The application wavelet transform algorithm in testing adc effective number of bits

International Journal of Computer Science & Information Technology (IJCSIT) Vol 5, No 5, October 2013

THE APPLICATION WAVELET TRANSFORM
ALGORITHM IN TESTING ADC EFFECTIVE
NUMBER OF BITS
Emad A. Awada
Department of Electrical and Computer Engineering,
Applied Science University,
Amman, Jordan

ABSTRACT
In evaluating Analog to Digital Convertors, many parameters are checked for performance and error rate.
One of these parameters is the device Effective Number of Bits. In classical testing of Effective Number of
Bits, testing is based on signal to noise components ratio (SNR), whose coefficients are driven via
frequency domain (Fourier Transform) of ADC’s output signal. Such a technique is extremely sensitive to
noise and require large number of data samples. That is, longer and more complex testing process as the
device under test increases in resolutions. Meanwhile, a new time – frequency domain approach (known as
Wavelet transform) is proposed to measure and analyze Analog-to-Digital Converters parameter of
Effective Number of Bits with less complexity and fewer data samples.
In this work, the algorithm of Wavelet transform was used to estimate worst case Effective Number of Bits
and compare the new testing results with classical testing methods. Such an algorithm, Wavelet transform,
have shown DSP testing process improvement in terms of time and computations complexity based on its
special properties of multi-resolutions.

KEYWORDS
Discrete Wavelet Transforms (DWT), Analog-to-Digital Converters (ADCs), Effective Number of Bits
(ENOB).

INTRODUCTION
The ability of transferring data information from analog into digital domain and vise versa is
highly desired [1] and needed. Therefore, in choosing the right data acquisition board, conversion
accuracy can be one of the most important factors. For proper system performance and digitizing
accuracy, output digitized data need to be as close to the analog input signal. At most basic level,
digitizer testing would seem simple matter; however, testing is extremely expensive, and time
consuming as in [2-5] for both static and dynamic parameter characterizations.
Dynamic parameters, based on Fast Fourier Transform (FFT), provide specific information that
shows the effect of noise and signal distortions, especially with applications of high-frequency
signals. Therefore, depending on the quality of digitizer and the continuous switching of signals
between multiple channels, dynamic parameters such as Effective Number of Bits (ENOB) can
drop noticeably. For example, a 16-bit ADC’s can drop to 12-bit or even more, which drop the
accuracy of system performance as a result. Therefore, ENOB measurements can be one of the
major dynamic error characteristics of any digitizer board, but this measurement can be lengthy
DOI : 10.5121/ijcsit.2013.5512

161


and complicated due to the large number of data samples acquired and the fact that it is heavily
based on errors caused by noise (SNR).
Significant research has been done to improve testing techniques of Analog to Digital Convertors
(ADC’s) parameters. For instance, some focused on improving classical methods such as Fourier
transform and Sinusoidal Histogram in testing AC and DC performance [5-9]. While Fourier
Transform is based on additive noise model and requires a large number of data samples [2],
histogram testing has most samples occur near the ends of histogram (large number of samples
must be collected to increase the height of bins around the center) [1,4].
In this paper, Wavelet based testing algorithms is proposed for evaluating an ADC’s performance
in term of instantaneous ENOB. With the focus on shortening testing process, reducing sample
size, simplifying testing complexity and improving ENOB estimation for higher ADCs
resolutions, Wavelet computation efficient was very suitable for this application in contrast to the
mean value measured by the conventional methods [2,4,10].

EFFECTIVE NUMBER OF BITS (ENOB)
The conventional method of testing ADC’s effective bits is based on the ratio of the fundamental
signal to the sum of all distortion and noise products in the output signal, after the DC term is
removed. Therefore, in testing for ENOB, clean analog sine-wave was used as stimulus input,
since perhaps it is the most popular method of evaluating ADC [11]. For an ideal ADC’s output,
signal (data) transformed into frequency domain will result in one spectral component. In real
ADC’s operation, quantization errors which cause nonlinearity effect on ADC’s transfer function
results into spectral frequencies other than the input frequency being tested. Quantization errors
caused by IC internal or external noise [1,4] appear as random noise spread across the frequency
spectrum of FFT, distort the conversion process, and result in harmonic distortions [10] and
higher noise floor. The relation between nonlinearity, harmonics, and noise floor are used to
address ADC’s ENOB and SNR. Effective resolution of ADC’s is one part in 2n (n is the ADC
number of bits). For an ideal 12 bits ADC, 4096 unique digital codes are produced and the
effective resolution is one part in 4096 codes as given by, [1,3],

1 
dB  20 Log  n 
2 

(1)

ENOB can be expressed in term of SNR as

SNR   ( ENOB)  6.02)  1.76  dB

(2)

By relating the noise ratio to ENOB and rearranging (2), ENOB can be expressed as

ENOB 

SNR[ db]  1.76
6.02

(3)

This method of estimating ENOB is extremely sensitive to errors caused by noise [1,2] and
requires a large number of samples. Therefore, the new proposed method of WT based testing
can be especially suited for ENOB estimations with fewer samples.

