DIGITAL MODULATION RECOGNITION A THESIS … · Modülasyon tanıma sistemlerinin başarım kriteri...

DIGITAL MODULATION RECOGNITION

A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF MIDDLE EAST TECHNICAL UNIVERSITY

BY

EREM ERDEM

IN PARTIAL FULLFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF MASTER OF SCIENCE IN

ELECTRICAL AND ELECTRONICS ENGINEERING

DECEMBER 2009

Approval of the thesis:


submitted by EREM ERDEM in partial fulfillment of the requirements for the

degree of Master of Science in Electrical and Electronics Engineering

Department, Middle East Technical University by,

Prof. Dr. Canan Özgen

Dean, Graduate School of Natural and Applied Sciences

Prof. Dr. Đsmet Erkmen

Head of Department, Electrical and Electronics Engineering

Prof. Dr. Yalçın Tanık

Supervisor, Electrical and Electronics Engineering Dept., METU

Examining Committee Members:

Prof. Dr. Mete Severcan

Electrical and Electronics Engineering Dept., METU

Prof. Dr. Yalçın Tanık


Assoc. Prof. Dr. Melek Yücel


Assoc. Prof. Dr. A. Özgür Yılmaz


Y. Müh. Enis Doyuran

ASELSAN Inc. Date: 04.12.2009

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I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work.

Name, Last Name : Erem ERDEM

Signature :

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ABSTRACT


Erdem, Erem

M. S., Department of Electrical and Electronics Engineering

Supervisor : Prof. Dr. Yalçın Tanık

September 2009, 144 pages

In this thesis work, automatic recognition algorithms for digital modulated signals

are surveyed.

Feature extraction and classification algorithm stages are the main parts of a

modulation recognition system. Performance of the modulation recognition system

mainly depends on the prior knowledge of some of the signal parameters, selection

of the key features and classification algorithm selection.

Unfortunately, most of the features require some of the signal parameters such as

carrier frequency, pulse shape, time of arrival, initial phase, symbol rate, signal to

noise ratio, to be known or to be extracted. Thus, in this thesis, features which do

not require prior knowledge of the signal parameters, such as the number of the

peaks in the envelope histogram and the locations of these peaks, the number of

peaks in the frequency histogram, higher order moments of the signal are

considered. Particularly, symbol rate and signal to noise ratio estimation methods

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are surveyed. A method based on the cyclostationarity analysis is used for symbol

rate estimation and a method based on the eigenvector decomposition is used for the

estimation of signal to noise ratio. Also, estimated signal to noise ratio is used to

improve the performance of the classification algorithm.

Two methods are proposed for modulation recognition:

1) Decision tree based method

2) Bayesian based classification method

A method to estimate the symbol rate and carrier frequency offset of minimum-shift

keying (MSK) signal is also investigated.

Keywords: digital modulation recognition, feature extraction, Bayesian blind

classification, MSK carrier frequency and symbol rate estimation.

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ÖZ

SAYISAL MODÜLASYON TANIMA

Erdem, Erem

Yüksek Lisans, Elektrik Elektronik Mühendisliği Bölümü

Tez Yöneticisi : Prof. Dr. Yalçın Tanık

Eylül 2009, 144 sayfa

Bu tez çalışmasında, sayısal modülasyona sahip sinyaller için otomatik modülasyon

tanıma algoritmaları araştırılmıştır.

Öznitelik çıkarımı ve sınıflandırma algoritması; modülasyon tanıma sistemlerinin

temel yapılarıdır. Modülasyon tanıma sistemlerinin başarım kriteri temel olarak

sinyale özgü bazı parametrelerin önceden bilinmesine, öznitelik seçimine ve

sınıflandırma algoritması seçimine dayanmaktadır.

Ne yazık ki, özniteliklerin çoğunluğu taşıyıcı frekans, darbe biçimi, sinyal geliş

zamanı, faz bilgisi, veri hızı, sinyal gücünün gürültü gücüne oranı gibi bazı sinyale

özgü parametrelerin bilinmesini veya bulunmasını gerektirir. Bu nedenle, bu tez

çalışması kapsamında genlik histogramında yer alan tepe sayısı ve bu tepelerin

konumları, frekans histogramında yer alan tepe sayısı, sinyalin yüksek dereceden

momenti gibi sinyale özgü parametre ihtiyacı gerektirmeyecek öznitelikler üzerinde

düşünülmüştür. Özellikle, veri hızı ve sinyal gücünün gürültü gücüne oranı kestirme

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metotları da araştırılmıştır. Dairesel-durağan analize dayalı bir metot sembol veri

hızı kestirmede ve öz-vektör dağılımına dayalı bir metot sinyal gücünün gürültü

gücüne oranını kestirmede kullanılmıştır. Bulunan sinyal gücünün gürültü gücüne

oranı, sınıflandırma algoritmasının performansının arttırılmasında da kullanılmıştır.

Modülasyon tanıma için 2 metot önerilmiştir:

1) Karar ağacı yöntemine dayalı bir metot

2) Bayes dayalı bir sınıflandırma metodu

MSK sinyalinin veri hızını ve taşıyıcı frekansını bulmaya yönelik bir metot da

incelenmiştir.

Anahtar Kelimeler: sayısal modülasyon tanıma, öznitelik bulma, Bayes dayalı kör

sınıflandırma, MSK taşıyıcı frekans ve veri hızı bulma.

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To My Parents

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ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to Prof. Yalçın Tanık for his guidance,

advice, criticism, patience and encouragement throughout the completion of the

thesis.

I would like to thank to all of my friends and colleagues for their support and

encouragements.

Finally, I would like to thank to my family for everything that I have. This thesis is

dedicated to them.

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TABLE OF CONTENTS

ABSTRACT....................................................................................................................................... IV

ÖZ ......................................................................................................................................................VI

ACKNOWLEDGEMENTS ............................................................................................................... IX

TABLE OF CONTENTS.................................................................................................................... X

LIST OF ABBREVIATIONS ...........................................................................................................XII

LIST OF TABLES .......................................................................................................................... XIV

LIST OF FIGURES......................................................................................................................... XVI

CHAPTERS

1. INTRODUCTION............................................................................................................................ 1

1.1 BACKGROUND ....................................................................................................................... 1

1.2 OUTLINE OF THESIS.............................................................................................................. 4

2. LITERATURE SURVEY ................................................................................................................ 5

2.1 INTRODUCTION..................................................................................................................... 5

2.2 RELEVANT PREVIOUS WORK............................................................................................. 6

3. BASIC FEATURE EXTRACTION BLOCKS .............................................................................. 21

3.1 SIGNAL TO NOISE RATIO (SNR) ESTIMATION BLOCK................................................. 21

3.2 BANDWIDTH AND SPECTRAL SHAPE ............................................................................. 26

3.2.1 LINEARLY MODULATED SIGNALS ......................................................................... 27

3.2.2 CPM SIGNALS............................................................................................................... 28

3.3 CYCLOSTATIONARY ANALYSIS...................................................................................... 31

3.3.1 LINEARLY MODULATED SIGNALS ......................................................................... 31

3.3.2 CPM SIGNALS............................................................................................................... 32

3.3.3 CYCLOSTATIONARITY IN MODULATION RECOGNITION ................................. 34

3.4 ENVELOPE FEATURES........................................................................................................ 39

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3.5 SPECTRAL ANALYSIS OF MOMENTS .............................................................................. 42

3.6 INSTANTANEOUS FREQUENCY FEATURES .................................................................. 43

3.7 CARRIER FREQUENCY OFFSET ESTIMATION............................................................... 46

3.7.1 INTRODUCTION ........................................................................................................... 46

3.7.2 THE PROPOSED METHOD .......................................................................................... 48

3.7.3 SIMULATIONS .............................................................................................................. 55

3.7.4 SELECTION OF BLOCK PARAMETERS.................................................................... 55

3.7.5 SIMULATION RESULTS .............................................................................................. 59

3.7.6 CONCLUSION ............................................................................................................... 61

4. SPECIFIC MODULATION RECOGNITION TOOLS................................................................. 62

4.1 OUTPUTS OF THE FEATURE EXTRACTION BLOCKS.................................................... 62

4.2 MASK RECOGNITION.......................................................................................................... 67

4.3 MPSK RECOGNITION .......................................................................................................... 74

4.4 MQAM RECOGNITION ........................................................................................................ 82

4.5 CW RECOGNITION............................................................................................................... 91

4.6 CPFSK RECOGNITION......................................................................................................... 95

4.7 AWGN RECOGNITION....................................................................................................... 101

4.8 CONCLUSION ..................................................................................................................... 103

5. THE RECOGNITION SYSTEM ................................................................................................. 105

5.1 THE DECISION TREE METHOD........................................................................................ 105

5.2 THE BAYESIAN BASED RECOGNITION SYSTEM ........................................................ 108

5.3 SIMULATION RESULTS .................................................................................................... 110

5.3.1 FEATURES OF SIMULATIONS ................................................................................. 110

5.3.2 TUNING OF THE RECOGNITION SYSTEM ............................................................ 112

5.3.3 CONDUCTION OF TESTS .......................................................................................... 130

5.3.4 RESULTS OF SIMULATIONS.................................................................................... 130

6. CONCLUSIONS AND FUTURE WORK................................................................................... 136

REFERENCES................................................................................................................................. 139

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LIST OF ABBREVIATIONS

AM Amplitude Modulation

ANN Artificial Neural Network

ASK Amplitude Shift Keying

COMINT Communication Intelligence

CPFSK Continuous Phase Frequency Shift Keying

CPM Continuous Phase Modulation

CW Continuous Wave

DFT Discrete Fourier Transform

DTC Decision Tree Classifier

ECM Electronic Counter Measure

ESM Electronic Support Measure

EW Electronic Warfare

FDD Frequency Difference Detector

FFT Fast Fourier Transform

FM Frequency Modulation

FSK Frequency Shift Keying

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HOC High Order Cumulant

IF Intermediate Frequency

LPF Low Pass Filter

MASK M-ary Amplitude Shift Keying

MPSK M-ary Phase Shift Keying

MQAM M-ary Quadrature Amplitude Modulation

MSK Minimum Shift Keying

NNC Neural Network Classifier

QAM Quadrature Amplitude Modulation

QLLR Quasi-Log-Likelihood Ratio

PLL Phase-Locked Loop

RBF Radial Basis Function

SLC Square Law Classifier

SNR Signal to Noise Ratio

SVC Support Vector Machine

PSD Power Spectrum Density

PSK Phase Shift Keying

RF Radio Frequency

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LIST OF TABLES

TABLES

Table 4-1: Peak Vectors for ASK2, ASK4 and ASK8 Signals................................ 68

Table 4-2: Peak Vectors for PSK2, PSK4 and PSK8 Signals.................................. 75

Table 4-3: Peak Vectors for QAM2, QAM4 and QAM8 Signals............................ 85

Table 4-4: The Outputs of Each Block .................................................................. 103

Table 4-5: Features and Modulation Type Relation .............................................. 104

Table 5-1: Probability of 4 1.5n ≤ .......................................................................... 114

Table 5-2: Probability of 41.5 3n< ≤ .................................................................... 114

Table 5-3: Probability of 43 n< ............................................................................. 115

Table 5-4: Probability of 3(1) 1pv = ........................................................................ 116

Table 5-5: Probability of 3(1) 2pv = ....................................................................... 116





Table 5-10: Probability of 3(2) 0pv = .................................................................... 119






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Table 5-16: Probability of 3(3) 5pv = ..................................................................... 122




Table 5-20: Probability of 0 1k = ........................................................................... 125




Table 5-24: Probability of 5 1.5n ≤ ........................................................................ 127

Table 5-25: Probability of 51.5 2.5n< ≤ ............................................................... 128

Table 5-26: Probability of 52.5 4.5n< ≤ ............................................................... 128

Table 5-27: Probability of 54.5 8n< ≤ .................................................................. 129

Table 5-28: Confusion Matrix of The Bayesian Based Recognition System at

SNR=6 dB ......................................................................................................... 131

Table 5-29: Confusion Matrix of The Decision Tree Method at SNR=6 dB......... 131


SNR=9 dB ......................................................................................................... 132

Table 5-31: Confusion Matrix of The Decision Tree Method at SNR=9 dB......... 132


SNR=12 dB ....................................................................................................... 133

Table 5-33: Confusion Matrix of The Decision Tree Method at SNR=12 dB....... 133


SNR=15 dB ....................................................................................................... 134

Table 5-35: Confusion Matrix of The Decision Tree Method at SNR=15 dB....... 134

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LIST OF FIGURES

FIGURES

Figure 3-1 Estimation Bias vs. SNR ........................................................................ 25

Figure 3-2 Estimation STD vs. SNR........................................................................ 26

Figure 3-3 Absolute Estimation Average Error Normalized to Symbol Rate (Fb) for

Linear Modulation Signals vs. SNR.................................................................... 38

Figure 3-4: Absolute Estimation Average Error Normalized to Symbol Rate (Fb) for

CPFSK (h=0.5; Fb=2kHz) vs. SNR .................................................................... 38

Figure 3-5: Absolute Estimation Average Error for CW (fs=50kHz, SNR=20 dB) vs.

Carrier Frequency Offset..................................................................................... 45

Figure 3-6: Block Diagram of The Frequency and Symbol Rate Estimator............ 49

Figure 3-7: Basic PLL Block Diagram .................................................................... 50

Figure 3-8: The baseband equivalent PLL Block Diagram ..................................... 52

Figure 3-9: PLL Block Diagram .............................................................................. 52

Figure 3-10: Logarithm of the Estimation Error at SNR=-5dB ............................... 57

Figure 3-11: Logarithm of the Estimation Error at SNR=0dB ................................ 58

Figure 3-12: Absolute Carrier Frequency Estimation Error at / 15b oE N dB= ....... 59

Figure 3-13: Absolute Symbol Rate Estimation Error at / 15b oE N dB= ............... 60

Figure 3-14: Average Number of Bit Errors at / 15b oE N dB= .............................. 60

Figure 4-1: Envelope Histograms for ASK8............................................................ 64

Figure 4-2: Histogram Output for ASK2 ................................................................. 69



Figure 4-5: Spectral Analysis of Moments Block Output for MASK vs. SNR ....... 72

Figure 4-6: Frequency Histogram Output ( )5 6,n n for MASK vs. SNR.................. 73

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Figure 4-7: Histogram Output For PSK4................................................................. 76

Figure 4-8: Histogram Output For PSK8................................................................. 77

Figure 4-9: Spectral Analysis of Moments Block Output for PSK2 vs. SNR ......... 79

Figure 4-10: Spectral Analysis of Moments Block Output for PSK4 vs. SNR ....... 80

Figure 4-11: Spectral Analysis of Moments Block Output for PSK8 vs. SNR ....... 80

Figure 4-12: Frequency Histogram Output ( )5 6,n n for MPSK vs. SNR ................ 81

Figure 4-13: Constellation Diagram of QAM2........................................................ 83

Figure 4-14: Constellation Diagram of QAM4........................................................ 83

Figure 4-15: Constellation Diagram for QAM8 ...................................................... 84

Figure 4-16: Histogram Output for QAM4 vs. SNR ............................................... 86

Figure 4-17: Histogram Output for QAM8 vs. SNR ............................................... 86

Figure 4-18: k2 for QAM2, QAM4 and QAM8 vs. SNR ........................................ 89

Figure 4-19: Frequency Histogram Output ( )5 6,n n for QAM8 vs. SNR................ 90

Figure 4-20: Histogram output for CW vs. SNR ..................................................... 92

Figure 4-21: 0 2 4 8, , ,k k k k for CW vs. SNR .............................................................. 93

Figure 4-22: Frequency Histogram Output ( )5 6,n n for CW vs. SNR..................... 94

Figure 4-23: Frequency Histogram Output ( 5 6,η η ) for M-level CPFSK vs. SNR .. 97

Figure 4-24: Modulation Index (h) Estimation Performance for CPFSK Signals

(h=0.5) ................................................................................................................. 99

Figure 4-25: A Plot of the Estimated Modulation Index (h ) vs. True Modulation

Index (h) at SNR=20dB..................................................................................... 100

Figure 5-1: The Proposed Decision Tree for Modulation Classification............... 107

Figure 5-2: The Functional Block Diagram of the Bayesian Based Recognition

System for Modulation Classification............................................................... 109

Figure 5-3: Block Diagram of the Transmitter ...................................................... 111

Figure 5-4: Block Diagram of the Receiver........................................................... 112

Figure 5-5: Block Diagram of the Fixed Tuned Receiver...................................... 112

1

CHAPTER 1

INTRODUCTION

1.1 BACKGROUND

There are lots of communication signals with different modulation types and

different frequencies. It is required to identify and monitor these signals for some

applications. Some of these applications are for civilian purposes such as signal

confirmation, interference identification and spectrum management. The other

applications are for military purposes such as electronic warfare (EW), surveillance

and threat analysis. In electronic warfare applications, electronic support measures

(ESM) system plays an important role as a source of information required to

conduct electronic counter measures (ECM), electronic counter-counter measures

(ECCM), threat detection, warning, target acquisition and homing. [1]

In the past, COMINT systems have relied on the manual modulation recognition of

measured parameters to provide classification of different emitters. However,

recently automatic modulation recognition systems have been developed. One of

the oldest versions of modulation recognizers uses a bank of demodulators, each

designed for only one type of modulation. Modulation type of the received signal

can be decided by listening to the demodulator outputs. This type of a recognizer

requires long signal durations and highly skilled operators. The automation of such

a recognizer is achieved by introducing a set of intelligent decision algorithms at the

demodulator outputs. However, the implementation of such systems is complex and

2

requires excessive computer storage. Moreover, the number of modulation types

that can be recognized is limited by the number of demodulators used. [1]

Automatic modulation recognition is more powerful than manual modulation

recognition because by integrating the automatic modulation recognizer into an

electronic support measurement (ESM) receiver, including an energy detector and a

direction finder (DF), would allow an operator to improve his efficiency and his

ability to monitor the different activities in the frequency band of interest. Thus, in

advanced ESM systems, the operator is replaced by sophisticated electronic

machines. The main objective of any surveillance system is threat recognition by

comparing the characteristics of the intercepted emitters against a catalogue of

reference characteristics or signal sorting parameters. One of the important

parameters is the signal modulation type. [1]

Generally, any surveillance system in Communication Intelligence (COMINT)

applications consists of three main blocks: receiver-front-ends (activity detection

and frequency down conversion), modulation recognizer (key features extraction

and classification), and output stage (normal demodulation and information

extraction). There are several types of receiver-front-ends such as channelized,

acoustic-optical spectrum analyzer, instantaneous frequency measurement (IFM),

scanning superheterodyne, and microscan superheterodyne receivers. At the output

stage, there are several functions performed and they are mainly related to

information extraction, recording and exploitations. All these functions are

preceded by signal demodulation. Once the correct modulation type of the

intercepted signal is determined, some of the following functions performed at the

output stage are straightforward classical functions, but some may not be trivial

(e.g. deciphering). Thus, the key functional block is the modulation recognizer. The

prior information required for any modulation recognizer is the signal bandwidth,

which can be determined through the use of an energy detector in the receiver front-

end stage. The information obtained by the energy detector, modulation recognizer,

and parameters estimator such as the carrier frequency, the signal bandwidth, the bit

3

rate, the modulation type…etc. are gathered to perform the signal demodulation and

information extraction. [1]

Modulation recognition is extremely important in COMINT applications for several

reasons. Firstly, applying the signal to an improper demodulator may partially or

completely damage the signal information content. Secondly, knowing the correct

modulation type helps to recognize the threat and to determine the suitable jamming

waveform. [1]

Without any knowledge of the transmitted data and many unknown parameters at

the receiver, blind identification is a difficult task. Particularly, classification

process is even more challenging in real world scenarios with multipath fading,

frequency selective, and time varying channels. Therefore, a better recognition

system can be constructed by dividing the classification problem into subclasses

and dealing with each of these subclasses separately. In this thesis, a blind

recognition system is introduced to discriminate the digitally modulated signals in

an AWGN channel. Also, methods for the extraction of some of the signal

parameters such as the symbol rate, signal to noise ratio (SNR) and the carrier

frequency offset (CFO) are investigated.

4

1.2 OUTLINE OF THESIS

The thesis consists of six chapters. A brief summary of chapters 2 through 6 is

given in the following.

In Chapter 2, relevant previous works for digitally modulated signal classification

are reviewed.

In Chapter 3, the basic blocks that can be used in the classification of digitally

modulated signals are discussed.

In Chapter 4, performances of the selected key features for each modulation type

are investigated.

In Chapter 5, two classification methods are given. The first one is the decision tree

based method and the second one is the Bayesian based method. In addition,

computer simulations are carried out to compare the proposed methods.

In Chapter 6, some concluding remarks are given and future works are discussed.

