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Modulation characterization using the wavelettransformLanier A. WatkinsClark Atlanta University

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THESIS TRANSMITTAL FORM

Name of Student: Lanier A. Watkins

Title of Thesis: Modulation Characterization Using the Wavelet Transform

We the undersigned members of the Committee advising this

thesis have ascertained that in every respect it acceptably fulfills the final requirement for

the degree of M.S. in Physics

Computer Science

Department

Phvsics

Date

jartment Date

Dr. R. Date 'Dr.4> Hawk Date

As Chair ofthe Department of Phvsics

I have verified that this manuscript meets the School's Department's standards ofform and

content governing theses forjthe degree sought.

Date

As Dean ofthe School of Arts and Sciences I have verified that this

manuscript meets the School's regulations governing the content and form of theses.

Date

As Dean of Graduate Studies I have verified that this manuscript meets the

University's regulations governing the content and form oftheses.

32Dean ofGraduate Stuflies Date

ABSTRACT

PHYSICS

WATKINS, LANffiR A. B.S., CLARK ATLANTA UNIVERSITY, 1996

MODULATION CHARACTERIZATION USING THE WAVELET TRANSFORM

Advisor: Dr. Kenneth Peny, Department of Computer Science

Thesis Dated: May, 1997

The focus of this research is to establish an Automatic Modulation Identifier

(AMI) using the Continuous Wavelet Transform (CWT) and several different classifiers.

A Modulation Identifier is of particular interest to the military, because it has the potential

to quickly discriminate between different communication waveforms. The CWT is used to

extract characterizing information from the signal, and an artificial Neural Network is

trained to identify the modulation type.

Various analyzing wavelets and various classifiers were used to assess comparative

performance. The analyzing wavelets used were the Mexican Hat Wavelet, the Morlet

Wavelet, and the Haar Wavelet. The variety of classifiers used were the Multi-Layer

Perceptron, the K-Nearest Neighbor and the Fuzzy Artmap. The CWT served as a

preprocessor, and the classifiers served as an identifier for Binary Phase Shift Keying

(BPSK), Binary Frequency Shift Keying (BFSK), Binary Amplitude Shift Keying (BASK),

Quadature Phase Shift Keying (QPSK), Eight Phase Shift Keying (8PSK), and Quadature

Amplitude Modulation (QAM) signals. Separation of BASK, BFSK and BPSK was

performed in part one of the research project, and separation of BPSK, QPSK, 8PSK,

BFSK, and QAM comprised the second part of the project. Each experiment was

1

performed for waveforms corrupted with Additive White Gaussian Noise ranging from 20

dB - 0 dB carrier to noise ratio (CNR). To test the robustness of the technique, part one

of the research project was tested upon several carrier frequencies oa/2, and co/3 which

was different from the carrier frequency co that the classifiers were trained upon. In the

separation of BASK, BFSK and BPSK, the AMI worked extremely well (100% correct

classification) down to 5 dB CNR tested at carrier frequency co, and it worked well (80%

correct classification) down to 5 dB CNR tested at carrier frequencies oaf2, and co/3. In

the separation of BPSK, QPSK, 8PSK, BFSK, and QAM, the AMI performed very well at

10 dB CNR (98.8% correct classification). Also a hardware design in the Hewlet Packard

Visual Engineering Environment (HP-VEE) for implementation of the AMI algorithm was

constructed and is included for future expansion of the project

CLARK ATLANTA UNIVERSITY THESIS

DEPOSITED IN THE ROBERT W. WOODRUFF LIBRARY

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The author of this thesis is:

Name: Larder A. Watkins ___^

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City, State and Zip: Marshallville. GA 30314

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Professors: Dr. K. Perrv/Dr. L. Lewis

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information requested.

NAME OF USER ADDRESS DATE TYPE OF USE

MODULATION CHARACTERIZATION USING THE WAVELET TRANSFORM

A THESIS

SUBMITTED TO THE FACULTY OF CLARK ATLANTA UNIVERSITY IN

PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

BY

LANIER A. WATKINS

DEPARTMENT OF PHYSICS

ATLANTA, GEORGIA

MAY 1997 "-

VI'

©1997

LANIER A. WATKINS

All Rights Reserved

ACKNOWLEDGMENTS

I would like to thank Dr. Kennth Perry, my advisor, for giving me a chance to

work with him, sparking my interest in wavelets/neural network technology, and for a

monthly stipend. Also I would like to thank the members of my committee: Dr. Lonzy

Lewis, Dr. Romain Murenzi and Dr. Denise Stephenson-Hawk for taking time to review

my work. I would like to thank Dr. Lance Kaplan, Dr. John Hurley and Dr. Raymond

Brown for serving as my "last minute saviors." A special thanks goes to Dr. Dan

Dudgeon, Dr. Richard Molnar, and Dr. Robert Baxter all of M.I.T Lincoln Laboratory for

advising me on most of this work during my summer internship. An even bigger thanks

goes to Prism-D for funding me during my five years at Clark Atlanta University; thanks

goes also to the CTSP for allowing me to use their facilities at my leisure. I would like to

thank Alpha Phi Alpha Fraternity, Inc. for making me realize earlier in my life that, "I am

the master of my fate, I am the captain of my soul." Last but not least, I would like to

thank "The Most High" for my very existence.

TABLE OF CONTENTS

ACKNOWLEDGMENTS "

LIST OF TABLES v

LIST OF FIGURES *

LIST OF ABBREVIATIONS ix

Chapter

1. INTRODUCTION l

Communication Signals 5

Wavelet Theory 10

Neural Network Theory 17

Matlab Implementation of the CWT 21

2. RESEARCH METHODOLOGY AND DESIGN 23

Design Issues 23

Approach For Resolving Design Issues 24

Additive White Gaussian Noise and Varied Carrier Frequency.25

Feature Extraction 26

Classifiers 33

3. IMPLEMENTATION 37

Automatic Modulation Identification Algorithm 37

in

4. RESULTS 41

Results From Separation of BPSK/BFSK/BASK 41

Results From Separation of BPSK/QPSK/8PSK/BFSK/QAM. 48

5. SUMMARY/CONCLUSION 50

Future Work 52

APPENDIX

L COMMUNICATION SIGNALS 54

EL MATLAB PROGRAMS 56

m. SIMULATION OF GAUSSIAN NOISE CHANNEL 60

IV. DECISION BOUNDARIES FOR NEURAL NETWORKS 62

BIBLIOGRAPHY M

IV

LIST OF TABLES

Table Page

1. Comparison of Algorithms 3

2. Mexican Hat Wavelet Modulation Classifier Results 42

3. Morlet Wavelet Modulation Classifier Results 43

4. Haar Wavelet Modulation Classifier Results 45

5. Morlet Wavelet Modulation Classifier Results for 0^2 46

6. Morlet Wavelet Modulation Classifier Results for a/3 48

7. Morlet Wavelet Modulation Classifier Results for Experiment #2.... 49

LIST OF FIGURES

Figure Page

1. Binary Phase Shift Keying Signal From C computer program 6

2. Binary Frequency Shift Keying Signal 6

3. Binary Amplitude Shift Keying Signal 7

4. Quadature Phase Shift Keying Signal 8

5. 8 Phase Shift Keying Signal 8

6. 16-Quadature Amplitude Modulation Signal 9

7. Binary Phase Shift Keying Signal from internet on left, and Binary Phase

Shift Keying Signal from C computer program on right 10

8. 1-D Morlet Wavelet and FT of 1-D Morlet Wavelet 11

9. 1-D Mexican Hat 13

10. 1-D Haar Wavelet 13

11. Real part of 1-D Morlet Wavelet at (0=5.5 14

12. Artificial Neuron 18

13. Network Containing 1 Hidden Node 19

14. Haar Feature Vector For BPSK 27

15. Haar Features For BFSK 28

16. Haar Features For BASK 28

17. Mexican Hat Features For BFSK ... 29

vi

18. Mexican Hat Features For BPSK 29

19. Mexican Hat Features for BASK 30

20. Morlet Features For BASK 20

21. Morlet Features For BFSK 31

22. Morlet Features For BPSK From The Internet 31

23. Morlet Features For BPSK From C Program 32

24. Morlet Features For PSK8 32

25. Morlet Features For 16-QAM 33

26. Morlet Features For QPSK 33

27. Diagram Of Nearest Neighbor Classifier 34

28. Diagram Of Multi-Layer Perceptron Classifier 35

29. Diagram OfFuzzy Artmap Classifier 36

30. Flowchart Of Modulation Characterization Algorithm 37

31. Results From Mexican Hat Modulation Classifier For Constant ca— 42

32. Results From Morlet Modulation Classifier For Constant (a 43

33. Results From Haar Modulation Classifier For Constant eo. 44

34. Results From Modulation Classifier For tall 46

35. Results From Morlet Modulation Classifier for cott 47

36. Results From Second Experiment For Morlet Modulation Classifier... 49

37. HP-VEE Program For Wavelet Transform 53

38. Communication Signals From Experiment #1 54

39. Communication Signals From Experiment #2 55

vii

40. Simulation of Gaussian Noise Channel 60

41. Feature Extraction From Noisy Signal 61

42. Example of Decision Boundaries for Neural Networks 63

viu

LIST OF ABBREVIATIONS

AMI Automatic Modulation Identifier

AWGN Additive White Gaussian Noise

CNR Carrier-To-Noise Ratio

CWT Continuous Wavelet Transform

BASK Binary Amplitude Shift Keying

BFSK Binary Frequency Shift Keying

BPSK Binary Phase Shift Keying

8PSK Eight Phase Shift Keying

FT Fourier Transform

FFT Fast Fourier Transform

HMC Haar Modulation Classifier

HP-VEE Hewlett Packard Visual Engineering Environment

IFFT Inverse Fast Fourier Transform

KNN K-Nearest Neighbor

LNK Richard Lippman, Dave Nation and Linda Kukolich

MHMC Mexican Hat Modulation Classifier

M.I.T. Massachusetts Institute of Technology

MLP Multi-Layer Perceptron

IX

MMC Morlet Modulation Classifier

QAM Quadature Amplitude Modulation

QPSK Quadature Phase Shift Keying

CHAPTER 1

INTRODUCTION

This thesis deals with the problem of modulation characterization in the presence

of varying noise and varying carrier frequency. The results show that when the Automatic

Modulation Identifier (AMI) is trained and tested upon a constant carrier frequency, an

inverse relationship between increasing noise level and percent of correct classification is

found. This response is quite logical, because at higher noise levels the signals that the

classifiers are tested upon become distorted. Some of these distorted signals can no

longer be identified by the classifiers, thus the signals are misclassified.

