Microsoft Word - 385-2422-2-LE-rev2

ACTA IMEKO
ISSN: 2221‐870X
November 2016, Volume 5, Number 3, 47‐54

ACTA IMEKO | www.imeko.org November 2016 | Volume 5 | Number 3| 47

Signal and zero padding to improve parameters estimation of
sinusoidal signals in the frequency domain

Dušan Agrež
1
, Damir Ilić

2
, Janko Drnovšek

1
University of Ljubljana, Faculty of Electrical Engineering, Trzaska 25, 1000 Ljubljana, Slovenia

2
University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia

Section: RESEARCH PAPER

Keywords: sampling process; sampling by averaging; signal padding; zero padding; estimation of parameters; interpolated DFT

Citation: Dušan Agrež, Damir Ilić, Janko Drnovšek, Signal and zero padding to improve parameters estimation of sinusoidal signals in the frequency domain,
Acta IMEKO, vol. 5, no. 3, article 8, November 2016, identifier: IMEKO‐ACTA‐05 (2016)‐03‐08

Section Editor: Konrad Jedrzejewski, Warsaw University of Technology, Poland

Received May 13, 2016; In final form July 11, 2016; Published November 2016

Copyright: © 2016 IMEKO. This is an open‐access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited

Corresponding author: Dušan Agrež, e‐mail: dusan.agrez@fe.uni‐lj.si

1. INTRODUCTION

In the last decade, considerable research has been carried out
on the analysis of efficient methods capable of accurately
estimating parameters of the frequency components of interest
[1], [2]. In many cases the problem of evaluating the spectral
performance of a given periodic signal reduces to the
estimation of parameters of each spectral component
(frequency, amplitude, and phase). Parameter’s estimations of
periodic signals mostly base on sampling and acquiring digital
values of samples by analog-to-digital converters. In this
procedure values of sampling points are results of averaging in
the aperture time – measurement time. This averaging (or
integration) gives reduction of noise but causes systematic
errors in estimations of the signal parameters [3]–[5]. In this
paper, algorithms for estimation of parameters by signal and
zero padding first and then interpolation in the frequency
domain are presented.

A sampling process can be modelled with four signals and
their frequency spectra in the time and the frequency domain:
measured signal )()( F fGtg  (Figure 1), impulse response

of the sampling channel )()( F fHth  (Figure 2), sampling

function )()( F fSts  (Figure 3), and the window function

of the measurement interval )()( F fWtw  (Figure 4)
where F stands for the Fourier transformation from time
domain to frequency domain and vice versa.

maxf maxft

 tg

 fG

Figure 1. Measured signal.

Figure 2. Impulse response of the sampling channel.

 th

 fH

ABSTRACT
This paper presents the procedure to improve the estimation of the basic sinusoidal signal parameters (frequency, amplitude, and
phase, respectively) in the case of signal sampling by averaging in the aperture time. Prior to estimation in the frequency domain by
the interpolated DFT algorithms the sampled signal is padded with the signal average values in the aperture times and zeroes in the
rest of the sampling interval. We can increase padding points and a number of the signal cycles in the whole measurement interval
and with this nearing the errors to the level as with estimation of the signal without average sampling even the sampling Nyquist
condition is not fulfill.

ACTA IMEKO | www.imeko.org November 2016 | Volume 5 | Number 3| 48

The measured signal )(tg is always filtered by the impulse
response of the input front circuit of the measurement channel
represented by )(th f in Figure 5 and after that sampled by the

real finit duration sampling pulses represented by )(th

convolved on the sampling function )(ts (Figure 3: typically
time uniformly distributed) and finally only a finite number of
samples is taken into consideration represented by the window
function )(tw (Figure 4) to get a filtered, sampled and

windowed signal )(* , tg wf (Figure 5).

