ee247 lecture 26 - university of california, berkeleyee247/fa05/lectures/l26_2_f05.pdfee247 lecture...
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![Page 1: EE247 Lecture 26 - University of California, Berkeleyee247/fa05/lectures/L26_2_f05.pdfEE247 Lecture 26 • Higher order ΣΔ modulators – Last lectureÆCascaded ΣΔ modulators (MASH)](https://reader033.vdocuments.site/reader033/viewer/2022042006/5e6fe594dddf594653104dd5/html5/thumbnails/1.jpg)
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 1
EE247Lecture 26
• Administrative– Final exam:
• Date: Tues. Dec. 13th
• Time: 12:30pm-3:30pm• Location: 285 Cory• Office hours this week: Tues: 2:30p to 3:30p
Wed: 1:30p to 2:30p (extra)Thurs: 2:30p to 3:30p
• Closed book/course notes• No calculators/cell phones/PDAs/Computers
• You can bring two 8x11 paper with your own notes• Final exam covers the entire course material
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 2
EE247Lecture 26
• Higher order ΣΔ modulators– Last lecture Cascaded ΣΔ modulators (MASH)
– This lecture Bandpass ΣΔ modulators– This lecture Forward path multi-order filter
• Example: 5th order Lowpass ΣΔ – Modeling– Noise shaping– Effect of various nonidealities on the ΣΔ
performance
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 3
Bandpass ΔΣ Modulator
+
_vIN
dOUT
∫
DAC
• Replace the integrator in 1st order lowpass ΣΔ with a resonator
2nd order bandpass ΣΔ
Resonator
∫
∫
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 4
Bandpass ΔΣ ModulatorMeasured output for a bandpass ΣΔ (prior to digital filtering)
Key Point:
NTF notch type shape
STF bandpass shape
Ref: Paolo Cusinato, et. al, “A 3.3-V CMOS 10.7-MHz Sixth-Order Bandpass Modulator with 74-dB Dynamic Range “, ΙΕΕΕ JSSCC, VOL. 36, NO. 4, APRIL 2001
Input SinusoidQuantization Noise
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 5
Bandpass ΣΔ Characteristics• Oversampling ratio defined as fs/2B where B =
STF -3dB bandwidth
• Typically, sampling frequency is chosen to be fs=4xfcenter where fcenter=bandpass filter center frequency
• STF has a bandpass shape while NTF has a notch shape
• To achieve same resolution as lowpass, need twice as many integrators
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 6
Bandpass ΣΔ Modulator Dynamic RangeAs a Function of Modulator Order (K)
• Bandpass ΣΔ resolution for order K is the same as lowpass ΣΔ resolution with order L= K/2
K=415dB/Octave
K=621dB/Octave
K=29dB/Octave
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 7
Example: Sixth-Order Bandpass ΣΔ Modulator
Ref: Paolo Cusinato, et. al, “A 3.3-V CMOS 10.7-MHz Sixth-Order Bandpass Modulator with 74-dB Dynamic Range “, ΙΕΕΕ JSSCC, VOL. 36, NO. 4, APRIL 2001
Simulated noise transfer function Simulated signal transfer function
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 8
Example: Sixth-Order Bandpass ΣΔ Modulator
Ref: Paolo Cusinato, et. al, “A 3.3-V CMOS 10.7-MHz Sixth-Order Bandpass Modulator with 74-dB Dynamic Range “, ΙΕΕΕ JSSCC, VOL. 36, NO. 4, APRIL 2001
Features & Measured Performance Summary
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 9
Higher Order Lowpass ΣΔ ModulatorsForward Path Multi-Order Filter
• Zeros of NTF (poles of H(z)) can be positioned to flatten baseband noise spectrum• Main issue Ensuring stability for 3rd and higher orders
( ) 1( ) ( ) ( )1 ( ) 1 ( )
H zY z X z E zH z H z
= ++ +
Σ
E(z)
X(z) Y(z)( )H z Σ
Y( z ) 1NTF
E( z ) 1 H( z )= =
+
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 10
Overview
• Building behavioral models in stages
• A 5th-order, 1-Bit ΣΔ modulator– Noise shaping – Complex loop filters– Stability– Voltage scaling– Effect of component non-idealities
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 11
Building Models in Stages• When modeling a complex system like a 5th-order ΣΔ modulator, model
