ETC AB-072

APPLICATION BULLETIN
®
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DYNAMIC TESTS FOR A/D CONVERTER PERFORMANCE
BEAT FREQUENCY TESTING
The beat frequency and envelope tests are qualitative tests
that provide a quick, simple visual demonstration of ADC
dynamic failures. An input frequency is selected that provides worst-case range changes and maximal input slew
rates that the ADC is expected to see in use. The output is
then viewed on a display in real time.
This article describes useful theory and techniques for evaluating the dynamic performance of A/D converters. Four
techniques are discussed: (1) beat frequency, (2) histogram
analysis, (3) sine wave curve fitting, and (4) discrete finite
Fourier transform.
The key to confidence in the quality of a waveform recorder
is assurance that the analog-to-digital converter (ADC) encodes the signal without degrading it. Dynamic tests that
cover the frequency range over which the converter is
expected to operate can provide that assurance. The results
of the dynamic tests give the user a model of resolution
versus frequency for the recorder. More elaborate models of
failure mechanisms can be obtained by varying the conditions of the tests.
All of the dynamic tests used for the 5180A Waveform
Recorder use sine waves as stimulus. Sine waves were
chosen primarily because they are the easiest to generate in
practice at the frequencies of interest with adequate fidelity.
While it may be possible to generate a square wave, for
example, whose function is known to the 10-bit resolution of
the 5180A, no square wave generators exist that can guarantee the same waveshape to 10-bit resolution at 10MHz from
unit to unit. Another motivation for choosing a sine wave
stimulus is the simple mathematical model a sine function
provides for analysis. This benefit greatly simplifies the
algorithms used for data analysis.
Four dynamic tests for waveform recorder characterizations
are presented here: beat frequency testing(1) histogram analysis (2) sine wave curve fitting,(3,4) and discrete finite Fourier
transform.(5) The last three tests operate in the same way. A
sine wave source is supplied to the waveform recorder and
one or more records of data are taken. A computer is then
used to analyze the data. The tests differ primarily in the
analysis algorithms and consequently in the sort of errors
brought to light. Critical to the success of these tests is the
purity of the sine wave source. Synthesized sources are
necessary to provide the short-term and long-term stability
required by the dynamic range of the ADC. Passive filters (a
six-pole elliptical filter is used for 5180A tests) are required
to eliminate harmonic distortion from the source.
These tests provide the most stressful conditions for the
ADC with the input signal amplitude at full scale. Generally
speaking, nonlinear effects increase more quickly than the
signal level increases because of the nonideal large-signal
DC behavior of the ADC components and the higher slew
rates large amplitudes imply.
©
))))Burr-Brown Corporation
Waveform Recorder
Under Test
Memory
DAC
HP 3320A
Synthesizer
ADC
CRT
(Playback)
Time Base
fs
fs + ∆f
Input
FIGURE 1. Beat Frequency Test Setup.
∆f
fS + ∆f
fS
FIGURE 2. When the Input Frequency is Close to the Sample
Rate fS, the Encoded Result is Aliased to the
Difference or Beat Frequency, ∆f.
The name “beat frequency” describes the reasoning behind
the test. The input sinusoid is chosen to be a multiple of the
sample frequency plus a small incremental frequency (Figure 1). Successive samples of the waveform step slowly
through the sine wave as a function of the small difference
or beat frequency (Figure 2). Ideally, the multiplicative
properties of sampling would yield a sine wave of the beat
frequency displayed on the waveform recorder’s CRT. Errors can be seen as deviations from a smooth sine function.
Missing codes, for example, appear as local discontinuities
in the sine wave. The oversize codes that accompany missing codes are seen as widening in the individual codes
appearing on the sine wave. By choosing an arbitrarily low
beat frequency, a slow accurate DAC may be used for
viewing the test output. For best results, the upper limit on
the beat frequency choice is set by the speed with which the
beat frequency walks through the codes. It is desirable to
have one or more successive samples at each code. This
AB-072
1
Printed in U.S.A. Month, Year
alleviates the settling constraint on the DAC and ensures that
the display covers all possible code failures. For a 20MHz
sample rate and a 10-bit ADC, this implies a 3kHz maximum beat frequency for a minimum of one sample per code
bin.
