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EE 508 Lecture 15 Filter Transformations Lowpass to Bandpass Lowpass to Highpass Lowpass to Band-reject Review from Last Time Theorem: If the perimeter variations and contact resistance are neglected, the standard deviation of the local random variations of a resistor of area A is given by the expression A R = RN A Theorem: If the perimeter variations are neglected, the standard deviation of the local random variations of a capacitor of area A is given by the expression C = CN AC A Theorem: If the perimeter variations are neglected, the variance of the local random variations of the normalized threshold voltage of a rectangular MOS transistor of dimensions W and L is given by the expression 2 VT VTN A 2VTO 2 VTN WL or as 2 VT VTN A 2VT WL Review from Last Time Theorem: If the perimeter variations are neglected, the variance of the local random variations of the normalized COX of a rectangular MOS transistor of dimensions W and L is given by the expression 2 COX COXN 2 ACOX WL Theorem: If the perimeter variations are neglected, the variance of the local random variations of the normalized mobility of a rectangular MOS transistor of dimensions W and L is given by the expression 2R N A 2 WL where the parameters AX are all constants characteristic of the process (i.e. model parameters) • The effects of edge roughness on the variance of resistors, capacitors, and transistors can readily be included but for most layouts is dominated by the area dependent variations • There is some correlation between the model parameters of MOS transistors but they are often ignored to simplify calculations Review from Last Time Statistical Modeling of dimensionless parameters - example K = 1+ R2 R1 VOUT R2 R1 Assume common centroid layout Aρ=.01µm area of R1 is 100u2 VIN Determine the yield if the nominal gain is 10 1% K N 1, 0.00095 KN 9.9 < K < 10.1 .99 < K < 1.01 KN K 1 KN N 0,1 0.00095 K -1 KN -10< < 10 .00095 The gain yield is essentially 100% -.01< K -1< .01 KN Could substantially decrease area or increase gain accuracy if desired Review from Last Time Statistical Modeling of dimensionless parameters - example K = 1+ R2 R1 VOUT Aρ=.025um AR1=10um2 VIN R PROC Determine the yield if the nominal gain is 10 1% K N 1, 0.0075 KN 9.9 < K < 10.1 .99 < -.01< K < 1.01 KN K -1< .01 KN R2 R1 0.2 RNOM K 1 KN N 0,1 0.0075 K -1 KN -1.33< < 1.33 .0075 Y = 2FN(0,1) 1.33 -1 = 2*.9082-1 = 0.8164 Dramatic drop from 100% yield to about 82% yield! Review from Last Time Statistical Modeling of Filter Characteristics The variance of dimensioned filter parameters (e.g. ω0, poles, band edges, …) is often very large due to the process-level random variables which dominate The variance of dimensionless filter parameters (e.g. Q, gain, …) are often quite small since in a good design they will depend dominantly on local random variations which are much smaller than process-level variations The variance of dimensionless filter parameters is invariably proportional to the reciprocal of the square root of the relevant area and thus can be managed with appropriate area allocation Review from Last Time Linearization of Functions of a Random Variable • Characteristics of most circuits of interest are themselves random variables • Relationship between characteristics and the random variables often highly nonlinear • Ad Hoc manipulations (repeated Taylor’s series expansions) were used to linearize the characteristics in terms of the random variables Y Y a x • This is important because if the random variables are uncorrelated the variance of the characteristic can be readily obtained n N i Ri i 1 Y2 ai2 x2 n i 1 2Y YN Ri 1 n 2 2 ai xRi YN2 i 1 • This approach was applicable since the random variables are small • These Ad Hoc manipulations can be formalized and this follows Review from Last Time Formalization of Statistical Analysis Consider a function of interest Y Y f x1N , x2 N ,...xnN ,: x1R , x2 R ,...xnR f X , X N R This can be expressed in a multi-variate power series as Y f X N f , X R X R 0 i 1 x Ri n n 2 f xRi i 1 xR j xRi X N , X R 0 j 1 n xRi xRj .... X N , X R 0 If the random variables are small compared to the nominal variables Y f X , X N R n f X R 0 i 1 xRi