interpolated dft for $\sin^{\alpha}(x)$ windows

$: Interpolated DFT for $\sin^{\alpha}(x)$ Windows$
754 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 4, APRIL 2014

Interpolated DFT for sinα(x) WindowsKrzysztof Duda and Szymon Barczentewicz

Abstract— This paper describes interpolated discrete Fouriertransform (IpDFT) for parameter estimation of sinusoidal anddamped sinusoidal signals analyzed with a sinα(x) window. Forα = 0, 2, 4, . . . sinα(x) windows are Rife–Vincent class I (RVI)windows, for which IpDFT algorithms are known. We present anew IpDFTs for α = 1, 3, 5, . . .. The bias-variance trade-off ofthe proposed IpDFT fits between results offered by RVI windows,e.g., for α = 1, we get higher noise immunity than Hann (RVIorder 1, α = 2) window and lower bias than rectangular (RVIorder 0, α = 0) window.

Index Terms— Discrete Fourier transform (DFT), frequencyand damping estimation, frequency domain measurements, inter-polated DFT, signal processing, windowing.

I. INTRODUCTION

ESTIMATION of frequency, damping, amplitude, andphase of a single frequency and multifrequency signal is

based on the Fourier analysis or parametric modeling [1], [2].Tutorial comparisons of those methods are available, e.g., in[3] and [4].

For nonperiodic signals, discrete Fourier transform (DFT)spectrum is affected by spectral leakage [5], [6], and estimatesof frequency, damping, amplitude, and phase are improvedby windowing and interpolated DFT (IpDFT) algorithms[7]–[17]. Typically, Rife–Vincent class I (RVI) windows (alsoknown as the maximum sidelobe decay windows) are used inIpDFT because it is easy to obtain IpDFT formulas for thosewindows. However, RVI windows are sinα(x) windows forα = 0, 2, 4, . . ..

In this paper, we extend known IpDFTs for RVI windows tosinα(x) α = 0, 1, 2, 3, . . . windows. The contribution of thispaper is the new set of IpDFTs for sinusoidal and dampedsinusoidal signals analyzed with sinα(x) α = 1, 3, 5, . . .windows.

II. SIGNAL MODEL AND sinα(x) WINDOWS

Let us consider a discrete time, damped sinusoidal signal inthe following form:

x[n] = A cos(ω0n + ϕ)e−dn d ≥ 0 (1)

where A > 0 is the signal’s amplitude, 0 < ω0 < π is itsangular frequency in radians, and ω0 = π rad corresponds to

Manuscript received May 24, 2013; revised September 5, 2013; acceptedSeptember 6, 2013. Date of publication February 20, 2014; date of currentversion March 6, 2014. This work was supported by the Polish NationalScience Centre under Grant DEC-2012/05/B/ST7/01218. The Associate Editorcoordinating the review process was Dr. Dario Petri.

The authors are with the Department of Measurement and Electronics,AGH University of Science and Technology, Kraków 30-059, Poland (e-mail:[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIM.2013.2285795

the half of the sampling rate in hertz, ϕ is the phase angle inradians, d ≥ 0 is the damping factor, n = 0, 1, 2, . . . , N − 1is the index of the sample, and N is the number of samples.For d = 0 (1) simplifies to undamped sinusoid x[n] =A cos(ω0n+ϕ). IpDFT algorithms designed for damped signal(1) are still valid for undamped sinusoid, but the reversestatement is not true, because the damping changes the shapeof the signal’s spectrum [13].

Before DFT analysis, the signal x[n] is multiplied by thetime window w[n]

v[n] = w[n]x[n]. (2)

DFT V [k] of signal v[n] is defined as

V [k] =N−1∑

n=0

v[n]e− j 2πN kn k = 0, 1, . . . , N − 1. (3)

Frequencies of DFT bins with indices k = 0, 1, 2, . . . , N − 1are ω[k] = (2π/N)k rad; thus we refer to the value 2π /N radas a DFT frequency step.

