effect of adc quantization errors on some periodic signal measurements

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-36, NO.4, DECEMBER 1987 983

Effect of ADC Quantization Errors on Some PeriodicSignal Measurements

MAHMOUD FAWZY WAGDY, MEMBER, IEEE

(3)

where e(m) is the quantization error, which, for mostpractical cases, is modeled as a random variable e, uniformly distributed in amplitude between ±~/2 as shownin Fig. 1. The noise power, or variance, of e is given by[1]

Abstract-Effect of analog-to-digital converter (ADC) quantizationerrors on some periodic signal measurements, namely the 'de component and the amplitudes and phases of harmonically related sinusoids,are investigated. These parameters are determined via the Fast FourierTransform (FFT) algorithm, which is used for manipulating the quantized samples in a data-acquisition system.

Mean values and variances of the measured parameters are derived.Then, upper bounds on measurement errors for any confidence limitsand worst-case errors are obtained. Finally, the reduction of measurement errors is briefly discussed.

I. INTRODUCTION

ADC QUANTIZATION errors affect measurement ac. curacies in sampled-data-acquisition systems. For a

signal dynamic range of 2D (peak-to-peak) and a k-bitADC, the quantization step size ~ is equal to one leastsignificant bit (LSB) of the digital quantity, and given by

.:l = ~~ = r(K-1)D. (1)

The ADC quantized output S Q for the m th data sample isgiven by

IX

_A. 0 1:a.. e2 2

Fig. 1. Probability density function of quantization error.

quantization noise) on the output coefficients of an N-pointFFT (N large) has been shown to have a Gaussian distribution.

Effects of quantization on multiparameter signal measurements are of interest and importance to system designers. Accuracies of analog measurements have beeninvestigated [6] via the interpolated FFT along with floating-point arithmetic. The effects of noise and jitter on theestimation of power spectrum and frequency errors [7]have also been investigated. However, not much work hasbeen done to study the statistical properties of multiparameter signal measurement errors due to ADC quantization. It is the intention of this paper to fill some of thesegaps.

The paper starts by determining the mean values andvariances of amplitudes and phase angles and of the sinusoidal components, and the de component as well.Then, upper bounds on measurement errors of these parameters for any probability (confidence limit) will be determined . Worst-case measurement errors are also estimated. At the end two methods for reducing measurementerrors are suggested; the first is based on having a nonintegral average number of samples per cycle, and thesecond is based on employing dither.

(2)SQ(m) = S(m) + e(m)

The impact of digital signal processing on instrumentation systems has greatly increased in the last ten years.In a multiparameter signal measurement system, quantized data samples are processed via the FFT algorithm.Effects of sample quantization and finite word length onthe Fourier coefficients after a FFT have been extensivelyinvestigated in the literature [2]-[4] in terms of noise-tosignal power ratio (NSR) at the FFT output bins. The FFTalgorithm can be carried out exactly using today's computers; in other words, finite word-length effects will benegligible. Very recently [5], the effect of noise (e.g.,

Manuscript received April 28, 1987; revised July 15, 1987.The author is with the Department of Electrical Engineering, University

of Lowell, Lowell, MA 01854.IEEE Log Number 8716874.

II. MEANS AND VARIANCES OF MEASURED PARAMETERS

The time-domain signal under consideration is given by

V(t) = Y(0) + ~ Y(n) · sin [nwt + (n )] (4 )n~1

where n is a finite integral frequency index and Y(0) isthe de component. To determine the amplitudes andphases we sample V(t), satisfying the Nyquist criterion,then use an N-point DFT for the quantized samples. Thenth Fourier coefficient G( n) is given by

N-l

G(n) = ~ SQ(m)m=O

· exp l-j (~ nm ) l 0 $ n $ N - 1 (5)

0018-9456/87/1200-0983$01.00 © 1987 IEEE

984 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-36, NO.4, DECEMBER 1987

Since e ( i ) and e ( j ) are uncorrelated for i =1= j, then thefirst term disappears. Also, E[ e2( h ) ] = a;, and N is typically a power of 2, thus, (9) reduces to

where SQ (m) is the m th quantized time-domain data sample given by (2). The DFT introduces a scale factor (N/2)for G(n), 1 :::; n -s N - 1, and a scale factor (N) forthe de coefficient G ( 0 ).

