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Choice of the acquisition parameters for frequency estimation of a sine wave by interpolated DFT method
Computer Standards & Interfaces 31 (2009) 962–968
Contents lists available at ScienceDirect
Computer Standards & Interfaces j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c s i
Choice of the acquisition parameters for frequency estimation of a sine wave by interpolated DFT method Daniel Belega a,⁎, Dominique Dallet b,1 a b
”Politehnica” University of Timişoara, Faculty of Electronics and Telecommunications, Bv. V. Pârvan, Nr. 2, 300223, Timişoara, Romania Laboratoire IMS, University of Bordeaux - ENSEIRB, 351 Cours de la Libération, Bâtiment A31, 33405, Talence Cedex, France
a r t i c l e
i n f o
Article history: Received 23 January 2008 Received in revised form 10 July 2008 Accepted 28 September 2008 Available online 17 November 2008 Keywords: Frequency estimation Interpolated DFT method Maximum side lobe decay windows
a b s t r a c t The accuracy of the frequency estimation of a sine wave corrupted by quantization noise using the Interpolated Discrete Fourier Transform (IpDFT) method is affected by systematic and random errors. The systematic errors are independent of quantization noise and are due to the interferences from the image parts of the sine wave spectrum. They depend of the acquisition parameter – integer number of recorded sine wave cycles. The random errors are due to the quantization noise and depend on the acquisition parameter – number of acquired samples. In this paper for the situations in which the IpDFT method uses the maximum side lobe decay windows a condition for the integer part of the number of recorded sine wave cycles is derived to ensure that the systematic errors are very small compared with the quantization errors. Also, a condition for the number of samples is derived to ensure that the absolute error of the normalized frequency due to the random errors is smaller than a desired value with a high confidence level. Carried out simulations confirm the validity of each derived condition. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Frequency estimation of a sine wave is very important in many engineering applications. The methods used for this purpose can be classified either in time-domain methods (parametric methods) or frequency-domain methods (non-parametric methods) [1]. Timedomain methods lead to high accuracy frequency estimation when the model order is well-known. Otherwise, intensive computationally algorithms are required to estimate the model order [1]. In the frequencydomain methods the spectrum of the sine wave is firstly computed by means of the Discrete Fourier Transform (DFT) which is then exploited to estimate the frequency [2–7]. These methods have the advantages of the robustness toward signal model inaccuracies and low computational effort. On the other hand they have inherent limitations such as frequency resolution and spectral leakage effects due to noncoherent sampling. A frequency-domain method which provides accurate frequency estimates in non-coherent sampling mode is the Interpolated Discrete Fourier Transform (IpDFT) method [3–10]. Its best performances are obtained using the maximum side lobe decay windows (or class I RifeVincent windows) because in this case the frequency could be estimated by analytical formulas [6–10].
Due to the digitizing process the quantization noise is always present in sampled signal. The influence of the quantization noise on the estimation of sine wave parameters by IpDFT method has been studied in scientific literature [7–9]. The accuracy of the sine wave parameters estimation is affected by systematic and random errors. The first errors type is due to the interferences from the image parts of the sine wave spectrum and depends on the acquisition parameter — integer number of recorded sine wave cycles. They are independent on the quantization noise. The second errors type is due to the quantization noise and depends on the acquisition parameter — number of acquired samples. Unfortunately, the published results in this field do not provide any mathematical condition for these acquisition parameters which must be satisfied to obtain accurate sine wave parameter estimates. The aim of this paper is to derive two mathematical conditions for the acquisition parameters – integer number of recorded sine wave cycles and number of acquired samples – in order to obtain accurate frequency estimation of a sine wave corrupted by quantization noise using the IpDFT method with maximum side lobe decay windows. The validity of each derived condition is then verified by means of computer simulations. 2. Theoretical background
⁎ Corresponding author. Tel.: +40 2 56 33 65; fax: +40 2 56 40 33 62. E-mail addresses: [email protected] (D. Belega), [email protected] (D. Dallet). 1 Tel.: +33 5 40 00 26 32; fax: +33 5 56 37 15 45. 0920-5489/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.csi.2008.09.028
Let us consider a sine wave sampled at fs frequency: f xðmÞ = A sin 2π in m + u ; fs
m = 0; 1 N ; M − 1
ð1Þ
D. Belega, D. Dallet / Computer Standards & Interfaces 31 (2009) 962–968
where A, fin and φ are respectively the amplitude, frequency and phase of the sine wave and M is the number of acquired samples. In order to satisfy the Nyquist criteria, fin is smaller than fs/2. The relationship between the frequencies fin and fs is given by: fin λ l+δ = 0 = fs M M
ð2Þ
in which l and δ are respectively the integer part and the fractional part of λ0. δ is related to the non-coherent sampling mode and −0.5 ≤ δ b 0.5. For δ = 0 the sampling process is coherent with the input sine wave (coherent sampling mode) and Eq. (2) represents the coherent sampling relationship between the frequencies fin and fs. For δ ≠ 0 the sampling process is non-coherent with the input sine wave (non-coherent sampling mode) and the coherent sampling relationship is not fulfilled. In this mode the sine wave spectrum is affected by spectral leakage errors. To reduce these errors the windowing approach is used leading to the spectral analysis of windowed signal xw (m) =x(m)·w(m), where w(m) is the window sequence. The Discrete-Time Fourier Transform (DTFT) of the xw(m) is given by: Xw ðλÞ =
M −1 X
xðmÞwðmÞe
m − j2πλM
;
λa½0; MÞ
ð3Þ
m=0
where λ represents the normalized frequency expressed in bin. After some calculus Xw(λ) becomes: i Ah ju − ju W ðλ − λ0 Þe − W ðλ + λ0 Þe ; λa½0; MÞ ð4Þ Xw ðλÞ = 2j where W(λ) is the DTFT of the window w(m). The second term in Eq. (4) represents the image part of the spectrum. δ can be very accurately estimated by the IpDFT method with the maximum side lobe decay windows [6–9]. The H-term maximum side lobe decay window (H ≥ 2) has the most rapidly decaying side lobes, equal to 6(2H − 1) dB/octave, from all the H-term cosine-class windows [11]. This window is defined by: wðmÞ =
m h ð − 1Þ ah cos 2πh ; M h=0
HX −1
m = 0; 1; N ; M − 1
ð5Þ
where ah are the window's coefficients. They have the following expressions [12]: H − 1
a0 =
H − h − 1
C2H − 2 C −2 ; ah = 2H2H 22H − 2 2 −3
;
h = 1; 2; N ; H − 1
ð6Þ
where p
Cm =
m! : ðm − pÞ!p!
