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EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 1
EE247Lecture 11
• Filters (continued)– Example: Switched-capacitor filters in
CODEC integrated circuits
– Switched-capacitor filter design summary
– Comparison of various filter topologies
• New Topic: Data Converters
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 2
Summary Last Lecture
• Switched-capacitor filters – Switched-capacitor integrators
• LDI integrators• Effect of parasitic capacitance• Bottom-plate integrator topology
– Resonators– Bandpass filters– Lowpass filters
• Termination implementation• Transmission zero implementation
– Switched-capacitor filter design considerations– Effect of non-idealities– Switched-capacitor filters utilizing double sampling technique
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 3
Switched-Capacitor Filter ApplicationExample: Voice-Band CODEC (Coder-Decoder) Chip
Ref: D. Senderowicz et. al, “A Family of Differential NMOS Analog Circuits for PCM Codec Filter Chip,” IEEE Journal of Solid-State Circuits, Vol.-SC-17, No. 6, pp.1014-1023, Dec. 1982.
fs= 1024kHz fs= 128kHz fs= 8kHz fs= 8kHz
fs= 8kHz fs= 128kHz fs= 128kHz
fs= 128kHz
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 4
CODEC Transmit Path Lowpass Filter Frequency Response
0 2000 4000 6000 8000-50
-40
-30
-20
-10
0
Frequency (Hz)
Mag
nitu
de (d
B)
Note: fs=128kHz
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 5
CODEC Transmit Path Highpass Filter
1000-50
-40
-30
-20
-10
0
Frequency (Hz)
Mag
nitu
de (d
B)
1000010010
Note: fs=8kHz
23060
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 6
CODEC Transmit Path Filter Overall Frequency Response
1000-50
-40
-30
-20
-10
0
Frequency (Hz)
Mag
nitu
de (d
B)
1000010010
Low Q bandpass (Q<1) filter shape Implemented with lowpass followed by highpass
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 7
CODEC Transmit Path Clocking & Anti-Aliasing Scheme
First filter (1st order RC type) performs anti-aliasing for the next S.C. biquad
The 1st & 2nd stage filters form 3rd order elliptic LPF with corner frequency @ 32kHz Anti-aliasing for the next lowpass filter
The stages prior to the high-pass perform anti-aliasing for high-pass
Notice gradual lowering of clock frequency Ease of anti-aliasing
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 8
SC Filter SummaryPole and zero frequencies proportional to
– Sampling frequency fs– Capacitor ratios
High accuracy and stability in responseLong time constants realizable without requiring large value R
Compatible with transconductance amplifiers– Reduced circuit complexity, power dissipation
Amplifier bandwidth requirements less stringent compared to CT filters (low frequencies only)Issue: Sampled-data filters require anti-aliasing prefiltering
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 9
Switched-Capacitor Filters versus Continuous-Time Filter Limitations
Considering overall effects of:
• Opamp finite slew rate
• Opamp finite unity-gain-bandwidth
• Opamp settling issues
• Clock feedthru
• Switch+ sampling cap. finite time-constant
Filter bandwidth
MagnitudeError
5-10MHz
S.C. Filter
Cont. Time Filter
Limited switched-capacitor filter performance frequency range
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 10
SummaryFilter Performance versus Filter Topology
_
1-5%
1-5%
1-5%
1-5%
Freq. tolerance+ tuning
<<1%40-90dB~ 10MHzSwitched Capacitor
+-40-60%40-70dB~ 100MHzGm-C
+-30-50%50-90dB~ 5MHzOpamp-MOSFET-RC
+-30-50%40-60dB~ 5MHzOpamp-MOSFET-C
+-30-50%60-90dB~10MHzOpamp-RC
Freq. tolerance w/o tuning
SNDRMax. Usable Bandwidth
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 11
Material Covered in EE247Where are We?
Filters – Continuous-time filters
• Biquads & ladder type filters• Opamp-RC, Opamp-MOSFET-C, gm-C filters• Automatic frequency tuning
– Switched capacitor (SC) filters
• Data Converters– D/A converter architectures– A/D converter
• Nyquist rate ADC- Flash, Pipeline ADCs,….• Oversampled converters• Self-calibration techniques
• Systems utilizing analog/digital interfaces
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 12
Data Converters
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 13
Suggested Reference Texts• R. v. d. Plassche, CMOS Integrated Analog-to-Digital and
Digital-to-Analog Converters, 2nd ed., Kluwer, 2003.
