ECE 6560 Multirate Signal Processing Chapter 3

Transcription

1 Multirate Sigal Processig Chapter 3 Dr. Bradley J. Bazui Wester Michiga Uiversity College o Egieerig ad Applied Scieces Departmet o Electrical ad Computer Egieerig 903 W. Michiga Ave. Kalamazoo MI,

2 Chapter 3: Digital Filters 3. Filter Speciicatios 40 Freq. domai rect uctio prototype ilter Coheret gai 3. Widowig 43 FFT widowig with ilter uctios 3.3 The Remez Algorithm 5 Optimal digital FIR ilter geeratio 3.3. Equiripple Vs, / Ripple Desigs Acceptable I-bad Ripple Levels 66 otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

3 Filterig eeds Multirate processig ivolved: ilter-decimatio ad iterpolatio-ilter operatios The deiitio o ilters appropriate or the sigals beig processes is ecessary or the sigal processig beig perormed. The desired or required passbad badwidths or the sigals o iterest must be ow. The ilter structure is also a importat cosideratio. Filter passbads, trasitio bads, stopbads, ad other shape characteristics must be deied ad uderstood! otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

4 Filter otes The Filter otes slides provide stadard deiitios, MATLAB programmig or classic aalog ilters, MATLAB programmig or classic aalog ilters as digital IIR ilters, ad a examples o MATLAB geeratig the ilters. The FIR Filter DSP otes provide isight ito digital ilters, properties o the our types o FIR ilters, traslated ad complimetary ilters, ad a brie IIR ilter discussio. Thigs to ow: Summary pages, Type (odd # coe., real symmetric) ad type FIR (eve # coe., real symmetric) otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

5 FIR Filter Types Filter Type # Coe., Symmetry Commets Type Filter: Odd, Symmetric It ca do all ilters. (the best optio) Type Filter: Eve, Symmetric It ca ot be a highpass ilter. Type 3 Filter: Odd, Ati-Symmetric It ca oly be a badpass ilter. (worthless?!) Type 4 Filter: Eve, Ati-Symmetric It ca ot be a lowpass ilter. ote: Type ad type 4 are ote related by (-) (chagig symmetric to ati-symmetric) otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

6 otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB Rectagular Low Pass Filter rect H t t h sic t t t h si t t t h si : t t zero First zero st zero st

7 Preservig Filter Gai i Samplig We wat to ormalize the itegral o the cotiuous time impulse respose (uity gai DC cosideratio). t t dt H K h T h 0 Based o Poisso s sum ormula (ot prove here) K T Thereore the ilter coeiciets should be h s sic t s otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

8 otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB Sampled Impulse Respose s s s s t t t t t h si s s s s h si s or si h h s s

9 Sampled Respose How may discrete sample are there betwee =0 ad the ist ull? s ote that a irst order estimate o a low pass ilter cuto requecy ca be made by coutig the sic uctio discrete poits betwee 0 ad the irst ull. s otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB s 9

10 Sampled Respose The umber o samples is see to be s/( ), the ratio o sample rate s to the two-sided badwidth, which or our speciic example is 5. Thereore the two-sided badwidth is /5 th the sample rate! The sigle sided badwidth is /0 th the sample rate! The relative badwidth ca be estimated by coutig. otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

11 Spectrum The samplig process, o course, causes the spectra to be periodically exteded with spectral replicates at all multiples o the sample rate. The expressio or the spectrum o the sampled data impulse respose is show i where the sampled data requecy variable ωt s is deoted by θ with uits o radias/sample. I this coordiate system, the spectrum is periodic i π. H hexp j otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

12 Cotiuous vs. Discrete The previous aalysis looed at the cotiuous-time cotiuous-requecy Fourier Trasorm ad the discretetime cotiuous-requecy Fourier Trasorm. What about the iverse trasorm o the discrete-time discrete-requecy Fourier Trasorm? otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

13 3 Ideal Fiite Spectrum Discrete Filter W rect h exp j h 0 exp exp j j h j j j h exp exp exp j j h si si 0.5 si 0.5 si ' Iiite sum math tric otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

14 otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB Sample Replicated Sie Fuctios or j j h si si 0.5 si 0.5 si si si h h sic sic sic sic For those who preer sic uctios

15 Comparig results otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB h sic sic s s s s s s h sic si s h sic The umerator is the same, while the deomiator is due to discretizig the requecy space. DT-CF Fourier Trasorm DT-DF Fourier Trasorm