The Wavelet Transform is very suitable for transient signal analysis.1 A transient is

an abrupt change in the signal. Each modulation pattern contains transients at the points

of phase, frequency or amplitude change. These transients are the main points of interest,

because they correspond to binary information that is imposed upon the carrier signal. In

the Wavelet Transform of the signal there exist local maxima at the points where transients

are detected, thus the Wavelet Transform is used to extract a signature pattern or a feature

vector from the modulation pattern. The classifiers are trained using the noise free feature

vectors taken from each modulation type, and the classifiers are tested upon noisy feature

vectors.

The approach for presenting the research will be as follows: Modulation

Characterization via algorithms other than the AMI will be discussed, background

information on the Wavelet Transform and on Neural Network Technology will be

discussed, the Research Methodology/System Design, and Implementation will be

discussed in Chapters 2,3, and 4 respectively. Finally in Chapter 5 the research will be

summarized and results will be presented.

Modulation Characterization has been accomplished by K.C. Ho, W. Prokopiw,

and Y.T. Chan.2 Their principle objective was to distinguish between Binary Frequency

Shift Keying (BFSK) and Binary Phase Shift Keying (BPSK) signals. The investigators

used Haar Wavelet analysis for feature extraction and statistical procedures to produce a

modulation identification scheme. Taking the desired features, the researchers used

median filtering and the variance to separate the two signals. Their system was found to

possess the Rican distribution, and it directly follows that this particular probability density

function reduces to the Rayleigh distribution at low Carrier to Noise Ratio (CNR) and

Gaussian at high CNR. Once the correct probability function was determined, a median

filter was used and the variance was calculated. They noticed that the variance would

have a Chi-squared distribution since the median filter output was Gaussian. The above

information combined with a probability of misclassification of BPSK allowed them to

determine the threshold for BPSK and BFSK separation. Their results were 100% correct

classification of BFSK patterns and 96.33% correct classification of BFSK patterns at 13

dBCNR.

Another modulation identification scheme was proposed by S.Z. Hsue and Samier

S. Loliman.3 Their approach to the problem was zero crossing. The algorithm starts with

sampling the signal with a zero crossing sampler. The zero crossing points are recorded

and information about the phase and frequency of the signal is recorded. Next the

probability density functions of the zero crossing points are generated. At high carrier to

noise ratio the probability density function is Gaussian, and because of this the probability

density of the zero crossing points are approximated by the Gaussian density. From their

result the method is accurate from 6.5 dB CNR to 17 dB CNR. The next step would be

to calculate either the phase or frequency histogram, depending upon variance of the

probability. This histogram information is used as a feature vector for a parallel

processing scheme that is programmed to characterize the information from the histogram.

They report results at 15 dB CNR and above. All of the algorithms mentioned in this

thesis are revisited in Table 1.

Table 1. Comparison of Algorithms

INVESTIGATOR

CNR

^CORRECT

METHOD

HSUE

15 dB

97%

ZERO-CROSS

K.C.HO

13 dB

9fi.lfi%

HAAR

WATKINS

PART 1 PART 2

5dB

97%

MORLET

MdB

98.8%

MORLET

In this thesis, a Modulation Characterization algorithm is presented that can

completely separate (100% correctly) the following signals at 10 dB CNR (At a constant

carrier frequency): Binary Phase Shift Keying (BPSK), Binary Frequency Shift Keying

(BFSK), and Binary Amplitude Shift Keying (BASK). At varied carrier frequency (a/3),

results are reported at 88.67% correct classification. In the second experiment the same

algorithm is used to separate the following signals: Quadature Phase Shift Keying

(QPSK), Eight Phase Shift Keying (8PSK), Quadature Amplitude Modulation (QAM),

Binary Phase Shift Keying (BPSK), and Binary Frequency Shift Keying (BFSK). The

algorithm is capable of achieving 98.8% correct classification for the separation of any of

the 5 signals at 10 dB CNR. The experiments are separated into two sections because of

the levels of difficulty. The first experiment is nontrivial; however the second experiment

is even more difficult This is because the signals in the first section are more dissimilar

than the signals in the second section, thus the first set of signals should be easier to

separate.

The three forms of modulation used in the first classification experiment were

created using C computer programs. These C programs alter the amplitude (A), the

frequency ( go), or the phase ((J)) of the carrier wave, Acos(cot + <J>), such that the following

modulation patterns are produced: Binary Phase Shift Keying (BPSK), Binary Frequency

Shift Keying (BFSK), and Binary Amplitude Shift Keying (BASK). There are five forms

of modulation used in the second classification experiment, Quadature Phase Shift Keying

(QPSK), Eight Phase Shift Keying (8PSK), Quadature Amplitude Modulation (QAM),

and Binary Phase Shift Keying (BPSK). All of these signals except BFSK were

downloaded via the internet from the website

www.ee.byu.edu/ee/class/ee492.44/ee492.44.html. The BPSK signal used in the first

experiment was created using a C computer program, and the BPSK signal used in the

second part was downloaded from the internet Each signal in the first experiment was

created using the same binary information, and the binary digits were sent every T seconds

where T is the period of the cosine carrier. The binary information is only four digits long

(1010). The cosine carrier is discretized to 27 or 128 points. This means that there are

128 points in each file that corresponds to each modulation type. In the first experiment

the main objective was to prove that the AMI could detect the transients in the signals,

and to show that these transients could characterize each signal. The signals in the second

experiment that were download from the internet were created using the same binary

information. The carrier frequency is 2 Hz, the symbol rate was 1 symbol/second, and the

sampling rate was 16 samples/second. Later, in the discussion of the results from

experiment one and experiment two it will be shown that even though these signals were

created using different methods the features extracted using the Morlet Wavelet are very

similar.

Communication Signals

Binary modulation is a method of altering a carrier waveform so that binary data

can be sent through an analog channel.4 Each modulation type has its own characteristics,

and the CWT exploits this fact to allow the classifier to discriminate between the

modulation types.

BPSK is the result of varying the phase (<|>), of a carrier wave in a manner that

corresponds to either a binary 1 or 0.4 It has the advantage of being less susceptible to

noise than the other two modulation types. BPSK is very desirable for applications such

as microwave radio (see Figure 1).

o.:0.6

A 0.4m

p 0.2

! •t -0.2u

d -0.4

8 -0.6

-0.8

HI• 1/ 1 / I/ y I/ W

0 -5

BPSK

A A A A

I Hill0

Time

A AH

15

A A]

11110

Fig. 1. Binary Phase Shift Keying Signal From C computer program.

BFSK is the second modulation pattern of interest. It is accomplished by switching

the frequency CO, of a carrier wave to send either a binary 1 or 0.4 This method is more

susceptible to noise than BPSK, but less susceptible to noise than BASK. BFSK is very

useful for multiplexing audio frequencies onto telephone channels for teletype or data (See

Figure 2).

Fig. 2. Binary Frequency Shift Keying Signal.

BASK is very similar to Morse Code. The amplitude A, of a carrier wave is varied in

order to send a binary 1 or 0.4 This type of modulation is very vulnerable to noise;

therefore it is less popular than BFSK or BPSK. Because it is such a simple method,

BASK is used primarily for optical fiber transmission (See Figure 3).

A

m

P1

t

u

d

e

0.8 R /0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

■-1io -5

BASK

A

M V0

Time5

•

•

-

•

•

•

-

■

10

Fig. 3. Binary Amplitude Shift Keying Signal.

The QPSK and 8PSK signals are very similar to BPSK. Instead of sending binary

1 or 0, binary 00,01,10, and 11 can be sent for QPSK.4 Each set of binary digits

correspond to a 0, n/2, n, or 3tc/2 phase change, respectively (See Figure 4). Since

BPSK, QPSK and 8PSK signals are related by the fact that all three use phase changes to

transmit binary information, then their separation should be more difficult than signals that

utilize amplitude or frequency changes. This is the main reason that the second

experiment is more difficult than the first.

Fig. 4. Quadature Phase Shift Keying Signal.

Similarly, the sets of binary digits 000,001,010,100,1101,101, Oil and 111 can be sent

for 8PSK.4 Each set of binary digits correspond to a 0, jc/4, tc/2, 3jc/4, tc, 5tc/4, 3tc/2, or

Ik/4 phase change respectively (See Figure 5).

Fig. 5. 8 Phase Shift Keying Signal.