• B. Razavi, Data Conversion System Design, IEEE Press, 1995.
• S. Norsworthy et al (eds), Delta-Sigma Data Converters, IEEE Press, 1997.Extensive treatment of oversampled converters including stability, tones, bandpass converters.
• J. G. Proakis, D. G. Manolakis, Digital Signal Processing, Prentice Hall, 1995.
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 14
Converter Applications
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 15
Data Converter Basics• DSPs benefited from device scaling
• However, real world signals are still analog:
– Continuous time– Continuous amplitude
• DSP can only process:– Discrete time– Discrete amplitude
Need for data conversion from analog to digital and digital to analog
Analog Postprocessing
D/AConversion
DSP
A/D Conversion
Analog Preprocessing
Analog Input
Analog Output
000...001...
110
Filters
Filters
?
?
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 16
A/D & D/A ConversionA/D Conversion
D/A Conversion
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 17
Data Converters
• Stand alone data converters– Used in variety of systems
– Example: Analog Devices AD9235 12bit/ 65Ms/s ADC- Applications:
• Ultrasound equipment
• IF sampling in wireless receivers
• Various hand-held measurement equipment
• Low cost digital oscilloscopes
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 18
Data Converters• Embedded data converters
– Integration of data conversion interfaces along with DSPs and/or RF circuits Cost, reliability, and performance
– Main issues• Feasibility of integrating sensitive analog functions in a
technology typically optimized for digital performance• Down scaling of supply voltage as a result of downscaling of
feature sizes• Interference & spurious signal pick-up from on-chip digital
circuitry and/or high frequency RF circuits• Portable applications dictate low power consumption
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 19
Example: Typical Cell PhoneContains in integrated form:• 4 Rx filters• 4 Tx filters
• 4 Rx ADCs• 4 Tx DACs• 3 Auxiliary ADCs• 8 Auxiliary DACs
Total: Filters 8
ADCs 7
DACs 12
Dual Standard, I/Q
Audio, Tx/Rx powercontrol, Battery chargecontrol, display, ...
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 20
D/A Converter Transfer Characteristics• An ideal digital-to-
analog converter:– Accepts digital inputs
b1-bn
– Produces either an analog output voltage or current
– Assumption (will be revisited)
• Uniform, binary digital encoding
• Unipolar output ranging from 0 to VFS
D/A
……..…b1 b2 b3 bn
Vo or Io
MSB LSB
FS
FSN
FS2
N # of bi ts
V ful l scale output
min. s tep size 1LSB
V2
Vor N log resolut ion
=
=
Δ = →
Δ =
= →Δ
Nomenclature:
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 21
D/A Converter Transfer Characteristics
FS
FSN
N # of bi ts
V ful l scale output
min. step size 1LSB
V
2
=
=
Δ = →
Δ =
N
0 FSi 1
N
i 1
i
N i
biV V
2
bi 2 , bi 0 or 1
=
=
−
=
= Δ × × =
∑
∑
D/A
……..…b1 b2 b3 bn
V0
MSB LSB
binary-weighted
( )imax
o FSmax
o FS N
Note : D b 1,al l iV V
11V V
2
=→ = − Δ
⎛ ⎞−→ = ⎜ ⎟⎝ ⎠
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 22
D/A Converter Exampe: D/A with 3-bit Resolution
D/A
b1 b2 b3
V0
MSB LSB
( )
( )
FS
0 1 2 3
3FS
0
0
NFS FS
2 1 0
2 1 0
Example : N 3
Assume V 0.8VInput code is 101
V b 2 b 2 b 2
Then : V / 2 0.1V
V 0.1V 1 2 0 2 1 2
V 0.5V
Note : MSB V / 2 & LSB V / 2
=
=
= Δ × + × + ×
Δ = =
→ = =× + × + ×
→ =
→ →
1 0 1
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 23
Ideal 3-Bit D/A Transfer Characteristic
• Ideal DAC introduces no error!