The sinα(x) windows are defined as [6]

wα[n] = sinα( π

Nn)

n = 0, 1, 2, . . . , N − 1 (4)

with α normally being an integer.Using the well-known formulas for powers of trigonometric

functions, see (5.72) and (5.70) in [18], definition (4) may berewritten separately for even and odd α.

For even α = 2M (with M being nonnegative integer)window (4) is

wα=2M [n] =M∑

m=0

(−1)ma2M [m]cos

(2π

Nmn

)(5)

where

a2M [0] = 1

22M

(2MM

)a2M [m] = 1

22M−1

(2M

M − m

)

m = 1, 2, . . . , M. (6)

Cosine windows defined by (5) are known as RVI windowsof order M . For M = 0, α = 0 we get a rectangular window,and for M = 1, α = 2 we get Hann window.

For odd α = 2M − 1 (with M being positive integer),window (4) is

wα=2M−1[n]=M∑

m=1

(−1)ma2M−1[m]cos

(2π

N(m−0.5)n+π

2

)

(7)

where

a2M−1[m] = 1

22M−2

(2M − 1M − m

)m = 1, 2, . . . , M. (8)

0018-9456 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

DUDA AND BARCZENTEWICZ: INTERPOLATED DFT 755

TABLE I

WINDOW COEFFICIENTS (6), (8)

Window coefficients aα[m] (6) and (8) for α = 1, . . . , 8 aregiven in Table I.

IpDFT algorithms described in the literature are designedfor the signal analyzed with RVI window (5).

In the following, we derive new IpDFT algorithms forundamped and damped sinusoid analyzed with window definedby (7), i.e., for sinα(x) window with odd α.

It is seen from (5) that for even α, the time window isa weighted sum of rectangular window and M rectangularwindows modulated (i.e., shifted in frequency) by cosines withfrequencies being integer multiplies of a DFT frequency step.

For odd α, the time window (7) is a weighted sum of Mfrequency-shifted rectangular windows but the frequency shiftis always in the middle between the two successive DFT bins(i.e., the shift always contains the half of the DFT frequencystep) because there is (m−0.5) in definition (7).

The spectrum of (5) is

W2M (e jω) =M∑

m=0

[(−1)m a2M [m]

2W0(e

j (ω−ω [m]))

+(−1)m a2M [m]2

W0(ej (ω+ω [m]))

](9)

where ω[m] = (2π/N)m rad, and W0(e jω) is the spectrum ofthe rectangular window

W0(ejω) =

N−1∑

n=0

e− jω(N−1)/2 sin(ωN/2)

sin(ω/2). (10)

The spectrum of (7) is

W2M−1(ejω)

= jM∑

m=1

[(−1)m a2M−1[m]

2W0(e

j (ω−ω [m]+π/N))

− (−1)m a2M−1[m]2

W0(ej (ω+ω [m]−π/N))

]. (11)

To derive IpDFT for the damped signal (1) we interpretthis signal as undamped sinusoid analyzed with the dampedwindow [13]

v[n] = w [n]x [n] = w [n]A cos(ω0n + ϕ)e−dn

= w [n]A cos(ω0n + ϕ) (12)

where w [n] = w [n]e−dn.

The spectrum of damped window (5) is

W2M (e j ω) =M∑

m=0

[(−1)m a2M [m]

2W0(e

j (ω−ω [m]))

+(−1)m a2M [m]2

W0(ej (ω+ω [m]))

](13)

and the spectrum of damped window (7) is

W2M−1(ej ω)

= jM∑

m=1

[(−1)m a2M−1[m]

2W0(e

j (ω−ω [m]+π/N))

−(−1)m a2M−1[m]2

W0(ej (ω+ω [m]−π/N))

](14)

where ω = ω − jd and W0(e j ω) is the spectrum of dampedrectangular window

W0(ej ω) = e− j ω(N−1)/2 sin(ωN/2)

sin(ω/2). (15)

In the Appendix, spectrum (14) is used for deriving newIpDFTs. For undamped signal ω = ω, and (13) and (14) reduceto (9) and (11).