Solving (2) and (5) for the Fourier coefficients G(n), 1-s n :5 (N /2 ), the amplitudes and phase angles of thesinusoidal constituents are, respectively, given by

(15)

(13)

(14)

(12)

a2E(Y) ~ Yl +~2YI

d 2

(18)= Yl +--12NYt

Y l = .JA 2 + B 2

~I = tan-I (~)

The means and variances of the measured parameters aredetermined as follows:

A. Amplitude Measurement

The amplitude Y of a sinusoidal component is given by

22 2 2 a e

aXl = aX2 = ax = (N/2)·

For the generality of the analysis of the spectral constituents of the signal let us denote Y(n ), ell (n ), A (n ), B (n ),Xl (n), and X2(n) by Y, ell, A, B, Xl, and X2, respectively. Let us also denote the quantization-free amplitudeand phase by Yl and ell 1, respectively, i.e.,

(17)

From, (17) along with (3), (14), and (15) it can be shownthat

The next step would be to determine the variances ofXl (n) and X2(n). Since e(m), 0 -s m ~ N - 1, of (8)are uncorrelated, then

2 a; N~l 2 (27rnm)aXl(n) = 2 L.J cos --

(N/2) m=O N

= a; r~ + ! N~I cos (47rnm)l(NI2)2 L2 2 m=O N' (11)

Y = .J(A + X1)2 + (B - X2)2. (16)

Using (A3), the mean value of Y is given by

E(Y) ~ [Y + ~ (all a~;2 + a12 a~;2)ll .Xl,X2 =0

which indicates that E( Y) is biased. For an 8-bit bipolarADC with reference voltages of ±5.12 V, i.e., d = 40mV, the change of the normalized mean E ( Y) / Yl versusY l for different values of N is plotted in Fig. 2.

For a radix-2 FFT, the second term disappears, thus2

2 a e

aXl(n) = (N/2)·

Similarly, it can be shown that2

2 a e

aX2(n) = (N/2)·

Since (12) and (13) do not depend on the frequency indexn, let us denote ail (n) and ai2(n) by ail and ai2' respectively. We can thus define

(8)

(7)

(6)

(10)

. (9)

E [Xl (n) · X2 (n)]

a; N~I. (47rnh)= 2(N12)2 h=O SIll N = O.

Using (10) along with the fact that E [Xl (n )] = 0 andE[X2 (n)] = 0, we conclude that Xl (n) and X2 (n) areuncorrelated.

Y(n) = .J[A(n) + Xl(n)t + [B(n) - X2(n)]2

iJ,.( ) -1 [B(n) - X2(n)l':l' n =tan

A(n) + X1(n)

A complete knowledge of the statistical properties of themeasured parameters Y( n) and ell (n) would be possibleonly if the probability density functions (pdf's) of theseparameters are evaluated. However, this is very complicated because Y (n) and ell (n) are functions of two random variables Xl (n) and X2 (n ), which requires usingthe Jacobian Transformation [8]. An easier and still highlyaccurate approach to determine the Y ( n ) and ell ( n ) meansand variances, which are sufficient for most practical purposes, would utilize the Taylor series-based analysis givenin the appendix.

It should be noted that we first have to prove that Xl (n)and X2 (n) are uncorrelated, as follows:

E[X1(n) · X2(n)]

= (N;2)2 E t~; ~~: e(i)

. e(j) cos (2;i) sin (2':j)+ N~I e2(h) cos (27rnh) sin (27rnh)

h=O N N

where A(n) and B(n) are the frequency-domain quantization-free deterministic real and imaginary parts, respectively, and Xl (n ) and X2 (n ) are random variables (RV' s)given by

1 N-l (27rnm)Xl(n) = (NI2) m~o e(m) · cos J:.I

1 N-l (27rnm)X2(n) = (NI2) m~o e(m) · sin J:.I

WAGDY: EFFECT OF ADC QUANTIZATION ERRORS ON SIGNAL MEASUREMENTS 985

(27)

(26)

(25)2

2 axaeJ> =2·Y l

ax dact> = - = --- rad

Y l y l J6Nwhich importantly indicates that the rms phase error isinversely proportional to the amplitude of the sinusoid .The change of act> versus Y l for N = 4, 16,64,·256 withd = 40 mV is illustrated in Fig. 3.

