delta-sigma analog-to-digital converters sigma adc... · 2016. 9. 17. · delta-sigma (ΔΣ) adc...
TRANSCRIPT
Department of Electrical and Computer Engineering
© Vishal Saxena -1-
Delta-Sigma Analog-to-Digital
Converters Brief Overview
Vishal Saxena, Boise State University ([email protected])
© Vishal Saxena -2-
Analog to Digital Converter Architectures
Bandwidth
(Hz)
Resolution
(bits)
5
10
15
20
1k 10k 100k 1M 10M 100M 1G 10G
25
S/H with 1ps
rms jitterOversampling
Integrating
SAR,
Algorithmic
Pipelined,
Folding,Flash,
Time-Interleaved
© Vishal Saxena -3-
Why larger overlap on SAR and OS?
Sometimes oversampling modulators are preferred over SAR. Why?
Nyquist rate SAR converters can’s support complete bandwidth due to stringent requirements on Anti-aliasing Filters.
Consider a data converter with 250 ksps output rate.
For oversampled modulator OSR=64, input is sampled at 16 MHz.
Nyquist rate SAR samples the input at 500 kHz.
Requirements on anti-aliasing filter in the second case is very hard compared to oversampling case.
© Vishal Saxena -4-
Quantization Basics
y v
QuantizerContinuous
amplitude
Discrete amplitude represented
by a binary code
N
y v
QuantizerAnalog representation of the
discrete amplitude
N
….11000101….
DAC
Modeling
Quantization error/noise: e[n] = v[n]–y[n]
Least significant bit (LSB): Δ
Quantizer has non-linear input-output characteristics
Exact analytical modeling is difficult Use a simplified model
© Vishal Saxena -5-
Quantizer Modeling
Quantization noise modeling can be simplified with the assumptions Input (y) stays within the no-overload input range
e[n] is uncorrelated with the input y
Spectrum of e[n] is white
Quantization noise is uniformly distributed
Linearized quantizer model AWUN noise
Quantization noise power:
SQNR = 6.02·N + 1.76
22
12e
y v
Quantizer
kq
e[n]
Linearized
Quantizer model
y v
© Vishal Saxena -6-
Quantizer Modeling
Linear quantizer model holds true for large number of quantization levels and when the input is varying fast
Linear quantizer model breaks down when Input (y) is not varying fast
Input (y) is periodic with a frequency harmonically related to fs
Quantizer overload
With quantizer overload, the effective gain of the quantizer drops as the input amplitude inscreases
y v
Quantizer
kq
e[n]
Linearized
Quantizer model
y v
© Vishal Saxena -7-
Mid-Rise Quantizer (even number of levels)
Step rising at y=0 (mid-rise).
In this figure (DSM toolbox model), LSB= Δ= 2
M = Number of steps, (M is odd here) Number of levels (nLev) = M+1, (even)
Input thresholds: 0, ±2, ……, ±(M–1).
Output levels: ±1, ±3, ……, ±M.
© Vishal Saxena -8-
Mid-Tread Quantizer (odd number of levels)
Flat part of the step at y=0 (mid-tread).
Here, LSB= Δ= 2
M = Number of steps, (M is even here)
• Number of levels (nLev) = M+1, (odd)
Input thresholds: 0, ±2, ……, ±(M–1).
Output levels: 0, ±2, ±4, ……, ±M.
© Vishal Saxena -9-
Quantizer Characteristics : Slow ramp input
0 100 200 300 400 500 600 700 800-40
-20
0
20
40
samples
Quantization
y[n]
v[n]
0 100 200 300 400 500 600 700 800-10
-5
0
5
10
samples
e[n]
© Vishal Saxena -10-
Quantizer Characteristics : Sine input
0 100 200 300 400 500 600 700 800-10
-5
0
5
10
samples
Quantization
y[n]
v[n]
0 100 200 300 400 500 600 700 800-1
-0.5
0
0.5
1
samples
e[n]
© Vishal Saxena -11-
Quantization Noise : Example 1
5 10 15 20 25 30-20
-10
0
10
20
samples
Quantization
y[n]
v[n]
5 10 15 20 25 30-1
-0.5
0
0.5
1
E(e2) = 0.304
samples
e[n]
50 100 150 200 2500
10
20
30
40
50
60
70
80
FFT bins (k)
20*l
og
10|V
(k)|,
dB
Quantized Signal Spectrum
V[k]
nLev=17, Δ=2, fin/fs = 9/256 :
•E(e2) = 0.304 ≈ Δ2/12
© Vishal Saxena -12-
Quantization Noise : Example 1 contd.