III. PROPOSED IPDFT ALGORITHMS

In this section, we propose IpDFTs for sinα(x) windowswith odd α.

The spectrum of the windowed sinusoidal signal v[n] =w[n]A cos(ω0n + ϕ) is

V (e jω) = A

2e jϕW (e j (ω−ω0)) + A

2e− jϕW (e j (ω+ω0)) (16)

where W (e jω) is the spectrum of the used time windoww[n] (undamped or damped). If the signal’s frequency ω0does not equal ω[k] = (2π/N)k rad, then the IpDFTsare used for the accurate estimation of signal’s parame-ters [i.e., for finding the maximum of V (e jω) (16) locatedbetween DFT bins, see Fig. 1]. Based on a few succes-sive DFT bins (3) of the windowed signal (2) frequencycorrection (displacement) δ is estimated that fulfils theequation

ω0 = (kmax + δ)(2π/N) − 0.5 < δ ≤ 0.5 (17)

where kmax is the index of the DFT bin with the highestmodulus (see Fig. 1).

According to (16), the main lobe of amplitude spectrumof windowed sinusoidal signal depicted in Fig. 1 hasapproximately the same shape as amplitude spectrum of thewindow alone.


Fig. 1. Amplitude spectrum of windowed sinusoidal signal. Illustration ofDFT interpolation problem.

Let us define the following relations:|V [kmax + 1]|

|V [kmax]| = |V (ω[kmax + 1])||V (ω[kmax])|

= |V (ω0 − δ2p2π/N + 2π/N)||V (ω0 − δ2p2π/N)|

≈ |W (−δ2p2π/N + 2π/N)||W (−δ2p2π/N)| (18)

|V [kmax]| + |V [kmax + 1]||V [kmax]| + |V [kmax − 1]|= |V (ω0 − δ3p2π/N)| + |V (ω0 − δ3p2π/N + 2π/N)|

|V (ω0 − δ3p2π/N)| + |V (ω0 − δ3p2π/N − 2π/N)|≈ |W (−δ3p2π/N)| + |W (−δ3p2π/N + 2π/N)|

|W (−δ3p2π/N)| + |W (−δ3p2π/N − 2π/N)| . (19)

Solving (18) for δ2p we obtain a two-point IpDFT (two DFTbins are used) for the signal analyzed with sinα(x) window

δ2p = sgn(|V [kmax + 1]| − |V [kmax − 1]|)· (α/2 + 1)|V [kmax ± 1]| − (α/2)|V [kmax]|

|V [kmax]| + |V [kmax ± 1]|α = 0, 1, 2, 3, . . . (20)

with |V [kmax + 1]| in (20) if |V [kmax + 1]| > |V [kmax − 1]|(see Fig. 1) and |V [kmax − 1]| otherwise, and sgn(·) denotingsign function that returns −1 for argument less than 0, 1 forargument greater than 0, and 0 for 0.

Solving (19) for δ3p we obtain a three-point IpDFT for thesignal analyzed with sinα(x) window

δ3p = sgn(|V [kmax + 1]| − |V [kmax − 1]|)· |V [kmax + 1]| + |V [kmax − 1]|2|V [kmax]| + ||V [kmax + 1]| − |V [kmax − 1]||

α = 0 (21a)

δ3p = (α/2 + 1)|V [kmax + 1]| − |V [kmax − 1]|

2|V [kmax]| + |V [kmax − 1]| + |V [kmax + 1]|α = 1, 2, 3 . . . . (21b)

Solutions (20) and (21) of (18) and (19) were only reportedfor RVI windows (5). In the Appendix, we prove that they arealso valid for windows defined by (7).

