Oversampled ADCs Last Lecture

31 Pages·2004·7 MB·English

EE247
Lecture 23
• Oversampled ADCs
– 1-Bit quantization
• Quantization error spectrum
• SQNR analysis
• Limit cycle oscillations
– 2nd order DS modulator
• Dynamic range
• Practical implementation
– Effect of various nonidealities on the SD performance
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 1
Oversampled ADCs
Last Lecture
• Why Oversampling?
– Allows trading speed for resolution
– Relaxed transition band requirements for analog anti-
aliasing filters
– Reduced baseband quantization noise power
– Utilizes low cost, low power digital filtering
– Issue of limit cycle oscillation (spurious inband tones)
• By simply increasing oversampling ratio: 2X increase in
sampling ratio (cid:224)0.5-bit increase in resolution
• To achieve greater improvement in resolution:
– Embed quantizer in a feedback loop
(cid:224)Predictive (delta modulation)
(cid:224)Noise shaping (sigma delta modulation)
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 2
Oversampling A/D Conversion
Signal
BW=f/2M
s 1-bit n-bit
@ f @ f/M
s s
• Analog front-end (cid:224)oversampled noise-shaping modulator
•Converts original signal to a 1-bit digital output at the high rate of
(2MX f )
signal
• Digital back-end (cid:224)digital filter
•Removes out-of-band quantization noise
•Provides anti-aliasing to allow re-sampling @ lower sampling rate
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 3
Oversampled ADC
Predictive Coding
+
v
IN ADC
_ d
OUT
Predictor
• Quantize the difference signal rather than the signal itself
• Smaller input to ADC (cid:224)Buy dynamic range
• Only works if combined with oversampling
• 1-Bit digital output
• Digital filter computes “average” (cid:224)n-Bit output
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 4
Oversampled ADC
f = M f f = f +d
s1 N s2 N
Signal
E.g. Decimator
“wide”
Pulse-Count “narrow” DSP
transition fs= MfN Modulator transition
Analog Sampler Modulator Digital
B Freq AA-Filter AA-Filter
1-Bit Digital N-Bit
Digital
Decimator:
• Digital (low-pass) filter
• Removes quantization error for f > B
• Provides most anti-alias filtering
• Narrow transition band, high-order
• 1-Bit input, N-Bit output (essentially computes “average”)
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 5
Modulator
• Objectives:
– Convert analog input to 1-Bit pulse density stream
– Move quantization error to high frequencies f >>B
– Operates at high frequency f >> f
s N
• M = 8 … 256 (typical)….1024
• Better be “simple”
(cid:224) SD = SD Modulator
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 6
Sigma- Delta Modulators
Analog 1-Bit SD modulators convert a continuous time
analog input v into a 1-Bit sequence d
IN OUT
f
s
+
v H(z)
IN
d
_ OUT
DAC
Loop filter 1b Quantizer (a comparator)
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 7
Sigma-Delta Modulators
• The loop filter H can be either a switched-capacitor or continuous time
• Switched-capacitor filters 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
f
s
+
v H(z)
IN
_
d
OUT
DAC
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 8
DS
1st Order Modulator
In a 1storder modulator, simplest loop filter (cid:224)an integrator
z-1
H(z) =
1 – z-1
+
(cid:242)
v
IN
d
_ OUT
DAC
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 9
DS
1st Order Modulator
Switched-capacitor implementation
f f
1 2 f
2
-
d
OUT
1,0
+
V
i
+D/2
-D/2
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 10
SD
1st Order Modulator
+
(cid:242)
v
IN
_
-D/2£v £+D/2 d
IN OUT
-D/2 or +D/2
DAC
• Properties of the first-order modulator:
– Analog input range is equal to the DAC reference
– The average value of d must equal the average value of v
OUT IN
– +1’s (or –1’s) density in d is an inherently monotonic function of v
OUT IN
(cid:224)linearity is not dependent on component matching
– Alternative multi-bit DAC (and ADCs) solutions reduce the quantization error
but loose this inherent monotonicity
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 11
DS
1st Order Modulator
Tally of 1-Bit
quantization error quantizer
Analog input
1 2 3
-D /2£ V £ +D /2 X Q Y
in
z -1
1-z -1 1-Bit digital
Sine Wave Integrator Comparator
output stream,
-1, +1
Instantaneous Implicit 1-Bit DAC
quantization error +D /2, -D /2
(D = 2)
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 12
1st Order Modulator Signals
1st Order Sigma-Delta X analog input
1.5 X Q tally of q-error
Q
Y Y digital/DAC output
1
e0.5
d Mean of Y approximates X
u
mplit 0
A
-0.5
-1 T = 1/f = 1/ (M f )
s N
-1.5
0 10 20 30 40 50 60
Time [ t/T ]
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 13
DS
Modulator Characteristics
• Quantization noise and thermal noise (KT/C)
distributed over –fs/2 to +fs/2
(cid:224) Total noise reduced by 1/M
• Very high SQNR achievable (> 20 Bits!)