162


In the next section, a brief description of the Discrete Wavelet Transform (DWT) illustrates the
function and advantage of using DWT.

DISCRETE WAVELET TRANSFORM
The objective of signal transformation is to have a different representation of a signal with no
changes to the signal information. The Fourier transform, which is based on summation series of
sine and cosine waves, expresses signals in the frequency domain to determine frequencies in the
signal with no time representations. Meanwhile, in Wavelet transform, the objective is to achieve
a localized space frequency with the ability to determine the position of frequency components
[12-14]. Wavelet transforms have been used in various fields of signal processing [10,12,17] due
to the functionality of multi-resolutions that allow pinpointing of signal components. With
special properties of dilation and translation [4,10,13], wavelets can create different scaled and
shifted functions of signal transformation. In other words, unlike the Fourier transform, wavelet
transform capability of dilation and translation allow the shift of a signal in the time domain (X
axis), rescaling (to expand or compress a signal on Y axis) and produces flexible windows for
analysis as shown in figure 1.
Scale

Time

Figure 1: Wavelet Windowing Signal Analysis at Different Frequencies

Large scaling, provide most information of the signal (the big picture), meanwhile, small scales
wavelet shows signal details by zooming into the signal components.
To better understand wavelets, the continuous wavelet transform (CWT) will be looked at first.
In [16], wavelet (Ψ) is a function of zero average as, i.e.


  (t )dt  0

(4)



where (Ψ) is base function known as mother wavelet used to drive wavelet transformation
function through dilation (s) and translation (u) as

 u ,s (t ) 

1 t u 


s  s 

(5)

163


And since Wavelet merely performs a convolution operation with a given signal, Wavelet
transform of continuous signal x (t) can be illustrated as


Wx(u, s )   X (t )


1
 t u 
 *

s  s 

(6)

However, in this work of analyzing digital (discrete) signal, Discrete Wavelet Transform (DWT)
was used for faster computation based on predetermined low and high-pass filters for a particular
Wavelet function in most cases. That is, ADC output signal was analyzed first by filter banks of
high-pass and low-pass to down-sampled by factor of 2 (decimation) in each scale
[10,12,13,15,17] as shown in figure 2. Low-pass filter coefficient produces approximation signal
information, while the high-pass filter coefficient produces detail information.

S

h
h

d

g

g

a

Figure 2: Wavelet multilevel decomposition

By applying filtering and decimation factors of 2 at each decomposition level, frequency
characterizations are passed and number of samples rate is reduced by half (half the frequency
band). Starting with the largest scale (the original signal), bandwidth becomes a multiple of half
at the high and low-pass filters.

WAVELET-BASED ESTIMATION FOR ENOB
In this work, DWT was implemented to test high-speed ADC. Theoretically, by applying clean
perfect sine wave, stimulus signal parameters such as (amplitude, frequency, phase, dc offset, etc)
can be determined [11] and the deviation of the ADC output signal from the ideal input signal is
an effect of ADC performance. For an ideal ADC, the deviation is negligible (zero). However,

realistic ADC performance produce quantization error that distort the output signal X [n] as


X [ n]  X [ n]  q

(7)

where X [ n ] original value and q is quantization error.
From equation (7), the output signal is the summation of both input signal and quantization error.
Quantization is directly related to ADC’s number of bits through quantization step size (  ) that

164


determine the distance between adjacent codes. The step size expresses the size of bin code and
can be determined as



Voltage[2n  1]  Voltage[0] FSR
 n
LSB
2n  1
2 1

(8)

where n is ADC number of bits, FSR is ADC full scale voltage, and n is ADC number of bits.

By capturing ADC output data X [n] and Appling DWT algorithms, using multi-resolution
techniques, a combination of instantaneous low frequency (approximation coefficients an ) and
high frequency (detail coefficients d n ). Data from the high-pass filter, detail coefficients were
obtained as
(9)

(dn1,1dn1,0 , dn1,1 , dn1,2 , dn1,3 , dn1,4 , dn1,5 , dn1,6 , dn1,7 ,)

then down sampled by 2 (by taking the odd values as shown in 12) to end with half of the original
data, i.e.
(10)

(dn1,1, dn1,1 , dn1,3 , dn1,5 , dn1,7 ,)

In [2], the largest components of DWT high-pass coefficient at scale 1 defines the Dynamic
Range (DR) that is used to estimate worst case ENOB. DR can be defined as
 1   
 1  
DR   20 log 10 
     20 log 10   B  0.5    dB  .
ˆ

 2  2 
 2

(11)


where B is the Effective Number Of Bits ENOB, and  Quantization step size.
By rearranging equation (11), worst case instantaneous ENOB can be directly estimated as [2].

