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by: Miss Flossie Collier

DigitalSignalProcess ECE8231

Miss Flossie Collier
GPA 3.96


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This 36 page Class Notes was uploaded by Miss Flossie Collier on Wednesday October 28, 2015. The Class Notes belongs to ECE8231 at Villanova University taught by Staff in Fall. Since its upload, it has received 21 views. For similar materials see /class/230590/ece8231-villanova-university in ELECTRICAL AND COMPUTER ENGINEERING at Villanova University.



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Date Created: 10/28/15
Department ul39 Iflccl ricul 21ml mnpulcr Engineering ECE823 1 Digital Signal Processing httpwww ece villanova eduzhangECE823 1 Yimin Zhang Department of Electrical amp Computer Engineering Villanova University Chapter 7 Filter Design Techniques 23 Design ul FIR Fillers h 7inl m inf l We have discussed techniques for the design of discretetime IIR lters based on the transformations of continuoustime IIR systems In contrast FIR design lters are almost entirely restricted to discretetime implementations The design techniques for lters are based on directly approximating the desired 39equency response of the discretetime system Most techniques for approximating the magnitude response of an FIR system assume a linear phase constraint thereby avoiding the problem of spectrum factorization that complicates the direct design of IR lters The simplest method for FIR design is called the Window method Design ul FIR Fillers h Hmlrm ing 3 CI We consider an ideal desired frequency response that can be represented as m Hdequot T Zhdnle m nz w In turn the impulse response sequence can be expressed as 1 a am hdn Hdej e dw CI Many idealized systems are de ned by piecewiseconstant or piecewisefunctional frequency responses with discontinuities at the boundaries between bands As a result these systems have impulse responses that are noncausal and in nitely long 2 The most straightforward approach to obtaining a causal FIR approximation to such systems is to truncate the ideal response lcsi1nnlquotlill Fillers h 7iml4m in 5 CI The simplest way to obtain a causal FIR lter from hdn is to de ne a new system with impulse response hn given by h 0 lt S M hln M quot 0 otherWISe El As can be seen in the LPF example we discussed it in Ch 2 there is the Gibbs phenomenon C More generally we can represent hn as the product of the desired impulse response and a niteduration window wn kin hdnWn where the window for the above example is l 0 S n S M Wln 0 otherwrse W M Mix A mth by VY rnde m 4 V v 7 m a v 1 5 15 1 1 7 7 7 339quot 3 fme 2 l 3 05 3 05 I O I O 05 05 1 0 1 1 0 a 7239 a 7239 1 5 15 1 A 1 nv 339quot a in 4f 33 05 33 0 5 NJj E E I O V I O V V 0 5 O 5 1 0 1 1 0 a 7239 a 7239 71v I 11L y V V9 L Dwigm f FER F tms by W mhwimg 5 From 159163 Intlca vdmhatmm 11 w umdxg 2 ivw img i h 1r ni1gt 1 AW 7 a 1 V 7amp0 I 7 i 7w 7M DW 397 WI H 7U 63 W Az 41 a LE 61 x j v j kfIn r 3 7y wa H mm 0M3 39en W 64 AW h U7 397 v AWN mam 5 9 5 MS mam Gamwa OMwLM 111 Z wA W quotom Y n r fr V39 quot71 1 a 8 m gaffquot8 39 quot75v 8 3 P VF if J1 quotVe m Thug gig wulh mi swam W m m Livrja 3 yn Wer39w9 I 1 Design ul FIR Fillers h Villdlm ill 739 El If We139 is concentrated in a narrowband of frequencies around o0 then Heiquot will look like Hde except where Hdequot changes very abruptly CI The choice of the window is to have wn as short as possible in duration so as to minimize computation in the implementation of the lter while having We approximate an impulse