16 Geeratig Digital Filters For ilters with miimal passbad ripple ad arrow trasitio bads:. Geerate a aalog ilter ad the trasorm it ito a digital ilter. (ot used i this textboo/class). Use a widowed sic uctio. (otes ollow) Assumes a ideal requecy bad ilter is covolved by a requecy domai widow uctio. I the time domai, the iiite sic samples are trucated i legth ad shaped by a time widow. 3. Use a digital ilter geeratig algorithm Pars-McClella is the most popular (remez Algorithm) Other iterative ad o-iterative digital ilter algorithms exist. otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

17 Widows vs. Filters () Widows ad ilters are sets o coeiciets that modiy the spectrum o the sigal beig processed. A ilter is applied to a sigal i time (cotiuous or sampled) to modiy the spectral cotet. A widow is applied to a etity with a deied umber o samples (a sample set) to ehace spectral compoets. Beore applyig a FFT, the data is widowed to shape the spectral bis The iiite umber o discrete ilter coeiciets had a widow applied to create a iite impulse respose (e.g. a rectagular widowed) Fourier Pair Cocers Fiite Time Iiite Frequecy Iiite Time Fiite Frequecy otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

18 Widows vs. Filters () Filter Applicatio Covolutio i the time domai Multiplicatio i the requecy domai Widow Applicatio y x h h x x h Multiplicatio i the time domai Covolutio i the requecy domai Y y Y X H x h X H otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

19 otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB Widowig a Filter (Covolutio i the requecy domai) Multiplyig a ilter with a ixed widow uctio h w h w 0 w W h w H 0 0 l l w W h W l W H 0 0 l l w W h W l W H 0 0 l l w W h l W H 0 l w l H l W H Circular covolutio DTCF Derivatio DTDF Derivatio

20 otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB Symmetric Rectagular Widow (o-causal Spectral Deiitio) Our irst cadidate or a widow is the symmetric rectagle, sometimes called the deault widow (odd legth, M eve +). rect W M W M M M j M j j W 0 exp exp exp j M j M j W exp exp exp M M M W si si si si si si M or

21 otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB Rectagular Widow (Causal Discrete Time Deiitio) Our ext coteder or a widow is aother rectagle, also thought o as a deault widow (eve legth widow, M eve ). 0 ) (, 0 to M or w where W w W j M j W W M exp exp 0 M M j W si si exp or si si exp M M j W

22 Odd vs. Eve Rectagular Widow otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB M W si si M M j W si si exp M M j W si si exp M Odd (o-causal) M Eve (Causal) M W si si Phase due to time delay ote magitude dierece i eve vs. odd legths

23 Symmetric Rectagular Widow From the text: The sampled rectagle weightig uctio has a spectrum described by the Dirichlet erel as show i (3.9). See previous derivatios This is called the Dirichlet erel. It is see to be the periodic extesio o the si( π Tsupport/)/( π Tsupport/) spectrum, the trasorm o a cotiuous time rectagle uctio. otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

24 Rectagular Widow eve legth otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

25 Rectagular Widow Low Pass Filter Applicable to all discretely sampled ad the trucated (rect widow) i legth ilter implemetatios! The requecy domai covolutio o the ilter by a widow uctio. ripple trasitio otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

26 Rectagular Widowed Perect Low Pass Filter otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

27 Are There Better Widows? Somethig smoother that does t have the ripples o a sic uctio! Chages will liely mae the mai-lobe wider. Reducig the high requecy cotet meas more low requecy spectral cotet. Smoother meas less zero crossig ripple ad smaller side lobes due to the sic ripples. Idea: use oe cycle o a cosie wave with a appropriate DC oset otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

28 Raised Cosie Widow Time domai Frequecy domai otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

29 Raised Cosie Widow This ca be thought o as: A DC bias trasorms to a delta uctio i the requecy domai A cotiuous cosie wave delta uctios at the positive ad egative requecies A rectagular widow about the zero time sic i requecy Fourier trasorm: The DC ad cosie are added The sic is covolved Sice the sic lobes are at the delta uctio spacig, the sigals add coheretly ear DC (wideig the mai lobe) ad ocoheretly at higher requecies (reducig the ripples) otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

30 Raised Cosie Widow Time domai Delta Cosie Frequecy domai otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

31 Are There Better Widows? red harris wrote oe o the origial widow papers For a paper describig widow perormace, see: Harris, F.J., "O the use o widows or harmoic aalysis with the discrete Fourier trasorm," Proceedigs o the IEEE, vol.66, o., pp.5,83, Ja see Table i the paper otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