For M NN 1, the DTFT of the w(m) can be approximated by [10]: W ðλÞ =
M sinðπλÞ − jπλ jMπ λ e e 22H − 2 π
ð2H − 2Þ! λ
HQ − 1
h2 − λ2
:
ð7Þ
h=1
For estimation of δ by means of IpDFT method the rapport α must be firstly evaluated: 8 j Xw ðlÞj > > ; if − 0:5Vδb0 < j Xw ðl − 1Þj ð8Þ α= j X ðl + 1Þj > > : w ; if 0Vδb0:5 j Xw ðlÞj From Eq. (4) it can be established: i A h ju − ju j j Xw ðl − 1Þj = j W ð−1 − δÞe − W ð2l − 1 + δÞe 2 i A h ju − ju j Xw ðlÞj = j W ð−δÞe − W ð2l + δÞe j 2 i A h ju − ju j Xw ðl + 1Þj = j W ð1 − δÞe − W ð2l + 1 + δÞe j: 2
963
For large l the image parts of Xw(l − 1), Xw(l) and Xw(l + 1) can be neglected and from Eq. (9) α becomes: 8 j W ðδÞj > > ; < j W ð1 + δÞj αi j W ð 1 − δ Þ j > > ; : j W ðδÞj
if − 0:5Vδb0
:
ð10Þ
if 0Vδb0:5
Based on Eq. (7) after some calculus we obtain: 8 H+δ > < ; αi H − 1 − δ H − 1 + δ > : ; H−δ
if − 0:5Vδb0
ð11Þ
if 0Vδb0:5
From the above expression it follows that δ can be estimated by: 8 ðH − 1Þα − H > < ; if − 0:5Vδb0 α+1 δˆ = Hα − H + 1 > : ; if 0Vδb0:5 α+1
ð12Þ
l is the index that corresponds to the maximum of the |Xw(λ)| for integer λ values (i.e. for λ = 0, 1,…, M − 1). This can be easily determined by means of a maximum search routine applied to these samples. From Eq. (2), it follows that the normalized frequency λ0 can be estimated by l + δˆ and the frequency fin by (l + δˆ)·fs/M. Thus, the accuracy of λ0 and fin estimation is related to the accuracy of δ estimation. The systematic errors which affects the estimation of δ are due to the image parts of Xw(l − 1), Xw(l) and Xw(l + 1). 3. Imposed condition to the integer number of recorded sine wave cycles The systematic errors are due to the spectrum image parts, which are represented in Eq. (9) by the following terms: W(2l −1 +δ), W(2l+δ) and W(2l + 1 + δ). For the M value used in practice (M NN 1) from Eq. (7) after some calculus it can be established that the ratios between the magnitudes of the image part of Xw(l), |Xwi(l)| and of the image part of Xw(l − 1), |Xwi(l − 1)| and of Xw(l + 1), |Xwi(l + 1)| are given by: j Xwi ðlÞj j W ð2l + δÞ j 2l + δ − H = = ; j Xwi ðl − 1Þj j W ð2l − 1 + δÞj 2l + δ + H − 1 j Xwi ðlÞj j W ð2l + δÞj 2l + δ + H = = : jXwi ðl + 1Þj j W ð2l + 1 + δÞj 2l + δ − H + 1
ð13Þ ð14Þ
From Eqs. (13) and (14) we have |Xwi(l − 1)| N |Xwi(l)| N |Xwi(l + 1)|, but these magnitudes are very close. It means that if one of these magnitudes is small, the others will be small too. From Eq. (8) it can be also observed that the magnitude |Xw(l)| appears in both cases. Based on these observations it follows that very small systematic errors are obtained when |Xwi(l)| is very smaller. The magnitude |Xwi(l)| can be considered equal to the magnitude corresponding to a sine wave of amplitude A1, phase φ1 (φ1≅ π− 2πδ + 2πδ/M + 2πl/M −φ) and frequency fin sampled at fs frequency. Thus, the following equality holds on: i A A h jðπ − uÞ ju − ju1 : = 1 W ðδÞe 1 − W ð2l + δÞe W ð2l + δÞe 2j 2j
ð15Þ
Because the image part is very small, the amplitude A1 can be approximated with very high accuracy by: ð9Þ
A1 iA
j W ð2l + δÞj : j W ðδÞ j
The systematic errors decrease as A1 decreases.
ð16Þ
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From Eq. (7) it can be established: HQ − 1
δ A1 iAj
2
2
h −δ
h=1
j:
2HQ −1
ð17Þ
ð2l + δ + h − HÞ
h=1
If the sine wave is corrupted by quantization noise and A1 is below this noise, the systematic noise will be neglected. This assumption is used to derive a condition for l. The quantization noise is considered, in general, a stationary white noise characterized by zero mean and standard deviation σq. This is uniformly distributed on [−q/2, q/2], where q is the quantization step of the used analog-to-digital converter (ADC). Let us consider that the sine wave is corrupted by quantization noise and the amplitude A is much greater than the quantization step q. To obtain very small systematic errors compared with the quantization errors it is necessarily to have for all δ an amplitude A1 smaller than q/(2 µ), where µ is a real number, µ N 1 A1 b
q ; for all δ: 2μ
ð18Þ
If the quantization noise is generated by an n-bit ADC, characterized by full-scale range — FSR (FSR ≥ 2A), then q = FSR/2n. Thus, from Eqs. (17) and (18) it can be established that: jδj
HQ − 1
h2 − δ2
h=1
j
2HQ −1
ð2l + δ + h − H Þj
b
1 FSR ; for all δ: 2n μ 2A
ð19Þ
h=1
As one can see the above condition is independent on M. For a given l and H the maximum of the left side expression in Eq. (19) is obtained for δ = −0.5 (see Fig. 1 for different values of l for H = 2 and H = 3). Thus, to satisfy Eq. (19) for all δ we must have: 2H −1 Y
n
ð4l + 2h − 2H − 1ÞN 2 μ
h=1
− 1 2A HY 2 4h − 1 : FSR h = 1
ð20Þ
To obtain very high accurate δ estimates, µ must be greater than 100, µ ≥ 100. For a given ADC (i.e. n and FSR are known) the above condition is a function of the minimum value of l, H and A. Thus, if H and A are also known, then using Eq. (20) the minimum value of l needed to obtain very small systematic errors compared with the quantization error is determined. 4. Imposed condition to number of samples
Fig. 1. Expression from the left side of Eq. (19) as a function of δ for different values of l when: (a) 2-term maximum side lobe decay window is used; (b) 3-term maximum side lobe decay window is used.
