Performance Comparison of Power Quality Evaluation Using Advanced High Resolution Spectrum Estimation Methods

DOI: 10.9790/1676-1201015767 www.iosrjournals.org 57 | Page
IOSR Journal of Electrical and Electronics Engineering (IOSR-JEEE)
e-ISSN: 2278-1676,p-ISSN: 2320-3331, Volume 12, Issue 1 Ver. I (Jan. – Feb. 2017), PP 57-67
www.iosrjournals.org
Performance Comparison of Power Quality Evaluation Using
Advanced High Resolution Spectrum Estimation Methods
Md. Ziaul Hoque1
, Md. Raju Ahmed2
1 2
Research Scholar, Professor
Department of Electrical and Electronic Engineering
Dhaka University of Engineering and Technology, Bangladesh
Abstract: Most of the conventional methods of power quality assesment in power systems are almost
exclusively based on Fourier Transform that suffer from various inherent limitations. First limitation of an FFT
based method is that of frequency resolution, whereas the second limitation is due to no coherent signal
sampling of the data which proves itself as a leakage in spectrum domain. These two performance limitations of
FFT or similar methods are particularly troublesome when analyzing short data records. To overcome from this
problem, high regulation spectrum estimation methods can be used where resolution problem is not found. In
this thesis, high resolution methods, such as MUSIC, root MUSIC and ESPRIT are discussed that use a different
approach to spectral estimation; instead of trying to estimate the power spectral density (PSD) directly from the
data, they model the data as the output of a linear system driven by white noise, and then attempt to estimate the
parameters of that linear system. Detail Matlab simulations are carried out in order to investigate the
performance of MUSIC, Root MUSIC and ESPRIT methods in estimating amplitude, power (squared amplitude)
and frequency estimation of synthetic power signal both in clean and noisy conditions. Using mean square
error (MSE) as the evaluation criterion, the variation of amplitude, power (squared amplitude) and frequency
estimation are shown with respect to data sequence length and SNR and their influences on MSE are compared
for the different methods as mentioned above.
Keywords: Conventional Methods, Parametric & Nonparametric Methods, High Resolution Spectral Methods,
MUSIC, Root MUSIC and ESPRIT, Data Length, Signal to Noise Ratio, Means Square Error.
I. Introduction
Now a days, the quality of power or voltage waveforms has become a matter of great importance for
power utilities, electrical energy customers and additionally for manufacturers of electrical and electronic
equipment. The energy markets will strengthen the competition and is expected to drive down the prices of
energy which is the main part for the requirements concerning the power quality. Normally the power and
voltage waveform is expected to be a pure sinusoidal with a given amplitude and frequency. Most of the
fashionable frequency power converters generate a large spectrum of harmonic elements that indicate the
standard of the delivered energy and increase the energy. The proliferation of nonlinear loads connected to
power systems has triggered a growing concern with power quality issues. The inherent operation characteristics
of these loads deteriorate the quality of the delivered energy, and increase the energy losses as well as decrease
the reliability of a power system [1], [2], [6]. In some cases, large convertor systems generate not solely
characteristic harmonics typical for the perfect convertor operation however additionally a substantial quantity
of non characteristic harmonics and inter harmonics, which can powerfully verify the standard of the power-
supply voltage [7], [8]. So, that higher management control and protection depend on the estimation of the
signal elements. Normally Power could be defined as the actual physical power, or more often, for convenience
with abstract signals, could be defined as the squared value of the signals. (Statisticians study the variance of a
set of data, but because of the analogy with electrical signals, still refer to it as the power spectrum).The total
power of the signals would be a time average since power has units of energy/time.
1 T
lim x(t)2
dt
T→∞ 2T −T
Also the power of a signal may be finite even if the energy is infinite. Power also could be defined as the squired
amplitude of a particular signal.
P =
1
N
Ai
2N
i=1
Where 𝐴𝑖 is the amplitude of a particular signal.
In power system frequency, amplitude and phasor are the most important parameters for monitoring,
control and protection. All of these can reflect the whole power system situation. In electrical power system it is
of utmost important to keep the frequency as close to its original value as possible. In order to control the power

Performance Comparison of Power Quality Evaluation Using Advanced High Resolution Spectrum Estimation Methods
system frequency it needs to be measured quickly and accurately. Normally power system voltage and current
waveforms are distorted by harmonic and inter harmonic components, particularly during system disturbance.