32 Widow Fuctios otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

33 Widow Fuctios See MATLAB code example: WidowTest.m The FFT o the widows zero padded See MATLAB code example: WidowTest_FFT.m Si wave widow FFT operatio. Dieret FFT legths. Comparig bi cetered ad betwee bi outputs. See widowed ilter example: WidowTest.m Trucated time domai sic uctio approximatio o ideal ilter Widowed sic uctios with various time-domai widows. otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

34 Empirical Trade-o: Mai-lobe vs. Side-lobe Scaig Table 3- we ca estimate the rate at which we ca trade side-lobe levels or mai-lobe width. This rate is approximately - db/spectral bi so that i order to obtai 60 db sidelobes, we have to icrease the mai-lobe badwidth to.7 s /. Rememberig that the widow s two-sided mai-lobe width is a upper boud to the ilter s trasitio badwidth, we ca estimate the trasitio badwidth o a ilter required to obtai a speciied side-lobe level. Equatio (3.0) presets a empirically derived approximate relatioship valid or widow-based desig while (3.) rearrages (3.0) to obtai a estimate o the ilter legth required to meet a set o ilter speciicatios. otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

35 Kaiser-Bessel Widow The primary reaso we examied widows ad their spectral descriptio as weighted Dirichlet erels was to develop a sese o how we trade widow mai-lobe width, or widow side-lobe levels ad i tur ilter trasitio badwidth ad side-lobe levels. Some widows perorm this exchage o badwidth or side-lobe level very eicietly while others do ot. The Kaiser-Bessel widow is very eective while the triagle (or Fejer) widow is ot. The Kaiser-Bessel widow is i act a amily o widows parameterized over β, the time-badwidth product o the widow. The mai-lobe width icreases with β while the pea side-lobe level decreases with β. otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

36 Kaiser-Bessel Widow otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

37 Kaiser-Bessel Widow Example See Matlab: KaiserBesselTest.m Beta varied rom pi/ to 4pi i steps o pi/ FFT o resultig widow Mai lobe width ad magitude o st sidelobe estimated ote: a widowed ilter is expected to have () lower sidelobe levels tha the widow ad () a wider mailobe tha the widow. See Matlab: KaiserBesselTest.m Sic o width 7 times KaiserBessel Beta varied rom pi/ to 4pi i steps o pi/ otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

38 Kaiser-Bessel Widow otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

39 Example 3. Estimate umber o Taps, try ad iterate! otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

40 Example 3. otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

41 Example 3. MATLAB Chap3_4.m script otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

42 Kaiserord ad Demo [,W,BTA,FILTYPE] = aiserord(f,a,dev,fs) geerate parameters, icludig beta uses ir.m Matlab s widowed ilter geerator uctio see h = ir(, W, FILTYPE, aiser(+, BTA), 'oscale') geerate a widowed ilter usig the deied widow. see Kaiser_FilterGe.m otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

43 The Pars-McClella Filter remez exchage algorithm otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

44 FIRPM vs. REMEZ MATLAB has chaged algorithm ames Old ame remez The Pars-McClella FIR ilter geeratio algorithm is based o the remez exchage algorithm. Thereore, the old implemetatio was called remez. ew ame FIRPM FIR Pars-McClella ilter geeratios. It wors almost idetically to remez. It has a ull complimet o order estimators that go with it (lie Butterworth Filters) otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

45 FIRPM Desig Domai The widow desig o a FIR ilter occurs i the time domai as the poit-by-poit product o a prototype impulse respose with the smooth widow sequece. The quality o the resultat desig is veriied by examiig the trasorm o the widowed impulse respose. By cotrast, the equiripple desig is perormed etirely i the requecy domai by a iterative adjustmet o the locatio o sampled spectral values to obtai a Tchebyschev approximatio to a desired spectrum. The desired, or target, spectrum has accompayig tolerace bads that deie acceptable deviatios rom the target spectrum i distict spectral regios. otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

46 Equiripple Frequecy Domai Desig otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

47 Weighted Error Filter Desig Whe the FIR ilter has M+ eve symmetric coeiciets, its spectrum ca be expaded as a trigoometric polyomial i θ as show i (3.5). Deiig a positive valued weightig uctio W(θ ) ad the target uctio T(θ ), we ca deie the weighted error uctio E(θ ) as show i (3.6). otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

48 Derivatio o REMEZ The Mth order polyomial H(θ ) is deied by M+ coeiciets a() or equivaletly by the M- local extrema H(θ ) ad the - boudary values at H(θ pass) ad H(θ stop). The problem is that we do t ow the locatios o the local extrema. The Remez multiple exchage algorithm rapidly locates these extremal positios by iteratig rom a iitial guess o their positios to their actual positios. A ubiquitous desig algorithm writte by McClella, Pars, ad Rabier expaded o the origial Pars ad McClella desig ad has become the stadard implemetatio o the Remez algorithm. It is embedded i most FIR ilter desig routies. Read the text or a explaatio otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