development proceeds in stages– Each stage builds on its predecessor
• Design goal detect and eliminate problems at the highest possible level of abstraction
– Each successive stage consumes progressively more engineering time
• Our ΣΔ model development proceeds in stages:– Stage 0 gets to the starting line: Collect references, talk to veterans– Stage 1 develops a practical system built with ideal sub-circuits & simulated– Stage 2 models key sub-circuit non-idealities and translates the results into
real-world sub-circuit performance specifications – Real-world model development includes a critical stage 3: Adding elements to
earlier stages to model significant surprises found in silicon
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 12
Stage 1• In stage 1, we’ll study a model for a practical ΣΔ
modulator topology built with ideal blocks
• Stage 1 model focus– Signal amplitudes– Stability
• Identifying worst-case inputs• Unstable systems can’t graduate to stage 2
– Quantization noise shaping • Verify performance and functionality for all regions of
operation, find and test worst-case inputs• Determine appropriate performance metrics and build
the software infrastructure
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 13
ΣΔ Modulator Design• Procedure
– Establish requirements– Design noise-transfer function, NTF– Determine loop-filter, H– Synthesize filter– Evaluate performance, – Establish stability criteria
Ref: R. W. Adams and R. Schreier, “Stability Theory for ΔΣ Modulators,” in Delta-Sigma Data Converters- S. Norsworthy et al. (eds), IEEE Press, 1997
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 14
Example: Modulator Specification
• Example: Audio ADC– Dynamic range DR 18 Bits– Signal bandwidth B 20 kHz– Nyquist frequency fN 44.1 kHz– Modulator order L 5– Oversampling ratio M = fs/fN 64– Sampling frequency fs 2.822 MHz
• The order L and oversampling ratio M are chosen based on– SQNR > 120dB
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 15
Modulator Block Diagram
( ) ( )STF( ) 1 ( )( ) 1NTF( ) 1 ( )
Y z H zX z H zY zE z H z
= =+
= =+
Approach:Design NTF and solve for H(z)
Loop FilterH(z)Σx(kT) y(kT)Comparator
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 16
Noise Transfer Function, NTF(z)% stop-band attenuation Rstop ...
Rstop = 80; [b,a] = cheby2(L, Rstop, 1/M, 'high');
% normalize b = b/b(1); NTF = filt(b, a, 1/fs);
10 4 106-100
-80
-60
-40
-20
0
20
Frequency [Hz]
NTF
[dB
]
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 17
Loop-Filter, H(z)
( ) 1NTF( ) 1 ( )
1( ) 1
Y zE z H z
H zNFT
= =+
→ = −
104 106-20
0
20
40
60
80
100
Frequency [Hz]
Loop
filte
r H
[dB
]
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 18
Modulator TopologySimulation Model
Q
I_5I_4I_3I_2I_1
Y
b2b1
a5a4a3a2a1
K1 z -1
1 - z -1
I1
gDAC Gain Comparator
X
-1
1 - z -1
I2
K2 z -1
1 - z -1
I3
K3 z -1
1 - z -1
I4
K4 z -1
1 - z -1
I5
K5 z
+1-1
Filter
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 19
Rounded Filter Coefficients
a1=1;
a2=1/2;
a3=1/4;
a4=1/8;
a5=1/8;
k1=1;
k2=1;
k3=1/2;
k4=1/4;
k5=1/8;
b1=1/1024;
b2=1/16-1/64;
g =1;
Ref: Nav Sooch, Don Kerth, Eric Swanson, and Tetsuro Sugimoto, “Phase Equalization System for a Digital-to-Analog Converter Using Separate Digital and Analog Sections”, U.S. Patent 5061925, 1990, figure 3 and table 1
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 20
5th Order Noise Shaping
• Mostly quantization noise, except at low frequencies
• Let’s zoom into the baseband portion…
0 0.1 0.2 0.3 0.4 0.5-160
-140
-120
-100
-80
-60
-40
-20
0
20
40
Frequency [ f / fs ]
Out
put S
pect
rum
[dB
WN
]/ In
t. N
oise
[dB
FS]
Output SpectrumIntegrated Noise (20 averages)
Signal
Notice tones around fs/2
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 21
5th Order Noise Shaping
0 0.2 0.4 0.6 0.8 1-160
-140
-120
-100
-80
-60
-40
-20
0
20
40
Out
put S
pect
rum
[dB
WN
] /
Int.
