University of CaliforniaBerkeleyCollege of EngineeringDepartment of Electrical Engineeringand Computer SciencesProfessors : N.Morgan / B.GoldEE225D Spring,1999Digital FiltersLecture 7EE 225D LECTURE ON DIGITAL FILTERSN.MORGAN / B.GOLD LECTURE 7 7.1 University of CaliforniaBerkeleyCollege of EngineeringDepartment of Electrical Engineeringand Computer SciencesProfessors : N.Morgan / B.GoldEE225D Spring,1999Digital FiltersLecture 7EE 225D LECTURE ON DIGITAL FILTERSN.MORGAN / B.GOLD LECTURE 7 7.21. Example of inverse z-transform use. - Let input be and filter be un() yn() ay n1–()xn()+=Xz() xn()z1–n0=∞∑=Yz()11az1––-----------------=nxn() un()=so yn()12πj--------zn1–dz1z1––()1az1––()------------------------------------------∫°=Basic theorem12πj--------zn1–dz1az1––-----------------∫°an for n0≥=0 for n 0<=This allows computation of the integral to be This result can be proved by iteration.yn()1an1+–1a–-----------------=n0≥EE 225D LECTURE ON DIGITAL FILTERSN.MORGAN / B.GOLD LECTURE 7 7.32. Steady state respond to a complex exponential ejwnun()yn() hm()xn m–()m0=n∑xm()hn m–()m0=n∑==If xn() ejwnun() , then yn() from above is =yn() hm()ejw n m–()m0=n∑ejwnhm()ej–wmm0=n∑== m0=n∑ mn1+=∞∑–m0=∞∑= , soyn() ejwnhm()ejwnm0=∞∑hm()ej–wnmn1+=∞∑–=Steady state value of yn() ejwnHz()[]zejw==So the Frequency response is the value of the z-transform evaluated on the unit circle.steady stateTransmithm()0⇒m ∞→thus sum 0→,as n ∞→EE 225D LECTURE ON DIGITAL FILTERSN.MORGAN / B.GOLD LECTURE 7 7.43. Geometric Interpretation of Steady State Frequency Response for simple first order diff equation :Hz()11az1––-----------------zza–-----------==ωefUnit circleHz() at zejw is ef--=General RuleGiven a collection of poles and zeros in the complex z-plane, the Frequency response at any is where is the product of all vectors to the zeros and is the product of all vectors to the poles.[special rules apply for multiple pole and zeros.] ND----NWDEE 225D LECTURE ON DIGITAL FILTERSN.MORGAN / B.GOLD LECTURE 7 7.5Preview of the Rest of the Material1. Filtering concepts. - approximate problem2. Sampling and Impulse Invariance3. Bilinear Transformation4. The DFT5. Circular Convolution and Linear Convolution6. Basic FFT Concept.7. DFT’s and Filter BanksEE 225D LECTURE ON DIGITAL FILTERSN.MORGAN / B.GOLD LECTURE 7 7.65. Approximation Problem Example of Ideal FiltersIdeal low passIdeal band passIdeal band stepIdeal band pass differentiatorImportant PointLinear Analog filters of R, L, Cmust have frequency responses that are rational functions in ω.Similary, linear digital filtermust have rational functions in ejωA ω()ωC–ωCωωωωEE 225D LECTURE ON DIGITAL FILTERSN.MORGAN / B.GOLD LECTURE 7 7.7Analog designers tackle the approximation problem by specifying a REAL function on the jω axis.Example : Hjω()211ωωc-----2n+-----------------------=Figure 7.1 : Butterworth Frequency Response for Different n.H ω()ωEE 225D LECTURE ON DIGITAL FILTERSN.MORGAN / B.GOLD LECTURE 7 7.8Figure 7.2 : Chebyshev Frequency Response for n=4.ωT ω()EE 225D LECTURE ON DIGITAL FILTERSN.MORGAN / B.GOLD LECTURE 7 7.9If a suitable is chosen, it can lead to a specification in the complex s-plane of and this function holds true Everywhere in the s-plane.Let’s normalize, so that and then let .Hjω()2Hs()γωωC------=sjγ=So Hs()H * s()11s2–()n+----------------------=EE 225D LECTURE ON DIGITAL FILTERSN.MORGAN / B.GOLD LECTURE 7 7.10Hs()Hs–()11s2–------------=Hs()Hs–()11s4+-------------=Hs()Hs–()11s6–------------=n1=n2=n3=Hs() Hs–()EE 225D LECTURE ON DIGITAL FILTERSN.MORGAN / B.GOLD LECTURE 7 7.11Figure 7.4 : Comparison of Group Delay for Four Types, A = Butterworth, B = Chebyshev, C = Elliptic, D = Bessel, for a low pass filter with a 500Hz corner frequency.EE 225D LECTURE ON DIGITAL FILTERSN.MORGAN / B.GOLD LECTURE 7 7.12Raderschannel vocoder experiment - Butterworth filter bank yielded better results than Chebyshev.Note : Bessel and Lenner filters have good phase response and were used in Vocoders.Question : How do we construct digital filters that give good frequency responses?* Impulse Invariance - Linear analog filters have a given impulse response.hˆt()timeTEE 225D LECTURE ON DIGITAL FILTERSN.MORGAN / B.GOLD LECTURE 7 7.13Figure 7.16 : Aliasing Effects of Hopped FFT’s.EE 225D LECTURE ON DIGITAL FILTERSN.MORGAN / B.GOLD LECTURE 7 7.14Construct a Digital Filter that has an impulse response that are the samples of . ht()hn() hˆnT()=hˆn() L1–A1SS1+-------------A1es1t–==hn() A1es1nT–=Hz() hn()zn–n0=∞∑A11es1T–z1––----------------------==es1T––Start with a simple analog filter.ands1 is realjωHˆs()s1–αEE 225D LECTURE ON DIGITAL FILTERSN.MORGAN / B.GOLD LECTURE 7 7.15Procedure- Find impulse response of suitable analog filter.- Sample it to find . - Take z-transform to find transfer function .hn()Hz()EE 225D