Considering that the equivalent of multiplication in the time
domain is convolution in the frequency domain





  d)()()()( 21

F
21 fGGtgtg and vice

versa )()(d)()( 21
F

21 fGfGtgg  



 [6]

the sampling procedure can be modelled as follows (Figure 5):

It is evident that the spectrum of the sampled signal starts to
change with modification of the sampling pulses

)()( F fHth  (Figures 6 and 7).

2. PADDING AND ESTIMATION OF PARAMETERS

Sampling by the frequency SS 1 tf  of the periodic band
limited signal  tg composed of M components can be
expressed as    






0 SS
2sin

m mmm
tnfAtnw  with mf , mA

and m as frequency, amplitude, and phase of a particular
component, respectively. In the estimation procedure one has
to take into account that values of samples (Figure 8: line d) are
representatives in the aperture time apt or typically average

values of the signal in the aperture integration interval (Figure 8:
line c). For a demonstration of sampling, in Figure 8 three
periods of the one component sine signal are presented with a
duty cycle of sampling 4.0Sap  ttD and with sampling

ratio of 6.1SS  tTffr mm .
With sampling representatives – average values – we lose

some information of the signal and especially in the cases where
the aperture time is so long that the signal changes significantly

 th
 ts

 tg
 thf 



 tg f

 ts



 tg f

 tw
 tg wf,

S2t tSt0S2t St

Figure 6. Signals in the sampling process.

Sf S2fS2f Sf f0

 fS
 fH

 fGf

 fG wf,

 fGf

 fS

 fG
 fHf

 fW








maxf maxf f

Nyqf
0 fSf S2fS2f Sf

Sf S2fS2f Sf f0

Figure 7. Spectra of signals in the sampling process.

 1  1 1  1  1 F

)()( Stntts
n





 

St S2t0S2t St f

 S1t  S1t S1t  S1t  S1t

  )(1
SS







n t

n
f

t
fS 

S1 t S2 t0S2 t S1t

Figure 3. Sampling function in the time and the frequency domain.

 tg

M1 T M1 T

MT
t f

 fW

 tw

Figure 4. Window function of the measurement interval.

 tg
 thf 

 ts
 th 

 tg f

 ts



 tg f

 tw
 tg wf ,

 fG f

 fG wf ,

 fG f

 fS 

 fG
 fH f

 fS
 fH

 fW








Figure 5. Procedure of sampling in the time and the frequency domain.

ACTA IMEKO | www.imeko.org November 2016 | Volume 5 | Number 3| 49

in this interval and where the sampling Nyquist condition is not
fulfilled ( 2S  mffr ). One possibility to overcome these
problems is to add zeroes (known as zero-padding technique
[7]) and average values in the aperture times with the duty ratio

Sap ttD  (Figure 8: line c). The aperture time is positioned
symmetrically around the known sampling instants.

By increasing the number of samples and acquired signal
cycles it is possible to detect signals below the Nyquist
condition (Figure 9: the whole acquired signal contains around
50 cycles of the sinusoid 50S  Ntfff mmm on

00020N points). An undersampled sine wave still appears as

a sampled sine wave but at a lower frequency mfkff  S

( ...,2,1k ) or expressed by the relative frequency

mk   S where S is the relative sampling frequency.
With padding points we apparently increase the sampling
frequency in relation to the signal frequency.

To estimate parameters of the time-dependent signals
containing any periodicity, it is preferable to use a
transformation of the signal in the frequency domain. The
discrete Fourier transformation (DFT) of the windowed signal
   kgkw  on N sampled points at the spectral line i is given

by:

      



M

m
mmm

mm iWiWAiG
0

j-j ee
2

j   , (1)

where mmmm iff   is the frequency component
related to the base frequency resolution SM 11 tNTf 
and consists of an integer part and the non-coherent sampling
displacement term 5.05.0  m .