Noi
se [d
BFS
]
Output SpectrumIntegrated Noise (20 averages)
Quantization noise -130dBFS at band edge!Signal
Band-Edge
Frequency [ f / fN ]
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 22
5th Order Noise Shaping
sigma_delta_L5.m
• SQNR > 120dB
• Sigma-delta modulators are usually designed for negligible quantization noise
• Other error sources dominate, e.g. thermal noise
0 0.2 0.4 0.6 0.8 1-160
-140
-120
-100
-80
-60
-40
-20
0
20
40
Frequency [f / fN]
Out
put S
pect
rum
[dB
WN
] /
Int.
Noi
se [d
BFS
]
Output SpectrumIntegrated Noise (20 averages)
Digital decimation filter removes out-of-band quantization noise
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 23
0 0.2 0.4 0.6 0.8 140
60
80
100
120
140M
agni
tude
[dB
] Loop Filter
0 0.2 0.4 0.6 0.8 1-100
0
100
200
300
400
Frequency [f/fN]
Phas
e [d
egre
es]
In-Band Noise Shaping
• Lot’s of gain in the pass-band
• Remember that• NTF ~ 1/H•STF=H/(1+H)
0 0.2 0.4 0.6 0.8 1-160
-120
-80
-40
0
40
Frequency [f/fN]
Out
put S
pect
rum
Output SpectrumIntegrated Noise (20 averages)
|H(z)| maxima align up with noise minima
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 24
Stability Analysis
• Approach: linearize quantizer and use linear system theory!• Effective quantizer gain
• Obtain Geff from simulation
222
yGeff q
=
H(z)Σ Σ
Quantizer Model
QuantizationError e(kT)
x(kT) y(kT)Geffq(kT)
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 25
Modulator Root-Locus
• As Geff increases, poles of STF move from
• poles of H(z) (Geff = 0) to • zeros of H(z) (Geff = ∞)
• Pole-locations inside unit-circle correspond to stable STF and NTF
• Geff > 0.45 for stability
Geff = 0.45
z-Plane Root Locus
0.6 0.7 0.8 0.9 1 1.1-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4 Increasing Geff
Unit Circle
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 26
Effective Quantizer Gain, Geff
• Large inputs comparator input grows• Output is fixed (±1)
Geff dropsmodulator unstable for large inputs
• Solution:• Limit input amplitude• Detect instability (long sequence
of +1 or -1) and reset integrators• Beware of “worst-case inputs”
(e.g. square waves near high-Q poles – attenuate with anti-aliasing filter)
• Note that signals grow slowly for nearly stable systems use long simulations
-40 -35 -30 -25 -20 -15 -10 -5 0 500.20.40.60.81
1.21.41.61.82
Input [dBV]
Effe
ctiv
e Q
uant
izer
Gai
n
stable unstable
Geff =0.45
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 27
-40 -35 -30 -25 -20 -15 -10 -5 0
-20
-15
-10
-5
0
5
10
i1i2i3i4i5q
Input [dBV]
Loop
filte
r pea
k vo
ltage
s [V
]
Internal Node Voltages
• Internal signal amplitudes are weak function of input level (except near overload)
• Maximum peak-to-peak voltage swing approach +-10V! Exceed supply voltage!