• One-to-one mapping from input to output
000 001 010 011 100 101 110 111
Step Height (1LSB =Δ)
Ideal Response
Digital InputCode
Analog OutputVFS
VFS /2
VFS /8
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 24
A/D Converter Transfer Characteristics
• An ideal analog-to-digital converter:– Accepts analog input in the
form of either voltage or current
– Produces digital output either in serial or parallel form
– Assumption (will be revisited) • Unipolar input ranging from 0
to VFS• Uniform, binary digital
encoding
A/D
……..…b1 b2 b3 bn
Vin
MSB LSB
FS
FSN
FS2
N # of bi t s
V ful l scale output
min. resoluvable input 1LSB
V
2V
or N log resolut ion
=
=
Δ = →
Δ =
= →Δ
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 25
Ideal A/D Transfer Characteristic
• Ideal ADC introduces error with max peak-to-peak:
(+-1/2 Δ) Δ = VFS /2 N
N= # of bits
• This error is called ``quantization error``
111
110
101
100
011
010
001
000
DigitalOutput
Analog input
0 Δ 2Δ 3Δ 4Δ 5Δ 6Δ 7Δ
1LSB
VFS
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 26
Non-Linear Data Converters
• So far data converter characterisitics studied are with uniform, binary digital encoding
• For some applications to maximize dynamic range non-linear coding is used e.g. Voice-band telephony, – Small signals larger # of codes– Large signals smaler # of codes
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 27
Example: Non-Linear A/D ConverterFor Voice-Band Telephony Applications
Non-linear ADC and DAC used in voice-band CODECs
• To maximize dynamic range without need for large # of bits
• Non-linear Coding scheme called A-law & μ-law is used
• Also called companding Ref: P. R. Gray, et al. "Companded pulse-code modulation voice codec using monolithic weighted capacitor arrays," IEEE Journal of Solid-State Circuits, vol. 10, pp. 497 - 499, December 1975.
-VFS/2-VFS -VFS/4
Coder Input(ANALOG)
Coder Output(DIGITAL)
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 28
Data Converter Performance Metrics• Data Converters are typically characterized by static, time-domain,
& frequency domain performance metrics :– Static
• Monotonicity• Offset• Full-scale error• Differential nonlinearity (DNL)• Integral nonlinearity (INL)
– Dynamic• Delay, settling time• Aperture uncertainty• Distortion- harmonic content• Signal-to-noise ratio (SNR), Signal-to-(noise+distortion) ratio (SNDR)• Idle channel noise• Dynamic range & spurious-free dynamic range (SFDR)
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 29
Typical Sampling ProcessCT ⇒ SD ⇒ DT
ContinuousTime
Sampled Data(e.g. T/H signal)
Clock
Discrete Time
time
PhysicalSignals
"MemoryContent"
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 30
Discrete Time Signals
• A sequence of numbers (or vector) with discrete index time instants
• Intermediate signal values not defined(not the same as equal to zero!)
• Mathematically convenient, non-physical
• We will use the term "sampled data" for related signals that occur in real, physical interface circuits
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 31
Uniform Sampling
• Samples spaced T seconds in time• Sampling Period T ⇔ Sampling Frequency fs=1/T• Problem: Multiple continuous time signals can yield
exactly the same discrete time signal (aliasing)
y(kT)=y(k)
t= 1T 2T 3T 4T 5T 6T ...k= 1 2 3 4 5 6 ...
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 32
Data Converters
• ADC/DACs need to sample/reconstruct to convert from continuous-time to discrete-time signals and back
• Purely mathematical discrete-time signals are different from "sampled-data signals" that carry information in actual circuits
• Question: How do we ensure that sampling/reconstruction fully preserve information?
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 33
Aliasing
• The frequencies fx and nfs ± fx, n integer, are indistinguishable in the discrete time domain
• Undesired frequency interaction and translation due to sampling is called aliasing
• If aliasing occurs, no signal processing operation downstream of the sampling process can recover the original continuous time signal!
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 34
Frequency Domain Interpretation
fs …….. f
Am
plitu
de
fin 2fs
Am
plitu
de
f/fs
Signal scenariobefore sampling
Signal scenarioafter sampling DT
Signals @nfS ± fmax__signal fold back into band of interest
Aliasing
fs /2
0.5
ContinuousTime
DiscreteTime
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 35
Brick Wall Anti-Aliasing Filter
Sampling at Nyquist rate (fs=2fsignal) required brick-wall anti-aliasing filters
ContinuousTime
DiscreteTime
0 fs 2fs ... f
Amplitude
0 0.5 f/fs
Filter
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 36
Practical Anti-Aliasing Filter
• Practical filter: Nonzero "transition band"• In order to make this work, we need to
sample faster than 2x the signal bandwidth• "Oversampling"
ContinuousTime
0 fs 2fs ... f
Amplitude Filter
fs/2
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 37
Practical Anti-Aliasing Filter
ContinuousTime
DiscreteTime
0 fs ... f
DesiredSignal
0 0.5 f/fs
fs/2B fs-B
ParasiticTone
B/fs
Attenuation
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 38
Data ConverterClassification
• fs > 2fmax Nyquist Sampling– "Nyquist Converters"– Actually always slightly oversampled (e.g. CODEC fsig
max =3.4kHz &ADC sampling 8kHz fs /fmax=2.35)
– Requires anti-aliasing filtering prior to A-to-D conversion
• fs >> 2fmax Oversampling– "Oversampled Converters"– Anti-alias filtering is often trivial– Oversampling is also used to reduce quantization noise, see later
in the course...