Although the usual input frequency for a beat frequency test
is near the sample rate, the analog bandwidth of the ADC
may be measured by setting the carrier to a number of
different multiples of the sample rate. The band limit is
observed as a rolloff in amplitude as the carrier frequency is
increased.
The envelope test differs from the beat frequency test in the
choice of input frequency that the ADC encodes. Instead of
a multiple of the sample frequency, an input frequency near
one-half the sample rate is used. Now the ideal output is two
out-of-phase sine waves at the beat frequency (Figure 3).
This means that successive samples can be at the extreme
ends of the ADC range, which is useful for examining slew
problems that might not appear when successive samples are
at adjacent codes. To avoid placing the same stress on the
DAC used for display, a bank of D flip-flops removes every
other sample before the data arrives at the DAC. Thus only
one phase of the beat frequency remains.
FIGURE 5. A Beat Frequency Display for a Commercially
Available 10MHz, 8-Bit ADC with a
10.0031MHz Input.
HISTOGRAM TESTING
A sine-wave-based histogram test provides both a localized
error description and some global descriptions of the ADC.
Using the histogram test, it is possible to obtain the differential nonlinearity of the ADC, to see whether any missing
codes exist at the test frequency, and to get a measure of gain
and offset at the test frequency. Of the sine-wave-based tests
presented here, the histogram test yields the best information
about individual code bin size at an arbitrary frequency.
∆f
fS/2 + ∆f
fS
A statistically significant number of samples of the input
sinusoid are taken and stored as a record (Figure 6). The
frequency of code occurrence in the record is then plotted as
a function of code. For an ideal ADC, the shape of the plot
would be the probability density function (PDF) of a sine
wave (Figure 7) provided that the input and sample frequencies are relatively independent. The PDF of a sine wave is
given by:
∆f
FIGURE 3. When the Input Frequency is Near One-half the
Sample Rate, the Envelope of the Difference
Frequency Results.
Figure 4 shows 5180A beat frequency test results for a
10.0031MHz input sine wave sampled at l0MHz. For comparison, Figure 5 shows a 10.0031MHz sine wave being
sampled at l0MHz by a commercially available 8-bit, 20MHz
ADC.
p( V ) =
1
π A – V2
2
Full-Scale Sine Wave
HP 3325A
Synthesizer
5180A
Waveform Recorder
98034A
HP-IB
Interface
HP-IB
Plotter
HP 9825
Controller
FIGURE 4. A Beat Frequency Display Produced by the
5180A Waveform Recorder with a 10.0031MHz
Input Frequency and a 10MHz Sample Rate.
The smooth sine wave indicates freedom from
dynamic errors.
FIGURE 6. Setup for Histogram Test.
2
Differential Nonlinearity =
4
πA
Where actual P(nth code) is the measured probability of
occurrence for code bin n, and ideal P(nth code) is the ideal
probability of occurrence for code bin n. The code bin
number n goes from 1 to 2N, where N is the number of ADC
bits. Using the probability of occurrence eliminates dependence on the number of samples taken. To calculate the
probability for each code in the actual data record, the
number of occurrences for each code is divided by the
number of samples in the record. The ideal probability of
occurrence is what an ideal ADC would generate with a sine
wave input. For each code bin, this is the integral of the
probability density function of a sine wave over the bin:
3
πA
2
πA
1
πA
V = –A
0
actual P ( nth code )
–1
ideal P ( nth code )
V=A
FIGURE 7. Sine Wave Probability Density Function.
Where A is the sine wave amplitude and V is the independent variable (voltage). For a real ADC, fewer than the
expected number of occurrences for a given code bin indicates that the effective code bin width is smaller than ideal
at the input frequency.(1) No occurrences indicate that the
code bin width is zero for that input. A greater-than-expected number of occurrences implies a larger-than-ideal
code bin width.
P( n ) =
(
N −1


1  –1 B n – 2
sin 

π
A2 N


)  – sin
–1


(
 B n – 1 – 2 N −1


A2 N

)  

 
Where n is the code bin number, B is the full-scale range of
the ADC, and N is the number of ADC bits. To avoid large
differences in code probability caused by the sinusoid cusp,
a sine wave amplitude A is chosen that slightly overdrives
the ADC.
