency response may express where ( ) [ ]j j n d d n H e h n eω ω ∞ − =−∞ = ∑ 1 [ ] ( ) 2 j j n d dh n H e e d π ω ω π ω π − = ∫
- 28. Feb.2008 DISP Lab 28 FIR Filter Design by Window function technique To get this kind of systematic causal FIR to be approximate, the most direct method intercepts its ideal impulse response! [ ] [ ] [ ]dh n w n h n= g ( ) ( ) ( )dH W Hω ω ω= ∗
- 29. Feb.2008 DISP Lab 29 FIR Filter Design by Window function technique Truncation of the Fourier series produces the familiar Gibbs phenomenon It will be manifested in , especially if is discontinuous. ( )H ω ( )dH ω
- 30. Feb.2008 DISP Lab 30 FIR Filter Design by Window function technique 1.Rectangular window 2.Triangular window (Bartett window) 1, 0 [ ] 0, n M w n otherwise ≤ ≤ = 2 , 0 2 2[ ] 2 , 2 0, n Mn M n Mw n n M M otherwise ≤ ≤ = − < ≤
- 31. Feb.2008 DISP Lab 31 FIR Filter Design by Window function technique 1.Rectangular window 2.Triangular window (Bartett window) 0 10 20 30 40 50 60 0 0.5 1 sequence (n) T(n) Rectangular window 0 10 20 30 40 50 60 0 0.5 1 sequence (n) T(n) Bartlett window 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -100 -50 0 50 100 pi units FrequencyresponseT(jw)(dB) Rectangular window 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -100 -50 0 50 100 pi units FrequencyresponseT(jw)(dB) Bartlett window
- 32. Feb.2008 DISP Lab 32 FIR Filter Design by Window function technique 3.HANN window 4.Hamming window 1 2 1 cos , 0 [ ] 2 0, n n M w n M otherwise π − ≤ ≤ = 2 0.54 0.46cos , 0 [ ] 0, n n M w n M otherwise π − ≤ ≤ =
- 33. Feb.2008 DISP Lab 33 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -100 -50 0 50 100 pi units FrequencyresponseT(jw)(dB) Hanning window 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -100 -50 0 50 100 pi units FrequencyresponseT(jw)(dB) Hamming window 0 10 20 30 40 50 60 0 0.5 1 sequence (n) T(n) Hanning window 0 10 20 30 40 50 60 0 0.5 1 sequence (n) T(n) Hamming window FIR Filter Design by Window function technique 3.HANN window 4.Hamming window
- 34. Feb.2008 DISP Lab 34 FIR Filter Design by Window function technique 5.Kaiser’s window 6.Blackman window 2 0 0 2 [ 1 (1 ) ] [ ] , 0,1,..., [ ] n I Mw n n M I β β − − = = 2 4 0.42 0.5cos 0.08cos , 0 [ ] 0, n n n M w n M M otherwise π π − + ≤ ≤ =
- 35. Feb.2008 DISP Lab 35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -100 -50 0 50 100 pi units FrequencyresponseT(jw)(dB) Blackman window 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -150 -100 -50 0 50 100 pi units FrequencyresponseT(jw)(dB) Kaiser window 5.Kaiser’s window 6.Blackman window 0 10 20 30 40 50 60 0 0.5 1 sequence (n) T(n) Blackman window 0 10 20 30 40 50 60 0 0.5 1 sequence (n) T(n) Kaiser window FIR Filter Design by Window function technique
- 36. Feb.2008 DISP Lab 36 FIR Filter Design by Window function technique ( / )s Mω Window Peak sidelobe level (dB) Transition bandwidth Max. stopband ripple(dB) Rectangular -13 0.9 -21 Hann -31 3.1 -44 Hamming -41 3.3 -53 Blackman -57 5.5 -74
