NISTIR 8073

Simulated Sinewave Testing of 

Data Acquisition Systems using 

Sine Fitting and Discrete Fourier 

Transform Methods 

Part 1: Frequency Offset, Random, 

Quantization, and Jitter Noise 

Jon Geist 

M. Yaqub Afridi 

This publication is available free of charge from: 

http://dx.doi.org/10.6028/NIST.IR.8073
 

NISTIR 8073

Simulated Sinewave Testing of 

Data Acquisition Systems using 

Sine Fitting and Discrete Fourier 

Transform Methods 

Part 1: Frequency Offset, Random, 

Quantization, and Jitter Noise 

Jon Geist 

M. Yaqub Afridi 

Semiconductor and Dimensional Metrology Division 

Physical Measurement Laboratory 

This publication is available free of charge from: 

http://dx.doi.org/10.6028/NIST.IR.8073

July 2015

U.S. Department of Commerce 

Penny Pritzker, Secretary 

National Institute of Standards and Technology 

Willie May, Under Secretary of Commerce for Standards and Technology and Director 
 

 

 

 

 

_____________________________________________________________ 

Simulated Sinewave Testing of Data
 

Acquisition Systems using Sine Fitting and 

Discrete Fourier Transform Methods
 

Part 1: Frequency Offset, Random, 

Quantization, and Jitter Noise
 

ABSTRACT 

This report studies the effect of frequency offset, quantization error, random 

additive noise, and random phase jitter on the results of sine fitting and 

performing Discreet Fourier Transforms (DFT) of measurements of 

sinewaves with Data Acquisition Systems (DAS). Quantization error is the 

only effect of non-linearity that is considered. 

The majority of the simulations are carried out for full-scale signals on a 16­

bit DAS, but the simulations do not simulate the effect of clipping on 

portions of noisy signals that require more than 16 bits. Instead, the signals 

are rounded in a way that provides the minimum number of bits required to 

represent the integer part of the signal. The results of this approach are 

easier to interpret and simple corrections for this shortcoming are given. 

Only quantization error is studied as a function of available bits, ranging 

from 8- to 64- bits in increments of 8 bits. 

Among other results, it is shown that probability plots of the residuals of fits 

of sinewaves to simulated distorted sinewaves can distinguish among 

random additive noise, random jitter, and frequency offset error when the 

effect of quantization noise is small compared to the effects of these other 

sources of distortion. 

Acknowledgements: 

This work was partially supported by the U. S. Army WIAMan Warrior 

Injury Assessment Manikin Project 

ii
 
Simulated Sinewave Testing of Data Acquisition Systems using

Sine Fitting and Discrete Fourier Transform Methods

Part 1: Frequency Offset, Random, Quantization, and Jitter

Noise

July 17, 2015 

Introduction 

The purpose of this report is to describe the results of simulations that compare frequency-domain and 

time-domain sinewave-testing metrics for embedded data acquisition systems (DAS) that will be used 

in anthropomorphic test devices (ATD) subjected to pyroshock. But these results are applicable in a 

much wider range of applications. The metrics are inspired by those described in Institute of Electrical 

and Electronics Engineers Standard 1057 (IEEE 1057) for Digitizing Waveform Recorders[1, 2], but have 

been modified slightly for harmonization with standard statistical concepts. The simulation results give 

insights into the interactions of different sources of error in DAS measurements of sinewaves and other 

signals. Part 1, which is given here, is restricted to frequency error, quantization error, random additive 

noise, and random phase jitter noise. Quantization is the only source of non-linearity that will be treated 

here, and are the only distortion investigate for available bits ranging from 8 to 64 bits. Non-linearities 

will be treated in more detail in a future Part. 

For the purposes of the simulations described here, consider an ideal sinewave described in terms 

of peak-amplitude analog-to-digital-converter (ADC) count A, frequency f in Hz, and phase angle φ in 

radians relative to zero phase at time t = 0 

v(t) = A cos(2πft + φ). (1) 

Also consider a non-ideal implementation of that ideal sinewave, 

s(t) = A cos(2πf1 + φ) + σN rN (t), (2) 

where t is time, f1 is the frequency of the non-ideal sinewave, rN (t) is a random number drawn from a 

standard normal distribution (zero mean, unit standard deviation), and σN is the standard deviation of 

the noise voltage level. 

Assume that the non-ideal sinewave is sampled by a DAS at a nominal sampling frequency fs, at 

times 

tn = nΔt + σJ rJ (n), (3) 

where n = 1, . . ., M'

, M' is the number of samples, Δt = 1/fs is the average time between two adjacent 

samples, rJ(n) is a random number drawn from a standard normal distribution, and σJ is the standard 

deviation of the time-base jitter. 

In this case, the discrete non-ideal sinewave is given by 

 x[n] = Qb(s(tn)) = Qb A cos(2π f1 tn + φ) + σN rN (tn)), (4) 

1 
2 

for n = 1, . . ., M'

, where b is the number of bits available to represent the peak-to-peak ADC count 2A, 

and Qb is the function that quantizes the signal into the appropriate bit pattern. 

Next, define the relative offset of the sinewave frequency f1 from f as 

f1 − f EF = , (5) f 

which is a useful dimensionless parameter to express f1 in eq. 4 in terms of f in eq. 1. With this 

definition, the parameter EF and the relative standard deviations σN /A and σJ /Δt are physically meaningful dimensionless parameters that are convenient for expressing the variation of IEEE 1057 noise and 

distortion metrics with frequency offset error, random noise, and jitter. 

For the purposes of this report, the quantization function Qb was implemented by rounding the 

argument s(tn) of Qb in eq. 4 to the nearest integer. The error introduced by this overly-simplified 

model of signal-clipping, which is easily corrected by signal scaling as shown later in this report, make 

it much easier to understand the relations among the errors caused by frequency error, additive noise, 

phase jitter, and number of bits. 

It is important to understand that the DFT is extremely sensitive to the frequency offset error EF . 

DFTs were calculated from sinewaves defined by eq. 4 and 3 for sinewaves having the same parameters 

except that EF = 0 in Fig. 1 and EF = 10−6 in Fig. 2. 

Figure 1: The DFT of a sine wave that was calculated with the MATLAB fft function. The sinewave 

and sampling frequencies were, respectively, f1 = 20, 000.0 and fs = 100, 000.0 Hz to within the precision 

of MATLAB double-precision numbers. Only the first half of the DFT, which was plotted at 50,000 

frequencies from 0 to 49,999 Hz, is shown. The inset shows the shape of the peak at higher frequency 

resolution. 
�

3 

Figure 2: The DFT of a sinewave having the same parameters as the sinewave whose DFT is shown in 

Figure 1 except that the frequency of the sinewave was 1 part in 106 greater than that of the sinewave 

used for that figure. Specifically, the parameters σJ /Δt, σN/A, and EF were all set to zero for both 

figures, and only EF was changed to from zero to 10−6, which increased the frequency f1 from 20,000 Hz 

to 20,000.02 Hz for this figure. The inset shows the shape of the peak at higher frequency resolution. 

Clearly a 1 part in 10−6 time-base error in measurements of the response of ATDs to pyroshock is 

negligible. So from the point of view of that application, the sinewaves whose DFTs are shown in Figs. 

1 and 2 are equally good, which is also the conclusion that would be obtained from four-parameter sinefitting in the time domain as described in Annex A of IEEE 1057[3]. However, because the DFT does 

not treat them equally, the DFT of Fig. 2 cannot be used to characterize DAS performance for this 

application. Windowed and zero-padded DFTs, as described in Section 4.5.2 of IEEE 1057[6], are the 

solution when EF is large enough to interfere with other pertinent information in the DFT. 

Noise and Distortion Metrics 

IEEE 1057 defines the noise and distortion metric (NAD) of a digitized sine wave in both the time[4] and 

the frequency[7] domains. For the simulations described here, the theoretical value of NAD in the time 

domain is approximated by 

 

M 1 [ ] 

2 NAD = x[n] − v[n] = σx,v (6) M' − 1 n=1 

where x[n] is the non-ideal digitized sinewave defined in eq. 4, and v[n] is given by 

v[n] = A cos(2π f nΔt + φ), (7) 
�

 �

 �

 �

 �

 

�

�

 

4 

which is the ideally sampled implementation of eq. 1, and σx,v is the estimated standard deviation of the ' distribution from which the values of rN in eq. 4 are randomly drawn. The prefactor 1/(M − 1) could ' be replaced by 1/M to be consistent with IEEE 1057[4], but establishing the identity with the standard 

deviation is preferable. 