5

CHAPTER 2

LITERATURE SURVEY

2.1 INTRODUCTION

In modern communication systems, digital modulation techniques are more

frequently used. Hence, the digital modulation recognizers have critical importance.

In the literature, the recognizers can be divided into two subsets according to

methods used in approaching classification problems:

1) A decision-theoretic approach,

2) A pattern-recognition or feature-extraction approach.

In a decision-theoretic framework, probabilistic arguments are employed to derive a

proper classification rule. This necessitates a statistical description of the signals (or

hypotheses). This approach affords an optimal procedure in the sense that a

discriminating test can be found which is the best under a specific performance

measure. Typically, this optimal rule will be hard to implement exactly. A simpler

way to derive a classifier structure is to rely upon the classic pattern-recognition

concept of “feature,” which assigns signatures to specific signal formats. The key

advantage of a well-chosen feature set is, of course, simplicity. In a broad sense,

decision theory can be perceived as the rigorous framework which explains and

justifies the selection of the “correct” features, whereas the more practical pattern-

6

recognition approach chooses these features on an ad hoc (perhaps intuitive) basis,

and is more concerned with issues of implementability. [2]

2.2 RELEVANT PREVIOUS WORK

In [3], an adaptive technique for classifying ASK2, PSK2, PSK4 and FSK2 signals

was introduced. The envelope of the signal, the spectra of the signal, the signal

squared and the signal quadrupled are used to derive the following key features of

the intercepted signal:

• The mean of the envelope

• The variance of the envelope

• Magnitude and location of the two largest peaks in the signal spectrum

• The magnitude of the spectral component at twice the carrier frequency

of the signal squared

• The magnitude of the spectral component at four times the carrier

frequency of the signal quadrupled

In [3], feature vector extraction, weight vector generation, and the modulation

classification are the main steps of the classifier. There exists a training stage where

weight vectors are generated for each class of signal considered using an adaptive

technique based on LMS algorithm. In this training stage known signals with 20 dB

SNR are used. The features of the intercepted signals are multiplied with the weight

vectors, and then the selection of the modulation type is made according to the

weighting function which produces the largest output. The discrimination of PSK2

and PSK4 signal at 5 dB SNR is the only performance criteria of the proposed

recognizer.

7

In [2], quasi-log-likelihood ratio (qLLR) was used as a classification rule to

discriminate PSK2 and PSK4. Also, the performance of the proposed method was

compared with the square law classifier (SLC) and the Phase-Based Classifier. It is

proved analytically that qLLR is better than the other methods even at low SNR.

However, it is assumed that all the signal parameters such as carrier frequency,

initial phase, symbol rate, SNR, pulse shape at the receiver front end are known.

In [4], a classification method for the constant amplitude signals such as CW,

MPSK, and MFSK was proposed based on the zero-crossing characteristics of the

intercepted signal. By using a zero-crossing sampler, it is easy to obtain information

related to the phase transitions of the received signal in a wide frequency dynamic

range. The proposed method consists of four main stages:

1. Extraction of zero-crossing sequence x(i), zero-crossing interval

sequence y(i), zero-crossing interval difference sequence z(i)

2. Inter-symbol transition (ISI) sample detection

3. CNR, carrier frequency estimation, variance of zero-crossing interval

sequence, G

4. Modulation classification based on G; frequency and phase difference

histograms.

From the simulation results, it is claimed that successful modulation classification is

achievable for SNR>15 dB. The performance of the recognizer to discriminate CW

and MPSK signals is dominated by the estimation accuracy of the carrier frequency;

whereas CW and MFSK discrimination performance is dominated by the ratio of

symbol rate to carrier frequency. The proposed classifier does not need prior

knowledge of the basic signal parameters; it is able to estimate the carrier frequency

and the symbol rate. However, its performance degrades for SNR<12dB; and it

cannot discriminate varying amplitude signals such as MASK.

8

In [5], a classification method for the MPSK signals was proposed based on the

statistical moments of the intercepted signal phase. The Tikhonov function is used

to approximate the asymptotic distribution of the phase of the intercepted signal. It

is shown that nth moment (n even) of the phase of the signal is a monotonic

increasing function of M for MPSK signals. This property is used to find a proper

decision rule for classification. The proposed classifier consists of three main

stages:

1. Phase Extraction

2. Even order moment computation

3. Threshold comparison

4. Decision

It is claimed that the second order moment is sufficient for discrimination of CW

and MPSK; whereas for BPSK eighth order moment, and for QPSK-8PSK higher

order moments are required for proper operation of the classifier at low SNR. The

performance of the classifier with eighth moment is also compared with quasi log-

likelihood ratio (qLLRC), square-law (SLC) and the phase-based (PBC) classifiers.

From the simulation results, it is claimed that qLLR classifier is better at low SNR;

however, proposed classifier outperforms PRC and SRC for SNR>0. All the signal

parameters such as carrier frequency, initial phase, symbol rate and CNR are

assumed to be known exactly.

In [6], a classification method for the CW, BPSK, QPSK, BFSK, QFSK signals was

proposed based on the autoregressive modeling. Instantaneous carrier frequency and

BW of the intercepted signal is obtained from the poles of the autoregressive

polynomial for each analysis frame of the intercepted signal then the following

feature vectors are derived:

• Mean of the instantaneous frequency

• Standard deviation of the instantaneous frequency (IF)

9

• Standard deviation of the instantaneous bandwidth (IB)

• Mean of the IB peaks (normalized to the largest peak)

• The height of the peaks of the differentiated IF

It is claimed that success rate is greater than 99% at a SNR of 15 dB. Only constant

envelope signals are investigated and it is assumed that only one signal exists at the

receiver input.

In [7], a classification method was proposed for multichannel systems. The

proposed recognizer consists of two main parts. The first one is the outer loop that

detects the individual signal components; the second part is a single tone classifier

that identifies each of the signal components as ASK2, BPSK, QPSK and CW.

BFSK signals are considered as two correlated ASK2 signals. In a single tone

classifier an amplitude histogram is used to distinguish ASK from BPSK, QPSK,

CW and differential phase histogram is used for discrimination of CW, PSK2 and

PSK4 signals. It is claimed that success rate for discrimination of CW, PSK2 and

PSK4 signals is greater than 98% at 10dB SNR; whereas success rate of the

recognizer for ASK2 signals is 87% at 10dB SNR. The basic problem regarding

classification of ASK2 signals is that the phase modulation introduced by the noise.

Carrier frequency offset (CFO) is removed by using instantaneous frequency

histogram and by performing differential phase computation.

In [8], a classification method for the MFSK signals was proposed based on the

higher-order correlations (HOC). Average log-likelihood function of the intercepted

signal in terms of HOC domain is given for both of the channelized and

nonchannelized classifier structures. It is indicated that the optimal rule in the

MFSK classification is the usage of bank of matched filters, each of which is tuned

to one of a prescribed set of frequency locations. However, in the case of the

received signal arriving at a frequency that is not coincident with the center

frequency of any of these filters, mismatch at the outputs of the matched filters

occurs. Hence, in order to avoid mismatch; nonchannelized structures are proposed

10

as a MFSK classifier. In the proposed nonchannelized classifier; signal is divided

into three subbands and for each subband a processor is assigned. The first-order

correlation-based classifier, the second-order correlation-based classifier of the first

kind and the second-order correlation-based classifier of the second kind are

proposed as classifier algorithms. All of the proposed algorithms based on the

comparison of the average log-likelihood function with a threshold to classify

MFSK signals. The signal arrival time is assumed to be known perfectly; hence the

proposed classifiers are all synchronous.


likelihood function (LLF) of the instantaneous phase. Classification of the MPSK

signals is performed by comparing their LLF differences by a threshold.

Quasioptimal rules have been proposed for the following environments:

• Environment where the carrier phase and the symbol timing offset are

assumed to be known.

• Environment where the carrier phase and the symbol timing offset are

assumed to be unknown.

• Environment where the symbol timing offset is assumed to be known

but the carrier phase offset is assumed to be unknown.

In this work, performance comparison of the Statistical-Moment-Based Classifier

(SMBC) [5], Phase-Based Optimal Classifier (PBOC) [11], Phase-Histogram

Classifier (PHC), Square Law Classifier (SLC) and the proposed

quasioptimal/optimal classifiers are made. It is claimed that the proposed

quasioptimal rules and the optimal rules have similar performance curves.

Moreover, SLC requires 1.9 dB more SNR than the qLLR classifier to discriminate

PSK2 and PSK4 and Fourth Law Classifier needs 2.6 dB more SNR than qLLR to

discriminate PSK4 and PSK8. Numerical comparisons and simulations are

performed for rectangular pulse shape.

11


statistical moments of the intercepted signal phase. Exact phase distribution of the

received signal is used instead of Tikhonov function proposed in [5]. The proposed

method is 2 dB superior to the method used in [5] for 99% success rate.


hypothesis testing. Test statistics are derived from the probability density function

of MPSK signals and a classifier is proposed to classify CW, BPSK, QPSK, 8PSK.

Discrimination of CW, BPSK, QPSK, 8PSK is performed by calculation of test

statistics for each modulation type and the modulation type with the largest test

statistics is selected. The performance of the proposed classifier is compared with

[2], and it is claimed that the proposed classifier has a higher probability of correct

classification. Carrier frequency, initial phase and symbol rate of the intercepted

signal are assumed to be known.

In [1], classification method for PSK2, PSK4, ASK2, ASK4, FSK2 and FSK4 was

proposed based on the feature-extraction approach. The intercepted signal is divided

into M segments then the key feature extraction and comparison with a threshold is

performed for each segment. Modulation decision is made by comparing the

decision of M segments. The key features used in [1] are as follows:

• The maximum value of the spectral power density of the normalized-

centered instantaneous amplitude of the intercepted signal

• The standard deviation of the absolute value of the centered non-linear

component of the instantaneous phase, evaluated over the non-weak

intervals of a signal segment

• The standard deviation of the centered non-linear component of the

direct (non absolute) instantaneous phase, evaluated over the non-weak

intervals of a signal segment

• The standard deviation of the absolute value of the normalized-centered

instantaneous amplitude of a signal segment

12

• The standard deviation of the absolute value of the normalized-centered

instantaneous frequency, evaluated over the non-weak intervals of a

signal segment

Five algorithms are proposed based on the above key features; two of them use

artificial neural network approach to decide modulation type. The carrier frequency

of the intercepted signal is assumed to be known.

In [26], a classification method for both analog and digital modulations was

proposed based on cyclic spectral feature extraction and neural network modulation

recognition approach. The cyclostationarity property of intercepted signal is used to

extract the key features. AM, USB, LSB, FM, ASK, FSK, CW, BPSK: QPSK and

SQPSK signals are used to analyze the performance of the proposed classifier. The

only result about the performance of the proposed classifier is given in Table-1 in

[26].

In [12], a classification method for MPSK signals was proposed based on the

decision theoretic approach. Tikhonov function is used to approximate the phase

probability density function (ppdf) of MPSK signals and classification is performed

by hypothesis-testing derived from the pddf’s of the intercepted signal. A

suboptimal classifier to discriminate CW, BPSK, QPSK and 8PSK is introduced

and the performance of the proposed classifier is tested for BPSK and QPSK

signals. It is claimed that the proposed classifier is 2.5 dB superior to the method

used in [5] for 90% success rate.

In [25] a classification method for both analog and digital modulations was

proposed based on the feature extraction approach. Following features are defined

which are not included in [1]:

• The standard deviation of the normalized-centered instantaneous

amplitude in the non weak segment of a signal

• The kurtosis of the normalized instantaneous amplitude

13

• The kurtosis of the normalized instantaneous frequency

A decision tree (DTC) and an artificial neural network (ANNC) architectures are

proposed as a modulation classifier and performance comparison of these classifiers

are given. It is claimed that the success rate of DTC is greater than 94%; while the

success rate of ANNC is greater than 96% at a SNR of 15 dB.

In [13], a classification method for QAM, PSK and FSK signals was proposed

based on the Haar Wavelet Transform (HWT). It is showed that different transients

in frequency, phase or amplitude are observable for different digital modulation

signals and wavelet transform can be used to extract transient information.

Regarding the classification method following observations are given:

• The HWT of QAM and FSK is a multi-step function

• The HWT of a PSK and amplitude normalized QAM is a constant

• The HWT of amplitude normalized PSK and FSK signals are the same

as the ones that are not normalized.

Success rate of the proposed classifier is about 97% at a SNR larger than 5 dB for

50 observation symbols. The assumptions about the intercepted signal are the prior

knowledge of CNR and noise power. No information is given to distinguish M-ary

signals.

In [14], a classification method for QAM and PSK signals was proposed based on

the maximum-likelihood approach for both the coherent and noncoherent cases. In

coherent case all the signal parameters and in noncoherent case, except for the

carrier phase, other parameters are assumed to be known a priori. The only thing

mentioned about the performance of the classifier is that coherent case is 3 dB

superior to the noncoherent case.

In [15] a classification method for digital modulated signals was proposed based on

the constellation shape of the intercepted signal. Firstly, a fuzzy c-means clustering

algorithm is used as the constellation recovery method then recovered constellations

14

are modeled by binomial nonhomogenous spatial random fields. Lastly, ML

approach is used as a constellation shape classification. The symbol timing offset is

assumed to be small and hence the effect of timing errors is neglected. It is claimed

that the success rate of the classifier to discriminate 8PSK and 8QAM is about 90%

at a SNR of 0dB when no carrier and phase offset exist. In the case of random

carrier phase lock error, which is assumed to be constant for the duration of a

symbol, the 45 degree phase offset performance of the classifier to discriminate

16PSK and 16QAM is about 90% at a SNR of 3dB. There is no information about

the performance degradation of the classifier when the carrier phase lock error is

high.

In [22], a classification method to discriminate PSK2, PSK4, PSK8, PSK16, FSK2,

FSK4, QAM8 and OOK signals was proposed based on the pattern recognition

approach. The Margenau-Hill (time-frequency) distribution (MHD) is used to

extract the phase information and higher order autoregressive model is used to

extract frequency and amplitude information of the intercepted signal. It is stated

that, no a priori knowledge about the signal parameters are needed. It is claimed that

the success rate is about 94% at a SNR of 10dB and the success rate of the proposed

classifier is about 97% for real world short-wave signals.

In [16], a classification method for digital quadrature modulations was proposed

based on the maximum-likelihood method. All the signal parameters and the noise

parameters are assumed to be known a priori. An error probability is obtained for

any type of quadrature modulations and it is claimed that as the number of available

data symbols goes to infinity, failure rate of the proposed classifier goes to zero.

In [17], a classification method for baseband digitally modulated signals was

proposed based on the feature extraction and fuzzy classification method. Carrier

frequency is the only parameter assumed to be known a priori. Basic features used

in the classification problem are as follows:

• Kurtosis of the envelope of the signal.

15

The envelope distribution of constant envelope signals is a Rician

distribution and for SNR>10dB it can be approximated as Gaussian

distribution; whereas varying amplitude signals envelope distribution is a

mixture of Rayleigh and Rician distribution. Therefore kurtosis ([17] eqn 6.)

of constant envelope and varying envelope signals will be different.

• Variance of the PSD derivative.

The probability density function is estimated by Tikhonov function and the

phase variance of the intercepted signal is used to discriminate ASK, FSK

from PSK, QAM.

• Mean of the absolute value signal frequency.

Burg method is used to estimate signal PSD. FSK, MSK like signals are

discriminated from ASK, QAM, PSK like signals by calculating this feature.

In this work, Mamdani fuzzy classifier is used as a classification method and ASK,

4DPSK, 16QAM, FSK are used to evaluate the performance of the proposed

classifier. It is claimed that the proposed classifier works properly for SNR>5dB.

In [18], a classification method for digital modulation signals was proposed based

on the feature extraction and artificial neural network approach. Both the statistical

features and the features proposed in [1] are used as a feature set and a multilayer

perception (MLP) is used as a classification method. ASK2, ASK4, BPSK, QPSK,

FSK2, FSK4, QAM16, V29 (16 point QAM signal constellation with 8 phases, 4

amplitudes, 9600bps), V32 (32 point signal constellation TCM with 9600bps) and

QAM64 are used to evaluate the performance of the proposed classifier and it is

claimed that the success rate is about 98% at a SNR of 0dB. The center frequency is

assumed to be known a priori.

In [20], a classification method based on elementary fourth-order cumulants (eqn 3)

was proposed. It is assumed that the carrier phase offset, symbol timing offset and

the pulse shape are known. The performance of the proposed classifier is analyzed

16

in the case of small phase offset, small frequency offset, residual channel effect,

small symbol timing offset, self-interference, cochannel interference and impulsive

nongaussian noise. It is claimed that there is a small decrease in the success rate of

the classifier at a SNR of 12 dB in the case of small frequency offset, phase offset,

and impulsive nongaussian noise when compared to ideal Gaussian environment.

In [19], a classification method, based on the relationships between the second and

higher moments of received signal and signal and noise power, was proposed. The

proposed classifier uses mixed moments of different orders of the intercepted signal

as a feature set. The classifier consists of two stages. In the first stage, the signal

power is estimated and most of the modulations (16QAM, 32QAM, 128QAM, V29,

7200bps, 9600bps) are classified. In the second stage, classification of 4ASK,

8QAM, BPSK, QPSK, 8PSK is performed. The performance of the proposed

classifier is compared with an ideal HOC classifier [19] and it is claimed that at

lower SNR ideal high order cumulant (HOC) is better; whereas at high SNR

performances of classifiers are similar. It is also shown that as the number of data

samples increase, probability of correct classification also increases. Carrier

frequency, time of arrival and pulse shape are assumed to be known, and there is

only one sample for each of the data symbol.

In [21], a classification method to discriminate MASK, MPSK, MQAM, MFSK

signals was proposed based on the signal envelope statistics which is a function of

the second order moment, the fourth order moment of the received signal and

average power of the modulated signal. The modulated signal power is estimated

from the singular value decomposition of the signal auto-correlation matrix and

there is no assumption about the signal parameters. 2ASK, 2PSK, 2FSK, 4QAM, 4-

AM signals are used to analyze the classifier performance and it is claimed that the

probability of correct classification is about 85% at a SNR of 2.5dB.

In [23], a classification method to discriminate ASK2, ASK4, PSK2, PSK4, FSK2,

FSK4 was proposed based on the feature extraction and support vector machine

(SVM) classification approach. The following features are used:

17

• Average value of the imaginary part of the multiplication of two

consecutive signal values.

• Kurtosis of the real part of the multiplication of two consecutive signal

values.

• Kurtosis of the imaginary part of the multiplication of two consecutive

signal values.

• Ratio of the second maximum to third maximum of the FFT of the

intercepted signal

It is claimed that the success rate of the proposed SVM classifier is 100% at a SNR

of 5dB when (the ratio of symbol rate to sampling rate) p=0.05. Moreover the

performance of the proposed SVM classifier is compared with maximum likelihood

(ML) classifier proposed in [16], the qLLR classifier proposed in [2], cumulant

based classifier proposed in [20], the proposed decision tree classifier with fixed

threshold and the proposed decision tree classifier with dynamic threshold. Monte-

Carlo simulations show that:

• The ML Classifier has the best performance among the compared

classifiers

• At low SNR qLLR is the second to the best classifier for small values of

p; but its performance degrades as p increases.

• Dynamic threshold classifier outperforms the fixed one.

• Performance of the cumulant based classifier is independent of p.

• The proposed SVM classifier outperforms fixed and dynamic threshold

classifiers for all modulation types at all SNR values.

• The proposed SVM classifier outperforms cumulant based classifier for

SNR>0dB.

18

In [24] a classification method to discriminate both the analog and digital

modulations types was proposed which is based on the feature extraction approach.

Seven key features are used; six of these features are from [25] and the other one is

the occupied bandwidth which is defined as the ratio of the number of FFT bins

with the 90% of total power spectrum density (PSD) for MC. The following

classifiers are proposed to classify CW, AM, LSB, USB, FM, 2PSK, 4PSK, 2FSK,

and 4FSK.

• Decision Tree Classifier-1 (DTC1) in which Mahalanobis distance is

used to settle the threshold levels.

• Decision Tree Classifier-2 (DTC2) in which the average probability of

correct decision is used to settle the threshold levels.

• The minimum distance classifier in which the minimum normalized

Euclidean distance between the unknown entry and the mean values of

each of the other classes is calculated.

• Neural Network Classifier (NNC) in which Feed Forward Network with

double hidden layers is used.

• The Support Vector Classifier (SVC), which employs SVM (support

vector machine) to find a hyperplane that separate samples from class 1

from those of class 2 in some higher dimension

It is claimed that all the classifiers has a success rate greater than 95% at a SNR of

10 dB. Moreover, the success rate of MDC is 96.2% and the success rate of SVC is

98.4% at a SNR of 5 dB. A hardware implementation of DTC2 is also given and it

is claimed that numerical simulation results and the performance of the prototype

are consistent.

