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EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 1
EE247Lecture 24
Oversampled ADCs Why oversampling? Pulsecount modulation Sigmadelta modulation
1Bit quantization Quantization error (noise) spectrum SQNR analysis Limit cycle oscillations
2nd order modulator Dynamic range Practical implementation
Effect of various nonidealities on the performance
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 2
The Case for OversamplingNyquist sampling:
Oversampling:
Nyquist rate fN = 2B Oversampling rate M = fs/fN >> 1
fs >2B +FreqB
Signal
narrowtransition
SamplerAAFilter
NyquistADC DSP
=MFreqB
Signal
widetransition
SamplerAAFilter
OversampledADC DSP
fs
fs fN
fs >> fN??
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 3
Nyquist v.s. Oversampled ConvertersAntialiasing
X(f)
frequency
frequency
frequency
fB
fB 2fs
fB fs
Input Signal
Nyquist Sampling
Oversampling
fS ~2fB
fS >> 2fB
fs
Antialiasing Filter
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 4
Oversampling Benefits No stringent requirements imposed on analog
building blocks Takes advantage of the availability of low
cost, low power digital filtering Relaxed transition band requirements for
analog antialiasing filters Reduced baseband quantization noise power Allows trading speed for resolution
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 5
ADC ConvertersBaseband Noise
For a quantizer with step size and sampling rate fs : Quantization noise power distributed uniformly across Nyquist
bandwidth ( fs/2)
Power spectral density:
Noise is aliased into the Nyquist band fs /2 to fs /2
2 2
es s
e 1N ( f )
f 12 f
= =
fB f s /2fs /2 fB
Ne(f)
NB
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 6
Oversampled ConvertersBaseband Noise
B B
B B
f f 2
B ef f s
2B
s
B s2
B0
B B0B B0
ss
B
1S N ( f )df df
12 f
2 f12 f
where for f f / 2
S12
2 f SS S
f Mf
where M oversampling rat io2 f
= =
= =
=
= =
= =
fB f s /2fs /2 fB
Ne(f)
NB
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 7
Oversampled ConvertersBaseband Noise
2X increase in M 3dB reduction in SB bit increase in resolution/octave oversampling
B B0B B0
ss
B
2 f SS S
f Mf
where M oversampling rat io2 f
= =
= =
To increase the improvement in resolution: Embed quantizer in a feedback loopPredictive (delta modulation)Noise shaping (sigma delta modulation)
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 8
PulseCount Modulation
Vin(kT)Nyquist
ADCt/T0 1 2
OversampledADC, M = 8
t/T0 1 2
Vin(kT)
Mean of pulsecount signal approximates analog input!
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 9
PulseCount Spectrum
f
Magnitude
Signal: low frequencies, f < B
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 11
Oversampled ADC
Decimator: Digital (lowpass) filter Removes quantization error for f > B Provides most antialias filtering Narrow transition band, highorder 1Bit input, NBit output (essentially computes average)
fs= MfN
FreqB
Signalwide
transition
SamplerAnalogAAFilter
E.g.PulseCountModulator
Decimatornarrowtransition
fs1= M fN
DSP
Modulator DigitalAAFilter
fs2= fN+
1Bit Digital NBitDigital
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 12
Modulator
Objectives: Convert analog input to 1Bit pulse density stream Move quantization error to high frequencies f >>B Operates at high frequency fs >> fN
M = 8 256 (typical).1024 Since modulator operated at high frequencies need to
keep circuitry simple
= Modulator
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 13
Sigma Delta ModulatorsAnalog 1Bit modulators convert a continuous time analog input vIN into a 1Bit sequence dOUT
H(z)+
_vIN dOUT
Loop filter 1b Quantizer (comparator)
fs
DAC
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 14
SigmaDelta Modulators The loop filter H can be either switchedcapacitor or continuous time Switchedcapacitor filters are easier to implement + frequency
