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 Theorem: If the perimeter variation and contact reitance are neglected, the tandard deviation of the local random variation of a reitor of area A i given by the expreion A = A R R Theorem: If the perimeter variation are neglected, the tandard deviation of the local random variation of a capacitor of area A i given by the expreion C C A = A C Theorem: If the perimeter variation are neglected, the variance of the local random variation of the normalized threhold voltage of a rectangular MOS tranitor of dimenion W and L i given by the expreion V V T T AVTO V WL T or a V V T T A VT WL

3 Review from Lat Time Theorem: If the perimeter variation are neglected, the variance of the local random variation of the normalized C OX of a rectangular MOS tranitor of dimenion W and L i given by the expreion C C OX OX A COX WL Theorem: If the perimeter variation are neglected, the variance of the local random variation of the normalized mobility of a rectangular MOS tranitor of dimenion W and L i given by the expreion R A WL where the parameter A X are all contant characteritic of the proce (i.e. model parameter) The effect of edge roughne on the variance of reitor, capacitor, and tranitor can readily be included but for mot layout i dominated by the area dependent variation There i ome correlation between the model parameter of MOS tranitor but they are often ignored to implify calculation

4 Review from Lat Time Statitical Modeling of dimenionle parameter - example R V I R V OUT R K = + R Aume common centroid layout area of R i 00u A ρ =.0µm K K Determine the yield if the nominal gain i 0, < K < 0. K.99 < <.0 K K -.0< -<.0 K K K K - K % -0< < 0 0, The gain yield i eentially 00% Could ubtantially decreae area or increae gain accuracy if deired

5 Review from Lat Time Statitical Modeling of dimenionle parameter - example R K K R V I R V OUT Determine the yield if the nominal gain i 0, < K < 0. K.99 < <.0 K K -.0< -<.0 K K K K - K.0075 % -.33< <.33 (0,) 0, K = + R A ρ =.05um A R =0um PROC OM Y = F.33 - = *.908- = Dramatic drop from 00% yield to about 8% yield! R R 0.

6 Review from Lat Time Linearization of Function of a Random Variable Characteritic of mot circuit of interet are themelve random variable Relationhip between characteritic and the random variable often highly nonlinear Ad Hoc manipulation (repeated Taylor erie expanion) were ued to linearize the characteritic in term of the random variable n Y Y ai xri i Thi i important becaue if the random variable are uncorrelated the variance of the characteritic can be readily obtained n Y i xri i a n Y i xri Y Y i a Thi approach wa applicable ince the random variable are mall Thee Ad Hoc manipulation can be formalized and thi follow

7 Review from Lat Time Formalization of Statitical Analyi Conider a function of interet Y,,...,:,,..., Y f x x x x x x f X X n R R nr R Thi can be expreed in a multi-variate power erie a n n f f Y f X, X x x x... n X R 0 i x Ri, 0 i xr x X XR j Ri X, X 0 R Ri Ri Rj j R If the random variable are mall compared to the nominal variable n f Y f X, X R 0 x X i xri X, XR0 R Ri If the random variable are uncorrelated, it follow that n f Y x i xri X, X 0 n f Y x Y Y i xri X, X 0 R R Ri Ri

8 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 Will focu on tranformation that map paband to paband and topband to topband

9 Filter Tranformation X I T X OUT T M f T f X I TM X OUT Claim: If the imaginary axi in the -plane i mapped to the imaginary axi in the -plane with a variable mapping function, the baic hape of the function T() will be preerved in the function F(T()) but the frequency axi may be warped and/or folded in the magnitude domain Preerving baic hape, in thi context, contitute maintaining feature in the magnitude repone of F(T()) that are in T() including, but not limited to, the peak amplitude, number of ripple, peak of ripple,

10 Example: Shape Preervation A Original Function B C A Shape Preerved B C A B Shape Preerved C

11 Example: Shape Preervation A Original Function B C A B Shape ot Preerved C

12 Flat Paband/Stopband Filter T j T j Lowpa Bandpa T j T j Highpa Bandreject

13 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 Will focu on tranformation that map paband to paband and topband to topband

14 LP to BP Filter Tranformation X I TLP X OUT f BP LP T T f X I TBP X OUT TLP j TBP j Lowpa Bandpa Will conider rational fraction mapping f = ot all rational fraction mapping will map Im axi to the Im axi ot all rational fraction mapping will map paband to paband and topband to topband Conider only that ubet of thoe mapping with thee propertie m T i=0 n T i=0 a Ti b Ti i i

15 LP to BP Tranformation Mapping Strategy: Conider firt a mapping to a normalized BP approximation T j T j LP BP f BW ormalized - A B BW B A A B

16 LP to BP Tranformation Mapping Strategy: Conider firt a mapping to a normalized BP approximation A mapping from f() will map the entire imaginary axi in the frequency domain Thu, mut conider both poitive and negative frequencie. Since function of, the magnitude repone on the negative axi will be a mirror image of that on the poitive axi T j T j f BP T j i a BW BW ormalized - - A - B - A B BW B A A B

