l19 lecture
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 1A/DDSP
Oversampled ADCs
• One-bit quantization
– Quantization noise shaping
• A first-order, 1-Bit sigma-delta modulator
– The name … sigma-delta, delta-sigma, Σ∆, ∆Σ, …
– Time domain model
– Small-signal model
– Oversampling
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 2A/DDSP
Oversampling
• Nyquist rate ADCs
– Sample at fs around 2x bandwidth
– Resolution set by number of decision levels of quantizer
• Oversampled ADCs
– Sample at fs >> bandwidth (16 … 500x) – Use few quantization levels (typical 1-Bit)
– Employ DSP to reduce quantization error
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 3A/DDSP
One-Bit Quantization
• Let’s examine some properties of one bit randomsequences
– Values in the sequence are constrained to be either +1 or –1
• By picking +1’s and –1’s at random (using MATLAB’srand.m random number generator), we generate asequence with zero mean
– The sequence values model outputs of a hypothetical 1-Bit,1MHz ADC
– Nothing stops us from doing a DFT of this sequence …
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 4A/DDSP
Zero Mean 1-Bit DFT
Frequency (kHz)
A m
( d B W N ) 0
-30
-60
-900 400300200100 500
30
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 5A/DDSP
One-Bit DFTs
• For 1-Bit DFT plots, we’ll use a different normalizationscheme
– The dBWN (dB White Noise) scale sets the 0dB line at thenoise/bin of a random +1, -1 sequence
– From the energy theorem,
Σn=0
N-1
an2 Σ
m=0
N-1
Am21
N= N = ⇒ Am = √N
(+1)2 or (–1)2 = 1
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 6A/DDSP
Zero Mean 1-Bit DFTaverage of 30 spectra
Frequency (kHz)
A m
( d B W N ) 0
-30
-60
-900 400300200100 500
30
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 7A/DDSP
Non-zero Mean Sequences
• Of course, 1-Bit sequences can represent dc
inputs from –1 to +1
• For example, an average value of +1/11 can
be generated by a sequence with
– 5/11 probability of –1
– 6/11 probability of +1
• Let’s look at DFTs of a non-zero mean
sequence …
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 8A/DDSP
+1/11 Mean 1 Bit DFT
Frequency (kHz)
A m
( d B W N ) 0
-30
-60
-900 400300200100 500
30
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 9A/DDSP
+1/11 Mean 1-Bit DFTaverage of 30 spectra
Frequency (kHz)
A m
( d B W N ) 0
-30
-60
-900 400300200100 500
30
averaging makes the dc component clearly visible
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 10A/DDSP
+1/11 Minimum Error Sequence
• No random number generator is required to producethe 1-Bit sequence which represents +1/11 with theminimum mean-squared quantization error
• This 11 term sequence averages to 1/11:
[–1 +1 –1 +1 –1 +1 –1 +1 –1 +1 +1] – The sequence in []’s repeats
– Its DFT follows …
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 11A/DDSP
+1/11 Minimum Error DFT
Frequency (kHz)
A m
( d B W N ) 0
-30
-60
-900 400300200100 500
30 fS
11
fS
11
fS
2
1
2
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 12A/DDSP
+1/11 Minimum Error Sequence
• Minimum error is periodic error with a period of fs /11
– Note that the fundamental term in this Fourier series is thesmallest (a bit unusual)
• A quantization noise model is completelyinappropriate for this type of sequence
• Let’s deliberately increase the quantization error alittle and attack its periodicity …
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 13A/DDSP
3 Pattern Options
• Generate another 1-Bit sequence by concatenation ofthe following sequences: – [–1 +1 –1 +1 –1 +1 –1 +1 +1]
– [–1 +1 –1 +1 –1 +1 –1 +1 –1 +1 +1]
– [–1 +1 –1 +1 –1 +1 –1 +1 –1 +1 –1 +1 +1]
• Selection of the above 9, 11, and 13 term sequencesat random yields an average of +1/11 and thefollowing DFT …
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 14A/DDSP