characteristics scale with clock rate Continuous time filters provide antialiasing protection Loop filter can also be realized with passive LCs at very high frequencies
H(z)+
_vIN
dOUT
fs
DAC
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 15
Oversampling A/D Conversion
Analog frontend oversampled noiseshaping modulator Converts original signal to a 1bit digital output at the high rate of
(2MXB) Digital backend digital filter
Removes outofband quantization noise Provides antialiasing to allow resampling @ lower sampling rate
[email protected] fs /M
Input Signal Bandwidth
B=fs /2MDecimation
FilterOversampling
Modulator
fs
fs = sampling rateM= oversampling ratio
fs /M
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 16
1st Order ModulatorIn a 1st order modulator, simplest loop filter an integrator
+
_vIN dOUT
H(z) =z1
1 z1
DAC
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 17
1st Order ModulatorSwitchedcapacitor implementation
Vi

+
1 2 2
1,0dOUT
+/2
/2
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 18
1st Order Modulator
Properties of the firstorder modulator: Analog input range is equal to the DAC reference The average value of dOUT must equal the average value of vIN +1s (or 1s) density in dOUT is an inherently monotonic function of vIN linearity is not dependent on component matching Alternative multibit DAC (and ADCs) solutions reduce the quantization error
but loose this inherent monotonicity & relaxed matching requirements
+
_vIN
dOUT
/2vIN+/2
DAC/2 or +/2
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 19
1st Order Modulator
Instantaneous quantization error
Tally of quantization error
1Bitquantizer
1Bit digital output stream,
1, +1
Implicit 1Bit DAC+/2, /2
( = 2)
Analog input/2Vin+/2
3Y
2Q
1X
Sine Wave
z 1
1z 1Integrator Comparator
M chosen to be 8 (low) to ease observability
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 20
1st Order Modulator Signals
T = 1/fs = 1/ (M fN)
X analog inputQ tally of qerrorY digital/DAC output
Mean of Y approximates X
0 10 20 30 40 50 60
1.5
1
0.5
0
0.5
1
1.5
Time [ t/T ]
Am
plitu
de
1st Order SigmaDeltaXQY
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 21
Modulator Characteristics Quantization noise and thermal noise (KT/C)
distributed over fs /2 to +fs /2 Total noise within signal bandwidth reduced by
1/M Very high SQNR achievable (> 20 Bits!) Inherently linear for 1Bit DAC To first order, quantization error independent
of component matching Limited to moderate & low speed
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 22
Output Spectrum Definitely not white!
Skewed towards higher frequencies
Notice the distinct tones
dBWN (dB White Noise) scale sets the 0dB line at the noise per bin of a random 1, +1 sequence
Input
0 0.1 0.2 0.3 0.4 0.550
40
30
20
10
0
10
20
30
Frequency [ f /fs]
Am
plitu
de
[ dB
WN
]
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 23
Quantization Noise Analysis
SigmaDelta modulators are nonlinear systems with memory difficult to analyze directly
Representing the quantizer as an additive noise source linearizes the system
Integrator
Quantizer
Model
QuantizationError e(kT)
x(kT) y(kT)1
1H( ) 1zz
z
=
+
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 24
Signal Transfer Function
( )
1
1
0
H( )1
zzz
H jj
=
+
=
Signal transfer function low pass function:
IntegratorH(z)
x(kT)y(kT)

Frequency
Magnitude
f0
( )
( )0
1
1 1
( ) ( ) Delay( ) 1 ( )
Sig
Sig
H j s
Y z H zH z zX z H z
=
+
= = = +
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 25
Noise Transfer FunctionQualitative Analysis
2eqv
22
0n
fv f
vi
vo0j
22 2
0eq n
fv v f
= vi 
vo0j
Frequencyf0
2nv
vi
vo0
j
2 2
0eq n
fv v f
=
Input referrednoise zero @ DC (splane)
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 26
STF and NTF
Signal transfer function:1( ) ( )STF Delay
( ) 1 ( )Y z H z zX z H z
= = = +
Noise transfer function:
erentiator Diff 1)(1
1)()(NTF 1 =
+==
zzHzE
zY
Integrator
Quantizer
Model
QuantizationError e(kT)
x(kT) y(kT)1
1H( ) 1zz
z
=
+
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 27
Noise Transfer Function
( )( )
( ) ( )
( ) ( )
1
/ 2 / 2/ 2
/ 2
/ 2 / 2
/ 2
( ) 1 1 ( ) 1 ( )
( ) (1 )=22
2 sin / 2
2 sin / 2
2sin / 2
where 1/Thus:
( ) =2 sin / 2 =2 sin /
( )
j T j Tj T j T
j T
j T j
j T
s
s
y
Y zNTF zE z H z
e eNTF j e e
e j T
e e T
T e
T f
NTF f T f f
N f
= = =
+
= =
= =
=
=2( ) ( )eNTF f N f
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 28
First Order ModulatorNoise Transfer Characteristics
( )2
2
( ) ( ) ( )
4 sin /
y e
s
N f NTF f N f
f f
=
=
Noi
se S
hapi
ng F
unct
ion
FrequencyfB fN fs /2
FirstOrder Noise Shaping
LowpassDigital Filter
Key Point:Most of quantization noise pushed out of frequency band of interest
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 29
First Order ModulatorSimulated Noise Transfer Characteristic
0 0.1 0.2 0.3 0.4 0.550
40
30
20
10
0
10
20
30
Frequency [f/fs]
Ampl
itude
[ d
BW
N ]
Simulated output spectrumComputed NTF
Signal
( ) 2( ) 4 sin /y sN f f f=
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 30
Quantizer Error For quantizers with many bits
Lets use the same expression for the 1Bit case
Use simulation to verify validity
Experience: Often sufficiently accurate to be useful, with enough exceptions to be very careful
( )12
22
=kTe
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 31
First Order ModulatorInBand Quantization Noise
( )( ) ( )
( ) ( )
( )
2
2
2
1
2 2
2
2 2
2 2
3
1
4 sin / for 1
1 2sin12
13 12
jfT
fsM
fsM
sB
Y Q z eB
s
Y
NTF z z
NTF f f f M
S S f NTF z df
fT dff
SM
=
=
= >>
=
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 32
Dynamic Range
M DR16 33 dB32 42 dB
1024 87 dB
2
2 2
3
32
32 2
peak signal power10log 10logpeak noise power
1 sinusoidal input, 12 2
13 129
29 910log 10log 30log
2 2
3.4 30l
X
Y
X
Y
X
Y
SDRS
S STF
SM
S MS
DR M M
DR dB
= =
= =
=
=
= = +
= + + og M
2X increase in M9dB (1.5Bit) increase in dynamic range
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 33
Oversampling and Noise Shaping
modulators have interesting characteristics Unity gain for input signal VIN Large inband attenuation of quantization noise injected at quantizer
input Performance significantly better than 1Bit noise performance possible
for frequencies > 1) improves SQNR
considerably 1st order : DR increases 9dB for each doubling of M To first order, SQNR independent of circuit complexity and accuracy
Analysis assumes that the quantizer noise is white Not true in practice, especially for loworder modulators Practical modulators suffer from other noise sources also
(e.g. thermal noise)
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 34
DC Input
DC input A = 1/11
Doesnt look like spectrum of DC at all
Tones frequency shaped the same as quantization noise More prominent at higher frequencies
Seems like periodic quantization noise
0 0.1 0.2 0.3 0.4 0.550
40
30
20
10
0
10
20
30
Frequency [ f /fs ]
Am
plitu
de
[ dB
WN
]
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 35
Limit Cycle
111+11019+1817+1615+1413+12+11
DC input 1/11 Periodic sequence:
0 10 20 30 40 50
0.4
0.2
0
0.2
0.4
0.6
Time [t/T]
Out
put
First order sigmadelta, DC input
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 36
Limit Cycle
Inband spurious tone with f ~ DC input
Problem: quantization noise is periodicSolution:
Use dithering: randomizes quantization noise Thermal noise acts as dither
Second order loop
Noi
se S
hapi
ng F
unct
ion
FrequencyfB fN fs /2
FirstOrder Noise Shaping
Ideal LowpassDigital Filter
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 37
1st Order Modulator
( )1 1( ) ( ) 1 ( ) Y z z X z z E z = +
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 38
2nd Order Modulator
Two integrators 1st integrator nondelaying Feedback from output to both integrators Tones less prominent compared to 1st order
