Second and Higher-Order
Delta-Sigma Modulators
MEAD March 2008
Richard Schreier
Richard.Schreier@analog.com
ANALOG
DEVICES
R. SCHREIER
ANALOG DEVICES, INC.
Overview
1
MOD2: The 2nd-Order Modulator
•
•
•
•
•
•
2
MOD2 from MOD1
NTF (predicted & actual)
SQNR performance
Stability
Deadbands, Distortion & Tones (audio demo)
Topological Variants
Higher-Order Modulators
•
•
•
•
•
MODN from MOD1
NTF Zero Optimization
Stability
SQNR limits for binary and multi-bit modulators
Topology Overview
2
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ANALOG DEVICES, INC.
1. MOD2 from MOD1
• Replace the quantizer in MOD1 with another copy of
MOD1 in a recursive fashion:
Replace Q with MOD1
E
E1
Q
X1
U
V
z-1
z-1
z-1
z-1
V =
V =
⇒E
⇒V
U + ( 1 – z –1 )E
X 1 + E = X 1 + ( 1 – z –1 )E 1
= ( 1 – z –1 )E 1
= U + ( 1 – z –1 ) 2 E 1
3
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ANALOG DEVICES, INC.
Simplified Diagram
• Combine feedback paths, absorb feedback delay into
second integrator…
E
U
z
z−1
1
z−1
Q
V
V ( z) = z – 1 U ( z) + ( 1 – z – 1 ) 2 E ( z)
• NTF( z) = ( 1 – z –1 ) 2 and the STF is STF( z) = z –1
• MOD2’s NTF is the square of MOD1’s NTF
4
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ANALOG DEVICES, INC.
NTF(e j2πf)
(dB)
NTF Comparison
0
−20
MOD1
−40
−60
MOD2
Twice as much attenuation
at all frequencies
−80
−100
10–3
10–2
10–1
Normalized Frequency
5
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ANALOG DEVICES, INC.
Predicted Performance
• In-band quantization noise power
1 ⁄ ( 2 ⋅ OSR )
∫
IQNP =
NTF(e j2πf ) 2 ⋅ S ee( f ) d f
0
1 ⁄ ( 2 ⋅ OSR )
≈
∫
( 2πf ) 4 ⋅ 2σ e2 d f
0
π 4 σ e2
= --------------------5 ( OSR ) 5
• Quantization noise drops as the 5th power of OSR!
SQNR increases at 15 dB per octave increase in OSR.
6
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ANALOG DEVICES, INC.
Predicted SQNR vs. OSR
140
120
SQNR (dB)
100
MOD2
80
MOD1
60
40
20
0 3
2
24
25
26
27
28
29
210
OSR
• For OSR = 128 and binary quantization, the predicted
SQNR of MOD2 is 94.2 dB
7
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ANALOG DEVICES, INC.
MOD2 Simulated PSD
Half-scale sine-wave input with
f ≈ f s ⁄ 500
0
dBFS/NBW
–20
SQNR = 85 dB
@ OSR = 128
–40
–60
Simulated spectrum
(smoothed)
Theoretical PSD
k=1
–80
–100
40 dB/decade
–120
–140 –3
10
NBW = 5.7×10−6
10–1
10–2
Normalized Frequency
• Observed PSD similar to theory (40 dB/decade slope)
But 3rd harmonic is visible and in-band PSD is slightly higher.
8
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ANALOG DEVICES, INC.
Gain of the Quantizer in MOD2
• The effective quantizer gain can be computed from the
simulation data using
⟨ v, y⟩
E[ y ]
k = --------------- = -------------[S&T Eq. 2.5]
⟨ y, y⟩
E[ y2 ]
• For the preceding simulation, k = 0.63 .
• k ≠ 1 alters the NTF:
1
NTF 1( z)
NTF k( z) = ----------------------------------------------k + ( 1 – k )NTF 1( z)
0
-1
-1
0
1
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ANALOG DEVICES, INC.
Revised PSD Prediction
0
Theoretical PSD
k = 0.63
–20
dBFS/NBW
–40
–60
Simulated Spectrum
(smoothed)
–80
–100
–120
NBW = 5.7×10−6
–140
10–3
10–2
10–1
Normalized Frequency
• Agreement is now excellent
10
R. SCHREIER
ANALOG DEVICES, INC.
