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Unformatted text preview: Realization of High Order Filters
HighOrder
Part I Boris Murmann
Stanford University
murmann@stanford.edu
Copyright © 2009 by Boris Murmann B. Murmann EE315A ― HO #5 1 Architectural Options
• Cascades of (active) first and secondorder sections (biquads) • Ladder filters (passive or emulated using active components)
(p
g
p
) B. Murmann EE315A ― HO #5 2 Example1: WCDMA Receiver
1 0.5 A B 0 0.5 [Yee, ESSCIRC 2000]
[Yee
1
1 A B. Murmann 0.5
Real 0 B EE315A ― HO #5 3 Example 2: CDMA/GPS Receiver • Channel select filters
(CSF)
– 640 kHz passband,
p
,
lowpass
– 0.5 dB passband ripple
– > 40 dB stopband
attenuation at 900 kHz • 5th order elliptical filter • Phase distortion can be
tolerated in this application Lim et al., “A Fully Integrated DirectConversion Receiver for
CDMA and GPS Applications,” IEEE JSSC, Nov. 2006
Applications, B. Murmann EE315A ― HO #5 4 Matlab Synthesis Result
204155.1855 (s^2 + 2.786e013) (s^2 + 5.715e013)
(s+1.89e006) (s^2 + 2.217e006s + 1.034e013) (s^2 + 5.315e005s + 1.664e013) (s^2/2.786e013 + 1) (s^2/5.715e013 + 1)
= (s/1.89e006 + 1) (s^2/1.034e013 + s/4.6640e+006 + 1) (s^2/1.664e013 + s/3.1308e+007 + 1) 0.5
Magnitude [dB
B] 0 10
20
30
40 1
1.5
2
2.5 50 4
10 5 10
10
Frequency [Hz] 6 10 7 3 5 10
Frequency [Hz] EE315A ― HO #5 B. Murmann 10 5 Pole and Zero Locations
1.5
1
I m a g [M H z ] Magnitude [dB
B] 0 ωP QP p1,2 176.45 ± j4.8030 kHz 511.68 kHz 1.4499 300.80 kHz
± j1203.2 kHz z3,4 1 7.6748 z1,2 0.5 649.21 kHz p5 0 42.30 ± j6.4783 kHz p3,4 0.5 ± j840.1 kHz 1.5
0.6 0.4 0.2 0
Real [MHz] B. Murmann EE315A ― HO #5 6 6 Pairing Options for p1,2 (HighQ)
• Pairing with nearby zero • (s^2/2.786e013 + 1)
(s^2/1.664e013 + s/3.1308e+007 + 1) (s^2/5.715e013 + 1)
(s^2/1.664e013 + s/3.1308e+007 + 1) 10 10
M
Magnitude [d
dB] M
Magnitude [d
dB] Pairing with remote zero 0
10
0
20 4
10 10 5 10
f [Hz] B. Murmann 6 10 7 0
10
10
20 4
10 10 5 10 6 10 f [Hz]
EE315A ― HO #5 7 PoleZero Pairing
• Pairing highQ poles with nearby zeros is desirable from a
dynamic range perspective
– Say that the amplifier at the output of the biquad can handle
a maximum signal of 1 Vpeak
– If the biquad response peaks 20 dB above unity, this means
that we can only process inputs with 100 mV amplitude near
the frequency of the peak (which lies in the passband)
– The signal is therefore reduced relative to the thermal noise
of the circuit, which is highly undesirable ± 1V B. Murmann EE315A ― HO #5 8 7 Response of the Individual Sections
Firstorder section LowQ biquad HighQ biquad 5 5 0
5
10
15
20 4
10 • M
Magnitude [dB] 10 M
Magnitude [dB] 10 5
M
Magnitude [dB] 10 0
5
10
15 10
f [Hz] 6 20 4
10 0
5
10
15 10
f [Hz] 6 20 4
10 10
f [Hz] 6 In which order should we cascade these sections?
EE315A ― HO #5 B. Murmann 9 Biquad Ordering
In Out Ordering the Biquads from lowQ to highQ generally yields “smooth” transfer functions from the
input to the intermediate nodes, and often helps minimize harmonic distortion, but the output will
have significant noise peaking near the corner frequency due to the last stage with highQ. In Out Reversing the ordering will allow the later stages to filter out the noise peaking near corner
frequency. May also filter out harmonics (but not intermodulation).
In practical filter design, it would be worthwhile giving some thoughts to the options that you
may have for the ordering of the biquads. In a nonlowpass filter application, inherent accoupling may also be used to your advantage to suppress offset accumulation.
(Some good systemlevel discussions in Schaumann/Ghausi/Laker.)