Faults or other switching transients may change the magnitude and phase angles of the waveforms. However,
voltage and current can also be distorted by non-linear loads, power electronic components and inherent non-
linear nature of the system elements [8]. Not only that the assessment of power quality can be done either by
calculating, measuring or estimating power quality indices (frequency, spectrum, harmonic distortion etc.). So
that the estimation of power quality indices is still an important and yet challenging part in power system. Only
the estimation techniques can improve the accuracy of measurement of spectral parameters of distorted
waveforms encountered in power systems, in particular the estimation of the power quality indices [9]. More
reliable methods are required for power quality monitoring and estimation.
II. The Proposed Methods
There are various kinds of methods that are used in power, amplitude and frequency estimation to
improve the accuracy of measurement of spectral parameters of distorted waveforms encountered in power
systems, in particular the estimation of the power quality indices but the conventional high resolution methods
are more effective. In this chapter, the description of the methods of high resolution especially MUSIC, ESPRIT
and ROOT MUSIC are provided.
A. MUSIC Method
The idea of MUSIC (Multiple Signal Classification) was developed in [15] where the averaging was
proposed for improvement of the performance of Pisarenko estimator [16]. Instead of using only one noise
eigenvector, the MUSIC method uses many noises Eigen filters. The number of computed eigen values M > K
+1. MUSIC method is a method which presents signal sub space and using this method we can get more
accurate result than normal methods.
Let a signal to describe the MUSIC method.
x= Ai
p
i=1 si + ŋ; Ai= Ai ejφi (1)
Where Si = [1 e
jω
1….ej(N−1)ω2 ]T
, 𝐴𝑖-amplitudes of the signal components, N-number of signal samples, p-
number of components,ŋ-noise,𝜑𝑖-components frequency;
Now let a signal samples to estimate autocorrelation matrix as:
Rx = E{AiAi
∗
}
p
i=1
sisi
T
+ σ0
2
I (2)
N-p smallest eigen values of the correlation matrix (matrix dimension N>p+1) correspond to the noise subspace
and p largest (all greater than 𝜎0
2
- noise variance) correspond to the signal sub space.
The matrix of noise eigenvectors of the above matrix (2) is used
Enoise =[ep+1 ep+2 … . . eN ] (3)
to compute the projection matrix for the noise subspace:
Pnoise = Enoise Enoise
∗T
(4)
Which, by using an auxiliary vector W=[1 ejωt … . . ej(N−1)ωt ]T
allows computation of projection of vector, w
onto the noise subspace as:
w∗T
Pnoise w = w∗T
Enoise Enoise
∗T
w =
= Ei(ejω
)N
i−p+1 Ei
∗
(ejω
)
z
→ Ei z Ei
∗
(1/z∗
)N
i−p+1 (5)
The last polynomial in (5) has p double roots lying on the unit circle, which angular positions correspond to the
frequencies of the signal components. This method of finding the frequencies is therefore called root-MUSIC.
After the calculation of the frequencies, the powers of each component can be estimated from the eigen values
and eigenvectors of the correlation matrix, using the relations
ei
∗T
Rxei = λi and Rx = Pisisi
∗Tp
i=1 + σ0
2
I (6)
And solving for pi − components power.
B. Root MUSIC Method
Root-MUSIC method facilitates the same ideas with MUSIC and differs only in the second step of the
MUSIC algorithm. The main advantage of Root-MUSIC over MUSIC is its lower computational complexity.
The MUSIC spectrum is an all pole function of the form
Pmu θ =
1
abs [F θ H UN UN
H F(θ)]
(7)
Let C=UNUN
H
using equation (7) written as
Pmu
−1
= exp
j m−1 2πdsin θb
λ
Cmn AM
n=1
M
m=1 (8)

Where A=𝑒𝑥𝑝
−𝑗 𝑚 −1 2𝜋𝑑𝑠𝑖𝑛 𝜃 𝑏
𝜆 , and 𝐶 𝑚𝑛 is the entry in the 𝑚 𝑡ℎ
row and 𝑛 𝑡ℎ
column of C. Combination of
two sums into one gives equation (9):
Pmu
−1
= C1
M
n=1 exp
−j2πdlsin θb
λ
(9)
Where C1 = Cmnm−n=l is the sum of the entries of C. Along the 𝑙 𝑡ℎ
diagonal polynomial representation D (z)
will
D(z)= C1z−1M+1
l=−M+1 (10)
If the eigen decomposition corresponds to the true spectral matrix, then MUSIC spectrum becomes equivalent to
the polynomial D(z) on the unit circle and peaks in the MUSIC spectrum exists as ROOTs of the D(z) lie close
to the unit circle [17]. A pole of D (z) at z=𝑧1 = │𝑧1│exp (jarg (𝑧1)) will result in a peak in the MUSIC
spectrum at θ = sin−1
({λ/2rd} arg [z1]).