The Equivalent Noise Band Width (ENBW) of an H-term maximum side lobe decay window is given by [9]:
In order to have negligible systematic errors due to the image parts of the sine wave spectrum it is assumed a large l. Due to the quantization noise the standard deviation of δ estimations obtained by IpDFT method with the H-term maximum side lobe decay window is given by [9]:
ENBW =
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 1 C 2H − 2 2πδ H − jδj HY 2 2 4H − 4 h −δ σδ = 2M A sinðπδÞ ð2H − 1Þ! h = 1
If the quantization noise is generated by an n-bit ADC, characterized by full-scale range — FSR (FSR ≥ 2A) then, the standard deviation σq is given by:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð4H − 3Þ δ2 − j δj + 2H2 − 1 σ q: 2H − 1
ð21Þ
It should be observed that σδ not depends on l. The maximum of σδ, σδmax is obtained for δ = 0 [9]. From Eq. (21) it can be established that: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2H − 2 2H 1 2H 2 − 1u t 1 C4H − 4 σ : ð22Þ σ δmax = H 2H − 1 A 2H − 1 2M C − 1 2 q 2H − 2
2H − 2 − 4 H−1 2 C2H −2
C4H
:
q FSR σ q = pffiffiffiffiffiffi = n pffiffiffiffiffiffi : 12 2 12
ð23Þ
ð24Þ
From Eqs. (23) and (24), σδmax becomes: σ δmax
2H = 2H − 1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2H 2 − 1 ENBW FSR 1 : 2H − 1 6M 2A 2n
ð25Þ
When the sine wave is corrupted by quantization noise the δ estimates exhibit an approximate normal (Gaussian) distribution [7].
D. Belega, D. Dallet / Computer Standards & Interfaces 31 (2009) 962–968
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Based on Eqs. (25) and (27) it can be established that M must satisfied the following condition: 6H 2 2H 2 − 1 FSR2 1 ENBW MN : 2A 22n Vδ2 ð2H −1Þ3
ð28Þ
In order to reduce the DFT computational time M it is recommended to be equal to the nearest integer power of two which satisfy Eq. (28). From Eq. (28) some important conclusions can be drawn: • for a given ADC (i.e. n and FSR are known) the minimum value of M increases as the amplitude A decreases as well as the window order H; • for a given A and H the minimum value of M decreases as FSR decreases and n increases; • for a given ADC, A and H the minimum value of M increases as Vδ decreases. For a given ADC Eq. (28) is a function of the minimum value of M, H and Vδ. Thus, if H and Vδ are known, then using Eq. (28) the
Fig. 2. Probability density function of δ estimations.
Thus, the estimation of δ is affected by random errors which have a normal distribution. Fig. 2 shows the probability density function (pdf) of δ estimations for theoretical value set at δ = − 0.2 (Fig. 2a) and δ = 0.01 (Fig. 2b). The parameters of simulated sine wave are: A = 1 and φ = π/3 rad. The value of l is set to 173 and M = 4096. The ADC resolution is n = 8 bits and its FSR is equal to 5. For δ = −0.2 the 2-term maximum side lobe decay window is used in IpDFT method and for δ = 0.01 the 3-term maximum side lobe decay window is used. The quantization noise is modelled by uniformly distributed additive noise. 20000 trials are used. Due to the fact that the estimated value of δ has an approximate normal distribution it follows that the module of the absolute error of δ, |Δδ| is smaller than cσδmax, with a probability done by coverage factor c. In order to obtain a high confidence level c is set to 3: jΔδ j V3 σ δmax :
ð26Þ
Since the above condition is satisfied for all δ to have an absolute error |Δδ| smaller than a desired value Vδ the following condition must be satisfied: 3σ δmax bVδ :
ð27Þ
Fig. 3. |Δδ|max as a function of δ obtained for different value of l when: (a) the 2-term maximum side lobe decay window is used; (b) the 3-term maximum side lobe decay window is used.
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D. Belega, D. Dallet / Computer Standards & Interfaces 31 (2009) 962–968
increment of 0.04. For each value of δ, the maximum of |Δδ|, |Δδ|max occurring during phase scan is retained. For M = 4096 from Eq. (28) it follows that for H = 2 the absolute errors |Δδ| are smaller than 7.3 ppm for all δ with a high confidence level and than 9 ppm for H = 3. Fig. 3 shows the |Δδ|max as a function of δ for different value of l when the 2-term maximum side lobe decay window is used (Fig. 3a) and when the 3-term maximum side lobe decay window is used (Fig. 3b). As it can see from Fig. 3a for l ≥ 32 the δ estimates are very close since in these cases the systematic errors are very small compared with the quantization errors, which practically establish the accuracy of δ estimation. The same behaviour is obtained for l ≥ 9 when the 3-term maximum side lobe decay window is used (see Fig. 3b). 5.2. Verification the validity Eq. (28)
Fig. 4. (a) |Δδ|max as a function of δ; (b) number of occurrences of absolute errors |Δδ| higher than Vδ as a function of δ. The 2-term maximum side lobe decay window is used and M = 2174.
The error Vδ is set to 10 ppm. From Eq. (28) is obtained M N 2173 for H = 2 and M N 3325 for H = 3. l is set to 327. δ varies in the range [−0.5, 0.5) with an increment of 0.02. For each value of δ, φ is uniformly distributed in the range [0, 2π) rad and the maximum of |Δδ|, |Δδ|max is retained. Each time 1000 runs are done to determine |Δδ|max. Fig. 4 shows |Δδ|max as a function of δ (Fig. 4a) and the number of occurrences of absolute errors |Δδ| higher than Vδ as a function of δ (Fig. 4b) when the 2-term maximum side lobe decay window is used. M is set to 2174. From Fig. 4b it follows that in the worst case the probability to have all the absolute errors |Δδ| smaller than Vδ is equal to 99.6%, which is very close to the one corresponding to the ideal normal distribution (99.73%). Hence, the probability to have, for all δ, absolute errors |Δδ| smaller than Vδ is higher than 99.6%. For M = 4096 all the absolute errors |Δδ| are smaller than Vδ (see Fig. 5). This behaviour is obtained since as follows from Eqs. (25) and (26) for M = 4096, the absolute errors |Δδ| are smaller than 7.3 ppm (b10 ppm) for all δ with a high confidence level. Fig. 6 shows the same results as in Fig. 4 but when the 3-term maximum side lobe decay window is used and M = 3326. From Fig. 6b it follows that in the worst case the probability to have all the absolute errors |Δδ| smaller than Vδ is equal to 99.5%, which is very close to the one corresponding to the ideal normal distribution (99.73%). For M = 4096 the probability to have, for all δ, absolute errors |Δδ| smaller than Vδ is higher than 99.7% (see Fig. 7b). The number of
minimum value of M needed to ensure that |Δδ| is smaller than Vδ with a high confidence level is determined. 5. Computer simulation results The aim of this section is to verify the validity of Eqs. (20) and (28) by means of computer simulations. The signal used in simulation is: l+δ m + u + eq ðmÞ; m = 0; 1; N ; M − 1 ð29Þ xðmÞ = A sin 2π M where A = 1 and eq(m) is the quantization noise generated by an 14bit bipolar ADC. The FSR of the ADC is set to 5. It is assumed that the quantization errors from sample to sample are statistically independent. 5.1. Verification the validity of Eq. (20) For µ = 100 the minimum value of l obtained from Eq. (20) is equal to 32 for H = 2 and to 9 for H = 3. M is set to 4096. Phase φ varies in the range [0, 2π) rad with an increment of π/50 rad. δ varies in the range [− 0.5, 0.5) with an
Fig. 5. |Δδ|max as a function of δ when the 2-term maximum side lobe decay window is used and M = 4096.