A judicious choice of frequency for the input sinusoid in this
test is necessary for realistic test results. An input frequency
that is a submultiple of the sample frequency violates the
relative independence criterion and will result in sampling of
the same few codes each input cycle. Using an input frequency that has a large common divisor with the sample
frequency generates similar problems since the codes repeat
after each cycle of the divisor frequency. Ideally, the period
of the greatest common divisor should be as long as the
record length.
A 5180A histogram is shown in Figure 8 for an input sine
wave at 9.85MHz. For comparison, Figure 9 shows data
from a commercially available, 8-bit 20MHz ADC for an
input sine wave at 9.85MHz, while Figure 10 shows data
from an 8-bit, l00MHz ADC taken at 9.85MHz.
What is a statistically significant number of samples? We
can determine significance from probability theory. For a
given input PDF and record size, each bin of an ideal ADC
has an expected number of occurrences and a standard
deviation around that expectation. The confidence that the
number of occurrences is close to the expectation is equal to
the probability that the occurrences fall within the appropriate number of standard deviations. The ratio of the standard
deviation to the expectation (and thus the error for a given
confidence) decreases with more samples. To get the confidence for the entire range, the probabilities for all codes
lying within the desired error are multiplied together.
For an ideal 10-bit ADC, 100,000 samples would give us a
12% confidence that the peak deviation from the input PDF
is less than 0.3LSB and a 99.9% confidence that the peak
deviation is less than 0.5LSB. The notion of confidence
relies on the input’s being a random process. We can model
the sine wave input as random process only if the input and
sample frequencies are relatively independent.
Number of Occurances
The specification of greatest interest that can be calculated
using the histogram test is differential nonlinearity. Differential nonlinearity is a measure of how each code bin varies
in size with respect to the ideal:
NOTE: (1) Histogram testing can be thought of as a process of sampling and
digitizing the input signal and sorting the digitized samples into bins. Each bin
represents a single output code and collects samples whose values fall in a
specific range. The number of occurrences or samples collected in each bin
varies according to the input signal. If N is the number of ADC bits, there are
2N bins. Ideally, if B is the full-scale range of the ADC in volts, each bin
corresponds to a range of sample sizes covering B/2N volts. In a real ADC, the
bins may not all have the same width.
0
Output Code
1023
FIGURE 8. A 100,000-sample Histogram for a 5180A with
a 9.85MHz Sine Wave Input. All Discontinuities
are Less Than 1LSB.
3
HP 9825A
Controller
Number of Occurances
Extraordinarily
Large Differential
Nonlinearity
HP 3325A
Synthesizer
HP-IB
Numerous
Missing Codes
5180A
Waveform Recorder
Input
0
Output Code
Low-Pass Filter
256
f
FIGURE 9. A 100,000-sample Histogram Plot for a Commercially Available 20MHz, 8-bit ADC with a
9.85MHz Input. Large differential nonlinearities
and numerous missed codes are apparent.
FIGURE 11. Setup for the Curve-fit Test and the Discrete
Finite Fourier Transform (DFT) Test.
Effective Bits = 8.51
Cal. Amplitude 0.81
Cal. Frequency 9850320.9
Cal. Phase –94.95
Cal. Offset 509.31
Number of Occurances
226
Missing Codes
Data Record
Missed
Codes
Error Residue
0
Output Code
256
FIGURE 12. The First 20 Points of the Curve-fit Data Record
and the Error Residue from a Fitted Sine Wave.
FIGURE 10. A 100,000-sample Histogram Plot for a
100MHz, 8-bit ADC with a 9.85MHz Input
Sampled at 20MHz. Extremely large differential nonlinearities and numerous missed codes
are apparent.
The number of effective bits is computed using expressions
for average errors as follows:
 actual rms error 
Effective bits = N – log 2 

 ideal rms error 
CURVE FITTING
The curve-fit test is a global description of the ADC. This
means that the errors measured by the test are averaged to
give a general measurement of the ADC transfer function.
The result of this test is a figure of merit called the number
of effective bits for the ADC. The effective bit number is a
general measure of how much an ADC’s nonlinearity has
impaired its usefulness at a given frequency.
The number of effective bits is obtained by analyzing a
record of data taken from a sine wave source (Figure 11).