nLev=17, Δ=2, fin/fs = 9.1/256 :
•E(e2) = 0.295 ≈ Δ2/12
•Notice the FFT leakage.
5 10 15 20 25 30-20
-10
0
10
20
samples
Quantization
y[n]
v[n]
5 10 15 20 25 30-1
-0.5
0
0.5
1
E(e2) = 0.295
samples
e[n]
50 100 150 200 2500
10
20
30
40
50
60
70
80
FFT bins (k)
20*l
og
10|V
(k)|,
dB
Quantized Signal Spectrum
V[k]
© Vishal Saxena -13-
Quantization Noise : Example 1 contd.
nLev=17, Δ=2, fin/fs = 32/256 = 1/8 :
•E(e2) = 0.235 < Δ2/12
•Quantization noise approximation not valid
50 100 150 200 2500
10
20
30
40
50
60
70
80
FFT bins (k)
20*l
og
10|V
(k)|,
dB
Quantized Signal Spectrum
V[k]
5 10 15 20 25 30-20
-10
0
10
20
samples
Quantization
y[n]
v[n]
5 10 15 20 25 30-1
-0.5
0
0.5
1
E(e2) = 0.235
samples
e[n]
© Vishal Saxena -14-
Oversampling and Noise-Shaping
© Vishal Saxena -15-
Oversampling
yv
fs
2
sf
OSR
LPF
OSR
Decimation
Dout
@fs @fs/OSR
0 50 100 150 200 250 300-8
-6
-4
-2
0
2
4
6
8
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-80
-70
-60
-50
-40
-30
-20
-10
0
10
e2=0.29136
V(f)
Oversampling ratio (OSR)
Conversion bandwidth: fB = fs/2·OSR
SQNR = 6.02·N + 1.76 + 10·log10(OSR)
0.5 bits increase in resolution per doubling in OSR
© Vishal Saxena -16-
Oversampling with Feedback
Can we use feedback with high loop-gain (A·kq) to reduce the error
e=|u-v| ?
Since quantizer output can not be equal to the input
The loop will be unstable as the error gets unbounded
Ay v
u
fs
High-gain
error
qA k e u v
© Vishal Saxena -17-
Oversampling with Feedback
Use large loop-gain in the signal band and small loop-gain at higher frequencies
At low frequencies
At high frequencies, low loop-gain stabilizes the loop
L(z) is the loop-filter
This feedback arrangement is called a (noise) modulator
0e u v
vu
fs
error2
sf
OSR
L(z)
y
© Vishal Saxena -18-
First-order Modulator
Differencing (Δ) followed by an accumulator (Σ) ΔΣ modulator
At low frequencies 0e u v
y vu
fs1
1-z-1
z-1
Δ
Σ
© Vishal Saxena -19-
First-order Noise Shaping
Linearized model for the modulator
Noise transfer function (NTF) (1-z-1) : first-order differentiator
High-pass shaping of quantization noise
Signal transfer function (STF) Unit delay
y vu
fs1
1-z-1
z-1
e[n]
1 1( ) 1 ( )V z z U z z E z
STF NTF
© Vishal Saxena -20-
First-order ΔΣ Modulator
y vu
fs1
1-z-1
z-1
e[n]
Sv(f)v[n]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-140
-120
-100
-80
-60
-40
-20
0DSM Output Spectrum
0 100 200 300 400 500 600 700 800 900 1000-8
-6
-4
-2
0
2
4
6
8DSM time-domain Simulation
© Vishal Saxena -21-
First-order ΔΣ Modulator SQNR