The signal 16-QAM is a combination of amplitude shift keying and phase shift keying,

therefore the binary sets 0000,0001, 0011, 0111,1001,1101,0101,0010,0110,1100,

1010,0100,1011,0100,1000,1111 can be sent.4 Each set of binary digits correspond to

a 0, rc/8, n/4, 3tc/8, tc/2, 5jc/8, 3jc/4, 7tc/8, n, 9tc/8, 5tc/4, 11tc/8, 3tc/2, 13tc/8, Ik/4, or

15tc/8 phase change, respectively. Each set of four phase changes are displayed at an

amplitude of 1,2,3, or 4, respectively (See Figure 6). See Appendix I for more

information about the amplitude, phase or frequency changes used in producing the

signals.

Fig. 6. 16-Quadature Amplitude Modulation Signal.

Earlier it was mentioned that part one and two used different BPSK signals. In Figure 7

the two signals are placed side by side for comparison. Visual inspection reveals little

difference, and it will be shown in Chapter 3 that the extracted features are similar as well.

Fig. 7. Binary Phase Shift Keying Signal from internet on left, and Binary Phase Shift

Keying Signal from C computer program on right.

Wavelet Theory

Wavelet analysis is quickly becoming a standard for a variety of applications. It

has proven to be very useful because of its ability to analyze signals at different scales. In

this procedure the signal is decomposed as a linear superposition of the analyzing wavelet

at a variety of scales.5 This gives the user the flexibility to analyze any particular feature

of the signal desired. Wavelet analysis can be thought of as an advanced filtering device.

The reader will be spared the mathematical derivations and proofs of wavelets; instead the

Wavelet Transform will be defined and some of its properties will be explored.

The Fourier Transform (FT) is another tool that has been used for analyzing

signals, but for feature extraction the Fourier Transform provides less information about

the signals than the Wavelet Transform.6 This can be explained by considering the

procedure that the Fourier Transform uses to analyze a signal. The Fourier Transform

10

decomposes a signal as a superposition of sine and cosine functions.5 This transformed

signal then exist in what is called the frequency domain. Therefore the Fourier Transform

analyzes a signal for its frequency content. This is very useful when only the frequency

content of a signal is considered, but if transients are of interest then the Fourier

Transform is not effective. This is one of the main differences between the Fourier

Transform and the Wavelet Transform.

Each analyzing wavelet is localized in the time domain and in the frequency

domain. This means that all of the energy associated with the mother wavelet is

concentrated within a finite interval in the time domain and also in the frequency domain.

Figure 8 displays a plot of the one-dimensional Morlet Wavelet and its Fourier

Transform.7 The idea of localization evident in the plot.

M orlet W avelet

A

m

P

I

i

t

u

d

e

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-10

Fourier T rasform of M orlet W avelet

8

A

m

P

i

t

u

d

e

10

7

6

5

4

3

2

1

0

-1 0 1 0

Frequency

Fig. 8. 1-D Morlet Wavelet and FT of 1-D Morlet Wavelet

Localization is one property of wavelets. Before discussing other properties of wavelets,

the definition of a wavelet must be given. There are several requirements that must be met

11

before a function can be classified as a wavelet These requirement can be reduced to the

following statements:8

1) The function \jf(t) and its Fourier Transform *F(Q) must be square

integrable.

2) \|f(t) must meet the admissibility requirement

Cv/2n = j mCl)\2dn/\Q\<oo (1.1)

Equation (1.1) implies that

0, (1.2)

and that \j/(t) is of zero mean as stated in Equation (1.3).

J \j/(t)dt = O (1.3)

In equation (1.1) Cv is a constant that results from the admissibility condition. The above

statements are necessary requirements for a function to be called a wavelet, but there are

other restrictions that can be placed upon the wavelet to achieve better results. For

example, the wavelet can be required to have a certain number of vanishing moments.

This requirement improves the ability of \|f(t) to detect singularities in the signal.

This research deals with three different types of one-dimensional analyzing

wavelets, the Mexican Hat Wavelet (See Figure 9), the Haar Wavelet (See Figure 10), and

the Morlet Wavelet (See Figure 11).

12

The Mexican Hat Wavelet is defined as:

= (l-t2)exp(-t2/2) (1.4)

Fig. 9. 1-D Mexican Hat Wavelet.

The Haar Wavelet is defined as: Y(t)= 1, [-5a, 0) (1.5)

\|/(t) = -1, [0,5a)

\|f(t) = 0, elsewhere

Here "a" denotes the chosen scale of the wavelet.

Haar Wavelet

0.4

A

m 02f

1

t

1-02e

-0.4

°5o -5 0 5 10timn

Fig. 10. 1-D Haar Wavelet

13

The Morlet Wavelet is defined as: \|f(t)=exp(ioot)exp(-t2/2)+A (1.6)

Fig. 11. Real part of 1-D Morlet Wavelet at CD =5.5.

A is called the correction term. It allows the Morlet Wavelet to meet the admissibility

requirement; however if © is chosen greater than 5 then the correction term is not needed.

The Wavelet Transform can be thought of as an advanced filtering device. Before the

Wavelet Transform is possible, the analyzing wavelet must be scaled and translated. The

result is the family of wavelets which is very controllable.

The wavelet \|/(t) is referred to as the analyzing wavelet or the mother wavelet

Scaling and translating this mother wavelet produces the baby wavelet Vab(t).

The Continuous Wavelet Transform (CWT) is defined as the inner product of the baby

wavelet and the signal to be analyzed.

CWT =<\|/ab(0ls(t)> (1-8)

\|/ab(t)*s(t)dt (1.9)

14

It is obvious that the result of the CWT is a function of the scale 'a' and the shift 'b'. If

\|/(t) is assumed to be a one-dimensional function then the CWT will be a two-dimensional

function. This means that the CWT maps a one-dimensional function into a two-

dimensional function. This two-dimensional function exists in what is called Wavelet

Space. Wavelet Space exists because of several properties of the CWT as seen below:8

1) Linearity: The CWT is linear because of the linearity of the inner product.

2) Shift Property: If a function f (t) is shifted such that f(t) = f(t-b'), then

CWTf(a,b) =CWTf(a,b-b') (1.10)

=a-l/2j y((t-b)/a)*f(t-b')dt (1.11)

=a-l/2j \j/((t'+b'-b)/a)*f(t')dt' (1.12)

=CWTf(a,b-b') (1.13)

Property 2 simply states that a shift in the function, f(t) yields a shift in the CWT,

CWTf(a,b).

3) Scaling Property: The function f (t) is scaled such that f(t) = s

CWTf(a,b) =(a*s)"1/2J \|f( (t-b) /a )*f(t/s) dt (1.14)

=s(a*s)"1/2J \|/((st'-b)/a)*f(t')dt' (1.15)

=s1/2a"1/2 J \|/( (st'-b) /a )*f(t') dt' (1.16)

= CWTf(a/s,b/s) (1.17)

Property 3 states that energy is spread by a factor of s in both dimensions in the

CWTf (a,b) or simply scaling the function f(t) also scales the CWT off(t).

15

4) Localization Property: Consider a Dirac pulse at time to, 8(t-to)

CWT8(a,b) =a-l/2j \|/( (t-b)/a )* 8(t-to) dt (1.18)

= a-l/2\|/((to-b)/a) (1.19)

The CWT has sharp time localization at high frequencies, therefore at small scales the

CWT will yield a constant value at every point t and will be a local function at t^

5) Energy Conservation: Consider a function f(t) and its CWTf (a,b)

J lf(t)|2dt=l/CvJ j ICWTf(a,b)|2(dadb)/aV2 (1.20)

Here Cv is a constant that results from the admissibility condition. Property 5 is similar to

Parseval's formula of the Fourier Transform. It simply states that the energy in the time

domain is equivalent to the energy in the wavelet domain or that the wavelet transform

conserves energy. Also partial energies can be computed by considering ICWTabl »

which is the energy density of the signal in position and scale. The partial integration of

this density in one variable gives the energy density in the remaining variable as seen

below.

E(b) = J da/a2 ICWTab!2 (L21)

E(a) = J dblCWTabl2 (L22>

Equation (1.21) is the energy density as a function of shift, and equation (1.22) is the

energy density as a function of scale. The energy density as a function of shift expresses

the energy of the signal over all possible scales. Similarly the energy density as a function

16

of scale expresses the energy of the signal at all possible positions. The Wavelet

Transform and all of its properties affect the modulation patterns in such a way that a

Neural Network is capable of recognizing these signals. An introduction to Neural

Network technology is appropriate for better understanding of its ability to classify

modulation patterns.

Neural Network Theory

An artificial Neural Network is an information system that processes data or

stimuli in the same manner as biological Neural Networks. Once data or stimuli are

encountered, the neuron makes a decision to respond or not to respond. In an artificial

Neural Network a neuron is nothing more than a decision maker. This decision making

occurs in biological neurons as well as artificial neurons. Neural Networks are

mathematical models based on the assumptions that:9

1). Information processing occurs at many simple elements called neurons.

2). Signals are passed between neurons over connection links.

3). Each connection link has an associated weight, which in a typical Neural

Net multiplies the signal transmitted.

4). Each neuron applies an activation function (usually nonlinear) to its net

input to determine its output signal.

Neural Networks are characterized by:

1). Pattern of connection between the neurons (architecture).

2). Method of determining the weights on the connection (training).

3). Activation function.

17

Consider the neuron in Figure 12, it has three input nodes Xi and one output node Y.