The question now is: what if we are measuring the phasedifference <1>/ between two independent sinusoids whoseactual (quantization-free) phases are a and <l>b' while theiractual amplitudes are Ya and Yb? It can be shown that

Thus

where 1 is given by (15), which indicates that E ( <1» isunbiased.

It is interesting to mention that truncating the Taylorseries representation of (23) after the first- or second-order terms gives the same result for the variance of <1>,namely

MILLI-VOLT (rnv)

N=16

N=64

N=4

0

u:.-

l1'l

~

~

..--<~

r-<,

a~~

r-

~ lJ)(Y)

Z....:ITWL:g

0"":wNlJ)>--<N

.-JIT-L:CCoONZ·

W0lJ)~-:f--.---..-J oeL_L:rr-

lJ)0

-g-

101 . '1(1 103

RCTURL RMPL I TUDE (Y 1)

Fig. 2. Plot of E ( Y) / Y1 versus Y1•

(22)

(30)

(28)2 2 ( 1 1 )a</>j = ax Y~ + Y~ .

The variance of DC is given by

abc = E{[DC - E(Dc)]2} = ~2 E[[t: e(m)TJ.(31)

It can be shown that

It is readily obvious that1 N-l

E(DC) = - ~ S(m).N m=O

C. DC Component Measurement

When measuring the de component Y(O), denoted DC,the scale factor of (5) is N, thus

1 N-l

DC = - ~ [S(m) + e(m)]. (29)N m=O

and

(21)

The error in using (21) as opposed to (20) is negligible aslong as ax « 2Yl . For a 4-point FFT, and Y l = 2d (say),the error in a} is less than 0.27 percent, which means that(21) is acceptable for most practical purposes . Now, using(3) and (14), (21) can be rewritten as

day ~ J6N

To find the variance of Yusing a Taylor series truncatedafter the second-order term, we first use (AS) to determineE ( y 2

). It can be shown that

E(y2) = yj + 2ai. (19)

Using (A4) along with (18) and (19) it can be shown that4

2 2 ax ( )a y ~ ax - 4 2· 20Y l

Alternatively, using (A7) to determine a} using a Taylorseries truncated after the first-order term leads to

(24)

(23)

(33)

(32)

III. UPPER BOUNDS ON MEASUREMENT ERRORS

One of the ways for expressing measurement accuraciesis to determine the maximum probable measurement errors for any confidence limit (i.e., probability p). For ameasured parameter R (where R stands for Y, <1>, or DC )the exact value of this error €R is obtained from

B. Phase Measurement

The phase of a sinusoidal component is given by

(B- X2)ep = tan -I A + Xl .

Using (A3) to find the mean value of <1>, it can be shownthat

which means that ay is independent of Y. When the biasof (18) is negligible, ay represents the root-mean-square(rms) amplitude error.

oo

~ DEGREES

oo

oo

o

.: mv'-Uo

o

N=4

N=16

----I:=>10

0 ' I--+-----ri------,-I-------.-- -, I j I I I I0.00 0.10 0.20 0.30 0.40 0.50 O.GO 0.70 0.80 0.90 1.00

PROBRBILITY (P)

Fig. 4. Upper bounds on amplitude errors as function of confidence percent.