• fs < 2fmax Undersampling (sub-sampling)
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 39
Sub-Sampling
• Sub-sampling sampling at a rate less than Nyquist rate aliasing• For signals centered @ an intermediate frequency Not destructive!• Sub-sampling can be exploited to mix a narrowband RF or IF signal down
to lower frequencies
ContinuousTime
DiscreteTime
0 fs ... f
Amplitude
0 0.5 f/fs
BP Filter
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 40
Nyquist Data Converter Topics• Basic operation of data converters
– Uniform sampling and reconstruction– Uniform amplitude quantization
• Characterization and testing• Common ADC/DAC architectures• Selected topics in converter design
– Practical implementations– Compensation & calibration for analog circuit non-idealities
• Figures of merit and performance trends
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 41
Where Are We Now?
• We now know how to preserve signal information in CT DT transition
• How do we go back from DT CT? Analog
Postprocessing
D/AConversion
DSP
A/D Conversion
Analog Preprocessing
Analog Input
Analog Output
000...001...
110
Anti-AliasingFilter
?
Sampling(+Quantization)
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 42
Ideal Reconstruction
• Unfortunately not all that practical...
∑∞
−∞=
−⋅=k
kTtgkxtx )()()(Bt
Bttgπ
π2
)2sin()( =
• The DSP books tell us:
⇒x(k) x(t)
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 43
Zero-Order Hold Reconstruction
• How about just creating a staircase, i.e. hold each discrete time value until new information becomes available?
• What does this do to the frequency content of the signal?
• Let's analyze this in two steps...
0 10 20 30-1
-0.6
-0.2
0.2
1
Time [μs]
Am
plitu
de
sampled dataafter ZOH
0.6
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 44
DT CT: Infinite Zero Padding
DT sequence
Time Domain Frequency Domain
......0.5
InfiniteZero paddedInterpolation:
CT Signal
......
f /fs
f /fs0.5fs 1.5fs 2.5fs
Next step: pass the samples through a sample & hold block (ZOH)
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 45
Hold Pulse Tp=Ts Transfer Function
p
p
s
p
fTfT
TT
fHπ
π )sin(|)(| =
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
f /fs
abs(
H(f)
)s
sTf
)Tfsin(|)f(H|π
π=
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 46
ZOH Spectral ShapingContinuous Time
Pulse Train Spectrum
ZOH Transfer Function
("Sinc Shaping")
ZOH output, Spectrum of
Staircase Approximation
f / fs
0 0.5 1 1.5 2 2.5 30
0.5
1
0 0.5 1 1.5 2 2.5 30
0.5
1
0 0.5 1 1.5 2 2.5 30
0.5
1
X(k)
ZOH
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 47
Smoothing Filter• Order of the filter
required is a function of oversampling ratio
• High oversampling helps reduce filter order requirement
f / fs
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
Filter out the high frequency content associated with staircase shape of the signal
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 48
Summary
• Sampling theorem fs > 2fmax, usually dictates anti-aliasing filter
• If theorem is met, CT signal can be recovered from DT without loss of information
• ZOH and smoothing filter reconstruct CT signal from DT vector
• Oversampling helps reduce order & complexity of anti-aliasing & smoothing filters
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 49
Next Topic
• Done with "Quantization in time"
• Next: Quantization in amplitude
Analog Postprocessing
D/AConversion
DSP
A/D Conversion
Analog Preprocessing
Analog Input
Analog Output
000...001...