- 37. Feb.2008 DISP Lab 37 FIR Filter Design by Frequency sampling technique For arbitrary, non-classical specifications of , the calculation of ,n=0,1,…,M, via an appropriate approximation can be a substantial computation task. It may be preferable to employ a design technique that utilizes specified values of directly, without the necessity of determining ' ( )dH ω ( )dh n ' ( )dH ω ( )dh n
- 38. Feb.2008 DISP Lab 38 FIR Filter Design by Frequency sampling technique We wish to derive a linear phase IIR filter with real nonzero . The impulse response must be symmetric where are real and denotes the integer part ( )h n [ /2] 0 1 2 ( 1/ 2) ( ) 2 cos( ) 1 M k k k n h n A A M π = + = + + ∑ kA [ / 2]M 0,1,...,n M=
- 39. Feb.2008 DISP Lab 39 FIR Filter Design by Frequency sampling technique It can be rewritten as where and Therefore, it may write where 1 / 2 / 0 /2 ( ) N j k N j kn N k k k N h n A e eπ π − = ≠ = ∑ 0,1,..., 1n N= − 1N M= + k N kA A −= / 2 / ( ) j k N j kn N k kh n A e eπ π = 1 0 /2 ( ) ( ) N k k k N h n h n − = ≠ = ∑ 0,1,..., 1n N= −
- 40. Feb.2008 DISP Lab 40 FIR Filter Design by Frequency sampling technique with corresponding transform where Hence which has a linear phase 1 0 /2 ( ) ( ) N k k k N H z H z − = ≠ = ∑ / 2 / 1 (1 ) ( ) 1 j k N N k k j k N A e z H z e z π π − − − = − ' ( 1)/2 sin / 2 ( ) sin[( / / 2)] j T N k k TN H A e k N T ω ω ω π ω − − = −
- 41. Feb.2008 DISP Lab 41 FIR Filter Design by Frequency sampling technique The magnitude response which has a maximum value at where ' sin / 2 ( ) sin[( / / 2)] k k TN H A k N T ω ω π ω = − kN A /k sk Nω ω= 2 /s Tω π=
- 42. Feb.2008 DISP Lab 42 FIR Filter Design by Frequency sampling technique The only nonzero contribution to at is from , and hence that Therefore, by specifying the DFT samples of the desired magnitude response at the frequencies , and setting '( )H ω kω ω= ' ( )kH ω '( )k kH N Aω = ' ( )dH ω kω ' ( ) /k d kA H Nω= ±
- 43. Feb.2008 DISP Lab 43 FIR Filter Design by Frequency sampling technique We produce a filter design from equation (5.1) for which The desired and actual magnitude responses are equal at the N frequencies ' '( ) ( )k d kH Hω ω= kω
- 44. Feb.2008 DISP Lab 44 FIR Filter Design by Frequency sampling technique In between these frequencies, is interpolated as the sum of the responses , and its magnitude does not, equal that of '( )H ω ' ( )kH ω ' ( )dH ω
- 45. Feb.2008 DISP Lab 45 FIR Filter Design by Frequency sampling technique Example: For an ideal lowpass filter from , we would choose The frequency samples are indeed equal to the desired ' 1, 0,1,..., ( ) 0, 1,...,[ / 2] d k k P H k P M ω = = = + ' ( ) /k d kA H Nω= ± ( 1) / ( 1), 0,1,..., 0, 1,...,[ / 2] k k M k P A k P M − + = = = + ' ( )kH ω ' ( )d kH ω
- 46. Feb.2008 DISP Lab 46 FIR Filter Design by Frequency sampling technique The response is very similar to the result form using the rectangular window, and the stopband is similarly disappointing. We can try to search for the optimum value of the transition sample would quickly lead us to a value of approximately , k p≠0.38( 1) /( 1)p pA M= − +