For damped signal (1) analyzed with sinα(x) window theIpDFT solution given in [13] is extended to the case of odd α

δd = −α + 1

2

R1 − R2

(α + 2)R1 R2 − R1 − R2 − α(22)

d1 = 2π

N

√(δ + α/2)2 − R1(δ − α/2 − 1)2

R1 − 1δ �= 0.5 (23)

d2 = 2π

N

√(δ − α/2)2 − R2(δ + α/2 + 1)2

R2 − 1δ �= −0.5 (24)

where α = 0, 1, 2, 3, . . . and the ratios of DFT bins are definedby

R1 = |V [kmax + 1]|2|V [kmax]|2

R2 = |V [kmax − 1]|2|V [kmax]|2 . (25)

Equations (22)–(25) were derived in [13] for RVI win-dows (5). In the Appendix, we prove that they are also validfor windows defined by (7).

Systematic errors of damping estimation by (23) and (24)are practically the same. As suggested in [15], the mean valued = (d1 + d2)/2 may be used for damping estimation.

IV. RESULTS

In this section, we analyze the systematic errors and noiseimmunity of the proposed IpDFTs and compare it with theresults obtained for RVI windows. In all the figures, thewindow is specified by α value. For even α, the order ofRVI window is given, and for odd α, it is stressed up that thenew IpDFT is used. For sinusoidal signal, a three-point IpDFT(21) was used. The lower subscript in ωE stands for estimatedfrequency.

A. Theoretical Systematic Error

For the case of a single undamped sinusoid expressions forerrors of δ estimation due to simultaneous spectral leakage andadditive noise are derived in [16] for even α and multipointinterpolation. E.g. for Hann (wα=2[n]) window and three-pointIpDFT, the maximum systematic error is [16]

�max2 = 2(kmax + δ)|δ|

2kmax + δ

· (1 − δ2)(4 − δ2)

[(2kmax + δ)2 − 1][(2kmax + δ)2 − 4] . (26)

Maximum systematic error of δ estimation for the three-point IpDFT is defined in [17]. For window (7), it is

�max2M−1 = max

ϕ|δ3p − δ| = −(α/2 + 1)

× |W2M−1(2kmax + δ + 1)| − |W2M−1(2kmax + δ − 1)|2|W2M−1(−δ)| + |W2M−1(−δ − 1)| + |W2M−1(−δ + 1)|

(27)


Fig. 2. Theoretical maximum systematic errors of δ estimation for α = 1,α = 2 (Hann), and α = 3.

where δ is the true value, and window’s spectrum may beapproximated from (A2) by

W2M−1(ejω) ≈ sin(ω N/2 + π/2)e− jω((N−1)/2)

·M∑

m=1

(−1)ma2M−1[m](2m − 1)(

ωN2π − m + 0.5

) (ωN2π + m − 0.5

) (28)

e.g., maximum systematic error for wα=1[n] is given by

�max1 =

−3

2

∣∣∣ 1(2kmax+δ+1.5)(2kmax+δ+0.5)

∣∣∣ −∣∣∣ 1(2kmax+δ−0.5)(2kmax+δ−1.5)

∣∣∣∣∣∣ 2(−δ+0.5)(−δ−0.5)

∣∣∣+∣∣∣ 1(−δ−0.5)(−δ−1.5)

∣∣∣+∣∣∣ 1(−δ+1.5)(−δ+0.5)

∣∣∣(29)

Fig. 2 depicts the theoretical maximum systematic errorsof δ estimation for α = 1, α = 2 (Hann), and α = 3.As expected, by increasing α, we gain smaller systematicerrors. We also note that theoretical errors presented in Fig. 2are practically the same as the errors obtained during simula-tions described next.

For even α, sinα(x) window is a linear combination ofcosines with frequencies being even multiplies of π /N rad, i.e.,harmonics of 2π /N rad. For odd α, sinα(x) window is a linearcombination of sines with frequencies being odd multiplies ofπ /N rad. It goes from the Fourier modulation theorem thatmultiplying coherently sampled signal by the window witheven α shifts its spectrum by integer multiplies of 2π /N radto the neighboring DFT bins, and the signal is still coherentlysampled. For odd α, the spectrum of the signal is shifted byodd multiplies of π /N rad in the middle between neighboringDFT bins, and the signal is not coherently sampled any more;but if the frequency correction is δ = 0.5 the signal windowedwith odd α window becomes coherently sampled. The abovediscussion explains negligible errors in Fig. 2 for even andodd α.

