estimating frequency for different time-frequency analysis ...€¦ · results obtained in the...
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Estimating frequency for different time-frequency analysis
methods using Matlab®
Minor Thesis Report
By
Muhamad Zuhaili Khairul Anuar
Id No: 110163661
Bachelor in Technology Electrical Engineering
Master in Engineering Electrical Power
Supervisor: Mr Kevin Rogers
School of Engineering
University of South Australia
i
ABSTRACT
Signal processing is crucial in the electrical industry as obtaining quality results from the
process is an important task. This leads to enhancing the proper methods to analyse variety
of signals and to utilise the information in the signals. One of the analysis in signal
processing is time-frequency analysis. This analysis provides two dimensional views in
time and frequency domains as representation of signals. Results obtained in the time-
frequency analysis need to be accurate in order to be used in a particular field. Thus, in this
thesis, an accuracy of four different methods of time-frequency analysis was investigated.
The methods are Wigner-Ville Distribution (WVD), Pseudo Wigner-Ville distribution
(PWVD), Smooth Pseudo Wigner-Ville distribution (SPWVD) and Spectrogram (SPEC).
These four methods are compared to each other and are analysed in terms of their
performances and capabilities referring to the peak frequency estimation. MATLAB®
software and time-frequency toolbox are applied to generate signals and help the simulation
process. This research has been conducted through simulations and tested using four
different types of signals which are sinusoidal sine wave, signal with linear frequency
modulation, signal with constant frequency modulation, and signal with sinusoidal
frequency modulation. These signals are tested in three conditions, which are signal with
no noise, signal with low standard deviation of noise (from 0.05 to 0.30 standard deviation
of noise) and signal with high standard deviation of noise (from 0.35 to 0.75 standard
deviation of noise). Each method is repeated a hundred times to make sure the results taken
were precise and reproducible. Analysis from the results are done by taking the difference
between the actual frequency value and the estimated frequency value using quadratic
interpolation of three adjacent points from the results obtained by each method. The
standard deviation of the distribution output is then compared against the best frequency
error by Cramer Roa Lower Bound. This analysis showed that for different signals, it has
different type of methods that gives accurate frequency estimation. It is dependent on the
condition and type of the signals. Thus, this research led to finding an appropriate method
for different types of signals in different conditions.
Keywords: time-frequency analysis, Fourier Transform, Wigner-Ville distribution, Pseudo
Wigner-Ville distribution, Smooth pseudo Wigner-Ville distribution, Spectrogram,
Quadratic interpolation three adjacent, estimated frequency.
ii
TABLE OF CONTENTS
Abstract ............................................................................................................................. i
Table of Contents ............................................................................................................. ii
Table of Figures .............................................................................................................. iv
List of Tables .................................................................................................................. iv
List of Equations .............................................................................................................. v
Acknowledgment ............................................................................................................ vi
Chapter 1 : Introduction ....................................................................................................... 1
1.1 Overview ................................................................................................................ 1
1.1.1 Signal processing ............................................................................................ 1
1.1.2 Time-frequency Analysis ................................................................................ 2
1.2 Problem Statement ................................................................................................. 2
1.3 Objectives of Study ................................................................................................ 2
1.4 Expected Contribution ........................................................................................... 3
1.5 Relevance of study ................................................................................................. 3
1.6 Structure of Thesis ................................................................................................. 3
Chapter 2 : Literature Review .............................................................................................. 4
2.1 Introduction ............................................................................................................ 4
2.2 History Spectral Analysis ...................................................................................... 4
2.3 Time-Frequency Analysis ...................................................................................... 5
2.4 Method of Time-Frequency Analysis .................................................................... 5
2.4.1 Fourier Transform ........................................................................................... 5
2.4.2 Short Time Fourier Transform ........................................................................ 5
2.4.3 Wigner-Ville Distribution ............................................................................... 6
2.5 Previous Studies ..................................................................................................... 7
2.6 Improvement Methods ........................................................................................... 8
2.7 Cross-Term ............................................................................................................ 9
2.8 Comparison Methods in term of Cross-term ......................................................... 9
2.9 Summary .............................................................................................................. 10
Chapter 3 : Methodology ................................................................................................... 11
3.1 Introduction .......................................................................................................... 11
3.2 Time-Frequency Analysis Methods ..................................................................... 11
3.2.1 Wigner-Ville Distribution ............................................................................. 12
iii
3.2.2 Pseudo Wigner-Ville Distribution ................................................................ 12
3.2.3 Smooth Pseudo Wigner-Ville Distribution ................................................... 12
3.2.4 Spectrogram .................................................................................................. 13
3.3 Estimating Frequencies ........................................................................................ 13
3.4 Error Analysis ...................................................................................................... 14
3.5 Flow Chart of Methodology................................................................................. 15
3.6 Summary .............................................................................................................. 15
Chapter 4 : Results And Discussions ................................................................................. 16
4.1 Introduction .......................................................................................................... 16
4.2 Flowchart of the Results ...................................................................................... 16
4.3 Sinusoidal Sine wave Signal ................................................................................ 17
4.4 Signal with Linear Frequency Modulation .......................................................... 18
4.5 Signal with Constant Frequency Modulation ...................................................... 19
4.6 Signal with Sinusoidal Frequency Modulation .................................................... 20
4.7 Graph for each signal ........................................................................................... 21
4.8 Discussions .......................................................................................................... 23
4.8.1 Comparing methods ...................................................................................... 23
4.9 Predicted Projection ............................................................................................. 25
4.9.1 Signal with Sinusoidal Sine Wave ................................................................ 25
4.9.2 Signal with Sinusoidal Linear Modulation ................................................... 26
4.9.3 Signal with Sinusoidal Constant Modulation ............................................... 27
4.9.4 Signal with Sinusoidal Frequency Modulation ............................................. 28
4.10 Summary ............................................................................................................ 28
Chapter 5 : Recommendations and Conclusion ................................................................. 29
5.1 Conclusion ........................................................................................................... 29
5.2 Recommendations for Future Work..................................................................... 30
References ...................................................................................................................... 31
Appendices ..................................................................................................................... 35
A1 Programming in MATLAB® .......................................................................... 35
A2 Sample result from MATLAB® for each signals............................................ 37
A2.1 Signal Sinusoidal Sine wave .................................................................... 37
A2.2 Signal with Linear frequency Modulation ............................................... 37
A2.3 Signal with constant frequency Modulation ............................................. 38
A2.4 Signal with sinusoidal frequency Modulation .......................................... 38
iv
TABLE OF FIGURES
Figure 1: View from time and frequency domain ................................................................ 1
Figure 2: Comparison Methods in term of Cross-Term..................................................... 10
Figure 3: Flowchart of Methodology ................................................................................. 15
Figure 4: Flowchart of Results ........................................................................................... 16
Figure 5: Graph for Sinusoidal Sine wave Signal .............................................................. 21
Figure 6: Graph for Signal with Linear Frequency Modulation ........................................ 21
Figure 7: Graph for Signal with Constant Frequency Modulation .................................... 22
Figure 8: Graph for Signal with Sinusoidal Frequency Modulation .................................. 22
Figure 9: Prediction until SD of noise at 2 for Sinusoidal Sine Wave .............................. 25
Figure 10: Prediction until SD of noise at 2 for linear frequency signal ........................... 26
Figure 11: Prediction until SD of noise at 2 for constant frequency signal ....................... 27
Figure 12: Prediction until SD of noise at 2 for sinusoidal frequency signal .................... 28
LIST OF TABLES
Table 1: Results for Sinusoidal Sine wave Signal ............................................................. 17
Table 2: Result for Signal with Linear Frequency Modulation ......................................... 18
Table 3: Result for Signal with Constant Frequency Modulation ..................................... 19
Table 4: Result for Signal with Sinusoidal Frequency Modulation ................................... 20
Table 5: Comparison in No Noise Condition .................................................................... 23
Table 6: Comparison in Low Standard Deviation of Noise ............................................... 24
Table 7: Comparison in High Standard Deviation of Noise .............................................. 24
v
LIST OF EQUATIONS
Equation 1: Fourier Transform ............................................................................................ 5
Equation 2: Inverse Fourier Transform ................................................................................ 5
Equation 3: Short Time Fourier Transform ......................................................................... 6
Equation 4: Cross-Term Wigner-Ville Distribution ............................................................ 9
Equation 5: Cross-Term Spectrogram ................................................................................. 9
Equation 6: Wigner-Ville Distribution .............................................................................. 12
Equation 7: Pseudo Wigner-Ville Distribution .................................................................. 12
Equation 8: Smooth Pseudo Wigner-Ville Distribution .................................................... 12
Equation 9: Spectrogram .................................................................................................... 13
Equation 10: Estimating frequency using Matrix .............................................................. 13
Equation 11: Estimating Maximum Frequency ................................................................. 14
Equation 12: Error Analysis using Cramer Roa Lower Bound ......................................... 14
vi
ACKNOWLEDGMENT
A journey of a thousand miles begins with the first step. All praises to God, finally my
journey to complete this minor thesis has arrived at the final stage. First and foremost, I
would like to express my gratitude to God for the blessings that were given to me. Without
His blessings I would not be able to complete this minor thesis. On this occasion, I would
like to express my appreciation to all those who were involved throughout this research,
especially to my supervisor, Mr. Kevin Rogers for his guidance and insights that had helped
me a lot throughout this research. He was very supportive and his continuous optimism in
helping me to accomplish the tasks that I needed to do. Also, my biggest appreciation goes
to my beloved parents, Mr. Khairul Anuar and Mdm. Noriah Jaafar, and my siblings who
have always been motivating and encouraging even though we are thousands of miles apart.