- 47. Feb.2008 DISP Lab 47 FIR Filter Design by MSE : The spectrum of the filter we obtain : The spectrum of the desired filter MSE= ( )H f ( )dH f ( ) ( )∫− − − 2/ 2/ 21 s s f f ds dffHfHf 0 0.1 0.2 0.3 0.4 0.5 -0.5 0 0.5 1 1.5
- 48. Feb.2008 DISP Lab 48 FIR Filter Design by MSE Larger MSE, but smaller maximal error Smaller MSE, but larger maximal error 0 0.1 0.2 0.3 0.4 -0.5 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 -0.5 0 0.5 H(F) H(F) - H (F)d 0 0.1 0.2 0.3 0.4 -0.5 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 -0.5 0 0.5 H(F) H(F) - H (F)d
- 49. Feb.2008 DISP Lab 49 FIR Filter Design by MSE 1. ( ) ( ) ( ) ( )∫∫ −− − −=−= 2/1 2/1 22/ 2/ 21 dFFHFRdffHfRfMSE d f f ds s s ( ) ( ) dFFHFnns d k n ∫ ∑− = −= 2/1 2/1 2 0 || 2cos][ π ( ) ( ) ( ) ( ) dFFHFnnsFHFnns d k n d k n ∫ ∑∑− == − −= 2/1 2/1 00 2cos][2cos][ ππ ( ) ( ) 1/2 1/2 0 0 [ ]cos 2 [ ]cos 2 k k n s n n F s F dF τ π τ π τ − = = = ∑ ∑∫ ( ) ( ) ( ) 1/2 1/2 2 1/2 1/2 0 2 [ ]cos 2 k d d n s n n F H F dF H F dFπ − − = − +∑∫ ∫
- 50. Feb.2008 DISP Lab 50 FIR Filter Design by MSE 2. when n ≠ τ, when n = τ, n ≠ 0, when n = τ, n = 0, 3. The formula can be repressed as: ( ) ( ) 02cos2cos 2/1 2/1 =∫− dFFFn τππ ( ) ( ) 2/12cos2cos 2/1 2/1 =∫− dFFFn τππ ( ) ( ) 12cos2cos 2/1 2/1 =∫− dFFFn τππ ( ) ( ) ( )dFFHdFFHFnnsnssMSE dd k n k n ∫∫ ∑∑ −− == +−+= 2/1 2/1 22/1 2/1 01 22 2cos][22/][]0[ π
- 51. Feb.2008 DISP Lab 51 FIR Filter Design by MSE 4. Doing the partial differentiation: 5. Minimize MSE: for all n’s ( )∫− −= ∂ ∂ 2/1 2/1 2]0[2 ]0[ dFFHs s MSE d ( ) ( )∫− −= ∂ ∂ 2/1 2/1 2cos2][ ][ dFFHFnns ns MSE dπ 0 ][ = ∂ ∂ ns MSE ( )∫− = 2/1 2/1 ]0[ dFFHs d ( ) ( )∫− = 2/1 2/1 2cos2][ dFFHFnns dπ [ ] [0] [ ] [ ]/ 2 for n=1,2,...,k [ ] [ ]/ 2 for n=1,2,...,k [ ] 0 for n<0 and n N h k s h k n s n h k n s n h n = + = − = = ≥
- 52. Feb.2008 DISP Lab 52 Conclusions FIR advantage: 1. Finite impulse response 2. It is easy to optimalize 3. Linear phase 4. Stable FIR disadvantage: 1. It is hard to implementation than IIR
- 53. Feb.2008 DISP Lab 53 Conclusions IIR advantage: 1. It is easy to design 2. It is easy to implementation IIR disadvantage: 1. Infinite impulse response 2. It is hard to optimalize than FIR 3. Non-stable
- 54. Feb.2008 DISP Lab 54 References [1]B. Jackson, Digital Filters and Signal Processing, Kluwer Academic Publishers 1986 [2]Dr. DePiero, Filter Design by Frequency Sampling, CalPoly State University [3]W.James MacLean, FIR Filter Design Using Frequency Sampling [4] 蒙以正 , 數位信號處理 , 旗標 2005 [5]Maurice G.Bellanger, Adaptive Digital Filters second edition, Marcel dekker 2001
- 55. Feb.2008 DISP Lab 55 References [6] Lawrence R. Rabiner, Linear Program Design of Finite Impulse Response Digital Filters, IEEE 1972 [7] Terrence J mc Creary, On Frequency Sampling Digital Filters, IEEE 1972

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