system on a chip - montefiore institute€¦ · differential second order filter (biquad) transfer...
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System on a Chip
Prof. Dr. Michael Kraft
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Lecture 4:Filters
Filters– General Theory– Continuous Time Filters
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Filters are used to separate signals in the frequency domain, e.g. remove noise, tune to a radio station, etc
5 types of filter– Low pass
– High pass
– Band pass
– Band stop/reject
– All pass
Background
w
|T(jw)|
w
|T(jw)|
w
|T(jw)|
w
|T(jw)|
w
w
|T(jw)|
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Ideal Filter
Brick-wall filters do not exist in reality Real filters can approximate brick-wall filters as close as
required by the filter specification
|T(jw)|
w
Pass-band
Stop-band
Brick-wall LP filter
|T(jw)|
w
LowerStop-band
Pass-band
UpperStop-band
wp
Brick-wall BP filter
wp1 wp2
1 1
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“Real” LP Filter
|T(jw)| [dB]
w
0
Pass-band
Stop-band
Transition-band
wp ws
Amax
maximum deviation from 0dB;“Bandpass ripple”
wp/ws: Measure for filter sharpness;filter selectivity
Aminminimum attenuation
Parameters required for filter synthesis: Amax, wp, Amin, ws
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Filter Types Using Biquads
01
2
2
01
2
2)(bsbsb
asasaKsT
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Biquadratic LP Transfer Function
2
00
2
2
0
)/()(
ww
w
sQsKsTLow-pass biquad TF
Magnitude Response Phase Response
- Diagrams normalized to w0 = K = 1- Asymptotic fall is -40 dB/dec
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Biquad Block Diagram
2
00
2
2
0
)/()(
)()(
ww
w
sQsK
sV
sVsT
IN
LP
-w0/s -w0/s+
-1/Q
- -K
Vin VLP
VBPVHP
(K either pos. or neg.)
Universal Active Filter: realizes LP, HP, and BP
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Tow-Thomas BiquadRealization
-
+-
+-
+R3
R2
C1
R1C2
R4R5
R5
w02 = 1/R3R4C1C2
Q = Sqrt(R12C1/R2R4C2)
K = -R2/R3
(R5 arbitrarily chosen)
Vin
VLP
VLP’
VLP’: Non-inverting LP Filter
VLP: Inverting LP Filter
VBP
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N-th Order Filter
)())((
)())(()(
21
21
N
M
pspsps
zszszsKsT
Number of poles determines order
zeros are obviously placed in stopband for stability: MN; N-M zeros at w = for stability: Re{pi} < 0 no general optimisation algorithms known
)())((
1)(
21 NpspspsKsT
Special Case: all zeros at w = ; all pole filter
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Real Axis
Imag A
xis
Pole-zero map
-500 -400 -300 -200 -100 0 100-2000
-1500
-1000
-500
0
500
1000
1500
2000
Example: 5-th Order Filter
1411293645
14284
1047.21064.81058.11017.21500
103.21044.150)(
sssss
sssT
Poles: P[1..5] = 1.0e+002 *[-3.0000 + 7.0000i
-3.0000 - 7.0000i
-2.0000 + 9.0000i
-2.0000 - 9.0000i
-5.0000 ]
Zeros: Z[1..4] = 1.0e+003 *[0 + 1.2000i
0 - 1.2000i
0 + 1.8000i
0 - 1.8000i]
Passband
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Example: 5-th Order Filter
Time (sec.)