Each input node is directly connected to the output node, and each connection has a

weight, wy associated with it. This particular single-layer net is feed forward, which

means that information flows from the input units to the output units in a forward

direction; however the flow of information is not necessarily limited to the forward

direction. In a Recurrent Net information may flow from node A to node B then back to

node A again. The flow of information has a magnitude associated with it, which is called

a weight. The weights are very important, because if a large value is given for wj, but

very small values are given for W2 and W3 then the output of the activation function is

going to be mostly dependent upon node Xj. This means that the final decision of the

neuron will be based almost entirely upon node Xj. This process of assigning values for

the weights is called training.

Fig. 12. Artificial Neuron.

The artificial neuron depicted in Figure 12 is capable of solving some simple problems.

The equations below model the behavior of this neuron.

Ym = wixi+W2X1 + W3xi (1.23)

Ye* =f(Yin), where f(x) = 1/(1+e"x) (1.24)

18

Here f(x) is the activation function of the Neural Network, which is called a sigmoid

function. This function is one of the most important concepts of Neural Network

technology. The type of activation function used actually limits the network to the type of

problems it can solve.10

Fig. 13. Network Containing 1 Hidden Node.

The network in Figure 13 contains a node in between the input and output nodes and a

nonlinear activation function f(x). This node denoted by Y is called a hidden node. A

hidden node allows the neural network to solve more problems than a network without a

hidden node such as the one illustrated in Figure 12. One of the unfavorable features of a

Neural Network with a hidden node is that it is harder to train than a network without a

hidden node. Neural Networks can train the weights associated with each node using

various methods. Two methods of training are supervised training and unsupervised

training. In supervised training, a sequence of training vectors is presented to the

classifier, then the weights are adjusted according to a learning algorithm. From this

example, the neural network would be trained to classify an input vector as being in or

19

not being in the same class as one of the training vectors. Unsupendsed training occurs

when a sequence of input vectors is presented to the network, but no training data is

specified. The net then sets its weights such that similar input vectors arc classified

together. Then the net produces an exemplar vector which is representative of the entire

class or cluster. In this section the general theory of Neural Network technology is given.

Almost any network can be produced by taking the networks from Figure 12 and Figure

13 and making multiple connections. An understanding of how each neuron works gives

an understanding of the network as a whole. Two of the networks used in this research

are feed forward, fully connected networks (The Multi-Layer Perceptron and the K-

Nearest Neighbor), and the other is a recurrent, fully connected network (the Fuzzy

Artmap). All three of the networks in this study used supervised training. These three

classifiers were chosen because their methods of classification are very dissimilar. The K-

Nearest Neighbor (KNN) classifier uses Euclidean distances from the training set to the

test set to determine the correct classification. The Multi-Layer Perceptron (MLP)

classifier uses the back propagation algorithm to train its weights to achieve optimal

classification. Finally the Fuzzy Artmap classifier uses match tracking to govern the

performance of the net The diversity of these classifiers should prove advantageous for

comparing and contrasting the results. More details about each individual network will be

given later in Chapter 2.

20

Matlab Implementation of the CWT

The CWT was used to analyze the six modulation patterns mentioned in Chapter 1.

The CWT is well suited for analyzing these six modulation types, because it is sensitive to

any change in phase, frequency or amplitude of the signal.

Matlab programs were written to calculate the CWT (See Appendix II). The logic

used in constructing the algorithm is given here. The received signal, r(t) is the sum of

the original signal and the noise.

r(t) = s(t) + n(t) (1.25)

Vab = a~1/2\|K(t-b)/a) (1.26)

\|fab is the scaled and translated mother wavelet The Continuous Wavelet Transform is

the inner product of the received signal and the scaled, translated mother wavelet

CWT= <Vab(t)lr(t)> (1.27)

CWT =} a-1/2yab(t)*r(t)dt (1.28)

For this research the CWT is a function of shift only, therefore the scale 'a' is taken out of

the integral and held constant.

CWT =a"1/2J Vab(t)*r(t)dt (1.29)

CWT =a1/2j ¥ab* R(co)eJo* da> (1.30)

21

Equation (1.30) is obtained by taking the Fourier Transform of equation (1.29)

(Justification follows from Parseval's Theorem). To simplify the calculation even further

the following substitution is made: P(aa>)=¥ab* R((0).

CWT =a1/2j P(a©)eJ(* dco (1.31)

Equation (1.31) reduces to the inverse Fourier Transform.

CWT =p(a,b) (1.32)

Taking the inverse Fourier Transform of equation (1.31) yields the CWT as a function of

shift (a) and scale (b); however for this research only one particular scale of the CWT was

considered. Matlab was used to calculate the FFT (Fast Fourier Transform) of the Mother

Wavelet and the signal. This process yields a fast calculation of the Wavelet Transform

using the power of the FFT and the IFFT (Inverse Fast Fourier Transform).

By definition the CWT is a convolution of the analyzing wavelet and the signal.

This means that the wavelet transform will be the decomposition of the signal as a linear

superposition of the analyzing wavelet This makes the scale factor "a" very important If

"a" is chosen large then the scale at which the wavelet will analyze the signal will be large;

however if "a" is chosen small then the analyzing wavelet will analyze the signal on a small

scale. The scale value is chosen manually such that narrow peaks arise at the points of

phase change, frequency change or amplitude change. Scale analysis is the main reason

that the CWT was used for feature extraction. Each modulation pattern in question has

been purposely altered to transmit data, and the CWT directly zooms in on these

alterations.

22

CHAPTER 2

RESEARCH METHODOLOGY AND DESIGN

This thesis presents a procedure for establishing a Automatic Modulation

Identification Algorithm. This algorithm is capable of simulating a simple surveillance

system or a simple wireless modem. The scenario for the surveillance system could be a

ground unit that is capable of receiving signals from the air, and the purpose of this

ground unit could be to determine the modulation type of the received signal for

comparison to a database of known hostile modulation types. The scenario for the

wireless modem could be two parties that are interested in one-way communication. A

modem could be used to send the binary data through the air to another modem. Once the

modulated data gets transmitted through the channel to the receiver, the data must be

demodulated, but before the data can be demodulated the method of modulation must be

known. These are just two of many applications that the AMI can simulate.

Design Issues

The first design issue was to gather a database of noiseless signals. Initially there

was no data available to begin the research. This was a problem, because without data

there was no way of generating results for the AMI algorithm.

The second design issue was modeling the noise channel. In each application,

communication through the air is mentioned, therefore the simulated channel was chosen

23

to be Additive White Gaussian Noise (AWGN). AWGN best models the random

processes that take place in the air. Also AWGN can be easily simulated in Matlab.

The third design issue was choosing the best analyzing wavelet for the AMI.

There exist many different types of analyzing wavelets, each with its own properties.

Examples would be, the Haar Wavelet, the Morlet Wavelet, the Mexican Hat Wavelet,

and the Daubechies Wavelets. Even though there were many analyzing wavelets to

choose from, several main ideas had to be considered: (1) the analyzing wavelet had to be

well localized in time and frequency, (2) the analyzing wavelet had to produce desired

results, and (3) the analyzing wavelet had to have an algorithm associated with it that was

not computationally intense.11

The final design issue dealt with selecting the types of artificial classifiers to use in

the AMI. This problem was similar to the dilemma with the analyzing wavelets. There are

many different types of artificial classifiers, each with its own properties. Whatever

classifier used had to have the ability to properly classify the corrupted signals, and it had

to be able to accept one-dimensional signals.

Approach For Resolving Design Issues

To resolve the design issues an intense literature search was done in the areas of

signals/systems, noise channels, wavelets, and artificial classifiers. To solve the problem

associated with the signals, a decision was made to manually produce general modulation

signals, and to continue to search for more signals. The C programming language was

chosen to simulate three general modulation types: Binary Phase Shift Keying, Binary

Amplitude Shift Keying, and Binary Frequency Shift Keying.

24

The issue corresponding to the wavelets was resolved by choosing three wavelets

for analyzing the signals: the Haar Wavelet, the Morlet Wavelet, and the Mexican Hat

Wavelet. Each of these wavelets have good localization properties and fast algorithms

(This means that the number of mathematical operations is not of the order n2, where n is

the total number of mathematical operations ) associated with them.

The classifier problem was resolved by choosing three classifiers for identification

of the signals: the Multi-Layer Perceptron, the K-Nearest Neighbor, and the Fuzzy

Artmap. Each of these classifiers are compatible with one-dimensional data and very

capable of classifying the signals.

The system design was as follows: (1) obtain a noiseless signal for each

modulation pattern, (2) construct signals that reflected corruption by Additive White

Gaussian Noise, (3) extract characterizing features from each signal (corrupted signal and

uncorrupted signal) using the CWT, (4) use features taken from uncorrupted signals to

train artificial classifiers, (5) use features taken from corrupted signals to test the artificial

classifiers (6) use confusion matrix taken from artificial classifiers to display results in the

form of graphs.

Additive White Gaussian Noise And Varied Carrier Frequency

Additive White Gaussian Noise (AWGN) was used to corrupt each modulated pattern.

Each classifier was trained with noiseless feature vectors, and each test set contained equal

amounts of AWGN. The noise model was assumed to have a zero mean and a finite

power. AWGN was simulated by using the random number function in Matlab (This

function produces random numbers that are normally distributed, of zero mean and of unit

25

variance.). A Matlab program was written that produces the desired amount of noise by

changing the variance of the randomly distributed numbers. As a safety feature the

program tests the random numbers to see if they indeed possess the desired variance.

