data converter basics

– 1 –

Data Converters Data Converter BasicsProfessor Y. Chiu

EECT 7327Fall 2014

Data Converter Basics

A/D and D/A Conversion

– 2 –


EECT 7327Fall 2014

S/H

Analogin

Digitalout

Quantization

DSP

AAF

S/H

Analogout

Digitalin

D/A

DSP

Smoothingfilter

A/D Conversion

D/A Conversion

Quantization

– 3 –


EECT 7327Fall 2014

N in

outFS

VDivision : D = 2V

• Quantization = division + normalization + truncation • Full-scale range (VFS) is determined by Vref

A/Dbn

Digital outputAnalog input

b1

...

Vref

Quantization Error

– 4 –


EECT 7327Fall 2014

FSout in out inN

Vε =D Δ - V =D - V

2

FSN

VΔ = =LSB

2

in FSV 0, V

Δ Δ- ε2 2

Dout

0

Vin

Δ 2Δ 3Δ

1

3

5

0

2

4

6

7

VFS

2

-Δ-2Δ-3Δ

VFS

2

“Random” quantization error is usually regarded as noise

ε

Vin0

-Δ/2

Δ/2

Δ 2Δ 3Δ0-Δ-2Δ-3Δ

N = 3

Quantization Noise

– 5 –


EECT 7327Fall 2014

ε

Vin

Δ 2Δ 3Δ

0

4Δ 5Δ 6Δ 7Δ-Δ/2

Δ/2

VFS

Δ/2 2

2 2ε

-Δ/2

1 Δσ = ε dε =Δ 12

Pε

0-Δ/2 Δ/2ε

1/Δ

Assumptions:• N is large• 0 ≤ Vin ≤ VFS and Vin >> Δ• Vin is active• ε is Uniformly distributed• Spectrum of ε is white

Ref: W. R. Bennett, “Spectra of quantized signals,” Bell Syst. Tech. J., vol. 27, pp. 446-472, July 1948.

Signal-to-Quantization Noise Ratio (SQNR)

– 6 –


EECT 7327Fall 2014

2N22NFS

22ε

2 Δ / 8V / 8SQNR = = =1.5 2 ,Δσ12

Assume Vin is sinusoidal with Vp-p = VFS,

SQNR = 6.02 N+1.76 dB

N(bits)

SQNR(dB)

8 49.9

10 62.0

12 74.0

14 86.0

• SQNR depicts the theoretical performance of an ideal ADC• In reality, ADC performance is limited by many other factors:

– Electronic noise (thermal, 1/f, coupling/substrate, etc.)– Distortion (measured by THD, SFDR, IM3, etc.)

FFT Spectrum of Quantized Signal

– 7 –


EECT 7327Fall 2014

• N = 10 bits• 8192 samples, only

f = [0, fs/2] shown

• Normalized to Vin

• fs = 8192, fin = 779

• fin and fs must beincommensurate

0 500 1000 1500 2000 2500 3000 3500 4000-120

-100

-80

-60

-40

-20

0

PSD

Frequency

dB

SQNR -1.76 dBENOB =6.02 dB

SQNR = 61.93 dBENOB = 9.995 bits

Ref: W. R. Bennett, “Spectra of quantized signals,” Bell Syst. Tech. J., vol. 27, pp. 446-472, July 1948.

Commensurate fs and fin

– 8 –


EECT 7327Fall 2014

0 500 1000 1500 2000 2500 3000 3500 4000-120

-100

-80

-60

-40

-20

0

PSD

Frequency

dB

0 500 1000 1500 2000 2500 3000 3500 4000-120

-100

-80

-60

-40

-20

0

PSD

FrequencydB

fs = 8192fin = 256

fs = 8192fin = 2048

• Periodic sampling points result in periodic quantization errors• Periodic quantization errors result in harmonic distortion

Spectrum Leakage

– 9 –


EECT 7327Fall 2014

0 500 1000 1500 2000 2500 3000 3500 4000-120

-100

-80

-60

-40

-20

0

PSD

FrequencydB

0 500 1000 1500 2000 2500 3000 3500 4000-120

-100

-80

-60

-40

-20

0

PSD

Frequency

dB

fs = 8192fin = 779.3

fs = 8192fin = 779.3

• TD samples must include integer number of cycles of input signal• Windowing can be applied to eliminate spectrum leakage• Trade-off b/t main-lobe width and sideband rejection for different windows

w/Blackmanwindow

FFT Spectrum with Distortion

– 10 –


EECT 7327Fall 2014

0 500 1000 1500 2000 2500 3000 3500 4000-120

-100

-80

-60

-40

-20

0

PSD

Frequency

dB

• High-order harmonics are aliased back, visible in [0, fs/2] band• E.g., HD3 @ 779x3+1=2338, HD9 @ 8192-9x779+1=1182