They have kept me in their prayers through thick and thin. They have supported me a lot,
spiritually and financially, so that I could arrive to where I am now. I also want to dedicate
my thanks and appreciation to all my course mates, who were willing to share their
knowledge, information and other forms of assistance that I needed, especially in producing
this minor thesis. All forms of help that they provided are very much appreciated. Finally,
I would like to express my deepest thanks to those who were directly or indirectly involved
in the making of this minor thesis. Your willingness is very much appreciated.
1
CHAPTER 1 : INTRODUCTION
1.1 Overview
A signal is a variation of a quantity by which information is delivered in terms of the
characteristics, the composition, the trajectory, the evolution and other parameters (Vaseghi
2008). It can be described as a physical quantity that can be measured. Based on IEEE
Transactions on Signal Processing, the term signal includes audio, image, video, sound,
speech, communication, geophysical, sonar, radar, medical, and musical signals (Reddy,
Rayel & Rao 2015). These signals can be categorized into several classes based on some
criterion such as continuous or discrete signal, periodic or non-periodic signal,
deterministic or random signal and stationary or non-stationary signal.
1.1.1 Signal processing
It is significant to process these signal contents to utilise its information. This process is
called signal processing. Signal processing is a broad topic of discussion. It includes, among
others, image processing, video processing, audio processing, medical electronic, voice
recognition, automation system, etc. One of the most important branches of signal
processing is time-frequency analysis. This analysis produces a signal representation in the
time and frequency domain concurrently. As seen in Figure 1, the signal can be represented
from two different viewpoints. One is in the time domain and the other is in the frequency
domain. Another way of representing signals that is widely used is spectral analysis.
Spectral analysis shows the spectral density of signal variations with respect to frequencies
(Brown 1997). In summary, time-frequency analysis represents both of the domains while
spectral analysis only shows the frequency domain.
Figure 1: View from time and frequency domain (Brown 1997)
2
1.1.2 Time-frequency Analysis
There has been a significant number of methods developed to optimise the performance of
time-frequency analysis. In this analysis, it can be classified into two classes, which are
linear time-frequency representation and bilinear time-frequency representation or known
as quadratic representation (Auger et al. 1996). These classes have their advantages and
disadvantages depending on the application being used. Therefore, it is crucial to identify
the abilities of each methods and their performance in each field to ensure that the results
obtained is applicable. There are several terms that are similar yet different such as time-
frequency representation, time-frequency analysis, and time-frequency distribution. Time-
frequency representation (TFR) is a view of a signal, taken by various methods, to be a
function of time and represented in time and frequency domain (Sejdić, Djurović & Jiang
2009). Time-frequency analysis (TFA) is the analysis of the time-frequency domain
provided by time-frequency representation (Sejdić, Djurović & Jiang 2009). To analyse the
domain, the method used is called time-frequency distribution (TFD) (Cohen 1995).
1.2 Problem Statement
This research is important because the transformation of a signal from one domain to
another domain is often subjected to losses or corrupted results. Yet very little research is
done that compares the methods in time-frequency analysis which produces the result that
comes from this representation. Therefore, it is crucial to investigate whether the result can
be proven as valid.
1.3 Objectives of Study
Objectives of this study are:
1. To determine the accuracy of four types of time-frequency analysis in estimating
signal frequencies.
2. To obtain an estimated frequency of signals with and without noise for the four
different time-frequency analysis methods being analysed.
3. To analyse and recommend the suitable method for different level of noise
conditions and different type of signals.
3
1.4 Expected Contribution
The findings from this thesis are hoped to compare the performances of WVD, PWVD,
SPWVD, and SPEC on four types of signals at three different conditions of the signal. This
comparison will give an idea as to which method is best suited for each signal in different
conditions.
1.5 Relevance of study
As mentioned in the problem statement, it is important to know how close the results shown
in the representation are to the actual value. Thus, this thesis examines the accuracy of the
methods at three different conditions of the noise in the signals. From that, it can lead to
achieving the expected contribution.
1.6 Structure of Thesis
This minor thesis consists of five chapters. The structure of this thesis is as outline below:
Chapter 1 provides an introduction with research background, problem statement,
objectives of study, expected contribution of the thesis and the relevance of the study.
Chapter 2 provides a literature review of time frequency analysis, a brief history of spectral
analysis, previous study that compares some of the method that are used in this thesis, and
improvements of the time-frequency analysis method that had been done previously.
Chapter 3 explains the methodology and experimental procedures of this thesis to do the
analysis on estimating frequency, description of four methods are used and flowchart of
programming in Matlab®.
Chapter 4 contains the results obtained from the simulation of the time-frequency analysis
by four methods for the signals and their discussions.
Chapter 5 presents the conclusion of this thesis and recommendations for future work.
4
CHAPTER 2 : LITERATURE REVIEW
2.1 Introduction
Analysing the signal with varying time-frequency is one of the important tasks performed
in signal processing. It is not sufficient to have one dimensional view of the signal contents.
The main purpose of a time-frequency analysis is to develop a two dimensional display in
time and frequency domains that can reveal the information of the signal (Preis &
Georgopoulos 1999). However, these two domains have some trade-off that has to be
considered. This is crucial in time-frequency representation as when the resolution in time
domain increases, the resolution in frequency will decrease, and vice-versa (Boashash,
White & Imberger 1986; Reddy, Rayel & Rao 2015). Thus, measuring frequency content
and energy density of signals are widely studied and there are many proposed methods to
accomplish this task. For example, measuring frequency by extraction of the peak from
time-frequency representation at certain time is been made by Andria, Savino & Trotta
(1994). They compare Short Time Fourier Transform with Wigner-Ville Distribution using
this methods on estimating frequency. Other than that, this section are presented more
comparison that have been made previously.