xRi X N , X R 0 If the random variable are uncorrelated, it follows that 2 f x2Ri Y2 i 1 xRi X , X 0 N R 2 n 1 f 2 2 xRi Y 2 YN i 1 xRi X , X 0 YN N R n Filter Transformations Lowpass to Bandpass Lowpass to Highpass Lowpass to Band-reject (LP to BP) (LP to HP) (LP to BR) Approach will be to take advantage of the results obtained for the standard LP approximations Will focus on flat passband and zero-gain stop-band transformations Will focus on transformations that map passband to passband and stopband to stopband Filter Transformations XIN T s XOUT s f s TMs Tf s XIN TM s XOUT Claim: If the imaginary axis in the s-plane is mapped to the imaginary axis in the s-plane with a variable mapping function, the basic shape of the function T(s) will be preserved in the function F(T(s)) but the frequency axis may be warped and/or folded in the magnitude domain Preserving basic shape, in this context, constitutes maintaining features in the magnitude response of F(T(s)) that are in T(s) including, but not limited to, the peak amplitude, number of ripples, peaks of ripples, Example: Shape Preservation A Original Function B C ω ω A Shape Preserved B C ω A B Shape Preserved C ω Example: Shape Preservation A Original Function B C ω A Shape Not Preserved B C ω Flat Passband/Stopband Filters T j T j Lowpass ω Bandpass ω T j T j ω ω Highpass Bandreject Filter Transformations Lowpass to Bandpass Lowpass to Highpass Lowpass to Band-reject (LP to BP) (LP to HP) (LP to BR) • Approach will be to take advantage of the results obtained for the standard LP approximations • Will focus on flat passband and zero-gain stop-band transformations • Will focus on transformations that map passband to passband, stopband to stopband, and Im axis to Im axis LP to BP Filter Transformations TLPN s XIN XOUT s f s TBP s XIN XOUT TBP s TLPN f s T j T j BP LP Lowpass ω Bandpass ω mT Will consider rational fraction mappings f s = a Ti si b Ti si i =0 nT i =0 • Not all rational fraction mappings will map Im axis to the Im axis • Not all rational fraction mappings will map passband to passband and stopband to stopband • Consider only that subset of those mappings with these properties LP to BP Transformation Mapping Strategy: Consider first a mapping to a normalized BP approximation T LPN j T BPN 1 s f s j 1 BWN Normalized -1 1 ω ωAN ωBN 1 BW N BN AN 1 AN BN ω LP to BP Transformation Mapping Strategy: Consider first a mapping to a normalized BP approximation A mapping from s → f(s) will map the entire imaginary axis Thus, must consider both positive and negative frequencies. Since T jω is a function of ω2, the magnitude response on the negative ω axis will be a mirror image of that on the positive ω axis T j T j BP s f s 1 1 BWN BWN Normalized -1 1 ω - ωAN -1 - ωBN ωAN ωBN 1 BW N BN ω AN 1 AN BN Standard LP to BP Transformation Normalized LP to Normalized BP mapping Strategy: T j T j BP s f s 1 1 BWN BWN Normalized 1 -1 ω - ωAN -1 - ωBN ωAN ωBN 1 BW N BN ω AN 1 AN TLPN(s) TLPN(f(s)) BN TBPN(s) Variable Mapping Strategy to Preserve Shape of LP function: consider: s-domain ω-domain map s=j0 to s= j1 TLPN(f(s)) map ω=0 to ω=1 map s=j1 to s=jωBN map ω=1 to ω=ωBN map s= –j1 to s= jωAN map ω= –1 to ω=ωAN This mapping will introduce 3 constraints Standard LP to BP Transformation Mapping Strategy: TLPN(s) TLPN(f(s)) TBPN(s) s-domain ω-domain map s=0 to s= j1 map s=j1 to s=jωBN map s= –j1 to s= jωAN TLPN(f(s)) map ω=0 to ω=1 map ω=1 to ω=ωBN map ω= –1 to ω=ωAN aT2s2 aT1s+aT0 f s = bT1s+bT0 Consider variable mapping With this mapping, there are 5 D.O.F and 3 mathematical constraints and the additional constraints that the Im axis maps to the Im axis and maps PB to PB and SB to SB Will now show that the following mapping will meet these constraints s +1 f s s•BW N s +1 s s•BW 2 2 or equivalently N This is the lowest-order mapping that will meet these constraints and it doubles the order of the approximation Standard LP to BP Transformation s-domain ω-domain map s=0 to s= j1 map s=j1 to s=jωBN map s= –j1 to s= jωAN TLPN(f(s)) map ω=0 to ω=1 map ω=1 to ω=ωBN map ω= –1 to ω=ωAN s +1 s s•BW 2 Verification of mapping Strategy: N s +1 s•BW 2 N 0 j1 AN ω -1 ω -1 j j j ω -1 ω -ω ω AN ω -1 ω -1 j j -j 1-ω ω ω -ω 1-ω s +1 s•BW jω jω ω -ω 2 