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967
situations in which the absolute errors |Δδ| are higher than Vδ is 14, smaller than 36, which is obtained for M = 3326. This behaviour is obtained since as follows from Eqs. (25) and (26) for M = 4096, the absolute errors |Δδ| are smaller than 9 ppm (which is very close to 10 ppm) for all δ with a high confidence level. 6. Conclusion The accuracy of δ (and also of normalized frequency) estimations of a sine wave corrupted by quantization noise obtained using the IpDFT method is affected by systematic and random errors. The systematic errors are due to the interferences from the image parts of the sine wave spectrum. They depend on the integer number of recorded sine wave cycles, l. The random errors are due to the quantization noise. They depend on the number of samples, M. Until now in the scientific literature focused on this field is not given any mathematical condition concerning l or M which must be satisfied to obtain accurate δ estimates. In this paper for the situations in which the IpDFT method uses the maximum side lobe decay windows two conditions for l and M are derived. The condition for l allows the determination of minimum value of l which ensures that the systematic errors which affect
Fig. 7. (a) |Δδ|max as a function of δ; (b) number of occurrences of absolute errors |Δδ| higher than Vδ as a function of δ. The 3-term maximum side lobe decay window is used and M = 4096.
Fig. 6. (a) |Δδ|max as a function of δ; (b) number of occurrences of absolute errors |Δδ| higher than Vδ as a function of δ. The 3-term maximum side lobe decay window is used and M = 3326.
the δ estimations are very small compared with the quantization errors. The condition for M allows the determination of minimum value of M which ensures that the absolute error of δ, |Δδ|, due to random errors are smaller than a desired value for all δ with a high confidence level. The validity of each condition has been proven by means of computer simulations. Based on these conditions the parameters l and M can be proper choosing to obtain high accurate δ estimates. In order to obtain small systematic errors for small l the windows with H ≥ 3 must be used. For high l the 2-term maximum side lobe decay window must be used because performed the best. The derived conditions can be also applied for the frequency estimation of a multi-frequency signal component. For example, the conditions derived in this paper can be used for a priori frequency estimation in the three or four-parameter sine-fit algorithms in order to obtain accurate sine wave parameters estimations. Therefore, in the authors' opinion the both derived conditions can be included as a clause in the existent standards for ADC characterization – IEEE standard 1241 [13] or European draft standard DYNAD [14] – concerning the testing of ADCs by three or four-parameter sinefit algorithms.
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References [1] S.L. Marple, Digital Spectral Analysis, Englewood Cliffs, Prentice-Hall, NJ, 1987. [2] D. Petri, Frequency-domain testing of waveform digitizers, IEEE Trans. Instrum. Meas. 51 (3) (2002) 445–453. [3] V.K. Jain, W.L. Collins, D.C. Davis, High-accuracy analog measurements via interpolated FFT, IEEE Trans. Instrum. Meas. IM-28 (2) (1979) 113–122. [4] T. Grandke, Interpolation algorithms for discrete Fourier transforms of weighted signals, IEEE Trans. Instrum. Meas. IM-32 (2) (1983) 350–355. [5] C. Offelli, D. Petri, Interpolation techniques for real-time multifrequency waveforms analysis, IEEE Trans. Instrum. Meas. 39 (1) (1990) 106–111. [6] G. Andria, M. Savino, A. Trotta, Windows and interpolation algorithm to improve electrical measurement accuracy, IEEE Trans. Instrum. Meas. 38 (4) (1989) 856–863. [7] C. Offeli, D. Petri, The influence of windowing on the accuracy of multifrequency signal parameters estimation, IEEE Trans. Instrum. Meas. 41 (2) (1992) 256–261. [8] M. Novotny, M. Sedlacek, Influence of quantization noise on DSP-based DSP algorithms, XVIII IMEKO World Congress, Metrology for a Sustainable Development, September, Rio de Janeiro, Brazil, 2006, pp. 17–22. [9] D. Belega, D. Dallet, Multifrequency signal analysis by Interpolated DFT method with maximum sidelobe decay windows, Measurement, vol. 42 (3) (2009), pp. 420–426. doi:10.1016/j.measurement.2008.08.006. [10] D. Belega, D. Dallet, Estimation of the multifrequency signal parameters by interpolated DFT method with maximum sidelobe decay windows, Proceedings of the IEEE International Workshop on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications, (IDAACS'2007), Dortmund, Germany, 2007, pp. 294–299. [11] A.H. Nuttall, Some windows with very good sidelobe behavior, IEEE Trans. Acoust. Speech Signal Process. ASSP-29 (1) (1981) 84–91. [12] D. Belega, The maximum sidelobe decay windows, L'Académie Roumaine, Revue Roumaine des Sciences Techniques, Série Electrotechnique et Energetique, vol. 50 (3), 2005, pp. 349–356. [13] IEEE Std. 1241, Standard for terminology and test methods for analog-to-digital converters, Tech. rep. IEEE-TC10 (December) (2000). [14] European Project DYNAD. Methods and draft standards for the dynamic characterization and testing of analog-to-digital converters [Online]. Available: http:// paginas.fe.up.pt/~hsm/dynad.
Daniel Belega received the engineer degree in electronics and telecommunications in 1994, the M.S. degree in electrical measurement in 1995, and PH.D. degree (cum laude) in electronics in 2001, all from the “Politehnica” University of Timisoara, Timisoara, Romania. He joined the Measurements and Optical Electronics Department, “Politehnica” University of Timisoara, Timisoara, Romania, in 1995, where he is currently an Associate Professor. His research interests are applications of digital signal processing to measurements, parameter estimation and analogto-digital converter testing.
Dominique Dallet was born in Rochefort/Mer, France, on July 3, 1964. He obtained his Ph.D. Degree in Electrical Engineering in 1995 from the University of Bordeaux 1, where he is currently a Professor at the Electronic Engineering School of Bordeaux (ENSEIRB). His main research activities, carriedout at the IMS laboratory, focus on mixed-signal circuit design and testing, digital and analogue signal processing, and programmable devices' applications. His interests include also digital design and its application in BIST structures for the characterization of embedded A/D converters, as well as, digital signal processing applied to nondestructive techniques based on time–frequency representation.