\N=16

\~G\~~'~~"I N=~5~ I ~~~~~~~~:~~,~iiiiii~iiiiii;iiiii~ 1 ~;'ll

11(f' 103 1CReT UR~ ri 'v1 PL ~ U0E ('( 11

Fig. 3. Variation of rms phase error with amplitude.

oo

'"

oo

LO

where R I is the quantization-free value of R, and fR(R )is the pdf of R. However, since our analysis employsvariances, and not pdf's, (33) will not be used here. Instead of the exact value of ER' we will derive an upperbound. It is obvious that when p = I, the problem boilsdown to determining the worst-case measurement errorsin R. The details of the new approach are given below asfollows:

oo

(\J DEGREES

oOJ

w(j)

cr~::r:: •CL.~

aRER < (36)

- ~l - P(R - ER < R < R + ER)

2

P(R+ER<R<R-ER)~af (34)ER

(37)

~=200 mV

~::I V

o=t'

o

oC\.J

o

go

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00PROBRB I L I T'I lPJ

Fig. 5. Upper bounds on phase errors as function of confidence percent.

B. Worst-Case Measurement Errors

We start by calculating the worst-case values of Xl (n)and X2(n) of (8). Since the convolution of RV's produces a new RV whose range of values is the sum of theindividual ranges of the constituent RV's, then it can beshown that

Xl (n) I - + A/2 N~I Icos (21rmn) Iworst-case - N /2 m=O N

X2(n) I - + A/2 N~I Isin (21rmn) I"

worst-case - N /2 m = 0 N

(35)= I - P(R - ER < R < R + ER)

then from (34) and (35) we get

A. Upper Bounds on Measurement Errors for AnyConfidence Limits

Chebyshev's inequality [9] states that regardless of theshape of fR(R ) we have

where R = E (R ), and P denotes probability. Now, since

P(R + ER < R < R - ER)

Figs. 4 and 5 illustrate the changes of Ey and Eel> with theconfidence limits P( Y - Ey < Y < Y + €y) and P( < ), respectively.

It is obvious that by letting peR - ER < R < R + ER )

= 1 in (36) we have ER -s 00, which means that we shouldresort to another method for worst-case analysis.

Since N is a power of 2, the worst-case values of Xl (n) f). Thusand X2 (n) are equal, and X ( n) will be denoted by ±C.When n = 1 it can be shown, using finite series tables[8], that

MNT= j' (45)

Neglecting the second-order term of (AI) leads to neglecting the second term of (39). Also, by letting IXIIIX21 = C, we get

A · Xl - B · X2y- y -- ------

1 -- (A 2 + B 2 ) 1/ 2

2+ (B · Xl + A · X2) (39)

(A2 + B 2 )3/2

For example, we get C = 0.5d, 0.604d, 0.628d,0.636Id, 0.63659d for N = 4, 8, 16, 64, and 256, respectively. It can also be shown that the upper bound onC, when N is very large, for all n, is (2d/7r), i.e.,0.63662d. C will be used in the estimation of worst-casemeasurement errors as follows:

1) Worst-Case Amplitude Error Wy : From (16) and(AI), it can be shown that

This can be achieved via phase-locked loops or digitalfrequency multipliers [11].

In the special case when (N / M) is an integer, and assuming that all noise sources in the measuring system arenegligible compared to ADC quantization noise, thequantized data sequence will be (N / M) periodic, and(N / M) points will only be needed for the FFT, thus reducing measurement accuracies. Also, if some spectralcomponents are missing in the range 1 ~ n < (N /2) in(4), this reduces the randomness of quantization errors andconsequently reduces measurement accuracies. To illustrate this last problem with an example, let M = 1, i.e.,let N be the number of samples per cycle, and let the signal be just a single sinusoid. Thus

(47)

(46)

e(m) = -e(m + ~).

SCm) = -s(m + ~).

Consequently

(38)2d (II)X( I) = ±c = ±N cot N ·

3) Worst-Case DC Voltage Error: It is readily obvious, from (29), that

2) Worst-Case Phase Error W¢: From (23) and thefirst-order approximation of (AI), we get

(49)

(48)

(50)

NM =1= integer.

Mn(max) < (~).