The analysis consists of generating a sine wave in software
that is a best fit to the data record. Any difference between
the data record and the best-fit sine wave is assumed to be
error (Figure 12). The standard deviation of the error thus
calculated is compared to the error an ideal ADC of the same
number of bits might generate. If the error exceeds the ideal,
the number of effective bits exhibited by the ADC is less
than the number of bits it digitizes. Errors that cause degradation in this test are nonlinear effects such as harmonic
distortion, noise, and aperture uncertainty. Gain, offset, and
phase errors do not affect the results since they are ignored
by the curve-fit process.
where N is the number of ADC bits. The ideal rms error is
not actually computed for the input waveform, but is assumed to be the quantization noise exhibited by an ideal
ADC with a uniform-probability-density (UPD) input such
as a perfect triangle wave. The ideal error is found from the
expectation of squared error for a rectangular distribution. A
rectangular distribution is used since that represents a UPD
taken over an ideal code bin. The result thus obtained is:
Q
Ideal rms error =
12
where Q is the ideal code bin width. Although the input sine
wave is not a UPD function, the UPD assumption is still
valid since it is locally applied over each code bin. The
deviation from a UPD over each code bin is very small, so
the errors in using sine waves to approximate UPD inputs
are negligible.
The actual rms error is simply the square root of the sum of
the squared errors of the data points from the fitted sine
wave. The actual rms error is given by:
4
[
k =1
E = ∑ x k – A cos(ωt k + P ) – C
m
]
2
(1)
where E is the actual rms error, Xk and tk are the data points,
m is the number of data points in the record, and the fitted
sine wave parameters are amplitude A, frequency ω, phase
P, and offset C.
Equation 1 is also used to find the best-fit sine wave by
minimizing the error E. The error is minimized by adjusting
the fit parameters: frequency, phase, gain, and offset. This is
done by taking the partial derivative of E in Equation 1 with
respect to each of the four parameters. The error minimum
occurs when all of the derivatives are equal to zero. This
gives the four simultaneous equations:
∑ x k cos(ωt k + P ) = A ∑ cos 2 (ωt k + P ) + C ∑ cos(ωt k + P )
m
m
m
k =1
k =1
k =1
(2)
∑ x k = A ∑ cos(ωt k + P ) + nC
m
m
k =1
k =1
(3)
∑ x k t k sin (ωt k + P ) =
m
k =1
A ∑ t k cos(ωt k + P ) sin (ωt k + P ) + C ∑ t k sin (ωt k + P )
m
m
k =1
k =1
(4)
∑ x k sin (ωt k + P ) =
m
k =1
A ∑ cos(ωt k + P ) sin (ωt k + P ) + C ∑ sin (ωt k + P )
m
m
k =1
k =1
(5)
Equations 2 and 3 result from gain and offset adjustments.
These are substituted into the other two equations, 4 and 5,
giving two nonlinear equations:
∑ ( x k – x )t k sin (ωt k + P )
m
k =1
m
∑ ( x k – x ) cos(ωt k + P )
m
=
k =1
[
]
∑ cos(ωt k + P ) – a t k sin (ωt k + P )
k =1
m
[
]
∑ cos(ωt k + P ) – a cos(ωt k + P )
k =1
(6)
[
]
∑ ( x k – x ) cos(ωt k + P ) ∑ [ cos(ωt k + P ) – a ] cos(ωt k + P )
k =1
k =1
∑ ( x k – x )sin (ωt k + P )
m
k =1
m
∑ cos(ωt k + P ) – a sin (ωt k + P )
m
=
k =1
m
(7)
Where a = ∑ cos(ωt k + P )
m
k =1
These are solved iteratively to give values for the parameters. The difference between the right and left sides of
Equation 6 is defined as error parameter R and the difference
between the right and left sides of Equation 7 is defined as
error parameter S. An approximation algorithm using a firstorder Taylor series expansion drives R and S to zero. This
approximation algorithm requires an initial guess for frequency and phase close to the solution to ensure conver-
gence to the best-fit sine wave. For frequency, the frequency
of the generator output in Figure 11 is used as a guess. For
phase, a guess is based on an examination of the data record
by a software routine.
Although the result of this process is a single figure of merit,
some enlightenment can be gained about the error components in the ADC by varying the test conditions. White noise
produces the same degradation regardless of input frequency
or amplitude. That is, the error term in Equation 1 is
independent of test conditions for this sort of error. Another
way of identifying noise in this test is by the randomness in
the error residue, or the difference between the best-fit sine
wave and the data taken.
