Efficient ADC Testing Condition

with Histogram Method

Yujie Zhao, Anna Kuwana, Shuhei Yamamoto, Yuto Sasaki,

Haruo Kobayashi, Takayuki Nakatani, Kazumi Hatayama

Keno Sato, Takashi Ishida,

Toshiyuki Okamoto, Tamotsu Ichikawa

Division of Electronics and Informatics

Gunma University

ROHM Semiconductor

Dec. 3, 2020

4th International Conference on Technology and Social Science

2/25

Outline

• Objective

• ADC Test with Histogram Method

• Input Sine Wave and Sampling Frequency

Relationship in ADC Histogram Test Method

Sine Wave Histogram and Waveform Missing

Metallic Ratio and Prime Number Ratio

• Conclusion

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Outline

• Objective

• ADC Test with Histogram Method

• Input Sine Wave and Sampling Frequency

Relationship in ADC Histogram Test Method

Sine Wave Histogram and Waveform Missing

Metallic Ratio and Prime Number Ratio

• Conclusion

4/25

Background

2020/12/4

IoT（Internet of Things）

High quality & Low cost test is required

The application of digital signal and

analog signal conversion is very

extensive.

Analog signal

(sound, light)

Digital signal

(Binary number)

5/25

Research Objective & Approach

Analog-to-Digital Converter (ADC)

Linearity Test

Test cost is proportional to test time

the low-sampling-rate high-resolution ADC

● low-speed sampling

● high-resolution

take a long time

Increasing the sampling efficiency

Propose “short-time” Relationship Between Input Frequency and Sampling Frequency

This Work

6/25

Outline

• Objective

• ADC Test with Histogram Method

• Input Sine Wave and Sampling Frequency

Relationship in ADC Histogram Test Method

Sine Wave Histogram and Waveform Missing

Metallic Ratio and Prime Number Ratio

• Conclusion

7/25

■Histogram method (Ramp wave input)

7

Highly linear ramp signal generation is difficult

ADC output histograms for all bins are equal if

ADC is perfectly linear

t

Ramp wave

ADC

DUT

001010011100101110111

Output

Code

Number

of Samples

INL

DNL

Conventional Linearity Testing 1

DNL

INL

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Conventional Linearity Testing 2

8

■Histogram method(Single sine wave input)

High accuracy sine wave can be generated

using an analog filter

The number of samples is small

around the middle of output codes

t

Sine wave

Sine wave

GeneratorBPF ADC

DUT

Output

Code

Number

of Samples

INL

DNL

f

Remove

DNL

INL

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DNL & INL

9

001 010 011 100 101 110 111

Output

Code

Number

of Samples

DNL

Important testing for ADCs

DNL : Difference between an actual step width

and the ideal value

INL : Deviation from ideal conversion line

k

i

iDNLkINL1

)()(

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Outline

• Objective

• ADC Test with Histogram Method

• Input Sine Wave and Sampling Frequency

Relationship in ADC Histogram Test Method

Sine Wave Histogram and Waveform Missing

Metallic Ratio and Prime Number Ratio

• Conclusion

11/25

Repetitive waveform sampled with asynchronous

Sampling CLK

Measured waveform

Compose a 1-period waveform0

50

100

150

200

250

0 128 256 384 512 640 768 896 1024N

um

ber

of

Sam

ple

s

Output Code

Probability Distribution Function

𝑝 𝑣 =1

𝜋𝐴2−𝑣2

The sampled histogram

is compared with

the PDF.

The histogram is

measured, DNL and INL

are calculated.

Sine Wave Histogram

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A large amount of data is required to reproduce the waveform Test time: long

Waveform Missing

𝑻𝑪𝑳𝑲

TSIG

Waveform Missing occurs in

Special ratio(TCLK and Tsig)

Sampling CLK

Measured waveform

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𝛼 = 1,1

2,1

3,2

3,⋯ ,

1

6,⋯

Τ1 6

1

Τ1 1024

CLK

sig

Waveform Missing

𝑓𝐶𝐿𝐾 ≈1

𝛼𝑓𝑠𝑖𝑔𝑓𝐶𝐿𝐾 ≫ 𝑓𝑠𝑖𝑔

𝑓𝐶𝐿𝐾 ≈ 𝑓𝑠𝑖𝑔

Yuto Sasaki, Yujie Zhao, Anna Kuwana and Haruo Kobayashi, "Highly Efficient Waveform Acquisition Condition in