A finite measurement time is a source of dynamic errors,
which are shown as leakage parts of the measurement window
spectrum convolved on the spectrum of the measured-sampled
signal (Figure 9). The long-range leakage contributions can be
reduced in more ways: by increasing the measurement time




meas
1 Tf , by using windows with a faster reduction of the

side lobes (like the Rife-Vincent windows class I - RV1, etc.
[8]), or by using the multi-point interpolated DFT algorithms.
Dedicated algorithms are needed to obtain the correct
parameters of the sinusoidal components in the signal.
Parameters of the measurement component can be estimated
by means of interpolation 9. From a comparative study 10 it
can be concluded that the key for estimating the three basic
parameters is in determining the position of the measurement
component mmm i between the DFT coefficients )( miG
and )1( miG surrounding the component m . In estimations,
the well-known expressions for the three-point estimations for
frequency (2), amplitude (3), and phase (4) were used. The
three-point DFT interpolation gives optimal results owing to:
symmetry around the local peak amplitude DFT coefficient;
equal suppression of leakage coming from both sides; equal
minimal error curves as with one-, five- and multi-point
interpolations. Only the order P of the windows has to be
changed using RV1 windows [9].

     
     121

11
13 




mmm

mm
m

iGiGiG

iGiG
P (2)

   
 

      121
πsin

!22

2
2

22
2






 




mmm

l m
m

m
P

iGiGiG

l
P

A 





(3)

The single phase can be estimated with the arguments of the
three largest local DFT coefficients   mi iGm arg [9], [11]:

   
2

141 11
3 


  mimiimm

a
mmm


 . (4)

3. EVALUATION OF THE ESTIMATIONS

3.1. Systematic behaviour

We can estimate three basic parameters of a particular
component (true or apparent on Figure 9) by the three-point
interpolations since we need only the local largest DFT
coefficients. The estimation errors were compared for the
frequency (2), amplitude (3), and phase (4) estimations using the
Hann window ( 1P ). The absolute errors of the frequency
estimation trueest.)(  E and the phase estimation

trueest.)(  E , and the absolute relative errors of the

amplitude estimation 1)( trueest.  AAAe are first checked
for one sine component in the signal with a double scan,
varying specific sampling parameters and the phase of the signal

1

  maxgtg

apt

d
c

Figure 8. Signals of sampling by averaging in the aperture time: a – original
signal, b – truncated signal in the aperture intervals, c – average signal
values in the aperture integration interval, d – average value
representatives of the sampled signal in the aperture integration interval.



 G

50 100 150 200

10-6

10-4

10-2

s s2 s3

c
d

m

ms  
ms  

Figure 9. Spectra of signals from Figure 8: a – original signal, b – truncated
signal in the aperture intervals, c – signal with average values in the
aperture integration interval, d – signal with average values in the aperture
integration interval with sinc correction.

ACTA IMEKO | www.imeko.org November 2016 | Volume 5 | Number 3| 50

at particular relative frequencies because the long-range
leakages are frequency- and phase-dependent (Figures 10 to 23:

1mA , and 2π2π   , 18π ).
First, the sampling ratio mffr S was changed to find

intervals where the interpolation algorithm can be used (Figures
10 to 19). The absolute maximum error values (from 19
iterations changing phase) at a given sampling ratio are
compared using the duty ratio of 4.0Sap  ttD . In Figure
10, we can see that it is possible to estimate the frequency of
one component even better than if we have a complete signal
also when the sampling ratio mffr S is in a interval
between 1 and 2 (the sampling condition is not fulfilled). The
vicinities around the integer values 1 and 2 where we cannot
estimate parameters as in the case of the original signal depend
on the number of signal cycles in the whole measurement
interval MT . The largest value  gives a better frequency
resolution and borders come closer to integer values 1 and 2
(Figures 10, 11, and 12). The width of the error estimation
main-lobe around integer values depends on the position of the
investigated component and interspacing between neighbouring
components if we have enough sampling points in one period
(400 points are used in simulations from analysis in Figures 22
and 23). If we have 50 cycles in the measurement interval
(Figure 10) we get a basic bin resolution 501 and this
resolution gives error main-lobe borders 15.185.0  r and

3.27.1  r around integers where the frequency estimation
errors increase due to leakage influence on the investigated
component m from its replicas mk  S 2,1k (Figures 9

and 12).
It can be also noticed that there are error peaks which

number increases below 32r due to replicas mk  S with
higher values of ..,4,3k (Figure 12).