2.2 History Spectral Analysis
The history of the spectral analysis came about when Sir Isaac Newton performed an
experiment in 1704. He tested a glass of prism to resolve the sunbeam into the colours in a
rainbow and found the images of frequencies in the sunlight (Sandsten 2013). This spectral
analysis showed a spectrum of the signal. In 1807, Jean Baptiste Joseph Fourier found a
formula to handle the discontinuities function which can be expressed as the sum of a
continuous frequency function. This idea has been agreed by many scientists at that time
and was called Fourier expansion (Sandsten 2013). Later in the 19th century, Robert Bunsen
showed that every material has their own spectrum with different frequency contents.
Therefore, to discover frequency content in a particular material, spectral analysis need to
be conducted.
5
2.3 Time-Frequency Analysis
Time-Frequency analysis focuses on a distribution of the total energy of the signal at a
specific time and frequency (Staszewski, Worden & Tomlinson 1997). There are also
different parameters and properties which can be obtained simultaneously when time-
frequency analysis are performed such as the energy of the signal per unit time at specific
time can be seen when the time domain is used, and at the same time the energy density of
the signal per unit frequency can be seen when the frequency domain is used. This means
it is hard to determine precisely the signal in both time as well as frequency domain because
of the trade-off between frequency and time. If accuracy in frequency domain is increased,
it will decrease the information in time domain and vice-versa. Thus, it depend on the
application which parameter is more important. In this thesis, the frequency is interest
parameter to estimate.
2.4 Method of Time-Frequency Analysis
2.4.1 Fourier Transform
Fourier transform is a function to decompose a function of time into frequency and is also
called the frequency domain representations. Spectral analysis is used in Fourier Transform
to get the spectrum of the signals. Fourier transform of a continuous signal x(t) can be
define as indicate in Equation 1 (Rioul & Vetterli 1991).
𝑋(𝑓) = 𝐹{𝑥(𝑡)} = ∫ 𝑥(𝑡)𝑒−𝑗2𝜋𝑓𝑡𝑑𝑡+∞
−∞
Equation 1: Fourier Transform
And by doing an inverse Fourier Transform, the original signal x(t) is converted back as
shown in Equation 2 (Sandsten 2013).
𝑥(𝑡) = 𝐹−1{𝑋(𝑓)} = ∫ 𝑋(𝑓)𝑒𝑗2𝜋𝑓𝑡𝑑𝑓+∞
−∞
Equation 2: Inverse Fourier Transform
These two equations will bring the representation of time and frequency of the signal.
2.4.2 Short Time Fourier Transform
Fourier Transform only can show representation in frequency domain and cannot localized
signals in the time domain. In light of this, an improvement has been made which is called
the Short Time Fourier Transform (STFT) which is able to represent both time and
6
frequency domain (Wright 1999). STFT takes a linear approach for a time-frequency
representation and it decomposes the signal into elementary components called atoms. Each
atom is obtained from the window by translation in time and frequency. STFT can be define
as indicate in Equation 3 (Yun et al. 2013).
𝐹𝑥(𝑡, 𝜐; ℎ) = ∫ 𝑥(𝑢) ℎ∗ (𝑢 − 𝑡)𝑒−𝑗2𝜋𝜐𝑢𝑑𝑢+∞
−∞
Equation 3: Short Time Fourier Transform
The window function h(t) is centred at time (t), then multiplied with signal x(t) before the
Fourier Transform. Thus, Fourier Transform will estimate the frequency around time (t)
due to the window function.
2.4.3 Wigner-Ville Distribution
Wigner Distribution is the most popular distribution because it can concentrate on the signal
properties and widely been used such as in fault diagnosis (Wu & Chiang 2009). This
distribution has been develop by Eugene Wigner in quantum mechanics (Wigner 1932)
then implemented by J de Ville in signal processing (Ville 1948). The analysed signal was
described as the real value signal which is converted into a positive frequency signal. This
was also mentioned by Staszewski, Worden & Tomlinson (1997). The Wigner Distribution
which is used for analytic signal can be called as Wigner-Ville Distribution. It can be define
as a special signal transformation that shows an excellent time and frequency resolution
(Andria, Savino & Trotta 1994; Khadra 1988). It is very suitable for time-frequency
analysis of non-stationary signals and various papers proves that WVD can gives better
frequency concentration and less phase dependence than Fourier transform (Andria, Savino
& Trotta 1994; Velez & Absher 1992; Waldo & Chitrapu 1991). After the implementation,
this distribution was widely use in signal processing and became one of the popular topic
in frequency analysis research. Thus, there are lot of developments that have been made
such as Claasen & Mecklenbrauker (1980) who introduced a discrete approach to the
distribution and etc. Sandsten (2013) said that Wigner Distribution always gave good
resolution for mono component signal such as chirp signal and sinusoidal signal. It will
give an accurate instantaneous frequency for these signals. It also been used in other
application shown in (Baydar & Ball 2001; Hafeez, Zaidi & Siddiqui 2013; Iqbal et al.
2009; Staszewski, Worden & Tomlinson 1997; Szmajda, Górecki & Mroczka 2010).
7
2.5 Previous Studies
Comparison of methods is a popular study topic among researchers in frequency analysis
techniques. One of the research is estimating the instantaneous frequency of a propeller
blade rate using time-frequency analysis during a passage overhead of a turbo-prop aircraft
which showed that the WVD and STFT produced nearly the same results in estimating the
frequency with different error is 0.01Hz which proved that WVD is slightly closer to the
actual frequency (Ferguson & Quinn 1994; Stankovic, Djurovic & Stankovic 2002).
Another research on signal with linear frequency (chirp signal) using LabView software
has been done and the research used STFT with length of the window at 64 points and
PWVD method to analyse the signal. That research proved that both methods, STFT and
PWVD are nearly identical in results for both methods. The accuracies were similar but
when the signals were included with noise, PWVD performed better than STFT because
PWVD’s have low signal to noise ratio (SNR) (Andria, Savino & Trotta 1994). Comparison
amongst WVD, PWVD and Hilbert-Huang transform (HHT) were made by (Reddy, Rayel
& Rao 2015). They found that HHT was a good approach for feature extraction in non-
stationary signal because it can exactly reflect instantaneous frequency components and
produce physically meaningful representations of the data from nonlinear and stationary
processes (Reddy, Rayel & Rao 2015). Based on the research, HHT can exactly express the
local information of non-stationary signal in high time frequency resolution and overcome
the irreconcilable contradiction between time-frequency aggregation and cross term. It was
also shown that PWV distribution was smoother than WVD and this is also supported in
(Chandra Sekhar & Sreenivas 2003; Lee et al. 1999). Other than that, a spectrogram
modified by Moss (1989) using Kodera modification showed an increased performance of
the Spectrogram. In a more recent study, Moss & Hammond (1994) compared PWVD with
Spectrogram and showed PWVD provided better resolution. The results were more
accurate in estimating the instantaneous frequency in mono component signals. Andria,
Savino & Trotta (1994) said measuring the instantaneous frequency of particular non-
stationary signal is a crucial task in signal processing. Thus, this research is important to
improve visualization and accuracy of the results whereby the methods produce valuable
information to relevant applications.