• Solutions:• Reduce Vref ??• Node scaling
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 28
Internal Node Voltage Scaling
• If we scale filter k1 by 0.1,– All state variables and Q scale by 0.1– But since the comparator output is fixed and input is
decreased by 10, G increases 10X
• The change in k1 doesn’t change the shape of the root locus, either– The effective gain for each root position is increased 10X– Gnew=10XGold Gnew > 4.5 is thus required to ensure stability
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 29
5th Order Modulator – ScalingOnly the sign of Q matters,so we can make k1 whatever we wantwithout changing the 1-Bit data at all
Q
I_5I_4I_3I_2I_1
Y
b2b1
a5a4a3a2a1
K1 z -1
1 - z -1
I1
gDAC Gain Comparator
X
-1
1 - z -1
I2
K2 z -1
1 - z -1
I3
K3 z -1
1 - z -1
I4
K4 z -1
1 - z -1
I5
K5 z
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 30
Loop Voltage Scaling (cont.)• Note that ∫3, ∫4, and ∫5 have substantially
larger swings than ∫1 and ∫2
• Just about any filter topology allows node scaling which change internal state variable amplitudes without changing the filter output (recall filter node scaling)– The next slide shows an example
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 31
Node Scaling Example:3rd Integrator Output Voltage Scaled by α
K3 * α, b1 /α, a3 / α, K4 / α, b2 * αVnew=Vold* α
Q
I_5I_4I_3I_2I_1
Y
b2b1
a5a4a3a2a1
K1 z -1
1 - z -1
I1
gDAC Gain Comparator
X
-1
1 - z -1
I2
K2 z -1
1 - z -1
I3
K3 z -1
1 - z -1
I4
K4 z -1
1 - z -1
I5
K5 z
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 32
Voltage Scaling
-40 -35 -30 -25 -20 -15 -10 -5 0-1.5
-1
-0.5
0
0.5
1
1.5
Input [dBV]
Loop
filte
r pea
k vo
ltage
s [V
]
k1=1/10;k2=1;k3=1/4;k4=1/4;k5=1/8;a1= 1; a2=1/2;a3=1/2; a4=1/4;a5=1/4;b1=1/512;b2=1/16-1/64;g =1;
• Integrator output range is fine now• But: maximum input signal limited to -5dB (-7dB with safety) – fix?
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 33
Input Range ScalingIncreasing the DAC levels by g reduces the analog to digital conversion gain:
Increasing vIN & g by the same factor leaves 1-Bit data unchanged
gzgHzH
zVzD
IN
OUT 1)(1
)()()( ≅
+=
Loop FilterH(z)ΣvIN
dOUT+1 or -1Comparator
g
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 34
Input Range Scaling• Scaling the DAC output levels adjusts the
modulator input range
– If VIN and the DAC outputs are scaled up by the same factor g, the 1-Bit data is completely unchanged
– Note that increasing the range also increases the quantization noise the dynamic range and peak SQNR remain constant!
– If the DAC output levels are increased and the analog full scale is held constant, the stability margin improves … at the expense of reduced SQNR
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 35
Scaled Stage 1 Model
g = 2.5;
-40 -35 -30 -25 -20 -15 -10 -5 0-1.5
-1
-0.5
0
0.5
1
1.5
Input [dBV]
Loop
filte
r pea
k vo
ltage
s [V
]
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 36
Scaled Stage 1 Model
2dB safety margin for stability
-40 -35 -30 -25 -20 -15 -10 -5 0 50
1
2
3
4
5
6
7
8
Input [dBV]
Effe
ctiv
e Q
uant
izer
Gai
n
Geff=4.5stable unstable
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 37
5th Order ModulatorFinal Parameters
±2.5V
Stable input range ~ ±1V
1/10 1 1/4 1/4 1/8
1/512 1/16-1/64
1 12 1/2 1/4 1/4
Stable input range~ ±1V
Q
I_5I_4I_3I_2I_1
Y
b2b1
a5a4a3a2a1
K1 z -1
1 - z -1
I1
gDAC Gain Comparator
X
-1
1 - z -1
I2
K2 z -1
1 - z -1
I3
K3 z -1
1 - z -1
I4
K4 z -1
1 - z -1I5
K5 z
1
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 38
Summary• Stage 1 model verified –
stable and meets SQNR specification
• Stage 2 issues in 5th order ΣΔ modulator– DC inputs– Spurious tones– Dither– kT/C noise
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 39
0 0.5 1 1.5-150
-100
-50
0
50
Frequency [MHz]
Out
put S
pect
rum
[dB
WN
] /
Int.