Decreasing the bin resolution to 201 increases unusable
intervals to 2.18.0  r and 5.25.1  r (Figure 11). Going
in opposite direction by increasing the bin resolution to 1001
(Figure 12) reduces unusable intervals to 1.19.0  r and

15.285.1  r , and the frequency can be accurately estimated
also in the interval of sampling ratio 85.11.1 S  mff what is
below the sampling condition.

In the case of the amplitude estimation (Figures 13 and 15)
we need to correct the estimated amplitude or the complete
amplitude DFT spectrum (Figure 9, line d) by the well-known
sinc correction    mm TtTtk apapsinc-corr πsinπ  [3]. We see
the same behaviour of the errors in the cases of amplitude and
phase estimations as with the frequency estimation (Figures 13
to 16). A bin resolution of 501 reduces unusable intervals

15.185.0  r and 2.28.1  r for the phase estimation as
for frequency estimation (Figure 14) and even better for the
amplitude estimation (Figure 13: unusable intervals

1.19.0  r and 3.27.1  r ). A resolution of 1001
further reduces unusable intervals to 08.192.0  r and

15.285.1  r (Figures 15 and 16).

mffS

1 1.5 2 2.5

10-5

10-3

10-1

a b

1k2k3k...
 maxE

Figure 12. Absolute maximum values of errors of the frequency estimation
in relation to the sampling ratio: a – original signal, b – signal with average

values; k40N , Hz320Hz10S f , 100m .

mffS

 Aemax

1 1.5 2 2.5

10-5

10-3

10-1

Figure 13. Absolute maximum values of relative errors of the amplitude
estimation in relation to the sampling ratio: a – original signal, b – signal

with average values; k20N , Hz160Hz5S f , 50m .

mffS

 maxE

1 1.5 2 2.5

10-5

10-3

10-1

a b

Figure 10. Absolute maximum values of errors of the frequency estimation
in relation to the sampling ratio: a – original signal, b – signal with average

values; k20N , Hz160Hz5S f , 50m .

mffS

 maxE

1 1.5 2 2.5

10-5

10-3

10-1
a b

Figure 11. Absolute maximum values of errors of the frequency estimation
in relation to the sampling ratio: a – original signal, b – signal with average

values; k8N , Hz64Hz2S f , 20m .

ACTA IMEKO | www.imeko.org November 2016 | Volume 5 | Number 3| 51

Algorithms were analysed also by the multi-component
signal. The second harmonic component as the closest and the
most disturbing component was added with amplitude

1012 AA  ( 1.0THD ) and phase 02  , other parameters
of simulations are the same as for Figure 8. We get the most
disturbing replicas of the second harmonic at positions 5.1r
and 3 due to the leakage effect mfkff 2S  ( 2,1k )
(Figures 17 to 19). Simulation results show that the proposed
estimation procedures give very good results when the sampling

condition is fulfilled, but beyond the Nyquist frequency the
THD has to be low ( 01.0 ).

If we change the duty ratio Sap ttD  in the sampling
interval (changing the aperture time at a fixed sampling
frequency) the estimation errors do not change very much in
comparison to the estimation of the original signal if we correct
the estimation by a sinc correction [12]. In Figures 20 and 21 the
duty ratio was changed almost in the whole possible interval

998.0001.0 D at 6.1S  mffr with 400P N

mffS

  degmax E

1 1.5 2 2.5

10-2

1 a b

102

Figure 14. Absolute maximum values of errors of the phase estimation in
relation to the sampling ratio: a – original signal, b – signal with average

values; k20N , Hz160Hz5S f , 50m .