110
Anti-AliasingFilter
Sampling(+Quantization)
SmoothingFilter
D/A+ZOH
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 50
Ideal ADC ("Quantizer")• Accepts & analog input &
generates it’s digital representation
• Quantization step:
Δ (= 1 LSB)
• Full-scale input range:-0.5Δ … (2N-0.5)Δ
• E.g. N = 3 Bits
VFS= -0.5Δ to 7.5Δ
-1 0 1 2 3 4 5 6 7 801234567
Dig
ital O
utpu
t Cod
e
ADC characteristicsIdeal converter with infinite # of bits
ADC Input Voltage [ Δ]VFS
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 51
Quantization Error• Quantization error Difference between analog input and
output of the ADC converted to analog via an ideal DAC
• Called:
Quantization error
Residue
Quantization noise
Vin ADCIdeal DAC Σ
Residue
+- εq (Vin ).....
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 52
Quantization Error
• For an ideal ADC:
• Quantization error is bounded by –Δ/2 … +Δ/2for inputs within full-scale range
+
εq (Vin )
Vin Dout
ADC Model
-1 0 1 2 3 4 5 6 7 801234567
Dig
ital O
utpu
t Cod
e ADC characteristicsideal converter with infinite bits
-1 0 1 2 3 4 5 6 7 8
-0.5
0
0.5
Qua
ntiz
atio
n er
ror
[LS
B]
ADC Input Voltage [Δ]
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 53
ADC Dynamic Range• Assuming quantization noise is much larger
compared to circuit generated noise:
• Crude assumption: Same peak/rms ratio for signal and quantization noise!
Question: What is the quantization noise power?
10
20
20 20 2 6 02
Maximum
Maximum
NFS
Full Scale Signal PowerD.R. logQuantization Noise Power
Peak Full ScaleD.R. logPeak Quantization NoiseVlog log . N [ dB ]
=
=
= = = ×Δ
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 54
Quantization Error
0
Δ/2
Quantizationerror [LSB]
Let us assume Vin is a ramp signal with amplitude equal to ADC full-scale
Vin_Ramp
Time
Time
VFS
Note: Quantization error waveform periodic and also ramp
−Δ/2
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 55
Quantization ErrorNeed to find the rms value for quantization error waveform:
Quantizationerror
0
Δ/2
Time
−Δ/2
εq=k.t
−Δ/2kΔ/2k
( ) ( )2 22eq
22
22
eq
eq
T / 2 / 2k
T / 2 / 2k
/ 2k
/ 2k
1k t dt k t dt
T k
kt dt
k
12
12
ε
ε
ε
+ +
+
Δ
− −Δ
Δ
−Δ
Δ= × = ×
Δ×=
Δ→ =
Δ→ =
∫ ∫
∫
Independent of k
In general above equation applies if:• Input signal much larger than 1LSB• Input signal busy• No signal clipping
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 56
Quantization Error PDF
• Probability density function (PDF) Uniformly distributed from–Δ/2 … +Δ/2 provided that:
– Busy input– Amplitude is many LSBs– No overload
• Not Gaussian!
• Zero mean• Variance
Ref: W. R. Bennett, “Spectra of quantized signals,” Bell Syst. Tech. J., vol. 27, pp. 446-72, July 1988.
B. Widrow, “A study of rough amplitude quantization by means of Nyquist sampling theory,” IRE Trans. Circuit Theory, vol. CT-3, pp. 266-76, 1956.-Δ/2
error
1/Δ
+Δ/2
2 22
/ 2
/ 2
ee de
12
+Δ
−Δ
Δ= =
Δ∫
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 57
Signal-to-Quantization Noise Ratio• If certain conditions the quantization error can be viewed as being
"random", and is often referred to as “noise”
• In this case, we can define a peak “signal-to-quantization noise ratio”, SQNR, for sinusoidal inputs:
• Real converters do not quite achieve this performance due to other sources of error:
– Electronic noise– Deviations from the ideal quantization levels
2N
2N2
1 22 2
SQNR 1.5 2
12
6.02N 1.76 dB Accurate for N>3
⎛ ⎞Δ⎜ ⎟⎜ ⎟⎝ ⎠= = ×
Δ
= +
e.g. N SQNR8 50 dB
12 74 dB16 98 dB20 122 dB
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 58
SQNR Measurement
20log(SQNR)
Vin [dB]0dB
6dB/octaveRealistic
peakSQNR 6.02N 1.76 dB= +
Dynamic Range
Ideal
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 59
Static Ideal Macro Models
ADC
+DoutVin
εq
DACVoutDin
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 60
Cascade of Data ConvertersADC
+Vin
εq
DACVout
ADC
+εq
DACDout
Din
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 61
Static Converter Errors
Deviation of converter characteristics from ideal:– Offset
– Full-scale error
– Differential nonlinearity DNL
– Integral nonlinearity INL
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 62
Offset ErrorADC DAC
Ref: “Understanding Data Converters,” Texas Instruments Application Report SLAA013, Mixed-Signal Products, 1995.