Variable Quantizer Gain
• When the input is small (below -12 dBFS), the effective
gain of the quantizer is k = 0.75
• The “small-signal NTF” is thus
( z – 1 )2
NTF( z) = ------------------------------------z 2 – 0.5 z + 0.25
• This NTF has 2.5 dB les quantization noise suppression
than the ( 1 – z –1 ) 2 NTF derived from the assumption
that k = 1
Thus the SQNR should be about 2.5 dB lower than
.
• As the input signal increases, k decreases and the
suppression of quantization noise degrades
SQNR increases less quickly than the signal power, and eventually the
SQNR saturates and then decreases as the signal power is increased.
11
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ANALOG DEVICES, INC.
Simulated SQNR of MOD2
100
15 A 2 ( OSR ) 5
-------------------------------2π 4
SQNR (dB)
80
High-frequency input
Low-frequency input
60
40
SP = A 2 ⁄ 2
20
0
−100
−80
−60
−40
−20
Input Amplitude (dBFS)
0
π4( 1 ⁄ 3 )
IQNP = ---------------------5 ( OSR ) 5
SP
SQNR = --------------IQNP
• Well-modeled by ideal formula; less erratic than MOD1
Saturation at high signal levels due to decreased quantizer gain and
altered NTF. (Worse with low-frequency inputs.)
12
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ANALOG DEVICES, INC.
Stability of MOD2
• Known to be stable with DC inputs up to full-scale,
but the state bounds blow up as u → 1
Hein [ISCAS 1991]: u ≤ 1 ⇒
x 1 ≤ u + 2 (output of 1st integrator)
( 5 – u )2
x 2 ≤ ----------------------- (output of 2nd integrator)
8(1 – u )
• However, with a hostile input (whose magnitude is less
than 30% of full-scale) MOD2 can be driven unstable!
0.5
u
u
0
-0.5
∞
= 0.3
0
100
200
• As a result of this input-dependent stability, it is wise to
keep the input below 70-90% of full-scale
13
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ANALOG DEVICES, INC.
Deadbands, Distortion & Tones
Audio Comparison of MOD1 and MOD2
1
1
1
-1
-1
-1
0
0
0
-20
-20
-20
-40
-40
-40
-60
-60
-60
-80
-80
-80
NBW = 3.0 Hz
-100
10
100
1kHz
-100
2
10
NBW = 0.8 Hz
-100
10
14
4
10
100
1kHz
R. SCHREIER
ANALOG DEVICES, INC.
Observations
Tones
• Quantization noise of MOD1 is distinctly non-white
Audible tones when input is near zero, or near other simple rational
fractions of full-scale.
• MOD2 is better than MOD1 in terms of its tendency
toward tonal behavior
Dead-bands
• MOD1 has dead-bands whose widths are proportional
to 1/A, where A is the gain of the internal op-amp
• MOD2 has dead-bands whose widths are proportional
to 1/A2
• Dead-band behavior is less problematic in MOD2
15
R. SCHREIER
ANALOG DEVICES, INC.
Topological Variant–
Delaying Integrators
E
1
z−1
U
-1
1
z−1
Q
V
-2
NTF( z) = ( 1 – z –1 ) 2
STF( z) = z –2
+ Delaying integrators reduce the settling requirements
Can decouple the integration phase from the driving phase
16
R. SCHREIER
ANALOG DEVICES, INC.
Topological Variant–
Feed-Forward
E
U
1
z−1
z
z−1
NTF( z) = ( 1 – z –1 ) 2
Q
V
STF( z) = 2 z –1 – z –2
+ Output of first integrator has no DC component
Dynamic range requirements of this integrator are relaxed.
– Although STF ≈ 1 near ω = 0 , STF = 3 for ω = π
Instability is more likely.
17
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ANALOG DEVICES, INC.
Topological Variant–
Feed-Forward with Extra Input Feed-In
E
U
1
z−1
z
z−1
NTF( z) = ( 1 – z –1 ) 2
Q
STF( z) = 1
+ No DC component in either integrator’s output
Reduced dynamic range requirements in both integrators.
+ Perfectly flat STF
No increased risk of instability.
18
V
R. SCHREIER
ANALOG DEVICES, INC.
Topological Variant–
Error Feedback
E
Q
U
V
2
z-1
z-1
–E
NTF( z) = ( 1 – z –1 ) 2
STF( z) = 1
+ Simple
– Very sensitive to gain errors
Only suitable for digital implementations.
19
R. SCHREIER
ANALOG DEVICES, INC.