Aperture uncertainty is identifiable because it generates an
error that is a function of input slew rate. When this is the
dominant error causing a low number of effective bits, the
number of effective bits will vary linearly with both input
frequency and amplitude. If the input waveform is sampled
only at points of constant slew rate, such as zero crossings,
then the aperture uncertainty corresponds to the amount that
the effective bits decline as a function of slew rate.
Harmonic distortion is usually a nonlinear function of amplitude and frequency. Its distinguishing characteristic is the
presence of the harmonics (or aliased harmonics if the
fundamental is close to the Nyquist frequency) in the error
residue. The amplitudes of the harmonics can be extracted
by fitting the error residue with best-fit sine waves of the
important harmonic frequencies. The impact of noise and
aperture uncertainty in the presence of large distortion errors
can be assessed by effective bit values and error residues
with the fitted harmonics removed.
The greatest pitfall in the curve-fit test is using an input
frequency that is a submultiple of the sample frequency.
Since the same codes are sampled at exactly the same
voltage each cycle, the locally uniform probability distribution assumption is violated. In the worst case, a submultiple
of one-half, the quantization error would not be measurable
at all. From a practical standpoint, this also defeats the
global description of the test by sampling only a handful of
codes.
Another potential pitfall is lack of convergence of the curvefit algorithm. There are a few occasions where this can
become a problem, such as when the data is very poor or the
computational resolution is inadequate.
Figure 12 shows the error plot for a 5180A curve-fit test
taken at a 9.85MHz input frequency. The number of effective bits associated with this error is 8.51.
FFT TESTING
The fast Fourier transform (FFT) is used to characterize an
ADC in the frequency domain in much the same way that a
spectrum analyzer is used to determine the linearity of an
analog circuit. The data output for both techniques is a
presentation of the magnitude of the Fourier spectrum for
the circuit under test. Ideally, the spectrum is a single line
5
that represents the pure sine wave input and is devoid of
distortion components generated by the circuit under test.
There are, however, significant differences between the
spectrum analyzer and ADC spectra because of the sampling
operation of the ADC.
The Fourier transform of a signal x(t) that is continuous for
all time is defined as:
form of the rectangular function. W(f) is the familiar sinx/x
function (see Figure 13 for |W(f)|), consisting of a main lobe
surrounded by a series of sidelobes whose amplitudes decay
at a 6dB-per-octave rate. It is these sidelobes that are
responsible for leakage. Even if the spectrum of X(f) is a
single line, the sidelobes of W(f) during the convolution
smear the energy in the single line into a series of spectral
lines spaced 1/M∆t apart whenever the frequency of x(t) is
not an integral multiple of 1/ M∆t.
Leakage can be reduced by multiplying the data in the record
by a windowing function that weights the points in the center
of the record heavily while smoothly suppressing the points
near the ends. Many different windowing functions exist
that offer various tradeoffs of amplitude resolution versus
frequency resolution. A function commonly used with sine
waves is the Hanning window, defined by | (1/2)(1 – cos 2πt/
M∆) |. Notice in Figure 13 that both the window and its
derivative approach zero at the two ends of the record and
that the transform’s main lobe is twice as wide as that of the
rectangular function, while the amplitudes of the sidelobes
decay by an additional 12dB per octave. The reduced level
of the sidelobes reduces leakage, but the wider main lobe
limits the ability to resolve closely spaced frequencies.
Furthermore, the shape of the main lobe can attenuate the
spectral amplitudes of X(f) by as much as 1.5dB. However,
for the DFT testing to be described here, the Hanning
window was selected as a good compromise between frequency and amplitude resolution.
The third difference between the spectra of X(f) and XD(f)
is the limited range of frequencies displayed for XD(f). The
sampling process causes the two-sided spectrum of X(f),
symmetrical about the origin, to be replicated as the sampling frequency L and at all of its harmonics. If X(f) contains
components that exceed fS/2, then these components are
folded back, or aliased, onto spectral lines below fS, causing
aliasing errors. The frequency fS/2 is sometimes called the
Nyquist frequency, referring to the Nyquist criterion, which
requires the sampling rate to be twice the highest frequency
present in the input signal to define the waveform uniquely.