8
2.6 Improvement Methods
Due to problems related to distribution, a lot of improvement has been suggested. This
section will portray some of the improvements that have recently been discovered. One of
the improvements in WVD made by Boashash (1993) is called the Wigner-Ville
Trispectrum. It has very efficient time-frequency representation on the signal because it
operates at a higher order of spectra. He strongly recommended the use of Wigner-Ville
Trispectrum for frequency modulated signals with or without the presence of noise. This
method surpassed the performance of the WVD based on the estimation of the
instantaneous frequency at peak extraction (Boashash, B. & Ristich 1993; Boashash,
Boualem & Ristich 1993). Another improvement that was made in WVD is the Cross
Wigner-Ville distribution (XWVD) (Boashash & O’Shea 1993; Boles & Boashash 1988;
Guanghua et al. 2008). The estimation from the peak of the XWVD is more accurate
compared to the estimation from the peak of the WVD which was done for chirp signal. If
the instantaneous frequency of the signal has a low signal to noise ratio (SNR), then XWVD
gives a more accurate estimation compared to WVD. Moreover, Boashash & O’Shea
(1993) also compared the XWVD with the Spectrogram at an instantaneous frequency and
found that the results of XWVD was more favourable. Next, another implementation was
made on Spectrogram based on Kodera modification and tested by Moss & Hammond
(1994). The comparison between Spectrogram and modified Spectrogram showed that the
modified spectrogram is significantly less dependent on the window selection and it has a
better resolution in the time and frequency domain compared to PWVD method. Moreover,
another implementation that has generated great interest amongst researchers is creating a
model that can duplicate the function of human ears to have better performance in analysing
sound. The creation of the human ear is a perfect recognising system that recognises the
different components in signals such as the amplitude and frequency that changes over time
(Hut 2004). Van Hengel (1996) has developed a model to simulate the response of the
cochlear membrane which imitates the performance of the human ears.
9
2.7 Cross-Term
Cross-term refers to interference in time frequency analysis. This cross-term appears in
WVD method, which makes the interpretation of WVD harder. The effect of this cross-
term can be seen if there are 2 components in a signal which can be represented as such
that x(t) = x1(t) +x2(t) and WVD becomes as in Equation 4.
𝑊𝑥(𝑡, 𝑓) = 𝑊𝑥1(𝑡, 𝑓) + 𝑊𝑥2(𝑡, 𝑓) + 2ℜ[𝑊𝑥1,𝑥2(𝑡, 𝑓)]
Equation 4: Cross-Term Wigner-Ville Distribution
The 1st and 2nd term of the equation is an auto term which is obtained from the WVD of
signal x1(t), x2(t) respectively and the 3rd term is a cross term that is always present in
between the auto-term. This cross-term is twice the amount of the auto-terms and it
oscillates proportionally to the distance between the auto-terms (Sandsten 2013). This
cross-term also appears in spectrograms but it only appears when the 2 components of a
signal are close to each other. It can be define when x(t) = x1(t) +x2(t) and spectrogram
becomes as in Equation 5.
|𝑋(𝑓)|2 = |𝑋1(𝑓)|2 + |𝑋2(𝑓)|2 + 𝑋1(𝑓)𝑋2∗(𝑓) + 𝑋1
∗(𝑓)𝑋2(𝑓)
Equation 5: Cross-Term Spectrogram
The 1st and 2nd term of the equation is an auto term which is obtain from the spectrogram
of the signal x1(t), x2(t) respectively and the 3rd and 4th term is a cross term affected by
different phases of the signal and this term is also called leakage. But this term will only
appear when there are components of a signal that overlaps or close to each other. Due to
this cross term, many researchers are keen to reduce these interferences by applying the
window function called smoothing kernel. The selection of the smoothing window length
has significant effects in Time-Frequency representation. If there is a different component
in the signal, it is not possible to choose one value of the window for all of the signals
(Eoza, Canagarajah & Bull 2003). Thus, a method needs to be developed to automatically
choose the appropriate window to the signals.
2.8 Comparison Methods in term of Cross-term
This section compared four methods of Time-Frequency analysis which are WVD, PWVD,
SPWVD and SPEC. These comparisons are made based on reducing the cross-term. Two
signals have been generated to show the effects between these distributions at cross-term
(Khan & Sandsten 2016). The signal is generated with combination of signals which have
a constant frequency and a Gaussian amplitude modulation set at a constant frequency.
10
Figure 2: Comparison Methods in term of Cross-Term
Based on Figure 2 WVD has interferences term between these two signals and this
interferences become less or degrades due to frequency smoothing that is applied in PWVD.
Further implementation of smoothing windows results in Smooth Pseudo Wigner-Ville.
This nearly reduces all of the interference terms by adding a time smoothing window. In a
spectrogram there are no interference term present since these two signals were not close
to each other. Thus it shows that there were effects on smoothing in time and frequency on
interferences term which degrades the interference terms. Unfortunately, the consequences
of applying this smoothing window include decreasing the time and frequency resolution
of the signal (Auger et al. 1996). Thus, it is important to investigate further this issue and
be aware of the capabilities of each method.
2.9 Summary
There are many research has been done in order to have better comparison of the methods
in time-frequency analysis. Some of the research has been discussed in this chapter. Thus,
this research are estimating frequency based on these research.
11
CHAPTER 3 : METHODOLOGY
3.1 Introduction
This thesis are been done in simulation by using MATLAB® software and time frequency
toolbox made by Auger et al. (1996). There are four signals are generated. There are
sinusoidal sine wave signal, linear frequency modulation, constant frequency modulation,
and sinusoidal frequency modulation. For sinusoidal sine wave signal it need to be
converted into analytic form to give a positive spectrum using Hilbert transform (Cohen
1989; Selesnick 2002) but the rest of the signals do not need to convert because it is already
analytic. Each of the signal are tested with four different time-frequency analysis methods
which are WVD, PWVD, SPWVD, and SPEC. Estimated frequency that obtained from the
methods at particular time using quadratic interpolation three adjacent on curve will be
compare to actual frequency and get the differences. These differences are converted to
standard deviation to compare with best frequency error using Roa Cramer Lower bound.
Then, there are three conditions of signals that are tested which are signal with no noise,
signal with low standard deviation of noise (from 0.05 to 0.30 standard deviation of noise)
and signal with high standard deviation of noise (from 0.35 to 0.75 standard deviation of
noise). For condition signal with noise, the variation of noise is increased by a factor of
0.05 until 0.75 of standard deviation of noise. To obtain proper result when dealing with
noise, the coding will be set to produce a same noise sequence and are simulated by 100
times and take the average value of the result. This is to make sure that the results are
consistent with other methods because the generated noise can be varied. Thus, the closest
result from the best frequency error indicates the best methods to use.
3.2 Time-Frequency Analysis Methods
In this section, time-frequency analysis methods that are used in this research are been
describes further. Methods that are be describe is Wigner-Ville distribution, Pseudo
Wigner-Ville distribution, Smooth Pseudo Wigner-Ville distribution and Spectrogram.
Moreover, its formula and variable in these distribution will also be explained.
12
3.2.1 Wigner-Ville Distribution
The WVD is one of the best distributions that gives high resolution and concentration of
the signal. This WVD is as indicated in Equation 6 (Branch 2016; Loutridis 2006; Luo et
al. 2012).
𝑊𝑥(𝑡, 𝜐) = ∫ 𝑥(𝑡 + 𝜏 2⁄ )𝑥∗(𝑡 − 𝜏 2⁄ )𝑒−𝑗2𝜋𝜐𝜏𝑑𝜏+∞
−∞
Equation 6: Wigner-Ville Distribution
Where x(t) is an analytic signal. By using analytic signal it can reduce a bit effects of the
cross term (Sandsten 2013). But the interferences still appear and hence the PWVD was
introduced.
