Hilbert Based Testing of ADC Differential Non-linearity Using Wavelet Transform Algorithms

International Journal of Electrical and Computer Engineering (IJECE)
Vol.8, No.6, December2018, pp. 5071~5079
ISSN: 2088-8708, DOI: 10.11591/ijece.v8i6.pp5071-5079  5071
Journal homepage: http://iaescore.com/journals/index.php/IJECE
Hilbert Based Testing of ADC Differential Non-linearity Using
Wavelet Transform Algorithms
Emad A. Awada
Electrical and Computer Engineering, Applied Science Private University, Jordan
Article Info ABSTRACT
Article history:
Received Dec 25, 2017
Revised Jul 28, 2018
Accepted Aug 11, 2018
In testing Mixed Signal Devices such as Analog to Digital and Digital to
Analog Converters, some dynamic parameters, such as Differential Non-
Linearity and Integral Non-linearity, are very critical to evaluating devises
performance. However, such analysis has been notorious for complexity and
massive compiling process. Therefore, this research will focus on testing
dynamic parameters such as Differential Non-Linearity by simulating
numerous numbers of bits Analog to Digital Converters and test the output
signals base on new testing algorithms of Wavelet transform based on Hilbert
process. Such a new testing algorithm should enhance the testing process by
using less compiling data samples and prompt testing results. In addition,
new testing results will be compared with the conventional testing process of
Histogram algorithms for accuracy and enactment.
Keyword:
Differential Non-Linearity
Digital-to-Analog Converters
Discrete Wavelet Transform
Hilbert Transform
Copyright © 2018 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Emad A. Awada,
Department of Electrical and Computer Engineering,
Applied Science Private University, Amman, Jordan.
Email: e_awada@asu.du.jo
1. INTRODUCTION
The notion of converting analog to digital waveform (digitizing) is very critical task for Digital
Signal Processing (DSP) applications to enable gathering, analyzing, and interpolating analog data in digital
domain. However, this conversion of analog to digital representation requires mixed signal Analog to Digital
Converters (ADCs), which comes in many types and speed for fast and instantaneous digitizing process.
Though, to ensure the proper performance of DSP devices, thorough testing parameters are required to
eliminate ADCs false representation outputs. For example, parameter such as Differential Non-Linearity
(DNL) must be checked for any abnormality output characteristics due to error added or deviation from the
original analog waveform [1], [2].
As a result, such testing will ensure proper performance of applications as expected. However, at
manufacture level, testing of such devicescan be very lengthy and complicated especially with higher number
of bits ADCs [2]. For instance, in [1]-[3], conventional testing of such parameters requires large number of
data samples which make testing and compiling process very tedious task and lengthy process. That is, all
output waveform data codes need to be included in compiling process and compared with ideal codes for
DNL. Each code contains a large number of samples that forming a fixed length of codes known as Least
Significant Bits (LSB) [2], [4].
Therefore, an increase in ADCs number of bits results in code increases by 2n
(n is number of ADC
bits) and increase number of data samples making the testing process very lengthy. In addition, conventional
methods of testing waveform characteristics such as FFT and Sinusoidal Histogram have been known to
include error in testing compile process and estimation results [2], [6]-[8]. For instance, while FFT is based
on additive noise model and require a large number of samples bit [1], [2], [9], [10], Sinusoidal Histogram
majority of samples collected are localized at both ends of the Histogram to produced larger error near the
peaks [2]. Therefore, deviation produced by faulted bits or external error will be included in estimating code

 ISSN:2088-8708
Int J Elec & Comp Eng, Vol. 8, No. 6, December 2018: 5071 - 5079
5072
length.As a result, testing process must be enhanced to shorten testing period of time with importance of high
sensitivity of data analysis.
Large number of researches has investigated classical testing algorithms to enhance testing
algorithms. While some works have performed in the area of Fourier Transform and Sinusoidal Histogram
[11]-[14], others have investigated new testing algorithms such as Wavelet transforms [1], [2], [9], [15], [16].
While, In previous works, [2]-[5], DNL errors for numerous ADC devices have been tested using Wavelet
transform, this work will focus on the benefit of combining Hilbert Transform and Wavelet transform to
shorten testing time significantly by reducing the total number of required collected data samples. That is,
using Hilbert Transform as a base signal module and Wavelet transform as a data extracting processto
achieve faster testing process and of higher testing sensitivity of bits error analysis.
2. THEROTICAL BACKGROUND
Ideally, to measure ADCs output waveform in term of DNL, the device under test (ADCs) output
voltage range need to be determined for step size estimation (Least Significant Bit (LSB)) (1). That is, the
device output full voltage range (Full Scale Rang (FSR)) is divided into equal segments of voltage. Each
division will contain a specific range of voltage values
[2 1] [0]
1
2 1
n
n
Voltage Voltage
LSB
 