C. ESPRIT Method
The original ESPRIT (Estimation of Signal Parameter via Rotational Invariance Technique) was
developed by another one as in example [13].It is based on a naturally existing shift invariance between the
discrete time series which leads to rotational invariance between the corresponding signal subspaces. The shift
invariance is illustrated below. After the Eigen-decomposition of the autocorrelation matrix as:
Rx = U∗T
A (11)
It is possible to partition a matrix by using special selector matrices which select the first and the last (M-1)
columns of a (M ×M) matrix, respectively:
Г1 = [IM−1|0(M−1)×1] M−1 ×M (12)
Г1 = [0(M−1)×1 |IM−1] M−1 ×M
By using of matrices Г two subspaces are defined, spanned by two subsets of eigenvectors as follows:
S1 = Г1U
(13)
S2 = Г2U
For the matrices defined as 𝑆1and 𝑆2 in (13), for every 𝜔 𝑘 ; 𝐾𝜖𝑁, representing different frequency components,
and matrix ф, defined as:
Ф =
ejω1
0
0 ejω2
⋯ 0
0 0
⋮ ⋮
0 0
⋮ ⋮
⋯ ejωk
(14)
The following relation can be proven [14]:
[Г1U]ф=Г2 (15)
The matrix ф contains all information about frequency components. In order to extract this information, it is
necessary to solve (15) for ф. By using a unitary matrix (denoted as T), the following equations can be derived:
Г1(UT)ф=Г2(UT)
Г1U TфT∗T
= Г2 (16)
In the further considerations the only interesting subspace is the signal subspace, spanned by signal
eigenvectors 𝑈𝑠. Usually it is assumed that these eigenvectors correspond to the largest Eigen values of the
correlation matrix and Us = [u1, u2, … … … . , uk]. ESPRIT algorithm determines the frequencies 𝑒 𝑗𝜔𝑘
as the
eigenvalues of the matrix ф. In theory, the equation (15) is satisfied exactly. In practice, matrices 𝑆1 and 𝑆 1are
derived from an estimated correlation matrix, so this equation does not hold exactly, it means that (15)
represents an over-determined set of linear equations.
III. Simulation Results And Performance Comparison
A. Simulation Conditions
>> Signal y = .99 ∗ sin 2 ∗ π ∗ t ∗ 100 + .97 ∗ sin 2 ∗ π ∗ t ∗ 150 is used.
>> Random noise is generated by using randn function in Matlab.
>> Sampling frequency 50< Fs <1000 Hz.
>> SNR varies from 100 to 0 dB.
>> Signal length from 50 to 400 samples is considered.
B. Performance Evaluation Criterion
The criterion used for evaluating the MUSIC, ESPRIT and Root MUSIC is mean square error (MSE)
with respect to data length and SNR. The equation of MSE is given by,

For amplitude estimation, AMSE = √[(
1
N
) (Aiest − Aorg )2
]N
i=1 .
For Frequency estimation, FMSE = √[(
1
N
) (fiest − forg )2
]N
i=1 .
And for power estimation, PMSE = √[(
1
N
) (Aiest
2
− Aorg
2
)]N
i=1 .