2 BN N 0 j1 BN BN 2 2 BN BN 2 2 BN AN BN j1 jω BN BN BN 1-ω s +1 s•BW jω jω ω -ω 2 2 AN N AN AN BN AN 2 2 AN AN BN 2 2 AN AN j1 jω Must still show that the Im axis maps to the Im axis and maps PB to PB and SB to SB AN Standard LP to BP Transformation s-domain ω-domain map s=0 to s= j1 map s=j1 to s=jωBN map s= –j1 to s= jωAN TLPN(f(s)) map ω=0 to ω=1 map ω=1 to ω=ωBN map ω= –1 to ω=ωAN s +1 s s•BW 2 Verification of mapping Strategy: N s +1 jω = s•BW 2 Image of Im axis: N solving for s, obtain s jω•BWN ± BW jω - 4 N 2 2 ω•BW ± N j ω + 4 N 2 BW 2 this has no real part so the imaginary axis maps to the imaginary axis Can readily show this mapping maps PB to PB and SB to SB s +1 s s•BW 2 The mapping N is termed the standard LP to BP transformation Standard LP to BP Transformation s +1 s s•BW 2 The standard LP to BP transformation N If we add a subscript to the LP variable for notational convenience, can express this mapping as s +1 s s•BW 2 x N Question: Is this mapping dimensionally consistent ? • The dimensions of the constant “1” in the numerator must be set so that this is dimensionally consistent • The dimensions of BWN must be set so that this is dimensionally consistent Standard LP to BP Transformation T LP j 1 TLPN(s) ω 1 -1 s s +1 s•BW 2 N T BP j 1 TBPN(s) BWN BWN - ωAN -1 - ωBN ωAN 1 ωBN ω Standard LP to BP Transformation Frequency and s-domain Mappings (subscript variable in LP approximation for notational convenience) s2 +1 sX s•BWN TLPN(sx) ω2 -1 ωX ω•BWN sX solving for s or ω s2 +1 s•BWN s TBPN(s) ω s X •BWN ± BWN s X - 4 2 ω X •BWN ± BWN ω X 4 2 2 2 Exercise: Resolve the dimensional consistency in the last equation Standard LP to BP Transformation Denormalized Mapping T LP j 1 TLPN(s) ω 1 -1 s s 2 +ω M2 s•BW T BP j 1 TBP(s) BW BW - ωA - ωM - ωB ωA ωM ωB ω Standard LP to BP Transformation Frequency and s-domain Mappings - Denormalized (subscript variable in LP approximation for notational convenience) s2 +ωM2 sX s•BW ω2 - ωM2 ωX ω•BW TLPN(sx) sX s2 +ωM2 s•BW s TBP(s) ω s X •BW ± BW s 2 X - 4ωM2 2 ω X •BW ± BW ω X 2 4ωM2 2 Exercise: Resolve the dimensional consistency in the last equation Standard LP to BP Transformation Frequency and s-domain Mappings - Denormalized (subscript variable in LP approximation for notational convenience) TLPN(sx) sN s WN TLP(s) s 2 s +1 s•BWN TBP(s) TLPN(sx) TLPN(sx) sX sX sN2 +1 sN •BWN s 2 +ωM2 s•BW TBP(s) TBPN(sN) sN s WN TBP(s) All three approaches give same approximation Which is most practical to use? Often none of them ! Standard LP to BP Transformation Frequency and s-domain Mappings - Denormalized (subscript variable in LP approximation for notational convenience) TLPN(sx) sX sN2 +1 sN •BWN TBPN(sN) Often most practical to synthesize directly from the TBPN and then do the frequency scaling of components at the circuit level rather than at the approximation level Standard LP to BP Transformation Frequency and s-domain Mappings (subscript variable in LP approximation for notational convenience) Poles and Zeros of the BP approximations sX f s2 +1 s•BWN f 1 s s X •BWN ± solving for s BWN sX - 4 2 2 TBP s TLPN f s TLPN px 0 TLPN f p 0 TBP p TLPN f p 0 Since this relationship maps the complex plane to the complex plane, it also maps the poles and zeros of the LP approximation to the poles and zeros of the BP approximation Standard LP to BP Transformation Pole Mappings Claim: With a variable mapping transform, the variable mapping naturally defines the mapping of the poles of the transformed function p2 +1 pX p•BWN TLPN(sx) sX s2 +1 s•BWN TBPN(s) p p X •BWN ± BW N pX - 4 2 2 Exercise: Resolve the dimensional consistency in the last equation Standard LP to BP Transformation Pole Mappings p p X •BWN ± BWN p X - 4 2 2 ω0BPH,QLBPH ω0BPL ,QLBPL Im Im ω0LP ,QLP Re Re Image of the cc pole pair is the two pairs of poles Standard LP to BP Transformation Pole Mappings ω0BPH,QLBPH ω0BPL ,QLBPL Im Im ω0LP ,QLP Re Re Can show that the upper hp pole maps to one upper hp pole and one lower hp pole as shown. Corresponding mapping of the lower hp pole is also shown Standard LP to BP Transformation Pole Mappings p p X •BWN ± BW pX - 4 N 2 2 Im Im Re Re multipliity 6 Note doubling of poles, addition of zeros, and likely Q enhancement End of Lecture 15