In-band quantization noise (IBN)
SQNR = 6.02·N –3.4 + 30·log10(OSR)
1.5 bits increase in resolution per doubling in OSR
Out-of-band noise is filtered out using a digital filter
0
2
0
22
0
22
0
2 3
0
2 2
3
( )
( )12
4sin / 212
12
12 3
36
OSRj
v
OSRj
OSR
OSR
OSR
IBN S e d
NTF e d
d
d
OSR
2 23
12 3OSR
ππ
OSR
|NTF(ejω
)|
0
2
© Vishal Saxena -22-
Delta-Sigma (ΔΣ) ADC
Use oversampling (fs=2·OSR·BW) to shape the quantization noise out of the signal band
Use low-resolution ADC and DAC to higher much higher resolution
Digitally filter out the out-of band shaped (modulated) noise
Trades-off SQNR with oversampling ratio (OSR)
+
–
vinL(z)
DAC
ADCvDSM Decimation
Filter
vout
ΔΣ Modulator
E(z)
f
STF
NTF•E
E
fs/2•OSR fs/2
|VDSM(f)|vin
ffs/2
|Vout(f)|vin
fs/2•OSR
© Vishal Saxena -23-
Second-Order Noise Shaping
2
1 1( ) 1 ( )V z z U z z E z
STF NTF
Linearized model for the modulator
Second-order noise-shaping
In-band quantization noise (IBN):
2.5 bits increase in resolution per doubling in OSR
Can be extended to higher orders
2 45
12 5OSR
v1
1-z-1
z-1
u 1
1-z-1
y
e[n]
© Vishal Saxena -24-
Second-order ΔΣ Modulator
v1
1-z-1
z-1
u 1
1-z-1
y
e[n]
Sv(f)v[n]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-180
-160
-140
-120
-100
-80
-60
-40
-20
0
20
0 100 200 300 400 500 600 700 800 900 1000-8
-6
-4
-2
0
2
4
6
8
© Vishal Saxena -25-
-1 -0.5 0 0.5 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
X2
NTF(z)
NTF
zeroes
Second-order ΔΣ Modulator
10-4
10-3
10-2
10-1
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
SNDR = 73.8 dB
ENOB = 11.97 bits
@OSR = 32
Frequency
dB
Second-Order KD1S Output Spectrum
40 dB/dec
Sv(f)
NTF(z) = (1-z-1)2
Two zeroes at DC
Out-of-Band Gain (OBG) (i.e gain at ω≈π) = 4
© Vishal Saxena -26-
Higher-Order ΔΣ Modulators
© Vishal Saxena -27-
Higher-Order NTFs
Higher order noise shaping Reduced in-band noise, higher SQNR
For NTF=(1-z-1)-N, in-band noise (IBN):
Ideally (N+1/2) bits increase in resolution per doubling in OSR
y vu L(z)
( ) 1
( ) ( )1 ( ) 1 ( )
L zV z U z E z
L z L z
STF NTF
2 2(2 1)
12 2 1
NNOSR
N
© Vishal Saxena -28-
Higher-Order NTFs
NTF gain increases at high frequencies (around ω≈π)
Can we go on increasing the order?
0.5 1 1.5 2 2.5 3-200
-150
-100
-50
0
50
|NT
F(e
j )|
Bode Diagram
Frequency (rad/s)
© Vishal Saxena -29-
Third-order ΔΣ Modulator Example
NTF(z) = (1-z-1)3
OBG = 8, Full-scale input.
Unstable after few samples (look at quantize input (y) blowing up!). Signature for ΔΣ instability
Worst case for a single-bit quantizer.