Am
plitu
de
Step Response
0 0.005 0.01 0.0150
0.2
0.4
0.6
0.8
1
1.2
1.4From: U(1)
To:
Y(1
)
1411293645
14284
1047.21064.81058.11017.21500
103.21044.150)(
sssss
sssT
Frequency (rad/sec)
Ph
ase
(d
eg
);
Ma
gn
itu
de
(dB
)
T(S
)
Bode Diagrams
-50
-40
-30
-20
-10
0
102
103
-300
-200
-100
0
-3
755
Z1/2 Z3/4
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Butterworth LP Filter
- N: Filter order
- All pole filter
w0: |T(jw)| has dropped by 3 dB
For w0=1:
N Denominator of T(s)1 (s+1)
2 (s2+1.1414s+1)
3 (s+1)(s2+s+1)
4 (s2+0.765s+1)(s2+1.848s+1)
5 (s+1)(s2+0.618s+1)(s2+1.618s+1)
NjT
2
0
2
1
1)(
w
ww
- Make T(jw) so that:
Normalized Butterworth Polynomials:
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BW-LP Frequency Response
Frequency (rad/sec)
|T(jw)| (dB)
maximally flat in passband i.e. the first 2N-1 derivatives of |T(jw)| are 0 at w=0
• |T(jw)| monotonically falling
• not steepest roll-off
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BW-LP Design
Design a filter so that in the passband |T(jw)| has fallen not more than by amax and in the stopband the minimum attenuation is amin
Find w0 and N
a/dB
amin
amax
wp wsw/rad/s
s
p
a
a
N
w
wlog2
110
110log
10/
10/
max
min
Na
p
2
110/
0
110 max
w
w
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BW Pole Locations
• Poles located on a circle around the origin
• k = 90 (2k + N - 1)/N k = 1,2,…,2N
• If N is odd, then there is a pole at = 0, if N is even there are poles at = 90 /N
• Poles are separated by = 180 /N
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Chebychev LP Filter
• N: Filter order• All pole filter• Normalized for w0 = 1• e: design parameter;
determines ripple
N e0.3493; (0.5 dB ripple) e0.5089; (1 dB ripple)1 (s+2.863) (s+1.965)
2 (s2+1.425s+1.516) (s2+1.098s+1.103)
3 (s+0.626)(s2+0.626s+1.142) (s+0.494)(s2+0.494s+0.994)
4 (s2+0.351s+1.064)(s2+0.845s+0.356) (s2+0.297s+0.987)(s2+0.674s+0.279)
5 (s+0.362)(s2+0.224s+1.036)(s2+0.586s+0.477) (s+0.289)(s2+0.179s+0.988)(s2+0.468s+0.429)
)(1
1)(
22
2
wew
NCjT
• Make T(jw) so that:
Chebychev Polynomials; Denominator of T(s):
1))(coscos()( 1 www forNCN
1))(coshcosh()( 1 www forNCN
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Frequency (rad/sec)
Magnitude (
dB
)
Bode Diagrams
-80
-60
-40
-20
0
1 100.1
CC-LP Frequency Response
N = 5; e = 0.5 dB
e = 3 dB
Properties: Ripples in Bandpass between w = 0 and w = 1/(1+e2)0.5
|H(j1)| = 1/(1+e2)0.5 for all N |H(0)| = 1 for N odd = 1/(1+e2)0.5 for N even steeper roll-off than
Butterworth Implementation: see
Butterworth example
Frequency (rad/sec)
Magnitude (
dB
)
Bode Diagrams
-80
-60
-40
-20
0
1 100.1
N = 6; e = 0.5 dB
e = 3 dB
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Chebychev Pole Locations
Poles located on an ellipse around the origin; narrow ellipse means poles closer to imag. axis larger ripples. Wider ellipse small ripples; approaches Butterworth filter
• sk = -sinh(1/N sinh-1(1/ e sin((2k-1)p/2N)
• wk = -cosh(1/N sinh-1(1/ e cos((2k-1)p/2N)
Real Axis
Imag A
xis
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
• Minor axis:
a = sinh(1/N sinh-1(1/e))
• Major axis:
b = cosh(1/N cosh-1(1/e))
a
b
N = 6; e = 1 dB
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Motivation
Switched Capacitor Filters– Pro: Accurate transfer-functions– Pro: High linearity, good noise performance– Con: Limited in speed
• Clock rate must be greater than twice the signal frequency
– Con: Requires anti-aliasing filters
Continuous-time filters– Con: Moderate transfer-function accuracy (requires tuning circuitry)– Con: Moderate linearity– Pro: High-speed– Pro: Good noise performance
Required building blocks:– Integrators, summers and gain stages
• Allow to realise any rational function, hence any integrated continuous-time filter
• Any rational transfer function with real-valued coefficients may be factored into first- and second-order terms
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First Order Filter
first-order continuous-time filter requires one integrator, one summer, and up to three gain elements
In general: One integrator is required for each pole in an analogfilter
block diagram of a first-order continuous-time filter
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Second Order Filter
Two integrators are required to realise the two poles For stability: w0/Q must be positive
– One integrator must have feedback around it, hence the integrator is “lossy”
– A large feedback coefficient w0/Q results in a very lossy integrator, hence the Q is low• Q<1/2: both poles are real; Q>1/2: poles are complex-conjugate pairs
block diagram of a second-order continuous-time filter
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Gm-C Integrators
Use a transconductor (or OPA) to build an integrator: 𝑖𝑜 = 𝐺𝑚𝑣𝑖 Output current is linearly related to input voltage Output impedance is ideally infinite OTA (operational transconductance amplifier) has a high Gm value
but is not usually linear
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Multiple Input Gm-C Integrators
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Example
What Gm is needed for an integrator having a unity gain frequencyof wti= 20 MHz when C=2 pF?