With this definition, in the case where the contribution of all other sources of distortion are negligible 

compared to that of random noise, 

M 1 [ ]2 NAD = lim σN rN (tn) = σN (8) M →∞ M' − 1 n=1 

because the random variable rN (tn) in eq. 4 is drawn from a normally-distributed random distribution 

whose standard deviation is 1. 

1 In the time domain: Define NADx as

⎛ ⎞ M 1 )2 NADx = min ⎝ x[n] − x ' [n] ⎠ = σx,x M' − 3 n=1 

⎛ ⎞ M 1 )2 

≈ min ⎝ x[n] − x ' [n] ⎠ = NAD1057 (9) M'

n=1 

' where x[n] are the M values of the sine wave defined in eq. 4, and NAD1057 is the definition for NAD 

that is used in IEEE 1057. 

The quantity σx,x in eq. 9 is the standard deviation of the distribution of the residuals of a least 

squares fit of 

) x ' [n] = Qb A1 cos(2π f1 tn) + B1 sin(2π f1 tn) + C , (10) 

to the sampled sine wave x[n] defined in eqs. 4. The quantities A1, B1, and C are the free parameters 

that are varied to minimize NADx. The constant term C is needed even though no constant term occurs 

in eq. 4 because the mean of a finite sample of the normally distributed random numbers rN (tn) is not 

identically zero even though the mean of the distribution from which the rN (tn) are drawn is identically 

zero. ' The prefactor 1/(M − 3) in the first line of eq. 9 is appropriate because three degrees of freedom 

have been used in fitting. Therefore, the theoretical relation of NADx to NAD and NAD1057 is given by 

J J M' − ' 1 M NADx = NAD = NADx1057. (11) M' − ' 3 M − 3 

The quantity b' » b in eq. 10 is the number of bits available for representing x '

[n] in the computer 

program being used to carry out the least-squares fitting procedure. Since b in eq. 4 is accurately known 

for a modern DAS, it is not adjusted to improve the fit, nor is this likely to prove practical for determining 

the effective number of bits (ENOB) available from the DAS. 

The results of the fitting procedure give time-domain estimates, denoted here by 

 

Ax = A2 + B2 (12) 1 1 

from eq. 10 and NADx from eq. 9. Later in this report, these will be compared with the theoretical 

values of A and NAD from eqs. 7 and 8 for various values of EF , σN /A, σJ /Δt, and b. 

1The subscript x will be used to identify quantities that are calculated in the time domain. 
 

 

 

 

 

  

 

5 

In the frequency domain: Start by adopting the MATLAB definition and normalization of the DFT, 

which is given by 

M ( ) −j2π[n − 1][k − 1] Y [k] = x[n] exp , (13) M n=1 

where k = 1, . . ., M, where M is an even positive integer. With this normalization, the peak amplitude 

AY = |Y [kA]| is M|Y [kA]|/2. 

To proceed, set Y [1] = Y [kA] = Y [M − kA] = Y [M] = 0, and define 

 M 1 2 NAD Y = Y [k] ≈ NADY 1057. (14) M[M − 6] k=1

Note that NADY 1057 differs from NADY in that it has M − 3 instead of M − 6 in its prefactor. The 

use of M − 6 is justified on statistical grounds because six degrees of freedom have been removed from 

the sum in NADY , namely the first and last points of the DFT (which in general are non-zero for the 

same reason that C was needed in x ' [n] in eq. 10), the fundamentals at k' and M − k' A A, and two degrees 

of freedom for the two identical sums from k = 1, . . ., N = M/2 and k = N + 1, . . ., M in eq. 14. 

It is often inconvenient for AY to depend upon the number data points M in the time series. In this 

case, it is convenient to use only the first half of the DFT in eq. 13 and express as 

M M/2 N 2 1 2 2 2 2M2 

 2Y [k] NADY = Y [k] = Y [k] = M [M − 6] M[M − 6] 4M[M − 6] M

k=1 k=1 k=1

N 2 N M 2Y [k] N 2 = = X [k] = NADX , (15) 

 2[M − 6] M 2[N − 3] k=1 k=1

where 

M ( ) 2Y [k] 1 −j2π[n − 1][k − 1] X[k] = = x[n] exp (16) M N M n=1 

for k = 1, . . ., N and 

 AX = |X[kA]| = Xre[kA]2 + Xim[kA]

2 (17) 

as desired, where the re and im subscripts denote the real and imaginary parts of the DFT. This is a 

particularly convenient normalization when M = fs/U is an even integer, f/U is an integer, where U is 

the frequency unit being used (Hz, kHz, etc.). These two constraints on f, fs, and M are necessary to 

avoid DFT artifacts that can obscure important information contained in the DFT even when EF = 0. 

NAD calculated from the DFT noise floor: The inset in Fig. 3 shows the effect (blue curve in 

inset) of noise with σN = 1 on a 1 ms segment of the digitized sinewave of eq. 4 (black curve in inset) at 

frequency f1 = 20 kHz with fs = 100 kHz, σJ = EF = 0, and b = 16. The main figure shows the effect 

of the same noise on the scaled DFT of the same sinewave segment (blue curve in main figure). Notice 

how the clean sinewave in the time domain (black curve in inset) is obscured by the noise when the SNR 

is 1, in striking contrast to how the amplitude of the same sine wave rises out of the noise floor of the 

DFT (blue curve in main figure). This illustr

√ 

ates the important fact that the noise floor of the DFT 

is not NADX, but is approximately NADX / M, which can result in major underestimates of NAD if 

forgotten. The green curve is the noise floor of the scaled DFT increased by the factor required to make 

it comparable to NADX . 
6 

Figure 3: Comparison of the effect of noise giving a SNR equal to 1 on the display of a sinewave in the 

time domain (inset) and frequency domain (main figure). The green curve is the noise floor of the scaled 

DFT increased by the factor required to make it comparable to the noise on the sinewave as discussed 

above. When x[n] − v[n] (eqs. 4 and 7) is drawn from a normally-distributed random distribution, the 

DFT noisefloor is drawn from a Rayleigh distribution. 

SINAD versus Sources of Distortion 

This section summarizes the results of setting the additive noise, jitter noise, and frequency offset to zero 

and the number of available bits to 16 and then independently varying each of the distortion parameters 

and b to determine the effect on SINAD. In agreement with IEEE standard IEEE 1057[1], define signalNADX 

to-noise-and-distortion ratios. 

A SINAD = , NAD (18) 

SINADx = , Ax 

NADx 

(19) 

SINADX = AX (20) 

for the distorted sinewave in eq. 4, where the theoretical amplitude A is defined in eq. 7, and the 

estimated amplitudes Ax and AX are defined in eqs. 12 and 17, respectively. Similarly, the theoretical 
7 

Figure 4: A: Theoretical variation of SINAD as a function of the relative, normally-distributed additive 

random noise parameter σN /A. B: Ratio of time- and frequency domain estimates of SINADx and 

SINADX, respectively, to SINAD as a function of σN /A, both for a 16-bit DAS. C and D: The same as 

A and B, but the relative normally-distributed random phase noise σJ /Δt is the parameter being varied. 

E and F: The same as A and B, but the relative frequency offset from the target sampling frequency is 

the parameter being varied. G and H: The same as A and B, but the number of bits b is the parameter 

being varied. 
8 

noise-and-distortion metric NAD is defined in eq. 8, and the estimated metrics NADx and NADX are 

defined in eqs. 9 and 15, respectively. 

The quantities A, Ax/A, AX/A, SINAD = A/NAD, SINADx = Ax/NADx, and SINADX = AX /NADX 

were independently calculated for a range of values of the additive noise σN , the jitter σJ , the frequency 

offset EF , and the number of available bits b. The results are summarized in Fig. 4 on the previous page. 

As pointed out previously, the simulations described above do not treat clipping properly. The proper 

treatment of clipping is described in Appendix ??. 

Frames A and B of Fig. 4 plot the theoretical values of SINAD and the ratios SINADx/SINAD and 

SINADX/SINAD, respectively, as a function of σN /A from 10−9 to 1 in factors of 10 with b = 16 and 

σJ = EF = 0. Each time the program that calculates the values in Fig. 4 is run with new random values 

for rN (tn) in eq. 4, new values of Ax/A and AX /A that differ from the old values by much less than 

σN /A are obtained. Both SINADx/SINAD and SINADX /SINAD overestimate SINAD by a factor of 

approximately 1.9 below σN = 10−7 and approximate SINAD quite well above σN = 10−5 . 