2. MODN from MOD2
U
z
z−1
z
z−1
1
z−1
Q
V
1
z
z
V = E + -----------  – V + -----------  – V + … ----------- ( – V + U ) 

z–1
z – 1
z – 1
( 1 – z –1 ) N V = ( 1 – z –1 ) N E – ( ( 1 – z –1 ) N – 1 + ( 1 – z –1 ) N – 2 + … + 1 )z –1 V + z –1 U
( 1 – z ) – 1 –1
- z V + z –1 U
( 1 – z –1 ) N V = ( 1 – z –1 ) N E –  -------------------------------- 1 – z –1 – 1 
( 1 – z –1 ) N – 1 –1
- z V + z –1 U
( 1 – z –1 ) N V = ( 1 – z –1 ) N E –  --------------------------------

– z –1
–1 N
∴V ( z) = z –1 U( z) + ( 1 – z –1 ) N E( z)
• NTF of MODN is the Nth power of MOD1’s NTF
20
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ANALOG DEVICES, INC.
NTF Comparison
20
-0
-20
-40
MOD1
-60
-80
D2
MO
3
D
O
M
-100 -3
10
M
O
D
M 4
OD
5
NTF(e j2πf)
(dB)
40
-2
10
-1
10
Normalized Frequency
21
R. SCHREIER
ANALOG DEVICES, INC.
Predicted Performance
• In-band quantization noise power
1 ⁄ ( 2 ⋅ OSR )
∫
IQNP =
NTF(e j2πf ) 2 ⋅ S ee( f ) d f
0
1 ⁄ ( 2 ⋅ OSR )
≈
∫
( 2πf ) 2N ⋅ 2σ e2 d f
0
π 2N
-σ2
= ------------------------------------------------( 2N + 1 ) ( OSR ) 2N + 1 e
• Quantization noise drops as the (2N+1)th power of
OSR!
SQNR increases at (6N+3) dB per octave increase in OSR.
22
R. SCHREIER
ANALOG DEVICES, INC.
Improving NTF Performance–
Zero Optimization
• Minimize the rms in-band value of H by finding the ai
which minimize the integral of H 2 over the passband.
Normalize passband edge to 1 for ease of calculation.
H( f )
i.e. Find the ai which minimize the integral
2
∫
1
( x 2 – a 12 ) 2 d x ,
n = 2
x 2 ( x 2 – a 12 ) 2 d x ,
n = 3
( x 2 – a 12 ) 2 ( x 2 – a 22 ) 2 d x ,
n = 4
–1
∫
1
a
1 f ⁄ fB
–1
∫
1
–1
…
-1 -a
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ANALOG DEVICES, INC.
Solutions Up to Order = 8
Order
Optimal Zero Placement Relative to fB
SQNR Improvement
1
0
0 dB
2
1
± ------3
3.5 dB
3
3
0, ± --5
8 dB
4
3 2 3
3
± --- ±  --- – -----
7
7
35
13 dB
5
5 2 5
5
0, ± --- ±  --- – ------ [Y. Yang]
 9
9
21
18 dB
6
±0.23862, ±0.66121, ±0.93247
23 dB
7
0, ±0.40585, ±0.74153, ±0.94911
28 dB
8
±0.18343, ±0.52553, ±0.79667, ±0.96029
34 dB
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ANALOG DEVICES, INC.
Topological Implication
• Apply feedback around pairs integrators:
Non-delaying + Delaying
Integrators (LDI Loop)
2 Delaying Integrators
-g
1
z−1
-g
1
z−1
z
z−1
Poles are the roots of
1
1 + g  -----------
 z – 1
i.e.
1
z−1
Poles are the roots of
gz
1 + ------------------ = 0
( z – 1 )2
2
= 0
i.e. z = e ± jθ, cos θ = 1 – g ⁄ 2
z = 1± j g
Not quite on the unit circle,
but fairly close if g<<1.
Precisely on the unit circle,
regardless of the value of g.
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ANALOG DEVICES, INC.
Problem: High-Order Modulators Want
Multi-bit Quantizers
e.g. a 3rd-Order Modulator
with an Infinite Quantizer and Zero Input
7
Quantizer input gets large,
even if the input is small.
5
3
6 quantizer levels are
used by a small input.
v1
-1
-3
-5
-70
10
20
30
Sample Number
26
40
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ANALOG DEVICES, INC.