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 39
2nd Order Modulator
( )( )
1 1 221 1 1
Recursive drivation: 2
Using the delay operator : ( ) ( ) 1 ( )
n n n n nY X E E E
z Y z z X z z E z
= + +
= +
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 40
2nd Order ModulatorInBand Quantization Noise
( )
( ) ( )( )
( )
1
1
21
2
44
11
1
2 sin / for 1s
zH zz
G
NTF z z
NTF f
f f M
=
=
=
=
= >>
( ) ( )
( )
2
2
2
2
24
4 2
5
1 2sin12
15 12
jfT
fsM
fsM
B
Y Q z eB
s
S S f NTF z df
fT dff
M
=
=
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 41
Quantization Noise2nd Order Modulator
Noi
se S
hapi
ng F
unct
ion
FrequencyfB fs /2
FirstOrder Noise Shaping
Ideal LowpassDigital Filter
2nd Order Noise Shaping
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 42
2nd Order Modulator Dynamic Range
M DR16 49 dB32 64 dB
1024 139 dB
2
4 2
5
54
54 4
peak signal power10log 10logpeak noise power
1 sinusoidal input, 12 2
15 1215
215 1510log 10log 50log
2 2
11.1
X
Y
X
Y
X
Y
SDRS
S STF
SM
S MS
DR M M
DR dB
= =
= =
=
=
= = +
= + 50log M+
2X increase in M15dB (2.5bit) increase in DR
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 43
2nd Order Modulator Example
M 256=28 to allow some margin & also for ease of digital filter implementation Sampling rate (2x20kHz + 5kHz)M = 12MHz
Digital audio applicationSignal bandwidth 20kHzResolution 16bit
min
16 98 Dynamic Range
11.1 50log153
bit dB
DR dB MM
= + +
=
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 44
Higher Order Modulator Dynamic Range
2X increase in M(6L+3)dB or (L+0.5)bit increase in DR
( )
( )
( )
( )
1 1
2
2 2
2 1
2 12
2 12
2
( ) ( ) 1 ( ) , L order
1 sinusoidal input, 12 2
12 1 123 2 1
23 2 1
10log2
3 2 110log 2
2
L
X
LY L
LXL
Y
LL
L
Y z z X z z E z
S STF
SL M
LS MS
LDR M
LDR
+
+
+
= +
= =
=
+
+=
+=
+= + ( )1 10 logL M+
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 45
Modulator Dynamic RangeAs a Function of Modulator Order
Potential stability issues for L >2
L=2
L=3
L=1
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 46
Tones in 1st Order & 2nd Order Modulator
Higher oversampling ratio lower tones
2nd order much lower tones compared to 1st
2X increase in M decreases the tones by 6dB for 1st order loop and 12dB for 2nd order loop
Ref: B. P. Brandt, et al., "Secondorder sigmadelta modulation for digitalaudio signal acquisition,"
IEEE Journal of SolidState Circuits, vol. 26, pp. 618  627, April 1991.R. Gray, Spectral analysis of quantization noise in a singleloop sigmadelta modulator with dc
input, IEEE Trans. Commun., vol. 37, pp. 588599, June 1989.
6dB
12dB 2nd Order Modulator
1st Order Modulator
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 47
2nd Order ModulatorSwitchedCapacitor Implementation
IN Dout
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 48
SwitchedCapacitor Implementation 2nd Order Phase 1
Sample inputsCompare output of 2nd integratorAt the end of phase1, S3 opens prior to S1 opening
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 49
SwitchedCapacitor Implementation 2nd Order Phase 2
Enable feedback from output to input of both integrators Integrate Reset comparator At the end of phase2 S4 opens before S2
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 50
ImplementationPractical Design Considerations
Internal nodes scaling & clipping
Finite opamp gain & linearity
Capacitor ratio errors
KT/C noise
Opamp noise
Power dissipation considerations
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 51
SwitchedCapacitor Implementation 2nd Order Nodes Scaled for Maximum Dynamic Range
Modification (gain of in front of integrators) reduce & optimize required signal range at the integrator outputs ~ 1.7x input fullscale ()
Ref: B.E. Boser and B.A. Wooley, The Design of SigmaDelta Modulation A/D Converters, IEEE J. SolidState Circuits, vol. 23, no. 6, pp. 12981308, Dec. 1988.