Substituting (47) into (8), it ultimately can be shown that

2) Using Dither: Dither signals have been successfullyused in processing visual and speech signals [12], but notextensively used in sampled-data signal measurements. Todecorrelate quantization errors of analogous samples,dither may be added to the sampled data before quanti-

which means that ai has doubled.To avoid the above problems, two methods are sug

gested:1) Using a Nonlntegral Average Number of Samples

Per Cycle: To avoid the dependency of the quantizationerrors of some samples on the quantization errors of othersamples, as given by (47), we must select M and N suchthat

Taking the FFT points from many cycles, i.e., M > 1,requires a smaller sampling frequency than in the case ofM = 1, which allows sampling a periodic signal withhigher frequencies. It should be noted that a harmoniccomponent with frequency nf corresponds to the (Mn )th

frequency index of the N-point FFT. Thus, to performmultiparameter measurements for all harmonic constituents, we should satisfy

(40)

(43)

(44)d

2

JicWq, = ---y;- rad.

which is a very accurate result as long as C « Y1• It canbe shown that the worst-case value of IY - Y1 Imax occurswhen IA I = IB I, thus

Wy ~ -Ii C. (41)

B · Xl + A · X2~ - ~l ~ - 2 2 (42)

A + B

IV. REDUCING MEASUREMENT ERRORS

In order to measure the above signal parameters for acontinuous (nondiscrete) frequency range, while avoidingleakage effects [10], the observation time NT, Tbeing thesampling interval, must be an integral multiple M of theduration of the fundamental component (with frequency

Again, by letting IXII = IX21 = C, it can be shownthat


a result that is independent of the signal harmonic composition. Obviously, a; can be reduced to any arbitrarylimit by increasing N.

zation via the ADC. For effective dithering we may use,say a ±~/2 uniform dither with a pdf similar to that ofe, i.e., a~ = a;. In general, the overall noise variancea; of any quantized sample would be given by

(A5)

(A3)

(A2)

(AI)

Z = [(Xl, X2) = [(ex, (3)

+ ~ I (Xl - o ) Of(Ci, (3) + (X2 - (3) Of(Ci, (3)JI! L aXI aX2

+ ~ I (Xl _ Ci)2 02f(Ci, (3)2! L aXl 2

+ 2(XI - Ci)(X2 - (3) O'i(Ci, (3)aXI aX2

+ (X2 - (3)2 a2[( ex, (3)J + ...aX2 2 .

Assuming that Xl and X2 are uncorrelated, then E(XI ·X2) = E(XI) · E(X2) = O. Also, since ex = 0, (3 = 0then E(XI 2) = ail and E(X2 2)

= ai2' where all andai2 are the variances of Xl and X2, respectively. Consequently, (A2) reduces to

E(Z) ~ [(0 0) + ! I02f(0, 0) 2, 2 L aXl 2 aXI

a2j (0, 0) 2 J

+ aX2 2 aX2'

a~ = E{[Z - E(Z)]2} = E{[Z - f(O, 0)]2}

"'" EnXl of(O, 0) + X2 of(O, 0)l2] (A6)cL aXI aX2'

The variance of Z is given by

a~ = E(Z2) - [E(Z)( (A4)

To determine E(Z2), let Z2 be defined as X(XI, X2).Following a similar analysis to (AI) through (A3) leadsto

Assuming that Xl and X2 have zero means, i.e., ex = 0,(3 = 0, the mean or expected value of Z, i.e., E(Z), isgiven by

E(Z) = f(O, 0) + ~ l02~~;2°) E(XI 2)

a2f (0 0)+ 2 'E(XI · X2)

aXI aX2

+ 02f(0, 0) E(X2 2)J + ...aX2 2 .

The above equations ignore Taylor series terms beyondthe second-order one, which seemed to be an excellentapproximation throughout the paper. There is, however,a simpler formula for a~, which is derived by ignoringTaylor series terms beyond the first one in (AI) as follows:

(51)

(52)

(53)

222as = ae + ad·

2 2 22 aa ae + adax = (NIM)/2 = (NI2)

Finally, we work with an (NI M )-point FFT, thus

Although it may be difficult to prove that ergodicityconstitutes a reasonable assumption in any physical situation [8], we shall assume that quantization noise, afterdithering, is ergodic to utilize the fact that temporal characteristics are similar to the statistical characteristics. Byaveraging every M analogous samples throughout the Ndata points (NI M = integer), the variance a~ of the estimate of the average sample will be given by