Or equivalently: Gm=1/3.98kW
This is related to the unity gain frequency by:
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Fully Differential Integrators
Use a single capacitor between differential outputs Requires some sort of common-mode feedback to set output
common-mode voltage Needs some extra caps for compensating common mode feedback
loop
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Fully Differential Integrators
Use two grounded capacitors Still requires common-mode feedback but compensation caps for
common-mode feedback can be the same grounded capacitors
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Fully Differential Integrators
Integrated capacitors have top and bottom plate parasitic capacitances
To maintain symmetry, usually 2 parallel caps used as shown above Note that parasitic capacitance affects time-constant and cause
non-linearity
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Gm-C Opamp Integrator
Use an extra Opamp to improve linearity and noise performance Also known as a “Miller Integrator” The gain of extra Opamp reduces the effect of parasitic capacitances Cross coupling of output wires to maintain positive integration
coefficient
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Gm-C Opamp Integrator
Advantages Effect of parasitic caps reduced by opamp gain — more accurate
time-constant and better linearity Less sensitive to noise since transconductor output is low
impedance (due to opamp feedback) cell drives virtual Gnd — output-impedance of Gm cell can be
lower and smaller voltage swing neededDisadvantages Lower operating speed because it now relies on feedback Larger power dissipation Larger silicon area
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First Order Filter
General first-order transfer-function:
Built with a single integrator and two feed-ins branches w0 sets the pole frequency
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Can show that the transfer function is given by (using a current equation at the output node):
Equating with the block diagram transfer function:
First Order Filter
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Same equations as single-ended case but cap sizes doubled
Can realize k1<0 by cross-coupling wires at Cx
Fully-Differential First-Order Filter
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Find fully-diff values when dc gain = 0.5, a pole at 20 MHz and a zero at 40MHz. Assume CA=2pF
So:
Example
K1=0.25, k0=2p×107, w0=4p×107
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Second Order Filter
Block diagram: see lecture on switched capacitor circuits– Modified to have positive integrators
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Differential Second Order Filter(Biquad)
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Differential Second Order Filter(Biquad)
Transfer function:
Note that there is a restriction on the high-frequency gain coefficient k2 as in the first-order case
Note that Gm3 sets the damping of this biquad Gm1 and Gm2 form two integrators with unity-gain frequencies of w0/s
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Example
Find values for a bandpass filter with a centre frequency of 20 MHz, a Q value of 5, and a centre frequency gain of 1
Assume CA = CB = 2 pF
where G =1 Is the gain at the center frequency
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Example
Since w0 = 2p × 20MHz and Q = 5, we find:
Since k0 and k2 are zero, we have Cx = CmA = 0 The transconductance values are:
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CMOS Tranconductors
A large variety of methods Best approach depends on application Two main classifications: triode or active transistor based
Triode vs. Active Triode based tends to have better linearity Active tend to have faster speed for the same operating current
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Triode Tranconductors
A large variety of methods Best approach depends on application Two main classifications: triode or active transistor based
Triode vs. Active Triode based tends to have better linearity Active tend to have faster speed for the same operating current
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Triode Tranconductors Recall n-channel triode equation
Conditions to remain in triode
or equivalently:
Above models are only reasonably accurate– Higher order terms are not modelled
Not nearly as accurate as exponential model in BJTs Use fully-differential architectures to reduce even order distortion
terms — also improves common mode noise rejection– The third order term dominates
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Fixed Bias Triode Tranconductors Use a small vDS voltage so v2
DS term goes to zero– Drain current is approximately linear with applied vDS. Transistor in triode
becomes a linear resistor
Resulting in:
Can use a triode transistor where a resistor would normally be used — resistance value is tunable
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Fixed Bias Triode Tranconductors