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Computer Standards & Interfaces j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c s i
Choice of the acquisition parameters for frequency estimation of a sine wave by interpolated DFT method Daniel Belega a,⁎, Dominique Dallet b,1 a b
”Politehnica” University of Timişoara, Faculty of Electronics and Telecommunications, Bv. V. Pârvan, Nr. 2, 300223, Timişoara, Romania Laboratoire IMS, University of Bordeaux - ENSEIRB, 351 Cours de la Libération, Bâtiment A31, 33405, Talence Cedex, France
a r t i c l e
i n f o
Article history: Received 23 January 2008 Received in revised form 10 July 2008 Accepted 28 September 2008 Available online 17 November 2008 Keywords: Frequency estimation Interpolated DFT method Maximum side lobe decay windows
a b s t r a c t The accuracy of the frequency estimation of a sine wave corrupted by quantization noise using the Interpolated Discrete Fourier Transform (IpDFT) method is affected by systematic and random errors. The systematic errors are independent of quantization noise and are due to the interferences from the image parts of the sine wave spectrum. They depend of the acquisition parameter – integer number of recorded sine wave cycles. The random errors are due to the quantization noise and depend on the acquisition parameter – number of acquired samples. In this paper for the situations in which the IpDFT method uses the maximum side lobe decay windows a condition for the integer part of the number of recorded sine wave cycles is derived to ensure that the systematic errors are very small compared with the quantization errors. Also, a condition for the number of samples is derived to ensure that the absolute error of the normalized frequency due to the random errors is smaller than a desired value with a high confidence level. Carried out simulations confirm the validity of each derived condition. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Frequency estimation of a sine wave is very important in many engineering applications. The methods used for this purpose can be classified either in time-domain methods (parametric methods) or frequency-domain methods (non-parametric methods) [1]. Timedomain methods lead to high accuracy frequency estimation when the model order is well-known. Otherwise, intensive computationally algorithms are required to estimate the model order [1]. In the frequencydomain methods the spectrum of the sine wave is firstly computed by means of the Discrete Fourier Transform (DFT) which is then exploited to estimate the frequency [2–7]. These methods have the advantages of the robustness toward signal model inaccuracies and low computational effort. On the other hand they have inherent limitations such as frequency resolution and spectral leakage effects due to noncoherent sampling. A frequency-domain method which provides accurate frequency estimates in non-coherent sampling mode is the Interpolated Discrete Fourier Transform (IpDFT) method [3–10]. Its best performances are obtained using the maximum side lobe decay windows (or class I RifeVincent windows) because in this case the frequency could be estimated by analytical formulas [6–10].
Due to the digitizing process the quantization noise is always present in sampled signal. The influence of the quantization noise on the estimation of sine wave parameters by IpDFT method has been studied in scientific literature [7–9]. The accuracy of the sine wave parameters estimation is affected by systematic and random errors. The first errors type is due to the interferences from the image parts of the sine wave spectrum and depends on the acquisition parameter — integer number of recorded sine wave cycles. They are independent on the quantization noise. The second errors type is due to the quantization noise and depends on the acquisition parameter — number of acquired samples. Unfortunately, the published results in this field do not provide any mathematical condition for these acquisition parameters which must be satisfied to obtain accurate sine wave parameter estimates. The aim of this paper is to derive two mathematical conditions for the acquisition parameters – integer number of recorded sine wave cycles and number of acquired samples – in order to obtain accurate frequency estimation of a sine wave corrupted by quantization noise using the IpDFT method with maximum side lobe decay windows. The validity of each derived condition is then verified by means of computer simulations. 2. Theoretical background
⁎ Corresponding author. Tel.: +40 2 56 33 65; fax: +40 2 56 40 33 62. E-mail addresses: [email protected] (D. Belega), [email protected] (D. Dallet). 1 Tel.: +33 5 40 00 26 32; fax: +33 5 56 37 15 45. 0920-5489/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.csi.2008.09.028
Let us consider a sine wave sampled at fs frequency: f xðmÞ = A sin 2π in m + u ; fs
m = 0; 1 N ; M − 1
ð1Þ
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where A, fin and φ are respectively the amplitude, frequency and phase of the sine wave and M is the number of acquired samples. In order to satisfy the Nyquist criteria, fin is smaller than fs/2. The relationship between the frequencies fin and fs is given by: fin λ l+δ = 0 = fs M M
ð2Þ
in which l and δ are respectively the integer part and the fractional part of λ0. δ is related to the non-coherent sampling mode and −0.5 ≤ δ b 0.5. For δ = 0 the sampling process is coherent with the input sine wave (coherent sampling mode) and Eq. (2) represents the coherent sampling relationship between the frequencies fin and fs. For δ ≠ 0 the sampling process is non-coherent with the input sine wave (non-coherent sampling mode) and the coherent sampling relationship is not fulfilled. In this mode the sine wave spectrum is affected by spectral leakage errors. To reduce these errors the windowing approach is used leading to the spectral analysis of windowed signal xw (m) =x(m)·w(m), where w(m) is the window sequence. The Discrete-Time Fourier Transform (DTFT) of the xw(m) is given by: Xw ðλÞ =
M −1 X
xðmÞwðmÞe
m − j2πλM
;
λa½0; MÞ
ð3Þ
m=0
where λ represents the normalized frequency expressed in bin. After some calculus Xw(λ) becomes: i Ah ju − ju W ðλ − λ0 Þe − W ðλ + λ0 Þe ; λa½0; MÞ ð4Þ Xw ðλÞ = 2j where W(λ) is the DTFT of the window w(m). The second term in Eq. (4) represents the image part of the spectrum. δ can be very accurately estimated by the IpDFT method with the maximum side lobe decay windows [6–9]. The H-term maximum side lobe decay window (H ≥ 2) has the most rapidly decaying side lobes, equal to 6(2H − 1) dB/octave, from all the H-term cosine-class windows [11]. This window is defined by: wðmÞ =
m h ð − 1Þ ah cos 2πh ; M h=0
HX −1
m = 0; 1; N ; M − 1
ð5Þ
where ah are the window's coefficients. They have the following expressions [12]: H − 1
a0 =
H − h − 1
C2H − 2 C −2 ; ah = 2H2H 22H − 2 2 −3
;
h = 1; 2; N ; H − 1
ð6Þ
where p
Cm =
m! : ðm − pÞ!p!