3.2.2 Pseudo Wigner-Ville Distribution
PWVD is an improvement from WVD, whereby a window function is applied to the WVD.
This function is shown in Equation 7 (Luo et al. 2012; Pei & Yang 1992).
𝑃𝑊𝑥(𝑡, 𝜐) = ∫ ℎ(𝜏)𝑥(𝑡 + 𝜏 2⁄ )𝑥∗(𝑡 − 𝜏 2⁄ )𝑒−𝑗2𝜋𝜐𝜏𝑑𝜏+∞
−∞
Equation 7: Pseudo Wigner-Ville Distribution
Where h is a regular window which it function is to smoothing frequency window of WVD.
By windowed the WVD it will attenuate the interferences term and can improved
readability of the properties. But interferences in PWVD are not completely removed and
therefore the SPWVD was subsequently introduced.
3.2.3 Smooth Pseudo Wigner-Ville Distribution
Limitations occur if smoothing function can only be controlled by one degree of window
(h). With that regards, some implementation were made by adding another degree of
freedom for having two separate smoothing functions. By applying these two degree of
smoothing functions which are smoothing in time and smoothing in frequency will further
improve of Wigner-Ville Distribution and this distribution is called Smooth Pseudo
Wigner-Ville. This function is described in Equation 3 (Luo et al. 2012; Pereira De Souza
Neto et al. 2001) where g is time smoothing window and h is frequency smoothing window.
𝑆𝑃𝑊𝑥(𝑡, 𝜐) = ∫ ℎ(𝜏) ∫ 𝑔(𝑠 − 𝑡)(𝑠 + 𝜏 2⁄ )𝑥∗(𝑠 − 𝜏 2⁄ )𝑑𝑠 𝑒−𝑗2𝜋𝜐𝜏𝑑𝜏+∞
−∞
+∞
−∞
Equation 8: Smooth Pseudo Wigner-Ville Distribution
13
3.2.4 Spectrogram
If take a square of the magnitude of STFT it will give a power and can represent a spectrum.
This representation is called Spectrogram. Function of Spectrogram can be shown as in
Equation 9 (Auger et al. 1996).
𝑆𝑥(𝑡, 𝜐) = |∫ 𝑥(𝑢) ℎ∗ (𝑢 − 𝑡)𝑒−𝑗2𝜋𝜐𝑢𝑑𝑢+∞
−∞
|
2
Equation 9: Spectrogram
In spectrogram, the signal will be divided into several pieces whereby every piece gives
information about the time and frequencies occurring. Spectrogram advantages include fast,
easy interpretation and clear connection to the periodogram.
3.3 Estimating Frequencies
Estimating the frequencies need to be understand to utilize the notion of instantaneous
frequency of the signal (Boashash, B. 1992; Boashash, Boualem 1992). In this paper,
Quadratic interpolation of three adjacent applies three points at the curve to estimate the
maximum frequency occurrences. If there is a curve, for example half parabolic curve, its
equation is similar to quadratic equation which is y = ax2 + bx + c. This theory is applied to
the graph that represents the signal to estimate the frequency at a particular time. For
instance, there are three point of amplitude Am, A1, and A2. These points are a function of
quadratic and can be represented in the matrix form as shown in Equation 10.
[𝐴𝑚
𝐴1
𝐴2
] = [
𝑎𝑓𝑚2 + 𝑏𝑓𝑚 + 𝑐𝑚
𝑎𝑓12 + 𝑏𝑓1 + 𝑐1
𝑎𝑓22 + 𝑏𝑓2 + 𝑐2
]
[𝐴𝑚
𝐴1
𝐴2
] = [
𝑓𝑚2 𝑓𝑚 𝑐𝑚
𝑓12 𝑓1 𝑐1
𝑓22 𝑓2 𝑐2
] 𝑥 [𝑎𝑏𝑐
]
[𝑎𝑏𝑐
] = [
𝑓𝑚2 𝑓𝑚 𝑐𝑚
𝑓12 𝑓1 𝑐1
𝑓22 𝑓2 𝑐2
]
−1
𝑥 [𝐴𝑚
𝐴1
𝐴2
]
Equation 10: Estimating frequency using Matrix
14
Then, the values of a, b, and c are obtained. Taking differential on quadratic equation will
equal to zero and the maximum estimated frequency occur at that curve can be find as
indicated in Equation 11.
𝜕𝐴 𝜕𝑓 = 0 = 2𝑎𝑓 + 𝑏⁄
𝑓𝑚𝑎𝑥 = −[𝑏 2𝑎⁄ ]
Equation 11: Estimating Maximum Frequency
Thus, this method has developed as a function in programming name ‘qint’ and ‘maxfreq’.
These two functions will be used to estimate the frequency in the MATLAB® software.
3.4 Error Analysis
Estimating the frequency using quadratic interpolation of three adjacent curves need to be
verified using error analysis (Andria et al. 1996). With presence of noise in the signal, the
estimated frequencies become less accurate. Thus, it is important to estimate the best
frequency error present and compare with estimated results using the quadratic
interpolation of three adjacent curve. This section shows fisher information to estimate the
best frequency error using Cramer Roa Lower Bound method.
𝐼𝜔𝜔 =𝑛𝐴𝑚
2 𝑇𝑤2
24𝜎𝑛2
𝜎𝑓2 =
24𝜎𝑛2
𝑙𝐴𝑚2 (𝑇2𝜋)2
𝜎𝑓 = √24𝜎𝑛
2
4𝑙(𝐴𝑚𝑇𝜋)2
𝜎𝑓 = √6𝜎𝑛
2
𝑙(𝐴𝑚𝑇𝜋)2
Equation 12: Error Analysis using Cramer Roa Lower Bound
Therefore, Equation 12 is final formula to estimated the best frequency error in presence of
noise (𝜎𝑛) in the signal. In conclusion, results on error for estimating the frequency in every
distribution will be compared to the best frequency error. When the error in the distribution
is closest to the best frequency error, the method is chosen as the optimum method in
estimating the frequencies for the particular condition.
15
3.5 Flow Chart of Methodology
Figure 3 shows flow of the programming in MATLAB® software. There are four types of
programming for different signals. Once the selection of signal has been computed, the
programming will begin with Spectrogram function, followed by WVD, PWVD and
SPWVD functions. The process is repeated for different signals.
Figure 3: Flowchart of Methodology
3.6 Summary
Estimation of the frequencies is important and can be done by the proposed methods. Thus,
this thesis will lead to find performances of selected methods in estimating frequencies of
signals.
Start
Select
Signal
s
Compute
Spectrogram
Compute
WVD
Compute
PWVD
Compute
SPWVD
End
16
1. Sinusoidal Sine Wave Signal
2. Linear Frequency Signal
3. Constant Frequency Signal
4. Sinusoidal Frequency Signal
SPEC
No Noise Condition
Low STD Noise
High STD Noise
WVD
No Noise Condition
Low STD Noise
High STD Noise
PWVD
No Noise Condition
Low STD Noise
High STD Noise
SPWVD
No Noise Condition
Low STD Noise
High STD Noise
CHAPTER 4 : RESULTS AND DISCUSSIONS
4.1 Introduction
In this chapter, results of the estimated frequencies on four different signals that have been
tested using four types of time-frequency analysis are shown. Each signal will be presented
in a graph to show the comparison between the different methods used against the best
frequency error and the results from the simulation are tabulated.
