EE123 Digital Signal Processing

Transcription

1 EE123 Digital Sigal Processig Lecture 20 Filter Desig

2 Liear Filter Desig Used to be a art Now, lots of tools to desig optimal filters For DSP there are two commo classes Ifiite impulse respose IIR Fiite impulse respose FIR Both classes use fiite order of parameters for desig We will cover FIR desigs, briefly metio IIR

3 What is a liear filter Atteuates certai frequecies Passes certai frequecies Effects both phase ad magitude IIR Mostly o-liear phase respose Could be liear over a rage of frequecies FIR Much easier to cotrol the phase Both o-liear ad liear phase

4 FIR Desig by Widowig Give desired frequecy respose, H d (e jω ), fid a impulse respose h d [] = 1 2 Z H d (e j! )e j! d! ideal Obtai the M th order causal FIR filter by trucatig/widowig it h[] = hd []w[] 0 apple apple M 0 otherwise

5 FIR Desig by Widowig We already saw that, periodic H(e j! )=H d (e j! ) W (e j! ) For Boxcar (rectagular) widow H d (e j! ) W (e j! )=e j! M 2 W (e j! ) si(w(m + 1)/2) si(w/2) H(e j! ) * = \W (e j! )

6 FIR Desig by Widowig pass-bad ripple H(e j! ) ideal trasitio width stop-bad ripple ωc ωc

7 Tapered Widows Name(s) Defiitio MATLAB Commad Graph (M = 8) ha(m+1), M = 8 1 Ha w 1 1 cos 2 M 2 0 M M 2 2 ha(m+1) w[] haig(m+1), M = 8 1 Haig w 1 1 cos 2 M M M 2 2 haig(m+1) w[] hammig(m+1), M = 8 1 Hammig w cos M 2 0 M M 2 2 hammig(m+1) w[]

8 Tradeoff - Ripple vs Trasitio Width M = 16 M = Boxcar Triagular 0-10 Boxcar Triagular ω M = log 10 W (e j ) ω M = Haig Hammig 0-10 Haig Hammig ω Pytho: scipy.filter.firwi log 10 W (e j ) ω

9 FIR Filter Desig Choose a desired frequecy respose H d (e jω ) o causal (zero-delay), ad ifiite imp. respose If derived from C.T, choose T ad use: Widow: H d (e j! )=H c (j T ) Legth M+1 effect trasitio width Type of widow trasitio-width/ ripple Modulate to shift impulse respose H d (e j! )e j! M 2

10 FIR Filter Desig Determie trucated impulse respose h 1 [] h 1 [] = 1 2 R H d(e j! )e j! M 2 e j! 0 apple apple M 0 otherwise Apply widow h w [] =w[]h 1 [] Check: Compute H w (e jω ), if does ot meet specs icrease M or chage widow

11 Example: FIR Low-Pass Filter Desig 1 H d (e j! )=! apple!c 0 otherwise Choose M Widow legth ad set H 1 (e j! )=H d (e j! )e j! M 2 h 1 [] = ( si(!c ( M/2)) ( M/2) 0 apple apple M 0 otherwise! c sic(! c ( M/2))

12 Example: FIR Low-Pass Filter Desig The result is a widowed sic fuctio h w [] =w[]h 1 [] High Pass Desig: Desig low pass hw[] Trasform to h w [](-1)! c sic(! c ( M/2)) Geeral badpass Trasform to 2h w []cos(ω 0 )

13 Characterizatio of Filter Shape Time-Badwidth Product, a uitless measure T(BW) = (M+1)ω/2π also, total # of zero crossigs TBW=2 TBW=4 TBW=8 TBW=12 Larger TBW More of the sic fuctio hece, frequecy respose looks more like a rect fuctio

14 TBW = 2, M=16 TBW = 8, M=16 TBW = 16, M =32

15 Frequecy Respose Profile Q: What are the legths of these filters i samples? TBW=2 TBW=12 -π -π/12 π/12 -π/2 π/2 π 2 = (M+1)*(π/6) / (2π) M=23 12= (M+1)*(π) / (2π) M=23 Note that trasitio is the same!

16 Alterative Desig Through FFT To desig order M filter: Over-Sample/discretize the frequecy respose at P poits where P >> M (P=15M is good) H 1 (e j! k )=H d (e j! k )e j! k M 2 Sampled at:! k = k 2 P Compute h1[] = IDFTP(H1[k]) Apply M+1 legth widow: k =[0,,P 1] h w [] =w[]h 1 []

17 Example: sigal.firwi2 sigal.firwi2(m+1,omega_vec/pi, amp_vec) taps1 = sigal.firwi2(30, [0.0,0.2,0.21,0.5, 0.6, 1.0], [1.0, 1.0, 0.0,0.0,1.0,0.0]) impulse respose magitude frequecy respose

18 Example: Desig usig FFT For M+1=14 P = 16 ad P = 1026 magitude frequecy respose

19 Optimal Filter Desig Widow method Desig Filters heuristically usig widowed sic fuctios Optimal desig Desig a filter h[] with H(e jω ) Approximate Hd(e jω ) with some optimality criteria - or satisfies specs.

20 Optimality H d (e j! ) Do t Care! p! s Least Squares: miimize Z!2care H(e j! ) H d (e j! ) 2 d! Variatio: weighted least-squares miimize Z W (!) H(e j! ) H d (e j! ) 2 d!

21 Optimality Chebychev Desig (mi-max) miimize!2care max H(e j! ) H d (e j! ) Parks-McClella algorithm - equi-ripple Also kow as Remez exchage algorithms (sigal.remez)

22 Example of Complex Filter Larso et. al, Multibad Excitatio Pulses for Hyperpolarized 13C Dyamic Chemical Shift Imagig JMR 2008;194(1): Need to desig 11 taps filter with followig frequecy respose: 1 Do t Care D.C. D.C. D.C. Do t Care Hz