∞ x t e –i2 πft ∆t
X( f ) = ∫–∞
()
and includes the amplitude and phase of every frequency in
x(t). The Fourier transform cannot be used in this form for
an ADC, however, because x(t) is only digitized at a finite
number of points, M, spaced ∆t apart. Instead, the discrete
finite transform (DFT) must be used. It is defined as:
M –1
XD( f ) = ∑ x( m∆t )e – i2 πf ( m∆t ) ∆t
m=0
There are significant differences between X(f) and XD(f).
While X(f) has infinite spectral resolution, XD(f) has a
discrete frequency resolution of ∆f = 1/m∆t because of the
finite number of points in the data record. The finite record
size also accounts for another difference between X(f) and
XD(f) whenever a nonintegral number of cycles of X(t) is
contained in the record. Since the DFT assumes that the
record repeats with a period of M∆t (to satisfy the Fourier
transform condition that x(t) be continuous for all time)
sharp discontinuities at the points where the start of one
record joins the end of the preceding record cause the
spectral components of X(f) to be spread or smeared in
XD(f).
The smearing, called leakage, can be explained as follows.
The finite record size of x(t) can be considered the consequence of multiplying x(t) by a rectangular function having
unity amplitude during the time period M∆t that the record
is acquired and zero amplitude elsewhere. Since multiplication of two functions in one domain (time, in this case) is
equivalent to convolution in the other, the spectrum of XD(f)
is derived by convolving X(f) with W(f), the Fourier trans-
w(t)
Fourier Transform
of Time Window
Rectagular
Window
|sine(πfT)|
|w(f)|
6d
1
1
B/O
cta
ve
t
–f
+f
M∆t
1/M ∆t
w(t)
|w(f)|
Hanning
Window
1
1
t
18d
B/O
cta
ve
–f
+f
1/M∆t
M∆t
2/M∆t
FIGURE 13. Time-domain and Frequency-domain Representations of Rectangular and Hanning Windows.
6
The result is that the spectrum of XD(f) is displayed only
from DC to fS/2 and the maximum input frequency must be
limited to less than fS/2 to avoid aliasing.
Figure 14 presents the magnitude of the spectra derived from
the DFT for perfect 10-bit and 6-bit ADC's given a pure
sinusoidal input. Useful information about the ADC's performance can be derived from three features of the spectra:
the noise floor, the harmonic level, and the spurious level.
0dB
transfer function. Furthermore, localized code errors can
increase in amplitude and in the number of codes affected
under dynamic input conditions. Aperture jitter is another
major source of dynamic error; the magnitude of this localized code error is dependent upon the slew rate of the input
at the time of sampling. Each of these localized code errors
can be modeled as a sharp discontinuity in the time domain
that when transformed into the frequency domain results in
a broad spectrum that raises the height of the noise floor
above that caused by quantization noise alone.
The second feature of the DFT-derived spectrum that indicates an ADC’s level of dynamic performance is the harmonic content. Static and dynamic integral nonlinearities
cause curvature in the ADC’s transfer function. If the input
frequency fIN is much lower than the Nyquist frequency (fS/
2), then the harmonic components will be in the expected
locations: 2fIN, 3fIN, etc. If, on the other hand, the harmonics
of fIN are greater than fS/2, then these frequencies will be
aliased onto components below fS/2. Take, for instance, a
20-megasample-per-second (fS) ADC with an input of
9.85MHz. The second harmonic at 19.7MHz is aliased to
0.3MHz, the third harmonic at 29.55MHz is aliased to
9.55MHz, the fourth at 39.4MHz is aliased to 0.6MHz, and
so on.
Care must be exercised in selecting the input frequency for
the DFT test. An incorrectly chosen frequency can alias one
of its harmonic components on to the fundamental and
thereby understate the harmonic distortion (in the example
above, an input of exactly 5MHz would place the third
harmonic at the fundamental frequency). The input frequency should be chosen so that the harmonics are far
enough away to be easily resolvable from the fundamental,
whose energy has been spread into several adjacent bins (1/
M∆t locations) by the Hanning window. This accounts for
the 0.15MHz offset from 10MHz used in the example of
Figure 14.