B. Simulations

Figs. 3–6 depict systematic errors of frequency estimationfor undamped and damped sinusoid. Test signal has N = 512samples. The frequency of the test signal was changed from(2π /N) rad to 128 · (2π /N) rad with the step 0.1 · (2π /N) rad.

Fig. 3. Systematic errors of frequency estimation for undamped sinusoidanalyzed with sinα(x) α = 0, . . . , 8 window. Three-point IpDFT (21).

Fig. 4. Systematic errors of frequency estimation for undamped sinusoidanalyzed with sinα(x) α = 0, . . . , 8 window. Zoomed-in view of Fig. 3.

For the given frequency value, the phase was changed from−π /2 rad to π /2 rad with the step π /20 and the highestabsolute value of the frequency estimation error was takenas a systematic error.

The lower OX axis is labeled by frequency in radians,and the upper OX axis is scaled in frequencies of DFTbins k. The frequency of the kth DFT bin is k·(2π /N) radand sinusoidal signal with frequency k contains k periods.For integer k, sinusoidal signal is coherently sampled andfrequency correction δ is zero.

In Fig. 3, bold lines highlight envelops of the error showingthe worst case. The same errors are depicted in Fig. 4 withzoomed axes. It is seen from Fig. 4 that for even α (i.e., forRVI windows) estimation error is practically zero for integer k,whereas for odd α, the error is negligible for integer k plus 0.5.Results of simulations presented in Fig. 4 are practically thesame as theoretical errors shown in Fig. 2, i.e., max|ωE −ω| ≈�max

2M−12π/N .Figs 5 and 6 show systematic errors of frequency estimation

for damped sinusoid. Damped signal is not periodic, andeven for integer number of cycles the estimation error is notnegligible, although local minima are observed in Fig. 6.

It is seen in Figs. 2–6 that systematic errors of the pro-posed IpDFTs are placed between errors obtained with RVIwindows for both undamped and damped sinusoid. Thus


Fig. 5. Systematic errors of frequency estimation for damped sinusoid (1),d = 0.001, analyzed with sinα(x) α = 0, . . . , 8 window. IpDFT (22).

Fig. 6. Systematic errors of frequency estimation for damped sinusoid (1)analyzed with sinα(x) α = 0, . . . , 8 window. Zoomed-in view of Fig. 5.

Fig. 7. Standard deviation of frequency estimation divided by CRLB forundamped sinusoid analyzed with sinα(x) α = 0, . . . , 8 window. Three-pointIpDFT (21).

we gained possibility of additional adjustment of the IpDFTalgorithm.

Figs. 7 and 8 show standard deviation of frequencyestimation for undamped and damped sinusoid respectively asa function of signal-to-noise ratio (S/N) divided by Cramér–Rao Lower Bound (CRLB) [19], [20] (i.e., the optimalestimator would be on the level 1). For each S/N 10 000 test,

Fig. 8. Standard deviation of frequency estimation divided by CRLB fordamped sinusoid (1) analyzed with sinα(x) α = 0, . . . , 8 window. IpDFT(22). Rectangular window (α = 0) is above the scale (it starts with 6.3 for10 dB).

Fig. 9. Errors of frequency estimation for undamped sinusoid with additiveGaussian 60 dB noise analyzed with sinα(x) α = 0, . . . , 8 window. Three-point IpDFT (21).

signals were generated with the frequency equal 10.2 · (2π /N)rad and random phase with uniform distribution from −π toπ rad.

For undamped signal (Fig. 7) rectangular (RV0) windowgives the best results for the S/N from about 15 to 35 dB.Proposed IpDFT with sinα(x) α = 1 window performs betterthan Hann (RV1) window for S/N from 10 to 55 dB.