0 100 200 300 400 500 600 700 800 900 1000-10
-5
0
5
10DSM time-domain Simulation
u
v
0 100 200 300 400 500 600 700 800 900 1000-4
-2
0
2
4x 10
7
y
© Vishal Saxena -30-
Third-order ΔΣ Modulator Example
Stable for 50% of full-Scale amplitude
Signal dependent stability Need to develop intuition for modulator stability
Reference: Stability theory from the Yellow Bible of delta-sigma
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
20
DSM Output Spectrum
0 100 200 300 400 500 600 700 800 900 1000-10
-5
0
5
10DSM time-domain Simulation
u
v
0 100 200 300 400 500 600 700 800 900 1000-10
-5
0
5
10
y
© Vishal Saxena -31-
Systematic NTF Design
NTFs of the form (1-z-1)N have stability issues The OBG (2N) are too high
A larger OBG causes more wiggling at the quantizer input This saturates the quantizer for even smaller inputs
Irrecoverable quantizer saturation causes loop instability
For high-OBG the maximum stable (input) amplitude (MSA) is small
The stability is worse for low quantizer resolutions
Thus we need to reduce OBG while maintaining high in-band noise shaping
© Vishal Saxena -32-
Systematic NTF Design Procedure
Introduce poles into the NTF
NTF realizability criterion No delay-free loops in the modulator
• First sample of the NTF impulse response (i.e. h[0])=1
NTF()=1
D(z=)=1
Commonly used pole positions: Butterworth, Inverse Chebyshev and maximally flat poles (maxflat)
1(1 )( )
( )
NzNTF z
D z
-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
1Pole-Zero Map
Real Axis
Imagin
ary
Axis
X3
© Vishal Saxena -33-
NTF Response with Poles
Select appropriate OBG for the NTF to assure stability
Trade-off between stability and increased in-band noise MSA vs SQNR for a given order and quantizer resolution
0 50 100 150 200 250 300 350 400 450 500-10
-5
0
5
10OBG=8
u
v
0 50 100 150 200 250 300 350 400 450 500-10
-5
0
5
10OBG=3
u
v
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
1
2
3
4
5
6
7
8
9
/
|NT
F(e
j )|
NTF response
© Vishal Saxena -34-
Systematic NTF Design Example
Specifications SQNR > 120 dB
A signal bandwidth which results in an OSR = 64
• Study optimal clock rate for the given process and quantizer design.
Designer’s Choice Order = 3
Quantizer levels (nLev) = 16
Butterworth high-pass response for the NTF
Use MATLAB for finding coefficients of the HPF response. [b,a] = butter(order, ω3dB, ‘high’) The cutoff frequency ω3dB specifies the transfer function.
© Vishal Saxena -35-
Systematic NTF Design contd.
Start with cutoff frequency ω3dB=π/8, for the butterworth HPF H(z).
Derive a realizable NTF using NTF(z)=H(z)/b0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
/
Magnitude
H(z)
NTF(z)
© Vishal Saxena -36-
Systematic NTF Design contd.
Map the NTF response to a loop-filter architecture (details later)
Simulate the modulator for all possible amplitudes and input tone frequencies.
Compute the peak SQNR and MSA. simulateDSM function in the toolbox.
Can use Risbo’s method shown later
© Vishal Saxena -37-
Systematic NTF Design contd.
Peak SNR = 107 dB
MSA = 0.9
-100 -80 -60 -40 -20 00
10
20
30
40
50
60
70
80
90
100
110
120
130
140
Input Level, dB
SN
R d
B
peak SNR = 107.2dB
© Vishal Saxena -38-
Systematic NTF Design contd.
If SNR is not enough, repeat the entire procedure with a higher cutoff frequency for the Butterworth HPF IBN ↓, SQNR ↑
OBG ↑ and MSA ↓
If SNR is too high, repeat the entire procedure with a lower cutoff frequency for the Butterworth HPF IBN ↑, SQNR ↓
OBG ↓ and MSA ↑
© Vishal Saxena -39-
Systematic NTF Design contd.
ω3dB=π/4.
Peak SNR = 119 dB, OBG = 2.25, MSA = 0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
/
Magnitude
H(z)
NTF(z)
-100 -80 -60 -40 -20 00
10
20
30
40
50
60
70
80
90
100
110
120
130
140
Input Level, dB
SN
R d
B
peak SNR = 119.7dB
© Vishal Saxena -40-
Systematic NTF Design contd.
ω3dB=2π/7.
Peak SNR = 121 dB, OBG = 2.54, MSA = 0.8. Design closed !
