Upload
b-naresh-kumar-reddy
View
234
Download
1
Tags:
Embed Size (px)
DESCRIPTION
Second and Higher-OrderDelta-Sigma Modulators
Citation preview
Second and Higher-OrderDelta-Sigma Modulators
MEAD March 2008
Richard [email protected]
ANALOGDEVICES
R. SCHREIER ANALOG DEVICES, INC.
2
Overview1 MOD2: The 2nd-Order Modulator
MOD2 from MOD1 NTF (predicted & actual) SQNR performance Stability Deadbands, Distortion & Tones (audio demo) Topological Variants
2 Higher-Order Modulators MODN from MOD1 NTF Zero Optimization Stability SQNR limits for binary and multi-bit modulators Topology Overview
R. SCHREIER ANALOG DEVICES, INC.
3
1. MOD2 from MOD1 Replace the quantizer in MOD1 with another copy of
MOD1 in a recursive fashion:
z-1
Q
z-1
Replace Q with MOD1
z-1
z-1
U V
E
X1
V U 1 z 1( )E+=
E1
V X1 E+ X1 1 z 1( )E1+= =E 1 z 1( )E1=V U 1 z 1( )2E1+=
R. SCHREIER ANALOG DEVICES, INC.
4
Simplified Diagram Combine feedback paths, absorb feedback delay into
second integrator
and the STF is
MOD2s NTF is the square of MOD1s NTF
Q1z1z
z1U V
E
V z( ) z 1 U z( ) 1 z 1( )2E z( )+=
NTF z( ) 1 z 1( )2= STF z( ) z 1=
R. SCHREIER ANALOG DEVICES, INC.
5
NTF Comparison
103 102 101100
80
60
40
20
0N
TFe
j2
f(
)(d
B)
Normalized Frequency
MOD1
MOD2Twice as much attenuationat all frequencies
R. SCHREIER ANALOG DEVICES, INC.
6
Predicted Performance In-band quantization noise power
Quantization noise drops as the 5th power of OSR!SQNR increases at 15 dB per octave increase in OSR.
IQNP NTF e j2f( ) 2 See f( ) fd0
1 2 OSR( )
=2f( )4 2e2 fd
0
1 2 OSR( )
4e2
5 OSR( )5----------------------=
R. SCHREIER ANALOG DEVICES, INC.
7
Predicted SQNR vs. OSR
For and binary quantization, the predictedSQNR of MOD2 is 94.2 dB
23 24 25 26 27 28 29 2100
20
40
60
80
100
120
140
OSR
SQNR
(dB)
MOD1MOD2
OSR 128=
R. SCHREIER ANALOG DEVICES, INC.
8
MOD2 Simulated PSDHalf-scale sine-wave input with
Observed PSD similar to theory (40 dB/decade slope)But 3rd harmonic is visible and in-band PSD is slightly higher.
f f s 500
103 102 101Normalized Frequency
dBFS
/NBW
140
120
100
80
60
40
20
0SQNR = 85 dB@ OSR = 128
40 dB/decade
Theoretical PSDk = 1
Simulated spectrum(smoothed)
NBW = 5.7106
R. SCHREIER ANALOG DEVICES, INC.
9
Gain of the Quantizer in MOD2 The effective quantizer gain can be computed from the
simulation data using[S&T Eq. 2.5]
For the preceding simulation, . alters the NTF:
k v y, y y, ---------------E y[ ]E y2[ ]---------------= =
k 0.63=k 1
-1 0 1-1
0
1
NTFk z( ) NTF1 z( )k 1 k( )NTF1 z( )+------------------------------------------------=
R. SCHREIER ANALOG DEVICES, INC.
10
Revised PSD Prediction
Agreement is now excellent
103 102 101
Theoretical PSDk = 0.63
Normalized Frequency
dBFS
/NBW
NBW = 5.7106140
120
100
80
60
40
20
0
Simulated Spectrum(smoothed)
R. SCHREIER ANALOG DEVICES, INC.
11
Variable Quantizer Gain When the input is small (below -12 dBFS), the effective
gain of the quantizer is The small-signal NTF is thus
This NTF has 2.5 dB les quantization noise suppressionthan the NTF derived from the assumptionthat
Thus the SQNR should be about 2.5 dB lower than . As the input signal increases, k decreases and the
suppression of quantization noise degradesSQNR increases less quickly than the signal power, and eventually theSQNR saturates and then decreases as the signal power is increased.
k 0.75=
NTF z( ) z 1( )2
z2 0.5z 0.25+--------------------------------------=
1 z 1( )2k 1=
R. SCHREIER ANALOG DEVICES, INC.
12
Simulated SQNR of MOD2
Well-modeled by ideal formula; less erratic than MOD1Saturation at high signal levels due to decreased quantizer gain andaltered NTF. (Worse with low-frequency inputs.)