mffS

 Aemax

1 1.5 2 2.5

10-5

10-3

10-1

Figure 15. Absolute maximum values of relative errors of the amplitude
estimation in relation to the sampling ratio: a – original signal, b – signal

with average values; k40N , Hz320Hz10S f , 100m .

mffS

  degmax E

1 1.5 2 2.5

10-2

1 a b

102

Figure 16. Absolute maximum values of errors of the phase estimation in
relation to the sampling ratio: a – original signal, b – signal with average

values; k40N , Hz320Hz10S f , 100m .

mffS

1 1.5 2 2.5

10-5

10-3

10-1

a b

mfk ;1 mfk 2;1mfk ;2 mfk 2;2
 maxE

Figure 17. Absolute maximum values of errors of the frequency estimation
in relation to the sampling ratio: a – original signal, b – signal with average

values; k20N , Hz160Hz5S f , 50m .

mffS

 Aemax

1 1.5 2 2.5

10-5

10-3

10-1

Figure 18. Absolute maximum values of relative errors of the amplitude
estimation in relation to the sampling ratio: a – original signal, b – signal

with average values; k20N , Hz160Hz5S f , 50m .

mffS

  degmax E

1 1.5 2 2.5

10-2

1
a b

102

Figure 19. Absolute maximum values of errors of the phase estimation in
relation to the sampling ratio: a – original signal, b – signal with average

values; k20N , Hz160Hz5S f , 50m .

ACTA IMEKO | www.imeko.org November 2016 | Volume 5 | Number 3| 52

samples in one period (other parameters were the same as in
Figure 8).

The estimation also much depends on the number of points
in the period PN and in the whole measurement interval.
Padding with more points (average signal values and zero
values) will improve estimations since interpolation equations
(2), (3), and (4) are derived for large number of points 1N
[9]. In Figures 22 and 23, the number of points in the period

PN was changed from 10P N to more than 512P N
(other parameters were the same as in Figure 8). We can see
that the frequency estimation does not differ significantly if we
have reduced the information of the signal (Figure 22: curves a
and b) and both estimations errors decrease with increasing
number of points.

The amplitude estimation much more depends on the
number of points (Figure 23) but after having more than

128P N points per period the estimation errors drop to the
level as with the estimation of the original signal without
averaging in the aperture time.

3.2. Noise propagation

The price for the effective leakage reduction is in the
increase of the estimation uncertainties related to the unbiased
Cramér-Rao bounds 13 fixed by the signal-to-noise-ratio for a
particular component  22 2 tmm ASNR  corrupted by a white
noise with standard uncertainty t 14. In Figures 24 and 26,
there are standard uncertainties of the frequency, amplitude,
and phase estimations related to the CR bounds, respectively.

 
 CRB,

113


NSNR
, (5)

AA
NSNR

CRB,

11
  , (6)

  CRB,
11

2 
NSNR

. (7)

Moving away from integers of the relative frequency, what is
the case when the sampling ratio is below 2, the standard
uncertainties increase in relation to the minimal attainable
values (Figure 24 for the frequency estimation, Figure 25 for
the amplitude estimation, and Figure 26 for the phase
estimation) but these changes can be neglected. In the case of
amplitude estimation the standard deviations even decrease if
the frequency  is estimated first.

  CRB,

6 1 5432

Figure 24. Standard uncertainty of the three‐point displacement estimation
(2) related to the CR bound (5).

Figure 20. Absolute maximum values of errors of the frequency estimation
in relation to the duty ratio: a – original signal, b – signal with average
values in the aperture interval.

Sap tt

  5max 10Ae

0 0.1 0.5

2.2

2.6

3.0
a

3.4

Figure 21. Absolute maximum values of relative errors of the amplitude
estimation in relation to the duty ratio: a – original signal, b – signal with
average values in the aperture integration interval with sinc correction.

Figure 22. Absolute maximum values of errors of the frequency estimation
in relation to the number of sampling points in the period: a – original
signal, b – signal with average values in the aperture interval.