The third feature of the DFT-based spectrum that is indicative of the ADC’s level of dynamic performance is the
spurious content. Spurious components are spectral components that are not harmonically related to the input. For
example, a strong signal near the ADC may contaminate the
ADC’s analog ground somehow and thereby appear in the
spectrum. The nearby signal will not only appear as itself,
but because of nonlinearities within the ADC, can combine
with the input signal to form sum and difference terms
resulting in intermodulation distortion.
The combined effects of noise floor, harmonic distortion and
spurious errors are reflected in the ADC’s rms signal-tonoise ratio, which can be derived from the DFT magnitude
spectrum. The signal energy is determined by summing the
energy in all the bins associated with the fundamental. The
noise energy is the sum of the energy in all other bins. By
taking the logarithm of the ratio of signal energy to noise
energy and multiplying by 20, the signal-to-noise ratio for
the ADC can be calculated. An ideal N-bit ADC having
quantization noise only is theoretically known to have a
signal-to-noise ratio equal to (6N + 1.8)dB, which sets an
Perfect 10-Bit ADC
S/N Ratio = 61.9
Theoretical S/N = 61.8
–60dB
(A)
0
1
2
3
4
5
6
7
8
9
10
9
10
Frequency (MHz)
0dB
Perfect 6-Bit ADC
S/N Ratio = 37.9
Theoretical S/N = 37.8
–60dB
(B)
0
1
2
3
4
5
6
7
Frequency (MHz)
8
FIGURE 14. FFT Plots for 0.85MHz Data Quantized by
Perfect 10-bit (A) and 6-bit (B) ADCs. The
signal-to-noise ratio computed in each case
agrees closely with the theoretical value of 6N
+ 1.8dB where N is the number of ADS bits.
Two classes of noise sources determine the level of the noise
floor. The first is called quantization noise. This is the error,
bounded by ±1/2LSB, that is inherent in the quantization of
the input amplitude into discrete levels. As can be seen in
Figure 14, even ideal ADCs have noise floors determined by
quantization noise. The higher the number of bits, the
smaller the error bound and, therefore, the lower the noise
floor.
All real-life ADCs have noise floors that are higher than that
solely from quantization noise. The second class of noise
source includes wideband noise generated within the ADC,
along with other sources. In a parallel-ripple ADC, for
example, such things as misadjustment between the firstpass and second-pass ranges (exceeding the redundancy
range) or inadequate DAC settling can cause localized code
errors or differential nonlinearities in the ADC’s status
7
severe harmonic distortion errors resulting from integral
nonlinearity in the transfer functions of both of these ADCs.
A rule of thumb has evolved that uses the DFT-based
spectrum as a quick overview of an N-bit ADC’s dynamic
performance. If all harmonic and spurious components are at
least 6N dB below the full-scale amplitude of the fundamental, then the ADC is performing satisfactorily, since each
error component has a peak-to-peak amplitude smaller than
an LSB. If, on the other hand, harmonic or spurious components are less than 6N dB down, or if the noise floor is
elevated, then other tests can be performed that are better at
isolating the particular integral and differential nonlinearity
errors. In particular, the FFT test an be followed by the
histogram test or the beat frequency test (or envelope test),
as conditions warrant.
upper bound. A signal-to-noise ratio below this ideal limit is
indicative of errors of all types that the ADC produces.
The FFT test setup is presented in Figure 11. A full-scale
sine wave of a properly chosen frequency is applied to the
ADC under test. The low-pass filter ensures a spectrally pure
input. A 1024-point record sampled at the maximum sampling rate is then taken and given to the computer, which
calculates the DFT using an FFT algorithm. The spectral
magnitude is plotted as a function of frequency.
Figure 15 shows the graphical outputs for the 5180 for fullscale sine wave input at 0.95MHz and 9.85MHz. As might
be expected, the distortion increases with increasing frequency. Harmonic and spurious components are typically
better than –60dBc below 1MHz and –54dBc at 9.85MHz.
The latter spectrum at 9.85MHz is the frequency-domain
representation for one of the most demanding tests of an
ADC, called the envelope test which was described earlier.