B. Murmann [Unku Moon]
10 Intermediate Outputs for LowQ 0
5 10 4
10 • 5 10
f [Hz] 10 6 5
Magnit
tude [dB] 5
Magnit
tude [dB] Magnit
tude [dB] 5 0 5 10 4
10 5 10
f [Hz] 10 0 5 10 4
10 6 5 10
f [Hz] 10 6 This d i i
Thi ordering is most frequently used in practice
tf
tl
di
ti
EE315A ― HO #5 B. Murmann 11 Intermediate Outputs for HighQ 5
0
5
5
10 4
10 5 10
f [Hz] 10 6 LowQ 10
Magnitude [dB
B] 10
Magnitude [dB
B] Magnitude [dB
B] 10 • HighQ 5
0
5
5
10 4
10 5 10
f [Hz] 10 6 5
0
5
5
10 4
10 5 10
f [Hz] 10 6 At first glance this looks bad, but the noise from the highQ
biquad is filtered before it reaches the output
– We will revisit this situation in the context of noise analysis B. Murmann EE315A ― HO #5 12 Dynamic Range Scaling
• Suppose we decided that the second ordering is what we want
to use for our design • In this case, we need to think about a proper gain distribution
that avoids “clipping” in the individual amplifiers • For this purpose, we introduce gain scale factors for each
purpose
section, while keeping the overall gain constant (K1K2K3 = 1 in
this example) (s^2/5.715e013 + 1)
(s^2/2.786e013 + 1)
 (s^2/1.664e013 + s/3.1308e+007 + 1) (s^2/1.034e013 + s/4.6640e+006 + 1) 1
(s/1.89e006 + 1) K2*(s^2/5.715e013 + 1)
K1*(s^2/2.786e013 + 1)
 (s^2/1.664e013 + s/3.1308e+007 + 1) (s^2/1.034e013 + s/4.6640e+006 + 1) K3
(s/1.89e006 + 1) EE315A ― HO #5 B. Murmann 13 Analysis (1)
ˆ
Vmax 1⋅ H2 1⋅ H1 1⋅ H 3 • Suppose we chose K1=K2=K3=1 and assume that we will apply
K K 1
single sine waves with arbitrary frequencies to the input • Since H1 has significant peaking (H1max ≅ 3.19 ≅ 10 dB), we can
guarantee proper operation only for input amplitudes up to
ˆ
Vmax
H1 max • e.g. 1
V
= 321mV
3.12 Since the overall gain is unity (with no peaking above 1), this
means Vout swings only 312mV, meaning that we are “wasting”
available signal range B. Murmann EE315A ― HO #5 14 Analysis (2)
• A more desirable outcome may be to scale K1, K2, K3 such that all
stages utilize the maximum available swing as the input tone is
swept across all frequencies
– Note that in general, the maximum output swings for each
stage may not occur at the same frequency ˆ
Vmax K 2H2 K1H1 ˆ
Vmax K 3H3 ˆ
Vmax EE315A ― HO #5 B. Murmann 15 Analysis (3)
• This is achieved for
K1 H1 max = K1K 2K 3 H1H2H3 max
K1K 2 H1H2 max = K1K 2K 3 H1H2H3 max • In our example
K1K 2K 3 = 1 H1 max = 3.19 H1H2 max = 2.3 H1H2H3 max = 1 and therefore
K1 = B. Murmann 1
H1 max = 1
3.19 K2 = 1
3.19
=
K1 H1H2 max
2.3 EE315A ― HO #5 K3 = 1
3.19 ⋅ 2.3
=
K1K 2
3.19 16 Intermediate Outputs After DR Scaling 0 5 5 5 10 15
15 20 4
10 B. Murmann Magnitude [dB]
e 0 Magnitude [dB]
e Magnitude [dB]
e 0 10 15
15 5 10
f [Hz] 10 6 10 15
15 20 4
10 5 10
f [Hz] 10 6 20 4
10 5 10
f [Hz] 10 6 EE315A ― HO #5 17 Arguments Against “Sinusoidal” DR Scaling • If the input signal is wideband (as in many telecommunication
p
g
(
y
systems), the node with peaking may not saturate due to limited
signal power in that frequency region
– May want to optimize the g
y
p
gain distribution based on a p
power
spectral density “template” of the incoming signal • Aligning the peaks for each output perfectly will require noninteger component ratios
– But we may want to use integer ratios to improve matching • For a discussion on why sinusoidal dynamic range scaling may
not always the best choice, see Behbahani, JSSC 4/2000 B. Murmann 18 Expressions for Implementation
1.3865*(s^2/5.715e013 + 1)
0.3133*(s^2/2.786e013 + 1)
 (s^2/1.664e013 + s/3.1308e+007 + 1) (s^2/1.034e013 + s/4.6640e+006 + 1) 2.3021
(s/1.89e006 + 1) 2 T Th
2x TowThomas Vo 2 b2s 2 + b1s + b0
= 2
Vin
s + a1s + a0 b1 = 0 EE315A ― HO #5 B. Murmann 19 TowThomas Component Values (b1=0)
R1 = 1
a1C1 R4 = 1 1 1
k2 a1b2 C1 ωZ = R2 = R6
R3R5R7C1C2 k1 R3 = a0 C2
R5 = k1 a0
b0C2 ωP = 1
k1k 2
R6 = R8
R2R3R7C1C2 1
a0 C1
R8
b2 R7 = k 2R8 QP = ωP R1C1 • a0, a1, b0, b1, b2 are known; can pick k1, k2, C1, C2 and R8