 Aemax

100 200 300 400

10-5

10-3

10-1

a
b

5000

Figure 23. Absolute maximum values of relative errors of the amplitude
estimation in relation to the number of sampling points in the period: a –
original signal, b – signal with average values in the aperture integration
interval with sinc correction.

Sap tt

  5max 10E

0 0.1 0.5

a b

 maxE

20 40 60 80

10-4

10-3

a
b

100 0 120

ACTA IMEKO | www.imeko.org November 2016 | Volume 5 | Number 3| 53

4. EXPERIMENTAL RESULTS

To demonstrate the proposed algorithms in reducing both
estimation errors (phase and noise contributions at different
frequencies) in a real measurement environment we use a
digitizing voltmeter to acquire signal, Agilent 3458A [15] and a
stable voltage generator, Keysight 33500B [16] to generate a
nominal sine voltage with amplitude V1n, uA

( %04.0THD ) and changing frequency. In opposite to
simulations, the sampling frequency was set at kHz10S f

μs100S  t and the signal frequency was changed from
kHz100m f down to kHz125.3m f to achieve the same

sampling ratio 2.31.0mS  ffr . The fixed aperture time
μs40ap t determined the duty cycle of sampling

4.0Sap  ttD and with 80s.p. n acquired sampling points

the measurement time was ms8Ss.p.M  tnT with a frequency

resolution of Hz1251 M  Tf . As in the simulations,
400padd N padding points (for signal and zero padding) was

used around the acquired points, and altogether
k32padds.p.  NnN was used in the interpolation DFT

algorithms. Each sampling sequence of 80 points was
synchronized using the ‘sync out’ terminal of the generator and
the external trigger input of the voltmeter.

Maximal estimation errors are shown in Figures 27 to 29,
where reference values were those set on the generator ( 1A ,

kHz125.3kHz100 f and 2π2π   , 18π )

and with 50 trials at each frequency and phase. The estimation
error behaviours are very close to those in the simulations.

It can be noticed that systematic error contributions behaves
as expected and a very small second harmonic component is
presented in the measurement system. The frequency
estimation (Figure 27) can be compared with the simulation
results from Figure 10 except the noise floor is higher and at
the level of 4max 102)(

E even in the interval below the
sampling condition 8.115.1 S  mff . The results of the

6.1

4.1

2.1

1
4 51 2 3 6 

AA CRB,

Figure 25. Standard uncertainty of the amplitude three‐point estimation (3)

related to the CRB (6); a –  is estimated, b –  is known.

  CRB,m

53 61 2 4



Figure 26. Ratios of the uncertainties of the phase three‐point estimation

(4) related the CRB (7); a –  is estimated, b –  is known.

mffS

10-2

 maxE

1 1.5 2 2.5

10-4

10-3

10-1

Figure 27. Absolute maximum values of errors of the frequency estimation

in relation to the sampling ratio; 80s.p. n , k32N , kHz10S f ,
25800mm  ff .

mffS

10-2

 Aemax

1 1.5 2 2.5

10-4
10-3

10-1
1

Figure 28. Absolute maximum values of the relative errors of the amplitude

estimation in relation to the sampling ratio; 80s.p. n , k32N ,
kHz10S f , 25800mm  ff .

mffS

  degmax E

1 1.5 2 2.5

10-1

101

Figure 29. Absolute maximum values of errors of the phase estimation in

relation to the sampling ratio; 80s.p. n , k32N , kHz10S f ,
25800mm  ff .

ACTA IMEKO | www.imeko.org November 2016 | Volume 5 | Number 3| 54

amplitude (Figure 28) and the phase estimations (Figure 29) are
worse in comparison to the simulation results (Figures 13 and
14) due to inaccurate values at the output of the generator and
inaccuracy of the sampling voltmeter, but the systematic
contributions of errorss confirm the expected behaviour like
with the frequency estimation.