The Carrier-to-Noise Ratio (CNR) is defined as follows:12

CNR = 101ogio(Pc/Pn) (2-1)

In this equation Pc is the average power of the carrier and Pn is the average power of the

noise. The average power of the carrier is computed from its power spectrum, and the

average power of the noise is computed from its variance.12 In essence the power of the

noise is chosen such that a desired Carrier to Noise ratio (CNR) is produced. Once the

desired noise level is obtained the characterizing features are extracted from each signal.

See Appendix IE for a demonstration of the noise channel.

Also as a subset of experiment one, the carrier frequency of BPSK, BASK and

BFSK test patterns are altered to see if the classifiers will detect the change. This means

the classifiers are trained on signals with a carrier frequency (0 and tested upon signals

with carrier frequency 0^2 and atf3. The results from this experiment are discussed in

Chapter 4.

Feature Extraction

The different modulation types were characterized by extracted features. These

features were used by the classifier to identify the modulation pattern. The CWT at a

particular scale "ao" was performed on each signal. The scale was chosen such that very

distinct peaks arose from each signal. These peaks directly corresponded to either a

change in frequency, amplitude or phase of the carrier. This method is not the traditional

26

CWT, because for a 1-Dimension CWT the result would be a function of two variables

(scale and shift); however these peaks are a function of shift only. The CWT at a

particular scale ao is simply a linear filtering operation. This particular scale at ao

characterizes each modulation pattern. The use of the CWT for feature extraction was

motivated by the research done by K.C. Ho, W. Prokopiw and Y.T. Chan; however the

methodology of this project is unique, because of the use of Neural Networks and various

wavelets.

Three mother wavelets were used to extract information from the different

modulation patterns. Haar Wavelet analysis will be discussed first, next Mexican Hat

Wavelet analysis will be discussed, then Morlet Wavelet analysis will be discussed. The

Haar Wavelet analysis of BPSK resulted in double peaks at the points where phase

changes occurred (See Figure 14).

Haar Analysis of BPSK

M

a

g 12n

I 1

t

u 0.8d

e 0.6

of0.4

C

W 0.2

T ■

■

•

■

•

■

-

•

-10 -5 0 5 10Shift(b)

Fig. 14. Haar Feature Vector For BPSK.

27

Haar Wavelet analysis ofBFSK resulted in single peaks at the points where frequency

changes occurred (See Figure 15). Close visual inspection shows the Wavelet Transform

of the carrier wave, which is represented by the smaller peaks.

M

a

Sn

1

t

u

d

e

of

C

w

T

2

1.8

1.6

1 .4

1.2

1

0.8

0.6

0.4

0.2

?1

Haar Analysis of BFSK

11 .

fl 1II 1lln n n n n nItll II II II 11 II iIllillflllflll

• |ii! 1! If lillmf\ f\ 1 \ \1 \l \l \/^\/~\/0-5 0

Shift(b)

1 in

i || *

In n n n n IIIII II II 11 II fl

1 |f |f If |f HI1 U U U U Ul /\ A

\J \] '

5 10

Fig. 15. Haar Features For BFSK.

Haar Wavelet analysis of BASK resulted in two broad peaks separated into four smaller

peaks at the points where the amplitude was nonzero, and the CWT yielded a value of

zero where the signal's amplitude was zero (See Figure 16)

M

a 3 ■[

9

t

u 2

d

e 1.5

of1 \

C

W 0.5 •T

°10

Haar Analysis of BASK

i n

k AA A

1

u

./-5 0

Shift(b)

5 10

Fig. 16. Haar Features For BASK.

28

Mexican Hat Wavelet analysis of BFSK (See Figure 17) yielded double peaks and

Mexican Hat Wavelet analysis of BPSK (See Figure 18) yielded single peaks.

Mexican Hat Analysis of BFSK

Fig. 17. Mexican Hat Features For BFSK.

M 3

a

I 2S1

t 2

u

e 1'5

of 1

C 0.5W

T

Mexican Hat Analvsis of BPSK

A A A A

■ A A A A•

«■■A I I 1 I \ "

■/» u U iwi «v ■0 -5 0 5 10

Shift(b)

Fig. 18. Mexican Hat Features For BPSK.

29

Mexican Hat wavelet analysis ofBASK produced two broad peaks that were separated

into three smaller peaks at the nonzero amplitude points, and zero at the zero amplitude

points of the signal (See Rgure 19). The features extracted from the signals using the Haar

Wavelet, and the features extracted from the signals using the Mexican Hat Wavelet are

somewhat visually dissimilar.

x 10-1< Mexican Hat Analysis ofBASK4.5 I

M La 4ft A

d 25re 2

of 1.5

C 1W

T °-5

■

"10

(I (IM I1

-5 0Shift(b)

A :•

"1 :5 10

Fig. 19. Mexican Hat Features for BASK.

Morlet Wavelet analysis produced results that were similar to Mexican Hat analysis for

BASK (See Figure 20).

Morld Analysis ol BASK

S hllt(b )

Fig. 20. Morlet Features For BASK.

30

For BFSK (See Figure 21) and BPSK (See Figure 22) the Morlet Wavelet sensed the

frequency and phase changes by double peaks. Note that Figure 22 is the Wavelet

Transform of the signal downloaded from the internet and Figure 23 is the Wavelet

Transform of the signal created by the C program mentioned earlier. Both graphs display

double peaks that mark phase changes; however Figure 22 has three peaks and Figure 23

has four. This difference comes from the fact that the carrier frequency of the two signals

are not the same.

M

a

Sn

1

t

u

d

0

of

C

w

T

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

P1

M

.1

11

AnIII!

■

i ii ii ii i1II111111111 \J 1 w\r

0 -5

1

L II

11 I

0Shift(b)

1

1

1 *I /

5

■

■

■

1 0

Fig. 21. Morlet Features For BFSK.

Fig. 22. Morlet Features For BPSK From The Internet

31

M

a

9n

I

t

u

d

e

of

C

W

T

0.8 II .

0.7 lA [IVI II

Oi | |

0.5

0.4

0.3

0.2

0.1\ I

-10

M oriel Analysis of BPSK

I k\1 II1

1 111

v / \S 0

S hift(b)

illIII

I

I ' 1

5

;

M

If1

1 0

Morlet Analysis ol BPSK

Fig. 23. Morlet Features For BPSK From C Program.

The chosen feature vectors are distinct for each modulation pattern, therefore it is possible

to visually classify each pattern according to its feature vector, however visual comparison

is not the method proposed for this thesis. Morlet Wavelet Analysis for the other signals

that were obtained from the internet are also illustrated. Morlet Wavelet Analysis of

PSK8 is given in Figure 24. Morlet Wavelet analysis of 16-QAM is given in Figure 25.

Finally, Morlet Wavelet analysis ofQPSK is given in Figure 26.

M

a

9n

I

u

d

ol

C

W

T

Morlst Analysis of PSK8

0

Shilt(b)1 0

Fig. 24. Morlet Features For PSK8

32

2.5

Morlet Analysis of QAM

0

Shilt(b)

Fig. 25. Morlet Features For 16-QAM.

1

0.8

0.6

0.4

0.2

-1

M orlet Analysis of

I ilk iik1

D -S 0

Shift(b)

QPSK

ll

Lu

5

■

j:

<s\s\yJ1 0

Fig. 26. Morlet Features For QPSK.

Classifiers

Two of the classifiers used in this research were part of a classification software

packet called LNKnet.13 The other classifier, the Fuzzy Artmap was supplied by Dr.

33

Robert Baxter of M.I.T Lincoln Laboratory. LNKnet is a collection of twenty classifiers

that have a graphical user interface which makes it easy to use. The acronym LNK is

attributed to the three principal programmers, Richard Lippman, Dave Nation and Linda

Kukolich, who are all employees at M.I.T Lincoln Laboratory. The Multi-Layer

Perceptron (MLP), the K Nearest Neighbor (KNN), and the Fuzzy Artmap were used to

identify the feature vectors.

The K-Nearest Neighbor is a simple classifier that stores all training data that is

given to it Then it measures the Euclidean distance from the K stored patterns to the test

data closest to it. The KNN then takes a vote among the K neighbors and the class that

occurs most is assigned to the test pattern. The KNN is a fully connected feed forward

network that uses a threshold as the activation function for each neuron (See Figure 27).

K-NEAREST NEIGHBOR

Feedforwsrd

3 Outputs

3 Hidden Nodes

128 Inputs

Fig. 27. Diagram Of Nearest Neighbor Classifier.

34

The Multi-Layer Perceptron is a feed forward net with one or more layers of nodes

between the input and output nodes. Until recently this type of net was not used for lack

of efficient training algorithms. This network can be used to solve complex problems, but

it cannot be proven that this network will converge to optimal weight values. The MLP is

a fully connected network that uses the sigmoid activation function (See Figure 28). The

training mechanism of this network is the back propagation algorithm. To optimize the

weights it performs a gradient decent algorithm which minimizes the error of the output

according to some cost function.

MULTI-LAYER PERCEPTRON

3O«psU

25 Hidden Nodei

12SInpull

Fig. 28. Diagram Of Multi-Layer Perceptron Classifier.

The Fuzzy Artmap is a fully connected net that performs match tracking. Match

tracking allows continuous communication between the hidden layer and the output layer.

In a sense, the hidden layer takes feedback from the output layers to determine if the data

35

are being correctly classified. If the data are not being correctly classified then the weights

are changed to produce the desired classification (See Figure 29).