Where
Aiest = Estimated amplitude
Aorg = Original amplitude
fiest = Estimated frequency
forg = Original frequency
C. Amplitude Estimation
Amplitude is the most important parameter in power system monitoring, control, and protection. It can
reflect the whole power system situation. In electrical power system it is of utmost importance to keep the
amplitude as close to its nominal value as possible. In order to control the power system amplitude it needs to be
measured quickly and accurately. For this the estimation of amplitude is still an important and yet challenging
part. The effects of data length and SNR on MSE for amplitude estimation are following:
D. Power Estimation
In general, power system voltage and current waveforms are distorted by harmonic and inter harmonic
components, particularly during system disturbance. Faults or other switching transients may change the
magnitude and phase angles of the waveforms. However, voltage and current can also be distorted by non-linear
loads, power electronic components and inherent non-linear nature of the system elements [3].Not only that the
assessment of power quality can be done either by calculating, measuring or estimating power quality indices
(frequency, spectrum, harmonic distortion etc.).Only the estimation techniques can improve the accuracy of
measurement of spectral parameters of distorted waveforms encountered in power systems, in particular the
estimation of the power quality indices [4].For these reason the power estimation is very important. The effects
of data length and SNR on power estimation are following:
E. Frequency Estimation
Frequency is the most important parameter in power system monitoring, control, and protection. It can
reflect the whole power system situation. In electrical power system it is of utmost importance to keep the
frequency as close to its nominal value as possible. In order to control the power system frequency it needs to be
measured quickly and accurately. But in general, power system voltage and current waveforms are distorted by
harmonic and inter harmonic components, particularly during system disturbance. Faults or other switching
transients may change the magnitude and phase angles of the waveforms. So the estimation of frequency is still
an important and yet challenging part in power system. The effect of data length and SNR on MSE for
frequency estimation is shown below:
F. Effect of Data Length on MSE for Amplitude Power and Frequency Estimation
The data sequence length influences the mean square error and therefore, the accuracy of high
resolution methods depends on data length. The performance of the high resolution methods (Root MUSIC,
MUSIC and ESPRIT) could be identified by comparing the mean square error of Amplitude, Power and
Frequency estimation both for shorter and higher data length. Both for clean and noisy signal, the performance
of the mean square error of amplitude, Power and Frequency estimation of the high resolution methods (Root
MUSIC, MUSIC, and ESPRIT) are shown in Figure (1, 2, 3, 4, 5 and 6). When roughly summarizing different
results from the Figure. (1, 2, 3, 4, 5 and 6) a list of data of Amplitude, Power and Frequency estimation both for
clean and noisy signal in terms of MSE for increasing data length can be represented, as show that in Table (1,
2,3, 4, 5 and 6).

Fig. 1: Amplitude estimation (MUSIC & Root MUSIC) in terms of MSE with respect to data length for clean
signal in log scale.
Table 1: The amplitude estimation for clean signal in terms of MSE for increasing data length
Data Length MUSIC Method Root MUSIC Method
50 10.17 -0.04498
100 8.989 -0.08449
150 7.142 -0.12770
200 6.557 -0.17530
250 6.183 -0.22820
300 6.742 -0.28760
350 5.973 -0.35490
400 4.307 -0.43100
Fig. 2: Amplitude estimation (MUSIC & Root MUSIC) for noisy signal in terms of MSE with respect to data
length in log scale.
Table 2: The amplitude estimation for noisy signal in terms of MSE for increasing data length
50 9.685 -0.04504
100 8.981 -0.08450
150 7.142 -0.12770
200 6.557 -0.17530
250 6.183 -0.22820
300 6.742 -0.28760
50 100 150 200 250 300 350 400
10
0
10
1
10
2
X: 50
Y: 10.17
X: 100
Y: 8.989
X: 150
Y: 7.142
X: 200
Y: 6.557
X: 250
Y: 6.183
X: 300
Y: 6.742
X: 350
Y: 5.973
MSE of amplitude estimation (MUSIC)depending on data length
Data length
log10(MSE)
X: 400
Y: 4.307
50 100 150 200 250 300 350 400
-10
0
-10
-1
-10
-2
X: 50
Y: -0.04504 X: 100
Y: -0.0845 X: 150
Y: -0.1277 X: 200
Y: -0.1753
X: 250
Y: -0.2282
X: 300
Y: -0.2876
X: 350
Y: -0.3549
MSE of amplitude estimation (RootMUSIC) depending on data length
Data length
log10(MSE)
X: 400
Y: -0.4315
50 100 150 200 250 300
10
0.8
10
0.9
X: 50
Y: 9.685
X: 100
Y: 8.981
X: 150
Y: 7.142
X: 200
Y: 6.557 X: 250
Y: 6.183
MSE of amplitude estimation (MUSIC) depending on data length
Data length
log10(MSE)
50 100 150 200 250 300
-10
0
-10
-1
-10
-2
X: 300
Y: -0.2876
X: 250
Y: -0.2282
X: 200
Y: -0.1753
X: 150
Y: -0.1277
X: 100
Y: -0.08455
MSE of amplitude estimation (RootMUSIC) depending on data length
Data length
log10(MSE)
X: 50
Y: -0.02793

Fig. 3: Power estimation (MUSIC & Root MUSIC) for clean signal in terms of MSE with respect to data length
in log scale.