5. CONCLUSIONS

The paper proposes algorithms for the estimation of basic
sinusoidal parameters (frequency, amplitude, and phase of the
frequency component), when the acquired sampling points do
not fulfil the sampling condition and some signal information is
lost due to signal averaging in the aperture time. In the
proposed procedure, the empty space between successive real
sampling points is virtually padded by average values of the real
sampling point in the interval of knowing aperture and zeroes
in the rest of the sampling interval. This procedure with suitable
large acquired cycles in the whole measurement interval
improves the frequency resolution, and the interpolated DFT
estimation algorithms can be adopted for particular frequency
components.

In many cases the number of sampling points is limited but
by performing the algorithms on a computer we can increase
padding points and with this nearing the errors to the level as
with estimation on the theoretically original signal without
averaging in the aperture time and with the number of points
equal to all padding points. Simulation and experimental results
show that the parameters’ estimations are possible also even in
the interval below the sampling condition.

REFERENCES

[1] G. D’Antona, A. Ferrero, Digital Signal Processing for
Measurement Systems. Theory and Applications, Springer
Science, 2006.

[2] D. Agrež, "Dynamics of frequency estimation in the frequency
domain", IEEE Trans. Instr. Meas. 56 (2007) pp. 2111-2118.

[3] R. L. Swerlein, "A 10 ppm accurate digital AC measurement
algorithm", Proc. NCSL Workshop Symp., 1991, pp. 17–36.

[4] G. A. Kyriazis, "Extension of Swerlein’s Algorithm for AC
Voltage Measurement in the Frequency Domain", IEEE Trans.
Instr. Meas. 52 (2003) pp. 367-370.

[5] H. E. van den Brom, E. Houtzager, S. Verhoeckx, Q. E. V. N.
Martina, G. Rietveld, "Influence of Sampling Voltmeter
Parameters on RMS Measurements of Josephson Stepwise-
Approximated Sine Waves", IEEE Trans. Instr. Meas. 58 (2009)
pp. 3806-3812.

[6] W. McC. Siebert, Circuits, Signals and Systems, The MIT Press,
McGraw-Hill, Cambridge, New York, 1986, pp. 497-502.

[7] G. Leus, M. Moonen, "Semi-blind channel estimation for block
transmissions with non-zero padding", Records of Conf. on
Signals, Systems and Comp., 2001, pp. 762-766.

[8] F. J. Harris: "On the use of windows for harmonic analysis with
the discrete Fourier transform", Proc. IEEE 66 (1978) pp. 51-83.

[9] J. Štremfelj, D. Agrež, "Nonparametric estimation of power
quantities in the frequency domain using Rife-Vincent windows",
IEEE Trans. Instr. Meas. 62 (2013) pp. 2171-2184.

[10] J. Schoukens, R. Pintelon, H. Van Hamme, "The interpolated
fast Fourier transform: A comparative study", IEEE Trans. Instr.
Meas. 41 (1992) pp. 226-232.

[11] T. J. Stewart, "Spectral leakage errors when using an Agilent
3458A to measure phase at mains power frequencies", Proc.
NCSL Workshop Symp. 1991, pp. 17–36.

[12] D. Agrež, D. Ilić, J. Drnovšek, "Estimation of periodic signal
parameters by signal and zero padding", in Proc. XXI IMEKO
World Congress/2015, Prague, Czech Republic, 2015, TC4: 474-
479.

[13] A. Moschitta, P. Carbone, "Cramer-Rao lower bound for
parametric estimation of quantized sinewaves", Proc. IEEE
IMTC/2004, Como, Italy, 2004, pp. 1724–1729.

[14] Widrow B, Kollar I, Quantization Noise, Cambridge Univ. Press,
2008.

[15] 3458A Multimeter - User's Guide, Ed. 5, 1988-2012, 2011,
Agilent Technologies.

[16] 33500B Series Waveform Generators – Data Sheet, 5991-
0692EN, 2015, Keysight Technologies.