Frames C and D of Fig. 4 plot the same quantities for σJ /Δt from 10−9 to 0.1 in factors of 10 with 

b = 16 and σN /A = EF = 0. These results are qualitatively similar to those in frames A and B, the most 

notable difference occurring in the transition region around σN /A = σJ /Δt = 10−6 . No result is shown 

for σJ /Δt = 1 because these results were not reproducible, and in many cases the least squares fitting 

program did not even converge. 

Frames E and F of Fig. 4 plot the same quantities for E = 10−9 to 0.1 in factors of 10 with b = 16 and 

σN /A = σJ /Δt = 0. Comparison of these results with those in A and C shows that SINAD is much more 

sensitive to relative frequency-offset errors than to relative amplitude errors and relative jitter errors. 

Also, SINADx and SINADX both overestimate SINAD for EF < 10−12 by a factor of 1.9 as is the case for 

random additive noise and random jitter below their transition regions at σN /A = σJ /Δt = 10−6 . The 

reason for this is that NAD is dominated by the same quantization error associated with b = 16 available 

bits below all three transition regions. 

Frames G and H of Fig. 4 plot the same quantities for b = 8 to 64 bits in increments of 8 bits with 

σN /A = σJ /Δt = EF = 0. SINAD is exponential as expected, but the behavior of SINADx and SINADX 

as a function of the number of available bits in frames G and H is very different from that shown in 

frames BD, and F because the number of bits is a completely different type of parameter. Increases 

in this parameter increase SINAD whereas increases in the other parameters decrease SINAD. Also the 

number of available bits was set at 16 for the other simulations, but was varied in this one. Over the 

range from b = 8 to b = 24 SINADx/SINAD and SINADX/SINAD decrease from approximately 3 to 

approximately 1, which is also its value at b = 32 bits. Above the transition region b = 24 − 32 bits, the 

relative accuracy of these ratios deteriorates rapidly. 

Given a measured value of SINADx or SINADX and based only on the information provided in Fig. 

4, it would not be possible to distinguish among any of the source of distortion considered here. However, 

a closer look at the measured data from which SINADx and SINADX are calculated provides much more 

information as shown below. 

SINAD versus Additive Noise 

Table 1 in Appendix B compares the Ax/A, AX/A, SINAD = A/NAD, SINADx = Ax/NADx, and 

SINADX = AX/NADX for times equal to 1 s, 10 s, and 100 s as a function of the relative additive 

noise σN /A from 10−9 to 1 with b = 16 and σJ /Δt = EF = 0. This table shows that A/NAD = 1/σN 

(1/sigN in the table) to within a small fraction of σN when σN > 10−5 and that SINADX is an excellent 

approximation to SINADx for all conditions. Also, the excellent agreement between the corresponding 

quantities at the three different simulated times shows that NAD is independent of time for this type of 

error, which justifies estimating the mean noise as the time average of NAD and the standard deviation 

uncertainty in the mean noise as NAD/M. This is just one of the reasons for adopting the constraints 

on f, fs, and M for the DFT. 
 

  

 

  

9 

A closer look at the σN /A = 1 results: For this simulation, Fig. 5 presents in A: a probability plot 

of the set of differences x[n] − v[n] for n = 1, . . ., M from eqs. 4 and 7, respectively, in B: a probability 

plot of the set of residuals x[n] − x ' [n] for n = 1, . . ., M from the fit defined in eq. 9, in C: a plot of 

the noisefloor of the DFT of x[n] defined as the set of X[k] for k = 1, . . ., N in eq. 15 with X[kA] and 

X[1] set to zero, and in D: a probability plot of the noisefloor of the DFT of x[n]. 

Figure 5: From Top Left moving clockwise: Probability plot of the theoretical difference between the 

noisy sinewave x[n] and the near perfect sinewave v[n]. Top Right: Probability plot of the residual after 

a least squares fit of x '

[n] to x[n]. Bottom Left: Standard plot of the DFT noisefloor obtained from X[k] 

following removal of the fundamental and the DC component. Bottom Right: Probability plot of the 

DFT noise floor. 

The probability plots of the set of differences x[n]−v[n] and the residuals of the fit x[n]−x ' [n] are very 

similar and normally distributed for all practical purposes. This is expected since the data were simulated 

with normally distributed random noise with a standard deviation of 1, which rendered the quantization 

noise completely negligible. Similarly, the probability plot of the noisefloor of the DFT follows a Rayleigh 

distribution very well, as expected for magnitudes calculated from normally distributed data. 

A closer look at the σN /A = 10−9 results: For this simulation, Fig. 6 presents the same type of 

graphs used in Fig. 5 above. These four graph types, which will be used in the remainder of this report 
10 

to examine some instructive examples of the simulation results in detail, will be referred to as the four 

standard graphs. 

Figure 6: From Top Left moving clockwise: Probability plot of the theoretical difference between the 

quantization-noise limited sinewave x[n] and the near perfect sinewave v[n]. Top Right: Probability 

plot of the residual after a least squares fit of x ' [n] to x[n]. Bottom Left: Standard plot of the DFT 

noisefloor obtained from X[k] following removal of the fundamental and the DC component. Bottom 

Right: Probability plot of the DFT noise floor. 

None of the probability plots in Fig. 6 contain normally or Rayleigh-distributed data. Because the 

20 kHz sinewave is sampled at 100 kHz, the upper left probability plot contains five samples per cycle 

of the distorted minus ideal sinewave data, x[n] − v[n], which is referred to here as the difference data. 

The upper right probability plot contains five samples per cycle of the distorted minus fit sinewave data, 

x[n]−x '

[n], which is referred to here as the residual data. Because the sampling and sinewave frequencies 

are very well synchronized, the same points of the difference and residual sine waves are sampled each 

cycle, which explains why both probability plots contain many duplicates of only five different values 

of the sine wave. Because the fit can never perfectly reconstruct the ideal sinewave from the distorted 

sinewave, the values of the five different residuals in the upper right plot will in general never duplicate 

the five different values of the differences in the upper left plot. The fact that the range of the residuals is 

about a factor of two smaller than that of the differences is consistent with SYNADx ≈ 2SINAD in Fig. 
11 

4 B. Finally, because the phase of the sinewave relative to the sampling signal is 0.3 radians, for which π 

is not an integer multiple, none of the five points that are sampled on the sinewave have the same value. 

For special values of φ, some of the points being sampled will be duplicates, and the probability plot will 

contain fewer different values. For instance, only three values occur when φ = 0. 

The DFT noisefloor plotted in the lower left of Fig. 6 doesn’t even look like a DFT noisefloor. 

Instead, it consists of an easily resolved signal at 40 kHz, which is the first harmonic of the 20 kHz 

sinewave. The true noisefloor is negligible at the scale of this figure. The fact that all of the energy in 

the noisefloor occurs at a harmonic of the sinewave frequency shows that quantization noise is caused by 

the non-linearity of the quantization process rather than a random-noise-like process. 

A closer at look the σN /A = 10−5 results: For this simulation, Fig. 7 presents the standard graphs. 

Figure 7: Top Left: Probability plot of the theoretical difference between the noisy sinewave x[n] and the 

near perfect sinewave v[n]. Top Right: Probability plot of the residual after a least squares fit of x '

[n] 

to x[n]. Bottom Left: Standard plot of the DFT noisefloor. Bottom Right: Probability plot of the DFT 

noise floor obtained from X[k] following removal of the fundamental and the DC component. 

This simulation is on the quantization-noise side of the transition from quantization noise limited 

behavior to normally distributed, additive, random-noise-limited behavior. The probability plots in Fig. 

7 still retain the qualitative characteristics of quantization noise, but the range has increased by about 
12 

a factor of 30, and 14 different values instead of just 5 are present in the differences and the residuals. 

The strong signal at the 40 kHz is still evident in the DFT noisefloor data, but unlike the case where 

σN /A = 10−9, the true noisefloor, which is Rayleigh distributed, is resolved. 

A closer at look the σN /A = 10−4 results: For this simulation, Fig. 8 presents the standard graphs. 

This simulation is on the random-noise side of the transition from quantization noise limited behavior to 

normally distributed, additive, random-noise-limited behavior. 

