EE 508 Lecture 16. Filter Transformations. Lowpass to Bandpass Lowpass to Highpass Lowpass to Band-reject

Transcription

1 EE 508 Lecture 6 Filter Tranformation Lowpa to Bandpa Lowpa to Highpa Lowpa to Band-reject

2 Review from Lat Time Flat Paband/Stopband Filter T j T j Lowpa Bandpa T j T j Highpa Bandreject

3 Review from Lat Time Standard LP to BP Tranformation Verification of mapping Strategy: Image of Im axi: The mapping -domain map =0 to = j map =j to =j BN map = j to = j AN olving for, obtain + BW N + j = BW T LPN (f()) N + BW N -domain map =0 to = map = to = BN map = to = AN j BW ± BW j - 4 BW ± BW + 4 N N N N j thi ha no real part o the imaginary axi map to the imaginary axi Can readily how thi mapping map PB to PB and SB to SB i termed the tandard LP to BP tranformation

4 Review from Lat Time T LPN () Standard LP to BP Tranformation TLP j - + BW N TBP j T BPN () BW N BW N - AN - BN - AN BN

5 Review from Lat Time Standard LP to BP Tranformation Pole apping p p BW ± BW p - 4 X N N X,Q 0LP LP Im,Q 0BPH,Q 0BPL LBPH LBPL Im Re Re Image of the cc pole pair i the two pair of pole

6 Review from Lat Time Standard LP to BP Tranformation Pole apping,q 0LP LP Im,Q 0BPH,Q 0BPL Im LBPH LBPL Re Re Can how that the upper hp pole map to one upper hp pole and one lower hp pole a hown. Correponding mapping of the lower hp pole i alo hown

7 Review from Lat Time Standard LP to BP Tranformation Pole apping p p BW ± BW p - 4 X N N X Im Im Re Re multipliity 6 Note doubling of pole, addition of zero, and likely Q enhancement

8 LP to BP Tranformation Claim: Other variable mapping tranform exit that atify the imaginary axi mapping propertie needed to obtain the LP to BP tranformation but are eldom, if ever, dicued. The Standard LP to BP tranform I by far the mot popular and mot author treat it a if it i unique. T LPN ( x ) X f () T BPN ()

9 LP to BP Tranformation Pole Q of BP Approximation TLPN j TBP j BW BW = - H L L H H L Conider a pole in the LP approximation characterized by { 0LP,Q LP } It can be hown that the correponding BP pole have the ame Q (i.e. both bp pole lie on a common radial line),q 0LP LP Im,Q 0BPH,Q 0BPL LBPH LBPL Im Re Re

10 Pole Q of BP Approximation LP to BP Tranformation TLPN j TBP j BW Im,Q 0BPH,Q 0BPL BPH BPL Im BW = - H H L L L H,Q 0LP LP Re Re Define: BW It can be hown that 0LP Q Q Q LP BPL BPH QLP For d mall, It can be hown that 0BP Q BP Q LP Q BP Q BP 4 QLP Q LP Note for d mall, Q BP can get very large

11 LP to BP Tranformation Pole Q of BP Approximation BW 0LP Q Q Q LP BPL BPH QLP

12 LP to BP Tranformation Pole location v Q LP and BW 0LP

13 LP to BP Tranformation TLPN j TBP j BW L H Claical BP Approximation Butterworth Chebychev Elliptic Beel Obtained by the LP to BP tranformation of the correponding LP approximation

14 Standard LP to BP Tranformation + BW Standard LP to BP tranform i a variable mapping tranform ap j axi to j axi ap LP pole to BP pole Preerve baic hape but warp frequency axi Double order Pole Q of reultant band-pa function can be very large for narrow pa-band Sequencing of frequency caling and tranformation doe not affect final function N

15 Example : Obtain an approximation that meet the following pecification A A R A S AL A B BH BW= - B = A B A Aume that AL, BH and atify - - AL BH BW BW AL BH