3 Pattern Options DFT
Frequency (kHz)
A m
( d B W N ) 0
-30
-60
-900 400300200100 500
30
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 15A/DDSP
3 Pattern Options
• The randomized concatenation of longer patternscertainly breaks up periodic error
– Some concentration of energy near fs /11 and its harmonics(especially 5fs /11) is still visible
• Noise below 50kHz is significantly lower than that ofthe sample-by-sample random sequences of slide 9
– The “noise shaping” obtained with 3 pattern options benefitslow frequencies at the expense of increased quantization
error at high frequencies
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 16A/DDSP
3 Pattern Options DFTaverage of 30 spectra
Frequency (kHz)
A m
( d B W N ) 0
-30
-60
-900 400300200100 500
30
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 17A/DDSP
5 Pattern Options
• Let’s add two more patterns to our set of pattern options: – [–1 +1 –1 +1 –1 +1 –1 +1 –1 +1 –1 +1 –1 +1 +1]
– [–1 +1 –1 +1 –1 +1 +1]
• Selection of 7, 9, 11, 13, and 15 term sequences atrandom preserves the +1/11 average
– Think of this process as a sort of time-variant dither
– DFT follows …
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 18A/DDSP
3/5 Pattern Optionsaverage of 30 spectra
Frequency (kHz)
A m
( d B W N ) 0
-30
-60
-900 400300200100 500
30
3 options5 options
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 19A/DDSP
5 Pattern Noise Shaping
• Noise vs. frequency still follows the general upwardtilt of the periodic error harmonics
– 5 pattern options further reduce the fs /11 bump
– The 5f s /11 component remains large
• The model that quantization error is uncorrelated withthe input signal becomes reasonable with only 5pattern options
– Such reasonableness is required for small-signal analysis ofthe sigma-delta modulator …
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 20A/DDSP
Sigma- Delta Modulators
Analog 1-Bit Σ∆ modulators convert a continuous timeanalog input vIN into a 1-Bit sequence dOUT
H(z)+
_ vIN
dOUT
+1 or -1
Loop filter 1b Quantizer (a comparator)
fs
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 21A/DDSP
Sigma-Delta Modulators
• The loop filter H can be either a SC or continuous time• SC’s are “easier” to implement and scale with the clock rate
• Continuous time filters provide anti-aliasing protection
• Can be realized with passive LC’s at very high frequencies
H(z)+
_ vIN
dOUT
+1 or -1
fs
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 22A/DDSP
1st Order Σ∆ Modulator
In a 1st order modulator, the loop filter is an integrator
+
_ vIN
dOUT
+1 or -1∫
H(z) =z-1
1 – z-1
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 23A/DDSP
1st Order ∆Σ Modulator
• Properties of the first-order modulator: – Analog input range is the dout times the DAC reference
– The average value of dOUT must equal the average value of v IN
– +1’s (or –1’s) density in dOUT is an inherently monotonic function of v INà linearity is not dependent on component matching
– Alternative multi-bit DAC (and ADCs) solutions reduce the quantization errorbut loose this inherent monotonicity
+
_ vIN
dOUT
+1 or -1
∫ -∆/2≤vIN≤+∆/2
DAC-∆/2 or +∆/2
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 24A/DDSP
Simulation
0 10 20 30 40 50 60-1.5
-1
-0.5
0
0.5
1
1.5
Time [ t/T ]
A m p l i t u d e
XQY
3
Y
2
Q
1
X
Sine Wave
z-1
1-z-1
Discrete Filter Comparator
see L19_level1
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 25A/DDSP
1st Order Σ∆, +1/11 dc Input
• Let’s continue our +1/11 dc input example with themodulator sampling frequency of 1MHz
• A 1024 sample DFT plot appears on the followingslide…
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 26A/DDSP
1st Order Σ∆, +1/11 dc Input
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
Frequency [ f/fs
]
A m p l i t u d e
[ d B W N ]
Looks familiar?