Figure 8: Top Left: Probability plot of the theoretical difference between the noisy sinewave x[n] and the 

near perfect sinewave v[n]. Top Right: Probability plot of the residual after a least squares fit of x '

[n] 

to x[n]. Bottom Left: Standard plot of the DFT noisefloor. Bottom Right: Probability plot of the DFT 

noise floor obtained from X[k] following removal of the fundamental and the DC component. 

Increasing σN /A = 10−5 by one order of magnitude to σN /A = 10−4 completely changed the character 

of the probability plots as shown in Fig. 8. Not only do all four plots now bear a strong resemblance 

to the σN /A = 1.0 plots in Fig. 5, but the probability plots of the differences and of the residuals are 

identical for all practical purposes. Even the fine-grain structure of these two probability plots, which is 

the contribution of the quantization, looks very similar. 

Before reporting the results of a similar simulation as a function of the jitter parameter σJ /A with 

σN /A = 0, there is one more important point to consider. Fig. 4B and reference to Table 1 show that 
13 

SINADx and SINADX agree very well for 10−9 ≤ σN /A ≤ 1 even though the shapes of the probability 

plots are very different because one is normally distributed and the other Rayleigh distributed. This is 

an approximate application of Parseval’s energy theorem. It is not exact because different prefactors are 

used in the definitions of NADx and NADX . But the differences are quite small for large M or N. On the 

other hand, even though the probability plots of the differences between the distorted and ideal sinewaves 

and the residuals of the fit to the distorted sinewave look much more similar than do the probability plots 

of the DFT noisefloor and the residuals of the fit to the distorted sinewave, they agree very well only for 

additive noise-limited data, differing by almost a factor of two for quantization “noise”-limited data. 

This space intentionally left blank . 

SINAD versus jitter 

Table 2 in Appendix B compares the Ax/A, AX/A, SINAD = A/NAD, SINADx = Ax/NADx, and 

SINADX = AX /NADX for times equal to 1 s, 10 s, and 100 s as a function of the relative jitter σJ /Δt 

from 10−9 to 0.1 with b = 16 and σN /A = EF = 0. The excellent agreement between the corresponding 

quantities at the three different simulated times shows that NAD is independent of time for this type of 
14 

error, which is also true for relative additive noise and which justifies estimating the mean noise as the 

time average of NAD and the standard deviation uncertainty in the mean noise as NAD/M for jitter as 

well as random additive noise and also for their combination. However, no correction for clipping such as 

that given in Appendix A for random noise is required for jitter because jitter is not additive. Instead, 

it samples the signal at times that are randomly different from when it was supposed to be sampled, but 

doesn’t change its value as additive noise does. 

A closer look at the σJ /Δt = 0.1 results: For this simulation, Fig. 9 presents the standard graphs. 

Figure 9: Top Left: Probability plot of the theoretical difference between the noisy sinewave x[n] and the 

near perfect sinewave v[n]. Top Right: Probability plot of the residual after a least squares fit of x '

[n] 

to x[n]. Bottom Left: Standard plot of the DFT noisefloor obtained from X[k] following removal of the 

fundamental and the DC component. Bottom Right: Probability plot of the DFT noise floor. 

As was the case with the SINAD simulation at σN /A = 1, the probability plots of the set of differences 

x[n] − v[n] and the residuals of the fit x[n] − x '

[n] for σJ /Δt = 0.1 are very similar to each other. 

Furthermore, the probability plots of the noisefloors of the σN /A = 1 and σJ /Δt = 1 DFTs are very 

similar. On the other hand, the probability plots of the SINAD(σJ /Δt) and SINAD(σN /A = 1) timeseries data are quite different, the latter being normally distributed while the former has a small central 

region that is normally distributed between two nominally symmetric long-tailed distributions. This is a 
15 

result of the non-additive nature of jitter. Even though the sampling times tn are normally distributed, 

the resulting errors in the signal amplitude are not normally distributed. Adding a random time to a 

sampling time produces a very different signal-amplitude error depending upon whether it occurs near a 

extremum or a zero crossing of the signal. 

The other major difference between the simulation results for jitter and for additive noise is that 

above the transition region from quantization-dominated behavior, SINAD = A/NAD ≈ 0.8Δt/σJ for 

the former and SINAD = A/NAD ≈ A/σN for the latter. 

A closer look at the σJ /Δt = 10−9: The results for σN /ΔJ = 10−9, cannot be distinguished from 

the results for σN /A = 10−9 that are shown in Fig. 6 because the 16-bit quantization noise overwhelms 

the effects of σN /ΔJ = 10−9 in Fig. 4 (C) just as it overwhelms the effect of σN /A = 10−9 in Fig. 4 (A). 

Figure 10: Top Left: Probability plot of the theoretical difference between the noisy sinewave x[n] and 

the near perfect sinewave v[n]. Top Right: Probability plot of the residual after a least squares fit of x '

[n] 

to x[n]. Bottom Left: Standard plot of the DFT noisefloor. Bottom Right: Probability plot of the DFT 

noise floor obtained from X[k] following removal of the fundamental and the DC component. 

A closer look at the σJ /Δt = 10−5 results: For this simulation, Fig. 11 presents the standard graphs. 

This simulation is on the quantization-noise side of the transition from quantization noise limited behavior 
16 

to normally distributed, random, additive, noise-limited behavior. 

Figure 11: Top Left: Probability plot of the theoretical difference between the noisy sinewave x[n] and 

the near perfect sinewave v[n]. Top Right: Probability plot of the residual after a least squares fit of x '

[n] 

to x[n]. Bottom Left: Standard plot of the DFT noisefloor. Bottom Right: Probability plot of the DFT 

noise floor obtained from X[k] following removal of the fundamental and the DC component. 

The probability plots in Fig. 11 have progressed a little further toward the random-noise limit than 

have those for σN /A in Fig. 7. This is also evident in Fig. 4 B versus Figs. 4 D. However, if these plots 

were shown without a description of how they were generated, one might erroneously conclude that Fig. 

11 was created by additive noise and that Fig. 7 was created by jitter based on their similarities to Figs. 

5 and 9, respectively. 

A closer look at the σJ /Δt = 10−4 results: For this simulation, Fig. 12 presents the standard 

graphs. This simulation is on the random-noise side of the transition from quantization noise limited 

behavior to normally distributed, additive, random-noise-limited behavior. In contrast to the case when 

quantization “noise” is the dominant noise source, random additive noise can be distinguished from 

random jitter when one of the latter two is the only other source of noise. 
17 

Figure 12: Top Left: Probability plot of the theoretical difference between the noisy sinewave x[n] and 

the near perfect sinewave v[n]. Top Right: Probability plot of the residual after a least squares fit of x '

[n] 

to x[n]. Bottom Left: Standard plot of the DFT noisefloor. Bottom Right: Probability plot of the DFT 

noise floor obtained from X[k] following removal of the fundamental and the DC component. 
18 

SINAD versus relative frequency offset 

Table 3 in Appendix D compares the Ax/A, AX/A, SINAD = A/NAD, SINADx = Ax/NADx, and 

SINADX = AX/NADX for times equal to 1 s, 10 s, and 100 s as a function of the relative frequencyoffset error EF from 10−9 to 0.1 with b = 16 and σN /A = σJ /Δt = 0. Examination of this table shows 

that SINAD(EF , t1) =� SINAD(EF , t2) when t2 �= t1 for EF above the transition region, which means that 

errors of this type should not be thought of as noise and cannot be extrapolated over time by statistical 

means. However, no correction for clipping such as that given in Appendix A for random noise is required 

for frequency offset error because this source of distortion is not additive. 

A closer look at the EF = 10−10 results: For this simulation, Fig. 13 presents the standard graphs. 

Figure 13: Top Left: Probability plot of the theoretical difference between the noisy sinewave x[n] and 

the near perfect sinewave v[n]. Top Right: Probability plot of the residual after a least squares fit of x '

[n] 

to x[n]. Bottom Left: Standard plot of the DFT noisefloor obtained from X[k] following removal of the 

fundamental and the DC component. Bottom Right: Probability plot of the DFT noise floor. 