;
p • Reasonable starting values
– k1 = k2 = 1
– Set C1 = C2 to a reasonable value that is easily
implemented, e.g. 1pF
– Set R8 to a reasonable value that is easily implemented and
represents an integer multiple or fraction of R2, R3 or R7
B. Murmann EE315A ― HO #5 20 Implementation Steps
• First cut component calculation using reasonable starting values
for k1, k2, C1, C2 and R8 • Dynamic range scaling of internal amplifier outputs by adjusting
k1 and k2 • Thermal noise scaling using ideal amplifiers
– Increase all capacitors and reduce all resistors until noise
specification is met
p • Design amplifiers • Repeat thermal noise scaling to accommodate amplifier noise • Analyze sensitivity to component variations and devise tuning
mechanism (if needed) EE315A ― HO #5 B. Murmann 21 Dynamic Range Scaling of Internal Nodes −k 2 ( b2a1 − b1 ) s + ( b2a0 − b0 )
s + a1s + a0
2 b2s 2 + b1s + b0
s 2 + a1s + a0 − • 1 ( b0 − b2a0 ) s + ( a1b0 − a0b1 )
s 2 + a1s + a0
k1 a0 Scale k1 and k2 such that peak magnitude at Vo1 and Vo2
corresponds to maximum available amplifier swing
p
p
g B. Murmann EE315A ― HO #5 22 Sensitivity Analysis
• Ideally, we would like to have an
analytical expression th t
l ti l
i that
relates “interesting points” of the
response to variations in all
components
– E.g. calculate variations in
the passband ripple as a
function of the percent error
in R2 • This is almost impossible or at
least impractical to do in
practice 0 M
Magnitude [dB
B] 0.5
1
1.5
2
2.5
3 5 10
Frequency [Hz] B. Murmann 10 6 EE315A ― HO #5 23 Sensitivity Analysis – Monte Carlo
• Monte Carlo Analysis
– Have a statistical model for all components
– Run a large number of simulations (Matlab or Spectre) to
capture many statistical outcomes and create overlay plot
from all runs MAXIM APPLICATION NOTE 738: Minimizing
ComponentVariation Sensitivity in Single Op Amp Filters
http://www.maximic.com/appnotes.cfm/an_pk/738/ • Such an analysis is very useful for validation, but perhaps too
much work intuition building and/or design guidance B. Murmann EE315A ― HO #5 24 Basic Sensitivity Analysis
• Say we just want to get a basic feel for the sensitivities • Look at impact of
p
– Global process variations
– Component mismatch • For global process variations, we have already seen that
ωZ = R6
1
∝
R3R5R7C1C2
RC ωP = R8
1
∝
R2R3R7C1C2
RC QP = ωP R1C1 ∝ 1 • If all R and C vary by the same percentage, the filter “shape” is
preserved and shifted back and forth along the frequency axis • If this is a problem for the application, we can “tune” either R or
C to b g the filter response bac to the des ed location
o bring e e espo se back o e desired oca o
EE315A ― HO #5 B. Murmann 25 Mismatch Analysis
ωZ = R6
1
∝
R3R5R7C1C2
RC ωP = R8
1
∝
R2R3R7C1C2
RC QP = ωP R1C1 ∝ 1 • Suppose we had resistors and capacitors that deviate from their
nominal component value (which is subject to global variations)
by a standard deviation of 1% • Since
ω
ω
ω
ω
ωP
ω
SRP = SRP = SRP = −SR P = SC1 = SC P = −
2
3
3
8
2 this means σ Δω P / ωP = 1
2 1
6 ⋅ 1% = 1.22%
1 22%
2 3σ ΔωP /ωP = 3.67% ≅ 4% B. Murmann EE315A ― HO #5 26 Passband with Pole Errors (1)
• ± 4% change in ωP of first order section
4
+4%
4% Magnitude [dB] 2
0
2
4
6
8
8
10 4
10 5 10
Frequency [ ]
q
y [Hz] 10 6 EE315A ― HO #5 B. Murmann 27 Passband with Pole Errors (2)
• ± 4% change in ωP of lowQ section
4 Magni
itude [dB] 2
0
2
4
6
8
8
10 4
10 B. Murmann Worse. +4%
4%
4%
5 10
Frequency [Hz] EE315A ― HO #5 10 6 28 Passband with Pole Errors (3)
• ± 4% change in ωP of highQ section
4 Magnit
tude [dB] 2
0
2
4
6
8
10 4
10 B. Murmann Bad ! +4%
4%
4%
5 10
Frequency [Hz] EE315A ― HO #5 10 6 29 ...
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This note was uploaded on 08/13/2009 for the course EE 315 taught by Professor Borismurmann during the Spring '09 term at Stanford.
 Spring '09
 BorisMurmann

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