For damped signal, rectangular (RV0) window has very highstandard deviation of frequency estimation and is not depictedin Fig. 8 (for 10 dB it has 6.3 times higher standard deviationthan CRLB, for 40 dB 98.5 times, and for 60 dB 983.9 times).Proposed IpDFT with sinα(x) α = 1 window gives betterresults than rectangular window for the considered S/N range,and better results than Hann (RV1) window for S/N lower thanabout 40 dB.

Results depicted in Figs. 7 and 8 describe errorssimultaneously caused by spectral leakage and noisepropagation. A well-known bias-variance trade-off is observed,i.e., the window immune to noise (e.g., α = 1) is notrobust against leakage which begins to dominate for high S/Nvalues.

Figs. 9 and 10 show maximum frequency estimation errorsfor the signal disturbed by additive Gaussian 60 dB noise. This


Fig. 10. Errors of frequency estimation for damped sinusoid (1), d = 0.001,with additive Gaussian 60 dB noise analyzed with sinα(x) α = 0, . . . , 8window. IpDFT (22).

is another look on bias-variance trade-off depicted in Figs. 7and 8. Comparing Figs. 4 and 6 with Figs. 9 and 10, we seesignificant increase in maximum frequency estimation errordue to noise for higher α windows.

V. CONCLUSION

In this paper, we showed a second, unknown part of IpDFTsfamily for (un)damped sinusoids for the signal analyzed withsinα(x) α = 1, 3, 5, . . . windows. The main idea of the paperis that whenever RVI window (sinα(x) with even α) is usedin IpDFT then sinα(x) with odd α may also be used. Asan illustration, we showed results of frequency estimationfor damped signals and three-point IpDFT for undampedsignals.

From practical point of view, the most important in thenew family seems to be sinα(x) α = 1 window, becauseits performance is placed between very popular rectangularwindow and Hann window. It was verified that the systematicerror (bias) of IpDFTs with sinα(x) α = 1 is lower than forrectangular window and the variance is lower than for Hannwindow.

Additional attractiveness of our observation is that IpDFTalgorithms for odd α fit into IpDFT formulas for RVI orderM windows. Thus the IpDFT software for the signal analyzedwith RVI windows is also valid for sinα(x) we just have toput M = α/2.

IpDFT inherits properties of the applied time window.Sinα(x) windows have the fastest decay of the sidelobes [6],but they also have wide main lobe. By selecting α we controlbias-variance trade-off of estimation.

For other arbitrary windows, IpDFT is realized by thepolynomial approximation of the window’s spectrum as, e.g.,shown in [11].

APPENDIX

Let us rewrite the spectrum (14) of damped window (7) inthe following form:

W2M−1(ej ω)

= j1

2sin(ωN/2 + π/2)

·M∑

m=1

[e− j

(ω N−1

2 −mπ N−1N +π/2 N−1

N

)

× a2M−1[m]sin(ω/2 − mπ/N + π/(2N))

+e− j

(ω N−1

2 +mπ N−1N −π/2 N−1

N

)

× a2M−1[m]sin(ω/2 + mπ/N − π/(2N))

]. (A1)

Assuming (N−1)/N ≈ 1 we get approximate spectrum ofdamped window (7)

W2M−1(ej ω) ≈ 1

2sin(ωN/2 + π/2)e− j ω((N−1)/2)

·M∑

m=1

[(−1)ma2M−1[m]

sin(ω/2 − m(π/N) + π/(2N))

+ (−1)m+1a2M−1[m]sin(ω/2 + m(π/N) − π/(2N))

]. (A2)

We solve (18) for δ2p by inserting (A2) into (18) and thenapproximate sine functions by their arguments

|V [kmax + 1]||V [kmax]|≈ |W2M−1(−δ2p2π/N + 2π/N)|

|W2M−1(−δ2p2π/N)|

=

∣∣∣∣M∑

m=1

[(−1)ma2M−1[m]

sin((−δ2p−m+1.5)(π/N)) + (−1)m+1a2M−1[m]sin((−δ2p+m+0.5)(π/N))