These results are clearly on the quantization noise side of the transition from quantization-noiselimited behavior to frequency-offset-limited noise behavior. These plots share some similarities with 

those for quantization-noise-limited behavior with σN /A = 10−9 in Fig. 6, but the plots here also have 
 

  

19 

some noticeable differences. Both time-domain probability plots have only a few values that are repeated 

over multiple cycles, but unlike those in Fig. 6, which each have five values repeated in multiple cycles 

with a factor of two difference in range, these have six values that are repeated in multiple cycles with 

only a 30 % difference in range. Also, both noisefloor DFTs have large peaks at the first harmonic 

frequency, but the DFT noisefloor here also has double peaks (not resolvable at the resolution of the 

figure) at the fundamental frequency. These are caused by the frequency offset EF /Δt = 10−10, which 

is already large enough to put a non-negligible quantity of energy into the frequencies in the vicinity of 

the fundamental at kA, as is shown more clearly in the next two figures and in Fig. 2. However, only 

X[kA] was set to zero when calculating NAD, which splits the fundamental into two peaks bracketing 

the missing fundamental in the plot of the DFT noisefloor. 

A closer look at the EF = 10−8 results: For this simulation, Fig. 14 presents the standard graphs. 

Figure 14: Top Left: Probability plot of the theoretical difference between the quantization-noise limited 

sinewave x[n] and the near perfect sinewave v[n]. Top Right: Probability plot of the residual after a 

least squares fit of x ' [n] to x[n]. Bottom Left: Standard plot of the DFT noisefloor obtained from X[k] 

following removal of the fundamental and the DC component. Bottom Right: Probability plot of the 

DFT noise floor. 

These results are on the frequency-offset side of the transition from quantization-noise-limited behavior 
20 

to frequency-offset-limited noise behavior. These plots share some similarities with those for noise and 

jitter with σN /A = σJ /δt = 10−4 in Fig. 8. But there is a factor of two difference in the ratio of the 

range of the residuals to the range of the differences. Also the shape of the probability plots of the time 

domain data is very different from that for random noise limited data and that for random jitter limited 

data. 

A closer look at the EF = 10−6 results: For this simulation, Fig. 15 presents the standard graphs. 

Figure 15: Top Left: Probability plot of the theoretical difference between the noisy sinewave x[n] and 

the near perfect sinewave v[n]. Top Right: Probability plot of the residual after a least squares fit of x '

[n] 

to x[n]. Bottom Left: Standard plot of the DFT noisefloor. Bottom Right: Probability plot of the DFT 

noise floor obtained from X[k] following removal of the fundamental and the DC component. 

These results, which are for the same parameters used in Fig. 1, and which are strongly on the 

frequency-offset side of the transition from quantization-noise-limited behavior to frequency-offset-limited 

noise behavior, continue the trends apparent in Fig. 14. The contribution to the first harmonic from 

frequency offset is now larger than that from the quantization noise, so the latter cannot be detected in 

the noisefloor DFT. 

The probability plot of the residuals of the fit of a sinewave to the distorted sinewave data is nominally identical to the probability plot of the differences between the distorted and undistorted sinewave. 
21 

Furthermore the data in these probability plots appears consistent with random distributions that are 

approximately normal near their means but that become extremely short-tailed at their extremes. However, this does not mean that these results can be extrapolated to different size samples collected over 

different times as described above with reference to the simulation results in Table 3. 

No closer look at the EF = 10−16 , 10−14, and 10−12 results are shown because they are similar to the 

σN /A = σJ /Δ = 10−9 quantization-noise-limited results in Fig. 13. 

This space intentionally left blank . 
� = = �

22 

SINAD versus Available Bits 

Table 4 in Appendix E compares the Ax/A, AX/A, SINAD = A/NAD, SINADx = Ax/NADx, and 

SINADX = AX /NADX for times equal to 1 s, 10 s, and 100 s as a function of the available number 

of bits from b = 8 to b = 64 and σN /A = σJ /Δt = EF = 0. Examination of this table shows that 

SINADx(b, t1) = SINAD(b, t2) SINADx(b, t1) � SINAD(b, t2) when � t1 = t2, which means that errors 

of this type should not be thought of as noise and cannot be extrapolated over time by statistical means. 

However, no correction for clipping such as that given in Appendix A for random noise is required for 

frequency offset error because this source of distortion is not additive. 

A closer look at the b = 8 results: For this simulation, Fig. 16 presents the standard graphs. 

Figure 16: Top Left: Probability plot of the theoretical difference between the noisy sinewave x[n] and 

the near perfect sinewave v[n]. Top Right: Probability plot of the residual after a least squares fit of x '

[n] 

to x[n]. Bottom Left: Standard plot of the DFT noisefloor obtained from X[k] following removal of the 

fundamental and the DC component. Bottom Right: Probability plot of the DFT noise floor. 

These results are qualitatively similar to those in Fig. 6 for σN /A = 10−9 . Specifically, the probability 

plot of the difference between the ideal v[n] and the simulated distorted data x[n] has only five values 

that are repeated synchronously over the data set, and the probability plot of the residuals of the fit of 
23 

the sinewave x ' [n] to the x[n] data is similar to the probability plot of x[n] − v[n], also having five values 

repeated over many cycles, but with a reduced range. Finally, the noise floor DFT has only one spike, 

which shows that the effect of quantization is synchronous with the ideal sinewave v[n]. 

A closer look at the b = 16 results: For this simulation, Fig. 17 presents the standard graphs. 

Figure 17: Top Left: Probability plot of the theoretical difference between the quantization-noise limited 

sinewave x[n] and the near perfect sinewave v[n]. Top Right: Probability plot of the residual after a 

least squares fit of x ' [n] to x[n]. Bottom Left: Standard plot of the DFT noisefloor obtained from X[k] 

following removal of the fundamental and the DC component. Bottom Right: Probability plot of the 

DFT noise floor. 

These results are qualitatively similar to those for 8 bits, but the ratio of the range of the residuals 

to that of the differences is reduced. 
24 

A closer look at the b = 24 results: For this simulation, Fig. 18 presents the standard graphs. 

Figure 18: Top Left: Probability plot of the theoretical difference between the noisy sinewave x[n] and 

the near perfect sinewave v[n]. Top Right: Probability plot of the residual after a least squares fit of x '

[n] 

to x[n]. Bottom Left: Standard plot of the DFT noisefloor. Bottom Right: Probability plot of the DFT 

noise floor obtained from X[k] following removal of the fundamental and the DC component. 

These results are qualitatively similar to those for 8 and 16 bits, but the ratio of the range of the 

residuals to that of the differences is further reduced. 
25 

A closer look at the b = 32 results: For this simulation, Fig. 19 presents the standard graphs. 

Figure 19: Top Left: Probability plot of the theoretical difference between the noisy sinewave x[n] and 

the near perfect sinewave v[n]. Top Right: Probability plot of the residual after a least squares fit of x '

[n] 

to x[n]. Bottom Left: Standard plot of the DFT noisefloor. Bottom Right: Probability plot of the DFT 

noise floor obtained from X[k] following removal of the fundamental and the DC component. 

These results are qualitatively different from those for 8, 16, and 24 bits. First, the ratio of the range 

of the residuals to that of the differences is nominally 1. More significantly, the effect of the quantization 

error is no longer synchronous with the sinewave as is evident both in the probability plots in the top left 

and top right and in the noise floor DFT in in the lower left. This is surprising. Presumably, it is caused 

by the interaction of the rounding of x[n] at 32 bits which was programmed into the simulation with 

the rounding at 51 bits intrinsic to the MATLAB double (precision) arithmetic implemented according 

to the IEEE 754 standard for double precision. But there is no obvious reason why the synchronization 

that produced five discrete values for b = 8, 16, and 24 bits should be broken. Therefore, these and the 

following results must be viewed with suspicion. 
26 

A closer look at the b = 40 results: For this simulation, Fig. 20 presents the standard graphs. 

Figure 20: Top Left: Probability plot of the theoretical difference between the noisy sinewave x[n] and 

the near perfect sinewave v[n]. Top Right: Probability plot of the residual after a least squares fit of x '

[n] 

to x[n]. Bottom Left: Standard plot of the DFT noisefloor. Bottom Right: Probability plot of the DFT 

noise floor obtained from X[k] following removal of the fundamental and the DC component. 

These results are qualitatively different from those for 32 bits as well as those for 8, 16, and 24 bits. 

The shape of the probability plot of the residuals is similar to that of the differences, but the former is 

much more asymmetric with respect to zero than any of the residual probability plots for 8 to 32 bits. 

The shape of all of the probability plots are remarkably similar to those for EF = 10−8 and EF = 10−6 , 

for no obvious reason. 
27 

A closer look at the b = 48 results: For this simulation, Fig. 21 presents the standard graphs. 