(1)
Where n is ADC number of bits
Segments deviation can be measure among consecutive codes of LSB. Therefore, DNL can be
determinate by (2)
[ 1] [ ]
1
V i V i
DNL
LSB
 
  (2)
3. TESTING METHODOLOGY
In this simulation of testing ADCs DNL, testing is based on generating digital waveform codes to
measures output corresponding voltages. As shown in [16], by capturing output voltage code, Hilbert
Transform and Wavelet decomposition will be applied respectively to transfer signal and eliminate extra
unrequired data.
In this proposed testing algorithms, Hilbert Transform is used to define the real and imaginary parts
of a signal (3). Such a process will enhance the ability to detect function peak through interpolation [16],
[17]. As a result, the computation of waveform samples can be based on one part of divide complex signal.
z(t) = g(t) + j ( )g t (3)
However, in this work, the imaginary part of tested waveform  ˆg t will be collected (a summation of input
cosine wave and other non-input components (noises  n ) (4).
   ˆ [ ]g t g n n  (4)
and the modulated output waveform can be define as in (5)
   0
ˆ sin(2 )g m A f n m     (5)
Yet, modulated signal will undergo further decomposition and filtering process by implementing
Discrete Wavelet Transform (DWT). With two special properties of convolution and down-sampling process,
Wavelet transform can produce a unique algorithm in analyzing the output tested waveform data. Assuming a
discrete signal (sn={snk}), Wavelet decomposition for approximation and detailing coefficient can be obtained
respectively by (6) and (7) respectively

Int J Elec& Comp Eng ISSN: 2088-8708 
Hilbert Based Testing of ADC Differential Non-linearity Using Wavelet Transform… (Emad A. Awada)
5073
1, 2n j k j nk
k
s h s   (6)
1, 2n j k j nk
k
d g s   (7)
This process is done by convolution as shown (8) and (9)
( )( ( )* )n j j k nk
k
h s h s    (8)
( )( ( )* )n j j k nk
k
g s g s    (9)
Followed by down-sampling as in (10) and (11)
1 ( 2)( ( )* )n ns h s    (10)
1 ( 2)( ( )* )n nd g s    (11)
As a result, number of processed data samples will be reduced by half (half the bandwidth) starting
from the largest energy level and down into a sub-level of signal energies decomposition [18]-[23]. This
particular property of Wavelet transform is possible by the advantage of group of filter banks of low-pass and
High-pass filters. However, based on Wavelet filter banks, which determine translation and scaling properties
of the waveform, there are many types of Wavelet uniqueness known as Mother Wavelets [9], [18].
Therefore, the selection of specific type family Wavelet can be very serious based on Wavelet attribute such
as width of frequency window, decaying, symmetry, regularity, orthogonal, biorthogonal, etc.
In this work, Daubechies Wavelets (dbxx) and Haar Wavelet will be used since they are widely used
in engineering applications and for matching the properties of tested signal data form [5]. Yet, keeping in
mind the shape of wavelet families, orthogonality (such as Daubechies), and biorthogonality (such as biorxx)
of Wavelet transform will have a serious impact on testing analysis and results [5], [24]-[27].
4. COMPUTATION TECHNIQUE
Ideally, ADCs output waveform is zero deviation from the original waveform. That is, based on
signal full scale range and ADC number of bits, the output digital waveform should have the same LSB
codes throughout the waveform voltage division [2], [4]. Yet, practically, ADCs digital output signal ( )x t
contains error build into the output waveform (12) due to noise, heat, and other effect that may cause
waveform distortion.
( ) ( )x t x t e  (12)
Where ( )x t = Original signal values, and e = error-values
That is, output signal is a combination of both signal data bits and error bits. Some of these errors are
due to quantization process which depends on ADCs number of bits that determine quantization step size
( ) . As in shown Figure 1, the analog input waveform is digitized by sampling and quantization process to
provide an output digital waveform as in Figure 2.
Codes
111
110
101
100
011
010
001
000
V-in
Figure 1. Analog Input Signal