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 63
Full-Scale ErrorADC DAC
Actual full-scale point
Ideal full-scale point Ideal full-scale
point
Full-scale error
Actual full-scale
point
Full-scale error
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 64
Offset and Full-Scale Error
-1 0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
6
7
Dig
ital O
utpu
t Cod
e
ADC Input Voltage [LSB]
ADC characteristicsideal converter
Offset error
Full-scale error
Note:For further measurements (DNL, INL) connecting the endpoints & deriving ideal codes based on the non-ideal endpoints elliminates offset and full-scale error
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 65
Offset and Full-Scale Errors
• Alternative specification in % Full-Scale = 100% * (# of LSB value)/ 2N
• Gain error can be extracted from offset & full-scale error
• Non-trivial to build a converter with extremely good full-scale/offset specs
• Typically full-scale/offset is most easily compensated by the digital pre/post-processor
• More critical: Linearity measures DNL, INL
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 66
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
6
7
8ADC Transfer CurveReal Ideal
ADC Differential Nonlinearity
DNL = deviation of code width from
Δ (1LSB)
+0.4 LSB DNL error
-0.4 LSB DNL error
1. Endpoints connected
2. Ideal characteristics derived elliminating offset & full-scale error
3. DNL measured
0 LSB DNL errorDig
ital O
utpu
t Cod
e
ADC Input Voltage [Δ]
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 67
ADC Differential Nonlinearity
• Ideal ADC transitions point equally spaced by 1LSB
• For DNL measurement, offset and full-scale error is eliminated
• DNL [k] (a vector) measures the deviation of each code from its ideal width
• Typically, the vector for the entire code is reported
• If only one DNL # is reported that would be the worst case
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 68
ExampleCompute Offset, Full-Scale Error, & DNL
A 3bit ADC is designed to have an ideal:LSB=0.1V
The measured transitions levels for the end product is shown in the table below, compute offset, full-scale, gain error, & DNL
1- Offset: (real transition-ideal)= -0.03V, in LSB -0.03/0.1=-0.3LSB
2- Full-scale error (real last transition-ideal)= 0.68-0.65=0.03Vin LSB 0.03/0.1=+0.3LSB
3- LSB after correcting for offset & full-scale error: LSB=(Last transition-first transition)/(2N-2)
LSB=(0.68-0.02)/6=0.11V 0.680.657
0.50.556
0.420.455
0.370.354
0.20.253
0.150.152
0.020.051
Real transition point [V]
Ideal transition point [V]
Transition #
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 69
ADC Differential NonlinearityExample
VFS= 2N.0.11V=0.88V4-Gain relative to ideal Gain=0.8/0.88=0.9
Find all code widthsWidth[k]=Transition[k+1]-Transition[k]-Divide code width by LSB W[k]
5- Find DNL:DNL[k]=W[k]-LSB 1.64
0.73
0.45
1.55
0.45
1.18
Width [LSB]
--0
--7
0.640.186
-0.270.085
-0.550.054
0.550.173
-0.550.052
0.180.131
DNL[LSB]
Code Width [V]
Code #
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 70
ADC Differential NonlinearityExample
-0
-7
0.646
-0.275
-0.554
0.553
-0.552
0.181
DNL[LSB]
Code #
Code #
DN
L [L
SB
]
0 1 2 3 4 5 6 7
1
0.5
0
-0.5
-1
Max.DNL
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 71
-1 0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8ADC characteristicsideal converter
ADC Differential NonlinearityExamples
-1 0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8ADC characteristicsideal converter
Non-monotonic(> 1 LSB DNL)
Missing code(+0.5/-1 LSB DNL)
Dig
ital O
utpu
t Cod
eADC Input Voltage [Δ]
Dig
ital O
utpu
t Cod
e
ADC Input Voltage [Δ]
EECS 247 Lecture 11: S.C. Filter Example/ Introduction to Data Converters © 2007 H. K. Page 72
ADC DNL
• DNL=-1 implies missing code• For an ADC DNL < -1 not possible undefined• Can show:
• For a DAC DNL < -1 possible
al l iDNL[i] 0=∑