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 52
2nd Order ModulatorSwitchedCapacitor Implementation
C2=2C1
The loss in front of each integrator implemented by choice of:
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 53
2nd Order Effect of Integrator Maximum Signal Handling Capability on SNR
Effect of 1st Integrator maximum signal handling capability on converter SNRRef: B.E. Boser and B.A. Wooley, The Design of SigmaDelta Modulation A/D Converters,
IEEE J. SolidState Circuits, vol. 23, no. 6, pp. 12981308, Dec. 1988.
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 54
2nd Order Effect of Integrator Maximum Signal Handling Capability on SNR
Effect of 2nd Integrator maximum signal handling capability on SNRRef: B.E. Boser and B.A. Wooley, The Design of SigmaDelta Modulation A/D Converters,
IEEE J. SolidState Circuits, vol. 23, no. 6, pp. 12981308, Dec. 1988.
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 55
2nd Order Effect of Integrator Finite DC Gain
Vi

+
1 2
aVo
Cs
CI( )
( )
1
1
1
1
1
1
111
ideal
Finit DC Gain
Cs zH zCI z
a zCsaCs CIH zCI
a zCsaCI
=
+ +
= +
+ +
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 56
2nd Order Effect of Integrator Finite DC Gain
Low integrator DC gain Increase in total inband noise Can be shown: If a > M (oversampling ratio) Insignificant degradation in
SNR Normally DC gain designed to be >> M in order to suppress nonlinearities
f0 /a
a
0P1 a
=
0
a
( )log H s Ideal Integ. (a=infinite)
Integrator magnitude response
Max signal level
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 57
2nd Order Effect of Integrator Finite DC Gain
Example: a =2M 0.4dB degradation in SNRRef: B.E. Boser and B.A. Wooley, The Design of SigmaDelta Modulation A/D Converters,
IEEE J. SolidState Circuits, vol. 23, no. 6, pp. 12981308, Dec. 1988.
M / a
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 58
2nd Order Effect of Integrator Overall Integrator Gain Inaccuracy
Gain of in front of integrators determined by ratio of C1/C2
Effect of inaccuracy in ratio of C1/C2 inspected by simulation
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 59
2nd Order Effect of Integrator Overall Gain Inaccuracy
Simulation show gain can vary by 20% w/o loss in performanceConfirms insensitivity of to component variations
Note that for gain >0.65 system becomes unstable & SNR drops rapidly
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 60
2nd Order Effect of Integrator Nonlinearities
2 32 3
2 32 3
v( kT T ) u(kT ) v( kT )
With nonlinearity added:
v( KT T ) u(kT ) .....u(kT ) u(kT )v( kT ) ....v( kT ) v( kT )
+ = +
+ = + +
+ + + +
Ref: B.E. Boser and B.A. Wooley, The Design of SigmaDelta Modulation A/D Converters, IEEE J. SolidState Circuits, vol. 23, no. 6, pp. 12981308, Dec. 1988
Delay
Ideal Integrator
u(kT) v(kT)
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 61
2nd Order Effect of Integrator Nonlinearities
Ref: B.E. Boser and B.A. Wooley, The Design of SigmaDelta Modulation A/D Converters, IEEE J. SolidState Circuits, vol. 23, no. 6, pp. 12981308, Dec. 1988.
Simulation for singleended topology Even order nonlinearities can be significantly attenuated by using
differential circuit topologies
2 20.01, 0 .02,0.05, 0.1%
= =
EECS 247 Lecture 24: Oversampling Data Converters 2005 H. K. Page 62
2nd Order Effect of Integrator Nonlinearities
Ref: B.E. Boser and B.A. Wooley, The Design of SigmaDelta Modulation A/D Converters, IEEE J. SolidState Circuits, vol. 23, no. 6, pp. 12981308, Dec. 1988.
Simulation for singleended topology Odd order nonlinearities (3rd in this case)
3 30.05, 0.2, 1%
= =
6dB 1=Bit