In [27], a classification method to discriminate 2ASK, 4ASK, 2FSK, 4FSK, 2PSK,

4PSK, 16QAM, 64QAM, AWGN, OFDM (DVB-T, 802.11a) signals, under

19

multipath conditions, was proposed. The classification method is based on the

feature extraction and the bootstrap approach. The selected features are:

• The maximum frequency of the spectrum of the signal

• The variance of the filtered Haar wavelet transform [50] of the signal

(V1) and the variance of the filtered twice Haar wavelet transform of the

signal (V2).

• The standard deviation of the absolute value of the real part of the signal,

evaluated over the non-weak intervals of a signal segment.

• Variance of the discrete Fourier transform (DFT) of the autocorrelation

function of the signal.

• 21 1 log ( ( ) / (2 ))DBF d d= + ∆ ∆ and 22 1 log ( ( ) / (2 ))DBF b d= + ∆ ∆

where

o 1

11

( ) ( ) ( )N

i i

i

d f t f t−

+=

∆ = −∑ ,

o 1

11

( ) ( ) ( )N

i i

i

b a t a t−

+=

∆ = −∑ ,

o ( 1)/2

2 1 2 2 1 2 1 2 2 11

(2 ) (max{ ( ), ( ), ( )} min{ ( ), ( ), ( )})N

i i i i i i

i

d a t a t a t a t a t a t−

− + − +=

∆ = −∑

1( ),..., ( )Nf t f t are defined as the samples of the signal and

1( ),..., ( )Na t a t are defined as the amplitude samples of the signal.

It is claimed that DBF2 can be used to discriminate 2FSK from 2PSK, 4PSK,

4FSK, and DVB-T from 802.11a.

Radial basis function (RBF) based artificial neural network is used to classify

signals. It is claimed that the success rate of the classifier is greater than 80% at a

SNR of 0 dB; and 76% at a SNR of 20dB.

20

In the literature, there are many techniques for digital modulation recognition.

Methods, using a decision-theoretic approach, have better performance than the

ones using feature-extraction approach. However, most of the signal parameters are

assumed to be known a priori in the case of a decision-theoretic approach. In a

feature-extraction approach, the performance of the recognizer mainly depends on

the selected features. In the literature, many features are proposed for the

classification of different types of modulations. Among these features, the ones

which are extracted from the envelope histogram, the frequency histogram, the

higher order statistics, the wavelet transform, the autocorrelation function, the

cyclic autocorrelation function, and the discrete Fourier transform do not require the

knowledge of the most of the signal parameters, such as symbol rate, carrier

frequency offset, pulse shape, time of arrival, initial phase. Moreover, the SNR

value is one of the most important parameters affecting the reliability of the selected

features.

21

CHAPTER 3

BASIC FEATURE EXTRACTION BLOCKS

In this chapter, SNR estimation, bandwidth and spectral estimation, cyclostationary

analysis, envelope extraction, spectral analysis of moments, instantaneous

frequency extraction blocks, which are the selected blocks to be used to extract

features for classification of digital modulation signals, are introduced and

investigated. Moreover, performances of these selected blocks are investigated for

different modulation types.

3.1 SIGNAL TO NOISE RATIO (SNR) ESTIMATION BLOCK

SNR is one of the most critical parameters affecting the performance of a

communication system. In the literature, most of the proposed methods, which can

discriminate digital modulation signals, use a prior knowledge or estimation for

SNR to get better classification performance. In [28], a blind SNR estimation

method for digital modulation IF signals in AWGN channel was proposed based on

the eigenvector decomposition and subspace approach. We employ this method in

our work because it does not require any knowledge about the intercepted signal

and accurate SNR estimation of different digital modulation types can be performed

in a simple way.

Let the signal model be ( ) ( ) ( )r k s k n k= + , where ( )r k is a down-converted

baseband low-pass equivalent (complex) signal and ( )n k is AWGN with zero-mean

and variance 2nσ .

22

Assuming that the information signal samples )(ks and noise samples )(kn are

uncorrelated then the autocorrelation matrix of the received signal vector

TNrrr )](),...,1([= can be expressed as

{ } { } { }H H H

r s nR E rr E ss E nn R R= = + = + (3.1)

where {.}E denotes the expectation operator, H denotes Hermitian transposition

and and N is the number of samples. Since rR is conjugate and symmetric

[ ] * *{ ( ) ( )} ( ) ( )r r rijR E r i r j r i j r j i= = − = − (3.2)

where Mkkrr ,...,2,1),( = are the samples of the autocorrelation function of the

received signal.

1

* *

1

1( ) { ( ) ( 1)} ( ) ( 1)

N k

r

n

r k E r n r n k r n r n kN

− +

=

= + − = + −∑ (3.3)

and the autocorrelation matrix of noise

2n n nR I σ= (3.4)

where nI is an MxM identity matrix. By using the eigenvalue decomposition of

matrices, the autocorrelation matrices of the received signal and the information

signal are given by

H

sR UQU= , (3.5)

H

rR UPU= (3.6)

where U is an MxM orthogonal matrix containing eigenvectors and

),...,,( 21 MbbbdiagP = , )0,...,0,,...,,( 21 pdiagQ λλλ= are MxM diagonal matrices

containing corresponding eigenvalues. Then

2H H

r n nR UPU UQU Iσ= = + (3.7)

23

and eigenvalues

2

2

1,...,

1,...,i n

i

n

i pb

i p M

λ σσ

=+=

= + (3.8)

where )...( 21 pi λλλλ ≥≥≥ , (p<M) is the power of the signal along the thi

eigenvector.

From (3.7) and (3.8), it can be seen that p eigenvalues spans the signal subspace and

M-p eigenvalues spans the noise subspace. Hence, if the signal subspace dimension

is found then the total power of the received signal can be grouped into power of the

desired signal plus noise and power of the noise [28].

In [28], a minimum description length (MDL) criteria was proposed to determine

the signal subspace dimension. The proposed MDL function is

1/( )

1

1

1( ) ln (2 ) ln( )

1 2

MM k

i

i k

M

i

i k

b

MDL k k M k N

bM k

−

= +

= +

= − + − −

∏

∑ (3.9)

Then, the signal subspace dimension is given by

arg min( ( ))k

p MDL k= (3.10)

Since the signal subspace dimension was determined, then the noise power is

2

1

1 M

n i

i p

bM p

σ= +

=− ∑ (3.11)

and the estimated signal power is

' '

'

, if 0

0, if 0

s s

s

s

P PP

P

≥=

< (3.12)

24

where ( )' 2

1

1 p

s i n

i

P bM

σ=

= −∑ .

Hence, the estimated SNR becomes

2s

n

Psnr

σ= (3.13)

The following bias and standard deviation terms were defined to analyze the

proposed method [28]:

1010log ( / )bias mean snr= (3.14)

1/2

210

1

1(10log ( / ))

tN

it

STD snr meanN =

=

∑ (3.15)

Where, tN is the number of trials, snr is the estimated SNR value and

∑=

=tN

it

snrN

mean1

1.

We tested the performance of the proposed method, which is given in [28], for

ASK2, ASK4, ASK8, PSK4, PSK8, QAM4, QAM8, CW, MSK, FSK2, FSK4 and

AWGN signals. In the case of AWGN, indicated SNR value is taken as the noise

power. For 2500 symbols with autocorrelation matrix dimension M=100, estimated

bias and STD is given in Figure 3-1 and Figure 3-2.

From Figure 3-1 and Figure 3-2, it can be seen that as the SNR increases, the

absolute value of the estimated bias and the STD also increase. This is due to the

fact that noise power gets smaller as SNR increases; hence, a small estimation error

of noise power, which may be neglected in the low SNR, causes a large SNR

estimation error for a high SNR [28]. The type of modulation has little influence on

the STD. However, the estimation error is quite large for the linear modulation

types (ASK2, ASK4, ASK8, PSK4, PSK8, QAM8) compared to the nonlinear

25

modulations, continuous wave and noise. Moreover, the bias quickly increases as

SNR increases.

In [28], the performance of the proposed method was investigated for different

number of received symbols and different dimensions of the autocorrelation matrix.

It is claimed that the estimated bias and the STD improve little as the dimension of

the autocorrelation matrix, M, increases. Since the performance of the proposed

method is quite good for M=100, we preferred to use M=100 in our analysis to

reduce the simulation time.

The proposed method is considerably accurate for SNR>-10dB and for SNR<20dB

which may provide a satisfactory feedback to a recognition system.

-25 -20 -15 -10 -5 0 5 10 15 20 25-2.5

-2

-1.5

-1

-0.5

0

0.5

SNR (dB)

Estimation Bias (dB)

ASK2

ASK4

ASK8

PSK4

PSK8

QAM4

QAM8

MSK

FSK4

FSK8

CW

NOISE

Figure 3-1 Estimation Bias vs. SNR

26

-25 -20 -15 -10 -5 0 5 10 15 20 250

0.5

1

1.5

SNR (dB)

Estimation STD (dB)

ASK2

ASK4

ASK8

PSK4

PSK8

QAM4

QAM8

MSK

FSK4

FSK8

CW

NOISE

Figure 3-2 Estimation STD vs. SNR.

3.2 BANDWIDTH AND SPECTRAL SHAPE

Bandwidth and spectral shape of an intercepted signal are one of the key features in

the classification of signals. Specifically, spectral symmetry is one of the features

proposed in [1] to discriminate analog modulated signals. In [24], the occupied

bandwidth, which is defined as the ratio of the number of FFT bins with the 90% of

total power spectrum density (PSD), is used as a feature to discriminate analog and

digital modulated signals. Bandwidth estimation is also important in the pre-

processing stage of a classification system. We investigate the bandwidth and

spectral shape characteristics of linearly modulated and CPM signals separately.

27

3.2.1 LINEARLY MODULATED SIGNALS

A linearly modulated signal can be expressed as

( )

{ }2

( ) ( ) cos 2

Re ( ) c

k c

k

j f t

s t I g t kT f t

v t eπ

π

= −

=

∑ (3.16)

where T is the symbol rate, g is the pulse shape, cf is the carrier frequency, { }kI is

a wide-sense stationary symbol sequence and the low-pass equivalent of the signal

is

( ) ( )k

k

v t I g t kT= −∑ (3.17)

Then, the autocorrelation function of ( )s t is [29]

{ }2( , ) Re ( , ) cj f

ss vvR t t R t t eπ ττ τ+ = + (3.18)

where ( , )vvR t tτ+ is the autocorrelation function of the equivalent low-pass signal

( )v t . Hence, the power spectral density is

{ }1( ) ( ) ( )

2ss vv c vv cS f S f f S f f= − + − − (3.19)

In [29], the power spectral density of ( )v t is given as

21

( ) ( ) ( )vv iiS f G f S fT

= (3.20)

where ( )iiS f is the power spectral density of the information sequence { }kI and it

is defined as

28

2( ) ( ) j fmT

ii ii

m

S f R m e π=∑ (3.21)

where

*( ) { }ii n n mR m E I I += (3.22)

From (3.20), it can be seen that the spectral shape and the bandwidth of the linearly

modulated digital signal depend on the symbol rate, the pulse shape and the

information sequence; whereas they are independent of the type of the linear

modulation.

3.2.2 CPM SIGNALS

A simple representation of a constant envelope continuous-phase modulated (CPM)

signal can be expressed as

( ) cos(2 ( , ))cs t f t t Iπ φ= + (3.23)

where cf is the carrier frequency and ( , )t Iφ is

( , ) 2 ( )k

k

t I h I g kT dτ

φ π τ τ= −∑ ∫ (3.24)

where { }kI is a wide-sense stationary symbol sequence, ( )g t is the pulse shape

with '( ) ( )q t g t= and ( ) 0; 0q t t= < ; 1

( ) ;2

q t t T= > .

And the low-pass equivalent of the signal is

( , )( ) j t Iv t e φ= (3.25)

In [29], power density spectrum of the equivalent low-pass signal of full response

CPM with rectangular pulse shape is given as

29

2

21 1 1

1 2( ) ( ) ( ) ( ) ( )

M M M

ss n c nm n m

n n m

S f T A f f B f A f A fM M= = =

= − +

∑ ∑∑ (3.26)

where

( )

( )

1sin 2 1

2( )

12 1

2

n

fT n M h

A f

fT n M h

π

π

− − − = − − −

(3.27)

( )

2

cos 2 cos( )( )

1 2 cos(2 )nm nm

nm

fTB f

fT

π α ψ αψ ψ π

− −=

+ − (3.28)

( 1 )nm h m n Mα π= + − − (3.29)

sin( )

( )sin( )

M hjh

M h

πψ ψ

π= = (3.30)

For the minimum-shift keying signal, where 1

2h = and { 1,1}kI ∈ − with a

rectangular pulse shape, the phase process of the carrier ( , )t Iφ in the symbol

interval is

1

( , ) , nT t (n+1)T2 2

, nT t (n+1)T2

n

k n

k

n n

t nTt I I I

T

t nTI

T

π πφ

πθ

−

=−∞

− = + ≤ ≤

− = + ≤ ≤

∑ (3.31)

Then, the intercepted MSK signal is

1

( ) cos 2 , nT ( 1)4 2c n n ns t f I t nI t n TT

ππ θ

= + − + ≤ ≤ +

(3.32)

In [29], the power spectral density of the equivalent low-pass signal of MSK is

given as

30

22

2 2 2

cos(2 ( ) )16( )

1 16( )c

ss

c

f f TA TS f

f f T

ππ

−=

− − (3.33)

From (3.26) and (3.33) it can be seen that the bandwidth of the CPM signal depends

mainly on the symbol rate, information sequence alphabet size (M) and modulation

index ( )h . For MSK, since the modulation index and alphabet size is fixed, the

bandwidth depends only on the symbol rate.

Bandwidth information can be used to discriminate analog and digital modulated

signals. However, bandwidth alone is not a feature to classify digital modulation

types.

31

3.3 CYCLOSTATIONARY ANALYSIS

The term “cyclostationary” is mainly used for a special class of non-stationary

random signals which exhibit periodicity in their statistics. Cyclostationary spectral

analysis gives more information about the signal, compared to the conventional

spectral analysis. Therefore, we investigate the cyclostationary characteristics of the

digitally modulated signals and use the cyclic autocorrelation function to estimate

the symbol rate of the intercepted signal. Also, we tested the performance of the

symbol rate estimator for different modulation types.

3.3.1 LINEARLY MODULATED SIGNALS

The autocorrelation function of a low pass equivalent of a linearly modulated signal

can be expressed as

{ }

{ }

*

* *

( , ) ( ) ( )

( ) ( )

vv

n m

n m

R t t E v t v t

E I I g t nT g t mT

τ τ

τ∞ ∞

=−∞ =−∞

+ = +

= − + −∑ ∑ (3.34)

where ( )g t is the pulse shape and { }kI is a wide-sense stationary sequence with an

autocorrelation function ( )iiR m given in (3.22). Then, (3.34) can be expressed as

*

*

( , ) ( ) ( ) ( )

( ) ( ) ( )

vv ii

n m

ii

m n

R t t R m n g t nT g t mT

R m g t nT g t mT

τ τ

τ

∞ ∞

=−∞ =−∞

∞ ∞

=−∞ =−∞

+ = − − + −

= − + −

∑ ∑

∑ ∑ (3.35)

and

32

*

*

( , ) ( ) ( ) ( )

( ) ( ( 1) ) ( ( 1) )

vv ii

m n

ii

m n

R t T t T R m g t nT T g t mT T

R m g t n T g t m T

τ τ

τ

∞ ∞

=−∞ =−∞

∞ ∞

=−∞ =−∞

+ + + = − + + − +

= − − + − −

∑ ∑

∑ ∑ (3.36)

Since summation in (3.36) is from −∞ to ∞ ;

*

*

( , ) ( ) ( ( 1) ) ( ( 1) )

( ) ( ) ( )

( , )

vv ii

m n

ii

m n

vv

R t T t T R m g t n T g t m T

R m g t nT g t mT

R t t

τ τ

τ

τ

∞ ∞

=−∞ =−∞

∞ ∞

=−∞ =−∞

+ + + = − − + − −

= − + −

= +

∑ ∑

∑ ∑ (3.37)

Also, in a similar way { }( )E v t can be obtained as follows

{ } { }

{ }

{ }

( ) ( )

( ( 1) )

( )

k

k

k

k

E v t T E I g t kT T

E I g t k T

E v t

∞

=−∞

∞

=−∞

+ = − +

= − −

=

∑

∑ (3.38)

From (3.37) and (3.38), it is seen that both the autocorrelation function and the

mean of ( )v t is periodic with a period T; that is low pass equivalent of a linearly

modulated signal ( ( )v t ) is a cyclostationary process.

3.3.2 CPM SIGNALS

From (3.24) and (3.25), the autocorrelation function of a low pass equivalent of a

CPM signal can be expressed as [29]

[ ]

[ ]

2 ( ) ( )

2 ( ) ( )

( , )

k

k

k

j h I q t kT q t kT

vv

j hI q t kT q t kT

k

R t t E e

E e

π τ

π τ

τ

∞

=−∞

+ − − −

∞+ − − −

=−∞

∑ + =

=

∏

(3.39)

33

Since { }kI is a statistically independent sequence with equally likely symbols,

1

2 [ ( ) ( )]

( 1),

1( ; )

Mj hn q t kT q t kT

vv

n M oddk

R t t eM

π ττ∞ −

+ − − −

=− −=−∞

+ =

∑∏ (3.40)

and

[ ]

[ ]

12 ( ) ( )

( 1),

12 ( ( 1) ) ( ( 1) )

( 1),

1( , )

1

Mj hn q t kT T q t kT T

vv

n M oddk

Mj hn q t k T q t k T

n M oddk

R t T t T eM

eM

π τ

π τ

τ∞ −

+ − + − − +

=− −=−∞

∞ −+ − − − − −

=− −=−∞

+ + + =

=

∑∏

∑∏ (3.41)

Since, the product in (3.41) is from −∞ to ∞

[ ]

12 ( ) ( )

( 1),

1( , )

( , )

Mj hn q t kT q t kT

vv

n M oddk

vv

R t T t T eM

R t t

π ττ

τ

∞ −+ − − −

=− −=−∞

+ + + =

= +

∑∏ (3.42)

In a similar way { }( )E v t can be obtained as follows

{ }

{ }

2 ( )

2 ( ( 1) )

12 ( ( 1) )

( 1),

( )

( )

k

k

k

k

j h I q t kT T

j hI q t k T

k

Mj hI q t k T

n M oddk

E v t T E e

E e

e

E v t

π

π

π

∞

=−∞

− +

∞− −

=−∞

∞ −− −

=− −=−∞

∑ + =

=

=

=

∏

∑∏

(3.43)

Therefore, similar to linearly modulated signals, we can conclude that low pass

equivalent of a CPM signal is a cyclostationary process.

34

3.3.3 CYCLOSTATIONARITY IN MODULATION

RECOGNITION

Since autocorrelation function of low pass equivalent of digital signal is generally a

periodic function of t then it will have a Fourier series representation

2( / 2, / 2) ( ) j t

vv xR t t R eα πα

α

τ τ τ+ − =∑ (3.44)

where ( )vvRα τ is the cyclic autocorrelation function and can be expressed as

/2

2

/2

1( ) lim ( / 2, / 2)

z

j t

vv vvz

z

R R t t e dtz

α πατ τ τ −

→∞−

= + −∫ (3.45)

Where α is an integer multiple of fundamental frequencies such as carrier

frequency, symbol rate, hop rate and their sums and differences. If there is only one

period, say T, and then (3.45) can be written as [30]

/2

2

/2

1( ) ( / 2, / 2)

T

j t

vv vv

T

R R t t e dtT

α πατ τ τ −

−

= + −∫ (3.46)

• If ( ) 0vvRα τ = for all 0α ≠ and ( ) 0vvR τ ≠ then v(t) is said to be purely

stationary.

• If ( ) 0vvRα τ ≠ only for 0integer/Tα = for some period 0T then v(t) is said to

be purely cyclostationary.

• If ( ) 0vvRα τ ≠ for values of α that are not all integer multiples of some

fundamental frequency 01/T then v(t) is said to exhibit cyclostationarity.

Also, cyclic spectral density function is defined in [30] as

2( ) ( ) j f

vv vvS f R e dα α π ττ τ∞

−

−∞

= ∫ (3.47)

35

In the case of 0α = (3.47) turns out to be conventional power spectral density

function; but for 0α ≠ , ( )vvS fα can be considered as the density of correlation

between spectral components at the frequencies 2

fα

+ and 2

fα

− .