4.2 Flowchart of the Results
Results from this study are demonstrated in three conditions with four different methods on
four different signals. Presentation of this results is summarised in
Figure 4, this to ensure that the readers can identify the situation of the results. The
following subsections will discuss about the result as in Figure 4.
Figure 4: Flowchart of Results
17
4.3 Sinusoidal Sine wave Signal
For this signal all of the result for three conditions are tabulated in Table 1. At zero noise
condition, WVD produces the largest error of 3.4 x10-5 while the other three methods give
an error of equal value of 1.0 x10-6. As the standard deviation of noise increases, the
frequency error of each method increases at different rates. This is observed by the gradient
value of the straight line equation of each method as shown in Figure 5. The WVD method
has the lowest gradient value of 0.0002 and the lowest frequency error starting at a standard
deviation of noise of 0.075. The other method continues to increase in error from that point
onwards as shown in Figure 5.
Table 1: Results for Sinusoidal Sine wave Signal
Sigma_n Best frequency
error
Error
Spectrogram
Error
WVD
Error
PWVD
Error
SPWVD
0 0.00E+00 1.00E-06 3.40E-05 1.00E-06 1.00E-06
0.05 3.00E-06 3.20E-05 3.60E-05 3.20E-05 2.90E-05
0.1 7.00E-06 6.40E-05 4.00E-05 6.40E-05 5.70E-05
0.15 1.00E-05 9.60E-05 4.50E-05 9.60E-05 8.60E-05
0.2 1.30E-05 1.28E-04 5.20E-05 1.30E-04 1.16E-04
0.25 1.70E-05 1.61E-04 6.00E-05 1.65E-04 1.45E-04
0.3 2.00E-05 1.93E-04 6.90E-05 2.02E-04 1.75E-04
0.35 2.40E-05 2.26E-04 7.80E-05 2.40E-04 2.05E-04
0.4 2.70E-05 2.59E-04 8.80E-05 2.79E-04 2.35E-04
0.45 3.00E-05 2.93E-04 1.00E-04 3.21E-04 2.65E-04
0.5 3.40E-05 3.26E-04 1.12E-04 3.64E-04 2.96E-04
0.55 3.70E-05 3.60E-04 1.25E-04 4.10E-04 3.26E-04
0.6 4.00E-05 3.94E-04 1.38E-04 4.57E-04 3.57E-04
0.65 4.40E-05 4.28E-04 1.53E-04 5.06E-04 3.88E-04
0.7 4.70E-05 4.63E-04 1.68E-04 5.58E-04 4.19E-04
0.75 5.00E-05 4.97E-04 1.85E-04 6.11E-04 4.51E-04
18
4.4 Signal with Linear Frequency Modulation
For this signal all of the result for three conditions are tabulated in Table 2. At zero noise
conditions, Spectrogram method produces the largest error of 2.51 x10-04 while the other
method showed an error of 3.70 x10-05, 3.20 x10-05, and 1.00 x10-06 for SPWVD, WVD,
and PWVD respectively. The spectrogram method displayed the highest error throughout
all ranges in the standard deviation of noise shown in Figure 6. At a standard deviation of
noise from approximately 0.07 until 0.75, WVD method performs better than the rest as it
is the closest to the best frequency error followed by PWVD, and SPWVD. It can also be
seen that WVD has the smallest gradient from all the straight line equation of the four
methods shown in Figure 6.
Table 2: Result for Signal with Linear Frequency Modulation
Sigma_n Best frequency
error
Error
Spectrogram
Error
WVD
Error
PWVD
Error
SPWVD
0 0.00E+00 2.51E-04 3.20E-05 1.00E-06 3.70E-05
0.05 3.00E-06 3.33E-04 3.60E-05 2.10E-05 6.50E-05
0.1 7.00E-06 4.79E-04 3.60E-05 4.10E-05 1.15E-04
0.15 1.00E-05 6.30E-04 3.90E-05 6.20E-05 1.69E-04
0.2 1.30E-05 7.97E-04 4.50E-05 8.40E-05 2.25E-04
0.25 1.70E-05 9.81E-04 4.90E-05 1.05E-04 2.82E-04
0.3 2.00E-05 1.17E-03 5.50E-05 1.27E-04 3.40E-04
0.35 2.40E-05 1.35E-03 6.20E-05 1.48E-04 3.99E-04
0.4 2.70E-05 1.54E-03 7.00E-05 1.71E-04 4.59E-04
0.45 3.00E-05 1.75E-03 7.80E-05 1.94E-04 5.20E-04
0.5 3.40E-05 1.97E-03 8.70E-05 2.17E-04 5.83E-04
0.55 3.70E-05 2.18E-03 9.40E-05 2.41E-04 6.47E-04
0.6 4.00E-05 2.40E-03 1.02E-04 2.66E-04 7.14E-04
0.65 4.40E-05 2.68E-03 1.12E-04 2.91E-04 7.82E-04
0.7 4.70E-05 2.90E-03 1.21E-04 3.17E-04 8.54E-04
0.75 5.00E-05 3.12E-03 1.31E-04 3.43E-04 9.29E-04
19
4.5 Signal with Constant Frequency Modulation
For this signal all of the result for three conditions are tabulated in Table 3Table 2. At zero
noise condition WVD method revealed a high error of 7.70 x10-05 whilst the rest of the
methods have the same level of performance with a standard deviation of noise equal to 0.0
x100. For signals with noise, at low standard deviation of noise from 0.05 until 0.3,
spectrogram method performs better but the frequency error increases exponentially from
that point onwards. At high standard deviation of noise level of 0.35 to 0.75, WVD gives a
more accurate result compared to others method. WVD is nearest to best frequency error
followed by SPWV and PWVD as seen in Figure 7.
Table 3: Result for Signal with Constant Frequency Modulation
Sigma_n Best frequency
error
Error
Spectrogram
Error
WVD
Error
PWVD
Error
SPWVD
0 0.00E+00 0.00E+00 7.70E-05 0.00E+00 0.00E+00
0.05 3.00E-06 7.00E-06 7.70E-05 1.60E-05 1.50E-05
0.1 7.00E-06 1.50E-05 8.00E-05 3.20E-05 2.90E-05
0.15 1.00E-05 2.00E-05 8.40E-05 4.90E-05 4.40E-05
0.2 1.30E-05 2.00E-05 8.80E-05 6.60E-05 5.90E-05
0.25 1.70E-05 2.50E-05 9.40E-05 8.40E-05 7.50E-05
0.3 2.00E-05 8.00E-05 1.01E-04 1.02E-04 9.10E-05
0.35 2.40E-05 3.84E-04 1.08E-04 1.21E-04 1.06E-04
0.4 2.70E-05 4.91E-04 1.16E-04 1.41E-04 1.23E-04
0.45 3.00E-05 5.46E-04 1.25E-04 1.62E-04 1.39E-04
0.5 3.40E-05 5.87E-04 1.35E-04 1.84E-04 1.56E-04
0.55 3.70E-05 6.30E-04 1.45E-04 2.07E-04 1.73E-04
0.6 4.00E-05 7.04E-04 1.56E-04 2.31E-04 1.90E-04
0.65 4.40E-05 7.48E-04 1.67E-04 2.57E-04 2.08E-04
0.7 4.70E-05 7.72E-04 1.79E-04 2.83E-04 2.26E-04
0.75 5.00E-05 7.94E-04 1.92E-04 3.12E-04 2.44E-04
20
4.6 Signal with Sinusoidal Frequency Modulation
For this signal all of the result for three conditions are tabulated in Table 4. From Figure 8,
sinusoidal frequency modulation signal exhibits a different result from the rest of the
signals. This can be seen by the results for each method showing a high error in frequency
at zero noise condition when compared to the rest of the signal. SPWVD has the least error
compared to the other three methods at zero noise condition, with a value of 1.81 x 10-3
until a standard deviation of noise of approximately 0.36, where PWVD displayed the least
frequency error of 2.45 x 10-3. SPWVD method continues to increase in its frequency error
since the straight line graph has the highest gradient value of 0.0026 as shown in Figure 8.
