1. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN
0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 4, July-August (2013), © IAEME
126
ESTIMATION OF ENOB OF A/D CONVERTER USING HISTOGRAM TEST
TECHNIQUE
*
Manish Jain1
and R S Gamad2
1
Research Scholar, Department of Electronics &Communication Engineering, PAHER University,
Udaipur Rajasthan, India – 313024
2
Department of Electronics & Instrumentation Engineering, Shri G. S. Institute of Technology and
Science, 23, Park Road, Indore, M.P., India – 452003
ABSTRACT
This paper reports a new application for one of the widely known Analog to Digital Converter
(ADC) dynamic testing methods, namely the histogram method. After estimating code transition levels
and applying corrections of the ADC transfer characteristics, the Effective Number of Bits (ENOBs)
are computed with standard deviation and overdrive effect. ENOB is determined by taking deviation of
corrected rms error from ideal rms error. Simulation results for 5 and 8 bit ADC are presented which
show effectiveness of the proposed method. Finally results of this method are compared with earlier
reported work and improvements are obtained in present results.
Keywords: Effective number of bits; Transfer Characteristics; Code bin width; Error estimation;
Transition levels.
1. INTRODUCTION
Testing and characterizing ADC is still a challenging issue for mixed signal device
manufacturers and designers, both in terms of speed, resolution and cost. Generally the goal of such
procedure is to verify in a short time whether a given ADC meets its performance requirements. As
known, many techniques in the time, frequency and amplitude domains have been proposed for ADC
testing. ADC is an important device widely used in many electronics applications like: Instrumentation
systems, communication system, medical Instrumentation, radar system and military applications for
interfacing analog electronics with digital electronics. If the aim is to select a better device for an
application then data sheet specifications are sufficient for comparison. But if a selected device is to be
used in a design then determination of its functional parameters over application condition is must.
Selection of an ADC is done based upon resolution, speed, power consumption, conversion accuracy
required and interfacing to the system. In this work method is developed for dynamic testing of an
ADC using different inputs like sine wave, triangular wave and application mode signal. ENOB of an
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2. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN
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ADC is a function of test frequency. It is necessary to determine the value of ENOB at this test
frequency. With increase in input frequency nonlinearity error increases which results in decreases in
value of ENOB. Sine wave and triangular wave based histogram methods are popular in determining
these errors of an ADC [1]-[3]. First of all code transition levels of ADC transfer characteristics are
computed by collecting large number of samples of full scale sine wave by test ADC. Recently work
has been reported for computation of DNL, INL, gain error, offset error and ENOB in performance of
ADC transfer characteristics [4]. In this proposed work, we have used existing method of error
computation in code transition levels and based upon this, error in estimate of nonlinearity and ENOB
of an ADC are computed. In addition to existing method of finding error in code transition levels we
have also computed error by estimating difference in code transition levels and best fit code transition
levels. Dynamic testing using sine wave input based on histogram method is an important activity for
characterization of an ADC.
ADC is usually characterized by its figures of merits like Effective Number of Bits (ENOB),
Signal to Noise and Distortion ratio (SINAD), DNL and INL [5] [6].
2. COMPUTATION OF ERROR IN CODE TRANSITION LEVELS
Estimate of code transition level ( )T i
∧
for level i is given by [1] [7]:
( )
[ 1]
cos
Ch i
T i C A
X
π∧ − 
= −   
, i = 1… 2n
(1)
Where A is amplitude and C is offset of sine wave applied to input of an ADC with X number of
samples and [ ]Ch i is the cumulative histogram defined by:
1
0
[ ] [ ]
i
j
C h i H j
−
=
= ∑ , i = 1……… 2n
(2)
Where, H [0] = 0 and H [j] is histogram for code j.
Error in code transition level can also be computed by taking difference of estimated transition level
( )T i
∧
from best fit transition level Tb(i).
( ) ( ) ( )be T i T i T i
∧ ∧
 
= −  
(3)
Where, best fit code transition levels ( ) 0
1
b
i a
T i
a
−
=
and best fit parameters
( ) ( ) ( )
( ) ( )( )
0 22
i T i T i T i i
a
n T i T i
   −   =
   −    
∑ ∑ ∑ ∑
∑ ∑
( ) ( )
( ) ( )( )
1 22
n T i i T i i
a
n T i T i
   −   =
   −    
∑ ∑ ∑
∑ ∑
Here 1
2N
n −
= , where N is number of bits of an ideal ADC. Limit for all summation is 0 to n.
The real life ADC can be modeled as gain error, offset error, nonlinearity and followed by ideal
quantization process is presented in Figure 1 and implementation steps of algorithm for determination
of code transition error with estimated code transition levels and best fit code transition levels are given
in Figure 2.

3. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN
0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 4, July-August (2013), © IAEME
128
Figure 1 Real life model of an ADC with application mode input
Start
Initialization and Selection of test conditions
Simulate ideal ADC transfer characteristics and introduce
arbitary DNL error
Simulate full scale sine wave and take samples with test
ADC
Compute histogram and estimate code transition levels from
cumulative histogram
Apply corrections in code transition levels and determine
best fit transfer characteristics
Take difference of estimated and best fit code transition
levels, determine code transition error
Determine code transition error in each code levels
Stop
or
Take difference of Mean of the estimated transition levels
and normalished code transition levels
Figure 2 Implementation steps of algorithm for determination of code transition error with
estimated code and best fit code transition levels
3. DETERMINATION OF ENOB WITH SLIGHT OVERDRIVE AND CORRECTION
The values of corrected ENOB considering corrected code transition level can be expressed as cENOB .
( )
2log
( )
c
rm serror C orrected
EN O B N
rm serror ideal
 
= −  
 
(4)

4. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN
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Corrected rms error is computed by considering corrected code transition levels in general formula for
computing rms error and is given as [7] [10]:
The rms value of actual and ideal noise can be computed by
rms noise (K) =
2
1
1]X[K
X[K]
2
][]1[
e














−+
∫
+
KxKx
dx
(5)
Where e is the error signal between output and input of transfer characteristic and is expressed as
e = p x + q (6)
equation (10) passes through two points ( x(K), e(K) ) and ( x(K+1), e(K +1) ). The parameters p and
q can be obtained as
p= ]1[][
]1[][
+−
+−
KxKx
KeKe
(7)
q =
1
2
[e (K) + e (K+1) - p{x (K) + x (K+1)}] (8)
For ideal case:
e (K) = Vcbf (K) – Tb (K) (8a)
X (K) = Tb (K) (8b)
And for actual case:
e (K) = Vcbf (K) – T (K) (9a)
X (K) = T (K) (9b)
The centre values of best fit transition level Vcbf (K) is given by
Vcbf (K) =
2
)()1( KTbKTb ++
(10)
4. SIMULATION RESULTS AND DISCUSSIONS
Ideal ADC transfer characteristics for 5 and 8 bit resolution are simulated and arbitrary
nonlinearity error is introduced in their transfer characteristics. Simulated full scale signals with 0.98
MHZ frequency with slight over derive is applied to the ADC and large number of samples at 25 MHZ
sampling frequency are collected. In first case using standard histogram technique code transition
levels are computed after introducing DNL error in ADC transfer characteristics. Further error in code
transition level estimation is done. After that author have determined rms quantization noise error
would be equal to total rms error from all sources in the ADC under test. Three types of input are
applied to ADC, full scale sine wave in first case and triangular wave in second case and simulated
application mode input in third case. ENOB for all the three cases for 5 and 8 bit ADC are estimated by
proposed method and plotted in Figure 3 and 4 respectively. It is observed that estimated ENOB for
application mode input and sine wave input are close to each other while for triangular wave input
estimated value of ENOB is less. The reason for this is due to more than one component in application
input and only one component is present in sine wave input. So that the estimated value of ENOB for

5. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN
0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 4, July-August (2013), © IAEME
130
these two are very close to exact value. Because triangular wave input is made-up of different
sinusoidal component so higher values of error are obtained and due to this estimated value of ENOB
are less then sine wave and application mode input. Comparison of the estimated ENOB for 5 and 8 bit
ADC with earlier reported works are given in table 1 and 2 respectively.
Table 1: Comparison of ENOB estimation for 5 bit ADC with earlier reported work
Figure 3 Graphical representation of ENOB for 5 bit ADC
Number
of
samples
ENOB
earlier
Ref.
[8]
ENOB
earlier
Ref. [10]
ENOB
earlier
Ref. [2]
ENOB
earlier
Ref.
[5]
ENOB estimation by proposed method
using different input signals
Application
mode input
(Sum of
two signals)
Triangular
wave
input
Single Sine
wave
input
512 - 4.717892 4.597796 4.613421 4.669743 4.641890 4.832065
1024 4.342 4.709405 4.607100 4.615422 4.712700 4.646127 4.769381
2048 4.353 4.685809 4.607080 4.626420 4.724758 4.676413 4.768702
4096 4.356 4.682797 4.604679 4.628864 4.729836 4.679168 4.778661
8192 4.375 4.649188 4.604537 4.637960 4.731202 4.681242 4.782439
11200 4.379 - - 4.642014 4.737856 4.688672 4.780181
16384 4.378 4.654399 4.605245 4.648931 4.748938 4.681429 4.787243
20000 4.3757 - - - 4.747349 4.689567 4.781189
25000 - - - - 4.746698 4.678036 4.787588
30000 - 4.651987 4.606943 4.649932 4.746971 4.677637 4.782033
64K - - - - 4.726521 4.656431 4.764341
128K - - - - 4.725985 4.657654 4.765542
256K - - - - 4.726531 4.658926 4.766321
512K - - - - 4.736582 4.658864 4.772875
1024K - - - - 4.726574 4.658896 4.768943

6. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN
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Table 2: Comparison of ENOB estimation for 8 bit ADC with earlier reported work
Figure 4 Comparative graphical representation of ENOB for 8 bit ADC
Number
of
Samples
ENOB
earlier
Ref. [8]
ENOB
earlier
Ref. [10]
ENOB
earlier
Ref. [2]
With
train.
Wave
ENOB
Earlier
Ref. [13]
ENOB
earlier
Ref. [3]
ENOBc
earlier
Ref. [12]
Proposed
work
With
triangular
wave input
Proposed
work
With
application
input
1024 7.3705 6.928920 6.112926 7.7 max.
& 7.0 min
With
5 – 40
KHz freq.
6.887535 7.657624 7.532015 7.664230
2048 7.4701 7.347980 6.107440 7.448753 7.667821 7.534660 7.668931
4096 7.5092 7.491235 6.129020 7.560021 7.686620 7.548934 7.687821
8192 7.5120 7.582269 6.172342 7.589882 7.698766 7.557810 7.699320
16384 7.5193 7.608350 6.266685 7.599012 7.699864 7.560241 7.712430
30K - 7.631215 6.366832 - 7.600021 7.723002 7.563452 7.735682
64K - - - - - 7.647827 7.558651 7.656341
128K - - - - - 7.646182 7.557650 7.658302
256K - - - - - 7.648210 7.558649 7.658321
512K - - - - - - 7.558652 7.668320
1024K - - - - - - 7.557751 7.658328

7. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN
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5. EFFECTS OF STANDARD DEVIATION OVERDRIVE ON ENOB
In addition effects of overdrive and standard deviation on ENOB estimation of an ADC are also
presented with and without corrections applied in code transition levels. During this process the
number of samples was 64k, input frequency was 0.98 MHz and sampling frequency was 25 MHz.
Table 3 shows the effects of overdrive and standard deviation. When fixed the values of standard
deviation (σn = 0.25, 0.5, 0.75 and 1.25) with increasing values of overdrive voltage. It is observed that
the values of ENOB are better in the range of σn = 0.25 to 0.75 and overdrive range 0.88 to 1.1 LSB
after that values of ENOB is deteriorates.
Table 3: Representation of overdrive and standard deviation effects on ENOB with sine wave
6. CONCLUSION
In this paper effects of error, overdrive and standard deviation on ENOB have been studied.
Mean of transition voltage is calculated and error in transition voltage is computed by taking difference
from estimated transition voltages after that correction is applied in the estimated code transition
levels. ENOB is determined by taking deviation of actual rms error from best fit (ideal) rms error of
ADC transfer characteristics. Simulation results are reported for five bit ADC with different amount of
overdrive and additive noise and their effects on ENOB. This work will be useful for error
minimization and testing A/D converter from device manufacturer point of view as well as circuit
designer. Because this paper mainly concentrated on triangular wave and application mode input based
results for higher bits. A test algorithm developed using simulation is equally suitable for testing real
life ADC in application conditions.
Standard Deviation » σn = 0.25 σn = 0.5 σn = 0.75 σn = 1.25
S.
No
Over
drive
Voltage
(LSB)
ENOB
without
correction
ENOB with
correction
ENOB with
correction
ENOB with
correction
ENOB with
Correction
1. 0.11 4.641100 3.591714 3.862237 4.285664 4.502580
2. 0.22 4.618717 3.749800 3.857520 4.182453 4.381658
3. 0.33 4.629631 3.954577 3.956024 4.184426 4.393105
4. 0.44 4.606220 4.590521 4.102843 4.378797 4.488625
5. 0.55 3.783103 4.682992 4.246127 4.463543 4.569603
6. 0.66 3.580804 4.717010 4.592254 4.585825 4.663260
7. 0.77 3.618924 4.674049 4.693146 4.744065 4.884715
8. 0.88 3.589631 4.630507 4.858720 4.862046 4.985870
9. 3.563010 5.062183
10. 3.543120 5.064202

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