-100 -80 -60 -40 -20 00
10
20
30
40
50
60
70
80
90
100
110
120
130
140
Input Level, dB
SN
R d
B
peak SNR = 121.1dB
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
/
Magnitude
H(z)
NTF(z)
© Vishal Saxena -41-
Systematic NTF Design contd.
An advanced version of this iterative process is implemented as the function synthesizeNTF in the delta-sigma Toolbox. Several ‘opt’ params for NTF zero (and pole) optimization
Use synthesizeChebyshevNTF for low OSR and low OBG designs.
© Vishal Saxena -42-
NTF-Zero Optimization
Spread zeros in the signal band to minimize in-band noise Complex zeros on the unit circle
8dB increase in SQNR for 3rd order modulator
Bandwidth normalized NTF-zero locations obtained by toolbox function ds_optzeros(order, 1)
Already implemented in synthesizeNTF function for opt=1
10-3
10-2
10-1
-120
-100
-80
-60
-40
-20
0
20
/
|NT
F(e
j )|
NTF response
-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
1Pole-Zero Map
Real Axis
Imagin
ary
Axis
© Vishal Saxena -43-
Estimating MSA (Maximum Stable Amplitude)
MSA is found through extensive simulation
Simulate for input sinusoids of varying amplitudes for all possible signal frequencies in the signal band. For every input amplitude compute in-band SNR.
Beyond the MSA, the NTF poles move out of the unit circle.
Noise shaping is disrupted and the in-band SNR drops.
At this point the quantizer input (y[n]) blows up.
simulateSNR function in the toolbox does exactly the same
Time consuming and often impractical for iterative design
© Vishal Saxena -44-
Estimating MSA using Risbo’s Method
Use a slow ramp input from 0 to FS value. Plot log10|y[n]|. Observe where this plot blows up.
Take 90% of the input amplitude where log10|y[n]| blows up as a conservative estimate for MSA.
Estimated MSA is close to that predicted by the sinewave input method.
Much quicker than the sinewave technique (simulateSNR function)
y v
u
L
t0
FS
Slow ramp input
© Vishal Saxena -45-
Estimating MSA using Risbo’s Method
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-6
-4
-2
0
2
4
6
8
10
12
14
log |y|
Input, u
MSA = 0.843
MSA Estimation by Simulation
© Vishal Saxena -46-
Loop-Filter Architectures
Several loop-filter discrete-time architectures possible
Toolbox function realizeNTF maps the synthesized NTF to loop-filter co-efficients [a,g,b,c] = realizeNTF(H, form);
Dynamic Range Scaling (DRS) performed to scale loop-filter states to a bounded value Scaling performed using ABCD matrix representation of the loop-
filter
See any introductory text on Linear Systems ABCD = stuffABCD(a,g,b,c,form);
[ABCDs umax] = scaleABCD(ABCD, nLev, f0, xLim);
[a,g,b,c] = mapABCD(ABCDs, form);
© Vishal Saxena -47-
CIFB (Cascade of Integrators with Distributed Feedback)
Cascade of delaying integrators: Feedback coefficients a’s realize the zeros of L1 and thus the
NTF and STF poles.
Feed-in coefficients b’s determine zeros of L0 and thus the STF zeros.
State scaling coefficients c’s are used for dynamic range scaling.
-g1
c1
-a1 -a2
x1(n) x2(n) y(n)c2
z-1
1-z-1
z-1
1-z-1
b2b1
DAC
v(n)
-a3
x3(n)c2
z-1
1-z-1
b3
-a4
b4
u(n)
© Vishal Saxena -48-
CRFB (Cascade of Resonators with Distributed Feedback)
Combine a non-delaying and a delaying integrator with local feedback around them, to form a stable resonator Local feedback coefficients g’s realize the complex zeros in the
NTF.
Implements NTF with complex zeros.