HD3HD9

Dynamic Performance

– 11 –


EECT 7327Fall 2014

SNDR[dB]

Vin

[dB]0 VFS

Overload

• Peak SNDR limited by large-signal distortion of the converter• Dynamic range implies the “theoretical” SNR of the converter

2in

10 2 2N

in

SNR

V / 2=10LOGΔ /12+σ

V dB

PeakSNDR

Dynamic range

Circuitnoise

Dynamic Performance Metrics

– 12 –


EECT 7327Fall 2014

• Signal-to-noise ratio (SNR)• Total harmonic distortion (THD)• Signal-to-noise and distortion ratio (SNDR or SINAD)• Spurious-free dynamic range (SFDR)• Two-tone intermodulation product (IM3)• Aperture uncertainty (related to the frontend S/H and clock)

• Dynamic range (DR) – misleading (avoid it if possible!)• Idle channel noise or pattern noise in oversampled converters

Evaluating Dynamic Performance

– 13 –


EECT 7327Fall 2014

0 500 1000 1500 2000 2500 3000 3500 4000-120

-100

-80

-60

-40

-20

0

PSD

Frequency

dB

• Signal-to-noiseplus distortion ratio(SNDR)

• Total harmonicdistortion (THD)

• Spurious-freedynamic range(SFDR)

SNDR = 59.16 dBTHD = 63.09 dBSFDR = 64.02 dBENOB = 9.535 bits

HD3HD9

SNDR -1.76 dBENOB =6.02 dB

Static Performance Metrics

– 14 –


EECT 7327Fall 2014

• Offset (OS)• Gain error (GE)• Monotonicity• Linearity (unique to converters)

– Differential nonlinearity (DNL)– Integral nonlinearity (INL)

– 15 –


EECT 7327Fall 2014

Static Performanceof DAC

DAC Transfer Characteristic

– 16 –


EECT 7327Fall 2014

Note: Vout (bi = 1, for all i) = VFS - Δ = VFS(1-2-N) ≠ VFS

N N

N-iiout FS ii

i=1 i=1

bV = V = Δ b 22

D/Abn

Digital input

Vout

Analog output

b1

...

Vref

• N = # of bits• VFS = Full-scale input

• Δ = VFS/2N = 1LSB

• bi = 0 or 1• Multiplication

Ideal DAC Transfer Curve

– 17 –


EECT 7327Fall 2014

Vout

000Din

001 011 101010 100 110 111

VFS-Δ

VFS

2

Offset

– 18 –


EECT 7327Fall 2014

Vout

000Din

001 011 101010 100 110 111

VFS

2

VFS-Δ

Vos

Gain Error

– 19 –


EECT 7327Fall 2014

Vout

000Din

001 011 101010 100 110 111

VFS

2

VFS-Δ

Monotonicity

– 20 –


EECT 7327Fall 2014

Vout

000Din

001 011 101010 100 110 111

VFS

2

VFS-Δ

Differential and Integral Nonlinearities

– 21 –


EECT 7327Fall 2014

• DNL = deviation of an output step from 1 LSB (= Δ = VFS/2N)• INL = deviation of the output from the ideal transfer curve

DNL < -1 ?

Vout

000Din

001 011 101010 100 110 111

VFS

2INL

VFS-Δ

th

ii Step Size - ΔDNL =

Δ

DNL and INL

– 22 –


EECT 7327Fall 2014

Vout

000Din

001 011 101010 100 110 111

VFS

2

VFS-Δ

i

i jj=0

INL = DNL

INL = cumulative sum of DNL

DNL and INL

– 23 –


EECT 7327Fall 2014

• DNL measures the uniformity of quantization steps, or incremental (local) nonlinearity; small input signals are sensitive to DNL.

• INL measures the overall, or cumulative (global) nonlinearity; large input signals are often sensitive to both INL (HD) and DNL (QE).

Vout

000Din

001 011 101010 100 110 111

VFS

2

VFS-Δ

Vout

000Din

001 011 101010 100 110 111

VFS

2

VFS-Δ

Smooth Noisy

Measure DNL and INL (Method I)

– 24 –


EECT 7327Fall 2014

Vout

000Din

001 011 101010 100 110 111

VFS

2

VFS-Δ

Endpoints of the transfer characteristic are always at 0 and VFS-Δ

Endpointstretch

Measure DNL and INL (Method II)

– 25 –


EECT 7327Fall 2014

Vout

000Din

001 011 101010 100 110 111

VFS

2

VFS-Δ

Least-squarefit and stretch

(“detrend”)

Endpoints of the transfer characteristic may not be at 0 and VFS-Δ

Measure DNL and INL

– 26 –


EECT 7327Fall 2014

Method I (endpoint stretch)