Clearly there are con icting requirements Cl For the rectangular window the side lobes are large and as M increases the peak amplitudes of the main lobe and the side lobes grow in a manner such that the area under lobe is a constant while the width of each lobe decreases with M Dm gm Df FER F mrs by W mhw mg 7 Q Er qmamcy om m1 EKGW i jqu ng m9 3 74L WK jm firm M 7 g ZWM SEEM V7 1 T 3 Ig J C My x M W W sin tn2 M 7 Peak sidelobe M1 M1 Awm lt Mainlobe width Q Ner mm by mas Wilnldbw smm y it mh mg lama ming m g dia zsm Tb ant f a 1m aim I mpcrlics ul 39mnnmn l39ml Vintltms I CI Commonly used windows 1 0 lt n S M Recmngular Mn 0 otherwise 211 M O S n lt M 2 Bartlett triangular wn 2 2nM M2 s n s M 0 otherwise Hanning WM 05 0SCos27rnM OSnSM 0 otherwise amm O54 046cos27rnM OSnSM H mg WM 0 otherwise Blackman 042 05 cos27rnM 008 cos47r nM 0 s n s M WM 0 otherwise f h mmwm ylbd5W mdW 2 Pr pgr mg 0 W 37144 7 NUS w n Rectangular 10 Hamming Hanning 08 Blackman Bartlett Pmpwt m lt 1TC mrm m y Usmdl W mdmws 3 40 i in J F J ma m We 80 AWE359 ii J x 0 02n 0411 0617 087r 77 Radian frequency w 20 logw Wequot 20 740 pm in I Q 60 Ajp 2010ngWef g39 xCM 081r 71 477 1r Radian frequency w Pmpwt m lt 1TC mrm m y Usmdl W mdlqms 4 20 logm IWe 20 logm mm 20 logm wwn 021r 0471 1677 087r Radian freuuencv an 0277 0477 0607 031139 Radian frequency w 760 780 rum NM I m u 027 047 0071 087 Rude l39rcqucncy l HIEAum n n g mi 39ngeam 397 39k 1 Ida Emir i 2W5 r Properties nl nmnmn soil Windows 5 Incorporation of Generalized Linear Phase 2 All the windows have the property that wM n OSnSM WIn 0 otheI39WlSC ie they are symmetric about MZ El If the desired impulse response is also symmetric about M2 ie if hdn hdM n then the windowed impulse response will also have that symmetry and the resulting frequency response will have a generalized linear phase that is Hejo Aeejwe ij2 where Aeei is real and is an even function of co Properties nl mnnmn sell Vimlmu v Incorporation of Generalized Linear Phase cont II If the desired impulse response is antisymmetric about M2 ie if hdn hdM n then the windowed impulse response will also be antisymmetry about M2 and the resulting frequency response will have a generalized linear phase that is He 1A em equot quot 2 where Aoefquot is real and is an odd mction of a The Kaiser Vt imlrm Filter Design Mcllmd I CI The tradeoff between the main lobe width and sidelobe area can be quanti ed by seeking the window function that is maximally concentrated around m0 in the equency domain This was considered and the solution shows that the lter design involves prolate spheroidal wave functions and thus is di icult El However Kaiser found that a nearoptimal window could be formed using the zerothorder modi ed Bessel function of the rst kind 106 a function that is much easier to compute D The Kaiser window is de ned as 2 12 Iowa to com 1 OsnsM W 1009 0 otherwise where XM2 The Kaiser Vt imlim Filter Design Mcllmrl 3 CI The Kaiser window has two parameters the window length M1 and a shape parameter B By adjusting M1 and B the sidelobe amplitude and mainlobe width can be properly designed El B0 becomes the rectangular window Increasing B will reduce the sidelobe level but the mainlobe becomes wider Cl Increasing M while holding B constant causes the main lobe to reduce in width but does not affect the amplitude of the side lobes El Through extensive numerical experiments Kaiser obtained a pair of formulas to predict the values of M and B needed to meet a given lter speci cation The Kaiser Window Filter Design Method 3 a 1 E i E lt 10 Sanlplcs 