Cyclic spectral function for analog modulated signals is given in [31] and for digital

modulated signals is given in [32]. It is claimed that the signals with the same

power spectral density but with different modulations may have distinct cyclic

spectrum, stationary noise and interference have no cyclic correlation, phase and

frequency information in the intercepted signal can be extracted by the usage of

spectral correlation function. Hence spectral correlation function can be used for

classification of different modulation types and extraction of different signal

parameters.

Symbol rate is one of the important parameters in digital modulated signals.

Estimated symbol rate gives information about the presence of a digital modulation

and the bandwidth of the intercepted signal. Therefore, the estimated symbol rate

can be used as a key feature in a digital modulation recognition system. Moreover,

cyclostationarity property of the digital modulated signals can be used to estimate

the symbol rate.

In Section 3.4.1 it is shown that low pass equivalent of a linearly modulated signal

( ( )v t ) is cyclostationary with cyclic frequencies , intk

k egerT

α =

, then from

(3.46) the cyclic autocorrelation function corresponding to cyclic frequency α is

/22

/2

/2* 2

/2

/2* 2

/2

1( ) ( / 2, / 2)

1 ( ) ( / 2 ) ( / 2 )

1 ( ) ( / 2 ) ( / 2 )

T

j t

vv vv

T

T

j t

ii

m nT

T

j t

ii

m nT

R R t t e dtT

R m g t nT g t mT e dtT

R m g t nT g t mT e dtT

α πα

πα

πα

τ τ τ

τ τ

τ τ

−

−

∞ ∞−

=−∞ =−∞−

∞ ∞−

=−∞ =−∞−

= + −

= + − − −

= + − − −

∫

∑ ∑∫

∑ ∑∫

(3.48)

36

Assuming { }kI is white with zero mean, that is ( ) 0; 0iiR m m= ≠ , then (3.48) turns

out to be

/2* 2 ( )

/2

* 2

1( ) (0) ( / 2) ( / 2)

1 (0) ( / 2) ( / 2)

T nT

j t nT

vv ii

n T nT

j t

ii

R R g t g t e dtT

R g t g t e dtT

α πα

πα

τ τ τ

τ τ

−∞− +

=−∞ − −

∞−

−∞

= + −

= + −

∑ ∫

∫ (3.49)

From (3.18), (3.48) and (3.49), the cyclic autocorrelation function of the intercepted

signal ( )s t can be obtained as

( )

( )

( )

/22

/2

/22

/2

2* 2

/222

/2

1( ) ( / 2, / 2)

1 ( ( / 2, / 2)

2

( / 2, / 2) )

1 ( / 2, / 2)

2

c

c

vv

c

T

j t

ss ss

T

T

j f

vv

T

j f j t

T

j fj t

vv

T

R R t t e dtT

R t t eT

R t t e e dt

R t t e dt eT

α πα

π τ

π τ πα

π τπα

τ τ τ

τ τ

τ τ

τ τ

−

−

−

− −

−

−

= + −

= + −

+ + −

= + −

∫

∫

∫

( )

( )

{ }

/22* 2

/2

*2 2

2

1 ( / 2, / 2)

2

1 1 ( ) ( )

2 2

Re ( )

c

vv

c c

c

T

j fj t

T

j f j f

vv vv

j f

vv

R t t e dt eT

R e R e

R e

π τπα

π τ π τα α

π τα

τ τ

τ τ

τ

−−

−

−

+ + −

= +

=

∫

(3.50)

Hence, the intercepted signal and the low pass equivalent signal have the same

cyclic frequencies.

Cyclostationary analysis involves too much calculation; however, in [33] a fast and

simple algorithm was proposed to estimate the symbol rate of linear digital

modulated signals. In that work, both the rectangular pulse shape and pulse shape

which is obtained from the response of the linear time invariant (LTI) low pass filter

excited by the square wave are considered and it is found out that ( )ssRα τ is

37

maximum when 1

2 2

Tτ

α= ± = ± . Therefore in [33], the following method was

proposed to estimate the symbol rate

[ ]1 2

0,

1 1arg max

2ssRT

α

α α αα

α∈

= =

(3.51)

We tested the performance of the proposed method, which is given in [33], for

ASK2, ASK4, ASK8, PSK4, PSK8, QAM8, CW, and AWGN signals. In the case

of AWGN, indicated SNR value is taken as the noise power. For 2500 symbols, the

estimated absolute average error normalized to symbol rate b

b

mean Fe

F

−= is given

in Figure 3-3 where �

1

1 tN

b

nt

mean FN =

= ∑ , bF is the symbol rate, �bF is the estimated

symbol rate and tN is the number of trials.

During trials if �bF is found to be less than 100 Hz, then � 0bF = is taken.

From Figure 3-3, it can be seen that the proposed symbol rate estimation method

gives satisfactory results even at low SNR values. Moreover, the proposed method

can be used to discriminate linearly modulated signals from continuous wave

signals and noise.

The proposed method can also be used for CPFSK signals. Also, we tested the

performance of the proposed method applied to the instantaneous frequency of the

CPFSK signal and the results are given in Figure 3-4.

38

-15 -10 -5 0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Absolute Estimation Average Error Nor. Fb

Absolute Estimation Average Error Nor. Fb vs. SNR

ASK2

ASK4

ASK8

QAM8

PSK4

PSK8

CW

NOISE

Figure 3-3 Absolute Estimation Average Error Normalized to Symbol Rate (Fb) for Linear Modulation Signals vs. SNR

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Absolute Estimation Average Error Nor. Fb vs. SNR

SNR (dB)

Absolute Estimation Average Error Nor. Fb

FSK2

FSK4

FSK8

Figure 3-4: Absolute Estimation Average Error Normalized to Symbol Rate (Fb) for CPFSK (h=0.5; Fb=2kHz) vs. SNR

39

3.4 ENVELOPE FEATURES

The envelope of low pass equivalent of MASK, MQAM signals are time varying

and cyclostationary; noise envelope is also time varying but not periodic. If MPSK

signals are bandlimited, then amplitude variations at the transitions between

successive symbols can be observable. CW, CPM signals have constant envelope.

Therefore envelope information of the intercepted signal is one of the key features

that can be used to discriminate digitally modulated signals.

In the literature, there are different methods that use envelope information to extract

features for classification of digitally modulated signals. However, most of these

methods require some of the signal parameters such as carrier frequency, pulse

shape, time of arrival, initial phase, symbol rate, signal to noise ratio, to be known

or to be extracted.

In [14], an ML classification method was proposed and the matched filter output is

used in the analysis. In [1] instantaneous amplitude information is used to extract a

feature; but the carrier frequency offset is assumed to be known. In [21] a blind

classification method was proposed based on the envelope function ( )J of the

signal

( )24 22

4 22 2

[ ] 2 [ ]2( )

4 4s s

v n v nm mJ

P P

< > − < >−= = (3.52)

where { }1

[ ]N

nv n

= is the sample sequence of the equivalent low-pass signal envelope.

Then 4[ ]v n< > can be found by time averaging the samples of the equivalent low-

pass of the intercepted signal:

40

44

1

4 4

1

/( / ) /4 4

1 1

/( / )4 4

_1

4 4_

1[ ] [ ]

1 ( )

/ ( )

/

[ ]

s s

s

N

n

N

k s

n k

N T T T T

s sk s

k n

N T T

sk g av

k

g av

v n v nN

I g nT kTN

T T TI g nT kT

N T

T TI P

N

I k P

=

=

= =

=

< >=

= −

= −

=

=< >

∑

∑ ∑

∑ ∑

∑

(3.53)

where g(t) is a pulse shape with g(t)=0 for t<0 and for t>T, T is the symbol period

and sT is the sampling period. Also, 2[ ]v n< > can be found as

22

1

2 2

1

/( / ) /2 2

1 1

/( / )2 2

_1

2 2_

1[ ] [ ]

1 ( )

/ ( )

/

[ ]

s s

s

N

n

N

k s

n k

N T T T T

s sk s

k n

N T T

sk g av

k

g av

v n v nN

I g nT kTN

T T TI g nT kT

N T

T TI P

N

I k P

=

=

= =

=

< >=

= −

= −

=

=< >

∑

∑ ∑

∑ ∑

∑

(3.54)

Then, the envelope function J is obtained as follows:

( )

( )

( )

24 2

2

24 4 2 4_ _

22 2_

4 4_

22 2_

2( )

4

[ ] 2 [ ]

4 [ ]

[ ] 1

24 [ ]

s

g av g av

g av

g av

g av

m mJ

P

I k P I k P

I k P

I k P

I k P

−=

< > − < >=

< >

< >= −

< >

(3.55)

41

Since the ratio ( )

4_

22_

g av

g av

P

P is not equal to 1 for every pulse shaping, the envelope

function J depends on the pulse shape of the intercepted signal. Hence by using J ,

a fixed threshold can not be defined to classify the signals.

Histogram based methods require no prior knowledge about the signal parameters

and are able to classify MASK signals at high SNR. However, at low SNR

histogram based methods fail to discriminate constant envelope and varying

envelope signals. Therefore, extra features are needed to classify ASK signals.

42

3.5 SPECTRAL ANALYSIS OF MOMENTS

Spectral analysis of moments of the intercepted signal gives important information

about the modulation type. In [5], [10], [19] and [20] higher order moments of the

signal are used to classify digitally modulated signals. In [3], the signal squared and

the signal quadrupled are used to classify BPSK and QPSK signals. By increasing

the order of the signal, modulation can be removed from the signal and this

information can be used to classify signals. However, as the order increases the

corrupted noise power increases also.

Let define the following Fourier Transforms;

[ ] { [ ]}vF k DFT v n= (3.56)

2

2[ ] { [ ]}v

F k DFT v n= (3.57)

4

4[ ] { [ ]}v

F k DFT v n= (3.58)

8

8[ ] { [ ]}v

F k DFT v n= (3.59)

where { }1

[ ]N

nv n

= is the sample sequence of the equivalent low-pass signal. Then the

following features, obtained from higher orders of the low-pass equivalent signal

v[n], can be defined to be used in the classification process;

• 0k =0 if the difference between the successive DFT bins satisfying

{ }2

2

[ ]

max [ ]

v

vk

F k

F kξ> is lower than a predefined value '

dF , otherwise 0k =1.


{ }2

2

2

2

[ ]

max [ ]

v

vk

F k


dF ; otherwise 2k =1.

43


{ }4

4

2

2

[ ]

max [ ]

v

vk

F k




{ }8

8

2

2

[ ]

max [ ]

v

vk

F k



where ξ is a pre-defined threshold, which can be determined according to the

simulation results. Hence, 1ik = means modulation is removed from the carrier and

0ik = is the indication of an existing modulation on the carrier.

3.6 INSTANTANEOUS FREQUENCY FEATURES

The instantaneous frequency information of the intercepted signal is another key

feature in the classification of signals. Instantaneous frequency of CPM signals is

time varying and cyclostationary; for MPSK, frequency variations at the transitions

between successive symbols can be observable. CW and MASK signals have

constant instantaneous frequency. Moreover, the carrier frequency offset (CFO) can

be extracted from the instantaneous frequency information. Therefore, we proposed

a method to extract the instantaneous frequency information and use this method to

estimate the CFO. The performance of the proposed method in estimating the CFO

is also tested.

The instantaneous frequencies of a low pass equivalent of a signal can be found as:

44

( )

( )

( )

' '

2 2

' '

2 2

( ) ( )

Im( ( ))arctan

Re( ( ))

( )1 1( ) ( ) arctan

2 2 ( )

( ) ( ) ( ) ( )1 1

2 ( )( )1

( )

( ) ( ) ( ) ( )1

2 ( ) (

j t

v t

Ii

R

I R R I

RI

R

I R R I

R I

v t A t e

v t

v t

v td df t t

dt dt v t

v t v t v t v t

v tv t

v t

v t v t v t v t

v t v t

φ

φ

φπ π

π

π

=

=

= =

−=

+

−=

+ )

(3.60)

where ( ) Im( ( ))Iv t v t= , ( ) Re( ( ))Rv t v t= . ( ) ( )k

k

A t I g t kT= −∑ , ( ) 2t ftφ π= ∆ for

linearly modulated signals, and ( )A t is a random amplitude variation,

( ) 2 2 ( )k

k

t ft h I g kT dτ

φ π π τ τ= ∆ + −∑ ∫ for CPM signals. f∆ is the CFO, g(t) is a

pulse shape with g(t)=0 for t<0 and for t>T and T is the symbol period.

For discrete time differentiation, the following formula is used [47]:

' [ 3] 9 [ 2] 45 [ 1] 45 [ 1] 9 [ 2] [ 3][ ]

60

x i x i x i x i x i x ix i

− − + − − − + + − − + −= (3.61)

To estimate the CFO, firstly the received signal is down-converted to IF and filtered

such that the signal is in the passband of the filter and then moved to baseband. The

baseband signal has a CFO of f∆ . The baseband signal is sampled at a rate equals

to the IF bandwidth to make noise samples independent. Then, the instantaneous

frequency of the low-pass equivalent signal is extracted by using (3.60) and (3.61).

A simple smoothing filter is used to remove abrupt changes in the obtained

instantaneous frequency waveform and the CFO is estimated by averaging the

filtered instantaneous frequency samples.

For CW signal, the performance of the proposed instantaneous carrier frequency

offset estimation method is given in Figure 3-5. Sampling frequency is fs=50kHz,

second order Chebychev type-II filter with a cut-off 20kHz is used as a smoothing

45

filter, SNR is 20dB and estimated absolute average error is defined as

ce mean F= − . Where �

1

1 tN

c

nt

mean FN =

= ∑ and tN is the number of trials.

Figure 3-5: Absolute Estimation Average Error for CW (fs=50kHz, SNR=20 dB) vs. Carrier Frequency Offset

From Figure 3-5, it is seen that the estimation error increases as the carrier

frequency offset increases due to the parabolic noise power spectral shape at the

output of limiter discriminator [48] and the filter effect of the differentiation given

in (3.61). It can be concluded that, the estimation error is increased dramatically for

10s

c

fF > .

46

3.7 CARRIER FREQUENCY OFFSET ESTIMATION

3.7.1 INTRODUCTION

In digital modulation recognition, in general, we do not have a priori information on

the center frequencies of the signals. Even if the center frequency is known

nominally, by the published standards or by intelligence, carrier frequency offset

estimation should be performed to compensate the local oscillator drifts and

Doppler shifts induced by the relative motion of mobile systems and channel

effects. Channel estimation and synchronization can be made by the usage of set of

known data symbols. However, data-aided or timing-aided techniques reduce the

effective transmission rate and may not be feasible in many applications.

In the literature, there are many frequency offset estimation techniques for digitally

modulated signals; some of them are given in [34], [35],[36], [43], [44], [45], [46].

In this thesis work, since our scope is limited, only MSK frequency offset and

symbol rate estimation are investigated. However, the symbol rate and the coarse

carrier frequency offset of digitally modulated signals can be estimated as described

in section 3.3 and 3.6 respectively.

Minimum-shift keying (MSK) is very attractive for transmission in a mobile radio

environment, because of three major properties [40]:

• 99.5% of the signal energy is contained within a bandwidth of 1.5 times the

data rate.

• The envelope of the signal is constant, therefore power efficient nonlinear

amplifiers can be used.

• MSK may be interpreted as linear modulation. Hence, simple MLSE

receiver structures exist in this case

47

In electronic warfare (EW) applications, automatic modulation recognition systems

should be able to classify the modulation type of the intercepted signal and also be

able to extract some of the signal parameters for burst-type transmission. In the

literature, there are blind carrier frequency offset estimators for MSK signals based

on the frequency difference detector (FDD) technique [38], [39]. The acquisition

range of these types of estimators is limited to the symbol rate and they require

longer acquisition time. In [40], feedforward demodulator structure with decimation

for timing control was proposed. Although, the proposed method requires shorter

acquisition time compared to FDDs, its performance is also limited to the symbol

rate and the acquisition time.

48

3.7.2 THE PROPOSED METHOD

Sunde’s FSK is a special form of CPFSK with modulation index 1h = . In this FSK,

two tones appear whose difference equals to the symbol rate and this information

helps in the demodulation of the signal without external timing information. In [37],

Sunde’s FSK was obtained from MSK signal and it is shown that the carrier

frequency offset and the symbol rate of a MSK signal can be extracted by using a

frequency doubler and two phase-locked loops (PLL) as follows:

( )1 2

1 2

41

c

f ff

f fT

+=

= −

(3.62)

where 1 2,f f are the output frequencies of the PLLs.

The block diagram of the proposed MSK frequency offset and symbol rate

estimator is illustrated in Figure 3-6.

Firstly, the low-pass equivalent of the Sunde’s FSK, y[n], is obtained by squaring

the low-pass equivalent of MSK signal, v[n]. Then, Discrete Fourier Transform

(DFT) is carried out to find the tones in the Sunde’s FSK. From the squared signal,

another sequence x[n], whose length is an integer multiple of the length of the

sequence y[n], is constructed to improve the performance of the phase-lock loops.

The new sequence x[n] and the tones found by DFT are the inputs of the phase-lock

loops. Finally, outputs of these phase-lock loops are used to estimate the carrier

frequency offset, f∆ , and the symbol rate, bf , of the MSK signal.

49

Figure 3-6: Block Diagram of The Frequency and Symbol Rate Estimator

3.7.2.1 DFT BLOCK

Let [ ]Y k be the DFT of the squared signal [ ]y n and it is given as

2 ( 1)( 1)1

1

[ ] [ ]n kN jN

n

Y k y n e

π − − − −

=

=∑ (3.63)

Then, the two frequency components with largest energy are selected as the tones

( 1 2,o of f ) in Sunde’s FSK. By using DFT block false-locking of the PLL’s can be

prevented.

3.7.2.2 ACCUMULATOR BLOCK

In the case of burst-type transmission, the duration of the signal is so short that PLL

may not lock to the signal; hence, it is required to feed the squared signal to PLL

more than once. Thus, larger sequence [ ] [ (mod N)]x n y n= is constructed from the

sequence y[n].

fb

fc

x[n]

fo1, fo2 y[n] v[n]

DFT

Accumulator

PLL-I

PLL-II

D E C I S I O N

( )2

50

3.7.2.3 PLL BLOCK

The phase-lock loop (PLL) is used to improve the performance of the estimator. A

basic PLL block diagram is shown in Figure 3-7 [48]. A phase detector is used to

generate a signal that is proportional to the phase difference between the reference

input and the feedback from the voltage controlled oscillator (VCO). Hence, a

simple multiplier can be used as a phase detector. The output of the phase detector

is low pass filtered and used as a control signal to drive VCO.

Figure 3-7: Basic PLL Block Diagram

The bandpass input signal is

1( ) sin(2 ( ))s cs t A f t tπ φ= + (3.64)

and the VCO output signal is

2( ) sin(2 ( ))r cr t A f t tπ φ= + (3.65)

VCO

s(t) Phase

Detector

Loop

Filter

r(t)

p(t)

51

where

2 ( ) ( )t

vt K v dφ τ τ−∞

= ∫ (3.66)

and vK is the VCO gain constant. Then phase detector output is

'1 2

1 2 1 2

( ) sin(2 ( ))cos(2 ( ))

sin( ( ) ( )) sin(4 ( ) ( ))2 2

m s r c c

m s r m s rc

e t K A A f t t f t t

K A A K A At t f t t t

π φ π φ

φ φ π φ φ

= + +

= − + + + (3.67)

where mK is the gain of the multiplier circuit. Since the high frequency term will be

filtered by the loop filter, the loop filter output is

( ) sin( ( ))* ( )d ep t K t h tφ= (3.68)

where 1 2( ) ( ) ( )e t t tφ φ φ= − is the phase error, 2

m s rd

K A AK = is the equivalent phase

detector constant, for the multiplier-type phase detector, and h(t) is the impulse

response of the loop filter. The describing equation for the baseband PLL model can

be derived by taking the derivative of the phase error:

1

0

( ) ( ) sin( ( )) ( )t

e d v e

d dt t K K h t d

dt dtφ φ φ τ τ τ= − −∫ (3.69)

Then, the baseband PLL block diagram is given in Figure 3-8.

52

Figure 3-8: The baseband equivalent PLL Block Diagram

The discrete low-pass equivalent of the PLL can be obtained by sampling the

equivalent low-pass signal at a rate larger than the nyquist sampling rate and

replacing all the blocks in the PLL with their discrete equivalent models. The block

diagram of the discrete low-pass equivalent PLL including the decision block,

which is used to decide whether the PLL is in the lock stare or not, is given in

Figure 3-9. The equivalence of the system blocks are explained in each subsections

below.

Figure 3-9: PLL Block Diagram

p[n]

e[n]

KO

σd

d[n]

Q[n]

e1[n]

r[n]

x[n] x Im() LPF-L

VCO exp()

LPF

var{}

Decision

Conj()

LPF

f0

Phase

Detector

1 ( )j te φ

Im(.) LPF

(h(t))

p(t)

sin(.)

