Sigma Delta Modulation
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Transcript of Sigma Delta Modulation
Integrated Systems Laboratory ETH Zürich
Introduction to
Sigma-Delta Modulation
ESPRIT - Mixed Signal Design Cluster - Work Shop
Prof. Dr. Qiuting Huang
Seedamm-Plaza, Pfäffikon, SwitzerlandOctober 22nd, 2001
Integrated Systems Laboratory ETH Zürich
Outline
q ADC as a Tracking Loop
q Linearity and Quantization Noise
q First Order Σ∆-Modulator
q Higher Order Loops
q Cascaded Modulators
q Non-Idealities
q Conclusions
Integrated Systems Laboratory ETH Zürich
Motivation for Oversampled Converters
q Need for spectral efficiency in communications systems (ADSL, UMTS, etc. ) => high requirements for A/D and D/A converters (12-14 bit of resolution and linearity)
q VLSI technologies: Low component accuracy, decreasing analog signal dynamic range => difficult for Nyquist rate converters and anti-aliasing filters
q IDEA: Exchange of speed + complexity vs. analog resolution
q Solution: Oversampled data converters, spectral shaping of
quantization noise
Integrated Systems Laboratory ETH Zürich
Published Sigma-Delta Converters
PrecisionMeasurement
Speech
Audio
MobileCommunications
GSMADSL
Broadband WirelineCommunications
ADAMS86
BOSER88 LONGO93
KERTH94
HAIRAPETIAN96
FUJIMORI96
LEUNG97
RABII97
FELDMAN98BURGER98
GEERTS00
NAIKNAWARE00
GEERTS00
BALMELLI00BURGER01
10 100 1k 10k 100k 1M 10M 100M50
150
120
110
140
130
100
80
90
70
60
Dyn
amic
Ran
ge [d
B]
Input Signal Bandwidth [Hz]
Speed vs. ResolutionTrade-Off
Integrated Systems Laboratory ETH Zürich
Generic ADC as Tracking Loop
q D/A converter determines gain and linearity of entire A/D converter
Examples:- Dual-Slope- SAR- Σ∆ - Modulation
( )xfyfqA
xqAy
qA1band signalin
1
1−
>>
= →+
=o
oooo
A(u)
A/D (Quantizer)
D/A
q(w)
f(y)
x u wy
n
- n
Integrated Systems Laboratory ETH Zürich
D/A Converter Linearity and Quantizer Levels
q Dynamic Range Is Determined by Linearity and Noise
q Static Linearity Depends on Matching Accuracy of ComponentsExcept for a 1-bit D/A
q Quantization Noise Decreases with Increasing Quantizer Levels
q Noise Power Is Spread Between DC and fs/2
q DR Can Be Traded with Speed
Integrated Systems Laboratory ETH Zürich
Spectral Behavior of Quantization Error (Noise)
0
-40
-20
-60
-80
-1000 Fs/2Frequency [Hz]
Mag
nitu
de [d
BFS]
10bit Quantizer (Quantization Error):
0
-40
-20
-60
-80
-1000 Fs/2Frequency [Hz]
Mag
nitu
de [d
BFS]
10bit Quantizer (Output Signal):
0
-40
-20
-60
-80
-100 Fs/2Frequency [Hz]
Mag
nitu
de [d
BFS]
5bit Quantizer (Quantization Error):
0
-40
-20
-60
-80
-1000 Fs/2Frequency [Hz]
Mag
nitu
de[d
BFS
]
1bit Quantizer (Quantization Error):
Integrated Systems Laboratory ETH Zürich
Av.
0.26
-1
v[k]
0.25
0.24
0.23
0.22
0.21
1-0.1-0
ε[k]x[k]u[k]K
First Order Σ∆ ModulatorBlock Diagram Example:
First Order Shaped Noise
x[k] = x[k-1] + ε[k-1]v[k] = Q(x[k])ε[k] = u[k] – v[k]
Low-Pass
Integrator 1bitquantizer
D/A
u(kTs) v(kTs) d(kTs)
∫ dt
Fs/2000 Fs/2-100
-80
-60
-40
-20
0
Fs/200 Fs/20Frequency [Hz]
Mag
nitu
de [d
B FS]
D
v[k]x[k]
-1
+1
u[k] ε[k]Q(.)
0.9 1 - 0.8
0.1 1 - 0.8
- 0.7 -1 1.2
0.5 1 - 0.8
- 0.3 -1 1.2
0.9 1 - 0.8
0.2
Integrated Systems Laboratory ETH Zürich
General Single-Path Σ∆ Modulator
D/A
x(kTs) y(kTs)
A/DH(z)
w(kTs)
e(kTs)
The power spectrum of quantizer error e(kTs) = y(kTs) – w(kTs) can be assumed white and uncorrelated with x(kTs), if the latter is sufficiently active.