For M NN 1, the DTFT of the w(m) can be approximated by [10]: W ðλÞ =
M sinðπλÞ − jπλ jMπ λ e e 22H − 2 π
ð2H − 2Þ! λ
HQ − 1
h2 − λ2
:
ð7Þ
h=1
For estimation of δ by means of IpDFT method the rapport α must be firstly evaluated: 8 j Xw ðlÞj > > ; if − 0:5Vδb0 < j Xw ðl − 1Þj ð8Þ α= j X ðl + 1Þj > > : w ; if 0Vδb0:5 j Xw ðlÞj From Eq. (4) it can be established: i A h ju − ju j j Xw ðl − 1Þj = j W ð−1 − δÞe − W ð2l − 1 + δÞe 2 i A h ju − ju j Xw ðlÞj = j W ð−δÞe − W ð2l + δÞe j 2 i A h ju − ju j Xw ðl + 1Þj = j W ð1 − δÞe − W ð2l + 1 + δÞe j: 2
963
For large l the image parts of Xw(l − 1), Xw(l) and Xw(l + 1) can be neglected and from Eq. (9) α becomes: 8 j W ðδÞj > > ; < j W ð1 + δÞj αi j W ð 1 − δ Þ j > > ; : j W ðδÞj
if − 0:5Vδb0
:
ð10Þ
if 0Vδb0:5
Based on Eq. (7) after some calculus we obtain: 8 H+δ > < ; αi H − 1 − δ H − 1 + δ > : ; H−δ
if − 0:5Vδb0
ð11Þ
if 0Vδb0:5
From the above expression it follows that δ can be estimated by: 8 ðH − 1Þα − H > < ; if − 0:5Vδb0 α+1 δˆ = Hα − H + 1 > : ; if 0Vδb0:5 α+1
ð12Þ
l is the index that corresponds to the maximum of the |Xw(λ)| for integer λ values (i.e. for λ = 0, 1,…, M − 1). This can be easily determined by means of a maximum search routine applied to these samples. From Eq. (2), it follows that the normalized frequency λ0 can be estimated by l + δˆ and the frequency fin by (l + δˆ)·fs/M. Thus, the accuracy of λ0 and fin estimation is related to the accuracy of δ estimation. The systematic errors which affects the estimation of δ are due to the image parts of Xw(l − 1), Xw(l) and Xw(l + 1). 3. Imposed condition to the integer number of recorded sine wave cycles The systematic errors are due to the spectrum image parts, which are represented in Eq. (9) by the following terms: W(2l −1 +δ), W(2l+δ) and W(2l + 1 + δ). For the M value used in practice (M NN 1) from Eq. (7) after some calculus it can be established that the ratios between the magnitudes of the image part of Xw(l), |Xwi(l)| and of the image part of Xw(l − 1), |Xwi(l − 1)| and of Xw(l + 1), |Xwi(l + 1)| are given by: j Xwi ðlÞj j W ð2l + δÞ j 2l + δ − H = = ; j Xwi ðl − 1Þj j W ð2l − 1 + δÞj 2l + δ + H − 1 j Xwi ðlÞj j W ð2l + δÞj 2l + δ + H = = : jXwi ðl + 1Þj j W ð2l + 1 + δÞj 2l + δ − H + 1
ð13Þ ð14Þ
From Eqs. (13) and (14) we have |Xwi(l − 1)| N |Xwi(l)| N |Xwi(l + 1)|, but these magnitudes are very close. It means that if one of these magnitudes is small, the others will be small too. From Eq. (8) it can be also observed that the magnitude |Xw(l)| appears in both cases. Based on these observations it follows that very small systematic errors are obtained when |Xwi(l)| is very smaller. The magnitude |Xwi(l)| can be considered equal to the magnitude corresponding to a sine wave of amplitude A1, phase φ1 (φ1≅ π− 2πδ + 2πδ/M + 2πl/M −φ) and frequency fin sampled at fs frequency. Thus, the following equality holds on: i A A h jðπ − uÞ ju − ju1 : = 1 W ðδÞe 1 − W ð2l + δÞe W ð2l + δÞe 2j 2j
ð15Þ
Because the image part is very small, the amplitude A1 can be approximated with very high accuracy by: ð9Þ
A1 iA
j W ð2l + δÞj : j W ðδÞ j
The systematic errors decrease as A1 decreases.
ð16Þ
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From Eq. (7) it can be established: HQ − 1
δ A1 iAj
2
2
h −δ
h=1
j:
2HQ −1
ð17Þ
ð2l + δ + h − HÞ
h=1
If the sine wave is corrupted by quantization noise and A1 is below this noise, the systematic noise will be neglected. This assumption is used to derive a condition for l. The quantization noise is considered, in general, a stationary white noise characterized by zero mean and standard deviation σq. This is uniformly distributed on [−q/2, q/2], where q is the quantization step of the used analog-to-digital converter (ADC). Let us consider that the sine wave is corrupted by quantization noise and the amplitude A is much greater than the quantization step q. To obtain very small systematic errors compared with the quantization errors it is necessarily to have for all δ an amplitude A1 smaller than q/(2 µ), where µ is a real number, µ N 1 A1 b
q ; for all δ: 2μ
ð18Þ
If the quantization noise is generated by an n-bit ADC, characterized by full-scale range — FSR (FSR ≥ 2A), then q = FSR/2n. Thus, from Eqs. (17) and (18) it can be established that: jδj
HQ − 1
h2 − δ2
h=1
j
2HQ −1
ð2l + δ + h − H Þj
b
1 FSR ; for all δ: 2n μ 2A
ð19Þ
h=1
As one can see the above condition is independent on M. For a given l and H the maximum of the left side expression in Eq. (19) is obtained for δ = −0.5 (see Fig. 1 for different values of l for H = 2 and H = 3). Thus, to satisfy Eq. (19) for all δ we must have: 2H −1 Y
n
ð4l + 2h − 2H − 1ÞN 2 μ
h=1
− 1 2A HY 2 4h − 1 : FSR h = 1
ð20Þ
To obtain very high accurate δ estimates, µ must be greater than 100, µ ≥ 100. For a given ADC (i.e. n and FSR are known) the above condition is a function of the minimum value of l, H and A. Thus, if H and A are also known, then using Eq. (20) the minimum value of l needed to obtain very small systematic errors compared with the quantization error is determined. 4. Imposed condition to number of samples
Fig. 1. Expression from the left side of Eq. (19) as a function of δ for different values of l when: (a) 2-term maximum side lobe decay window is used; (b) 3-term maximum side lobe decay window is used.