Figure 21: Top Left: Probability plot of the theoretical difference between the noisy sinewave x[n] and 

the near perfect sinewave v[n]. Top Right: Probability plot of the residual after a least squares fit of x '

[n] 

to x[n]. Bottom Left: Standard plot of the DFT noisefloor. Bottom Right: Probability plot of the DFT 

noise floor obtained from X[k] following removal of the fundamental and the DC component. 

Three of the plots are qualitatively similar to the corresponding three for 40 bits, but the probability 

plot of the residuals has changed dramatically as discussed in a little more detail below. 
28 

A closer look at the b = 56 results: For this simulation, Fig. 22 presents the standard graphs. 

Figure 22: Top Left: Probability plot of the theoretical difference between the noisy sinewave x[n] and 

the near perfect sinewave v[n]. Top Right: Probability plot of the residual after a least squares fit of x '

[n] 

to x[n]. Bottom Left: Standard plot of the DFT noisefloor. Bottom Right: Probability plot of the DFT 

noise floor obtained from X[k] following removal of the fundamental and the DC component. 

The shape of the probability plot of the residuals appears almost identical to that for 48 bits, which 

suggests that the residuals of the sinewave fit are smaller than a threshold in the program that carries 

out the fit. The shape of the probability plot of the differences is entirely new. 
29 

A closer look at the b = 64 results: For this simulation, Fig. 23 presents the standard graphs. 

Figure 23: Top Left: Probability plot of the theoretical difference between the noisy sinewave x[n] and 

the near perfect sinewave v[n]. Top Right: Probability plot of the residual after a least squares fit of x '

[n] 

to x[n]. Bottom Left: Standard plot of the DFT noisefloor. Bottom Right: Probability plot of the DFT 

noise floor obtained from X[k] following removal of the fundamental and the DC component. 

The shape of the probability plot of the residuals appears almost identical to that for 48 and 56 bits, 

which lends support to the idea that the shape for 48, 56, and 64 bits is determined by a threshold in the 

sinewave fitting program. All of the differences are now zero because the simulation program is rounding 

the simulated distorted data to a number of bits that is greater than the 51 bits in the mantissa of a 

MATLAB double (precision) number. In retrospect, it is clear that the differences x[n] − v[n] and the 

residuals x[n] − x ' [n] were approaching this situation as the number of bits was increased from 8 to 64 in 

the simulation, but a computation-terminating threshold in the fitting function prevented the residuals 

from reaching zero. However, this explanation is at best incomplete because it does not explain why the 

differences were not all zero for the 56 bit simulation, which also exceeds the 51 bit length of the mantissa 

in the MATLAB double (precision) numbers, nor does it explain the detailed changes in shape observed 

when adding 8 bits to the simulated number of available bits to represent the distorted sinewave data. 
30 

Conclusion 

This report described results of simulating the effect of random noise σN , random jitter σJ , the number 

of bits b, and frequency-offset error EF between the sampling frequency and the sinewave frequency on the 

time and frequency domain estimates of the amplitude A and the signal-to-noise-and-distortion metric 

SINAD of a sinewave sampled by a 16-bit DAS. For these simulations, the theoretical value of SINAD 

was defined as the standard deviation of the difference x[n] − v[n] between an ideal sinewave v[n] and a 

distorted version x[n] of the ideal sinewave. 

A suitably normalized discrete Fourier transform (DFT) was used to determine the frequency domain 

estimates AX and SINADX of A and SINAD, respectively, where SINADX was defined as the ratio of 

AX to the standard deviation of the noisefloor of the DFT of x[n], taking proper account of the loss of 

degrees of freedom when calculating the noisefloor from the DFT. 

Fitting a three parameter sinewave x ' [n] to x[n] was used to determine the time domain estimates 

Ax and SINADx of A and SINAD, respectively, where SINADx was defined as the standard deviation of 

the residuals of the fit, taking proper account of the loss of degrees of freedom when fitting the sinewave 

x '

[n] to the distorted sinewave x[n]. 

It was also shown that these tools work best when the sinewave and the sampling frequency are 

integers, and that SINADx and SINADX are extremely sensitive to violations of this requirement, which 

may be much more stringent than what is required in many applications. In this case, a four parameter 

non-linear fit (which may be implemented iteratively with three-parameter linear fits) is required in the 

time domain and suitably windowed DFTs are required in the frequency domain. A future entry in this 

series will describe the relations among SINAD, SINADx, and SINADX when analyzed with these more 

sophisticated tools. 

It was shown that SINADx and SINADX, as well as Ax and AX , are identical for all practical purposes. 

It was also shown that 

• Ax and AX are excellent approximations to A for all cases studied, 

• SINADX ≈ SINADx ≈ 1.9 SINAD when quantization is the dominant source of error, 

• Quantization error and frequency-offset error cannot be treated as random errors by standard 

statistical tools, 

• Random additive noise and random jitter can be treated as random errors by standard statistical 

tools, and for these sources of error: 

– SINADx and SINADX are excellent estimates of SINAD for σN /A = σJ /Δt ≥ 10−4 , 

– SINADX(σN /A) ≈ SINADx(σN /A) and SINADX(σJ /Δt) ≈ SINADx(σJ /Δt). 

It was further shown that probability plots of the residuals of the fits of three parameter sinewaves 

to distorted sinewaves have distinctively different shapes for normally distributed random additive noise, 

normally distributed random jitter noise, and frequency-offset error if the effect of quantization-induced 

error is negligible compared to the effect of these types of distortion. This suggests that it may be 

possible to quantify the separate contribution from each of these types of distortion in the outputs of 

data acquisition systems acquiring sinewave data. 

Finally, it was shown that this type of analysis can also be applied to 24-bit, and possibly 32-bit DAS, 

but that some anomalous behavior starting at 32 bits has the potential to produce errors of unknown 

magnitude in the estimates of SINAD. 
REFERENCES 31 

References 

[1] IEEE Std 1057tm-2007 Standard for Digitizing Waveform Recorders, The Institute of Electrical and 

Electronics Engineers, Inc., 3 Park Avenue, New York, NY 10016-5997, USA (2008). 

[2] Certain commercial equipment, instruments, or materials are identified in this report in order to 

specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended 

to imply that the materials or equipment identified are necessarily the best available for the purpose. 

[3] Reference [1], Annex A, page 112. 

[4] Reference [1], eq. 66 on page 77. 

[5] Reference [1], page 21. 

[6] Reference [1], page 25. 

[7] Reference [1], eq. 68 on page 78. 

[8] Nien Fan Zhang, Calculation of the uncertainty of the mean of autocorrelated measurements, 

Metrologia 43 (2006) S276S281, doi.org/10.1088/0026-1394/43/4/S15 

[9] Nien Fan Zhang,The batched moving averages of measurement data and their applications in data 

treatment, Measurement 39 (2006) 864875, www.elsevier.com/locate/measurement 
 

  

 

  

 

  

 

  

 

 

  

 

Appendix A: Headroom for Signal Clipping 32 

This appendix describes how to account for signal clipping in the simulation and in practice. The goal 

is to derive a conservative effective peak amplitude AE < A2b−1 to be used to assure that the probability 

that no x[n] in eq. 4 for n = 1, . . ., M will exceed A is less than some user defined threshold Q. Assume 

that A = 2b−1 is the peak amplitude of the maximum noise-free sinewave that can be measured without 

clipping. Further assume that the dominant source of noise is normally distributed, random, additive 

noise characterized by σN . To start, approximate Q by 

M 

) Q = Qn AEv[n] + rN [n] σN − A

n=1 

M ( ) A − AEv[n] = Qn rN [n] < σN n=1 

M � ∞ 1 

√ 

2 exp(−t /2)dt A− AEv[n] 2π n=1 σN 

M ( ) 1 A − AEv[n] = erfc √ « 1, (21) 

n=1 

2 2σN

where v[n], which is given in eq. 7, is the expectation value of x[n], Qn is the probability that its 

argument is negative, | . . . | denotes the absolute value function, and the « symbol emphasizes that this 

approximation is accurate only when Q « 1. Finally, let AˆE be the maximum value of AE for which eq. 

21 is true. 

Unfortunately, eq. 21 is very difficult to solve exactly for AˆE, but a very conservative estimate is 

given by 

M ( ) ( ) 1 A − AE M A − AE Q = erfc = erfc √ , (22) 

n=1 

2 σN 2 2σN

which is readily solved as 

( ) A − AE 2Q

√ = inverfc , (23) 2σN M 

which in turn gives 

( ) ( ) √ 2Q √ 2Q AE = A − 2σN inverfc ≤ 6 2σN inverfc . (24) M M 

As an example, let M = 105 , σN /A = 10−3, and Q = 0.1, which means that in a set of 100 

measurements of a sinewave, each of which consists of M samples of the sinewave, 10 measurements 

containing one or more values of x[n] > A will be considered acceptable. In this case AE/A = 0.9952. 