16 Recall from lat lecture Standard LP to BP Tranformation Frequency and -domain apping - Denormalized (ubcript variable in LP approximation for notational convenience) T LPN ( x ) X + BW T BP () X X + BW - BW BW ± BW - 4 X X BW ± BW 4 X X Exercie: Reolve the dimenional conitency in the lat equation

17 Example : Obtain an approximation that meet the following pecification A A R A S AL A B BH A A RN SN +ε A = A R A = A S A = A R BW= - B = A B A A RN A SN S - - AL BH BW BW AL BH S A A R AL - - AL BW (actually A and AL that map to and S repectively but how A and AL for convenience)

18 Example : Obtain an approximation that meet the following pecification A A R A SL A SH AL A B BH BW= - B = A B A In thi example, - - AL BH BW BW AL BH

19 Example : Obtain an approximation that meet the following pecification A A R A RN A = A +ε R A = A R A SL A SH AL A B BH A A SH A =min, SN A A SL A RN A SN BW= - B A A RN A SN S S A A R S S - - AL BW AL - BH BW BH = B A min, SN S S

20 A RN A SN Example : Obtain an approximation that meet the following pecification A RN A SN S S AR A = RN A A = +ε A A A SH A =min, SN R A A SL BW= - B A RN A SN A SN min, SN S S A R S S A - - AL BW BH AL - BH BW = B A min, SN S S

21 Filter Tranformation Lowpa to Bandpa Lowpa to Highpa Lowpa to Band-reject (LP to BR) (LP to BP) (LP to HP) Approach will be to take advantage of the reult obtained for the tandard LP approximation Will focu on flat paband and zero-gain top-band tranformation

22 LP to BS Tranformation Strategy: A wa done for the LP to BP approximation, will ue a variable mapping trategy that map the imaginary axi in the -plane to the imaginary axi in the -plane o the baic hape i preerved. X IN TLPN X OUT f X IN TBS X OUT BS LPN T T f f = m T i=0 n T i=0 a Ti b Ti i i

23 LP to BS Tranformation T j T j BS BW N Normalized BW N=BN -AN AN BN AN BN T j T j BS Normalized BW N BW N - - BN - AN AN - BN

24 apping Strategy: Standard LP to BS Tranformation T j T j BS Normalized BW N BW N - - BN - AN AN - BN Variable apping Strategy to Preerve Shape of LP function: F N () hould map =0 to =± j map =0 to = j0 map =j to =j A map =j to =-j B map =-j to =j B map =-j to =-j A map =0 to = ± map =0 to = 0 map = to = A map = to = - B map = to = B map = to = - A

25 Standard LP to BS Tranformation map =0 to = ± map =0 to = 0 map = to = A map = to = - B map = to = B map = to = - A T j - = - TBS j =

26 Standard LP to BS Tranformation T LPN () F N () T BSN () apping Strategy: conider variable mapping tranform map =0 to =± j map =0 to = j0 map =j to =j A map =j to =-j B map =-j to =j B map =-j to =-j A F N () hould map =0 to = ± map =0 to = 0 map = to = A map = to = - B map = to = B map = to = - A Conider variable mapping LPN T F ( ) =T N BW BSN N BW N

27 Comparion of Tranform LP to BP + BW LP to BS N BW N T j T j TLPN j BS TBP AN j BW N BN BW L H

28 Standard LP to BS Tranformation Frequency and -domain apping (ubcript variable in LP approximation for notational convenience) T LPN ( x ) X BW T BSN () N BWN BW - X X BW BW N N ± - 4 X X BW BW N N X X N 4

29 Standard LP to BS Tranformation Un-normalized Frequency and -domain apping (ubcript variable in LP approximation for notational convenience) T LPN ( x ) X BW T BS () BW BW - X X BW BW ± - 4 X X BW BW 4 X X

30 Standard LP to BS Tranformation Pole apping Im,Q 0BPH LBPH,Q 0LPN LP Im,Q 0BPL LBPL Re Re Can how that the upper hp pole map to one upper hp pole and one lower hp pole a hown. Correponding mapping of the lower hp pole i alo hown