Noi
se [d
BV
]
Output SpectrumIntegrated Noise (30 averages)
5th Order Noise Shaping
Input: 0.1V, sinusoid215 point DFT30 averages
Note: Large spurious tonesin the vicinity of fs/2
Let us check whether tones appear inband?
Tones at fs/2-Nfin exceed input level
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 40
0 10 20 30 40 50-150
-100
-50
0
50
Frequency [kHz]
Out
put S
pect
rum
[dB
WN
] /
Int.
Noi
se [d
BV
]
Output SpectrumIntegrated Noise (30 averages)
In-Band Noise
In-Band quantization noise:–120dB !
Note: No in-band tones!While Large spurious tones appear in the vicinity of fs /2
![Page 21: EE247 Lecture 26 - University of California, Berkeleyee247/fa05/lectures/L26_2_f05.pdfEE247 Lecture 26 • Higher order ΣΔ modulators – Last lectureÆCascaded ΣΔ modulators (MASH)](https://reader033.vdocuments.site/reader033/viewer/2022042006/5e6fe594dddf594653104dd5/html5/thumbnails/21.jpg)
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 41
0 0.5 1 1.5-150
-100
-50
0
50
Frequency [MHz]
Out
put S
pect
rum
[dB
WN
] /
Int.
Noi
se [d
BV
]
Output SpectrumIntegrated Noise (30 averages)
5th Order Noise Shaping
Input: 0.1V, sinusoid215 point DFT30 averages
Note: Digital filter required attenuation function of tones in the vicinity of fs/2 & in-band quantization noise
150dB stopband attenuation neededto attenuate unwanted fs/2-Nfin componentsdown to the in-band quantization noise level
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 42
Out-of-Band vs In-Band Signals• A digital (low-pass) filter with suitable
coefficient precision can eliminate out-of-band quantization noise
• No filter can attenuate unwanted in-band components without attenuating the signal
• We’ll spend some time making sure the components at fs/2-Nfin will not “mix” down to the signal band
• But first, let’s look at the modulator response to small DC inputs (or offset) …
![Page 22: EE247 Lecture 26 - University of California, Berkeleyee247/fa05/lectures/L26_2_f05.pdfEE247 Lecture 26 • Higher order ΣΔ modulators – Last lectureÆCascaded ΣΔ modulators (MASH)](https://reader033.vdocuments.site/reader033/viewer/2022042006/5e6fe594dddf594653104dd5/html5/thumbnails/22.jpg)
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 43
ΣΔ Tones Generated by Small DC Input Signals
5mV DC input(VDAC 2.5V)
Simulation technique:A random 1st sample randomizes the noise from DC input and enables averaging. Otherwise the small tones will not become visible.
0 10 20 30 40 50-150
-100
-50
0
50
Frequency [kHz]
Out
put S
pect
rum
[dB
WN
] /
Int.
Noi
se [d
BV
]
6kHz12kHz
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 44
Limit Cycles• Representing a DC term with a –1/+1 pattern … e.g.
• Spectrum:
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
++−+−+−+−+−→
444444444 3444444444 21
44444444 344444444 21321321321321321
111
0
54321
1 1 1 1 1 1 1 1 1 1 1111
K11
311
211
sss fff
![Page 23: EE247 Lecture 26 - University of California, Berkeleyee247/fa05/lectures/L26_2_f05.pdfEE247 Lecture 26 • Higher order ΣΔ modulators – Last lectureÆCascaded ΣΔ modulators (MASH)](https://reader033.vdocuments.site/reader033/viewer/2022042006/5e6fe594dddf594653104dd5/html5/thumbnails/23.jpg)
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 45
Limit Cycles• The frequency of the tones are indeed quite predictable
– Fundamental
– Tone velocity (useful for debugging)
– Note: For digital audio in this case DC signal>20mV generates tone with fδ >24kHz out-of-band no problem
DCs
DAC
Vf f
V5mV
3MHz2.5V
6kHz
δ =
=
=
s
DC DAC
DC
df f
dV V
df1.2kHz/mV
dV
δ
δ
=
=
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 46
1.47 1.48 1.49 1.5-150
-100
-50
0
50
Frequency [MHz]
Out
put S
pect
rum
[dB
WN
] /
Int.
