Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw Analog IIR Filter Deign Commonly ued analog filter : Lowpa Butterworth filter all-pole filter characterized by magnitude repone. Nfilter order Pole of HH- are equally paced point on a circle of radiu in -plane N c N c H H G j H j G + + c pole of H N4 c
Butterworth Filter Lowpa Butterworth filter monotonic in pa-band & top-band N c `maximum flat repone : N- derivative are zero at 0 and Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw
Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw 3 Analog IIR Filter Deign Commonly ued analog filter : Lowpa Chebyhev filter type-i all-pole filter characterized by magnitude repone Nfilter order i related to paband ripple are Chebyhev polynomial: H H G T j H j G c N + ε ε T N x... 0 x T x xt x T x x T x x T x T N N N
Chebyhev & Elliptic Filter Lowpa Chebyhev filter type-i All-pole filter, pole of HH- are on ellipe in -plane Equiripple in the pa-band Monotone in the top-band Lowpa Chebyhev filter type-ii Pole-zero filter baed on Chebyhev polynomial Monotone in the pa-band Equiripple in the top-band H j Lowpa Elliptic Cauer filter Pole-zero filter baed on Jacobian elliptic function Equiripple in the pa-band and top-band hence yield mallet-order for given et of pec + ε U N c Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw 4
Analog IIR Filter Deign Frequency Tranformation : Principle : prototype low-pa filter e.g. cut-off frequency rad/ec i tranformed to properly caled low-pa, high-pa, band-pa, band-top, filter example: replacing by move cut-off frequency to C C C example: replacing by turn LP into HP, with cut-off frequency C + example: replacing by turn LP into BP Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw 5
Principle : Analog -> Digital deign analog filter LP/HP/BP/, and then convert it to a digital filter. Converion method: convert differential equation into difference equation convert continuou-time impule repone into dicretetime impule repone convert tranfer function H into tranfer function Hz Requirement: the left-half plane of the -plane hould map into the inide of the unit circle in the z-plane, o that a table analog filter i converted into a table digital filter. Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw 6
Analog -> Digital I convert differential equation into difference equation : in a difference equation, a derivative dy/dt i replaced by a backward difference ykt-ykt-t/ty[k]-y[k-]/t, where Tampling interval. imilarly, a econd derivative, and o on. eventually detail omitted, thi correpond to replacing by -/z/t in H a analog tranfer function : z H a -plane jw z-plane H table analog filter are mapped into table digital filter, but pole location for digital filter confined to only a mall region o.k. only for LP or BP Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw 7 j z T 0 z z 0
Analog -> Digital II convert continuou-time impule repone into dicrete-time impule repone : given continuou-time impule repone h c t, dicrete-time impule repone i h [ k ] h kt where T d ampling interval. c d eventually detail omitted thi correpond to a many-to-one mapping z e T d -plane jw z-plane j 0 ± z jπ / T d z aliaing! if continuou-time repone ha ignificant frequency content above the Nyquit frequency i.e. not bandlimited Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw 8
Example: Filter Deign by Impule Invariance Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw 9
Many-to-one mapping Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw 0
Example: A Low-Pa Filter Then Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw
By uing Butterworth filter, then Conequently, Then N5.8858 Since N mut be integer. So, N6. And we obtain Ω c 0.703 We can obtain pole of H c. They are uniformly ditributed in angle on a circle of radiu Ω c 0.703 Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw
K 0 0.093 Dicrete Filter Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw 3
Analog -> Digital III convert continuou-time ytem tranfer function into dicrete-time ytem tranfer function : Bilinear Tranform mapping that tranform whole! jw-axi of the -plane into unit circle in the z-plane only once, i.e. that avoid aliaing of the frequency component. H z H a -plane T z + z jw z-plane 0 j. for low-frequencie, thi i an approximation of for high frequencie : ignificant frequency compreion `warping Non-linear tranform ometime pre-compenated by pre-warping Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw 4 z j z e z T
The bilinear tranformation avoid the problem of aliaing problem becaue it map the entire imaginary axi of the -plane onto the unit circle in the z-plane. The price paid for thi, however, i the nonlinear compreion the frequency axi warping. Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw 5
Concluion/Software IIR filter deign coniderably more complicated than FIR deign tability, phae repone, etc.. Fortunately IIR Filter deign abundantly available in commercial oftware Matlab: [b,a]butter/cheby/cheby/ellipn,,wn, IIR LP/HP/BP/BS deign baed on analog prototype, pre-warping, bilinear tranform, Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw 6