047 167 Radian frequency w R a r imS ip f 1116 Kaisw 33711111110111 m ltOgttrmr V W mdn vs W TABLE 71 COMPARISON OF COMMONLY USED WINDOWS Peak Transition Peak Appmximation Equivalent Width Sidc Lobc Approximate Ermr Kaiser of Equivalent Type 0139 Amplitudc Width 0139 20g105 Window Kaiser Window Relative Main Lobe dB Window Rectangular 13 47rM 1 lt21 0 1817M Bartlett 725 871114 725 133 237rrM Hannng 31 SnM 44 386 501 nM Hamming 741 SnM 753 486 6277M Blackman 757 IZHM 774 704 919nM Hmlqm Related Mullah l unctiunx Signal Processing TUUHNH WINDOW Window function gateway WINDOWWNAMEN returns an Npoint window of type speci ed by the lnction handle WNAME in a column vector WNAME can be any valid window function name for example bartlett Bartlett window blackman Blackman window hamming Hamming window hann Hann window kaiser Kaiser window rectwin Rectangular window triang Triangular window Mimllm Related lnlluh Functions Signal Processing Tnulhcn W BARTLETTN returns the N point Bartlett window WBLACKMAN N returns the Npoint symmetric Blackman window in a column vector WHAMMINGN returns the Npoint symmetric Hamming window in a column vector HANNN returns the N point symmetric Harm window in a column vector W RECTWINN returns the Npoint rectangular window W TRIANGN returns the Npoint triangular window W KAISERNbeta returns the BETAvalued Npoint Kaiser window HR Design Related Mullah Functions Signal Processing I milhm rl Window based FIR lter design low high band stop multi kaiserord Kaiser window design based lter order estimation Also the following two functions are available at the standard Specialized math functions BESSELI Modi ed Bessel function of the rst kind I BESSELINUZ is the modi ed Bessel function of the rst kind InuZ For more functions related to window and lter design use help signal For detailed information for each function use help funcname The Kaiser Wim qiw Filter Design Mmimdl 4 H M pHdww 05 39 31M mwbaurid mm 39 J Md The Kaiser imlmx filter Design Mt lllml 5 El De ning A 2010g108 Kaiser determined empirically that the value of 3 is given by 01102A 87 A gt 50 05842A 21 4 007886A 21 21 s A s 50 00 A lt 21 El When the transition bandwidth is Am cos cop M must satisfy A 8 39 m This equation predicts M to within i2 over a wide range of values of A0 and B The I1iserimlm Filter Design Mollmtl 1 LPF Design Example 61001 820001 up 041 ms 061 El Step 1 because the Kaiser window assume the same error bound for the passband and stopband we let 8 min81 820001 El Step 2 The cutoff frequency of the underlying ideal LPF is me 0 cog 2 051 El Step 3 Determine A and A0 A 2010g105 60 dB A0 cop op 021 Then we can obtain 3 5653 M 37 El Step 4 The impulse response of the lter is obtained as ocM2 sinwAn a Io 1 n aa2 2 Mn 7rn a 10 T 0 Ideal LPF window OtherWise OSnSM The Kaiser llllltl Filter Design Mellmtl 7 LPF Design ExaanIe 35001 820001 0 04x cos 061 E El cont Since M is an odd integer the resulting linearphase system would be of type II He is zero at 0310 The approximation error function is de ned as 1 H 1 s 3 EM e l 0 a 02 0Hej 025025 note that the error is not de ned in the transition region In this design example the peak approximation error is slightly greater than 50001 Increasing M to 38 results in a type I lter for which 500008 It is not necessary to plot the phase and group delay because the phase is precisely linear and the delay is M2 samples The Kaiser Window Filter Design Method 8 LPF Design Example 51001 620001 up 047 DS 067 0477 0677 Radian frequency 1 u v E T E lt 00010 3 011005 Sample I1umhurn Amplil ude 700005 1 