(.)dt∫

exp(j.)

2 ( )j te φ

( )ej te

φ ( )e tφ

2 ( )d

tdtφ

+

53

Loop Low Pass Filter (LPF-L)

The transfer function of the low-pass filter is [41]

2 2 2

1 1

11

( )s

sH s

s s

τ τ ττ τ

++= = (3.70)

By using Bilinear Transformation [49] 1

1

12

1s

zs F

z

−

−

−= +

, z transform of the filter

can be obtained as follows

1

122

11

1

1

2 221

1

11 21

( )1

21

1 12 2( )

2 1

s

s

s s

s

zF

zH z

zF

z

F F z

H zF z

τττ

τ τττ

−

−

−

−

−

−

−+ + = − +

+ + − =

−

(3.71)

Voltage Controlled Oscillator (VCO)

VCO angular frequency, ω , can be expressed as

0( ) ( )vt K p tω ω= + (3.72)

Where 0w is the free running (angular) frequency of the VCO and vK is the VCO

gain constant [41].

The phase of the VCO is

2

0

0

0 0

( ) ( )

( )

t

t t

v

t d

d K p d

φ ω τ τ

ω τ τ τ

=

= +

∫

∫ ∫ (3.73)

54

By using the trapezoidal rule, the integral ( )0

( )t

vK t K p dτ τ= ∫ can be expressed as

( )[ ] [ 1] [ ] [ 1]2

v

s

KK n K n p n p n

F= − + + − (3.74)

Hence, the phase in (3.73) becomes

2 2 0

0

[ ] ( ) [ ]

2 [ ]

s s

s

n nT nT K n

fn K n

F

φ φ ω

π

= = +

= + (3.75)

In [41], the natural frequency ( nω ) and damping factor (ξ ) of the PLL, for

multiplier type phase detector and the selected loop filter, is given as

1

v mn

K Kω

τ= (3.76)

2

2nω τ

ξ = (3.77)

where vK is the VCO gain and mK is the multiplier type phase detector gain.

PLL Decision Block:

When the PLL is in locked state; then the bandwidth of the loop filter can be

reduced, by reducing vK , to improve the estimator performance. To decide whether

the PLL is in locked state or not following thresholds are defined:

'

1

2 '1

[ ] [ 1]

1[ ]

d d

N

n

T n n

T d nN

σ σ

=

= − −

= ∑ (3.78)

where d[n] is the filtered error signal, [ ]d nσ is the filtered variance of the sequence

{ } 1

1[ ]

n

md m

−

= and 'N is the length of the sequence d[n]. If 1Τ is larger than a

55

predefined value ( 1ν ) then the VCO gain will be reduced to vK

K, and if 2Τ is larger

than a predefined value ( 2ν ) it is decided that PLL is in lock state hence the

estimated frequency and symbol rate are reliable. The threshold values v1 and v2 can

be determined according to simulation results.

If the PLL is in the lock state; that is, if 2 2νΤ > , then the PLL output frequency

(�PLLf ) is

[ ] [ ] [ ]( )

� [ ]_

12

1

_

sPLL

L

PLL PLL

n PLL Lock

Ff n Q n Q n

f f nL PLL Lock

π

=

= + −

=− ∑

(3.79)

where L is the length of the accumulated sequence [ ]x n , and _PLL Lock is the

index of the sequence when 1 1νΤ > is satisfied.

3.7.2.4 ESTIMATOR DECISION BLOCK

Output of each PLL Blocks _1PLLf and _ 2PLLf are used to estimate the carrier

frequency offset and symbol rate by using eqn (3.62).

3.7.3 SIMULATIONS

To determine the performance of the proposed estimator, computer simulations

were carried out by using the MATLAB package.

3.7.4 SELECTION OF BLOCK PARAMETERS

Selection of the block parameters in Figure 3-9 depends on the acquisition time of

symbols, maximum available symbol rate and the maximum available carrier

frequency offset. Therefore, it is assumed that

56

• Available maximum symbol rate is 10 kHz;

• Available maximum carrier frequency offset equals to the symbol rate, that

is 10 kHz;

• n=50 symbols are available.

Then, the selection of block parameters is given as follows:

• Squared MSK signal with symbol rate 10kHz and carrier frequency offset

10kHz can have tones at 2 / 2 25c bf f kHz+ = and 2 / 2 25c bf f kHz− − = − .

Hence IF bandwidth of the filter is selected as 50kHz and baseband signal is

sampled with the IF bandwidth to make noise samples independent, that is,

50sF kHz= .

• Since 50sF kHz= and n=50 symbols are available then the length of the

analytical signal [ ]v n will be 250s

b

Fxn

f= samples. Therefore the length of

the DFT is taken as 512.

• Length of the accumulated sequence [ ]x n is taken as 40 times the

intercepted signal, that is, the length of [ ]x n is taken as 10000.

To determine the natural frequency nw of the PLL block, average absolute

frequency error �2 / 2c b PLLe f f f= + − for different nw values, in the case of

0 100cf f Hz− = are given in Figure 3-10, Figure 3-11. During simulations

01;d sK K F= = are assumed and when PLL is in the lock state, which is determined

by eqn. (3.78), VCO gain vK is reduced by a factor K .

of is the free running frequency of the PLL and �PLLf is the PLL output frequency.

Since 50sF kHz= and the length of the DFT is 512, then the maximum frequency

57

error at the output of DFT block will be 50

98512

kHzHz≅ . Therefore

0 100cf f Hz− = is a reasonable assumption.

From these figures, it can be seen that for 900, 5nw K= = , logarithm of the average

absolute error (e) is low even at low SNR. Damping factor is taken as 0.7ξ = . In

these figures, the plots “b)” are the focused versions of the plots “a)”.

When the natural frequency nw and the damping factor ξ are obtained, then the

filter coefficients can be determined by using eqns. (3.71), (3.76), and (3.77).

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-20

0

20

40

60

a) wn (Hz)

10log10(e)

100 200 300 400 500 600 700 800 900 1000 1100 1200-10

-5

0

5

10

b) wn (Hz

10log10(e)

K=1

K=2

K=3

K=4

K=5

K=6

K=7

K=8

K=9

K=10

Figure 3-10: Logarithm of the Estimation Error at SNR=-5dB

58

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-20

0

20

40

60

a) wn (Hz)

10log10(e)

100 200 300 400 500 600 700 800 900 1000 1100 1200-15

-10

-5

0

5

10

b) wn (Hz))

10log10(e)

K=1

K=2

K=3

K=4

K=5

K=6

K=7

K=8

K=9

K=10

Figure 3-11: Logarithm of the Estimation Error at SNR=0dB

59

3.7.5 SIMULATION RESULTS

Absolute carrier frequency estimation error �c ce f f= − and absolute symbol rate

estimation error at �b be f f= − at / 15b oE N dB= is given in Figure 3-12 and Figure

3-13 respectively.

To determine the effect of the estimated frequency error on the demodulated

signals, MSK signals are coherently demodulated by the estimated carrier

frequency. For each carrier frequency estimation 100 independent MSK signals are

generated with n=50 symbols and demodulated by using the estimated carrier

frequency offset. The average number of bit errors are given in Figure 3-14. In these

figures, the plots “b)” are the focused versions of the plots “a)”.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 104

0

5

10

15x 10

4

a) Carrier Frequency Offset (Hz)

CFO Estimation Error

CFO Estimation Error vs CFO

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 104

0

0.02

0.04

0.06

0.08

0.1

b) Carrier Frequency Offset (Hz)

CFO Estimation Error

CFO Estimation Error vs CFO

Figure 3-12: Absolute Carrier Frequency Estimation Error at / 15b oE N dB=

60

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 104

0

5

10

15x 10

4

Carrier Frequency Offset (Hz)

Symbol Rate Estimation Error

Symbol Rate Estimation Error vs CFO

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 104

0

0.02

0.04

0.06

0.08

0.1

Carrier Frequency Offset (Hz)

Symbol Rate Estimation Error

Symbol Rate Estimation Error vs CFO

Figure 3-13: Absolute Symbol Rate Estimation Error at / 15b oE N dB=

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 104

0

10

20

30

a) Carrier Frequency Offset (Hz)

Average Number of Bit Errors

Average Number of Bit Errors vs CFO

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 104

0

0.1

0.2

0.3

0.4

0.5

b) Carrier Frequency Offset (Hz)

Average Number of Bit Errors

Average Number of Bit Errors vs CFO

Figure 3-14: Average Number of Bit Errors at / 15b oE N dB=

61

3.7.6 CONCLUSION

A blind method is investigated to estimate the carrier frequency offset and symbol

rate of a short duration MSK signal. The design procedure of the proposed method

has also been given.

The estimator is able to estimate the carrier frequency offset and the symbol rate of

the short duration MSK signals. As can be seen from Figure 3-14, probability of bit

error is about 0.002 when 50 bits are transmitted. Carrier frequency estimation limit

of the proposed estimator depends on the IF bandwidth and the sampling frequency

of the receiver, whereas it does not directly related to symbol rate of the signal.

The large error in Figure 3-12 and Figure 3-13 at 5 kHz is caused by the incorrect

output of the DFT block. Performance of the proposed estimator can be improved

by using a block which has a better frequency estimation performance than a DFT

block and by increasing the length of the accumulated sequence.

62

CHAPTER 4

SPECIFIC MODULATION RECOGNITION TOOLS

In Chapter 3, basic feature extraction blocks were introduced and in this chapter

selection of these blocks according to the modulation type is discussed. Also, we

tested the performances of the features, which are extracted from these blocks, for

ASK2, ASK4, ASK8, PSK2, PSK4, PSK8, QAM2, QAM4, QAM8, Noise, FSK2,

FSK4, FSK8 and CW signals.

4.1 OUTPUTS OF THE FEATURE EXTRACTION BLOCKS

SNR Estimation Block

SNR ( snr ) and the average signal power ( )sP estimation are carried out using the

method reviewed in section 3.1.

Symbol Rate Estimation Block

By using cyclostationarity detection block, described in section 3.3, the symbol rate

( )0α of the signal is estimated.

Envelope Feature Extraction Block

The average signal power of the received signal is

• 2 2_s g avP A P= for ASK2, MPSK, MFSK, CW, QAM2, QAM4

63

• 2 2_5s g avP A P= for ASK4

• 2 2_5.5s g avP A P= for QAM8

• 2 2_21s g avP A P= for ASK8

where A is the amplitude of the received signal, ( )g t is the pulse shape and 2_g avP is

the average power of the pulse. Therefore, following four envelope histograms are

constructed:

• In the 1st histogram, the width of each bin is sP and the centers of the bins

are located at integer multiples of sP

• In the 2nd histogram, the width of each bin is / 5sP and the centers of the

bins are located at integer multiples of / 5sP

• In the 3rd histogram, the width of each bin is / 21sP and the centers of the


• In the 4th histogram, the width of each bin is / 50sP and the centers of the


where sP is the estimated average signal power.

Then, the number of peaks 1 2 3 4( ), ( ), ( ), ( )η η η η in each histogram is obtained

according to the following rule:

max

ienv

F

Fς> (4.1)

64

Where iF is the frequency count of the ith peak, maxF is the maximum frequency

count of the histogram and envς is a predefined threshold, which can be determined

according to simulation results.

The location of each peak in the 1st, 2nd, 3rd histogram is normalized to sP ,

/ 5sP , / 21sP respectively and then these normalized peak locations are stored

as a vector for the decision process.

In Figure 4-1, four histograms for ASK8 with SNR=20dB, A=1 and g(t) is a square

pulse shape is shown.

0 2 4 6 8 10 120

5000

10000

15000sqrt(Ps)

a)

0 2 4 6 8 10 120

2000

4000

6000

8000sqrt(Ps/5)

b)

0 2 4 6 8 10 120

1000

2000

3000

4000

5000sqrt(Ps/21)

c)

0 2 4 6 8 10 120

1000

2000

3000sqrt(Ps/50)

d)

Figure 4-1: Envelope Histograms for ASK8

65

In Figure 4-1, red line in each histogram shows the threshold value.

• Plot a) shows the 1st envelope histogram of ASK8 signal. The centers of the

bins are located at 0, 21 4.6≈ , 2 21 9.2≈ and the width of each bin is

21 4.6≈ . In the 1st histogram, only one peak at the bin location 21 4.6≈

is found. Hence, the 1st peak vector is vp1=[1 0 0 0] and 1 1η = .

• Plot b) shows the 2nd envelope histogram of ASK8 signal. The centers of the

bins are located at integer multiples of 21/ 5 2≈ and the width of each bin

is 21/ 5 2≈ . In the 2nd histogram, only one peak at the bin location

21/ 5 2≈ is found. Hence, the 2nd peak vector is vp2=[1 0 0 0] and 2 1η = .

• Plot c) shows the 3rd envelope histogram of ASK8 signal. The centers of the

bins are located at integer multiples of 21/ 21 1≈ and the width of each

bin is 21/ 21 1≈ . In the 3rd histogram, four peaks are found; hence, the 3rd

peak vector is vp3= [1 3 5 7] and 3 4η = .

• Plot d) shows the 4th envelope histogram of ASK8 signal. The centers of the

bins are located at integer multiples of 21/ 50 0.65≈ and the width of each

bin is 21/ 50 0.65≈ . In the 4th histogram, four peaks are found; hence, the

4th peak vector is vp4= [2 5 8 11] and 4 4η = .

Since the intercepted signal is 8 level ASK, the peak locations in the 3rd histogram

coincides with the envelope information of the ASK signal as expected.

Spectral Analysis of Moments Block

By using spectral analysis of moments block 0 2 4 8, , ,k k k k are calculated as

described in section 3.5.

66

Instantaneous Frequency Extraction Block

The instantaneous frequency of the intercepted signal is obtained as described in

section 3.6. Then, the following three frequency histograms are constructed:

• In the 1st histogram the width of each bin is / 4δ and the centers of the bins

are located at integer multiples of / 4δ .

• In the 2nd histogram the width of each bin is / 8δ and the centers of the

bins are located at integer multiples of / 8δ .

• In the 3rd histogram the width of each bin is /100δ and the centers of the

bins are located at integer multiples of /100δ .

δ is the standard deviation of the filtered instantaneous frequency and δ is defined

as

�( )1

2 2

1

1( )

N

i i

n

f n fN

δ=

= −

∑ (4.2)

where �1

1( )

N

i i

n

f f nN =

= ∑ is the mean value of the instantaneous frequency.

Number of peaks 5 6 7, ,n n n in each histogram is determined according to following

rule:

max

ifreq

F

Fς> (4.3)

where iF is the frequency count of the ith peak, maxF is the maximum frequency

count of the histogram and freqς is a predefined threshold, which can be determined

according to simulation results.

67

4.2 MASK RECOGNITION

MASK signal can be expressed as [29]

( ) ( ) cos(2 )k c i

k

s t A I g t kT f tπ φ∞

=−∞

= − +

∑ (4.4)

where ( ){ }1, 3,..., 1kI M∈ −∓ ∓ ∓ is a wide-sense stationary symbol sequence, ( )g t

is a pulse shape with g(t)=0 for t>T and t<0, T is the symbol rate, cf is the carrier

frequency, iφ is the initial phase and A is the amplitude of the received signal.

Then, the sampled equivalent low-pass signal [ ] ( )sv n v nT= with a CFO, f∆ , is

2

[ ] ( )i

s

fj n

F

k

k

v n A I g t kT eπ φ∆∞ +

=−∞

= −

∑ (4.5)

SNR Estimation Block:

The average power of the low pass equivalent signal is

[ ]{ } { }22 2 2_s k g avP E v n A E I P= = (4.6)

For ASK2; 2 2_s g avP A P= , for { }1,1kI ∈ −

For ASK4; 2 2_5s g avP A P= , for { }3, 1,1,3kI ∈ − −

For ASK8; 2 2_21s g avP A P= , for { }7, 5, 3, 1,1,3,5,7kI ∈ − − − −

Then, the average signal power ( sP ) can be estimated as described in section 3.1.

68

Symbol Rate Estimation Block:


( )0α of a MASK signal can be estimated even for SNR>-9dB as seen from Figure

3-3.

Envelope Feature Extraction Block:

The envelope of the low-pass equivalent signal is

,

{ [ ]} [ ]

( )

k s

k

k n k

env v n v n

A I g nT kT

A I g

∞

=−∞

=

= −

=

∑ (4.7)

where , ( )n k sg g nT kT= −

Then, at high SNR, the peak vectors for ASK2, ASK4 and ASK8 are given in Table

4-1. As seen in Table 4-1, the peak vector vp3, obtained from the 3rd histogram, can

be used to classify ASK2, ASK4 and ASK8 signals.

Table 4-1: Peak Vectors for ASK2, ASK4 and ASK8 Signals

Peak Vector

\Modulation ASK2 ASK4 ASK8

vp1 [1 0 0 0] [1 0 0 0] [1 0 0 0]

vp2 [2 0 0 0] [1 3 0 0] [1 3 0 0]

vp3 [5 0 0 0] [2 6 0 0] [1 3 5 7]

69

Number of peaks 1 2 3 4, , ,n n n n , defined in section 4.1, for ASK2, ASK4 and ASK8

as a function of a SNR are given in Figure 4-2, Figure 4-3 and Figure 4-4

respectively. The 1 2 3 4, , ,n n n n values in the figures are the average values obtained

from simulations.

Figure 4-2: Histogram Output for ASK2

70



71

As seen from Figure 4-2, Figure 4-3, Figure 4-4; 4η , which is defined as the

number of peaks obtained from the 4th histogram, can be selected to be used in the

classification of ASK2, ASK4 and ASK8 signals.

Therefore, the values 4η and 3pv are one of the key features that can be used in the

classification process.

Spectral Analysis of Moments Block:

The square of the low-pass equivalent signal is

4 22

4 22 2

[ ] ( ) ( )

( )

is

is

fj n

F

k s l s

k l

fj n

F

k s

k

v n I g nT kT I g nT lT e

I g nT kT e

π φ

π φ

∆∞ ∞ +

=−∞ =−∞

∆∞ +

=−∞

= − −

= −

∑ ∑

∑ (4.8)

and the expected value is

{ }

4 22 2 2

4 2

[ ] ( )

is

is

fj n

F

k s

k

fj n

F

s

E v n E I g nT kT e

Pe

π φ

π φ

∆∞ +

=−∞

∆+

= −

=

∑ (4.9)

Hence from eqn. (4.8) and (4.9), it can be seen that there is a power at the twice of

the carrier frequency offset 2 f∆ [29]. Therefore, the value 2k for ASK signals,

defined in Section 3.5, is expected to be equal to 1 for a suitable threshold ξ

selection.

Spectral analysis of moments block output ( 0k , 2k ) for ASK2, ASK4, and ASK8 as

a function of a SNR is given in Figure 4-5. Values of 0k and 2k in the figures are

the average values obtained from the simulations and threshold values 0.5ξ = and

'd bF F= are taken during simulations.

72

-10 -5 0 5 10 15 20 25-0.5

0

0.5

1

1.5

SNR (dB)

k0 & k2

k0 & k2 for MASK vs. SNR

k0, ASK2

k2, ASK2

k0, ASK4

k2, ASK4

k0, ASK8

k2, ASK8

Figure 4-5: Spectral Analysis of Moments Block Output for MASK vs. SNR

From Figure 4-5, it can be seen that 2 1k = for all SNR values, whereas 0 0k = for

SNR larger than -3dB. Therefore, information obtained from 0k and 2k can be used

as a key feature to classify MASK signals even at low SNR.


The instantaneous frequency extraction block output can be used to estimate the

carrier frequency offset of MASK signals and to discriminate frequency modulated

signals from MASK. For ASK8, ASK4 and ASK2, instantaneous frequency

extraction block outputs ( )5 6,n n , defined in section 4.2, are given in Figure 4-6.

Since there is no frequency modulation in a MASK signal, ( )5 61, 1n n= = is

obtained as expected.

73

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4Frequency Histogram Output for MASK vs. SNR

SNR (dB)

n5,n6

ASK2,n5

ASK4,n5

ASK8,n5

ASK2,n6

ASK4,n6

ASK8,n6

Figure 4-6: Frequency Histogram Output ( )5 6,n n for MASK vs. SNR

74

4.3 MPSK RECOGNITION

MPSK signal can be expressed as [29]

( ) ( ) cos(2 )c k i

k

s t A g t kT f tπ φ φ∞

=−∞

= − + +

∑ (4.10)

where 1

0

2M

k

m

mM

πφ

−

=

∈

are the M possible equally likely phases of the carrier, ( )g t

is a pulse shape with g(t)=0 for t>T and t<0, T is the symbol rate, cf is the carrier

frequency; iφ is the initial phase and A is the amplitude of the received signal.