FUZZY ARTMAP

3 OotpnU

3 Hidden Node.

Fig. 29. Diagram Of Fuzzy Artmap Classifier.

These three classifiers were chosen because their methods of classification are very

dissimilar. The KNN uses distances from the training set to the test set to determine the

correct classification. The MLP uses the back propagation algorithm to train its weights

to achieve optimal classification. Finally the Fuzzy Artmap uses match tracking to govern

the performance of the net The diversity of these classifiers should prove advantageous

for comparing and contrasting the results. Appendix IV has some addition information

concerning the method that the Neural Networks use to classify signals.

36

CHAPTER 3

IMPLEMENTATION

The first stage in the implementation process was to link the subunits described in

Chapter 2 into one unit Each subunit models a particular part of a simple

communications system, and linking subunits will complete software implementation.

Automatic Modulation Identification Algorithm

The goal of this research is to introduce an Automatic Modulation Identification

Algorithm that is capable of identifying BPSK, BFSK, QPSK, 16-QAM, 8PSK and BASK

patterns in the presence of high levels of noise and varying carrier frequency (See Figure

30).

ALGORITHM FLOWCHART

RECEIVED SIGNAL r(t) = s(t) + n(t)

M LP. KNN . A RTM A P

FEATU RE VECTOR

ID ENTIFIC A TIO NExample of Noiseless Training Vector

BFSK BASK |q AMPSK 8 P s'k

Fig. 30. Flowchart Of Modulation Characterization Algorithm.

37

The first step in the algorithm is to simulate a received signal, which contains the

modulated carrier { Acos(cot + <)>)} and noise { n(t) }. For BPSK, the parameter <}> is

switched from 0° to 180° corresponding to either a binary 0 or 1, respectively. In BFSK,

to is switched between ti/8 to n/16 corresponding to a binary 0 or 1, respectively. To

accomplish BASK, A is switched between 0 and 1 to denote a binary 1 or 0, respectively.

The description of QPSK, 8PSK, and QAM are given in Chapter 1. The AWGN is

simulated by a Matlab program that uses the normally distributed random number

generator. The program inputs a desired amount of noise in decibels. Then the program

calculates what the variance of the random numbers would have to be to produce the

desired amount of noise. The square root of this variance (standard deviation) is

multiplied by the random number generator to give it the desired variance. The calculation

is provided below:

10Logio(Pc/Pn) = #dB (3.1)

In this derivation Pn is replaced by x, because this is the quantity of interest

= #dB/10 (3.2)

Logl0(PC)-Logl0(x)= ttffi/10 (3.3)

Logio(x) = -#dB/10 + Logio(Pc) (3.4)

10Log(x) = 10-#dB/10 + Log(Pc) (35)

(3.6)

(3.7)

38

The value of x is the variance that will produce the desired carrier to noise ratio. The

square root of cr is multiplied by the random number generator to change the unit

variance to the desired variance, thus giving the user control over the levels of noise added

to each carrier. Once the desired CNR has been obtained, the program creates fifty sets

of each modulation pattern, and each pattern is corrupted with the same amount of

AWGN. A file is created for these resulting patterns. The motivation for doing this is to

produce a test set that is representative of the noise model.

In the third step, the CWT at a particular scale is taken for each pattern in each set

This corresponds to the inner product of the scaled-shifted mother wavelet and the

received signal. The results are CWT coefficients taken at a particular scale ao as a

function of shift These CWT coefficients are distinct for each modulation pattern;

therefore these coefficients are used as the feature vectors. The program writes the

coefficients to a file for later use by the classifier.

In the fourth step, the classifiers are trained with clean feature vectors, and tested

upon noisy test vectors of various CNRs. The performance of each classifier is recorded,

and their results are compared. For example, in the first part the training vectors consist

of three sets with one pattern in each seL Each pattern corresponds to BPSK, BFSK and

BASK. The test vectors consists of three sets with 50 patterns in each set. Each of these

patterns contain the same CNR, and it is the task of the classifier to place these patterns in

the correct class. There are four tests done corresponding to 20,10,5, or 0 dB CNR,

respectively. Each of these four tests are done ten times at random presentation orders, so

the results will be the average performance of the classifier. The classifiers are trained

39

with only three patterns to try to achieve maximum classification with the least amount of

training data. The same procedure is done for the second part of the experiment, except

the MLP is provided with fifty training vectors to increase performance.

In the final step the classifier computes the confusion matrix and assigns each

pattern in each set to a particular class. Then the errors that occurred with class confusion

are given. The percent of correct classification is recorded so comparisons can be made

among the different classifiers.

The entire algorithm has been simulated in the Hewelett Packard Visual

Engineering Environment (HP-VEE). This HP-VEE implementation is a working

demonstration of a surveillance system. The program reads a noisy modulation pattern in

from a file. Then the Wavelet Transform at a particular scale is taken. HP-VEE displays

both the signal and the Wavelet Transform of the signal. Then the square of the wavelet

coefficients or the Energy Density is applied to a threshold. This threshold serves as a

classifier for the modulation types. This classification technique is different from the

classification technique presented in Chapter 2. For this reason the HP-VEE

implementation is still under development, and a more detailed discussion will be held until

Chapter 5.

40

CHAPTER 4

RESULTS

An Automatic Modulation Identification (AMI) algorithm has been created using

the wavelet transform and various artificial classifiers. The AMI models a simple

communications system. All of the logic and mathematical derivation have been

presented. In theory the AMI should be able to correctly classify any modulation type

(assuming it is trained upon the signal prior to testing) at low levels of noise. It is

desirable for the AMI to work well in high levels of noise as well, otherwise the previously

mentioned algorithms would be more favorable. Indeed the results prove that the AMI

works well in high noise regions.

Results From Separation of BPSK/BFSK/BASK

Features extracted using the Mexican Hat coefficients were the first data sets fed

into the classifiers. The results for the Mexican Hat are displayed in Table 2 and Figure

31.

41

Mexican Hat Modulation Classifier at Constant ©

■

-M.P

-KNN

-ARTMAP

15

Carrier to Noise Ratio (dB)

20

Fig. 31. Results From Mexican Hat Modulation Classifier For Constant ca

Table 2. Mexican Hat Wavelet Modulation Classifier Results

Mexican Hat Wavelet

dB

20

10

5

0

MLP

100

100

86.67

45.5

KNN

100

100

94

36.67

ARTMAP

100

100

91.33

34.67

At a CNR of 20dB all three classifiers identified the patterns 100% correctly. As the noise

level increased at lOdB the classifiers remained constant in their performance. At 5dB

CNR the MLP classified 86.7% of the patterns correctly, the KNN classified 94% of the

patterns correctly, and the Artmap classified 91.3% of the patterns correctly. The final

test for the Mexican Hat data was at OdB CNR, at this point there is just as much noise as

carrier power, and the classifiers' performance immediately dropped. The MLP was down

to 55.5% correct classification of the modulation patterns, the KNN was down to 36.7%

correct classification, and the Artmap was down to 34.7% correct classification. The

overall performance of the Mexican Hat Modulation Classifier (MHMC) was pleasing. It

42

performed poorest at 0 dB CNR, but this is exactly what is expected at such high levels of

noise (See Figure 31).

The next coefficients used for feature extraction were those of the Morlet Wavelet

The results for the Morlet Wavelet are displayed in Table 3.

Morlet W8velel Modulation C lasslfler at Constant «d

so --

40 -"

30 "-

20 --

10 "-

0

5 10 IS

Carrier to Nolsa Ratio (d B )

20

■MLP

-KNN

•ARTM AP

Fig. 32. Results From Morlet Modulation Classifier For Constant ca

Table 3. Morlet Wavelet Modulation Classifier Results

Morlet Wavelet

dB

20

10

5

0

MLP

86.6

84.9

77.1

60.47

KNN

100

100

96.67

71.33

ARTMAP

100

100

97.33

69.33

At a CNR of 20dB the MLP was able to recognize 86.6% of the modulation patterns

correctly. The KNN and the Artmap classified 100% of the modulation patterns correctly.

Once the noise level increased in the lOdB case the MLP was down to about 84.9%

43

correct classification, and the KNN and the Artmap were still performing at 100%. With a

steady increase in noise at 5dB CNR the MLP was getting about 77.1% of the patterns

correct while the KNN and Artmap were sensing 96.7% correctly and 97.3% correctly

respectively. At OdB the MLP classified 60.5% of the patterns correctly, the KNN

classified 71.3%, and the Artmap classified 69.3% of the patterns correctly. The Morlet

Modulation Classifier (MMC) shows an inverse relationship between percent of correct

classification and increased noise level (See Figure 32). The MMC's performance

decreased with increasing noise level in the same manner as the Mexican Hat Modulation

Classifier, however the MMC was able to classify more patterns correctly at high noise

levels (97.33% correct at 5 dB CNR) than the MHMC.

The Haar Wavelet coefficients were used next for feature extraction. The results

for Haar Wavelet are displayed in Table 4.

100

Haar Wavelet Modulation Classifier at

Constant co

■+■

5 10 15

Carrier to Noise Ratio (dB)

20

-♦—MLP

-A—ARTMAP

Fig. 33. Results From Haar Modulation Classifier For Constant CO.