The Equivalent Noise Band Width (ENBW) of an H-term maximum side lobe decay window is given by [9]:
In order to have negligible systematic errors due to the image parts of the sine wave spectrum it is assumed a large l. Due to the quantization noise the standard deviation of δ estimations obtained by IpDFT method with the H-term maximum side lobe decay window is given by [9]:
ENBW =
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 1 C 2H − 2 2πδ H − jδj HY 2 2 4H − 4 h −δ σδ = 2M A sinðπδÞ ð2H − 1Þ! h = 1
If the quantization noise is generated by an n-bit ADC, characterized by full-scale range — FSR (FSR ≥ 2A) then, the standard deviation σq is given by:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð4H − 3Þ δ2 − j δj + 2H2 − 1 σ q: 2H − 1
ð21Þ
It should be observed that σδ not depends on l. The maximum of σδ, σδmax is obtained for δ = 0 [9]. From Eq. (21) it can be established that: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2H − 2 2H 1 2H 2 − 1u t 1 C4H − 4 σ : ð22Þ σ δmax = H 2H − 1 A 2H − 1 2M C − 1 2 q 2H − 2
2H − 2 − 4 H−1 2 C2H −2
C4H
:
q FSR σ q = pffiffiffiffiffiffi = n pffiffiffiffiffiffi : 12 2 12
ð23Þ
ð24Þ
From Eqs. (23) and (24), σδmax becomes: σ δmax
2H = 2H − 1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2H 2 − 1 ENBW FSR 1 : 2H − 1 6M 2A 2n
ð25Þ
When the sine wave is corrupted by quantization noise the δ estimates exhibit an approximate normal (Gaussian) distribution [7].
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Based on Eqs. (25) and (27) it can be established that M must satisfied the following condition: 6H 2 2H 2 − 1 FSR2 1 ENBW MN : 2A 22n Vδ2 ð2H −1Þ3
ð28Þ
In order to reduce the DFT computational time M it is recommended to be equal to the nearest integer power of two which satisfy Eq. (28). From Eq. (28) some important conclusions can be drawn: • for a given ADC (i.e. n and FSR are known) the minimum value of M increases as the amplitude A decreases as well as the window order H; • for a given A and H the minimum value of M decreases as FSR decreases and n increases; • for a given ADC, A and H the minimum value of M increases as Vδ decreases. For a given ADC Eq. (28) is a function of the minimum value of M, H and Vδ. Thus, if H and Vδ are known, then using Eq. (28) the
Fig. 2. Probability density function of δ estimations.
Thus, the estimation of δ is affected by random errors which have a normal distribution. Fig. 2 shows the probability density function (pdf) of δ estimations for theoretical value set at δ = − 0.2 (Fig. 2a) and δ = 0.01 (Fig. 2b). The parameters of simulated sine wave are: A = 1 and φ = π/3 rad. The value of l is set to 173 and M = 4096. The ADC resolution is n = 8 bits and its FSR is equal to 5. For δ = −0.2 the 2-term maximum side lobe decay window is used in IpDFT method and for δ = 0.01 the 3-term maximum side lobe decay window is used. The quantization noise is modelled by uniformly distributed additive noise. 20000 trials are used. Due to the fact that the estimated value of δ has an approximate normal distribution it follows that the module of the absolute error of δ, |Δδ| is smaller than cσδmax, with a probability done by coverage factor c. In order to obtain a high confidence level c is set to 3: jΔδ j V3 σ δmax :
ð26Þ
Since the above condition is satisfied for all δ to have an absolute error |Δδ| smaller than a desired value Vδ the following condition must be satisfied: 3σ δmax bVδ :
ð27Þ
Fig. 3. |Δδ|max as a function of δ obtained for different value of l when: (a) the 2-term maximum side lobe decay window is used; (b) the 3-term maximum side lobe decay window is used.
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increment of 0.04. For each value of δ, the maximum of |Δδ|, |Δδ|max occurring during phase scan is retained. For M = 4096 from Eq. (28) it follows that for H = 2 the absolute errors |Δδ| are smaller than 7.3 ppm for all δ with a high confidence level and than 9 ppm for H = 3. Fig. 3 shows the |Δδ|max as a function of δ for different value of l when the 2-term maximum side lobe decay window is used (Fig. 3a) and when the 3-term maximum side lobe decay window is used (Fig. 3b). As it can see from Fig. 3a for l ≥ 32 the δ estimates are very close since in these cases the systematic errors are very small compared with the quantization errors, which practically establish the accuracy of δ estimation. The same behaviour is obtained for l ≥ 9 when the 3-term maximum side lobe decay window is used (see Fig. 3b). 5.2. Verification the validity Eq. (28)
Fig. 4. (a) |Δδ|max as a function of δ; (b) number of occurrences of absolute errors |Δδ| higher than Vδ as a function of δ. The 2-term maximum side lobe decay window is used and M = 2174.
The error Vδ is set to 10 ppm. From Eq. (28) is obtained M N 2173 for H = 2 and M N 3325 for H = 3. l is set to 327. δ varies in the range [−0.5, 0.5) with an increment of 0.02. For each value of δ, φ is uniformly distributed in the range [0, 2π) rad and the maximum of |Δδ|, |Δδ|max is retained. Each time 1000 runs are done to determine |Δδ|max. Fig. 4 shows |Δδ|max as a function of δ (Fig. 4a) and the number of occurrences of absolute errors |Δδ| higher than Vδ as a function of δ (Fig. 4b) when the 2-term maximum side lobe decay window is used. M is set to 2174. From Fig. 4b it follows that in the worst case the probability to have all the absolute errors |Δδ| smaller than Vδ is equal to 99.6%, which is very close to the one corresponding to the ideal normal distribution (99.73%). Hence, the probability to have, for all δ, absolute errors |Δδ| smaller than Vδ is higher than 99.6%. For M = 4096 all the absolute errors |Δδ| are smaller than Vδ (see Fig. 5). This behaviour is obtained since as follows from Eqs. (25) and (26) for M = 4096, the absolute errors |Δδ| are smaller than 7.3 ppm (b10 ppm) for all δ with a high confidence level. Fig. 6 shows the same results as in Fig. 4 but when the 3-term maximum side lobe decay window is used and M = 3326. From Fig. 6b it follows that in the worst case the probability to have all the absolute errors |Δδ| smaller than Vδ is equal to 99.5%, which is very close to the one corresponding to the ideal normal distribution (99.73%). For M = 4096 the probability to have, for all δ, absolute errors |Δδ| smaller than Vδ is higher than 99.7% (see Fig. 7b). The number of
minimum value of M needed to ensure that |Δδ| is smaller than Vδ with a high confidence level is determined. 5. Computer simulation results The aim of this section is to verify the validity of Eqs. (20) and (28) by means of computer simulations. The signal used in simulation is: l+δ m + u + eq ðmÞ; m = 0; 1; N ; M − 1 ð29Þ xðmÞ = A sin 2π M where A = 1 and eq(m) is the quantization noise generated by an 14bit bipolar ADC. The FSR of the ADC is set to 5. It is assumed that the quantization errors from sample to sample are statistically independent. 5.1. Verification the validity of Eq. (20) For µ = 100 the minimum value of l obtained from Eq. (20) is equal to 32 for H = 2 and to 9 for H = 3. M is set to 4096. Phase φ varies in the range [0, 2π) rad with an increment of π/50 rad. δ varies in the range [− 0.5, 0.5) with an
Fig. 5. |Δδ|max as a function of δ when the 2-term maximum side lobe decay window is used and M = 4096.