Noi
se [d
BV
]
Output SpectrumIntegrated Noise (30 averages)
ΣΔ Spurious TonesEffect of Small DC Input @ Vicinity of fs /2
6kHz
![Page 24: EE247 Lecture 26 - University of California, Berkeleyee247/fa05/lectures/L26_2_f05.pdfEE247 Lecture 26 • Higher order ΣΔ modulators – Last lectureÆCascaded ΣΔ modulators (MASH)](https://reader033.vdocuments.site/reader033/viewer/2022042006/5e6fe594dddf594653104dd5/html5/thumbnails/24.jpg)
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 47
ΣΔ Spurious Tones• In-band spurious tones look like signals
• Can be a major problem in some applications– E.g. audio even tones with power below the quantization
noise floor can be audible
• Spurious tones near fs/2 can be aliased down into the signal band– Since they are often strong, even a small amount of aliasing
can create a major problem– We will look at mechanisms that alias tones later
• First let’s look at dither as a means to reduce or eliminate in-band spurious tones
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 48
Dither• DC inputs can be represented by many possible bit
patterns
• Including some that are random (non-periodic) but still average to the desired DC input
• The spectrum of such a sequence has no spurious tones
• How can we get a ΣΔ modulator to produce such “randomized” sequences?
![Page 25: EE247 Lecture 26 - University of California, Berkeleyee247/fa05/lectures/L26_2_f05.pdfEE247 Lecture 26 • Higher order ΣΔ modulators – Last lectureÆCascaded ΣΔ modulators (MASH)](https://reader033.vdocuments.site/reader033/viewer/2022042006/5e6fe594dddf594653104dd5/html5/thumbnails/25.jpg)
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 49
Dither• The target DR for our audio ΣΔ is 18 Bits, or 113dB• Designed SQNR~-120dB allows thermal noise to
dominate at -115dB level• Let’s choose the sampling capacitor such that it limits
the dynamic range:
( )
( )
212
2
2 12
1
1μV
FSFS
n
n FSDR
VDR V Vp
v
v V
= =
→ = =
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 50
5mV DC input
• Thermal noise added at the input of the 1st
integrator
• In-band spurious tones disappear
• Note: they are not just buried
• How can we tell?0 10 20 30 40 50-150
-100
-50
0
50
Frequency [kHz]
Out
put S
pect
rum
[dBW
N]
No ditherWith dither (thermal noise)
Effect of Dither on In-Band Spurious Tones
![Page 26: EE247 Lecture 26 - University of California, Berkeleyee247/fa05/lectures/L26_2_f05.pdfEE247 Lecture 26 • Higher order ΣΔ modulators – Last lectureÆCascaded ΣΔ modulators (MASH)](https://reader033.vdocuments.site/reader033/viewer/2022042006/5e6fe594dddf594653104dd5/html5/thumbnails/26.jpg)
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 51
Effect of Dither on Spurious Tones Near fs/2
Key point: Dither at an amplitude which eliminate the in-band tones has virtually no effect on tones near fs/2
1.47 1.48 1.49 1.5-150
-100
-50
0
50
Frequency [MHz]
Out
put S
pect
rum
[dB
WN
]
No ditherWith dither
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 52
kT/C Noise
• So far we’ve looked at noise added to the input of the ΣΔ modulator, which is also the input of the first integrator
• Now let’s add noise also to the input of the second integrator
• Let’s assume a 1/16 sampling capacitor value for the 2nd integrator wrt the 1st integrator– This gives 4μV rms noise
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EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 53
kT/C Noise
• 5mV DC input
• Noise from 2nd integrator smaller than 1st integrator noise shaped
• Why?