 ISSN:2088-8708
5074
000
001
010
011
100
101
x0 x1 x2 x3 x4 x5
xn
Figure 2. Digital output codes
As a result, ADCs number of bits (n) will affect the size of such a code pattern LSB and any
deviation will result in DNL error. In this work, a real time (practical) digital waveform will be simulated as
an output signal based on ADC number of bits. The generated digital waveform will be modulated by Hilbert
Transform to identify the real and imaginary part of the signal. By capturing the imaginary part, a further
process will be conducted by DWT to fetch the tested waveform data by translation and dilation effect
through low-pass and high-pass filtersto produce approximation coefficients ( ns ) detail coefficients ( nd ) as
in (13) and (14) respectively.
1 0 1 21, 1 , 1
1,0 1 0 1 2 ,0
1,1 ,11 0 1 2
n n
n n
n n
h h h hs s
s h h h h s
s sh h h h
  
 
 
    
    
    
    
    
    
    
        
(13)
1 0 1 21, 1
1, 1 1 0 1 2
1,0 1 0 1 2
1,0 1 0 1 2
1,1 1 0 1 2
1 0 1 21,1
n
n
n
n
n
n
h h h hs
d g g g g
s h h h h
d g g g g
s h h h h
g g g gd
 
  
 
 
 

  
  
  
  
  
  
      
  
  
  
  
  
    
, 1
,0
,1
n
n
n
s
s
s

 
 
 
 
 
 
 
 
 
 
 
 
 
   
(14)
Then decomposed tested signal data by taking the detail coefficients ( nd ) and down sample by half
(15) and (16)
1, 1 1,0 1,1 1,2 1,3 1,4 1,5 1,6 1,7( , , , , , , , , )n n n n n n n n nd d d d d d d d d         
(15)
1, 1 1,1 1,3 1,5 1,7( , , , , , )n n n n nd d d d d     
(16)
To compute instantaneous DNL measurements (17), high-pass coefficients of second level
decomposition will be obtained to estimate the difference between the magnitudes of consecutive real codes
[1], [2].
 1, 1, 1max
( ) 1
n j n j
ideal
d d
DNL n
  
 

(17)
where ideal is ideal LSB [1]

5075
5. SIMULATION AND MEASURMENTS
A range of Analog to Digital Converts was simulated using Matlab simulation software. The range
was based on ADCs number of bits (2-20 Bits) to be tested for worst case instantaneous DNL. The generated
output signal was modulated by Hilbert Transform to obtain the characteristic of the signal in real and
imaginary part as shown in Figure 3.
Figure 3. Hilbert Transform of real and imaginary part
Upon processing the signal by Hilbert Transform, the imaginary part of the signal (sinewave) was
quantized and converted for sampling Interleave as shown in Figure 4.
Figure 4. Hilbert Transform imaginary signal part, quantized imaginary part, and sample time
Meanwhile, the imaginary converted part of the waveform was checked by the Fourier Transform
algorithms to verify the signal by time and frequency domain as shown in Figure 5.
Figure 5. Hilbert converted sinewave signal in time domain and power density

 ISSN:2088-8708
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Based on such algorithms, the quantized Interleave signal data form is acquired for further process
of decomposition to suppress the waveform data sample on multilevel as specified in Figure 6 and 7
consecutively for first level and second level DWT decomposition process.
Figure 6. Discrete Wavelet Transform (first level decomposition)
In first level of DWT decomposition, the Interleave signal was decomposed into detail and
approximation coefficientby splitting the waveform data into half. Meanwhile, the detail coefficient from
first decomposition process was used for a second round of decomposition as in Figure 7. The detail part of
the second coefficientwas used to estimate for instantaneous DNL.
Figure 7. Discrete Wavelet Transform (Second level decomposition)
6. RESULTS AND DISCUSSION
In instantaneous testing of digitized output waveform through proposed testing algorithms of
Hilbertand Wavelet transform decomposition strategy, it was clear that the new algorithm has shown an
enhancement in both testing results and techniques. Using Matlab to simulate various numbers of bits ADCs,
and several types of discrete Wavelet, results have shown close approximate to conventional testing method.
In this work, DNL was the focus of testing and verification in term of accuracy with conjunction of
short time testing process. That is, as wavelet distinct properties of dilation and translation, a unique type
wavelet must be used to give a close approximate match of tested waveform characteristics. For example,
generated under test sine-waveform was tested with several types of wavelets. With Haar wavelet
characteristics, as shown in Figure 8, the sharp structures edges and straight line of re-adjusting and shifting
window did not give close match to tested waveform.