For odd-order, use an integrator in the front to avoid noise coupling due to g
-g1
c1
-a1 -a2
x1(n) x2(n+1) y(n)c2
z-1
1-z-1
1
1-z-1
b2b1
DAC
v(n)
-a3
x3(n)c2
z-1
1-z-1
b3
-a4
b4
u(n)
© Vishal Saxena -49-
CIFF (Cascade of Integrators with Feed-Forward Summation)
Feedforward summation of states
For b1=bN+1=1 and b2 to bN=0, STF=1 Loop-filter only processes quantization noise, low power and
distortion
Feedforward loop-filters typically result in lower-power implementation
DAC
-g1
c2
-c1
x1(n) x2(n) y(n)c3
v(n)
x3(n)a3
a2
a1
z-1
1-z-1
z-1
1-z-1
z-1
1-z-1
b2b1 b3 b4
u(n)
© Vishal Saxena -50-
CRFF (Cascade of Resonators with Feed-Forward Summation)
Use resonators with feedforward summation Implements NTF with complex zeros
For odd-order, use an integrator in the front to avoid noise coupling due to g
-g1
c1
-a1 -a2
x1(n) x2(n) y(n)c2
z-1
1-z-1
1
1-z-1
-b2-b1
DAC
v(n)
-a3
x3(n)c2
z-1
1-z-1
-b3
-a4
-b4
u(n)
© Vishal Saxena -51-
ΔΣ Modulator Architectures
Cascade/ MASH architecture:
•Eg. Two first order modulators are used to implement second order
modulator.
•Stability concerns are relaxed but mismatch in the two forward paths
should be properly monitored.
© Vishal Saxena -52-
ΔΣ Modulator Architectures
Feedforward modulators
•Most popular architecture.
•Input signal is summed at Nth stage integrator output.
•Summation block may be required at higher order modulators.
•Multibit quantizer is necessary.
© Vishal Saxena -53-
Key Terminologies :
SQNR – Signal to quantization noise ratio Thermal/electrical noise are not included.
SNR - Signal to noise ratio Distortion is not included.
SDNR - Signal to noise and distortion ratio All noise sources are included.
ENOB – Effective number of bits (resolution) This is very important than actual number of output bits
Dynamic Range (DR) Measured with input of the modulator shorted.
Harmonic Distortion THD is usually total harmonic distortion. Or Third??
Spur Free Dynamic Range (SFDR) Very key parameter in communication systems
© Vishal Saxena -54-
Frequency Domain Measurements
Dynamic range
Peak SNR
Peak SNDR
© Vishal Saxena -55-
Spurious (tone) Free Dynamic Range (SFDR)
SFDR
© Vishal Saxena -56-
References
Delta-Sigma Data Converters B.1 R. Schreier, G. C. Temes, Understanding Delta-Sigma Data Converters,
Wiley-IEEE Press, 2005 (the Green Bible of Delta-Sigma Converters). B.2 S. R. Norsworthy, R. Schreier, G. C. Temes, Delta-Sigma Data Converters:
Theory, Design, and Simulation, Wiley-IEEE Press, 1996 (the Yellow Bible of Delta-Sigma Converters).
B.3 S. Pavan, N. Krishnapura, “Oversampling Analog to Digital Converters Tutorial,” 21st International Conference on VLSI Design, Hyderabad, Jan, 2008.
B.4 S. H. Ardalan, J. J. Paulos, “An Analysis of Nonlinear behavior in Delta-Sigma Modulators,” IEEE TCAS, vol. 34, no. 6, June 1987.
B.5 R. Schreier, “An Empirical Study of Higher-Order Single-Bit Delta-Sigma Modulators,” IEEE TCAS-II, vol. 40, no. 8, pp. 461-466, Aug. 1993.
B.6 J. G. Kenney and L. R. Carley, “Design of multibit noise-shaping data converters,” Analog Integrated Circuits Signal Processing Journal, vol. 3, pp. 259-272, 1993.
B.7 L. Risbo, “Delta-Sigma Modulators: Stability Analysis and Optimization,” Doctoral Dissertation, Technical University of Denmark, 1994 [Online].
B.8 R. Schreier, J. Silva, J. Steensgaard, G. C. Temes,“Design-Oriented Estimation of Thermal Noise in Switched-Capacitor Circuits,” IEEE TCAS-I, vol. 52, no. 11, pp. 2358-2368, Nov. 2005.