Then, the sampled equivalent low-pass signal [ ] ( )sv n v nT= with a CFO, f∆ , is

given as

2

2

[ ] ( )

( )

ik s

is

fj n

j F

k

fj n

F

k

k

v n A g t kT e e

A g t kT I e

π φφ

π φ

∆∞ +

=−∞

∆∞ +

=−∞

= −

= −

∑

∑ (4.11)

where kj

kI eφ= is a wide-sense stationary symbol sequence.


The average power of the low-pass equivalent signal is

22 2 2 2 2

_ _{ [ ]} { }s k g av g avP E v n A E I P A P= = = (4.12)




( )0α of a MPSK signal can be estimated even for SNR>-9dB as seen from Figure

3-3.

75


The envelope of the low-pass equivalent signal is

,

{ [ ]} [ ]

( )

k s

k

n k

env v n v n

A I g nT kT

A g

∞

=−∞

=

= −

=

∑ (4.13)

where , ( )n k sg g nT kT= − .

Then, at high SNR, the peak vectors for PSK2, PSK4 and PSK8 are given in Table

4-2.

Table 4-2: Peak Vectors for PSK2, PSK4 and PSK8 Signals

Peak Vector

\Modulation PSK2 PSK4 PSK8

vp1 [1 0 0 0] [1 0 0 0] [1 0 0 0]

vp2 [2 0 0 0] [2 0 0 0] [2 0 0 0]

vp3 [5 0 0 0] [5 0 0 0] [5 0 0 0]

Number of peaks 1 2 3 4, , ,n n n n , defined in section 4.1, for PSK4 and PSK8 as a

function of a SNR are given in Figure 4-7 and Figure 4-8 respectively. Since PSK2

and ASK2 have the same constellation diagram and the simulation results for ASK2

76

are given in Section 4.2, PSK2 is not mentioned here. 1 2 3 4( ), ( ), ( ), ( )n n n n values in

the figures are the average values obtained from the simulations.

From these figures, it can be seen that for SNR<0 4η value becomes irrelevant due

to high noise power.

Since every communication system has a definite bandwidth, there will be a minor

amplitude variations at the transitions between successive symbols [1]; however,

these variations can be eliminated by adjusting the threshold value envς given in

eqn. (4.1). During simulations 0.5envς = is taken.

Figure 4-7: Histogram Output For PSK4

77

Figure 4-8: Histogram Output For PSK8


For MPSK signal, Mth power of the low-pass equivalent signal is

2

[ ] ( )i

s

fM n

FM M j M

k

v n A g t kT e eπ φ

φ

∆∞ +

=−∞

= −

∑ (4.14)

Since { }1

1

00

22

MM

k mm

M mM mM

πφ π

−−

==

= =

then,

2

[ ] ( )i

s

fM n

FM M

k

v n A g t kT eπ φ

∆∞ +

=−∞

= −

∑ (4.15)

From (4.15), it can be seen that the information is removed from the signal and a

carrier at M f∆ is obtained.

78

For MPSK signal; 'M th power, where ' , intM

M k even egerk

∈

and 'M M≠

, of the low-pass equivalent signal is

'

' ' '2

[ ] ( )i

s

fM n

FM M j M

k


φ

∆∞ +

=−∞

= −

∑ (4.16)

Since

11 1

' ' '

'0 00

2 2 2M

M k

k k

m mm

M mM m mMM k

M

π π πφ φ

−− −

= ==

= = = =

then

'

' ' '2

[ ] ( )i

s

fM n

FM M j

k


φ

∆∞ +

=−∞

= −

∑ (4.17)

From (4.17), it can be seen that 'M th power of the low-pass equivalent signal is

another PSK signal with level '

Mk

M= .

From (4.15) and (4.17); it can be concluded that higher orders of the intercepted

signal can be used to discriminate MPSK signals. Therefore,

• For PSK2 signal, it is expected to get 0 2 4 81;k k k k≠ = = =1 at high SNR

• For PSK4 signal, it is expected to get 0 2 4 81; 1; 1k k k k≠ ≠ = = at high SNR

• For PSK8 signal, it is expected to get 0 2 4 81; 1; 1; 1k k k k≠ ≠ ≠ = at high SNR

Output of the spectral analysis of moments block 0 2 4 8( ; ; ; )k k k k for PSK2, PSK4,

and PSK8 as a function of a SNR is given in Figure 4-9, Figure 4-10 Figure 4-11.

Values of 0 2 4 8( ; ; ; )k k k k in the figures are the average values obtained from the

simulations and threshold values 0.5ξ = and 'd bF F= are taken during simulations.

79

From the figures it can be concluded that output of the spectral analysis of moments

block is a feature that does not require any prior knowledge about the signal

parameters and gives satisfactory result for SNR>9 dB.

-10 -5 0 5 10 15 20 25-0.5

0

0.5

1

1.5

SNR (dB)

k0, k2, k4, k8

k0

k2

k4

k8

Figure 4-9: Spectral Analysis of Moments Block Output for PSK2 vs. SNR

80

-10 -5 0 5 10 15 20 25-0.5

0

0.5

1

1.5

SNR (dB)

k0, k2, k4, k8

k0

k2

k4

k8


-10 -5 0 5 10 15 20 25-0.5

0

0.5

1

1.5

SNR (dB)

k0, k2, k4, k8

k0

k2

k4

k8


81

Instantaneous Frequency Extraction Block:


carrier frequency offset of MPSK signals and to discriminate frequency modulated

signals from MPSK. For PSK4 and PSK8 instantaneous frequency extraction block

output ( )5 6,n n , defined in section 4.2, are given in Figure 4-12. There are

frequency variations at the transitions between successive symbols [1]; however

these frequency variations are eliminated by the usage of a low pass filter and by

selecting 0.5freqς = . Therefore ( )5 61, 1n n= = is obtained and from Figure 4-12 it

can be seen that 5n is sufficient for a classification process.

0 5 10 15 20 25 300.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6Frequency Histogram Output for MPSK vs. SNR

SNR (dB)

n5,n6

PSK4, n5

PSK8, n5

PSK4, n6

PSK8, n6

Figure 4-12: Frequency Histogram Output ( )5 6,n n for MPSK vs. SNR

82

4.4 MQAM RECOGNITION

MQAM signal can be expressed as [29]

( ) ( ) cos(2 )k c k i

k

s t A A g t kT f tπ φ φ∞

=−∞

= − + +

∑ (4.18)

And the equivalent low-pass signal ( )v t with a CFO, f∆ , is

2( ) ( ) ij ft

k

k

v t A I g t kT eπ φ

∞∆ +

=−∞

= −

∑ (4.19)

where { } { }kj

k kI A eφ= is a wide-sense stationary information sequence, k kA I= ,

1 Im( )tan

Re( )k

k

k

I

Iφ −

=

, ( )g t is a pulse shape with g(t)=0 for t>T and t<0, T is the

symbol rate, cf is the carrier frequency, iφ is the initial phase and A is the

amplitude of the received signal. Then, the sampled equivalent low-pass signal

[ ] ( )sv n v nT= is given by

2

[ ] ( )i

s

fj n

F

k

k

v n A I g t kT eπ φ∆∞ +

=−∞

= −

∑ (4.20)


The average power of the low-pass equivalent signal is

[ ]{ } { }22 2 2_s k g avP E v n A E I P= = (4.21)

For QAM2; 2 2_s g avP A P= , for { }1,1kI ∈ −

For QAM4; 2 2_s g avP A P= , for

3 3

4 4 4 4, , ,j j j j

kI e e e eπ π π π

− − ∈

83

For QAM8; 2 2_5,5s g avP A P= , for

3 3

4 4 4 4, , , ,3 ,3 , 3 , 3j j j j

kI e e e e j j j jπ π π π

− − ∈ + − − + − −

Constellation diagrams for QAM2, QAM4 and QAM8 are given in Figure 4-13,

Figure 4-14 and Figure 4-15 respectively.

Figure 4-13: Constellation Diagram of QAM2

Figure 4-14: Constellation Diagram of QAM4

84

Figure 4-15: Constellation Diagram for QAM8




( )0α of a MASK signal can be estimated even for SNR>-9dB as seen from Figure

3-3.


For MQAM, the envelope of the low-pass equivalent signal is

,

{ [ ]} [ ]

( )

k s

k

k n k

env v n v n

A I g nT kT

A I g

∞

=−∞

=

= −

=

∑ (4.22)

where , ( )n k sg g nT kT= −

85

Then, at high SNR, the peak vectors for QAM2, QAM4 and QAM8 are given in

Table 4-3. As seen in Table 4-3, the peak vector vp3, obtained from the 3th

histogram, can be used to discriminate QAM8 from QAM2 and QAM4.

Table 4-3: Peak Vectors for QAM2, QAM4 and QAM8 Signals

Peak Vector

\Modulation QAM2 QAM4 QAM8

vp1 [1 0 0 0] [1 0 0 0] [1 0 0 0]

vp2 [2 0 0 0] [2 0 0 0] [1 3 0 0]

vp3 [5 0 0 0] [5 0 0 0] [3 6 0 0]

Number of peaks 1 2 3 4, , ,n n n n , defined in section 4.1, for QAM4 and QAM8 as a

function of a SNR are given in Figure 4-16 and Figure 4-17 respectively. Since

QAM2 and ASK2 have the same constellation diagram and the simulation results

for ASK2 are given in Section 4.1, QAM2 is not mentioned in here. 1 2 3 4, , ,η η η η

values in the figures are the average values obtained from simulations.

From these figures, it can be seen that for SNR<0 dB 4η value becomes irrelevant

due to high noise power. However, for QAM8 4η =2 for SNR > 9dB and 4η =1 for

QAM2 and QAM4 for SNR > 0dB. Hence 4η is another key feature to be used to

discriminate QAM8 signals from QAM4 and QAM2.

86

Figure 4-16: Histogram Output for QAM4 vs. SNR

Figure 4-17: Histogram Output for QAM8 vs. SNR

87


For MQAM signal, the square of the low-pass equivalent signal is

4 22

4 22 2

[ ] ( ) ( )

( )

is

is

fj n

F

k s l s

k l

fj n

F

k s

k

v n I g nT kT I g nT lT e

I g nT kT e

π φ

π φ

∆∞ ∞ +

=−∞ =−∞

∆∞ +

=−∞

= − −

= −

∑ ∑

∑ (4.23)

and the expected value is

{ }4 2

2 2 2[ ] ( )i

s

fj n

F

k s

k

E v n E I g nT kT eπ φ∆∞ +

=−∞

= −

∑ (4.24)

For QAM2;

• { }2 1kE I = , since { }2 1kI ∈

• { }4 1kE I = , since { }4 1kI ∈

• { }8 1kE I = , since { }8 1kI ∈

For QAM4;

• { }2 0kE I = , since 2 2 22 , 2j j

kI e eπ π

− ∈

• { }4 4kE I = − , since { }4 4 ,4j j

kI e eπ π−∈

• { }8 16kE I = , since { }8 16kI ∈

For QAM8

• { }2 4kE I = , since 2 2 22 ,2 ,8 6,8 6j j

kI e e j jπ π

− ∈ + −

88

• { }4 12kE I = , since { }4 4 , 4 ,28 96,28 96j j

kI e e j jπ π−∈ + −

• { }8 4208kE I = − , since { }8 16, 8432 5376, 8432 5376kI j j∈ − + − −

Hence, for QAM2 and QAM8, there is a power at 2 f∆ , 4 f∆ , 8 f∆ ; whereas for

QAM4 at 4 f∆ , 8 f∆ . Therefore, the value 2k for QAM2 and QAM8 is expected to

be equal to 1 and for QAM4 2 1k ≠ at high SNR for a suitable threshold, ξ ,

selection.

Output of the spectral analysis of moments block ( 2k ) for QAM2, QAM4, and

QAM8 as a function of a SNR is given in Figure 4-18. 2k in the figure is the

average value obtained from simulations and threshold values 0.5ξ = and 'd bF F=

are taken during simulations.

From Figure 4-18, it can be seen that 2k can be used as a feature to discriminate

4QAM signals from 8QAM and 2QAM.

89

-10 -5 0 5 10 15 20 25-0.5

0

0.5

1

1.5

SNR (dB)

k2

QAM2

QAM4

QAM8

Figure 4-18: k2 for QAM2, QAM4 and QAM8 vs. SNR



carrier frequency offset of MQAM signals and to discriminate frequency modulated

signals from MQAM. For QAM8 instantaneous frequency extraction block output

( )5 6,n n , defined in section 4.2, are given in Figure 4-19. Threshold value

0.5freqς = is taken during simulations. From Figure 4-19, it can be seen that 5n is

sufficient for a classification process.

90

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Frequency Histogram Output for QAM8 vs. SNR

SNR (dB)

n5,n6

n5

n6

Figure 4-19: Frequency Histogram Output ( )5 6,n n for QAM8 vs. SNR

91

4.5 CW RECOGNITION

CW signal can be expressed as [29]

( ) cos(2 )c is t A f tπ φ= + (4.25)

Where cf is the carrier frequency; iφ is the initial phase and A is the amplitude of

the received signal. Then, the equivalent low-pass signal [ ] ( )sv n v nT= with a CFO,

f∆ , is given as

2

[ ]i

s

fj n

Fv n Ae

π φ∆

+

= (4.26)



[ ]{ }2 2sP E v n A= = (4.27)



Except the carrier frequency there is no cyclostationary component in CW signal.

Hence the symbol rate ( )0α ; estimated by (3.51)), is expected to be 0 as given in

Figure 3-3.


For CW; the envelope of the analytical signal is;

[ ]{ } [ ]{ }env v n abs v n A= = (4.28)

Hence for CW it is expected to get 1 2 3 4 1η η η η= = = = and the peak vectors at high

SNR:

92

o vp1= [1 0 0 0]

o vp2= [2 0 0 0]

o vp3= [5 0 0 0]


Since there is no modulation induced on continuous wave signal, it is expected to

get 0 2 4 8 1k k k k= = = = at high SNR.

The envelope feature extraction block output and output of spectral analysis of

moments block for CW as a function of a SNR is given in Figure 4-20 and Figure

4-21 respectively.

Figure 4-20: Histogram output for CW vs. SNR

93

-10 -5 0 5 10 15 20 25-0.5

0

0.5

1

1.5

SNR (dB)

k0, k2, k4, k8

k0

k2

k4

k8

Figure 4-21: 0 2 4 8, , ,k k k k for CW vs. SNR



carrier frequency offset of CW signals and to discriminate frequency modulated

signals from CW signals. For CW signal, the instantaneous frequency extraction

block outputs ( )5 6,n n are given in Figure 4-22. Since there is no frequency

modulation on a CW signal, ( )5 61, 1n n= = is obtained. Moreover, from Figure 4-22

it can be seen that 5n is sufficient for classification process.

94

0 5 10 15 20 25 300.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Frequency Histogram Output for CW vs. SNR

SNR (dB)

n5,n6

n5

n6

Figure 4-22: Frequency Histogram Output ( )5 6,n n for CW vs. SNR

95

4.6 CPFSK RECOGNITION

CPFSK signal can be expressed as [29]

( ) cos 2 4 ( )c d k i

kt

s t A f t Tf I g t T dπ π τ τ φ∞ ∞

=−∞=−∞

= + − +

∑∫ (4.29)

Then the sampled equivalent low-pass signal [ ] ( )sv n v nT= with a CFO, f∆ , is

2 2 ( )

[ ] ( )k i

s kt

fj n h I g t T d

F

sv n v nT Aeπ π τ τ φ

∞ ∞

=−∞=−∞

∆ + − +

∑∫= = (4.30)

where ( ){ }1, 3,..., 1kI M∈ −∓ ∓ ∓ is a wide-sense stationary symbol sequence,

2 dh f T= is the modulation index, cf is the carrier frequency, iφ is the initial

phase, ( )g t is a pulse shape with g(t)=0 for t>T and t<0 and a peak value 2bF and

A is the amplitude of the received signal.



[ ]{ }2 2sP E v n A= = (4.31)



By using the cyclostationarity detection block, described in section 3.3, the symbol

rate ( )0α of a CPFSK signal can be estimated for SNR>-8dB as given in Figure

3-4.


For CPFSK; the envelope of the low-pass equivalent signal is

96

[ ]{ } [ ]{ }env v n abs v n A= = (4.32)

Hence for the CPFSK signal, it is expected to get 1 2 3 4 1η η η η= = = = and the peak

vectors at high SNR:

o vp1= [1 0 0 0]

o vp2= [2 0 0 0]

o vp3= [5 0 0 0]


Nth power of the CPFSK signal is another CPFSK signal with a different

modulation index (h); hence the output of the spectral analysis of moments block

can give misleading information for CPFSK signals.



carrier frequency offset, the level of the modulation and the modulation index of the

CPFSK signal.

Carrier Frequency Offset Estimation

Carrier frequency offset of the CPFSK signal can be estimated from the mean value

of the instantaneous frequency �1

1( )

N

i i

n

f f nN =

= ∑ as described in section 3.6.

Level of the Modulation

By using the histogram output,; obtained from the instantaneous frequency of the

intercepted signal, the level of the CPFSK signal can be determined at high SNR. It

is expected to get 5 6 2η η= = for 2FSK; 5 6 4η η= = for 4FSK and 5 66; 8n n= = for

8FSK at high SNR.

97

The instantaneous frequency extraction block outputs, 5 6,η η , for 2CPFSK, 4CPFSK

and 8CPFSK are given in Figure 4-23. CPFSK signals are generated by frequency

modulating the MASK signals. During simulations, the carrier frequency offset is

assumed to be 10kHz for a sampling frequency of fs=50kHz, symbol rate is 2kHz,

modulation index is 0.5 and 2nd order Chebychev-II filter with a cut-off frequency

20kHz is used as a smoothing LPF. 5 6,η η in the figure are the average values

obtained from multiple simulations. From Figure 4-23; it can be seen that both 5n

and 6η can be used for a classification process. Therefore, it is sufficient to use only

one of 5n and 6η .

0 5 10 15 20 251

2

3

4

5

6

7

8

SNR (dB)

Frequency Histogram Output n5,n6

Frequency Histogram Output for CPFSK vs. SNR

FSK2,n5

FSK2,n6

FSK4,n5

FSK4,n6

FSK8,n5

FSK8,n6

Figure 4-23: Frequency Histogram Output ( 5 6,η η ) for M-level CPFSK vs. SNR

98

Modulation Index (h):

The modulation index of a CPFSK signal can be estimated by using the symbol rate

and the cluster centers in the frequency histogram. Then, the estimated modulation

index (h ) is

( )11

1 1 N

i i

ib p

h L LF N

+=

= −

∑ (4.33)

where iL is the ith cluster center, pN is the level of the modulation, bF is the

symbol rate and h is the estimated modulation index.

To find the cluster centers; firstly an instantaneous frequency histogram, with bin

centers /100δ , is constructed and then the bins, whose frequency count is larger

than a predefined threshold, are used in a k-means clustering method.

In Figure 4-24, estimated absolute average error normalized to modulation index (h)

and in Figure 4-25, a plot of the estimated modulation index (h ) and the true

modulation index (h) is given for 2, 4 and 8 level CPFSK signals.

99

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Estimated Absolute Average Error Normalized to h

Modulation Index Estimation Performance

FSK2

FSK4

FSK8

Figure 4-24: Modulation Index (h) Estimation Performance for CPFSK Signals

(h=0.5)

100

Figure 4-25: A Plot of the Estimated Modulation Index (h ) vs. True Modulation

Index (h) at SNR=20dB.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4Estimated Modulation Index vs. True Modulation Index at SNR=20dB

True Modulation Index

Estimated Modulation Index

FSK2

FSK4

FSK8

101

4.7 AWGN RECOGNITION

In this work, the channel is assumed be simply an AWGN channel. Therefore, all

we need to do is to estimate the noise variance which is 2σ and it is defined as

{ }{ }

{ } { }{ }

{ } { }{ }

{ } { }{ }

* 2,

2

,

2

,

( ) ( )

( ) ( ) 0;

Re ( ) Re ( )2

Im ( ) Im ( )2

Re ( ) Im ( ) 0

t s

t s

t s

E n t n s

E n t n s

E n t n s

E n t n s

E n t n s

σ δ

σδ

σδ

=

=

=

=

=

(4.34)


Since there is no signal then, the average power of the low-pass equivalent signal is

0sP ≈ (4.35)



There is no cyclostationary component in AWGN signal. Hence the symbol rate

( )0α ; estimated by (3.51)), is expected to be 0 as given in Figure 3-3.


From (4.35) histogram output is expected to be 1 2 3 4 0η η η η= = = = .

102


Since the Nth power of white Gaussian noise is still another noise signal, the

outputs of the spectral analysis of moments block are expected to be

0 2 4 8 0k k k k= = = = .