0dB
Spurious Level (dBc)
16dB
(A)
Frequency = 9.85MHz Channel A (1V Range)
Peak Carrier: 4.5dBm
Peak Noise: –59.6dBc
Limit: –46dBc
Fundamental
10
0
–10
–20
–30
–40
–50
–60
–70
–80
–90
–60dB
Test Limit
2nd Harmonic
(A)
0
3rd Harmonic
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
8
9
Spurious Level (dBc)
9
10
9
10
0dB
10
Frequency (MHz)
10
0
–10
–20
–30
–40
–50
–60
–70
–80
–90
18dB
Frequency = 0.95MHz Channel A
Peak Carrier: 5.0dBm
Peak Noise: –59.3dBc
Limit: –59dBc
Fundamental
–60dB
Test Limit
0
(B)
0
(B)
8
Frequency (MHz)
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
Frequency (MHz)
8
FIGURE 16. DFT Plots for a 20MHz, 8-Bit ADC (a) and a
100MHz, 8-Bit ADC (b). Full-scale input sine
waves at 9.85MHz were sampled at a rate of
20MHz. The high levels of harmonic distortion
indicate severe integral nonlinearities.
10
Frequency (MHz)
FIGURE 15. DFT Plots for the 5180A with Input Frequencies of 9.85MHz (a) and 0.95MHz (b). The low
harmonic distortion indicates very low integral
nonlinearity.
Figure 16 presents, for comparison, the test results for
commercially available digitizers: a 20-megasample-persecond, 8-bit ADC and a 100-megasample-per-second, 10bit ADC with a full-scale, 9.85MHz sine wave input, sampled
at 20 megasamples-per-second. The numerous large harmonic components, both odd and even, are indicative of
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CONCLUSION
The four sine-wave-based ADC tests described provide
information about the quality of any recorder. The tests may
be used to isolate specific failures, even at high-speed and
ERROR
HISTORGRAM
fine resolution (Figure 17). The tests are simple to run,
requiring only a synthesized generator and an HP-1B computer.
DFT
SINE WAVE CURVE-FIT
BEST FREQUENCY TEST
Differential
Nonlinearity
Yes—shows up as spikes
Yes—shows up as elevated noise floor
Yes—part of rms error
Yes
Missing Codes
Yes—shows up as bins with
0 counts
Yes—shows up as elevated noise floor
Yes—part of rms error
Yes
Integral
Nonlinearity
Yes—(could be measured directly
with a highly linear ramp waveform)
Yes—shows up as harmonics on fundamental aliased into baseband
Yes—part of rms error
Yes
Aperture
Uncertainity
No—averaged out. Can be measured
with “locked” historgram
Yes—shows up as elevated noise floor
Yes—part of rms error
No
Noise
No—averaged out. Can be measured
with “locked” histogram
Yes—shows up as elevated noise floor
Yes—part of rms error
No
Bandwidth
Errors
No
No
No
Yes—used to measure
analog bandwidth
Gain Errors
Yes—shows up in peak-to-peak
spread of distribution
No
No
No
Other Errors
Yes—shows up in offset of
distribution average
No
No
No
FIGURE 17. Summary of the Errors Exposed by the Dynamic Tests.
REFERENCES
1. D.J. Packard, "Beat Frequency Testing of Real Time A/D Converters," Workshop
on High Speed A/D Conversion, Portland, Oregon, February, 1980.
2. H.U. Koller, “New Criterion for Testing Analog-to-Digital Converters for Statistical
Evaluation,” IEEE Transactions on Instrumentation and Measurement, Vol. 1M22, pp. 214-17, September, 1973.
3. R. Potter, “Least-Squares Estimation of Sinusoidal Parameters from Measured
Data,” Hewlett-Packard internal memo, June, 1974.
4. L. Ochs, “Measurement and Enhancement of Waveform Digitizer Performance,”
IEEE International Convention, Boston, May, 1976.
5. R.N. Bracewell, The Fourier Transform and Its Applications, McGraw-Hill, New
York, 1965.
The information provided herein is believed to be reliable; however, BURR-BROWN assumes no responsibility for inaccuracies or omissions. BURR-BROWN assumes
no responsibility for the use of this information, and all use of such information shall be entirely at the user’s own risk. Prices and specifications are subject to change
without notice. No patent rights or licenses to any of the circuits described herein are implied or granted to any third party. BURR-BROWN does not authorize or warrant
any BURR-BROWN product for use in life support devices and/or systems.
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