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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
 2R 
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
TMs Tf 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