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time [t/11]
A m p l i t u d e
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 27A/DDSP
1st Order Σ∆, +1/11 dc Input
• A check of the time samples confirms the
obvious: the 1st order ∆Σ modulator produces
the minimum error 11 term sequence:
[–1 +1 –1 +1 –1 +1 –1 +1 –1 +1 +1]
• Of course, with this modulator model we can
look at much more interesting inputs than
dc…
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 28A/DDSP
1st Order Σ∆, Sinewave Input
vIN(k) = 0.99sin(2π 0.100001 t)dOUT = [ +1 –1 +1 +1 +1 +1 –1 –1 –1 –1 … ]
0 0.1 0.2 0.3 0.4 0.5-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
30
Frequency [ f/fs
]
A m p l i t u d e
[ d B W N ]
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 29A/DDSP
1st Order Σ∆, Sinewave Input
• The modulator output for the undistorted sinewaveinput produces huge distortion, suggesting the needfor dither
• We’ll add a dither signal q at the comparator input:
+
_ vIN
dOUT∫ +
q
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 30A/DDSP
Dithered 1st Order Σ∆
0 0.1 0.2 0.3 0.4 0.5-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
30
Frequency [ f/fs
]
A m p l i t u d e
[ d B W N ]
single DFTaverage
q: σ = ∆/2
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 31A/DDSP
Dithered 1st Order Σ∆
• 1Vrms Gaussian dither is difficult if not impossible toproduce on mixed-signal ICs
– On-chip digital circuitry adds excessive non-Gaussianinterference to analog noise generators
• First-order modulators are too prone to limit cycles tobe of much practical use
– They do provide the basis for higher-order Σ∆’s
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 32A/DDSP
1st Order Noise ShapingIf q(k) and vIN(k) are uncorrelated, we can compute the signal and noisetransfer functions independently:
How do we model the quantizer?
+ _
vINdOUT∫ +
q
H(z) =z-1
1 – z-1
( )( )
( )( )
( )( )
inin
out
zV
zQ z NTF
zV
z D zSTF ==
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 33A/DDSP
• Nonlinear elements like comparators are frequentlymodeled by some sort of linearized “effective gain”, G
• One measure of effective gain that’s proven itselfuseful for Σ∆ analysis is:
• The value of G depends on the input signal and can
be determined with simulation – E.g. for the simulation in slide 30, G=0.7 (-3.1dB)
1st Order Noise Shaping
G ≡rms value of the comparator output
rms value of the comparator input
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 34A/DDSP
1st Order Noise Shaping
• The input q models both dither, if added, and theinput-referred noise of the comparator
+
_ vIN dOUT∫ +
q
H(z)
G
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 35A/DDSP
1st Order Noise Shaping
+
_ vIN dOUT∫ +
q
H(z)
G
)(1
)(1
)(
zGH
G
V
Q NTF
zGH
zGH
V
DSTF
inin
out
+==
+==
When H(z) is large, this is approximately 1.
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 36A/DDSP
1st Order Noise Shaping
+
_ vIN dOUT∫ +
q
H(z)
G
)(1
)(1
)(
zGH
G
V
Q NTF
zGH
zGH
V
DSTF
inin
out
+==
+==
For large H(z), this is ≈0.