Since there is no deterministic frequency component on noise signal, the output of

the instantaneous frequency extraction block will be noisy.

103

4.8 CONCLUSION

The outputs of the SNR estimation block, the symbol rate estimation block, the

envelope feature extraction block, the spectral analysis of moments block, the

instantaneous frequency extraction block were introduced and investigated for

different types of digitally modulated signals. Some of the outputs of these blocks

were selected as features. Also, the performances of the features extracted from

these blocks were tested for ASK2, ASK4, ASK8, PSK2, PSK4, PSK8, QAM2,

QAM4, QAM8, Noise, FSK2, FSK4, FSK8 and CW signals. In Table 4-4, the

outputs of each block and in Table 4-5, the expected values of the selected features

for each modulation type is given.

As it can be seen in Table 4-5, each of the ASK2, ASK4, ASK8, PSK4, PSK8,

QAM8, FSK2, FSK4, FSK8 and CW signals has different combinations and thus, it

is possible to distinguish them from each other. Moreover, noise and information

carrying signals can be discriminated by using the estimated average signal power,

sP . In Table 4-5, the term “X” is used for a feature, which may be misleading for

the specified modulation type.

Table 4-4: The Outputs of Each Block

Blocks The Outputs of The Blocks

SNR Estimation Block SNR , sP

Symbol Rate Estimation Block 0α

Envelope Feature Extraction Block vp3, 4η

Spectral Analysis of Moments Block k0, k2, k4, k8

Instantaneous Frequency Extraction

Block f∆ , h , 5η

104

Table 4-5: Features and Modulation Type Relation

Blocks

Mod.

Types

Features

ASK2

PSK2

QAM2

ASK4 ASK8 PSK4

QAM4 PSK8 QAM8 FSK2 FSK4 FSK8 CW NOISE

vp3 [5 0

0 0]

[2 6

0 0]

[1 3

5 7]

[5 0

0 0]

[5 0

0 0]

[3 6

0 0]

[5 0

0 0]

[5 0

0 0]

[5 0

0 0]

[5 0

0 0] X

Envelope

Feature

Extraction Block 4η 1 2 4 1 1 2 1 1 1 1 X

k0 0 0 0 0 0 0 X X X 1 X

k2 1 1 1 0 0 1 X X X 1 X

k4 1 1 1 1 0 1 X X X 1 X

Spectral

Analysis of

Moments Block k8 1 1 1 1 1 1 X X X 1 X

Instantaneous

Frequency

Extraction Block

5η 1 1 1 1 1 1 2 4 6 1 X

105

CHAPTER 5

THE RECOGNITION SYSTEM

In this chapter, a modulation recognition system based on the selected features,

described in Chapter-3 and 4, is introduced. Firstly, a basic decision tree based

method, then a classification method based on a Bayesian probability model is

described. Finally, the two methods are compared.

5.1 THE DECISION TREE METHOD

Decision tree methods are one of the basic classification procedures in which

decision at each stage is made according to predefined threshold values. In these

methods, time-order of the features and predefined threshold values are the main

parameters affecting the performance of the classifier. Therefore, threshold values

and time ordering of the selected features must be chosen carefully in order to

reduce the probability of wrong decision. Moreover, sequence order of the features

and the threshold values should be updated to recognize a modulation type, which is

not included in the classification list of that recognizer.

The proposed decision tree is given in Figure 5-1. In this classification method,

firstly, the estimated average signal power, sP , is compared with a threshold, 1δ ,

and the decision will be “Noise” if the condition 1sP δ< is satisfied. Otherwise, the

decision will be made by using the remaining features.

106

If the intercepted signal is not a noise, then k0 value is used to decide whether there

is a modulation on the signal or not. Then, the number of peaks in the envelope

histogram is used to discriminate ASK4, ASK8 and QAM8, whose classification is

made by using the peak vector obtained from the 3rd envelope histogram, from other

modulation types. The classification of CPFSK signals is made by using the number

of peaks in the frequency histogram and the determination of the level of the

modulation on the MPSK signal is made by using the k2, k4, k8 values. At the last

stage of the tree, if the intercepted signal does not match any of the modulation type

in the classification list, the output of the recognizer will be “Unknown (UN)”.

107

Figure 5-1: The Proposed Decision Tree for Modulation Classification

ASK2,

MPSK

MFSK

NOISE

PSK4

no yes

yes no

yes

no

no yes

no

yes

no

yes

yes

no

no yes

no yes

yes

no

yes

v[n]

1sP δ<

0 1k =

4 2n δ<

3

3

5or

7

p

p

v

v

∈

∈

n δ<

5 4n δ<

5 5n δ<

5 6n δ<

2 1k =

8 1k =

no

noise

CW QAM8

ASK8

FSK8

FSK4

FSK2

ASK2

UN

PSK8

3

3

3

6

p

p

v

vand∈

∈

ASK4

4 1k =

CW

ASK4, ASK8, QAM8

MFSK

MPSK, ASK2

108

5.2 THE BAYESIAN BASED RECOGNITION SYSTEM

We used the following features in the proposed recognition system:

1. Estimated SNR, snr

2. Estimated signal power, sP

3. Number of peaks in the envelope histogram, 4n and the peak vector vp3

4. Spectral analysis of moments output for the signal, 0k

5. Spectral analysis of moments output for the second power of the signal, 2k

6. Spectral analysis of moments output for the fourth power of the signal, 4k

7. Spectral analysis of moments output for the eight power of the signal, 8k

8. Number of peaks in the instantaneous frequency histogram, 5n

Bayesian based classifier is used in the decision block. The block diagram of the

proposed recognition system is given in Figure 5-2.

The following formula is used for classification process [42]

1 1 2 1 1 1( , ,..., ) ( ) ( ) ( , ) ( ,..., , )j n j j j n n jP c x x P c P x c P x x c P x x x c−= (6.1)

For our recognition procedure, assuming features ix , conditioned on a given

modulation jc , are independent of each other,

( , ) ( )i j k i jP x c x P x c= (6.2)

Hence eqn. (6.1) becomes

11

( , ,..., ) ( ) ( )n

j n j i j

i

P c x x P c P x c=

= ∏ (6.3)

and

109

1 1 1( , ,..., ) ( ,..., ) ( ,..., )j n n j nP c x x P x x P c x x= (6.4)

From eqns. (6.3) and (6.4):

111

( )( ,..., ) ( )

( ,..., )

nj

j n i j

in

P cP c x x P x c

P x x =

= ∏ (6.5)

Specifically, we use

{ }ASK2,ASK4,ASK8,QAM8,PSK4,PSK8,CW,FSK2,FSK4,FSK8,NOISEjc ∈

and { }4 3 0 2 4 8 5, , , , , , ,i p sx n v P k k k k n∈ .

Figure 5-2: The Functional Block Diagram of the Bayesian Based Recognition System for Modulation Classification

, ssnr P

v[n]

SNR Estimation Block

Envelope Feature Extraction Block

Spectral Analysis of Moments Block

Instantaneous Frequency

Extraction Block

D

E

C

I

S

I

O

N

B

L

O

C

K

4 3, pn v

0 2 4 8, , ,k k k k

5n

110

In the decision block, the following steps are executed to decide the modulation

type

1. If the estimated average signal power sP is lower than a threshold 1δ , then

the decision will be “noise”.

2. If the estimated average signal power sP is higher than the threshold, then

the decision is made according to eqn. (6.5) by selecting the modulation type

with maximum probability, for the estimated SNR value.

3. If the result of eqn. (6.5) equals to 0, then the decision will be “Unknown

(UN)”.

5.3 SIMULATION RESULTS

To determine the performance of the proposed recognition system, computer

simulations were carried out by using the MATLAB package.

5.3.1 FEATURES OF SIMULATIONS

Test Data Signals

Linearly modulated signals were generated at baseband and then shifted to give

them a carrier frequency and phase offset. CPFSK signals are generated by

frequency modulating the MASK signals. In Figure 5-3, the block diagram of the

transmitter is given, where { }ka is the data sequence, { }kI is the sequence of

symbols that results from mapping the data sequence into corresponding signal

points selected from the appropriate signal space diagram.

111

Signals with raised cosine pulse shape are obtained by filtering the generated

baseband signals. A 4th order lowpass linear-phase filter with a cut-off frequency,

equals to four times of the symbol rate of the signal, is used for filtering.

Figure 5-3: Block Diagram of the Transmitter

SNR

The channel SNR value, which is defined as the ratio of the average power of the

modulated signal to the average power of noise at the output of the IF bandwidth of

the receiver, is used for performance evaluation.

The Receiver Model

It is assumed that the receiver consists of a set of fixed frequency receivers with

their passbands set contiguously. In these fixed tuned receivers, the received signal

is down-converted to IF and filtered such that the signal is in the passband of the

filter and then moved to baseband with a possible offset. The baseband signal is

sampled at a rate equals to the IF bandwidth to make noise samples independent;

hence, the in-phase and quadrature components of the down-converted signal are

obtained. Then, the signal from each of the fixed tuned receivers is analyzed by the

Data Converter Pulse Shaping

LO ( cf f+ ∆ )

IQ Modulator { }ka { }kI ( )v t ( )s t

112

recognition system. In Figure 5-4, basic block diagram of the receiver and in Figure

5-5, a block diagram of the fixed tuned receiver is given.

Figure 5-4: Block Diagram of the Receiver

Figure 5-5: Block Diagram of the Fixed Tuned Receiver

5.3.2 TUNING OF THE RECOGNITION SYSTEM

The probability of each feature for the given modulation type ( ( )i jP x c ) at

different SNR levels should be determined prior to tests. For this reason, following

test signals were generated for training at SNR of 0dB, 3dB, 6dB, 9dB, 12dB, 15dB,

18dB, 21dB.

Mu

lt

ip

le

xe

r Fixed Tuned Receiver, fc1

Fixed Tuned Receiver, fcN

s(t)

v1[n]

vN[n]

The

Recognition

System

v[n]

Fs=BWIF RF Stage Mixer

LO1

IF Stage, BWIF IQ Detector s(t)

113

Linearly Modulated Signal Parameters

• Square pulse with symbol rate Fb=5kHz, carrier frequency offset CFO=0.

• Raised-cosine spectrum pulse with a roll of r=0.25, Fb=2kHz, CFO=0.1 kHz.

• Raised-cosine spectrum pulse with r=0.75, Fb=5kHz, CFO=0.5kHz.

CW Signal Parameters

• No carrier frequency offset.

• CFO=0.1 kHz.

• CFO=0.5 kHz.

CPFSK Signal Parameters

• Square pulse with Fb=5kHz, CFO=0Hz and modulation index h=0.5

• Raised-cosine spectrum pulse with r=0.25, Fb=2kHz, CFO=0.1kHz and

h=0.75

• Raised-cosine spectrum pulse with r=0.75, Fb=5kHz, CFO= 0.5kHz and

h=0.4

In Tables 5.1 – 5.27, the computed probabilities of each feature parameter for each

modulation type at different SNR levels are given.

114

Table 5-1: Probability of 4 1.5n ≤

( )4 1.5 jP n c≤

SNR ASK2 ASK4 ASK8 PSK4 PSK8 QAM8 CW FSK2 FSK4 FSK8

0 0,95 0,73 0,70 0,95 0,95 0,73 0,95 0,98 0,95 0,95

3 0,98 0,81 0,87 0,98 0,98 0,81 0,98 0,99 0,98 0,98

6 0,99 0,25 0,81 1,00 1,00 0,25 1,00 1,00 1,00 1,00

9 1,00 0,05 0,55 1,00 1,00 0,00 1,00 1,00 1,00 1,00

12 1,00 0,01 0,10 1,00 1,00 0,00 1,00 1,00 1,00 1,00

15 1,00 0,00 0,00 1,00 1,00 0,00 1,00 1,00 1,00 1,00

18 1,00 0,00 0,00 1,00 1,00 0,00 1,00 1,00 1,00 1,00

21 1,00 0,00 0,00 1,00 1,00 0,00 1,00 1,00 1,00 1,00

Table 5-2: Probability of 41.5 3n< ≤

( )41.5 3 jP n c< ≤


0 0,05 0,27 0,28 0,05 0,05 0,27 0,05 0,05 0,05 0,05

3 0,02 0,19 0,10 0,02 0,02 0,19 0,02 0,02 0,02 0,02

6 0,01 0,75 0,15 0,01 0,01 0,25 0,01 0,01 0,01 0,01

9 0,00 0,95 0,40 0,00 0,00 1,00 0,00 0,00 0,00 0,00

12 0,00 0,99 0,72 0,00 0,00 1,00 0,00 0,00 0,00 0,00

15 0,00 1,00 0,16 0,00 0,00 1,00 0,00 0,00 0,00 0,00

18 0,00 1,00 0,00 0,00 0,00 1,00 0,00 0,00 0,00 0,00

21 0,00 1,00 0,00 0,00 0,00 1,00 0,00 0,00 0,00 0,00

115

Table 5-3: Probability of 43 n<

( )43 jP n c<


0 0,00 0,00 0,02 0,00 0,00 0,00 0,00 0,00 0,00 0,00

3 0,00 0,00 0,03 0,00 0,00 0,00 0,00 0,00 0,00 0,00

6 0,00 0,00 0,04 0,00 0,00 0,00 0,00 0,00 0,00 0,00

9 0,00 0,00 0,05 0,00 0,00 0,00 0,00 0,00 0,00 0,00

12 0,00 0,00 0,18 0,00 0,00 0,00 0,00 0,00 0,00 0,00

15 0,00 0,00 0,84 0,00 0,00 0,00 0,00 0,00 0,00 0,00

18 0,00 0,00 1,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

21 0,00 0,00 1,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

In Table 5-1, Table 5-2 and Table 5-3, probabilities of the feature 4n , which is

defined as the number of the peaks in the 4th envelope histogram, are given. From

these tables, we can conclude that

• Probabilities of the feature 4n are similar for MPSK, MFSK, CW and

ASK2.

• The probability ( )4 1.5 jP n c≤ for ASK8 is higher than the ASK4 at low

SNR, hence this information can be used to discriminate ASK4 from ASK8

at low SNR values.

• The probability 4(3 )jP n c< equals to 1 for ASK8 at high SNR, whereas

this quantity is zero for other modulation types, as expected.

• The probability 4(1.5 3 )jP n c< ≤ equals to 1 for ASK4 at high SNR,

whereas this quantity is zero for other modulation types, as expected.

116

Table 5-4: Probability of 3(1) 1pv =

3( (1) 1 )p j

P v c=


0 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

3 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

6 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

9 0,00 0,02 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

12 0,00 0,01 0,35 0,00 0,00 0,00 0,00 0,00 0,00 0,00

15 0,00 0,00 0,94 0,00 0,00 0,00 0,00 0,00 0,00 0,00

18 0,00 0,00 1,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

21 0,00 0,00 1,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00


3( (1) 2 )p j

P v c=


0 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

3 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

6 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

9 0,00 0,93 0,86 0,00 0,00 0,00 0,00 0,00 0,00 0,00

12 0,00 0,96 0,48 0,00 0,00 0,00 0,00 0,00 0,00 0,00

15 0,00 1,00 0,05 0,00 0,00 0,00 0,00 0,00 0,00 0,00

18 0,00 1,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

21 0,00 1,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

117


3( (1) 3 )p j

P v c=


0 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

3 0,00 0,07 0,10 0,00 0,00 0,00 0,00 0,00 0,00 0,00

6 0,00 0,99 0,93 0,00 0,00 0,31 0,00 0,00 0,00 0,00

9 0,00 0,02 0,14 0,00 0,00 1,00 0,00 0,00 0,00 0,00

12 0,00 0,01 0,17 0,00 0,00 1,00 0,00 0,00 0,00 0,00

15 0,00 0,00 0,01 0,00 0,00 1,00 0,00 0,00 0,00 0,00

18 0,00 0,00 0,00 0,00 0,00 1,00 0,00 0,00 0,00 0,00

21 0,00 0,00 0,00 0,00 0,00 1,00 0,00 0,00 0,00 0,00


3( (1) 4 )p j

P v c=


0 0,00 0,21 0,35 0,00 0,00 0,05 0,00 0,00 0,00 0,00

3 0,00 0,89 0,90 0,00 0,00 0,66 0,00 0,00 0,00 0,00

6 0,00 0,00 0,07 0,00 0,00 0,65 0,00 0,00 0,00 0,00

9 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

12 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

15 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

18 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

21 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

118


3( (1) 5 )p j

P v c=


0 0,60 0,80 0,65 0,43 0,57 0,90 0,70 0,69 0,59 0,70

3 1,00 0,02 0,00 1,00 1,00 0,34 1,00 1,00 1,00 1,00

6 1,00 0,00 0,00 1,00 1,00 0,04 1,00 1,00 1,00 1,00

9 1,00 0,00 0,00 1,00 1,00 0,00 1,00 1,00 1,00 1,00

12 1,00 0,00 0,00 1,00 1,00 0,00 1,00 1,00 1,00 1,00

15 1,00 0,00 0,00 1,00 1,00 0,00 1,00 1,00 1,00 1,00

18 1,00 0,00 0,00 1,00 1,00 0,00 1,00 1,00 1,00 1,00

21 1,00 0,00 0,00 1,00 1,00 0,00 1,00 1,00 1,00 1,00


3( (1) 6 )p j

P v c=


0 0,40 0,01 0,00 0,53 0,43 0,05 0,25 0,31 0,41 0,30

3 0,00 0,01 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

6 0,00 0,01 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

9 0,00 0,03 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

12 0,00 0,02 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

15 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

18 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

21 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

119


3( (2) 0 )p j

P v c=


0 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

3 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

6 1,00 0,51 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

9 1,00 0,00 0,93 1,00 1,00 0,15 1,00 1,00 1,00 1,00

12 1,00 0,00 0,42 1,00 1,00 0,00 1,00 1,00 1,00 1,00

15 1,00 0,00 0,01 1,00 1,00 0,00 1,00 1,00 1,00 1,00

18 1,00 0,00 0,00 1,00 1,00 0,00 1,00 1,00 1,00 1,00

21 1,00 0,00 0,00 1,00 1,00 0,00 1,00 1,00 1,00 1,00


3( (2) 3 )p j

P v c=


0 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

3 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

6 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

9 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

12 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

15 0,00 0,00 0,90 0,00 0,00 0,00 0,00 0,00 0,00 0,00

18 0,00 0,00 1,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

21 0,00 0,00 1,00 0,00 0,00 0,17 0,00 0,00 0,00 0,00

120


3( (2) 5 )p j

P v c=


0 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

3 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

6 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

9 0,00 0,00 0,05 0,00 0,00 0,00 0,00 0,00 0,00 0,00

12 0,00 0,00 0,55 0,00 0,00 0,00 0,00 0,00 0,00 0,00

15 0,00 0,00 0,02 0,00 0,00 0,00 0,00 0,00 0,00 0,00

18 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

21 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00


3( (2) 6 )p j

P v c=


0 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

3 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

6 0,00 0,49 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

9 0,00 0,98 0,00 0,00 0,00 0,84 0,00 0,00 0,00 0,00

12 0,00 1,00 0,00 0,00 0,00 0,91 0,00 0,00 0,00 0,00

15 0,00 1,00 0,00 0,00 0,00 0,98 0,00 0,00 0,00 0,00

18 0,00 1,00 0,00 0,00 0,00 0,99 0,00 0,00 0,00 0,00

21 0,00 1,00 0,00 0,00 0,00 0,81 0,00 0,00 0,00 0,00

121


3( (2) 7 )p j

P v c=


0 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

3 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

6 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

9 0,00 0,02 0,00 0,00 0,00 0,01 0,00 0,00 0,00 0,00

12 0,00 0,00 0,03 0,00 0,00 0,09 0,00 0,00 0,00 0,00

15 0,00 0,00 0,07 0,00 0,00 0,02 0,00 0,00 0,00 0,00

18 0,00 0,00 0,00 0,00 0,00 0,01 0,00 0,00 0,00 0,00

21 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00


3( (3) 0 )p j

P v c=


0 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

3 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

6 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

9 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

12 1,00 1,00 0,93 1,00 1,00 1,00 1,00 1,00 1,00 1,00

15 1,00 1,00 0,05 1,00 1,00 1,00 1,00 1,00 1,00 1,00

18 1,00 1,00 0,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

21 1,00 1,00 0,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

122


3( (3) 5 )p j

P v c=


0 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

3 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

6 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

9 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

12 0,00 0,00 0,01 0,00 0,00 0,00 0,00 0,00 0,00 0,00

15 0,00 0,00 0,90 0,00 0,00 0,00 0,00 0,00 0,00 0,00

18 0,00 0,00 1,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

21 0,00 0,00 1,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00


3( (3) 7 )p j

P v c=


0 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

3 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

6 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

9 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

12 0,00 0,00 0,06 0,00 0,00 0,00 0,00 0,00 0,00 0,00

15 0,00 0,00 0,05 0,00 0,00 0,00 0,00 0,00 0,00 0,00

18 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

21 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

123


3( (4) 0 )p j

P v c=


0 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

3 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

6 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

9 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

12 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

15 1,00 1,00 0,21 1,00 1,00 1,00 1,00 1,00 1,00 1,00

18 1,00 1,00 0,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00

21 1,00 1,00 0,00 1,00 1,00 1,00 1,00 1,00 1,00 1,00


3( (4) 7 )p j

P v c=


0 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

3 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

6 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

9 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

12 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

15 0,00 0,00 0,79 0,00 0,00 0,00 0,00 0,00 0,00 0,00

18 0,00 0,00 1,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

21 0,00 0,00 1,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00

124

In Tables 5-4 – 5-19, probabilities of the feature 3pv , which is defined as the peak

vector obtained from the 3rd envelope histogram, are given. The following

probabilities are equal to zero for all modulation types: 3( (1) 0 )p jP v c= ,

3( (1) 7 )p jP v c= , 3( (1) 8 )p jP v c= , 3( (2) 1 )p jP v c= , 3( (2) 2 )p jP v c= ,




3( (4) 5 )p jP v c= , 3( (4) 6 )p jP v c= , 3( (4) 8 )p jP v c= .