ˆ
B

DR
 0.5  bit 
20log10 (2)

(12)

Simulation and Measurements
In addition to the actual lab testing using DWT algorithms for worst case ENOB, MATLAB
simulation was implemented to verify the actual testing results. With no extraneous noise, ADCs
(range from 10 bits to 18 bits) were tested based on FFT and DWT algorithms. Results are shown
in Tables 1 and 2 for both ideal and non-ideal performance testing.
Table 1. ENOB estimation (conventional method Vs. Wavelet) based on no extraneous noise simulation.

DWT/FFT
Haar
db4
db10
Coif1
FFT

10 Bits
ADC
10.14
10.17
10.11
9.78
9.97

12 Bits
ADC
12.16
11.92
12.01
12.07
11.84

14 Bits
ADC
14.27
13.97
14.06
14.03
14.09

16 Bits
ADC
16.1
16.02
16.07
15.72
15.89

18 Bits
ADC
18.34
17.8
18.06
18.2
17.97
165


Figure 3. ENOB estimation with no extraneous noise (Table 1).
Table 2. ENOB testing results (conventional method Vs. Wavelet) based on extraneous noise simulation.

DWT/FFT
Haar
db4
db10
coif1
FFT

10 Bits
ADC
9.4
9.3
8.7
9.14
8.5

12 Bits
ADC
11.5
11.1
10.72
11.21
10.38

14 Bits
ADC
13.3
13.5
12.68
12.96
11.35

16
Bits
ADC
15.1
15.7
14.84
15.16
12.55

18
Bits
ADC
16.4
17.81
16.6
17.45
13.8

Figure 4. ENOB estimation with extraneous noise (Table 2).

Actual Testing Setup and Measurements
Sets of ADCs testing boards (10-18 Bits) were used to verify the new testing algorithms. Actual
ADCs output data were transformed and analyzed by FFT and DWT (such as Haar, db4, db10,
166


and Coief1) algorithms. A summary of ENOB testing results are given in Tables 3 and 4 for
sampling frequency 50 MHz and 100 MHz.
Table (3): ENOB estimation (conventional method Vs. Wavelet) at 50 MHz sampling frequency.

DWT/FFT
Haar
db4
db10
coif1
FFT

10 Bits
ADC
9.3
10.56
10.29
9.87
9.31

12 Bits
ADC
12.44
11.56
11.91
11.89
10.57

14 Bits
ADC
14.2
14.24
13.57
13.48
11.51

16
Bits
ADC
15.9
16.97
15.19
15.22
12.36

18
Bits
ADC
17.6
19.53
16.91
17.10
13.62

Figure (5): ENOB estimation at 50 MHz sampling frequency (Table 3).
Table (4): ENOB estimation (conventional method Vs. Wavelet) at 100 MHz sampling frequency.

DWT/FFT
Haar
db4
db10
coif1
FFT

10 Bits
ADC
10.57
10.6
10.1
9.93
9.27

12 Bits
ADC
12.14
11.95
11.93
11.88
10.31

14 Bits
ADC
14.25
14.1
13.6
13.8
11.4

16
Bits
ADC
16.28
16.34
15.81
15.68
12.26

18
Bits
ADC
18.11
18.31
16.89
17.71
13.51

167


Figure (6): ENOB estimation at 100 MHz sampling frequency (Table 4).

As illustrated in Figures 4-6, FFT tend to overestimate ENOB especially as ADCs number of bits
increases. FFT test depend on noise summation (average) to all noises including quantization
noise and very small noises [2,17]. This fact has higher effect on higher bits ADCs since
quantization levels get smaller and any detection of noise offset ENOB estimation. Meanwhile,
DWT tend to localize into ADCs output data as a result of multi-resolution property. This
property allows obtaining the dynamic range of the output signal without noise summation.

CONCLUSION
By implementing Wavelet transform to analyze ENOB, testing results of several types of
Wavelets were used and compared with each other and conventional methods such as FFT for
better illustration. As a result, it was clearly observed that classical testing of FFT sums all noises
(large and small) over many sample point. Such a fact influence ENOB testing results as ADCs
number of bits increase, quantization step sizes decrease, and quantization noise averaged.
Meanwhile, DWT have provided well localized measurements of signal components, and
estimate ENOB based on the localized signal dynamic range. DWT were successful in measuring
ADCs performance without averaging noises and therefor; testing cost, duration, and complexity
can be reduced based on fewer computed samples. In addition, higher accuracy of ENOB
estimation, especially for higher bits ADCs was noticed, which can lead for better DSP testing
algorithms and built in self-test ADC parameters.

ACKNOWLEDGEMENTS
The authors are grateful to Applied Science University (ASU), Amman - Jordan, for the financial
support grated to cover the publication fee of this paper research articles.

168


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Author
Emad Awada is an Assistance Professor at Applied Science University in Amman
Jordan. His research interests are in the areas of Mixed Signals Systems, Signal and
Image Processing, Broadband / Communication Systems, and Power Systems. He
received B.S. degree in electrical engineering from Prairie View University, Prairie
View, TX. in 1998. He received M.S. degree and Ph.D in electrical engineering from
Prairie View University in 2006 and 2011 respectively.

.

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The application wavelet transform algorithm in testing adc effective number of bits

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