49 More Pars-McClella Reereces The iterested reader should read the material preseted i: IEEE Papers T.W. Pars ad J.H. McClella, Chebyshev approximatio or orecursive digital ilters with liear phase, IEEE Tras. Circuit Theory, vol. CT-9, o., pp , 97. Pars, T.W.; McClella, J.H.; A Program or the Desig o Liear Phase Fiite Impulse Respose Digital Filters, IEEE Trasactios o Audio ad Electroacoustics, vol. AU-0, o. 3, p , 97. McClella, J.H.; Pars, T.W.;, "A persoal history o the Pars- McClella algorithm," Sigal Processig Magazie, IEEE, vol., o., pp. 8-86, March 005 Boos Hadboo or Digital Sigal Processig edited by Mitra ad Kaiser. Digital Filter Desig by Pars ad Burrus otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

50 Parameters Sample Rate Satisy the yquist or desired sample rate criteria Passbad The highest requecy o iterest Stopbad The requecy at which the stopbad atteuatio must be attaied Passbad Ripple (typically % to 5%) The ripple allowed i the ilter passbad (ote ripple may appear as a modulatio distortio o the sigal) Stopbad Ripple (typically 60 to 80 db) The maximum sidelobes allowed i the stopbad otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

51 Parameter Selectio Relaxig costraits typically allows ewer FIR ilter coeiciets to be required Wider percetage passbads (pb/s) Wider trasitio bads (stop-pb) More passbad ripple More stopbad ripple (less stopbad atteuatio) Order estimators (Chap3_5b.m) remezord_harris remez_est remezord irpmord (ormerly remezord) otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

52 Remez Example 3., p. 58 Chap3_6.m h=remez(53,[0 pass stop (sample/)]/(sample/),[ 0 0],[ 0]); otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

53 Example 3. otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

54 Desig Characteristic Uiorm passbad ripple Uiorm stopbad ripple Questio Do we really wat a uiorm stopbad ripple i we are goig to decimate the output?! A ilter spectrum has a decay rate related to the order o ay discotiuities i the time domai sigal ad the sigal derivatives. 0 th order equiripple st order / d order /^ etc. otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

55 The REMEZ Discotiuity: Equiripple otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

56 Achievig / Ripple Fid ad remove the discotiuity Modiy the discotiuous coeiciets Widow REMEZ output Zero the outer coeiciets (mae the ilter + larger) Chap3_7.m Modiied Filter / behavior Side lobe icrease Passbad ripple icrease otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

57 Simple oise Power Example Chap3_8 otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

58 irpm Example Template see PMcC_FilterGe.m Also see: KaiserVsirpm.m otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

59 MATLAB Related uctios aiserord irpmord ir irpm Examples Kaiser_FilterGe.m PMcC_FilterGe.m KaiserVsirpm.m otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

60 Quatizatio All real-world FIR ilters are quatized to the available bit precisio Thereore, all real ilters appears as a combiatio o the desired ilter h() mius the quatizatio error h FIR h h Qerror Quatizatio ca (will) cause modiicatios to the desired ilter respose! otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

61 Quatizatio Error otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

62 Stairstep Weight Fuctio o loger executes i MATLAB otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

63 Tools or / Improvemet Harris myrr matlab script Chap3_9.m Chap3_0.m Harris web site iles otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

64 I-Bad Ripple or Why Passbad Ripple is Bad The desired sigal output is a delayed versio o the iput as show To accomplish this, the ilter passbad requires the spectral characteristics show (ote: this reers to the passbad, ot the etire ilter!) otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

65 Passbad Ripple With passbad ripple, the equivalet spectral results are: The equivalet ripple i requecy otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

66 Time Domai Ripple Developig the equivalet represetatio Estimatig the iverse trasorm The result shows the desired SOI ad two additioal terms, paired echoes, a pre-echo ad a post-echo! otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

67 Time Domai Echoes otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

68 Text Example: Magitude Ripple otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

69 Magitude Ripple () otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

70 Passbad with Phase Ripple otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

71 Phase Ripple () otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB

72 Phase Equalized otes ad igures are based o or tae rom materials i the course textboo: redric j. harris, Multirate Sigal Processig or Commuicatio Systems, Pretice Hall PTR, 004. ISB