Spectrogram, WVD and PWVD have a gradient less than 10 x10-3 which is negligible, thus
their frequency error do not varies much with increase in standard deviation of noise.
Table 4: Result for Signal with Sinusoidal Frequency Modulation
Sigma_n Best frequency
error
Error
Spectrogram
Error
WVD
Error
PWVD
Error
SPWVD
0 0.00E+00 4.29E-03 3.16E-03 2.42E-03 1.81E-03
0.05 3.00E-06 4.21E-03 3.16E-03 2.42E-03 1.82E-03
0.1 7.00E-06 4.21E-03 3.16E-03 2.42E-03 1.87E-03
0.15 1.00E-05 4.25E-03 3.13E-03 2.43E-03 1.94E-03
0.2 1.30E-05 4.29E-03 3.12E-03 2.43E-03 2.02E-03
0.25 1.70E-05 4.39E-03 3.09E-03 2.44E-03 2.12E-03
0.3 2.00E-05 4.42E-03 3.08E-03 2.45E-03 2.26E-03
0.35 2.40E-05 4.42E-03 3.09E-03 2.45E-03 2.42E-03
0.4 2.70E-05 4.45E-03 3.02E-03 2.45E-03 2.62E-03
0.45 3.00E-05 4.46E-03 3.02E-03 2.45E-03 2.81E-03
0.5 3.40E-05 4.48E-03 2.98E-03 2.45E-03 3.01E-03
0.55 3.70E-05 4.46E-03 2.95E-03 2.46E-03 3.11E-03
0.6 4.00E-05 4.46E-03 2.93E-03 2.46E-03 3.25E-03
0.65 4.40E-05 4.47E-03 2.94E-03 2.46E-03 3.33E-03
0.7 4.70E-05 4.48E-03 2.96E-03 2.45E-03 3.40E-03
0.75 5.00E-05 4.47E-03 3.00E-03 2.45E-03 3.53E-03
21
4.7 Graph for each signal
Figure 5: Graph for Sinusoidal Sine wave Signal
Figure 6: Graph for Signal with Linear Frequency Modulation
22
Figure 7: Graph for Signal with Constant Frequency Modulation
Figure 8: Graph for Signal with Sinusoidal Frequency Modulation
23
4.8 Discussions
4.8.1 Comparing methods
This section summarises the comparison of each methods and signals with specific standard
deviation of noise in each condition. Standard deviation of 0.2 is chosen for low standard
deviation of noise, and for high standard deviation of noise 0.6 is chosen. This comparison
will give a best methods for these three conditions. Each table shows signal and time-
frequency analysis methods and for each signal the best method to be used is being
highlighted. Thus for each signal will have different methods that can give best frequency
estimation and in certain condition more than one methods produce the same accuracy. This
comparison will be illustrates in Table 5, Table 6, and Table 7.
4.8.1.1 No Noise Condition
Based on Table 5, PWVD best for three types of signal which are sinusoidal sine wave
signal, linear frequency modulation signal, and constant frequency modulation signal.
SPWVD also best for three types of signal which are sinusoidal sine wave signal, constant
frequency modulation signal, and sinusoidal frequency signal. For SPEC, it perform best
on sinusoidal sine wave signal and constant frequency signal with same value of error to
PWVD and SPWVD. However, at no noise condition WVD did not give better result for
all types of signals.
Table 5: Comparison in No Noise Condition
Sine Wave Linear Freq. Constant Freq. Sinusoidal
Frequency
SPEC 1.00 x10-06 2.51 x10-04 0.00 x10+00 4.29 x10-03
WVD 3.40 x10-05 3.20 x10-05 7.70 x10-05 3.16 x10-03
PWVD 1.00 x10-06 1.00 x10-06 0.00 x10+00 2.42 x10-03
SPWVD 1.00 x10-06 3.70 x10-05 0.00 x10+00 1.81 x10-03
24
4.8.1.2 Low Standard Deviation of Noise at 0.2
Referring to Table 6, WVD provide less error for two types of signal which are sinusoidal
sine wave and linear frequency signal. However, SPEC provide good result when dealing
with constant frequency signal and for sinusoidal frequency signal SPWVD are the best
method to used. In low standard deviation of noise PWVD did not give accurate result for
all types of signals.
Table 6: Comparison in Low Standard Deviation of Noise
Sine Wave Linear Freq. Constant Freq. Sinusoidal
Frequency
SPEC 1.28 x10-04 7.97 x10-04 2.00 x10-05 4.29 x10-03
WVD 5.20 x10-05 4.50 x10-05 8.80 x10-05 3.12 x10-03
PWVD 1.30 x10-04 8.40 x10-05 6.60 x10-05 2.43 x10-03
SPWVD 1.16 x10-04 2.25 x10-04 5.90 x10-05 2.02 x10-03
4.8.1.3 High Standard Deviation of Noise at 0.6
From Table 7, at high standard deviation of noise WVD give result near to best frequency
error obtain by Cramer Roa Lower Bound for all types of signals. This means WVD is best
methods while dealing with high standard deviation of noise and the rest of the methods
just achieve 2nd or 3rd place on estimating frequency in high standard deviation of noise.
Table 7: Comparison in High Standard Deviation of Noise
Sine Wave Linear Freq. Constant Freq. Sinusoidal
Frequency
SPEC 3.94 x10-04 2.40 x10-03 7.04 x10-04 4.46 x10-03
WVD 1.38 x10-04 1.02 x10-04 1.56 x10-04 2.93 x10-03
PWVD 4.57 x10-04 2.66 x10-04 2.31 x10-04 2.46 x10-03
SPWVD 3.57 x10-04 7.14 x10-04 1.90 x10-04 3.25 x10-03
25
4.9 Predicted Projection
Due to limitations of the program, the highest standard deviation of noise was achieved is
0.75. Whereas in practicality, the value of noise can reach a higher level than that. In order
to better conclude the results of the simulation in practical perspective, a projection is made
in order to obtain a rough estimation for higher level of standard deviation of noise present
in the signal. Based on the results obtained from the simulation, using the straight line
equation from the linear trend line, a straight line projection has been made. This projection
is up until standard deviation of noise at 2.0. The following subsections present results of
this predicted projection.
4.9.1 Signal with Sinusoidal Sine Wave
From Figure 9, the graph shows that by increasing the standard deviation of noise, the
performance of the methods does not change. WVD method still is the best method to use
for sinusoidal sine wave signal as mentioned by Andria (1994).
Figure 9: Prediction until SD of noise at 2 for Sinusoidal Sine Wave
26
4.9.2 Signal with Sinusoidal Linear Modulation
From Figure 10, the graph shows that increasing standard deviation of noise will not change
the performance of the methods. This means that WVD is the best method to use for signal
with linear frequency modulation. This result supported by Andria (1994).
Figure 10: Prediction until SD of noise at 2 for linear frequency signal
27
4.9.3 Signal with Sinusoidal Constant Modulation
From Figure 11, the graph shows that increasing the standard deviation of noise will not
change the performance of the methods. This means for when standard deviation of noise
is 2.0 WVD still the best method to use for signal with constant frequency modulation.