From these tables, we can conclude that:

• Probabilities of the feature 3pv are consistent with the values given in Table

4-5 at high SNR.

• The probabilities 3( (1) 2 )p jP v c= , 3( (1) 3 )p jP v c= , 3( (3) 7 )p jP v c= and

3( (2) 5 )p jP v c= for ASK8 do not always equal to zero; hence, this

information can be used to classify ASK8 at low SNR.

• The probability 3( (2) 7 )p jP v c= for ASK8 and QAM8 does not always

equal to zero; hence, this information can be used to classify ASK8 and

QAM8 at low SNR.

• The probability 3( (1) 1 )p jP v c= , 3( (1) 6 )p jP v c= for ASK4 does not always

equal to zero; hence, this information can be used to classify ASK4 at low

SNR.

125

Table 5-20: Probability of 0 1k =

( )0 1 jP k c=


0 0,00 0,00 0,00 0,00 0,00 0,00 1,00 0,00 0,00 0,00

3 0,00 0,00 0,00 0,00 0,00 0,00 1,00 0,01 0,00 0,02

6 0,00 0,00 0,00 0,00 0,00 0,00 1,00 0,00 0,00 0,00

9 0,00 0,00 0,00 0,00 0,00 0,00 1,00 0,00 0,00 0,02

12 0,00 0,00 0,00 0,00 0,00 0,00 1,00 0,00 0,00 0,02

15 0,00 0,00 0,00 0,00 0,00 0,00 1,00 0,00 0,00 0,02

18 0,00 0,00 0,00 0,00 0,00 0,00 1,00 0,00 0,00 0,03

21 0,00 0,00 0,00 0,00 0,00 0,00 1,00 0,00 0,00 0,01


( )2 1 jP k c=


0 1,00 1,00 1,00 0,00 0,00 1,00 1,00 0,53 0,50 0,51

3 1,00 1,00 1,00 0,00 0,00 1,00 1,00 0,50 0,50 0,53

6 1,00 1,00 1,00 0,00 0,00 1,00 1,00 0,50 0,51 0,68

9 1,00 1,00 1,00 0,00 0,00 1,00 1,00 0,50 0,50 0,66

12 1,00 1,00 1,00 0,00 0,00 1,00 1,00 0,50 0,51 0,70

15 1,00 1,00 1,00 0,00 0,00 1,00 1,00 0,51 0,51 0,62

18 1,00 1,00 1,00 0,00 0,00 1,00 1,00 0,51 0,50 0,60

21 1,00 1,00 1,00 0,00 0,00 1,00 1,00 0,51 0,55 0,65

126


( )4 1 jP k c=


0 1,00 1,00 1,00 1,00 0,00 0,09 1,00 0,50 0,00 0,01

3 1,00 1,00 1,00 1,00 0,00 0,88 1,00 0,50 0,50 0,01

6 1,00 1,00 1,00 1,00 0,00 1,00 1,00 0,61 0,50 0,25

9 1,00 1,00 1,00 1,00 0,00 1,00 1,00 0,50 0,50 0,41

12 1,00 1,00 1,00 1,00 0,00 1,00 1,00 0,50 0,50 0,40

15 1,00 1,00 1,00 1,00 0,00 1,00 1,00 0,50 0,50 0,47

18 1,00 1,00 1,00 1,00 0,00 1,00 1,00 0,50 0,50 0,58

21 1,00 1,00 1,00 1,00 0,00 1,00 1,00 0,51 0,50 0,58


( )8 1 jP k c=


0 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,01

3 0,00 0,00 0,00 0,00 0,00 0,00 1,00 0,00 0,01 0,00

6 0,28 1,00 1,00 0,09 0,06 0,93 1,00 0,00 0,00 0,02

9 1,00 1,00 1,00 1,00 1,00 1,00 1,00 0,50 0,11 0,00

12 1,00 1,00 1,00 1,00 1,00 1,00 1,00 0,63 0,52 0,02

15 1,00 1,00 1,00 1,00 1,00 1,00 1,00 0,50 0,58 0,06

18 1,00 1,00 1,00 1,00 1,00 1,00 1,00 0,54 0,56 0,19

21 1,00 1,00 1,00 1,00 1,00 1,00 1,00 0,52 0,52 0,17

127

In Tables 5-20 – 5-23, probabilities of the features k0, k2, k4 and k8, which are the

outputs of the spectral analysis of moments block, are given. From these tables, we

can conclude that:

• Probabilities of the features are consistent with the values given in Table 4-5

at high SNR.

• For CPFSK signal, the key features k0, k2, k4, k8 depends on the modulation

index. Hence, these probabilities can be misleading for CPFSK.

• For CW, the key feature k8 value can be misleading especially at low SNR.

Table 5-24: Probability of 5 1.5n ≤

( )5 1.5 jP n c≤


0 1,00 1,00 1,00 1,00 1,00 1,00 1,00 0,90 0,80 0,70

3 1,00 1,00 1,00 1,00 1,00 1,00 1,00 0,80 0,70 0,60

6 1,00 1,00 1,00 1,00 1,00 1,00 1,00 0,70 0,60 0,50

9 1,00 1,00 1,00 1,00 1,00 1,00 1,00 0,20 0,05 0,05

12 1,00 1,00 1,00 1,00 1,00 1,00 1,00 0,00 0,00 0,00

15 1,00 1,00 1,00 1,00 1,00 1,00 1,00 0,00 0,00 0,00

18 1,00 1,00 1,00 1,00 1,00 1,00 1,00 0,00 0,00 0,00

21 1,00 1,00 1,00 1,00 1,00 1,00 1,00 0,00 0,00 0,00

128

Table 5-25: Probability of 51.5 2.5n< ≤

( )51.5 2.5 jP n c< ≤


0 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,10 0,05 0,02

3 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,20 0,10 0,05

6 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,30 0,15 0,10

9 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,80 0,25 0,20

12 0,00 0,00 0,00 0,00 0,00 0,00 0,00 1,00 0,01 0,00

15 0,00 0,00 0,00 0,00 0,00 0,00 0,00 1,00 0,00 0,00

18 0,00 0,00 0,00 0,00 0,00 0,00 0,00 1,00 0,00 0,00

21 0,00 0,00 0,00 0,00 0,00 0,00 0,00 1,00 0,00 0,00

Table 5-26: Probability of 52.5 4.5n< ≤

( )52.5 4.5 jP n c< ≤


0 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,15 0,03

3 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,20 0,07

6 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,25 0,15

9 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,70 0,25

12 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,99 0,35

15 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 1,00 0,18

18 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 1,00 0,05

21 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 1,00 0,00

129

Table 5-27: Probability of 54.5 8n< ≤

( )54.5 8 jP n c< ≤


0 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,25

3 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,28

6 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,25

9 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,50

12 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,65

15 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,82

18 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,95

21 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 1,00

In Tables 5-24 – 5-27, probabilities of the feature 5n , which is defined as the

number of peaks in the instantaneous frequency histogram, are given. From these

tables, we can conclude that probabilities of this feature are consistent with the

values given in Table 4-5.

130

5.3.3 CONDUCTION OF TESTS

For CPFSK signal, the key features k0, k2, k4, k8 depends on the modulation index;

hence, probabilities of these features are not considered for CPFSK signals.

Moreover k8 value for CW can be misleading especially at low SNR, therefore

probability of k8 is also not considered for CW signal.

To compare the decision tree based classifier and the bayesian based recognition

system the threshold values, given in Figure 5-1, are selected to be equal to the ones

used in the Bayesian based recognition system.

2500 symbols were used for each modulation type, and for each SNR value 100

independent signals were generated and tested. ASK2, ASK4, ASK8, QAM8,

PSK4, PSK8, FSK2, FSK4, FSK8 were generated by using both the square pulse

shaping and the raised cosine pulse shaping.

5.3.4 RESULTS OF SIMULATIONS

In Tables 5-28 – 5-35, performances of the decision tree method and the Bayesian

based recognition system at SNR of 0dB, 6dB, 9dB, 12dB and 15dB are given by a

confusion matrix, which is a matrix providing information about the output of the

recognition system for the given modulation type. Rows of the matrix are the

expected classification, while the columns are the resulting classification and the

values indicate the success rate of the resulting classification for the given

modulation type. The term “Unknown (UN)” is used for a modulation type that is

not included in the classification list of the recognizer.

131


SNR=6 dB

MOD. Type

UN noise ASK2 ASK4 ASK8 PSK4 PSK8 QAM8 CW FSK2 FSK4 FSK8

noise 0 100 0 0 0 0 0 0 0 0 0 0

ASK2 0 0 97 0 0 0 0 0 0 3 0 0

ASK4 10 0 0 52 18 0 0 20 0 0 0 0

ASK8 0 0 0 12 66 0 0 22 0 0 0 0

PSK4 0 0 0 0 0 90 0 0 0 10 0 0

PSK8 0 0 0 0 0 11 88 0 0 1 0 0

QAM8 0 0 0 2 20 0 0 78 0 0 0 0

CW 0 0 0 0 0 0 0 0 100 0 0 0

FSK2 0 0 64 0 0 0 0 0 0 36 0 0

FSK4 0 0 15 0 0 0 0 0 1 66 18 0

FSK8 0 0 1 0 0 0 0 0 0 26 73 0

Table 5-29: Confusion Matrix of The Decision Tree Method at SNR=6 dB

MOD. Type


noise 0 100 0 0 0 0 0 0 0 0 0 0

ASK2 0 0 100 0 0 0 0 0 0 0 0 0

ASK4 0 0 26 44 0 0 0 30 0 0 0 0

ASK8 0 0 80 20 0 0 0 0 0 0 0 0

PSK4 0 0 0 0 0 100 0 0 0 0 0 0

PSK8 88 0 0 0 0 12 0 0 0 0 0 0

QAM8 0 0 89 10 1 0 0 0 0 0 0 0

CW 0 0 0 0 0 0 0 0 100 0 0 0

FSK2 0 0 66 0 0 0 0 0 0 34 0 0

FSK4 0 0 15 0 0 0 0 0 1 66 18 0

FSK8 0 0 3 0 0 0 0 0 0 24 73 0

132

Table 5-30: Confusion Matrix of The Bayesian Based Recognition System at SNR=9 dB

MOD. Type


noise 0 100 0 0 0 0 0 0 0 0 0 0

ASK2 1 0 98 0 0 0 0 0 0 1 0 0

ASK4 2 0 0 97 0 0 0 1 0 0 0 0

ASK8 3 0 0 0 94 0 0 3 0 0 0 0

PSK4 0 0 0 0 0 100 0 0 0 0 0 0

PSK8 0 0 0 0 0 29 69 0 0 2 0 0

QAM8 0 0 0 1 2 0 0 97 0 0 0 0

CW 0 0 0 0 0 0 0 0 100 0 0 0

FSK2 0 0 0 0 0 0 0 0 0 100 0 0

FSK4 0 0 1 0 0 0 0 0 0 0 99 0

FSK8 0 0 0 0 0 0 0 0 0 0 15 85


MOD. Type


noise 0 100 0 0 0 0 0 0 0 0 0 0

ASK2 0 0 100 0 0 0 0 0 0 0 0 0

ASK4 0 0 0 99 0 0 0 1 0 0 0 0

ASK8 0 0 63 33 4 0 0 0 0 0 0 0

PSK4 0 0 0 0 0 100 0 0 0 0 0 0

PSK8 0 0 1 0 0 30 69 0 0 0 0 0

QAM8 0 0 1 2 0 0 0 97 0 0 0 0

CW 0 0 0 0 0 0 0 0 100 0 0 0

FSK2 0 0 0 0 0 0 0 0 0 100 0 0

FSK4 0 0 1 0 0 0 0 0 1 0 98 0

FSK8 0 0 0 0 0 0 0 0 0 0 15 85

133


MOD. Type


noise 0 100 0 0 0 0 0 0 0 0 0 0

ASK2 1 0 99 0 0 0 0 0 0 0 0 0

ASK4 1 0 0 99 0 0 0 0 0 0 0 0

ASK8 7 0 1 0 92 0 0 0 0 0 0 0

PSK4 0 0 0 0 0 100 0 0 0 0 0 0

PSK8 0 0 0 0 0 5 95 0 0 0 0 0

QAM8 0 0 0 0 0 0 0 100 0 0 0 0

CW 0 0 0 0 0 0 0 0 100 0 0 0

FSK2 0 0 0 0 0 0 0 0 0 100 0 0

FSK4 0 0 0 0 0 0 0 0 0 0 100 0

FSK8 0 0 0 0 0 0 0 0 0 0 1 99


MOD. Type


noise 0 100 0 0 0 0 0 0 0 0 0 0

ASK2 0 0 100 0 0 0 0 0 0 0 0 0

ASK4 0 0 1 99 0 0 0 0 0 0 0 0

ASK8 0 0 5 39 56 0 0 0 0 0 0 0

PSK4 0 0 0 0 0 100 0 0 0 0 0 0

PSK8 0 0 0 0 0 5 95 0 0 0 0 0

QAM8 0 0 0 0 0 0 0 100 0 0 0 0

CW 0 0 0 0 0 0 0 0 100 0 0 0

FSK2 0 0 0 0 0 0 0 0 0 100 0 0

FSK4 0 0 0 0 0 0 0 0 0 0 100 0

FSK8 0 0 0 0 0 0 0 0 1 0 1 98

134


MOD. Type


noise 0 100 0 0 0 0 0 0 0 0 0 0

ASK2 0 0 100 0 0 0 0 0 0 0 0 0

ASK4 0 0 0 100 0 0 0 0 0 0 0 0

ASK8 0 0 0 0 100 0 0 0 0 0 0 0

PSK4 0 0 0 0 0 100 0 0 0 0 0 0

PSK8 0 0 0 0 0 0 100 0 0 0 0 0

QAM8 0 0 0 0 0 0 0 100 0 0 0 0

CW 0 0 0 0 0 0 0 0 100 0 0

FSK2 0 0 0 0 0 0 0 0 0 100 0 0

FSK4 0 0 0 0 0 0 0 0 0 0 100 0

FSK8 0 0 0 0 0 0 0 0 0 0 0 100


MOD. Type


noise 0 100 0 0 0 0 0 0 0 0 0 0

ASK2 0 0 100 0 0 0 0 0 0 0 0 0

ASK4 0 0 0 100 0 0 0 0 0 0 0 0

ASK8 0 0 0 0 100 0 0 0 0 0 0 0

PSK4 0 0 0 0 0 100 0 0 0 0 0 0

PSK8 0 0 0 0 0 0 100 0 0 0 0 0

QAM8 0 0 0 0 0 0 0 100 0 0 0 0

CW 0 0 0 0 0 0 0 0 100 0 0 0

FSK2 0 0 0 0 0 0 0 0 0 100 0 0

FSK4 0 0 0 0 0 0 0 0 0 0 100 0

FSK8 0 0 0 0 0 0 0 0 0 0 0 100

135

From Tables 5-28 – 5-35, we can conclude that:

• At SNR<6 dB, success rate of the proposed Bayesian based recognition

system is superior to the decision tree method in classifying ASK8, PSK8

and QAM8 signals. The decision tree method fails completely for ASK8,

PSK8 and QAM8; whereas the success rate in the Bayesian method is 66%

for ASK8; 88% for PSK8 and 78% for QAM8. However, at SNR>9dB the

performances of both methods are similar in classifying QAM8 and PSK8

signals.

• At SNR<13 dB, success rate of the Bayesian method is superior to the

decision tree method in classifying ASK8 signals.

• Both of the proposed recognition systems are able to discriminate noise

signals from information carrying signals at SNR>-21dB.

• The Bayesian based recognition system fails completely for CW at SNR<-

3dB; whereas the success rate of the decision tree method is %100 at SNR>-

9dB. CW is the only modulation type that the performance of the decision

tree method is superior to the Bayesian based method because the value of

the feature k0 is consistent even at low SNR as shown in Figure 4-21 and

this feature is used at the second stage of the tree. For CW, the performance

of the Bayesian based method can be improved by calculating the feature

probabilities at SNR<0dB.

• The success rate of both methods are similar in classifying CPFSK signals.

• At SNR>15 dB, success rate of the Bayesian method and the decision tree

method are similar.

• Both of the proposed recognition systems are able to discriminate ASK2,

ASK4, ASK8, PSK4, PSK8, QAM8, FSK2, FSK4, FSK8, CW and NOISE

at SNR>15 dB with a success rate 100%.

136

CHAPTER 6

CONCLUSIONS AND FUTURE WORK

In this thesis, a recognition system, which does not require prior knowledge of the

signal parameters, has been developed. Moreover, the recognition system is able to

extract some of the signal parameters, such as the modulation index of CPFSK

signals the symbol rate, the carrier frequency offset and SNR value.

First of all, most of the feature extraction techniques found in the literature have

been surveyed, then the selected feature extraction techniques are tested for digitally

modulated signals. Estimated signal power is used to decide whether the intercepted

signal is noise or an information carrying signal, this feature gives satisfactory

results even at very low SNR values (SNR>-21 dB). Higher order moments of the

intercepted signal are used to remove the modulation on the signal, this feature is

used to discriminate MASK signals from MPSK signals and to determine the level

of modulation on MPSK signals. Other key features are extracted from the envelope

and frequency histograms of the intercepted signal. Cyclostationarity is one of the

important properties of the digitally modulated signals that can be used to classify

signals and to extract some of the signal parameters. However, cyclostationary

analysis require excessive calculations; hence, cyclostationarity based feature

extraction has not been used in our classification process; but it is used to estimate

the symbol rate of the intercepted signal and this estimated symbol rate is used in

the spectral analysis of moments block.

137

Classification algorithm is another important part of a modulation recognition

system. Two methods are developed for classification: first one is based on a

proposed decision tree and the second is based on the Bayesian a posteriori

probability computation. In the Bayesian based method, the probability of the each

key feature for each modulation type at different SNR values is obtained. The two

methods were compared and the results show that the performace of the Bayesian

based recognition system is superior to the decision tree based method at low SNR

values. Moreover, both of the proposed recognition systems are able to discriminate

ASK2, ASK4, ASK8, PSK4, PSK8, QAM8, FSK2, FSK4, FSK8, CW and NOISE

at SNR>15 dB with a success rate 100%.

The success of the Bayesian method relies mainly on the feature independence. In

the Bayesian based method, each selected key feature depends on a different

parameter of the signal and all the features are used at the same time.

The recognition system has been designed for the classification of the ASK2,

ASK4, ASK8, PSK4, PSK8, QAM8, FSK2, FSK4, FSK8, CW and NOISE.

However, the system can easily be modified to classify digital signals of other types

or with larger parameter ranges.

In addition to recognizing the modulation type, the system also yields:

• Level of the modulation.

• Signal to noise ratio at the output of the IF filter.

• Coarse estimation of the carrier frequency offset.

• Symbol rate of the signal.

• Modulation index of the CPFSK signal.

In addition to recognition works, we have also investigated a method to estimate the

symbol rate and the carrier frequency offset of MSK signals. The limit of the

estimated carrier frequency offset does not depend on the symbol rate and the

138

estimator can also be used even for very short duration of pulses. In addition, design

procedure of the proposed estimator is given and the performance of the estimator is

tested for short duration MSK signals.

Some of the topics that remain as future work are noted below:

• The effect of the symbol duration on the performance of the recognition.

• The effect of ISI.

• Pulse shapes with partial response.

• Nongaussian noise interference

139

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