Equivalent-Time Sampling System", 27th IEEE Asian Test Symposium, Hefei, Anhui, China (Oct. 2018)

Special ratio

TCLK and Tsig , 𝑓𝐶𝐿𝐾 and 𝑓𝑠𝑖𝑔

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Waveform Missing

Normal situation Waveform Missing

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Golden Ratio

Golden Ratio： 𝐥𝐢𝐦𝒏→∞

𝑭𝒏

𝑭𝒏−𝟏= 𝟏. 𝟔𝟏𝟖𝟎𝟑𝟑𝟗𝟖𝟖𝟕𝟒𝟗𝟖𝟗𝟓 = 𝝋

The most beautiful ratio

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Yuto Sasaki, Yujie Zhao, Anna Kuwana and Haruo Kobayashi, "Highly Efficient Waveform Acquisition Condition in

Equivalent-Time Sampling System", 27th IEEE Asian Test Symposium, Hefei, Anhui, China (Oct. 2018)

Τ1 𝜑

CLK

Golden ratio 𝝋

𝒇𝑪𝑳𝑲 = 𝝋 × 𝒇𝒔𝒊𝒈𝝋 = 1.6180339887…

Golden Ratio

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Outline

• Objective

• ADC Test with Histogram Method

• Input Sine Wave and Sampling Frequency

Relationship in ADC Histogram Test Method

Sine Wave Histogram and Waveform Missing

Metallic Ratio and Prime Number Ratio

• Conclusion

18/25

Metallic ratio

Golden Ratio： 𝐥𝐢𝐦𝒏→∞

𝑭𝒏

𝑭𝒏−𝟏= 𝟏. 𝟔𝟏𝟖𝟎𝟑𝟑𝟗𝟖𝟖𝟕𝟒𝟗𝟖𝟗𝟓 = 𝝋

n Decimal

0 1

11 + 5

21.6180339887… Golden ratio ∅

2 1 + 2 2.4142135623… Silver ratio

33 + 13

23.3027756377… Bronze ratio

4 2 + 5 4.2360679774…

… …

n𝑛 + 𝑛2 + 4

2

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Histogram of Saw wave

𝑇𝑠𝑖𝑔

ℎ𝑖 𝑘 =𝑀

𝑁, 𝑘 = 1,2,3, . . . , 𝑁ideal value error 𝑒 𝑘 =

𝑁 ∙ ℎ 𝑘

𝑀− 1

ideal value ℎ𝑖 𝑘 =

𝑀

𝑁

Output Code

Number of

Samples

Total number of samples is M and

ADC resolution(the number of the histogram) is N.

ℎ 𝑘

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ADC Resolution 3Bit N = 8, Increase MRMS

Number of Samples M

Root-Mean-Square of the errors between the actual and ideal histograms 𝑅𝑀𝑆 =

σ ⅇ 𝑘2

𝑁

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RMS between the actual and ideal

Resolution N

RMS

Total number of samples M = 2048, Increase resolution N. Compare RATIO

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RMS of Prime Number RatioRMS

Most of the RMS results are not as good as Metallic ratio.

Total number of samples M = 2048, Increase resolution N. Compare RATIO

Resolution N

𝒇𝑪𝑳𝑲 = 𝒇𝒔𝒊𝒈 × 𝑹𝑨𝑻𝑰𝑶

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RMS results within a range(1~4)

0

500

1000

1500

2000

2500

3000

3500

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4

RMS

RatioTherefore, we calculated the RMS within

a certain range (1~4) to find a good ratio.

RMS minimum point

Total number of samples M = 2048, Resolution 8Bit N=256

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Outline

• Objective

• ADC Test with Histogram Method

• Input Sine Wave and Sampling Frequency

Relationship in ADC Histogram Test Method

Sine Wave Histogram

Random Sampling and Waveform Missing

Metallic Ratio and Prime Number Ratio

• Conclusion

25/25

Conclusion

25

Find a ratio that is more efficient and has a

smaller RMS like the golden ratio

next issue

Golden ratio sampling

Efficiency: Highest

Sampling frequency: low

Metallic ratio sampling

Efficiency: Good

Sampling frequency: High

Prime number ratio sampling

Efficiency:Not Good

Sampling frequency: High