The power of quantization error is:
srms TdEe ωπ
π
=ΩΩΩ= ∫ ,)(21 2
0
22
+−
Integrated Systems Laboratory ETH Zürich
Linearized General Model
)()(1
1)(
)(1)(
)( zEzH
zXzH
zHzY
++
+=
D/A
X(z) Y(z)
A/DH(z)
W(z)
E(z)
+
NTF1)(1
1)(1)(1
)()(1
)(STF −=
+−
++
=+
=zHzH
zHzX
zHzH
NTFSTF
1STF(Ω)
<<1NTF(Ω)
>>1H(Ω)
In-bandX(z) Y(z)
H(z)
W(z)
E(z)
Integrated Systems Laboratory ETH Zürich
Higher Order Loops
L
zz
zH
−
= −
−
1
1
1)(
Output noise spectral density
)()1()()1(
)1()(1
1)()( 1 Ω−≈ →Ω
+−−
==+
Ω=Ω Ω−<<ΩΩ−Ω−
Ω−
Ω EeEee
eezH
EF LjLjLj
Lj
j
)2( ss fTT πω ==Ω
Total in-band noise
∫∫∫ΩΩ
Ω−Ω
Ω−
ΩΩ
Ω
=ΩΩ−=ΩΩΩ=ooo
o
dEedEedFFNL
rms
Ljo
0
22
2
0
22* )()2
sin(21
)(11
)()(21
πππ
)12(2
2
0
2221,)sin(
12)12(1 )12( +−
Ω<<≈
+=Ω
+=ΩΩ →
+
∫ LL
rmsormsL
rmsxxx OSR
Le
Le
deL
o πππ soo T
OSRω
ππ=
Ω=
z-1
H(z)
z-1z-1
Integrated Systems Laboratory ETH Zürich
RMS-Noise in Signal Band
O dB corresponds to that of PCM sampled at Nyquist rate
Integrated Systems Laboratory ETH Zürich
Stability of Higher Order Modulators
Solutions:
q Add feedforward and feedback paths to increase damping
⇒ Higher order single loop modulators
q Cascade 1st and 2nd order stages
⇒ Cascaded modulators
NTF(z) = (1-z-1)L implemented with a chain of integrators can lead to unstable modulator behavior for L ≥ 3
1
6 dB
00
2
sTje ω−−1
sTωπ
Integrated Systems Laboratory ETH Zürich
Design of Single Loop Modulators
1. Design of NTF(z) - Filter function (e.g. inverse Chebychev)- STF = 1 – NTF- Steeper noise shaping can be tolerated
with more quantizer levels
2. Compute H(z)
H(z) = 1 - 1/NTF
1. Choose appropriate topology toimplement H(z)
2. Compute coefficients
3. Simulate behavior with quantizer
⇒ ds-toolbox (R. Schreier, ftp://next242.ece.orst.edu/pub/delsig.tar.Z)
103
104
105
106
90
80
70
60
50
40
30
20
10
10
Frequency [Hz]
0
100
STF NTF
Pass-Band
d1 = 4/5
d2 = 3/5
b1 = 1/4
b2 = 1/2
a1 = 1/4
c2 = 0.0105
d3 = 3/5b3 = 1/6
d2a1
b3b2b1
d1
c2
d3
Integrated Systems Laboratory ETH Zürich
Achievable SNR
Butterworth Type NTF(all zeros at DC)
Optimized NTF(zeros optimally spread over pass-band)
0
20
40
60
80
100
120
140
160
L = 2
L = 3
L = 4
L = 5
25612864321684
OSR
0
20
40
60
80
100
120
140
160
L = 2
L = 3
L = 4
L = 5
25612864321684
OSR
Integrated Systems Laboratory ETH Zürich
Stability of Single Loop Modulators
H(z)X(z) Y(z)-
k +
E(z)
Quantizer Gain0 ≤ k ≤ 1
)(11
zkHNTF
+=
)(1)(zkH
zkHSTF
+=
q Single bit quantizer- Ardalan/Paulos (TrCAS, Jun87)- Maguire/Huang (ISCAS94)
q Multi bit quantizerv
u
v = k u
Integrated Systems Laboratory ETH Zürich
Cascaded Modulators
Single Loop Modulator
Cascaded Modulator
H(z) A/D
D/A
Analoginput
- Digitaloutput
H(z) A/D
D/A
Analoginput
- Digitaloutput 1
H(z) A/D
D/A
Digitalsignalprocessing
-
Digitaloutput 2
-
Finaldigitaloutpu t
Integrated Systems Laboratory ETH Zürich
1-1 Cascaded Modulator
⇒ Performance depends on matching of analog and digital transfer functions
)2())(1( 21'
3'
22 −−−−− +−−−−+≈ nnnnnnn eeeeegxy
X(z)
-z-1
z-1 z-1- Y(z)
-z-1
ne
'ne
)( '1
'−−+ nnn eex
'1−⋅ neg
1'
2 −− −+⋅ nnn eeeg
-
Integrated Systems Laboratory ETH Zürich
Σ∆ Modulator Non-Idealities I
q Nonlinear STF due to quantization (Huang/Maguire (ISCAS 93))
- Example: 3rd order modulator with inv. Chebychev NTF
10 6-140
-120
-100
-80
-60
-40
-20
0
89 dB
10 4 10 510 3
Frequency [Hz]
Output Spectrum at Maximum Input Level
89 dB
3rdHarmonics
Integrated Systems Laboratory ETH Zürich
Σ∆ Modulator Non-Idealities II
q Tones - Limit cycles in the pass-band - Tones near fs/2
q Circuit non-idealities → other presentations
5.7 5.8 5.9 6 6.1 6.2 6.3 6.4 6.5x 10 6
-140
-120
-100
-80
-60
-40
-20
0
Tones- 6 dBfull scale
30 kHz sinusoidal input@-43 dB full scale
Output Spectrum Near fs/2
Frequency [Hz]
-140
-120
-100
-80
-60
-40
-20
0
- DC-input @ -51 dBfull scale
Tones
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10
5
Output Spectrum of 2x Pass-Band Width
Frequency [Hz]
Integrated Systems Laboratory ETH Zürich
Conclusions
q Linearity of an A/D converter is determined by that of its D/A
q Matching accuracy is typically limited, which makes linearity a problem for monolithic integration of A/D converters
q One-bit D/A is potentially very linear, but too much noise is generated by a 1-bit quantizer
q Noise shaping, effected by feedback, enables S and N to be separated
q A Σ∆ converter is therefore a combination of 1-bit quantizer for linearity, noise shaping for S and N separation and digital filtering for noise removal