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 37A/DDSP
Noise Transfer Function, NTF
0 0.1 0.2 0.3 0.4 0.5-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
30
Frequency [ f/fs
]
A m p l i t u d e
[ d B W N ]
Output SpectrumNTF
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 38A/DDSP
“Integrated” Noise
• Just as we did for thermal noise, let’s look at
the integrated noise at the output of the
modulator
• In the discrete time case, noise integrals aresummations, but the result is called
“integrated noise” nonetheless
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 39A/DDSP
Noise Summations
• Integrated noise is computed from theenergy theorem (N even):
arms =A0/ N , M=0
arms = Σm=1
M
2Am21
NA0 +
2, 0<M<N/2
arms = Σm=1
N/2-12Am
21
NA0 +
2, M=N/2AN/2 +
2
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 40A/DDSP
Integrated Noise
0 0.1 0.2 0.3 0.4 0.5-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
30
Frequency [ f/fs ]
A m p l i t u d e [ d B
W N ] /
I n t e g r a t e d N o i s e [ d B V ] Output Spectrum
Integrated Noise
The total “noise” atthe modulatoroutput with no inputsums to 0dBà
This is consistentwith a binary signal
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 41A/DDSP
Integrated Output
The total power stillsums to 0dB
Only little“quantization noise”at low frequency
0 0.1 0.2 0.3 0.4 0.5-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
30
Frequency [ f/fs
]
A m p l i t u d e [ d B W N ] /
I n t e g r a t e d N o i s e [ d B V ] Output Spectrum
Integrated Noise
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 42A/DDSP
Oversampling and Noise Shaping
• Σ∆ modulators have interesting characteristics
– Unity gain for the the input signal VIN
– Large attenuation of quantization noise injected at q
– Much better than 1-Bit noise performance is possible if we’reonly interested in frequencies << fs
• ADCs which sample their inputs at much higherfrequencies than the Nyquist rate minimum are called“oversampling ADCs”
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 43A/DDSP
Oversampling and Noise Shaping
• Higher order loop filters consisting of several integratorsprovide much better noise shaping than 1st order realizations
• They are also less prone to limit cycles or need less dither
• If a 1-Bit DAC is used, the converter is inherently linear—independent of component matching
• References
J. C. Candy and G. C. Temes, “Oversampling Methods for A/D and D/A Conversion”, Oversampling
Delta-Sigma Data Converters: Theory, Design, and Simulation, 1992, pp. 1-25.
S. R. Norsworthy, R. Schreier, and G. C. Temes, “Delta-Sigma Data Converters, Theory, Design, andSimulation,” IEEE Press, 1997.
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 44A/DDSP
Estimating Quantization Noise
+
_ vIN Y∫ +
q
H(z)
G
( ))(1 zGH
G z NTF
+=
( )
( ) ( )∫ −
==
∆=
B
B
e zQY
s
Q
df z NTF f SS
f f S
jfT
2
2
2
12
1
π
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 45A/DDSP
Example: 1st Order Modulator
( )
( )
( ) ( ) ( )
( )( )
fT T
z z
z z
z NTF z NTF z NTF
z z NTF
G
z
z z H
π
ω
sin2cos22
11
11
1
1
1
1
1
12
1
1
1
= −=
+−−=
−−=
=
−=
=−
=
−
−
−
−
−
−
( ) ( )
( )
12
1
3
sin212
1
2
3
2
22
2
2
∆≈
∆≅
=
∫
∫
−
−=
M
df fT f
df z NTF f SS
M s f
M s f
jfT
s
B
B
e zQY
π
π
π
EECS 247 Lecture 19: Oversampling © 2002 B. Boser 46A/DDSP
Example: Dynamic Range
3
2
2
3
2
2
2
912
1
3
1 input,sinusoidal 22
1
powernoisepeak
powersignalpeak
M DR
M
S
STF S
S
S DR
Y
X
Y
X
π
π
=
∆=
=
∆=
==
M DR16 33 dB32 42 dB
1024 87 dB
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EECS 247 Lecture 19: Oversampling © 2002 B. Boser 47A/DDSP
Dynamic Range
• DR increases 9dB for each doubling of M
• 1st order modulators require very high M for >10-Bit resolutionà higher order filters improve this tradeoff substantially
• Analysis is based on assumption that the quantization noise is“white”à not true in practice, especially for low-order modulatorsà practical modulators suffer from other noise sources also
(e.g. thermal noise)
• Next time we’ll design an oversampled audio ADC with betterthan 16-Bit resolution