]∣∣∣∣∣∣∣∣

M∑m=1

[(−1)ma2M−1[m]

sin((−δ2p−m+0.5)(π/N)) + (−1)m+1a2M−1[m]sin((−δ2p+m−0.5)(π/N))

]∣∣∣∣

≈

∣∣∣∣M∑

m=1

[(−1)ma2M−1[m]−δ2p−m+1.5 + (−1)m+1a2M−1[m]

−δ2p+m+0.5

]∣∣∣∣∣∣∣∣

M∑m=1

[(−1)ma2M−1[m]−δ2p−m+0.5 + (−1)m+1a2M−1[m]

−δ2p+m−0.5

]∣∣∣∣

. (A3)

By rearranging summations in (A3), bringing fractions tothe common denominator and using a2M−1[m] values (A3)simplifies to (A4), and we obtain

|V [kmax + 1]||V [kmax]| ≈ −δ2p − M + 0.5

−δ2p + M + 0.5. (A5)

From (A5) with M = (α+1)/2 we get (20).

(−1)M a2M−1[M]−δ2p−M+1.5 + · · · + (−1)1a2M−1[1]

−δ2p+0.5 + (−1)1+1a2M−1[1]−δ2p+1.5 + · · · + (−1)M+1a2M−1[M]

−δ2p+M+0.5

(−1)M a2M−1[M]−δ2p−M+0.5 + · · · + (−1)1a2M−1[1]

−δ2p−0.5 + (−1)1+1a2M−1[1]−δ2p+0.5 + · · · + (−1)M+1a2M−1[M]

−δ2p+M−0.5

= −δ2p − M + 0.5

−δ2p + M + 0.5. (A4)


By going the same path as before, i.e., using (A2) andneglecting sine functions, the ratio (19) for the three-pointIpDFT reduces to

|V [kmax]| + |V [kmax + 1]||V [kmax]| + |V [kmax − 1]| ≈ δ3p + M + 0.5

−δ3p + M + 0.5. (A6)

From (A6) with M = (α+1)/2 we get (21).For damped signal (12) the ratios (25), after inserting (A2)

with ω = ω − jd , are

R1 ≈ (δd + M − 0.5)2 + D21

(δd − M − 0.5)2 + D21

R2 ≈ (δd − M + 0.5)2 + D22

(δd + M + 0.5)2 + D22

(A7)

where D1 = d1 N/(2π) and D2 = d2 N/(2π). From (A7) withM = (α + 1)/2 we obtain frequency correction δ (22) andtwo formulas for damping estimation (23) from R1 and (24)from R2.

Simplifications of DFT bin ratios (A5), (A6), and (A7) areeasy to verify with Matlab Symbolic Math Toolbox.

REFERENCES

[1] S. M. Kay, Modern Spectrum Analysis. Englewood Cliffs, NJ, USA:Prentice-Hall, 1987.

[2] L. Marple, Digital Spectrum Analysis with Applications. EnglewoodCliffs, NJ, USA: Prentice-Hall, 1987.

[3] T. P. Zielinski and K. Duda, “Frequency and damping estimationmethods—An overview,” Metrol. Meas. Syst., vol. 18, no. 4, pp. 505–528, 2011.

[4] K. Duda and T. P. Zielinski, “Efficacy of the frequency and dampingestimation of real-value sinusoid. Part 44 in a series of tutorials oninstrumentation and measurement,” IEEE Instrum. Meas. Mag., vol. 16,no. 2, pp. 48–58, Apr. 2013.

[5] A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time SignalProcessing, 2nd ed. Englewood Cliffs, NJ, USA: Prentice-Hall, 1999.

[6] F. J. Harris, “On the use of windows for harmonic analysis with thediscrete Fourier transform,” Proc. IEEE, vol. 66, no. 1, pp. 51–83,Jan. 1978.