17 Standard LP to BP Tranformation ormalized LP to ormalized BP mapping Strategy: T j T j f BP BW BW ormalized - - A - B - A B BW B A A B T LP () T LP (f()) T BP () Variable Mapping Strategy to Preerve Shape of LP function: conider: -domain -domain map =j0 to = j map =j to =j B map = j to = j A T LP (f()) map =0 to = map = to = B map = to = A Thi mapping will introduce 3 contraint

18 Standard LP to BP Tranformation Mapping Strategy: T LP () T LP (f()) T BP () -domain map =0 to = j map =j to =j B map = j to = j A T LP (f()) -domain map =0 to = map = to = B map = to = A Conider variable mapping f = a a +a b +b T T T0 With thi mapping, there are 5 D.O.F and 3 mathematical contraint and the additional contraint that the Im axi map to the Im axi and map PB to PB and SB to SB Will now how that the following mapping will meet thee contraint f + BW T or equivalently T0 + BW Thi i the lowet-order mapping that will meet thee contraint and it double the order of the approximation

19 Standard LP to BP Tranformation Verification of mapping Strategy: + BW j + BW 0 0 j B B B j j j j j BW j domain map =0 to = j map =j to =j B map = j to = j A T LP (f()) B B A B A B B j B A A A j j -j j j BW j A B A A B A A ja -domain map =0 to = map = to = B map = to = A B A Mut till how that the Im axi map to the Im axi and map PB to PB and SB to SB

20 Standard LP to BP Tranformation Verification of mapping Strategy: Image of Im axi: The mapping -domain map =0 to = j map =j to =j B map = j to = j A olving for, obtain + BW + j = BW T LP (f()) + BW -domain map =0 to = map = to = B map = to = A j BW ± BW j - 4 BW ± BW + 4 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

21 Standard LP to BP Tranformation The tandard LP to BP tranformation + BW If we add a ubcript to the LP variable for notational convenience, can expre thi mapping a x + BW Quetion: I thi mapping dimenionally conitent? The dimenion of the contant in the numerator mut be et o that thi i dimenionally conitent The dimenion of BW mut be et o that thi i dimenionally conitent

22 T LP () Standard LP to BP Tranformation TLP j - + BW TBP j T BP () BW BW - A - B - A B

23 Standard LP to BP Tranformation Frequency and -domain Mapping (ubcript variable in LP approximation for notational convenience) T LP ( x ) X + BW T BP () X X + BW - BW BW ± BW - 4 X X olving for or BW ± BW 4 X X Exercie: Reolve the dimenional conitency in the lat equation

24 T LP () Standard LP to BP Tranformation Denormalized Mapping TLP j - + BW M TBP j T BP () BW BW A - A - B - M M B

25 Standard LP to BP Tranformation Frequency and -domain Mapping - Denormalized (ubcript variable in LP approximation for notational convenience) T LP ( x ) X + BW M T BP () X X +M BW - BW M BW ± BW - 4 X X M BW ± BW 4 X X M Exercie: Reolve the dimenional conitency in the lat equation

26 Standard LP to BP Tranformation Frequency and -domain Mapping - Denormalized (ubcript variable in LP approximation for notational convenience) T LP ( x ) W T LP () T LP ( x ) X + BW T BP ( ) T LP ( x ) X + BW M T BP () + BW T BP () W T BP () All three approache give ame approximation Which i mot practical to ue? Often none of them!

27 Standard LP to BP Tranformation Frequency and -domain Mapping - Denormalized (ubcript variable in LP approximation for notational convenience) T LP ( x ) X + BW T BP ( ) Often mot practical to yntheize directly from the T BP and then do the frequency caling of component at the circuit level rather than at the approximation level

28 Standard LP to BP Tranformation Frequency and -domain Mapping (ubcript variable in LP approximation for notational convenience) Pole and Zero of the BP approximation X f + BW olving for 0 T p LP x f X X T T f BP LP TLP f p 0 T p T f p BP LP 0 BW ± BW - 4 Since thi relationhip map the complex plane to the complex plane, it alo map the pole and zero of the LP approximation to the pole and zero of the BP approximation

29 Standard LP to BP Tranformation Pole Mapping Claim: With a variable mapping tranform, the variable mapping naturally define the mapping of the pole of the tranformed function T LP ( x ) p X p + p BW X + BW T BP () p p BW ± BW p - 4 X X Exercie: Reolve the dimenional conitency in the lat equation

30 Standard LP to BP Tranformation Pole Mapping p p BW ± BW p - 4 X 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

31 Standard LP to BP Tranformation Pole Mapping,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

32 Standard LP to BP Tranformation Pole Mapping p p BW ± BW p - 4 X X Im Im Re Re multipliity 6 ote doubling of pole, addition of zero, and likely Q enhancement

33 End of Lecture 6