If Q is reduced to 0.001, AE /A decreases only slightly to 0.9943. Note that these very favorable results 

are crucially dependent upon the assumption of normally distributed noise. Deviations for normality will 

change the results drastically. On the other hand, eq. 24 is very conservative. The net result is that for 

real measurements AE/A = 1 − 6σN for σN /A ≤ 10−2 is probably safe from noise-derived clipping, and 

less limiting than other restrictions of AE for other sources of error such as gain non-linearity well below 

the clipping threshold. 

In the absence of clipping SINADx(AE) <SINADx(A) if AE < A. However, this relation may not 

be true in the presence of noise induced clipping. In this case an iteration based on eq. 24 will quickly 

converge to the value of AE that maximizes SINADx(AE). 
Appendix B: NAD Metrics as a Function of Simulation Time 33 

Table 1: Amplitudes (A, Ax, and AX), and SINADs (A/NAD, Ax/NADx,and AX /NADX) as a function 

of σN and time. 

SINAD_vs_Time_2015_07_10 run at 2015-4-22T22:41

Target sampling frequency: fs = 100000.000000 Hz

Target sinewave frequency: f = 20000.000000 Hz

Target sinewave phase: phi = 0.300000 rad

Sampling time: time = 1.000000 s

Relative Std. Dev. of jitter: sigJ = 0.000000e+00

Number of bits in ADC convertor: bits = 16 bits

Peak full-scale signal amplitude: A = 65536.000000 counts

Relative frequency offset: epsF = 0.000000e+00

Relative additive noise: sigN = 0.000000e+00

SigN Ax/A AX/A A/NAD Ax/NADx AX/NADX

Sampling time: time = 1.000000 s

1.000000e-09 1.000008e+00 1.000008e+00 9.168113e+04 1.726374e+05 1.726348e+05

1.000000e-08 1.000008e+00 1.000008e+00 9.168113e+04 1.726374e+05 1.726348e+05

1.000000e-07 1.000008e+00 1.000008e+00 9.168113e+04 1.726374e+05 1.726348e+05

1.000000e-06 1.000004e+00 1.000004e+00 7.858388e+04 9.634392e+04 9.634289e+04

1.000000e-05 1.000001e+00 1.000001e+00 5.865573e+04 5.920136e+04 5.920065e+04

1.000000e-04 1.000000e+00 1.000000e+00 9.956720e+03 9.956699e+03 9.956558e+03

1.000000e-03 1.000004e+00 1.000004e+00 9.993095e+02 9.993117e+02 9.992981e+02

1.000000e-02 9.999738e-01 9.999738e-01 1.000393e+02 1.000364e+02 1.000350e+02

1.000000e-01 1.000136e+00 1.000136e+00 1.000255e+01 1.000391e+01 1.000385e+01

1.000000e+00 1.002425e+00 1.002425e+00 1.002177e+00 1.004601e+00 1.004592e+00

Sampling time: time = 10.000000 s

1.000000e-09 1.000008e+00 1.000008e+00 9.168168e+04 1.726398e+05 1.726348e+05

1.000000e-08 1.000008e+00 1.000008e+00 9.168168e+04 1.726398e+05 1.726348e+05

1.000000e-07 1.000008e+00 1.000008e+00 9.168168e+04 1.726398e+05 1.726348e+05

1.000000e-06 1.000004e+00 1.000004e+00 7.861966e+04 9.643945e+04 9.628322e+04

1.000000e-05 1.000001e+00 1.000001e+00 5.861457e+04 5.915715e+04 5.900965e+04

1.000000e-04 1.000000e+00 1.000001e+00 9.911960e+03 9.911952e+03 9.912022e+03

1.000000e-03 1.000000e+00 1.000002e+00 1.000280e+03 1.000280e+03 1.000298e+03

1.000000e-02 1.000001e+00 1.000019e+00 9.996103e+01 9.996105e+01 1.001045e+02

1.000000e-01 9.999952e-01 9.996796e-01 9.989817e+00 9.989760e+00 9.951990e+00

1.000000e+00 1.000582e+00 1.002793e+00 9.991335e-01 9.997149e-01 1.000885e+00

Sampling time: time = 100.000000 s

1.000000e-09 1.000008e+00 1.000008e+00 9.168104e+04 1.726400e+05 1.726348e+05

1.000000e-08 1.000008e+00 1.000008e+00 9.168104e+04 1.726400e+05 1.726348e+05

1.000000e-07 1.000008e+00 1.000008e+00 9.168098e+04 1.726392e+05 1.726348e+05

1.000000e-06 1.000004e+00 1.000004e+00 7.865024e+04 9.652275e+04 9.607787e+04

1.000000e-05 1.000001e+00 1.000002e+00 5.866650e+04 5.921463e+04 5.930748e+04

1.000000e-04 1.000000e+00 1.000000e+00 9.926256e+03 9.926255e+03 9.933983e+03

1.000000e-03 9.999997e-01 9.999979e-01 9.999839e+02 9.999835e+02 9.994964e+02

1.000000e-02 9.999988e-01 9.999535e-01 9.997583e+01 9.997570e+01 9.980950e+01

1.000000e-01 9.999636e-01 1.000171e+00 1.000233e+01 1.000197e+01 9.996901e+00

1.000000e+00 1.000203e+00 1.004682e+00 9.998174e-01 1.000020e+00 1.004140e+00
Appendix B: NAD Metrics as a Function of Simulation Time 34 

Table 2: Amplitudes (A, Ax, and AX), and SINADs (A/NAD, Ax/NADx,and AX /NADX) as a function 

of σJ and time. 