[7] V. H. Jain, W. L. Collins, and D. C. Davis, “High accuracy analogmeasurement via interpolated FFT,” IEEE Trans. Instrum. Meas., vol. 28,no. 2, pp. 113–122, Jun. 1979.

[8] T. Grandke, “Interpolation algorithms for discrete Fourier transformof weighted signals,” IEEE Trans. Instrum. Meas., vol. 32, no. 2,pp. 350–355, Jun. 1983.

[9] G. Andria, M. Savino, and A. Trotta, “Windows and interpolationalgorithms to improve electrical measurement accuracy,” IEEE Trans.Instrum. Meas., vol. 38, no. 4, pp. 856–863, Aug. 1989.

[10] D. Agrež, “Weighted multipoint interpolated DFT to improve amplitudeestimation of multifrequency signal,” IEEE Trans. Instrum. Meas.,vol. 51, no. 2, pp. 287–292, Apr. 2002.

[11] K. Duda, “DFT interpolation algorithm for Kaiser–Bessel andDolph–Chebyshev windows,” IEEE Trans. Instrum. Meas., vol. 60, no. 3,pp. 784–790, Mar. 2011.

[12] M. Bertocco, C. Offeli, and D. Petri, “Analysis of damped sinusoidalsignals via a frequency-domain interpolation algorithm,” IEEE Trans.Instrum. Meas., vol. 43, no. 2, pp. 245–250, Apr. 1994.

[13] K. Duda, T. P. Zielinski, L. B. Magalas, and M. Majewski, “DFTbased estimation of damped oscillation’s parameters in low-frequencymechanical spectroscopy,” IEEE Trans. Instrum. Meas., vol. 60, no. 11,pp. 3608–3618, Nov. 2011.

[14] D. Agrež, “A frequency domain procedure for estimation of the expo-nentially damped sinusoids,” in Proc. Int. Instrum. Meas. Technol. Conf.,May 2009, pp. 1321–1326.

[15] D. Agrež, “Estimation of parameters of the weakly damped sinusoidalsignals in the frequency domain,” Comput. Standards Inter., vol. 33,no. 2, pp. 117–121, Feb. 2011.

[16] D. Belega, D. Dallet, and D. Petri, “Accuracy of sine wave frequencyestimation by multipoint interpolated DFT approach,” IEEE Trans.Instrum. Meas., vol. 59, no. 11, pp. 2808–2815, Nov. 2010.

[17] D. Belega and D. Dallet, “Efficiency of the three-point interpolated DFTmethod on the normalized frequency estimation of a sine wave,” in Proc.IEEE Int. Workshop IDAACS, Sep. 2009, pp. 2–7.

[18] M. R. Speigel, Mathematical Handbook of Formulas and Tables(Schaum’s Outline Series). New York, NY, USA: McGraw-Hill, 1968.

[19] S. M. Kay, Fundamentals of Statistical Signal Processing: EstimationTheory. Englewood Cliffs, NJ, USA: Prentice-Hall, 1993.

[20] Y. Yao and S. M. Pandit, “Cramér-Rao lower bounds for a dampedsinusoidal process,” IEEE Trans. Signal Process., vol. 43, no. 4,pp. 878–885, Apr. 1995.

Krzysztof Duda received the M.S. degree in automatics and metrology andthe Ph.D. degree in electronics from the AGH University of Science andTechnology, Kraków, Poland, in 1998 and 2002, respectively.

He has been an Assistant Professor with the Department of Measurementand Electronics, AGH University of Science and Technology, since 2002. Hiscurrent research interests include the applications of digital signal processingand analysis.

Szymon Barczentewicz was born in Poland in 1988. He received the B.S. andM.S. degrees from the AGH University of Science and Technology, Kraków,Poland, in 2011 and 2012 respectively, where he is currently pursuing thePh.D. degree in electrical engineering.

His current research interests include phasor estimation and its applicationin wide area power system measurements.

interpolated dft for $\sin^{\alpha}(x)$ windows

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