0 1 2 3 4 5x 104
-150
-100
-50
0
50
Frequency [Hz]
Out
put S
pect
rum
[dBW
N]
/ In
t. N
oise
[dBV
]
No noise1st Integrator2nd Integrator
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 54
Effect of Integrator kT/C Noise
• Noise from 1st integrator is referred directly to the input• Noise from 2nd integrator is first-order noise shaped• Noise from subsequent integrators attenuated even further
Especially for high oversampling ratios, only the first 1 or 2 integrators add significant thermal noise. This is true also for other imperfections.
Q
I_5I_4I_3I_2I_1
Y
b2b1
a5a4a3a2a1
K1 z -1
1 - z -1
I1
gDAC Gain Comparator
X
-1
1 - z -1
I2
K2 z -1
1 - z -1
I3
K3 z -1
1 - z -1
I4
K4 z -1
1 - z -1
I5
K5 z
![Page 28: EE247 Lecture 26 - University of California, Berkeleyee247/fa05/lectures/L26_2_f05.pdfEE247 Lecture 26 • Higher order ΣΔ modulators – Last lectureÆCascaded ΣΔ modulators (MASH)](https://reader033.vdocuments.site/reader033/viewer/2022042006/5e6fe594dddf594653104dd5/html5/thumbnails/28.jpg)
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 55
Dither
No practical amount of dither eliminates the tones near fs/2
1.47 1.48 1.49 1.5-150
-100
-50
0
50
Frequency [MHz]
Out
put S
pect
rum
[dBW
N]
/ In
t. N
oise
[dBV
]
No noise1st Integrator2nd Integrator
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 56
Full-Scale Inputs• With practical levels of thermal noise added,
let’s try a 5kHz sinusoidal input near full-scale
• No distortion is visible in the spectrum– 1-Bit modulators are intrinsically linear– But tones exist at high frequencies
To the oversampled modulator, a sinusoidalinput looks like two “slowly” alternating DCs …hence giving rise to limit cycles
![Page 29: EE247 Lecture 26 - University of California, Berkeleyee247/fa05/lectures/L26_2_f05.pdfEE247 Lecture 26 • Higher order ΣΔ modulators – Last lectureÆCascaded ΣΔ modulators (MASH)](https://reader033.vdocuments.site/reader033/viewer/2022042006/5e6fe594dddf594653104dd5/html5/thumbnails/29.jpg)
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 57
0 0.5 1 1.5-150
-100
-50
0
50
Frequency [MHz]
Out
put S
pect
rum
[dBW
N]
Output SpectrumIntegrated Noise (30 averages)
Full-Scale Inputs
0 10 20 30 40 50-150
-100
-50
0
50
Frequency [kHz]
Out
put S
pect
rum
[dBW
N]
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 58
Recap
• Dither successfully removes in-band tones that would corrupt the signal
• The high-frequency tones in the quantization noise spectrum will be removed by the digital filter following the modulator
• What if some of these strong tones are demodulated to the base-band prior to digital filtering?
• Why would this happen?Vref Interference
![Page 30: EE247 Lecture 26 - University of California, Berkeleyee247/fa05/lectures/L26_2_f05.pdfEE247 Lecture 26 • Higher order ΣΔ modulators – Last lectureÆCascaded ΣΔ modulators (MASH)](https://reader033.vdocuments.site/reader033/viewer/2022042006/5e6fe594dddf594653104dd5/html5/thumbnails/30.jpg)
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 59
Vref Interference via Modulationx2(t)
x1(t) y(t)
( ) ( )( ) ( )
( ) ( ) ( ) ( )[ ]ttttXXtxtx
tXtxtXtx
212121
21
222
111
coscos2
coscos
ωωωω
ωω
−++=×
==
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 60
Modulation via DAC
DACy(t) v(t)
Vref
( )
( ) ( )
out
sref
ref
y t D 1
V 2.5V 1mV f /2 square wave
v t y t V
= =±
= +
= ×
![Page 31: EE247 Lecture 26 - University of California, Berkeleyee247/fa05/lectures/L26_2_f05.pdfEE247 Lecture 26 • Higher order ΣΔ modulators – Last lectureÆCascaded ΣΔ modulators (MASH)](https://reader033.vdocuments.site/reader033/viewer/2022042006/5e6fe594dddf594653104dd5/html5/thumbnails/31.jpg)