5077
Figure 8. Haar Wavelet
This in return, did not show very accurate result as shown in Tables 1-3. Meanwhile, the Daubechies
family had out performed the Haar Wavelet due approximate similarity of the original tested sinewave in
pattern as shown in Figure 9.
Figure 9. Daubechies family Wavelet
That is, the similarity in waveform pattern allows the Daubechies family wavelet to imitatethe under
testing waveform and analyze through adjusted shifted windows with higher detection sensitivity. As a result,
instantaneous DNL measurements was obtained and compared as shown in Tables 1- 3 at several testing
frequencies.
Table 1. DNL estimation (conventional method FFT Vs. Wavelet based Hilbert Transform 100khz
2 Bits 4 Bits 6 Bits 8 Bits 10 Bits 12 Bits 14 Bits 16 Bits 18 Bits 20 Bits
HAAR 0.74 0.53 0.47 0.39 0.58 0.56 0.13 0.57 0.24 0.38
DB2 0.56 0.34 0.39 0.37 0.37 0.33 0.31 0.31 0.35 0.26
DB4 0.53 0.31 0.33 0.31 0.28 0.34 0.29 0.32 0.37 0.22
DB10 0.55 0.36 0.36 0.32 0.36 0.39 0.38 0.35 0.36 0.28
Conv. 0.49 0.44 0.31 0.31 0.33 0.38 0.39 0.32 0.38 0.28
HDWT/FFT 2 Bits 4 Bits 6 Bits 8 Bits 10 Bits 12 Bits 14 Bits 16 Bits 18 Bits 20 Bits
HAAR 0.77 0.39 0.47 0.55 0.52 0.50 0.15 0.53 0.28 0.35
DB2 0.56 0.32 0.40 0.36 0.39 0.38 0.35 0.33 0.39 0.27
DB4 0.50 0.30 0.32 0.23 0.24 0.36 0.27 0.23 0.50 0.36
DB10 0.58 0.37 0.33 0.30 0.33 0.41 0.40 0.34 0.36 0.28
Conv. 0.49 0.43 0.33 0.40 0.23 0.31 0.23 0.44 0.46 0.28
HDWT/FFT 2 Bits 4 Bits 6 Bits 8 Bits 10 Bits 12 Bits 14 Bits 16 Bits 18 Bits 20 Bits
HAAR 0.77 0.23 0.47 0.35 0.52 0.50 0.15 0.53 0.28 0.35
DB2 0.56 0.32 0.40 0.36 0.39 0.38 0.35 0.33 0.39 0.27
DB4 0.50 0.30 0.30 0.23 0.24 0.36 0.27 0.23 0.50 0.56
DB10 0.58 0.36 0.33 0.30 0.33 0.41 0.40 0.34 0.36 0.28
Conv. 0.58 0.41 0.53 0.40 0.23 0.31 0.20 0.44 0.46 0.28
On the other hand, using Hilbert Transform and Wavelet decomposition to eliminate unrequired data

 ISSN:2088-8708
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and detection of function peaks, helps reducing the testing duration time significantly. That is, using the
imaginary waveform part of Hilbert Transform with the decomposition process of Wavelet, permit the
reduction of number of compiled data samples by half at each decomposition level to end-up with nearly one
fourth collected sample bits for compiling process.
7. CONCLUSION
In this new work of testing Differential Non-Linearity performance for Analog to Digital Converts,
Hilbert Transform was implemented as an initial modulation test base. As a result, a particular data was
extracted from the original waveform to create platform for Discrete Wavelet Algorithms analysis. Such an
algorithm will help reducing the number of waveform data collected by the advantage of both Hilbert and
Wavelet transform. That is, less samples to collect and store, shorter compiling process, and accurate error
estimation (DNL) near to conventional estimation of Histogram technique.
ACKNOWLEDGEMENTS
The authors are grateful to Applied Science Private University, Amman - Jordan, for the financial
support grated to cover the publication fee of this paper research articles.
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