V. CONCLUSIONS

Effects of ADC quantization errors on measuring theamplitude and phase spectra of periodic signals containing a finite number of harmonically related sinusoids havebeen investigated. It was shown first that amplitude meansare biased, which slightly affects measuring amplitudesapproximately below ~, whereas phase means are unbiased. Then, it was interestingly shown that amplitudevariances are almost constant with changing amplitudes,while phase variances are inversely proportional to amplitudes. In both amplitude and phase measurements, theroot-mean-square error is directly proportional to ~, andinversely proportional to IN. Use was made of the abovevariances to determine upper bounds on measurement errors for a given confidence limit, i.e., probability p. Because of the meaningless upper bound obtained when p =I, approximate worst-case measurement errors were separately derived. All the above analysis has been appliedto the de component as well. Finally, two methods forreducing measurement errors were considered: one isbased on having a nonintegral average number of samplesper cycle, and the other is based on employing dither todecorrelate analogous samples from different cycles before temporally averaging them.

ApPENDIX

STATISTICAL ANALYSIS OF A FUNCTION WITH RANDOM

VARIABLES U SING TAYLOR SERIES

Let Z be a function of two random variables Xl and X2with mean values ex and {3, respectively. Then


REFERENCES

Since E(XI · X2) = 0, a = 0, and {3 = 0, (A6) reduces to

[1] w. R. Bennett, "Spectra of quantized signals," Bell. Syst. Tech. J.,vol. 27, pp. 446-472, July 1948.

[2] P. D. Welch, "A fixed-point fast Fourier transform error analysis,"IEEE Trans. Audio Electroacoust., vol. AU-17, no. 2, pp. 151-157,June 1969.

[3] A. V. Oppenheim and C. J. Weinstein, "Effects of finite register

ACKNOWLEDGMENT

The author would like to thank Kuo-Liang Lo, a graduate student at Northeastern University, for carrying outthe computer programming associated with the figures ofthis paper.

length in digital filtering and the fast Fourier transform, " Proc. IEEE,vol. 60,no. 8, pp. 957-976, Aug. 1972.

[4] D. V. James, "Quantization errors in the fast Fourier transform,"IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-23, no.3, pp. 277-283, June 1975.

[5] J. Schoukens and J. Renneboog, "Modeling the noise influence onthe Fourier coefficients after a discrete Fourier transform," IEEETrans. Instrum. Meas., vol. IM-35, no. 3, pp. 278-286, Sept. 1986.

[6] V. K. Jain, W. L. Collins, Jr., and D. C. Davis, "High-accuracyanalog measurements via interpolated FFT," IEEE Trans. InstrumMeas., vol. IM-28, no. 2, pp. 113-122, June 1979.

[7] F.-I. Tseng and T. K. Sarkar, "Tolerance of spectral estimation,"IEEE Trans. Instrum Meas., vol. IM-32, no. 4, pp. 484-490, Dec.1983.

[8] P. Z. Peebles, Jr., Probability, Random Variables, and Random Signal Principles, 2nd ed. New York: McGraw-Hill, 1987.

[9] M. Eisen, Introduction to Mathematical Probability Theory. NewYork: Pretice-Hall, 1969.

[10] N. Ahmed and T. Natrajan, Discrete Time Signals and Systems.Reston, VA: Reston, 1983.

[11] K. Muniappen and R. Kitai, "Digitial frequency multiplier for spectrum measurement of periodic signals," IEEE Trans. Instrum. Meas.,vol. 29, no. 3, pp. 195-198, Sept. 1980.

[12] J. Vanderkooy and S. P. Lipshitz, "Resolution below the least significant bit in digital systems with dither," J. Audio Eng. Soc., vol.32, no. 3, pp. 106-113, Mar. 1984.

(A7)2(JX2·

2 [a!(o, 0)12

(JX} + aX2~ [a!(o, 0)12

aXl2

(Jz

effect of adc quantization errors on some periodic signal measurements

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