SINAD_vs_Time_2015_07_14_sigJ run at 2015-7-14T10:47

Target sampling frequency: fs = 100000.000000 Hz

Target sinewave frequency: f = 20000.000000 Hz

Target sinewave phase: phi = 0.300000 rad

Sampling time: time = 1.000000 s

Relative Std. Dev. of jitter: sigJ = 0.000000e+00

Number of bits in ADC convertor: bits = 16 bits

Peak full-scale signal amplitude: A = 65536.000000 counts

Relative frequency offset: epsF = 0.000000e+00

Relative additive noise: sigN = 1.000000e-20

SigJ Ax/A AX/A A/NAD Ax/NADx AX/NADX

Sampling time: time = 1.000000 s

1.000000e-09 1.000008e+00 1.000008e+00 9.168113e+04 1.726374e+05 1.726348e+05

1.000000e-08 1.000008e+00 1.000008e+00 9.168113e+04 1.726374e+05 1.726348e+05

1.000000e-07 1.000008e+00 1.000008e+00 9.168113e+04 1.726374e+05 1.726348e+05

1.000000e-06 1.000006e+00 1.000006e+00 8.284128e+04 1.103313e+05 1.103297e+05

1.000000e-05 1.000003e+00 1.000003e+00 5.206340e+04 5.269339e+04 5.269293e+04

1.000000e-04 9.999997e-01 9.999997e-01 7.915009e+03 7.915255e+03 7.915188e+03

1.000000e-03 9.999923e-01 9.999923e-01 7.925129e+02 7.925277e+02 7.925259e+02

1.000000e-02 9.999237e-01 9.999237e-01 7.960788e+01 7.960329e+01 7.960213e+01

1.000000e-01 9.919425e-01 9.919425e-01 7.974859e+00 7.926913e+00 7.926838e+00

Sampling time: time = 10.000000 s

1.000000e-09 1.000008e+00 1.000008e+00 9.168168e+04 1.726398e+05 1.726348e+05

1.000000e-08 1.000008e+00 1.000008e+00 9.168168e+04 1.726398e+05 1.726348e+05

1.000000e-07 1.000008e+00 1.000008e+00 9.168168e+04 1.726398e+05 1.726348e+05

1.000000e-06 1.000006e+00 1.000006e+00 8.277603e+04 1.100729e+05 1.104472e+05

1.000000e-05 1.000003e+00 1.000003e+00 5.220942e+04 5.283870e+04 5.287260e+04

1.000000e-04 1.000000e+00 1.000000e+00 7.922270e+03 7.922266e+03 7.919426e+03

1.000000e-03 1.000000e+00 1.000002e+00 7.962786e+02 7.962778e+02 7.956851e+02

1.000000e-02 9.999256e-01 9.999166e-01 7.964489e+01 7.964037e+01 7.976714e+01

1.000000e-01 9.920834e-01 9.921038e-01 7.971927e+00 7.924611e+00 7.944234e+00

Sampling time: time = 100.000000 s

1.000000e-09 1.000008e+00 1.000008e+00 9.168104e+04 1.726400e+05 1.726348e+05

1.000000e-08 1.000008e+00 1.000008e+00 9.168104e+04 1.726400e+05 1.726348e+05

1.000000e-07 1.000008e+00 1.000008e+00 9.168104e+04 1.726400e+05 1.726348e+05

1.000000e-06 1.000006e+00 1.000006e+00 8.278869e+04 1.101255e+05 1.101523e+05

1.000000e-05 1.000003e+00 1.000003e+00 5.225219e+04 5.287804e+04 5.271312e+04

1.000000e-04 1.000000e+00 1.000000e+00 7.922765e+03 7.922766e+03 7.925654e+03

1.000000e-03 9.999993e-01 1.000000e+00 7.959639e+02 7.959634e+02 7.962958e+02

1.000000e-02 9.999196e-01 9.999474e-01 7.958379e+01 7.957901e+01 7.964762e+01

1.000000e-01 9.921077e-01 9.925145e-01 7.970749e+00 7.923536e+00 7.861342e+00
Appendix B: NAD Metrics as a Function of Simulation Time 35 

Table 3: Amplitudes (A, Ax, and AX), and SINADs (A/NAD, Ax/NADx,and AX /NADX) as a function 

of EF and time. 

SINAD_vs_Time_2015_07_14_epsF run at 2015-7-14T13:1

Target sampling frequency: fs = 100000.000000 Hz

Target sinewave frequency: f = 20000.000000 Hz

Target sinewave phase: phi = 0.300000 rad

Sampling time: time = 1.000000 s

Relative Std. Dev. of jitter: sigJ = 0.000000e+00

Number of bits in ADC convertor: bits = 16 bits

Peak full-scale signal amplitude: A = 65536.000000 counts

Relative frequency offset: epsF = 0.000000e+00

Relative additive noise: sigN = 1.000000e-20

epsF Ax/A AX/A A/NAD Ax/NADx AX/NADX

Sampling time: time = 1.000000 s

1.000000e-16 1.000008e+00 1.000008e+00 9.168113e+04 1.726374e+05 1.726348e+05

1.000000e-14 1.000008e+00 1.000008e+00 9.168113e+04 1.726374e+05 1.726348e+05

1.000000e-12 1.000008e+00 1.000008e+00 9.168113e+04 1.726340e+05 1.726348e+05

1.000000e-10 9.999971e-01 9.999971e-01 7.067603e+04 8.690616e+04 9.020434e+04

1.000000e-08 1.000000e+00 9.999999e-01 1.378113e+03 8.036879e+04 2.755136e+03

1.000000e-06 1.000000e+00 9.993427e-01 1.378864e+01 8.026442e+04 2.755476e+01

Sampling time: time = 10.000000 s

1.000000e-16 1.000008e+00 1.000008e+00 9.168168e+04 1.726398e+05 1.726348e+05

1.000000e-14 1.000008e+00 1.000008e+00 9.168168e+04 1.726397e+05 1.726348e+05

1.000000e-12 1.000006e+00 1.000008e+00 8.562573e+04 1.234753e+05 1.726348e+05

1.000000e-10 9.999994e-01 9.999971e-01 1.346802e+04 8.183719e+04 9.020434e+04

1.000000e-08 1.000000e+00 9.999999e-01 1.378325e+02 8.026292e+04 2.755136e+03

1.000000e-06 1.000000e+00 9.993427e-01 1.433928e+00 8.031384e+04 2.755476e+01

Sampling time: time = 100.000000 s

1.000000e-16 1.000008e+00 1.000008e+00 9.168104e+04 1.726400e+05 1.726348e+05

1.000000e-14 1.000008e+00 1.000008e+00 9.168104e+04 1.726367e+05 1.726348e+05

1.000000e-12 9.999971e-01 1.000008e+00 7.067575e+04 8.690867e+04 1.726348e+05

1.000000e-10 1.000000e+00 9.999971e-01 1.378116e+03 8.037023e+04 9.020434e+04

1.000000e-08 1.000000e+00 9.999999e-01 1.378866e+01 8.026561e+04 2.755136e+03

1.000000e-06 1.000000e+00 9.993427e-01 7.071067e-01 8.031513e+04 2.755476e+01
Appendix B: NAD Metrics as a Function of Simulation Time 36 

Table 4: Amplitudes (A, Ax, and AX), and SINADs (A/NAD, Ax/NADx,and AX /NADX) as a function 

of Number of Bits and time. 

SINAD_vs_Time_2015_07_14_sigJ run at 2015-7-14T18:17

Target sampling frequency: fs = 100000.000000 Hz

Target sinewave frequency: f = 20000.000000 Hz

Target sinewave phase: phi = 0.300000 rad

Sampling time: time = 1.000000 s

Relative Std. Dev. of jitter: sigJ = 0.000000e+00

Number of bits in ADC convertor: bits = 16 bits

Peak full-scale signal amplitude: A = 65536.000000 counts

Relative frequency offset: epsF = 0.000000e+00

Relative additive noise: sigN = 1.000000e-20

Bits Ax/A AX/A A/NAD Ax/NADx AX/NADX

Sampling time: time = 1.000000 s

8.000000e+00 9.978583e-01 9.978583e-01 4.107682e+02 1.178420e+03 1.178402e+03

1.600000e+01 1.000008e+00 1.000008e+00 9.168113e+04 1.726374e+05 1.726348e+05

2.400000e+01 1.000000e+00 1.000000e+00 1.974972e+07 2.369203e+07 2.369167e+07

3.200000e+01 1.000000e+00 1.000000e+00 6.851407e+09 7.318231e+09 7.286117e+09

4.000000e+01 1.000000e+00 1.000000e+00 1.358524e+12 5.974162e+11 1.886386e+11

4.800000e+01 1.000000e+00 1.000000e+00 3.447107e+14 6.599051e+11 1.902051e+11

5.600000e+01 1.000000e+00 1.000000e+00 1.967085e+17 6.599082e+11 1.902053e+11

6.400000e+01 1.000000e+00 1.000000e+00 Inf 6.599082e+11 1.902053e+11

Sampling time: time = 10.000000 s

8.000000e+00 9.978583e-01 9.978583e-01 4.107700e+02 1.178436e+03 1.178402e+03

1.600000e+01 1.000008e+00 1.000008e+00 9.168168e+04 1.726398e+05 1.726348e+05

2.400000e+01 1.000000e+00 1.000000e+00 1.974595e+07 2.369233e+07 2.369167e+07

3.200000e+01 1.000000e+00 1.000000e+00 5.567443e+09 6.404409e+09 6.832991e+09

4.000000e+01 1.000000e+00 1.000000e+00 1.347659e+12 3.663300e+10 1.741727e+11

4.800000e+01 1.000000e+00 1.000000e+00 3.447085e+14 3.664569e+10 1.752944e+11

5.600000e+01 1.000000e+00 1.000000e+00 1.967561e+17 3.664569e+10 1.752945e+11

6.400000e+01 1.000000e+00 1.000000e+00 Inf 3.664569e+10 1.752945e+11

Sampling time: time = 100.000000 s

8.000000e+00 9.978583e-01 9.978583e-01 4.107702e+02 1.178437e+03 1.178402e+03

1.600000e+01 1.000008e+00 1.000008e+00 9.168104e+04 1.726400e+05 1.726348e+05

2.400000e+01 1.000000e+00 1.000000e+00 1.976388e+07 2.368886e+07 2.369167e+07

3.200000e+01 1.000000e+00 1.000000e+00 5.100547e+09 2.144688e+09 6.832991e+09

4.000000e+01 1.000000e+00 1.000000e+00 1.346720e+12 2.379477e+09 1.741727e+11

4.800000e+01 1.000000e+00 1.000000e+00 3.446705e+14 2.379480e+09 1.752944e+11

5.600000e+01 1.000000e+00 1.000000e+00 1.968036e+17 2.379480e+09 1.752945e+11

6.400000e+01 1.000000e+00 1.000000e+00 Inf 2.379480e+09 1.752945e+11