100 80 60 40 20 00
20
40
60
80
100
Input Amplitude (dBFS)
SQNR
(dB)
Low-frequency inputHigh-frequency input
15A2 OSR( )524
--------------------------------
SP A2 2=
IQNP 4 1 3( )
5 OSR( )5----------------------=
SQNR SPIQNP---------------=
R. SCHREIER ANALOG DEVICES, INC.
13
Stability of MOD2 Known to be stable with DC inputs up to full-scale,
but the state bounds blow up asHein [ISCAS 1991]:
(output of 1st integrator)
(output of 2nd integrator)
However, with a hostile input (whose magnitude is lessthan 30% of full-scale) MOD2 can be driven unstable!
As a result of this input-dependent stability, it is wise tokeep the input below 70-90% of full-scale
u 1u 1
x1 u 2+
x25 u( )2
8 1 u( )-----------------------
0 100 200-0.5
0
0.5
u u = 0.3
R. SCHREIER ANALOG DEVICES, INC.
14
Deadbands, Distortion & TonesAudio Comparison of MOD1 and MOD2
-1
1
10 100 1kHz-100
-80
-60
-40
-20
0
-1
1
102
104
-100
-80
-60
-40
-20
0
NBW = 3.0 Hz
-1
1
10 100 1kHz-100
-80
-60
-40
-20
0
NBW = 0.8 Hz
R. SCHREIER ANALOG DEVICES, INC.
15
ObservationsTones
Quantization noise of MOD1 is distinctly non-whiteAudible tones when input is near zero, or near other simple rationalfractions of full-scale.
MOD2 is better than MOD1 in terms of its tendencytoward tonal behavior
Dead-bands
MOD1 has dead-bands whose widths are proportionalto 1/A, where A is the gain of the internal op-amp
MOD2 has dead-bands whose widths are proportionalto 1/A2
Dead-band behavior is less problematic in MOD2
R. SCHREIER ANALOG DEVICES, INC.
16
Topological VariantDelaying Integrators
+ Delaying integrators reduce the settling requirementsCan decouple the integration phase from the driving phase
Q1z1
1z1U V
E
-2-1
NTF z( ) 1 z 1( )2= STF z( ) z 2=
R. SCHREIER ANALOG DEVICES, INC.
17
Topological VariantFeed-Forward
+ Output of first integrator has no DC componentDynamic range requirements of this integrator are relaxed.
Although near , forInstability is more likely.
Qzz1
1z1U V
E
NTF z( ) 1 z 1( )2= STF z( ) 2z 1 z 2=
STF 1 0= STF 3= =
R. SCHREIER ANALOG DEVICES, INC.
18
Topological VariantFeed-Forward with Extra Input Feed-In
+ No DC component in either integrators outputReduced dynamic range requirements in both integrators.
+ Perfectly flat STFNo increased risk of instability.
Qzz1
1z1U V
E
NTF z( ) 1 z 1( )2= STF z( ) 1=
R. SCHREIER ANALOG DEVICES, INC.
19
Topological VariantError Feedback
+ Simple Very sensitive to gain errors
Only suitable for digital implementations.
QU V
E
NTF z( ) 1 z 1( )2= STF z( ) 1=
z-1z-1
2E
R. SCHREIER ANALOG DEVICES, INC.
20
2. MODN from MOD2
NTF of MODN is the Nth power of MOD1s NTF
Q1z1z
z1U Vz
z1
V E 1z 1----------- V
zz 1----------- V
zz 1----------- V U+( )+ + +=
V z( ) z 1 U z( ) 1 z 1( )N E z( )+=
1 z 1( )NV 1 z 1( )NE 1 z 1( )N 1 1 z 1( )N 2 1+ + +( )z 1 V z 1 U+=1 z 1( )NV 1 z 1( )NE 1 z
1( )N 1
1 z 1 1---------------------------------- z 1 V z 1 U+=
1 z 1( )NV 1 z 1( )NE 1 z1
( )N 1z 1
---------------------------------- z 1 V z 1 U+=
R. SCHREIER ANALOG DEVICES, INC.
21
NTF Comparison
NTF
ej2
f(
)(d
B)
Normalized Frequency10-3 10-2 10-1
-100
-80
-60
-40
-20
-0
20
40
MOD1
MOD2
MOD3
MOD4
MOD5
R. SCHREIER ANALOG DEVICES, INC.
22
Predicted Performance In-band quantization noise power
Quantization noise drops as the (2N+1)th power ofOSR!