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967
situations in which the absolute errors |Δδ| are higher than Vδ is 14, smaller than 36, which is obtained for M = 3326. This behaviour is obtained since as follows from Eqs. (25) and (26) for M = 4096, the absolute errors |Δδ| are smaller than 9 ppm (which is very close to 10 ppm) for all δ with a high confidence level. 6. Conclusion The accuracy of δ (and also of normalized frequency) estimations of a sine wave corrupted by quantization noise obtained using the IpDFT method is affected by systematic and random errors. The systematic errors are due to the interferences from the image parts of the sine wave spectrum. They depend on the integer number of recorded sine wave cycles, l. The random errors are due to the quantization noise. They depend on the number of samples, M. Until now in the scientific literature focused on this field is not given any mathematical condition concerning l or M which must be satisfied to obtain accurate δ estimates. In this paper for the situations in which the IpDFT method uses the maximum side lobe decay windows two conditions for l and M are derived. The condition for l allows the determination of minimum value of l which ensures that the systematic errors which affect
Fig. 7. (a) |Δδ|max as a function of δ; (b) number of occurrences of absolute errors |Δδ| higher than Vδ as a function of δ. The 3-term maximum side lobe decay window is used and M = 4096.
Fig. 6. (a) |Δδ|max as a function of δ; (b) number of occurrences of absolute errors |Δδ| higher than Vδ as a function of δ. The 3-term maximum side lobe decay window is used and M = 3326.
the δ estimations are very small compared with the quantization errors. The condition for M allows the determination of minimum value of M which ensures that the absolute error of δ, |Δδ|, due to random errors are smaller than a desired value for all δ with a high confidence level. The validity of each condition has been proven by means of computer simulations. Based on these conditions the parameters l and M can be proper choosing to obtain high accurate δ estimates. In order to obtain small systematic errors for small l the windows with H ≥ 3 must be used. For high l the 2-term maximum side lobe decay window must be used because performed the best. The derived conditions can be also applied for the frequency estimation of a multi-frequency signal component. For example, the conditions derived in this paper can be used for a priori frequency estimation in the three or four-parameter sine-fit algorithms in order to obtain accurate sine wave parameters estimations. Therefore, in the authors' opinion the both derived conditions can be included as a clause in the existent standards for ADC characterization – IEEE standard 1241 [13] or European draft standard DYNAD [14] – concerning the testing of ADCs by three or four-parameter sinefit algorithms.
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References [1] S.L. Marple, Digital Spectral Analysis, Englewood Cliffs, Prentice-Hall, NJ, 1987. [2] D. Petri, Frequency-domain testing of waveform digitizers, IEEE Trans. Instrum. Meas. 51 (3) (2002) 445–453. [3] V.K. Jain, W.L. Collins, D.C. Davis, High-accuracy analog measurements via interpolated FFT, IEEE Trans. Instrum. Meas. IM-28 (2) (1979) 113–122. [4] T. Grandke, Interpolation algorithms for discrete Fourier transforms of weighted signals, IEEE Trans. Instrum. Meas. IM-32 (2) (1983) 350–355. [5] C. Offelli, D. Petri, Interpolation techniques for real-time multifrequency waveforms analysis, IEEE Trans. Instrum. Meas. 39 (1) (1990) 106–111. [6] G. Andria, M. Savino, A. Trotta, Windows and interpolation algorithm to improve electrical measurement accuracy, IEEE Trans. Instrum. Meas. 38 (4) (1989) 856–863. [7] C. Offeli, D. Petri, The influence of windowing on the accuracy of multifrequency signal parameters estimation, IEEE Trans. Instrum. Meas. 41 (2) (1992) 256–261. [8] M. Novotny, M. Sedlacek, Influence of quantization noise on DSP-based DSP algorithms, XVIII IMEKO World Congress, Metrology for a Sustainable Development, September, Rio de Janeiro, Brazil, 2006, pp. 17–22. [9] D. Belega, D. Dallet, Multifrequency signal analysis by Interpolated DFT method with maximum sidelobe decay windows, Measurement, vol. 42 (3) (2009), pp. 420–426. doi:10.1016/j.measurement.2008.08.006. [10] D. Belega, D. Dallet, Estimation of the multifrequency signal parameters by interpolated DFT method with maximum sidelobe decay windows, Proceedings of the IEEE International Workshop on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications, (IDAACS'2007), Dortmund, Germany, 2007, pp. 294–299. [11] A.H. Nuttall, Some windows with very good sidelobe behavior, IEEE Trans. Acoust. Speech Signal Process. ASSP-29 (1) (1981) 84–91. [12] D. Belega, The maximum sidelobe decay windows, L'Académie Roumaine, Revue Roumaine des Sciences Techniques, Série Electrotechnique et Energetique, vol. 50 (3), 2005, pp. 349–356. [13] IEEE Std. 1241, Standard for terminology and test methods for analog-to-digital converters, Tech. rep. IEEE-TC10 (December) (2000). [14] European Project DYNAD. Methods and draft standards for the dynamic characterization and testing of analog-to-digital converters [Online]. Available: http:// paginas.fe.up.pt/~hsm/dynad.
Daniel Belega received the engineer degree in electronics and telecommunications in 1994, the M.S. degree in electrical measurement in 1995, and PH.D. degree (cum laude) in electronics in 2001, all from the “Politehnica” University of Timisoara, Timisoara, Romania. He joined the Measurements and Optical Electronics Department, “Politehnica” University of Timisoara, Timisoara, Romania, in 1995, where he is currently an Associate Professor. His research interests are applications of digital signal processing to measurements, parameter estimation and analogto-digital converter testing.
Dominique Dallet was born in Rochefort/Mer, France, on July 3, 1964. He obtained his Ph.D. Degree in Electrical Engineering in 1995 from the University of Bordeaux 1, where he is currently a Professor at the Electronic Engineering School of Bordeaux (ENSEIRB). His main research activities, carriedout at the IMS laboratory, focus on mixed-signal circuit design and testing, digital and analogue signal processing, and programmable devices' applications. His interests include also digital design and its application in BIST structures for the characterization of embedded A/D converters, as well as, digital signal processing applied to nondestructive techniques based on time–frequency representation.