Table 3: The power estimation for clean signal in terms of MSE for increasing data length
50 20.39 -0.02034
100 18.13 -0.02924
150 14.41 -0.04442
200 13.23 -0.06647
250 12.43 -0.09625
300 13.63 -0.13500
350 12.07 -0.18440
400 8.673 -0.24640
Fig. 4: Power estimation (MUSIC & Root MUSIC) for noisy signal in terms of MSE with respect to data length
in log scale.
Table 4: The power estimation for noisy signal in terms of MSE for increasing data length
50 17.09 -0.01910
100 18.12 -0.02921
150 14.41 -0.04441
200 13.23 -0.06647
250 12.43 -0.09625
300 13.63 -0.13500
350 5.973 -0.35490
400 4.307 -0.39690
50 100 150 200 250 300 350 400
10
0
10
1
10
2
X: 50
Y: 20.37 X: 200
Y: 13.23
X: 100
Y: 18.13
X: 150
Y: 14.41
X: 250
Y: 12.43
X: 300
Y: 13.63
X: 350
Y: 12.07
X: 400
Y: 8.673
MSE of Power estimation(MUSIC)depending on data length
Data length
log10(MSE)
50 100 150 200 250 300 350 400
-10
0
-10
-1
-10
-2 X: 100
Y: -0.02924 X: 150
Y: -0.04442 X: 200
Y: -0.06647
X: 250
Y: -0.09625
X: 300
Y: -0.135
X: 350
Y: -0.1844
X: 400
Y: -0.2464
X: 50
Y: -0.02034
MSE of Power estimation (RootMUSIC) depending on data length
Data length
log10(MSE)
50 100 150 200 250 300
10
1.1
10
1.2 X: 50
Y: 17.09
X: 300
Y: 13.63
X: 100
Y: 18.12
X: 150
Y: 14.41
X: 200
Y: 13.23
MSE of Power estimation(MUSIC)depending on data length
Data length
log10(MSE)
X: 250
Y: 12.43
50 100 150 200 250 300
-10
0
-10
-1
-10
-2
X: 50
Y: -0.0191
X: 250
Y: -0.09625
X: 300
Y: -0.135
X: 200
Y: -0.06647
X: 150
Y: -0.04441
MSE of Power estimation (RootMUSIC) depending on data length
Data length
log10(MSE)
X: 100
Y: -0.02921

Fig. 5(a): Frequency estimation (MUSIC & Root MUSIC) for clean signal in terms of MSE with respect to data
Fig. 5(b): Frequency estimation (MUSIC & Root MUSIC) for clean signal in terms of MSE with respect to data
length in linear scale.
Table 5: The Frequency estimation for clean signal in terms of MSE for increasing data length
Data Length MUSIC Root MUSIC ESPRIT
50 1.897 1.898 1.898
100 1.896 1.898 1.898
150 1.895 1.898 1.898
200 1.891 1.898 1.898
250 1.891 1.898 1.898
300 1.886 1.898 1.898
350 1.885 1.898 1.898
400 1.883 1.898 1.898
50 100 150 200 250 300 350 400
10
0.32326
10
0.32333
MSE of frequency estimation (MUSIC) depending on data length
Data length
log10(MSE)
50 100 150 200 250 300 350 400
10
0.323334
10
0.323339
MSE of frequency estimation (Root MUSIC) depending on data length
Data length
log10(MSE)
50 100 150 200 250 300 350 400
10
0.32333
10
0.323338
MSE of frequency estimation (ESPRIT) depending on data length
Data length
log10(MSE)
50 100 150 200 250 300 350 400
2.105
2.105
2.1051
2.1052
2.1052
2.1052
2.1053
2.1054
2.1054
2.1054
MSE of frequency estimation depending on data length
Data length
log10(MSE)
MUSIC
Root MUSIC
ESPRIT

Fig. 6(a): Frequency estimation (MUSIC & Root MUSIC) for noisy signal in terms of MSE with respect to data
Fig. 6(b): Frequency estimation (MUSIC & Root MUSIC) for noisy signal in terms of MSE with respect to data
length in linear scale.