31 Pole Q of BS Approximation LP to BS Tranformation T j TBP j BW N Im AN BN,Q 0BSH LBSH BW = BN - AN,Q 0LPN LP Im Re,Q 0BSL Re LBSL Define: BW It can be hown that ANBN 0LPN Q Q Q LP BSL BSH QLP For γ mall, It can be hown that 0BS Q BS Q LP Q BS Q BS 4 QLP Q LP Note for γ mall, Q BS can get very large

32 Standard LP to BS Tranformation p BW Pole apping N p BW ± N - 4 p X X Im Im multipliity 6 Re Re multipliity 6 Note doubling of pole, addition of zero, and likely Q enhancement

33 Standard LP to BS Tranformation BW X Standard LP to BS tranformation i a variable mapping tranform ap j axi to j axi in the -plane Preerve baic hape of an approximation but warp frequency axi Order of BS approximation i double that of the LP Approximation Pole Q and 0 expreion are identical to thoe of the LP to BP tranformation Pole Q of BS approximation can get very large for narrow BW Other variable tranform exit but the tandard i by far the mot popular

34 Filter Tranformation Lowpa to Bandpa Lowpa to Highpa Lowpa to Band-reject (LP to BR) (LP to BP) (LP to HP) Approach will be to take advantage of the reult obtained for the tandard LP approximation Will focu on flat paband and zero-gain top-band tranformation

35 LP to HP Tranformation Strategy: A wa done for the LP to BP approximation, will ue a variable mapping trategy that map the imaginary axi in the -plane to the imaginary axi in the -plane o the baic hape i preerved. X IN TLPN X OUT f X IN THP X OUT HP LPN T T f f = m T i=0 n T i=0 a Ti b Ti i i

36 LP to HP Tranformation T j T j HP Normalized T j T j HP Normalized - -

37 apping Strategy: Standard LP to HP Tranformation T j T j LP Normalized HP - - Variable apping Strategy to Preerve Shape of LP function: F N () hould map =0 to =± j map =j to =-j map = j to =j map =0 to = map = to =- map = to =

38 Standard LP to HP Tranformation T LPN () F N () T HPN () apping Strategy: conider variable mapping tranform F N () hould map =0 to =± j map =j to =-j map = j to =j map =0 to = map = to =- map = to = Conider variable mapping T ( ) =T LPN LPN F

39 Comparion of Tranform LP to BP + BW LP to BS N BW N T j T j TLPN j BS TBP AN j BW N BN BW L H LP to HP T j T j LP HP

40 LP to HP Tranformation (Normalized Tranform) T j - THP j = - = -

41 Standard LP to HP Tranformation Frequency and -domain apping (ubcript variable in LP approximation for notational convenience) T LPN ( x ) X T HPN () X X - X - X

42 Standard LP to HP Tranformation Pole apping Claim: With a variable mapping tranform, the variable mapping naturally define the mapping of the pole of the tranformed function T LPN ( x ) p X p X T HPN () p p X

43 Standard LP to HP Tranformation Pole apping T LPN ( x ) X p p X T HPN () If p X =α+jβ α-jβ p= = α+jβ α +β and p X =α-jβ α+jβ p= = α+jβ α +β

44 Standard LP to HP Tranformation Pole apping T LPN ( x ) X p p X T HPN () If p X =α+jβ α-jβ p= = α+jβ α +β and p X =α-jβ α+jβ p= = α+jβ α +β Highpa pole are caled in magnitude but make identical angle with imaginary axi HP pole Q i ame a LP pole Q Order i preerved

45 Standard LP to HP Tranformation (Un-normalized variable mapping tranform) 0 T j - THP j = - = - 0 0

46 Filter Deign Proce Etablih Specification - poibly T D () or H D (z) - magnitude and phae characteritic or retriction - time domain requirement Approximation - obtain acceptable tranfer function T A () or H A (z) - poibly acceptable realizable time-domain repone Synthei - build circuit or implement algorithm that ha repone cloe to T A () or H A (z) - actually realize T R () or H R (z) Filter

47 End of Lecture 6