Q9 is in the triode region– transconductor has a variable transconductance value that can be
adjusted by changing the value of Vgs9
Moderate linearity
[Welland, 1994]
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Fixed Bias Triode Tranconductors
Alternative approach with lower complexity and p-channel inputs– transconductor has a variable transconductance value that can be
adjusted by changing the value of Vgs9
[Kwan, 1991]
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Fixed Bias Triode Tranconductors
Circuit can be easily made with multiple scaled output currents Multiple outputs allow filters to be realized using fewer
transconductors
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Biquads Using Multiple Outputs
Can make use of multiple outputs to build a biquad filter– scale extra outputs to desired ratio
Reduces the number of transconductors– saves power and die area
Above circuit makes use of Miller integrators
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Varying-Bias Triode Transconductor
Linearizes MOSFET differential stage– Transistors primarily in triode region
[Krummenacher, 1988]
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Varying-Bias Triode Transconductor
gates of Q3 and Q4 connected to the differential input (and not to bias voltage)
Q3 and Q4 undergo varying bias conditions to improve linearity It can be shown that
With
Note, Gm is proportional to square-root of as opposed to linear relation for a BJT transconductor
Transconductance can be tuned by changing bias current Ii
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Drain-Source Fixed-Bias Transconductor
If vDS is kept constant, then iD varies linearly with vGS
– Model is too simple, neglecting second order effects such as velocity saturation, mobility degradation
Possible implementation using fully differential architecture
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Drain-Source Fixed-Bias Transconductor
Can realize around 50 dB linearity (not much better since model is not that accurate)
Requires a fully-differential structure to cancel even-order terms VC sets vDS voltage Requires a non-zero common-mode voltage on input Note that the transconductance is proportional to vDS
– For vDS small the bias current I1 is also approximately proportional to vDS
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Alternative: MOSTFET-C Filters
Gm-C filters are most commonly used but MOSFET-C have advantages in BiCMOS for low power applications
MOSFET-C filters similar to active-RC filters but resistors replaced with MOS transistors in triode
Generally slower than Gm-C filters since opamps capable of driving resistive loads required
Rely on Miller integrators Two main types — 2 transistors or 4 transistors
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Alternative: MOSTFET-C Filters
Gm-C filters are most commonly used but MOSFET-C have advantages in BiCMOS for low power applications
MOSFET-C filters similar to active-RC filters but resistors replaced with MOS transistors in triode– Knowledge and architecture of active RC filters can be transferred
Generally slower than Gm-C filters since Opamps capable of driving resistive loads required
Rely on Miller integrators Two main types — 2 transistors or 4 transistors
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Two Transistor Integrators
Banu 1983
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Two Transistor Integrators
For resistor integrator can be shown
– If negative integration is required — cross-couple wires
For MOSFET-C integrator, assuming transistors are biased in triode region, the small-signal resistance is given by:
Therefore, the differential output of the MOSFET-C integrator is:
With:
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General MOSFET-C Biquad Filter
Equivalent active RC half circuit
𝑣0 𝑠
𝑣𝑖 𝑠=
𝐶1𝐶𝐵
𝑠2 +𝐺2𝐶𝐵
𝑠 +𝐺1𝐺3𝐶𝐴 𝐶𝐵
𝑠2 +𝐺5𝐶𝐵
𝑠 +𝐺3𝐺4𝐶𝐴𝐶𝐵
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Tuning Circuitry
Tuning can often be the MOST difficult part of a continuous-time integrated filter design
Tuning required for continuous-time integrated filters to account for capacitance and transconductance variations — 30 percent time-constant variations
Must account for process, temperature, aging, etc. While absolute tolerances are high, ratio of two like components can
be matched to under 1 percent Note that SC filters do not need tuning as their transfer-function
accuracy set by ratio of capacitors and a clock-frequency
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Indirect Tuning
Most common method — build an extra transconductor and tune it Same control signal is sent to filter’s transconductors which are scaled
versions of tuned extra Indirect since actual filter’s output is not measured
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Constant Transconductance
Can tune Gm to off-chip resistance and rely on capacitor absolute tolerance to be around 10 percent
𝐺𝑚 =1
𝑅𝑒𝑥𝑡