Table 6: The amplitude estimation for noisy signal in terms of MSE for increasing data length
Data Length MUSIC ESPRIT Root MUSIC
50 1.897 1.898 1.898
100 1.896 1.898 1.898
150 1.895 1.898 1.898
200 1.891 1.898 1.898
250 1.891 1.898 1.898
300 1.886 1.898 1.898
350 1.885 1.898 1.898
400 1.883 1.898 1.898
In Fig 1,2,3 and 4, it is seen that there is a sharp decrease of the estimation error for increasing length
of the data sequence (pattern for MUSIC and Root MUSIC method results are similar). Root MUSIC method
performs better for amplitude and power estimation in terms of MSE. In Fig 5 and Fig 6, it is seen that there is a
sharp decrease of the estimation error for increasing length of the data sequence (pattern for MUSIC, Root
MUSIC and ESPRIT method results are similar). Though it is seen that MUSIC method performs better for
frequency estimation but for many simplifications, different assumptions and the complexity of the problem
ESPRIT method is better than MUSIC and Root MUSIC.
50 100 150 200 250 300 350 400
10
0.275
10
0.278
MSE of frequency estimation(MUSIC)depending on data length
Data length
log10(MSE)
50 100 150 200 250 300 350 400
10
0.27819
10
0.27826
MSE of frequency estimation (RootMUSIC) depending on data length
Data length
log10(MSE)
50 100 150 200 250 300 350 400
10
0.27819
10
0.27827
MSE of frequency estimation(Esprit)depending on data length
Data length
log10(MSE)
50 100 150 200 250 300 350 400
2.105
2.105
2.1051
2.1052
2.1052
2.1052
2.1053
2.1054
2.1054
2.1054
MSE of frequency estimation depending on data length
Data length
log10(MSE)
MUSIC
Root MUSIC
ESPRIT

G. Effect of SNR on MSE for Amplitude, Power and Frequency Estimation
There is a strong dependency of the accuracy of the frequency estimation on SNR. The performance of
the high resolution methods (MUSIC & Root MUSIC) could be identified by comparing the mean square error
of amplitude, power and frequency estimation both for very low and very high noise levels.Both for low and
very high noise level the performance of the mean square error of amplitude estimation of the high resolution
methods (MUSIC & Root MUSIC and ESPRIT) are shown in Figure (1, 2, 3, 4, 5 and 6). When roughly
summarizing different results from the Figure. (7,8 and 9) a list of data of Amplitude, Power and Frequency
estimation both for low and very high noise level the performance of the mean square error can be represented,
as show that in Table (7,8 and 9).
Fig. 7: Amplitude estimation (MUSIC & Root MUSIC) in terms of MSE with respect to SNR in log scale.
Table 7: Amplitude estimation (MUSIC & Root MUSIC) in terms of MSE for changing SNR
0 3.798 -0.5897
10 3.635 -0.5998
20 3.631 -0.6003
30 3.632 -0.6004
40 3.632 -0.6004
50 3.632 -0.6004
60 3.632 -0.6004
70 3.632 -0.6004
80 3.632 -0.6004
90 3.632 -0.6004
100 3.632 -0.6004
Fig. 8: Power estimation (MUSIC & Root MUSIC) in terms of MSE with respect to SNR in log scale.
0 10 20 30 40 50 60 70 80 90 100
10
0.56
10
0.57
X: 30
Y: 3.632
X: 10
Y: 3.635
X: 20
Y: 3.631
X: 40
Y: 3.632
X: 50
Y: 3.632
X: 60
Y: 3.632
X: 70
Y: 3.632
X: 80
Y: 3.632
X: 90
Y: 3.632
X: 0
Y: 3.798
MSE of Amplitude estimation (MUSIC)depending on SNR
SNR [db]
log10(MSE)
0 10 20 30 40 50 60 70 80 90 100
-0.61
-0.6
-0.59
-0.58
X: 0
Y: -0.5897
X: 10
Y: -0.5998
X: 20
Y: -0.6003
X: 30
Y: -0.6004
X: 40
Y: -0.6004
X: 50
Y: -0.6004
X: 60
Y: -0.6004
X: 70
Y: -0.6004
X: 80
Y: -0.6004
MSE of Amplitude estimation (Root MUSIC)depending on SNR
SNR [db]
log10(MSE)
X: 90
Y: -0.6004
0 10 20 30 40 50 60 70 80 90 100
10
0.85
10
0.88
X: 10
Y: 7.27
X: 0
Y: 7.69
X: 30
Y: 7.273
X: 50
Y: 7.274
X: 70
Y: 7.274
X: 90
Y: 7.274
X: 20
Y: 7.269
X: 40
Y: 7.274
X: 60
Y: 7.274
MSE of Power estimation(MUSIC)depending on SNR
SNR [db]
log10(MSE)
X: 80
Y: 7.274
0 10 20 30 40 50 60 70 80 90 100
-10
-0.411
-10
-0.406
X: 10
Y: -0.3913