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 61
Modulation via DAC
0 fs/2 fs
DOUT spectrum
Vref spectruminterferer
convolution yields sum of red and green,mirrored tones and noise appear in band
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 62
0 10 20 30 40 50-150
-100
-50
0
50
Frequency [kHz]
Out
put S
pect
rum
[dB
WN
]
0V1μV1mV
60dB (1 dB/dB)
Vref Interference via Modulation
Key Point:In high resolution ΣΔmodulators Vref interference via modulation can significantly limit the maximum dynamic range
![Page 32: EE247 Lecture 26 - University of California, Berkeleyee247/fa05/lectures/L26_2_f05.pdfEE247 Lecture 26 • Higher order ΣΔ modulators – Last lectureÆCascaded ΣΔ modulators (MASH)](https://reader033.vdocuments.site/reader033/viewer/2022042006/5e6fe594dddf594653104dd5/html5/thumbnails/32.jpg)
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 63
Symmetry of the spectra at fs/2 and DC confirm that this is modulation
1.47 1.48 1.49 1.5x 10 6
-150
-100
-50
0
50
Frequency [Hz]
Out
put S
pect
rum
[dB
WN
]
0V1e-006V0.001V
0 1 2 3 4 5x 104
-150
-100
-50
0
50
Frequency [Hz]
Out
put S
pect
rum
[dB
WN
] 0V1e-006V0.001V
Vref Interference via Modulation
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 64
0 10 20 30 40 50-150
-100
-50
0
50
Frequency [kHz]
Out
put S
pect
rum
[dB
WN
]
0.006V0.012V
Vref Spurious Tone Velocity vs Native Tone Velocity
0.6kHz/mV
Vin = 6mV / 12mV DCVref = 2.5V DC
& 1mV fs/2
40dB shift for readability
Native tone
Aliased tone
• Native tone velocity
1.2kHz/mV
• Aliased tone velocity
0.6kHz/mV
![Page 33: EE247 Lecture 26 - University of California, Berkeleyee247/fa05/lectures/L26_2_f05.pdfEE247 Lecture 26 • Higher order ΣΔ modulators – Last lectureÆCascaded ΣΔ modulators (MASH)](https://reader033.vdocuments.site/reader033/viewer/2022042006/5e6fe594dddf594653104dd5/html5/thumbnails/33.jpg)
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 65
• Simulations performed to verify the effect of the DAC referencecontamination via output signal interference particularly in the vicinity of fs /2
• Interference modulates the high-frequency tones
• Since the high frequency tones are strong, a small amount (1μV) of interference suffices to create audible base-band tones
• Stronger interference (1mV) not only aliases spurious tones but elevated raises noise floor by aliasing high frequency quantization noise
• Amplitude of modulated tones is proportional to interference
• The velocity of modulated tones is half that of the native tones
• Such differences help debugging of silicon
• How clean does the reference have to be?
Vref Interference via Modulation
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 66
0 10 20 30 40 50-150
-100
-50
0
50
Frequency [kHz]
Out
put S
pect
rum
[dB
WN
] /
Int.
Noi
se [d
BV]
Output Spectrum (1μV interference on Vref)Integrated Noise (30 averages)
Vref Interference
Tone dominates noise floorw/o thermal noise
![Page 34: EE247 Lecture 26 - University of California, Berkeleyee247/fa05/lectures/L26_2_f05.pdfEE247 Lecture 26 • Higher order ΣΔ modulators – Last lectureÆCascaded ΣΔ modulators (MASH)](https://reader033.vdocuments.site/reader033/viewer/2022042006/5e6fe594dddf594653104dd5/html5/thumbnails/34.jpg)
EECS 247 Lecture 26: Oversampling Data Converters © 2005 H. K. Page 67
Summary• Our stage 2 model can drive almost all capacitor sizing
decisions– Gain scaling– kT/C noise– Dither
• Dither quite effective in the elimination of native in-band tones
• Extremely clean & well-isolated Vref is required for high-dynamic range applications e.g. digital audio
• Next we will add relevant component imperfections:
Effect of component nonlinearities on ΣΔ performance