44

Table 4. Haar Wavelet Modulation Classifier Results

Haar Wavelet

dB

20

10

5

0

MLP

75.67

61.93

49.47

47

KNN

100

65.33

52.67

50

ARTMAP

66

64.67

62

41

The 20dB case reflected 75.7% correct classification for the MLP and 100% correct

classification for the KNN. The Artmap was totally confused, it could only classify 66%

of the patterns correctly in the low noise region. At lOdB the MLP could only pick out

61.9% of the patterns correctly while the KNN recognized 65.3% correct, and the Artmap

was down to 64.7% correct classification. A gradual increase in noise level at 5dB

dropped the MLP down to 49.5% correct classification, 52.7% correct classification for

the KNN, and 62.0% correct classification for the Artmap. At OdB the three classifiers

were performing poorly. The MLP recognized 47.0% of the patterns correctly while the

KNN recognized 50% of the patterns correctly, and the Artmap could only see 41.0% of

the patterns correctly.

From the results of the test, the Haar Wavelet would probably not make a good

analyzing wavelet for modulation characterization. At the low noise level (lOdB CNR)

the best that the Haar coefficients could be classified was 65.3%. The Haar Wavelet has

a blocky shape, and it probably would not be able to extract all of the curved features of

the modulation patterns. This could be one of the reasons that the Haar Modulation

45

Qassifier (HMC) performed so poorly as compared to the other classifiers (See Figure

33).

The results began to differ a little more in the presence of varying noise level and

varying carrier frequency. The results for the Morlet Wavelet at carrier frequency oV2 are

displayed in Table 4. Since the Morlet Wavelet outperformed the other wavelets at

constant carrier frequency, only the performance of the Morlet Wavelet at varied carrier

frequency was considered.

Morlet Wavelet Classifier for Varied Carrier

co/2

100

0 5 10 15 20

Carrier to Noise Ratio (dB)

•MLP

-KNN

-ARTMAP

Fig. 34. Results From Modulation Classifier For aJ2.

Table 5. Morlet Wavelet Modulation Classifier Results for

Morlet Wavelet

(carrier frequency co/2)

dB

20

10

5

0

MLP

71.09

67.4

40.06

55

KNN

33.33

51.33

61.33

56

ARTMAP

80

84

88

69.33

46

There was an immediate decrease in performance due to the change in carrier frequency

(The discussion will be limited to the performance of the Artmap, because it was the only

classifier that showed consistent performance during this part of the experiment). At 20

dB CNR and a carrier frequency change of (a/2 the Artmap was able to classify 80% of the

patterns correctly. As the noise level increased in the 10 dB CNR case, the Artmap

recognized 84.0% of the patterns correctly. At 5 dB, the Artmap was up to 88.0%

correct classification, and at 0 dB the percent of correct classification dropped to 69.3%

(See Figure 34). At 20 dB CNR and a carrier frequency change of oV3 the Artmap

classified 80.7% of the pattern correctly. In the 10 dB case, the percent of correct

classification was 88.7%, and at 5 dB the percent of correct classification was down to

75.3%. At 0 dB CNR the Artmap was only seeing about 33.3% of the patterns correctly

(See Figure 35).

Morlet Wavelet Classifier at Varied Carrier co/3

■+-

5 10 15

Carrier to Noise Ratio (dB)

20

-M.P

-KNN

-ARTMAP

Fig. 35. Results From Morlet Modulation Classifier for atf3.

47

Table 6. Morlet Wavelet Modulation Classifier Results for ojG

Morlet Wavelet

(carrier frequenc

dB

20

10

5

0

|MLP

68.4

63.67

54.53

38.47

foa/3)

KNN

76.67

92

79.33

35.33

ARTMAP

80.67

88.67

75.33

33.33

From the fact that the Artmap is at the least 80% accurate at varied carrier frequency, it

can be assumed that the Artmap is capable ofrecognizing modulation patterns in the

presence of varying noise level and carrier frequency.

Results From Separation of BPSK/QPSK/8PSK/BFSK/QAM

The Morlet Modulation Identification Algorithm was extended to include more

signals. The signals used were Binary Phase Shift Keying, Quadature Phase Shift Keying,

8 Phase Shift Keying, Binary Frequency Shift Keying and Quadature Amplitude

Modulation. They were downloaded from an internet web site. The same message was

embedded upon each of these signals. The carrier frequency is 2 Hz, the symbol rate is 1

symbol/second, and the sampling rate is 16 samples/second. The Matlab programs and

Neural Networks were modified to accommodate 5 signals instead of 3 signals. The same

Neural Networks mentioned in Chapter 2 were used to classify these signals, using the

Morlet Wavelet coefficients. The Wavelet Transform proved to be just as effective in the

second experiment as it was in the first experiment at detecting transients in each signal.

The performance of the Morlet Modulation Classifier is given in Figure 36, and the

numerical results are displayed in Table 7.

48

Morlet Wavelet Modulation Classifier

Experiment #2

0 5 10 15 20

Carrier to Noise Ratio (CNR)

-MLP

-KNN

ARTMAP

Fig. 36. Results From Second Experiment For Morlet Modulation Qassifier.

Table 7. Morlet Modulation Classifier Results for Experiment #2

Experiment #2 Morlet Wavelet

dB

20

10

5

0

MLP

100

96.8

55.6

28.4

KNN

100

98.8

60.8

22

ARTMAP

100

93.6

56.8

22.4

The results show that even though the second experiment is more difficult than the first,

the percent of correct classification is still better at a lower CNR (This means a higher

noise region) than the previously mentioned algorithms.

49

CHAPTER 5

SUMMARY/CONCLUSIONS

This paper takes the concepts of wavelet analysis, signal processing and pattern

classification and intermingles them into a well defined project The theory of each

concept was introduced and discussed. Then an effective algorithm was devised to solve

the problem of Modulation Characterization. This algorithm was applied and data was

collected. The data was analyzed and compared to other research projects of similar

nature.

The CWT was used with a fixed scale in this research project. The 1-Dimensional

CWT maps a one variable function into a two variable function. This creates a large

amount of redundant data. This redundant data can be very helpful in trying to

characterize the behavior of a particular signal, but for some applications large amounts of

data are not desired. The CWT decomposes the signal at different scales, and depending

upon the application, the scale can be large or small. In this project the focus is upon the

small scales. At these scales the analyzing wavelet is more responsive to phase changes,

frequency changes and amplitude changes, and this is exactly what is needed. These CWT

coefficients taken at a small scale become feature vectors for direct input into a classifier.

In today's technological society, artificial intelligence has proven to be very useful.

There are many hazardous tasks that human operators once had to perform. Now

50

machines do these jobs for less money and with no loss to human life. This has produced

a desire to automate just about anything that is harmful or tedious. Classifiers are a result

of the need to automate society. They are the nucleus of any type of pattern recognition

application. Human operators are normally used for discriminating between modulation

types either by using moment techniques, zero crossing techniques or some other manual

procedure. In electronic warfare there is a need to accurately distinguish between

waveforms in free space. If it is possible to teach a machine to do this job accurately then

the human operator could be replaced. The classifiers used in this project are trained with

only the minimum amount of information, and still manage to produce results that are

comparable to other methods.

This research exclusively deals with the problem of modulation characterization in

the presence of varying noise and varying carrier frequency. It has been shown that this

system at constant carrier frequency possesses an inverse relationship between increasing

noise level and percent of correct classification. This is quite logical, because at higher

noise levels the features that the classifiers are trained upon become distorted. Also the

chosen wavelet scale is small, therefore in the frequency domain the wavelet acts as a

bandpass filter that only allows certain frequencies to pass.14

The second variable in this project was changing carrier frequency. Taking the

Artmap for example (mainly because it outperformed the other classifiers), even though

the carrier frequency was altered the Artmap continued to do a goodjob of classifying the

signals. This can be explained by understanding how the classifiers operate. As noise is

added to a signal, the features extracted by the CWT keep a constant horizontal position,

51

but are distorted vertically. The power of the technique comes from the extraction of

peaks of different magnitudes and shape, but during addition of noise all the peaks become

distorted, therefore really the only features that the classifiers can depend on are those that

are vertical. In varying the carrier frequency in the presence of noise the extracted peaks

become shifted and sometimes even broadened. This broadening of the peaks sometimes

makes the features more prominent. There will definitely be a reduction in percent of

correct classification from the fact that the peaks are being shifted (from the change in

carrier frequency); however it should not be drastic as seen in the results (In the Morlet

case there was a change from 100% correct to 80% correct at 20 dB CNR). At the point

where the broadening peaks become unrecognizable, the percent of correct classification

begins to drop again. Taking all of this information into account the system remains well

behaved even in the presence of varying noise level and carrier frequency (where the

carrier frequency does not deviate more than 1/3 of the original carrier frequency).

FUTURE WORK

This paper reflects the initial progress of an AutomaticModulation Identifier

centered around the use of pattern recognition and wavelet theory. This paper integrates

two topics in today's technology into one project. In a system such as this there are many

variables, and it is not sensible to change all possible variables at once. This work is

evidence that wavelet theory and pattern recognition can be combined to produce a

working system that is comparable to other models. At present the only variables that

52

were explored were increasing noise level and varying carrier frequency. The Wavelet

Transform has been implemented in HP-VEE (See Figure 37).

Fig. 37. HP-VEE Program For Wavelet Transform.

This program allows any algorithm to be expressed in the form of modules. This

language is lower than Matlab, and it will be beneficial to others in my research group

who want to implement the Wavelet Transform on an Altera Field Programmable Gate

Array computer board. The HP-VEE implementation is still being studied, thus the reason

for placing it in this section. Also plans are underway for a Neural Network that can

accept imaginary input.