X: 70
Y: -0.3923
X: 20
Y: -0.3921
X: 30
Y: -0.3924
X: 40
Y: -0.3923
X: 50
Y: -0.3923
X: 60
Y: -0.3923
X: 80
Y: -0.3923
X: 0
Y: -0.3876
MSE of Power estimation(RootMUSIC)depending on SNR
SNR [db]
log10(MSE)
X: 90
Y: -0.3923

Table 8: Power estimation (MUSIC & Root MUSIC) in terms of MSE for changing SNR
0 7.690 -0.3876
10 7.270 -0.3967
20 7.273 -0.3943
30 7.274 -0.3933
40 7.274 -0.3923
50 7.274 -0.3923
60 7.274 -0.3923
70 7.274 -0.3923
80 7.274 -0.3923
90 7.274 -0.3923
100 7.274 -0.3923
Fig. 9(a): Frequency estimation (MUSIC & Root MUSIC) in terms of MSE with respect to SNR in log scale.
Fig. 9(b): Frequency estimation (MUSIC & Root MUSIC) in terms of MSE with respect to SNR in linear scale.
Table 9: Frequency estimation (MUSIC & Root MUSIC) in terms of MSE for changing SNR
SNR MUSIC Method Root MUSIC Method ESPRIT Method
0 2.1051 2.10543 2.10537
10 2.1045 2.10541 2.10537
20 2.1045 2.10541 2.10537
30 2.1045 2.10541 2.10537
40 2.1045 2.10541 2.10537
50 2.1045 2.10541 2.10537
60 2.1045 2.10541 2.10537
70 2.1045 2.10541 2.10537
80 2.1045 2.10541 2.10537
90 2.1045 2.10541 2.10537
100 2.1045 2.10541 2.10537
0 10 20 30 40 50 60 70 80 90 100
2.105
2.1051
2.1052
2.1052
2.1052
2.1053
2.1053
2.1054
2.1054
M SE of frequency estimation depending on SNR
SNR [db]
log(MSE)
MUSIC
Root MUSIC
ESPRIT

In Fig 7 and Fig 8, it is seen that there is a sharp decrease of the estimation error for changing SNR
(pattern for MUSIC and Root MUSIC method results are similar). Root MUSIC method performs better for
amplitude and power estimation in terms of MSE. In Fig 9, it is seen that there is a sharp decrease of the
estimation error for changing SNR (pattern for MUSIC, Root MUSIC and ESPRIT method results are similar).
MUSIC method performs better for frequency estimation in terms of MSE.
IV. Conclusion
The performance evaluation criterion and simulation results of the proposed methods are described in
details. And the conclusion is that both methods (Root MUSIC & ESPRIT) are similar in the sense that they are
both eigen decomposition based methods which rely on decomposition of the estimated correlation matrix into
two subspaces: noise and signal subspace. On the other hand, MUSIC uses the noise subspace to estimate the
signal components while ESPRIT uses the signal subspace. In addition, the approach in many points is different.
Root MUSIC method performs better for amplitude and power estimation in terms of MSE with respect to data
length and SNR. Due to many simplifications, different assumptions and the complexity of the problem,
simulation results represent the performance of ESPRIT method is better for frequency estimation in terms of
MSE with respect to data length. Finally the performance of MUSIC method is better for frequency estimation
with respect to SNR. The major contributions of this Paper are: Scopes for Future Works, In practical cases, it
could be investigated and compared the performance of MUSIC, Root MUSIC and ESPRIT methods in terms of
MSE with respect to the size of the correlation matrix. In practical cases, it could be investigated and compared
the performance of amplitude estimation of MUSIC and ESPRIT in terms of MSE with respect to data length
and SNR. The performance of amplitude, power and frequency estimation in terms of MSE of all the Methods
mentioned above would be compared with respect to other high resolution methods.
References
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International Conference on Harmonics and Quality of Power(ICHQP),
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[4] T.Lobs, Z.Leonowicz and J. Rezmer, “harmonics and interharmonics estimation using advanced signal processing methods” in
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physical quantity.
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