The 3rd-Order Modulator is Unstable
with a Binary Quantizer
Long
strings
of +1/-1
1
v
0
-1
0
10
20
30
40
200
HUGE!
100
y
0
-100
-200
0
10
20
Sample Number
30
40
• The quantizer input grows without bound
The modulator is unstable, even with an arbitrarily small input.
27
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ANALOG DEVICES, INC.
Solutions to the Stability Problem
Historical Order
1 Use multi-bit quantization
Originally considered undesirable because the inherent linearity of a
1-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 which
quantization noise is attenuated.
3 Use a multi-stage (MASH) architecture
More on this later in the course.
• Combinations of the above are possible
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ANALOG DEVICES, INC.
Multi-bit Quantization
• Can show that a modulator with NTF H and unity STF
is guaranteed to be stable if u < u max at all times,
∞
h(i)
where u max = nlev + 1 – h 1 and h 1 = ∑
i=0
• In MODN,
H( z) = ( 1 – z –1 ) N , so
h(n) = { 1, – a 1, a 2, – a 3, … ( – 1 ) N a N, 0… } , where a i > 0
and thus h 1 = H(– 1) = 2 N .
• Thus nlev = 2 N implies u max = nlev + 1 – h
1
= 1.
MODN is guaranteed to be stable with an n-bit quantizer if the input
magnitude is less than ∆/2.
This result is extremely conservative.
• Similarly, nlev = 2 N + 1 guarantees the modulator is
stable for inputs up to 50% of full-scale.
29
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ANALOG DEVICES, INC.
Proof of h
1
Criterion
By Induction
• Assume STF = 1 and ( ∀n ) ( u(n) ≤ u max ) .
• Assume e(i) ≤ 1 for i < n .
[Induction Hypothesis]
Then
y ( n) = u( n) + ∑
≤ u max + ∑
≤ u max + ∑
∞
∞
i=1
i=1
∞
i=1
h(i)e(n – i)
h(i) e(n – i)
h(i) = u max + h 1 – 1
• Thus
( u max ≤ nlev + 1 – h 1 ) ⇒ ( y(n) ≤ nlev ) ⇒ ( e(n) ≤ 1 )
• And by induction e(i) ≤ 1 for all i > 0.
30
QED
R. SCHREIER
ANALOG DEVICES, INC.
The Lee Criterion for Stability
in a 1-bit Modulator: H ∞ ≤ 2
[Wai Lee, 1987]
• The measure of the “gain” of H is
the maximum magnitude of H (over frequency),
otherwise known as the infinity-norm of H:
H ∞ ≡ max( H(e jω) )
ω ∈ [ 0, 2π ]
Q: Is the Lee criterion necessary for stability?
For MOD2, H( z) = ( 1 – z –1 ) 2 and so H ∞ = H ( – 1 ) = 4 .
Since MOD2 is known to be stable, the Lee criterion is not necessary.
Q: Is the Lee criterion sufficient to ensure stability?
No. There are lots of counter-examples, but H
∞
≤ 1.5 often works.
• Let’s look at some examples
31
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ANALOG DEVICES, INC.
The NTF Family Used by the ∆Σ Toolbox
• Poles chosen such that 1 ⁄ den(H( z)) is a
maximally flat transfer function.
Pole Locus for 5th-order NTFs
1
For lowpass modulators, the pole
placement is similar to a
Butterworth transfer function.
Yields a flat STF for both lowpass
and bandpass modulators
employing the CRFB topology with
one feed-in.
0
-1
-1
0
1
32
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ANALOG DEVICES, INC.
σe vs. H
∞
Binary modulators of order N = 3, 4, 5 with a small input
σe
1
1
------3 0.5
N=5
1
N=3
Value of H ∞ resulting
in instability depends
on modulator order, N.
Toolbox
default is
H ∞ = 1.5
0
N=4
1.5
2
H
2.5
∞
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ANALOG DEVICES, INC.
Two 5th-Order Binary Modulators
NTF Magnitude
0
H
∞
= 1.5
H
∞
= 1.75
-20
-40
dB
-60
-80
-100
-120
0
Higher H
∞
has more “aggressive” noise shaping
0.25
Normalized Frequency
34
0.5
R. SCHREIER
ANALOG DEVICES, INC.
σe vs. uDC
For the two 5th-order modulators
σe
2
1
NTF with higher H ∞ has a
lower stable input limit.
0
0
0.25
0.5
DC Input
0.75
1
35
R. SCHREIER
ANALOG DEVICES, INC.