Figure 11: Prediction until SD of noise at 2 for constant frequency signal
28
4.9.4 Signal with Sinusoidal Frequency Modulation
From Figure 12, the graph shows that increasing standard deviation of noise will have some
changes to the performance of the methods. As seen in Error! Reference source not
found., SPWVD frequency error increases linearly at higher standard deviation of noise.
Other than that, it can be seen that WVD method would give a lower frequency error than
PWVD at higher standard deviation of noise.
Figure 12: Prediction until SD of noise at 2 for sinusoidal frequency signal
4.10 Summary
This chapter presents all of the result that obtain in the simulation by Matlab® software.
From the results, comparison between methods has been made and lead to discussion which
methods give better frequency estimation in different conditions. Then, prediction for
higher level of standard deviation of noise also been predicted to make sure the performance
of each methods. Thus, it clear shows that each of signal have a different method that can
give result in different conditions.
29
CHAPTER 5 : RECOMMENDATIONS AND CONCLUSION
5.1 Conclusion
As a conclusion, performance of time-frequency analysis on Wigner-Ville distribution,
Pseudo Wigner-Ville distribution, Smooth Pseudo Wigner-Ville distribution and
Spectrogram has been investigated based on frequency estimation of peak measurement.
The accuracies of each methods has been simulated using MATLAB® software at three
different conditions of signals which are signal with no noise, signal with low standard
deviation of noise and signal with high standard deviation of noise. Each of the conditions
has a particular method that is best suited. Also, different types of signal may have different
methods that produces better result. For no noise condition, PWVD performs better because
in three different types of signal, the results are near to best frequency error. For signals
that have low standard deviation of noise, WVD performs best at two types of signal which
are sine wave and linear frequency. For signal with high standard deviation of noise, WVD
also performs better for sine wave, linear frequency, and constant frequency signals. It can
be concluded that WVD is nearly an ideal method to handle these four signals with noise
but it is not suitable for no noise signal.
The recommended practices for estimating frequency analysis from this thesis are:
1. For no noise condition, it is good to use PWVD as a method to analyse the signal
because it is dominant in giving better results.
2. For signals with low standard deviation of noise, WVD is recommended because
based on two signals, it performs better than other methods.
3. For signal with high standard deviation of noise, WVD is better suited since it gives
the result nearest to the best frequency error for 3 out of 4 signals. Besides that, the
result obtained is limited to a certain level which is up to a standard deviation of
noise of 0.75.
The objectives of this thesis have been achieved which is to determine the accuracy of the
four methods in estimating signals’ frequencies in several conditions, with and without
noise. Also the recommendation for the methods for each signal has been recommended to
get a useful result for application that has used time-frequency methods.
30
5.2 Recommendations for Future Work
Although in this study several findings have been achieved, there still remain some gaps
which can be further discovered to achieve a higher accuracy of frequency analysis
estimation. Therefore, the following recommendations are drawn for future research in this
field:
1. Modifying the capabilities of the coding can give an accurate result rather than
relying on the projection value based on the straight line equation of the trend line.
2. It is also recommended to use a smaller increment for the standard deviation of
noise in order to obtain a more accurate result of the frequency error each method
gives.
31
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APPENDICES
A1 Programming in MATLAB®
clear all;
rng(4); %generate same noise sequence Td = 2; %total duration in second Fs = 256; %sample frequency per second N = Td * Fs; %total no of point dt = 1/Fs; %step size between samples per second at time domain f0 = 0.195; %setting frequency in pu
f1 = 0.05; %setting starting frequency f2 = 0.3; %setting final frequency f01 = f0*Fs; %setting frequency in real no t_secs = dt*(0:N-1); %sampling times
Sigma_n = 0.75; ==> standard deviation of noise change from 0.05 to 0.75 l = N; %length of signal; a0 = 1; %amplitude of signal; Td = dt*N; %time of signal; Sigma_f = sqrt(6*Sigma_n^2/(l*(pi*a0*Td)^2))/Fs; %calc best freq error; fprintf('When error Sigma_n = %9.6f\n', Sigma_n) fprintf('Best frequency error = %9.6f\n', Sigma_f)
Signal = a0*sin(2*pi*f01*t_secs');%generate signal sinusoidal sine wave; [Signal, IFLAW] = fmlin(N,f1,f2); %generate signal with linear freq; [Signal, IFLAW] = fmconst(N,f1); %generate signal with constant freq; [Signal, IFLAW] = fmsin(N,f1,f2); %generate signal with sinudoidal freq;
Noise = Sigma_n * randn(1,N); %generate noise; Signalwithnoise = Signal.*window + Noise'; %sinusoidal signal with noise sig = hilbert (Signalwithnoise); %convert signal to complex signal
figure(1); subplot(2, 1, 1) plot (t_secs, Signal); title('Sine sig no noise vs time'); xlabel('Time'); ylabel('Signal');
subplot(2, 1, 2) plot (t_secs,Signalwithnoise); title('Sine sig with noise vs time'); xlabel('Time'); ylabel('Signal');
for test = 1:4 switch(test) case 1 disp('Spectrogram'); %using spectrogram method [TFR, T, F] = tfrsp (sig); %compute SPEC distribution case 2 disp('Wigner Ville'); %using wigner ville method [TFR, T, F] = tfrwv (sig); %compute WV distribution case 3 disp('Pseudo WV'); %using PWV method [TFR, T, F] = tfrpwv (sig); %compute PWV distribution case 4
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disp('Smooth Pseudo WV'); %using SPWV method [TFR, T, F] = tfrspwv (sig);%compute SPWV distribution end figure(10*test + 1); surf(T,F,TFR,'Edgecolor','None') %plot 3D view of results set(gca, 'Ydir', 'Normal'); title('Time frequency representation'); xlabel('seconds'); ylabel('Hz'); view(0,90);
IFLAW(1:N,1:1) = f0; %make matrix IFLAW 1 x 512 matrix
Data = []; %make empty matrix for frequency estimated looping = 100;
for loop = 1:looping for i = 1:N %Number of the loop that will be do t = i; %select time to look the freq vs Amplitude F_Exp = IFLAW(t); %expected freq which what stated earlier df = F(2); %step size between samples at freq domain At = TFR(:,t); %taking all amplitude value at time of t FFT_Power = At; %equal the amp value with power value
[y, Pdb_Max ] = maxfreq( F_Exp, df, FFT_Power ); F_Est(i) = y; %insert value of y into F_Est to be a matrix
figure(2) plot(F,At, y, 0, 'x'); %plot Amp vs freq at the time of t drawnow end TransposeF_EST = F_Est'; Data = [Data TransposeF_EST]; end S = sum(Data,2)/looping;
figure(10*test + 2); plot(T,IFLAW, T,S); %plot sinusoidal freq vs time title('Frequency Vs Time'); xlabel('seconds'); ylabel('Hz');
Diff = IFLAW - S; %get the difference between actual and estimated
id = T > 50 & T < 450; figure(10*test + 3); plot(T(id),Diff(id)); %plot sinusoidal freq vs time title('Difference Vs Time'); xlabel('seconds'); ylabel('diff'); fprintf('Standard Error = %7.6f\n', std (Diff(id))) end
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A2 Sample result from MATLAB® for each signals
A2.1 Signal Sinusoidal Sine wave
A2.2 Signal with Linear frequency Modulation
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A2.3 Signal with constant frequency Modulation
A2.4 Signal with sinusoidal frequency Modulation