4quot 0677 18quot Radian frequency Ln The Kaiser imlmx Filler Dcsiun Mclhml U H HPF Design Exanmle 61820021 co 0351 cop 051 El The frequency response of an ideal HPF is 0 wltw MM thejwe39me2 09 2 H1pejm wcltw57r and the corresponding impulse response is sin7rn M2 siann MZ 75n M2 7rn M2 III The HZPF impulse response a er applying the Kaiser window is hm nwn O S n S M wltnltw 0 otherwise The Kaiser indrm Filter Design Icllmll Ill HPF Design Exanmle 61Sz0021 to 0351 up 051 Cl Using the Kaiser s formula we obtain that B26 M24 The cutoff frequency of the ideal HPF is me mp cop2 04251t CI The actual approximation error of this type I lter is 800213 Cl If we increase M to 25 and keep B unchanged as we did in the LPF design the frequency response becomes highly unsatisfactory because the type H lter has a zero at co 1 Type II FIR linearphase systems are generally not appropriate approximation for the either highpass or bandstop lters El Increasing M to 26 will result in type I lter with narrower transition region and will satisfy the design requirement The Ka sm W mdqm F tm qus gxm Mce mqul 1L 1 EEFQQQDZ h a my ULSSS zE D CD 1 mm M72211 m 393 15 a um I l w x U 04 r 027 041 1611 087 7r 3 Radian frequency m TE 02 7 m4 lt n r l 0 7 7 1102 7 i 44 V E y 1 m E 0 Sample number n lt 7mm 7 r 1 x L17 041 umr 081 n Radian frequency In The Ka merr W mdww mm D s gm Mce mdl 12 om 0 40 E 4 06 760 04 7 780 U 02 mm I 1 1 g T I I T 0 02 041 0677 0817 E u o c 39 1 l Radianfrcqucncym 4 I 02 7 704 03 7 x x o 06 A a 10 20 30 Samplennmbcr 1 EL 04 c 02 7 o N I 0277 0411 0577 081r Radian frequency w The Ka gm W mdww F tw Dcag glm Methwd J13 ij w Ifxm j Q Tm m roamgs dim g m 139F 51m gm 1 1399 if mm ibipk and gmpbmdgn gwm mm mm bmcdl ngh mg mm Hm mxgg buighmsa Mmdipagg 1de bmmdgitm Hmbltefwgtl G1 I Nlnb 4 Gf 64 1 G3 1 44 E The Kaiser imhm Filter Design Icthml I4 Multiband lter cont El If such a magnitude function is multiplied by a linear phase factor e fm the corresponding ideal impulse response is N sina n MZ h G G quot mp7 1 kl P39Mz where Nmb is the number of bands and CNN 0 CI If hmbn is multiplied by a Kaiser window the type of approximations we have observed at the single discontinuity of the lowpass and highpass systems will occur at each of the discontinuities The approximation error will be scaled by the size of the jump The Kaiser indtm Fillt r Design Il illtll39 I5 DiscreteTim Dt 39erentiators D As we discussed before the discretetime differentiator system has a frequency response jco T for 1r lt m lt 1 III For an ideal discretetime di erentiator with a linear phase the appropriate equency response is the factor of NT is omitted Hwe jwe M D The corresponding ideal impulse response is 0057 n MZ sin7rn M2 hdiff n 2 n MZ 7rn M2 C If hawks is multiplied by a symmetric window of length M1 then it is easy to show that hn hM n The resulting system is either type IE or type IV general linearphase system The Kama Windlqu F tcer Dcas gm Mce mcdl 16 gmquot Q1 mangmi mk imeaa ity 01 ows mm 3311 z 4 A 3 E 1 1 U 02quot 08quot 1 g I Radian rcqucncy a E T I t v 1 l 17 01 I l A 2 4 a 8 n E Sample number n lt V 701 w w I 02quot D 081 77 Radian frequency w The Ka gm W mmw F fm Dces gm Maethqu 17 D 0351 gm beialimp f gigging igyp g W a Q uqmp V mml Ibg iw u a pr xjm3a m it 1131ng mm mm mm r 2 E1 Whig 13W 1 a u gt J QLJ 1 lt lm 1 Lagvzn 3 E E Z E lt 1 1 4 w w w x 3 7 r 7 I39 027 47 0m 8 17 E Radian requuncy m lt 1 006 004 r I 3 4 Snmplu numlmr m Amplimde P B DZ 1 w I 027 0 An 067 08quot 7 Radian frequuncy w


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