Σ(INL) ≠ 0

Method II (LS fit & stretch)

Σ(INL) = 0

Vout

000Din

001 011 101010 100 110 111

VFS

2

VFS-Δ

Vout

000Din

001 011 101010 100 110 111

VFS

2

VFS-Δ

– 27 –


EECT 7327Fall 2014

Static Performanceof ADC

Ideal ADC Transfer Characteristic

– 28 –


EECT 7327Fall 2014

Dout

000 Vin

001

011

101

010

100

110

111

VFSVFS/20

Note the systematic offset! (floor, ceiling, and round)

DNL and Missing Code

– 29 –


EECT 7327Fall 2014

Dout

000 Vin

001

011

101

010

100

110

111

VFSVFS/20

DNL = deviation of an input step width from 1 LSB (= VFS/2N = Δ)

• DNL = ?• Can DNL < -1?

th

ii Step Size - ΔDNL =

Δ

DNL and Nonmonotonicity

– 30 –


EECT 7327Fall 2014

Dout

000 Vin

001

011

101

010

100

110

111

VFSVFS/20

DNL = deviation of an input step width from 1 LSB (= VFS/2N = Δ)

• DNL = ?• How can we even

measure this?

INL

– 31 –


EECT 7327Fall 2014

Dout

000 Vin

001

011

101

010

100

110

111

VFSVFS/20

INL = deviation of the step midpoint from the ideal step midpoint(method I and II …)

Any code• Missing?• Nonmonotonic?

10-bit ADC Example

– 32 –


EECT 7327Fall 2014

0 200 400 600 800 1000-2

-1

0

1

2DNL

LSB

0 200 400 600 800 1000-2

-1

0

1

2INL

Code

LSB

• 1024 codes• No missing code!• Plotted against

the digital code, not Vin

• Code density test (CDT)

DNL must always be greater or equal to -1 LSB!

Code Density Test

– 33 –


EECT 7327Fall 2014

Cou

nt

000Vin

001 011 101010 100 110 111

VFS0

Uniformly distributed 0 ≤ Vin ≤ VFS

Δ

n

Δ

n

Δ

n

Δ

n

Δ

n

Δ

n

Δ

n

Δ

n

Ball casting problem: # of balls collected by each bin (ni) is proportional to the bin size (converter step size)

th

i ii

i

n - ni Step Size - ΔDNL =Δ n

Cou

nt

000Vin

001 011 101010 100 110 111

VFS0

Uniformly distributed 0 ≤ Vin ≤ VFS

>Δ

ni

CDT and Nonmonotonicity

– 34 –


EECT 7327Fall 2014

• Two transition steps for one code?! How to plot INL/DNL?• CDT can be misleading in determining the static nonlinearity

Dout

000 Vin

001

011

101

010

100

110

111

VFSVFS/20

– 35 –


EECT 7327Fall 2014

Nyquist-Rate ADC

Nyquist-Rate ADC

– 36 –


EECT 7327Fall 2014

• Digitizes input signal up to Nyquist frequency (fN=fs/2)

• Minimum sample rate (fs) for a given input bandwidth• Each sample is digitized to the maximum resolution of converter• Often referred to as the “black box” version of digitization

A/Dbn

Digital outputAnalog input

b1...

Vref

fs

Nyquist-Rate ADC (N-Bit, Binary)

– 37 –


EECT 7327Fall 2014

• Word-at-a-time (1 step)† ← fast– Flash

• Level-at-a-time (2N steps) ← slowest– Integrating (Serial)

• Bit-at-a-time (N steps) ← slow– Successive approximation– Algorithmic (Cyclic)

• Partial word-at-a-time (1 < M ≤ N steps) ← medium– Subranging– Pipeline

• Others (1 ≤ M ≤ N step)– Folding ← relatively fast– Interleaving (of flash, pipeline, or SA) ← fastest

† the number in the parentheses is the “latency” of conversion, not “throughput”

Accuracy-Speed Tradeoff

– 38 –


EECT 7327Fall 2014

0

Resolution[Bits]

5

10

15

20

1k 10k 100k 1M 10M 100M 1G 10G

Sample Rate [Hz]

Nyquist

Oversampling

Integrating Oversampling

Successive ApproximationAlgorithmic

SubrangingPipelineFolding & Interpolating

FlashInterleaving

1 level/Tclk

1 word/OSR*Tclk

1 bit/Tclk

Partial word/Tclk

1 word/Tclk

100G

Building Blocks for Data Converters

– 39 –


EECT 7327Fall 2014

• Sample-and-Hold (Track-and-Hold) Amplifier• Switched-Capacitor Amplifiers, Integrators, and Filters• Operational Amplifier• Comparators (Preamplifier and Latch)• Voltage and Current DAC’s• Current Sources• Voltage/Current/Bandgap References

data converter basics

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