SQNR vs. H
∞
SQNR
(dB)
5th-Order NTFs, Binary Quantization, OSR = 32
SQNR has a broad maximum
90
70
50
1
1.25
1.5
1.75
2
1.75
2
umax
(dBFS)
0
-10
-20
Stable input limit drops
as H ∞ increases.
1
1.25
1.5
H
36
∞
R. SCHREIER
ANALOG DEVICES, INC.
N=8
N=7
N=6
N=5
SQNR Limits for Binary Modulators
140
N=4
N=3
N=2
Peak SQNR (dB)
120
100
N=1
80
60
40
20
0
4
8
16
32
64
128
256
512
1024
OSR
37
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ANALOG DEVICES, INC.
N=8
N=7
N=6
N=5
SQNR Limits for 2-bit Modulators
140
N=4 N=3
N=2
Peak SQNR (dB)
120
100
N=1
80
60
40
20
0
4
8
16
32
64
OSR
38
128
256
512
1024
R. SCHREIER
ANALOG DEVICES, INC.
N=8
N=7
N=6
N=5
SQNR Limits for 3-bit Modulators
140
N=4
N=3
N=2
Peak SQNR (dB)
120
N=1
100
80
60
40
20
0
4
8
16
32
64
128
256
512
1024
OSR
39
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ANALOG DEVICES, INC.
SQNR (dB) @ OSR = 8
with 1 LSB (peak) input
Theoretical SQNR Limits
for Multi-Bit Modulators
H
120
H
H
100
∞
∞
∞
N=8
= 32
N=7
= 16
N=6
= 8
N=5
H
80
∞
= 4
N=4
60
40
20
H ∞
= 2
N=3
N=2
100
101
Total RMS Noise Power (LSBs)
40
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ANALOG DEVICES, INC.
Example Waveforms
7th-order 17-level modulator, H
∞
= 2
16
SQNR = 55 dB
@ OSR = 8
0
-16
0
50
100
Time Step
41
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ANALOG DEVICES, INC.
Example Waveforms
7th-order 17-level modulator, H
∞
= 8
16
SQNR = 105 dB
@ OSR = 8
0
-16
0
50
Time Step
42
100
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ANALOG DEVICES, INC.
Spectra
0
SNR = 105dB @ OSR=8
SNR = 55dB @ OSR=8
–20
dBFS/NBW
–40
–60
–80
–100
–120
–140
NBW=1.8x10–4
–160
0
0.1
0.2
0.3
0.4
Normalized Frequency
0.5
43
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ANALOG DEVICES, INC.
SNR Curves
120
peak SNR = 105dB
@ OSR = 8
SNR (dB)
100
stable input
limit = 0dBFS
80
peak SNR = 61dB
@ OSR = 8
60
40
stable input
limit = –6dBFS
20
0
–120
–100
–80
–60
–40
Input Level (dBFS)
44
–20
0
R. SCHREIER
ANALOG DEVICES, INC.
Example Topology– Feedback
-g
U
I
-a1
I
I
-a2
Q
V
Adjust ai to get
the desired NTF
-a3
• N integrators precede the quantizer
• Feedback from the quantizer to the input of each
integrator (via a DAC)
• Local feedback around pairs of integrators is possible
• Multiple input feed-in branches are also possible
45
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ANALOG DEVICES, INC.
Example Topology– Feedforward
a1
a2
U
I
I
I
-g
a3
Q
V
• N integrators in a row
• Each integrator output is fed forward to the quantizer
• Local feedback around pairs of integrators is possible
• Multiple input feed-in branches are also possible
46
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ANALOG DEVICES, INC.
General Single-Quantizer ∆Σ Modulator
• The input to the quantizer is some linear combination
of the input to the modulator and the fed-back output
E
U
L0
Y
V
L1
Y = L0 U + L1 V
V = GU + HE , where
1
H = -------------- & G = L 0 H
1 – L1
V = Y+E
Inverse Relations:
L1 = 1 – 1/H, L0 = G/H
47
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ANALOG DEVICES, INC.
Summary
• MOD2 is better than MOD1
Higher SQNR
Whiter quantization noise
Smaller deadbands
• MODN is better than MOD2
Even higher SQNR
Tonal behavior unlikely
Deadbands virtually eliminated
• BUT high-order modulators must deal with instability
Modify the NTF, reduce the input range, and/or use multi-bit
quantization
48