• Inherently linear for 1-Bit DAC
• Quantization error independent of component
matching
• Limited to moderate to low speed
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 14
Output Spectrum
First order sigma-delta, A=0.79, fx=0.0628319, offset=0.020944, N=1024, K=30 • Definitely not white!
30
20 Input • Skewed towards
higher frequencies
10
N ]
W 0 • Tones
B
mplitude [ d--2100
A • dBWN (dB White
-30 Noise) scale sets
the 0dB line at the
-40
noise per bin of a
-50 random -1, +1
0 0.1 0.2 0.3 0.4 0.5
Frequency [ f/sf ] sequence
sigma_delta_L1_sin.m
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 15
Quantization Noise Analysis
Quantization
Error e(kT)
Integrator
z- 1
x(kT) S H(z)=1+z- 1 S y(kT)
Quantizer
Model
• Sigma-Delta modulators are nonlinear systems with memory
(cid:224)very difficult to analyze directly
• Representing the quantizer as an additive noise source
linearizes the system
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 16
Signal Transfer Function
H(z)=1+z-z1- 1 x(kT) S - IntHeg(zr)ator y(kT)
H(jw)=w0
jw
Signal transfer function
(cid:224)low pass function:
e
d
u
nit
H (jw)= 1 ag
Sig 1+ s M
w
0
Hz ( )==Y=z(H)( )z (cid:222) z - 1 Delay f0 Frequency
Sig Xz(H)1(z) +
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 17
Noise Transfer Function
Qualitative Analysis
v2
n
vi S w0 vo v2
eq
- jw
v2 = v2· (cid:230)(cid:231) f (cid:246)(cid:247)
eq n Ł fł
v2· (cid:230)(cid:231) f (cid:246)(cid:247) 2 0
n Ł fł
vi S 0 w0 vo
- jw
f Frequency
0
v2 =v2· (cid:230)(cid:231) f (cid:246)(cid:247) 2
eq n Ł fł
0
S w0 vo
vi - jw
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 18
STF and NTF
Quantization
Error e(kT)
Integrator
z- 1
x(kT) S H(z)=1+z- 1 S y(kT)
Quantizer
Model
Signal transfer function:
STF==Y= z(H)( z) D(cid:222) elay z- 1
Xz(H)1(z) +
Noise transfer function:
Y(z) 1
NTF= = =1- z-1 (cid:222) Differentiator
E(z) 1+H(z)
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 19
Noise Transfer Function
NTF==Y=()z- 1 1 z- 1
Ez(H)1(z) +
NTFj(e)(w1)==2 - e - jwTj T - w /2(cid:230)(cid:231) ejwTj T/2/2- e- w (cid:246)(cid:247)
Ł 2 ł
=2esij-nj/w2T/2T (w )
=2esie-nj/wT2/2/2· - j p Tغ (w øß)
=غ 2sin/(2wT øß) e- j(wT- p)/2
where T=1/f
s
Thus:
( ) ( )
NTFf(T)=f2sfin/2=2wsin/ p
s
N (f)= NTFf(N)( 2)f
y e
EECS 247 Lecture 23: Oversampling Data Converters © 2004H. K. Page 20