53

APPENDIX I

COMMUNICATION SIGNALS

The purpose of this section is to reiterate the procedure used to produce each signal.

Figure 38 displays the signals used in the first experiment. These signals were created

using C programs. The values of the phase, frequency and amplitude changes are given in

the figure. From closely observing the figure, it should be evident the procedure used to

create each signal.

COMMUNICATION SIGNALS

EXPERIMENTS

CARRIER: A(Cos cot + 4>)

>=7C or

co=7i/8 or co=iz/l6

A = lorA=0

Fig. 38. Communication Signals From Experiment #1.

54

All of the signals used in the second experiment are Phase Shift Keying except for BFSK

which was described in Figure 38. Phase Shift Keying signals can be described by "Signal

Constellations."12 A Signal Constellation is a representation of the Cartesian Graphing

Plane (See Figure 39). The axis has labels that mark the angle of the phase change for

each signal, thus a binary digit or a set of binary digits correspond to a particular phase

change.

COMMUNICATION SIGNALS

EXPERIMENTS

BPSKQPSK

BFSK

eo=n/8 or co=7i/16

00=0

01=71/2

10=71

ll=3n/2

8PSK 16-QAM

A=l 0000=0 1001= tf2

A=2 0001=71/8 1101=571/8

A=3 0011=71/4 0101=3/4

A=4 0111=3708 O01O=7n/8

Fig. 39. Communication Signals From Experiment #2.

55

APPENDIX n

MATLAB PROGRAMS

This program produces the CWT of the 50 sets of the 3 (experiment 1) or 5

(experiment 2) signals.

Cwt5.m

function do_cwt(fni,fno,a,wltype,display)

% do_cwt(fni,fno,a,wltype,display)

% Reads noisy raw patterns and convert them to wavelet patterns

% Inputs: fni = noisy raw pattern file name

% fno = output (wavelet pattern) file name

% a = wavelet scale factors for each class

% wltype = wavelet type ('Morlet1, 'Mexican1, or 'Haar1)

% display = if display > 0, the display-th patterns are plotted

if nargin < 4, error('Requires 4 or 5 arguments'), end

if nargin = 4, display = 0, end

ndim = 128;

npat = 250;

nclass = 5;

% Create wavelet filter

tl=-10:20/(ndim-l):10;

y = zeros(nclass,ndim);

forc= lmclass

t2 = tl/a(c);

if wltype ='Morlef

y(c,:)=real(l/sqrt(a(c)) * exp(i.*5.6.*t2) .* exp(-t2.A2/2));

elseif wltype = 'Mexican'

y(c,:)=l/sqrt(a(c)) .* Q-t2.A2) .* exp((-t2.A2)*.5);

elseif wltype= 'Haar1

forr=l:ndim

if t2(r) >= -.5*a(c) & t2(r) <0

y(c,r)=l/sqrt(a(c));

elseif t2(r) >= 0 & t2(r) < .5*a(c)

y(c,r)=-l/sqrt(a(c));

else

y(c,r)=0;

end

end

else

56

errorCwltype not recognized');

end

end

% Read the raw noisy patterns

M=fscanf(fid,'%gt,[(ndim+1) npat]);

M=M";

status=fclose(fid);

C = M(:, 1); % C contains the class information for each input vector

x = M(:,2:ndim+1); % x contains the input vectors

clear M;

% Compute and store the wavelet patterns

fid = fopenCfho.W);

for g= lrnpat

c = round(C(g))+l;

multiply = conj(fft(y(c,:))) .* fft(x(g,:));

wt=real(ifft(multiply));

CWTstoring=g;

mag=sqrt(fftshift(wt).A2);

if display = g

subplot(2,l,l), plot(tl,x(g,:));

tide('Modulation Pattern');

subplot(2,l,2), plot(tl,mag);

title('CWT of Modulation Pattern');

end

fprintf(fid,'%d',c-l);

fork=l:ndim

fprintf(fid, '%5.3f',mag(k));

end

fprintf(fid,V);

end

fclose(fid);

This program produces the 50 sets of 3 or 5 signals that contain AWGN.

Noisemak5.m

function CNR_actual = do_cnr(CNR_desired,fho,seed)

% CNR_actual = do_cnr(CNR_desired,fno)

% Generates a pattern file for LNKnet classifier

% from wavelet routines

57

% Inputs: CNR_desired = desired CNR

% fno = output file name

if nargin < 2, errorCRequires 2 or 3 arguments'), end

if nargin = 3, seed = randn('seed',seed); end

global noise;

global variable;

% Loop over modtypes

noise = CNR_desired;

fnn=fno;

fpnsfopenCfnn/w1);

formodtype= 1:5

ifmodtype=l

fnm = 'opsk.dat';

elseif modtype = 2

fnm = 'qaml6.dat1;

elseif modtype = 3

fnm = 'qpsk.dat';

elseif modtype= 4

fnm ='bask.dat';

elseif modtype= 5

fnm ='bfsk.dat';

end

fpm=fopen(fnm,'rl);

x^fscanf(fpm,'%f1);status=fclose(fpm);

forv=l:128

h(v)=x(v);

end

x=h;

t=-10:20/127:10;

fore = 1:50

noisewriting=c

A=max(x);

loglO(AA2/2));

n=sqrt(B)*randn(l,128);

s=x+n;

fprintfCfpn.^d \modtype-l);

forb=l:128;

fprintf(fpn,1%5.3ft,s(b));

fprintf(fpn, '

end

58

end

end

fclose(fpn);

CNR_actual = -999;

This program performs multi-resolutional analysis of each modulation pattern to

help determine the scale that should be used to extract features.

Scales.m

% This program takes the wavlet transfrom of the wavelet for various values of a

clear

energy =0;

k=inputCPlease enter the file name of your signal? 7s1)

p=inputCPress 1 to print anything else not to.')

fid=fopen(k,'f);

x=fscanf(fid/%f1);

status=fclose(fid);

forv=l:128;

f(v)=x(v);

end

x=f;b=size(x)

l=b(U)

tl=-10:20/a-l):10;

t=-10:20/a-l):10;

forq=l:10;

a=2*(-q+l)

t2= tl./a;

y=real(l/sqrt(a).*exp(i.*5.6.*t2).*exp(-t2.A2/2));

multiply=conj(fft(y)).*fft(x);

inverse=real(ifft(multiply));

A(q,:)=inverse(:,:);

mag=abs(A(q,:));

figure(q),subplot(31 l),plot(t^c)

titleCModulation Pattern')

figure(q),subplot(312),plot(t,fftshift(mag.A2yaA3))

title(['Energy Density at scale = \num2str(a)]);

figure(q),subplot(313),plot(t4ftshift(mag))

titleCrMagnitude of CWT at scale = ',num2str(a)]);

ifp = l

print -dps slotps

!lpr -Php4 slotps

end

end

B=real(fftshift(A));

59

APPENDIX

SIMULATION OF GAUSSIAN NOISE CHANNEL

The noise channel simulation is a straight forward process. As seen in Figure 40 the

noiseless modulation pattern is added to the noise signal. This noise signal has a specific

CNR that was created using the method outlined in Chapter 3. The addition of the noise

signal and the noiseless modulation pattern yields the received signal. The received signal

is a corrupted version of the noiseless modulation pattern, and this is exactly what happens

when a signal is transmitted through a channel. Therefore the procedure used in Figure 40

is indeed a noise channel simulation.

SIMULATION OF GAUSSIANNOISE CHANNEL

Fig. 40. Simulation of Gaussian Noise Channel.

60

To go a step further, the Wavelet Transform of the received signal is provided. Note that

the Wavelet Transform reflects the fact that noise was added to the modulation pattern;

however it is still possible to see the maximas which correspond to frequency changes.

This is the whole idea behind the research. The Neural Networks can identify these

maxima even though the signal is noisy, because the noise only corresponds to shifts in the

original signal.

Wavelet Transform of (BFSK + 20 dB AWGN)

Fig. 41. Feature Extraction From Noisy Signal.

61

APPENDIX IV

DECISION BOUNDARIES FOR NEURAL NETWORKS

Figure 42 is an example of a Decision Region that a Neural Network could

possible form.7 The concept behind Neural Network Technology is to train the Neural

Network on a particular pattern, then test the Neural Network on corrupted versions of

the training pattern. As a result the Neural Network will either be able to classify all of the

patterns, some of the patterns or none of the patterns. The procedure that the Neural

Networks use to decide which patterns are in what class is outlined in Figure 42. The

Neural Network uses the training data to form boundaries in what is called a Decision

Region. Then the Neural Networks place the test patterns in that Decision Region. The

test patterns either fall within the boundary of a particular class, on the boundary of a

particular class or near the boundary of a particular class. Given this information the

Neural Networks make the final decision about the class of the test pattern via its training

algorithm.

62

EXAMPLEOFDEQSIONBOUNDARIES FORNEURALNETWORKS

Fig. 42. Example of Decision Boundaries for Neural Networks.

63

BIBLIOGRAPHY

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3. Hsue, S.Z., S. S. Soliman, "Automatic Modulation Classification Using Zero

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1982.

5. Cohen, Albert and Jelena Kovacevic. "Wavelets: The mathematical Background."

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11. Sweldens, Wim. "Wavelets: What Next?" Proceedings of the IEEE vol. 84: No. 4,

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12. Wilson, Stephen G. Digital Modulation and Coding, pp 62-65; Prentice Hall; New

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14. Guillemain, Philippe, and Richard Kronland-Martinet. "Characterization of Acoustic

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65