SQNR increases at (6N+3) dB per octave increase in OSR.
IQNP NTF e j2f( ) 2 See f( ) fd0
1 2 OSR( )
=2f( )2N 2e2 fd
0
1 2 OSR( )
2N
2N 1+( ) OSR( )2N 1+--------------------------------------------------e2
=
R. SCHREIER ANALOG DEVICES, INC.
23
Improving NTF PerformanceZero Optimization
Minimize the rms in-band value of H by finding the aiwhich minimize the integral of over the passband.
Normalize passband edge to 1 for ease of calculation.H 2
x2 a12
( )2 xd1
1 , n = 2x2 x2 a1
2( )2 xd
1
1 , n = 3x2 a1
2( )2 x2 a22( )2 xd
1
1 , n = 4
-1 1a-a
H f( ) 2 i.e. Find the ai which minimize the integral
f f B
R. SCHREIER ANALOG DEVICES, INC.
24
Solutions Up to Order = 8Order Optimal Zero Placement Relative to fB SQNR Improvement
1 0 0 dB
2 3.5 dB
3 0, 8 dB
4 13 dB
5 0, [Y. Yang] 18 dB
6 0.23862, 0.66121, 0.93247 23 dB7 0, 0.40585, 0.74153, 0.94911 28 dB8 0.18343, 0.52553, 0.79667, 0.96029 34 dB
13
-------
35---
37---
37---
2 335------
59---
59---
2 521------
R. SCHREIER ANALOG DEVICES, INC.
25
Topological Implication Apply feedback around pairs integrators:
1z1
zz1
-g
1z1
1z1
-g
2 Delaying Integrators Non-delaying + DelayingIntegrators (LDI Loop)
Poles are the roots of
1 g 1z 1-----------
2+ 0=
i.e. z 1 j g=Not quite on the unit circle,but fairly close if g
R. SCHREIER ANALOG DEVICES, INC.
27
The 3rd-Order Modulator is Unstablewith a Binary Quantizer
The quantizer input grows without boundThe modulator is unstable, even with an arbitrarily small input.
0 10 20 30 40-1
0
1
0 10 20 30 40-200-100
0100200
Sample Number
v
yHUGE!
Longstringsof +1/-1
R. SCHREIER ANALOG DEVICES, INC.
28
Solutions to the Stability ProblemHistorical Order
1 Use multi-bit quantizationOriginally considered undesirable because the inherent linearity of a1-bit DAC is lost when a multi-bit quantizer is used.Less of an issue now that mismatch-shaping is available.
2 Use a more general NTF (not pure differentiation)Lower the NTF gain so that quantization error is amplified less.A common rule of thumb is to limit the maximum NTF gain to ~1.5.Unfortunately, limiting the NTF gain reduces the amount by whichquantization noise is attenuated.
3 Use a multi-stage (MASH) architectureMore on this later in the course.
Combinations of the above are possible
R. SCHREIER ANALOG DEVICES, INC.
29
Multi-bit Quantization Can show that a modulator with NTF H and unity STF
is guaranteed to be stable if at all times,where and
In MODN,, so
, whereand thus .
Thus implies .MODN is guaranteed to be stable with an n-bit quantizer if the inputmagnitude is less than /2.This result is extremely conservative.
Similarly, guarantees the modulator isstable for inputs up to 50% of full-scale.
u umax
h 1 H 1( ) 2N= =nlev 2N= umax nlev 1 h 1+ 1= =
nlev 2N 1+=
R. SCHREIER ANALOG DEVICES, INC.
30
Proof of CriterionBy Induction
Assume STF = 1 and . Assume for . [Induction Hypothesis]Then
Thus
And by induction for all i > 0. QED
h 1
n( ) u n( ) umax( )e i( ) 1 i n